Research Article Deformation of a Circular Elastic Tube...
16
Research Article Deformation of a Circular Elastic Tube between Two Parallel Bars: Quasi-Analytical Geometrical Ring Models A. Van Hirtum GIPSA-lab, UMR CNRS 5216, Grenoble University, 38000 Grenoble, France Correspondence should be addressed to A. Van Hirtum; [email protected] Received 8 March 2015; Accepted 21 April 2015 Academic Editor: Francesco Tornabene Copyright © 2015 A. Van Hirtum. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Several engineering problems are confronted with elastic tubes. In the current work, homothetic quasi-analytical geometrical ring models, ellipse, stadium, and peanut, are formulated allowing a computationally low cost ring shape estimation as a function of a single parameter, that is, the pinching degree. e dynamics of main geometrical parameters due to the model choice is discussed. Next, the ring models are applied to each cross section of a circular elastic tube compressed between two parallel bars for pinching efforts between 40% and 95%. e characteristic error yields less than 4% of the tubes diameter when the stadium model was used. 1. Introduction An accurate description of a constricted channel’s geometry and related geometrical features is oſten crucial and, there- fore, a recurrent problem in different engineering disciplines. Geometrical features will affect main fluid flow characteristics related to inertia, boundary layer development, flow detach- ment, or jet formation downstream from the constricted channel portion [1]. e same way, the accuracy of wave related problems will depend on the channels or waveguides geometry since it will influence among others the propa- gation or evanescence of higher order propagation modes [2]. Knowledge of the channel geometry is in particular an obstacle when dealing with natural occurring fluid flow or wave propagation through elastic channels for which no geometrical design is possible and in addition the shape can rapidly vary such as is the case for biological channel flows (lower and upper airway respiratory or blood circulation system) and related biomechanical phenomena. Partly motivated by the large number of applications, a substantial amount of research is performed (see, e.g., [3– 8]) on elastic tube modeling for a variety of boundary and loading conditions since the pioneering work [9–11] on thin- walled elastic tubes. Despite those efforts and the increased insight in the stability and bifurcation of circular elastic tubes, no analytical model is capable of providing geometrical features for channels subjected to rapidly varying loading resulting in small as well as large deformations of thin as well as thick wall tubes of variable length. erefore, in the current work quasi-analytical parameterised geometrical ring models are used in order to provide geometrical features of a compressed elastic tube. Although those models do not give insight in the dynamics of the elastic ring, they do allow, due to their simplicity, to avoid or overcome some of the limitations of more advantaged analytical models depending on boundary conditions (loading, geometry, ...). erefore, when solely the ring shape (or an associated geometrical parameters) is searched, quasi-analytical geometrical models might be of interest in providing an estimation of the shape at a negligible computational cost while the parameteri- zation favors usage in combination with other analytical model approaches exploiting a limited number of input parameters or when an initialization of the ring shape is needed. is might, for example, be the case for real-life applications for which the boundary conditions can be either unknown or rapidly changing. Moreover, for experimental validation the need to control few well defined geometrical parameters is attractive. In addition, an analytical expression of the sensitivity of the modeled shape (or an associated geometrical parameter) to an uncertainty of the imposed control parameters might be of interest for experimental setup characterization or design purposes. Quasi-analytical ring models are considered using the minor axis as an input control parameter. e pinching Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 547492, 15 pages http://dx.doi.org/10.1155/2015/547492
Transcript of Research Article Deformation of a Circular Elastic Tube...
Research ArticleDeformation of a Circular Elastic Tube between Two ParallelBars Quasi-Analytical Geometrical Ring Models
A Van Hirtum
GIPSA-lab UMR CNRS 5216 Grenoble University 38000 Grenoble France
Correspondence should be addressed to A Van Hirtum annemievanhirtumgipsa-labgrenoble-inpfr
Received 8 March 2015 Accepted 21 April 2015
Academic Editor Francesco Tornabene
Copyright copy 2015 A Van Hirtum This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Several engineering problems are confronted with elastic tubes In the current work homothetic quasi-analytical geometrical ringmodels ellipse stadium and peanut are formulated allowing a computationally low cost ring shape estimation as a function of asingle parameter that is the pinching degree The dynamics of main geometrical parameters due to the model choice is discussedNext the ring models are applied to each cross section of a circular elastic tube compressed between two parallel bars for pinchingefforts between 40 and 95The characteristic error yields less than 4 of the tubes diameter when the stadiummodel was used
1 Introduction
An accurate description of a constricted channelrsquos geometryand related geometrical features is often crucial and there-fore a recurrent problem in different engineering disciplinesGeometrical featureswill affectmain fluid flowcharacteristicsrelated to inertia boundary layer development flow detach-ment or jet formation downstream from the constrictedchannel portion [1] The same way the accuracy of waverelated problems will depend on the channels or waveguidesgeometry since it will influence among others the propa-gation or evanescence of higher order propagation modes[2] Knowledge of the channel geometry is in particularan obstacle when dealing with natural occurring fluid flowor wave propagation through elastic channels for which nogeometrical design is possible and in addition the shape canrapidly vary such as is the case for biological channel flows(lower and upper airway respiratory or blood circulationsystem) and related biomechanical phenomena
Partly motivated by the large number of applications asubstantial amount of research is performed (see eg [3ndash8]) on elastic tube modeling for a variety of boundary andloading conditions since the pioneering work [9ndash11] on thin-walled elastic tubes Despite those efforts and the increasedinsight in the stability and bifurcation of circular elastictubes no analyticalmodel is capable of providing geometricalfeatures for channels subjected to rapidly varying loading
resulting in small as well as large deformations of thin aswell as thick wall tubes of variable length Therefore in thecurrentwork quasi-analytical parameterised geometrical ringmodels are used in order to provide geometrical features of acompressed elastic tube Although those models do not giveinsight in the dynamics of the elastic ring they do allowdue to their simplicity to avoid or overcome some of thelimitations of more advantaged analytical models dependingon boundary conditions (loading geometry ) Thereforewhen solely the ring shape (or an associated geometricalparameters) is searched quasi-analytical geometrical modelsmight be of interest in providing an estimation of the shapeat a negligible computational cost while the parameteri-zation favors usage in combination with other analyticalmodel approaches exploiting a limited number of inputparameters or when an initialization of the ring shape isneeded This might for example be the case for real-lifeapplications for which the boundary conditions can be eitherunknown or rapidly changing Moreover for experimentalvalidation the need to control few well defined geometricalparameters is attractive In addition an analytical expressionof the sensitivity of the modeled shape (or an associatedgeometrical parameter) to an uncertainty of the imposedcontrol parameters might be of interest for experimentalsetup characterization or design purposes
Quasi-analytical ring models are considered using theminor axis 119887 as an input control parameter The pinching
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 547492 15 pageshttpdxdoiorg1011552015547492
2 Mathematical Problems in Engineering
d
2(y + d) y = b0
2a0
2b0
P0
lB
(a) Circular 1198870 = 1198632
d2b
2a
P
lB
2(y + d) y ge b
(b) Pinched 119887 lt 1198632Figure 1 Illustration of a deformable ring (light grey) with constant wall thickness 119889 compressed between two parallel bars (dark grey) oflength 0 lt 119897119861 The internal central ring height yields twice the minor axis 2119887 and the internal central width yields twice the major axis 2119886 Theparallel bars are spaced by 2(119910 + 119889) with 119910 = 119887 for nonbuckled rings and 119910 gt 119887 for buckled rings (a) Circular (1198870 = 1198632 and 1198860 = 1198632) ringwith internal perimeter 1198750 = 120587119863 (internal diameter119863) and (b) elongated ring (119887 lt 1198632 and 119886 gt 1198632)
degree of the ring is expressed as 1minus 1198871198870 where the subscriptsdot0 refers to values associatedwith a circular ring For a circularring both the minor 1198870 and major 1198860 axis yield half theinternal diameter119863We consider a circular ring with internalperimeter 1198750 = 120587119863 which is compressed between twoparallel bars of length 0 lt 119897119861 as illustrated in Figure 1(a) It isassumed that the ring has constant wall thickness 119889 Forcingthe bars closer together (Figure 1(b)) deforms the ring so thatfor the ringrsquos minor axis 0 lt 119887 le 1198632 holds where 1198870 = 1198632
denotes the minor axis of the undeformed circular ring Thedistance between the pinching bars yields 0 lt 2(119910 + 119889) le
119863+2119889 Deforming the ring results in an elongated ring shapeso that the area 119860 enveloped by the ring reduces as the barsare forced closer together The deformed ring is assumed tobe symmetrical with respect to its center lines containing theminor and major axis
Parameterised geometrical ring models are presented inSection 2 for nonbuckled and buckled rings Next (Section 3)a simple data-driven procedure is proposed enabling to applythe ring models in order to obtain the geometry of an elastictube pinched between two parallel bars A concrete case studyfor 119897119861 ≫ 2119886 is discussed
2 Geometrical Ring Models
The internal ring shape is modeled using quasi-analyticalelastic ring models which are symmetrical with respect to thecenter lines The main input parameters are the minor axis119887 and internal diameter 119863 defining the perimeter 1198750 = 120587119863
of the unpinched circular ring Consequently the pinchingdegree 0 lt 1 minus 1198871198870 le 1 of the ring is assumed a knownquantity The considered ring shapes are invariant underhomothetic scaling Therefore in the following lengths arenormalised (sdot) by the internal diameter119863 of the circular ringso that 0 lt
119887 = 119887119863 le 05 1198750 = 120587 and
1198600 = 1205874 arequantities associated with a circle with unit diameter Notethat the pinching degree is not altered by the normalisationsince 1 minus 1198871198870 = 1 minus
119887
1198870 = 1 minus 2
119887 Furthermore sincethe wall thickness
119889 is assumed constant the shape of theexternal ring
119877(120579 1 minus 1198871198870) is derived from the shape of theinternal ring as 119877(120579 1minus1198871198870) = 119903(120579 1minus
119887
1198870)+ sgn(119903) sdot 119889 withsign function sgn and the ring expressed in polar coordinateswith normalised internal radius 119903 normalised external radius
119903 and angle 0 le 120579 le 2120587 Depending on the length ofthe parallel pinching bars 119897119861 compared to the length of themajor axis 119886 buckling of the elastic ring will occur 119897119861 ≫ 2119886
(Figure 1) or not 119897119861 ≪ 2119886 as the pinching degree 1 minus
119887
1198870 isincreased Therefore in Section 21 geometrical ring modelsare presented which does not allow to describe buckling InSection 22 a ring model is presented for which buckling ofthe elastic ring occurs when the pinching degree 1 minus
119887
1198870
increases
21 Geometrical Nonbuckling Ring Models Ellipse and Sta-dium The internal ring is pinched between long parallel barsso that 119897119861 ≫ 2119886 as depicted in Figure 1 The ring shape isapproximated as an ellipse or a stadium with rounded edgesas illustrated in Figure 2
211 Ellipse For given minor axis 119887 with 0 lt
119887 le 05 theelliptic ring shape is fully determined by considering a secondgeometrical parameter such as perimeter 119875 or major axis 119886
When the major axis 119886 and minor axis 119887 are known theelliptic ring shape in polar coordinates yields [12 13]
119903
119886119887(120579) =
radic
119887
2
1 minus 119890
2cos2 (120579)with 119890 =
radic
(1 minus (
119887
119886
)
2
)
(1)
where 120579 isin [0 2120587] and eccentricity 119890 le 1 The perimeter isthen obtained from the expression of Ramanujan [14]
119875
119886119887= 120587 [3 (119886 +
119887) minusradic(119886 + 3
119887) (3119886 +
119887)] (2)
and the area yields 119860119886119887
= 120587
119887119886with
119860 = 119860119863
2The sensitivityof the radius perimeter and area to a variation of the imposedparameters Δ119887 and Δ119886 yields
Δ119903
119886119887(120579) =
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119887
3cos2120579119886
3(1 minus 119890
2cos2 (120579))32
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ119886
+
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1 minus cos2120579(1 minus 119890
2cos2 (120579))32
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ
119887
Δ
119875
119886119887=
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus120587(
6119886 + 10
119887
2radic(119886 + 3
119887) (3119886 +
119887)
minus 3)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ119886
Mathematical Problems in Engineering 3
+
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus120587(
10119886 + 6
119887
2radic(119886 + 3
119887) (3119886 +
119887)
minus 3)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ
119887
Δ
119860
119886119887= 120587119886Δ
119887 + 120587
119887Δ119886
(3)
When the perimeter is used as a second model inputparameter instead of themajor axis 119886 we assume that its valueis conserved when the elastic ring deforms so that
119875
= 120587
holds for all pinching degrees 1 minus1198871198870 The elliptic ring shapein polar coordinates for 120579 isin [0 2120587] is again of the form (1) forwhich eccentricity 119890 is only a function of 119887 since using (2) itfollows that
119903
(120579) =
radic
119887
2
1 minus (1 minus
119887
2119886
2
) cos2 (120579)
with 119886
=
radic
minus20
119887
2+ 12
119887 + 3
6
minus
2
119887
3
+
1
2
(4)
The area
119860 yields again
119860
= 120587
119887119886
or
119860
= 120587
119887(
radic
minus20
119887
2+ 12
119887 + 3
6
minus
2
119887
3
+
1
2
) (5)
The sensitivity of the radius to a variation of the imposedparameter Δ119887 yields
Δ119903
(120579)
Δ
119887
=
1
119903
(120579)
minus
119887
2
119886
2
sdot
cos2 (120579)119903
3
(120579)
(1 +
119887
3119886
[
10
119887 minus 3
radic
minus20
119887
2+ 12
119887 + 3
+ 2])
(6)
where the last term expresses the sensitivity of the major axisto a variation of the imposed parameter Δ119887 since
Δ119886
Δ
119887
=
minus1
3
(
10
119887 minus 3
radic
minus20
119887
2+ 12
119887 + 3
+ 2) (7)
Finally the sensitivity of the area to a variation of theimposed parameter Δ119887 for a constant perimeter becomes
Δ
119860
Δ
119887
=
120587
2
minus
4120587
119887
3
+
120587 (minus20
119887
2+ 12
119887 + 3)
12
6
+
120587
119887 (minus20
119887
2+ 12
119887 + 3)
minus12
12
(8)
212 Stadium with Rounded Edges The stadium shapeillustrated in Figure 2 consists of two flat portions connectedby circular arcs with radius 119887 so that the minor axis of thestadium equals the minor axis of the ellipse The criticalangle 120593 isin [0 1205872] is defined by the critical distance
119871 =
119887 tan(120593) ge 0 which corresponds to half the extent of the flatportion in contact with the parallel pinching bars As for theellipse (Section 211) the stadium shape is fully determinedby considering the minor axis 119887 and a second geometricalparameter such as perimeter 119875 or major axis 119886
Firstly assume that the perimeter
119875 is conserved whenthe elastic rings deform so that 119875
= 120587 holds for all pinching
degrees 1minus1198871198870 In this case the stadium ring shape 119903(120579)withparameters (119887 119875 = 120587) in polar coordinates (119903 120579) is given as apiecewise function of 120579 isin [0 2120587]which depends explicitly onthe pinching degree 1 minus
119887
1198870 = 1 minus 2
119887
119903
(120579) =
119887
sin (120579) for120593 le 120579 le 120587 minus 120593
120587
4
sdot (1 minus 2
119887) sdot ((
16
120587
2sdot
119887
2
(1 minus 2
119887)
2minus sin2 (120579))
12
+ cos (120579)) for minus 120593 lt 120579 lt 120593
120587
4
sdot (1 minus 2
119887) sdot ((
16
120587
2sdot
119887
2
(1 minus 2
119887)
2minus sin2 (120579))
12
minus cos (120579)) for120587 minus 120593 lt 120579 lt 120587 + 120593
minus
119887
sin (120579) for120587 + 120593 le 120579 le 2120587 minus 120593
(9)
where critical angle 120593and critical distance 119871
are a function
of pinching degree 1 minus
119887
1198870 = 1 minus 2
119887
120593
= arctan( 4
120587
sdot
119887
1 minus 2
119887
)
119871
=
120587 (1 minus 2
119887)
4
(10)
The major axis 119886and area
119860
are then expressed by a
linear and respectively quadratic function of the imposedparameter 119887
119886
=
119887 +
120587 (1 minus 2
119887)
4
=
120587
4
+ (1 minus
120587
2
)
119887
119860
= 120587
119887
2+ 4120587
119887 (1 minus 2
119887) = 120587
119887 (1 minus 7
119887)
(11)
4 Mathematical Problems in Engineering
Ellipse (b P = 120587) or (a b)
b
a
(a)
L = btan(120593)
Stadium (b P = 120587) or (a b)
120593bb
a
b
(b)Figure 2 Overview geometrical nonbuckling ring models requiring two geometrical input parameters corresponding to a circular ring withperimeter
1198750 = 120587 and area
1198600 = 1205874 (a) elliptic cross-section with major axis 119886 and minor axis 119887 indicated and (b) stadium with roundededges with minor axis 119887 and critical angle 120593 indicated from which the critical distance 119871 =
119887 tan(120593) can be derived
Consequently the sensitivity of the major axis Δ119886and
area Δ
119860
to a variation of the imposed parameter Δ119887 results
in respectively a constant and a linear function of 119887
Δ119886
Δ
119887
= 1 minus
120587
2
Δ
119860
Δ
119887
= 2120587 (2 minus 7
119887)
(12)
whereas the sensitivity of the radius to a variation of theexperimental imposed parameter Δ119887 is again expressed by apiecewise function of 120579
Δ119903
(120579)
Δ
119887
=
1
sin (120579) for 120593 le 120579 le 120587 minus 120593
16
119887 minus 2120587
2(1 minus 2
119887) sin2 (120579)
4
radic
16
119887
2minus 120587
2(1 minus 2
119887)
2
sin2 (120579)minus
120587
2
cos (120579) for minus 120593 lt 120579 lt 120593
16
119887 minus 2120587
2(1 minus 2
119887) sin2 (120579)
4
radic
16
119887
2minus 120587
2(1 minus 2
119887)
2
sin2 (120579)+
120587
2
cos (120579) for 120587 minus 120593 lt 120579 lt 120587 + 120593
minus
1
sin (120579) for 120587 + 120593 le 120579 le 2120587 minus 120593
(13)
which 119903(9) depends explicitly on the pinching degree 1minus1198871198870
or 1 minus 2
119887Next the major axis 119886 instead of the perimeter
119875 is usedas a second known geometrical parameter in addition to the
minor axis 119887 The stadium ring shape 119903119886(120579) with parameters
(
119887 119886) in polar coordinates (119903 120579) is again a piecewise functionof 120579 isin [0 2120587]
119903
119886(120579) =
119887
sin (120579) for 120593 le 120579 le 120587 minus 120593
radic
119887
2minus 119886
2(1 minus
119887
2
119886
2) sin2 (120579) + (119886 minus
119887) cos (120579) for minus 120593 lt 120579 lt 120593
radic
119887
2minus 119886
2(1 minus
119887
2
119886
2) sin2 (120579) minus (119886 minus
119887) cos (120579) for 120587 minus 120593 lt 120579 lt 120587 + 120593
minus
119887
sin (120579) for 120587 + 120593 le 120579 le 2120587 minus 120593
(14)
Therefore 119903
119886(14) does not depend explicitly on the
pinching degree (1 minus 2
119887) as was the case for (9) but it doesdepend explicitly on the critical distance
119871 = 119886 minus
119887 and thestadium eccentricity 119890 =
radic
1 minus
119887
2119886
2 The perimeter
119875
119886and
Mathematical Problems in Engineering 5
area
119860
119886are respectively a linear and quadratic function of
119887 and depend linearly on 119886 as well
119875
119886= 2
119887 (120587 minus 2) + 4119886
119860
119886= 4119886
119887 +
119887
2(120587 minus 4)
(15)
Consequently the sensitivity of the perimeter Δ
119875
119886and
area Δ
119860
119886to a variation of the input parameters Δ119886 and
Δ
119887 is respectively constant and linearly dependent on
119887 and119886
Δ
119875
119886= 2120587Δ
119887 + 4Δ119886
Δ
119860
119886= 4
119887Δ119886 + (2 (120587 minus 4)
119887 + 4119886)Δ
119887
(16)
The sensitivity of the radius to a variation of the experi-mentally imposed parameter Δ119887 is a piecewise function of 120579
Δ119903
119886(120579) =
1
sin 120579 for 120593 le 120579 le 120587 minus 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119887cos2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
minus cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for minus 120593 lt 120579 lt 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119886sin2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
+ cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for 120587 minus 120593 lt 120579 lt 120587 + 120593
1
sin 120579 for 120587 + 120593 le 120579 le 2120587 minus 120593
(17)
which depends explicitly on the stadium eccentricity since(
119887
2minus 119886
2) = minus119890
2119886
2
213 Comparison of Ellipse and Stadium Ring Models withPerimeter 120587 The geometrical ellipse and stadium ring mod-els with the assumption of constant perimeter 119875 = 120587 are fullydetermined as a function of a single parameter We considerthe minor axis 119887 as a parameter since it is directly related tothe pinching degree 1minus1198871198870 and therefore a control parameterfor ring deformation regardless of the applied model Singleparameter models are in particular of interest to be used incombination with other analytical flow or acoustic modelswhich require a geometrical ring (or cross-section) parametersuch as major axis 119886 or enveloped area
119860 Therefore inFigure 3 we consider the impact of the applied geometricalring model in absence of buckling (ellipse or stadium) onthese parameters as a function of pinching degree 1 minus
1198871198870Following (11) the normalised major axis 119886 of the stadiumring model increases linearly (Figure 8(c)) as a function ofpinching degree 1minus1198871198870 whereas the normalised area
119860 of thestadium model decreases quadratically (Figure 8(d)) From(4) and (5) we deduce that the ring model results in the sametendencies for major axis 119886 (maximum difference betweenellipse and stadium for all pinching degrees lt01) and area
119860 (maximum difference between ellipse and stadium for allpinching degrees lt005) although the increase for 119886 is notlinear as is the case for the stadium model Major axis 119886
of the ellipse is longer than the value of stadium model sothat the area
119860 enveloped by the ellipse is smaller than thearea enveloped by the stadium model The relative differencebetween the stadium and ellipse model for major axis 119886 andarea
119860 is further illustrated in Figure 3(c) Where the models
are equivalent for both extreme pinching degrees associatedwith a circular cross section in absence of pinching (1 minus
1198871198870 = 0 and 119886 =
1198870 = 05) and a completely pinchedring (1 minus
1198871198870 = 100 and cross-section area
119860 = 0) themaximum difference (lt12) of major axis 119886 between theellipse and stadium model occurs for a pinching degree of52 whereas the maximum difference (lt7) between bothmodels for area
119860 occurs for a pinching degree of 66 Thesensitivity of major axis Δ119886 and area Δ
119860 to a variation ofminor axis Δ
119887 is illustrated in Figures 3(d) and 3(e) It isseen that Δ119886 le Δ
119887 so that for both the ellipse and stadium119886 varies less than
119887 (maximum difference for all pinchingdegrees between ellipse and stadium lt08) For the stadiumthe variation of 119886 is independent from the pinching degree 1minus
1198871198870 following (12) For the ellipse 119886 becomes less sensitive toa variation of 119887 as the pinching degree increases as expressedin (7) For small pinching degrees the impact of a variationof 119887 on the area
119860 is small (Δ
119860 le Δ
119887 for pinching degreesle17 for the ellipse and le32 for the stadium) As thepinching degree increases the impact of a variation of Δ119887 onthe area becomes more important In general the impact ofa variation of 119887 on the area is of the same order of magnitudefor the ellipse and the stadium (maximum difference forall pinching degrees lt07) Nevertheless the area using thestadium ring is less sensitive to a variation Δ
119887 than whenan ellipse is used for pinching degrees le60 for pinchingdegrees gt60 the opposite holds The ellipse and stadiumring shape 119903(120579) and associated sensitivity to a variation ofminor axis Δ119903(120579) are illustrated in Figure 4(a) for a pinchingdegree of 50 As expressed in (6) and in (13) the sensitivitydepends on the angle so that it is of interest to consider a
6 Mathematical Problems in Engineering
EllipseStadium
04
06
08
a(mdash
)
0 50 1001 minus bb0 ()
lt01
a0
(a) 119886
0
02
04
06
A(mdash
)
lt005
A0
EllipseStadium
0 50 1001 minus bb0 ()
(b) 119860
0
5
10
15
|middotsminusmiddote|middot 0
()
middot = Amiddot = a
lt7
52lt12
66
0 50 1001 minus bb0 ()
(c) |119886119904 minus 119886119890|1198860 and |119860119904minus 119860119890|1198600
0
05
1
ΔaΔ
b(mdash
)
lt08
91
0 50 1001 minus bb0 ()
EllipseStadium
(d) Δ119886Δ
60
lt07
0
2
4
ΔAΔ
b(mdash
)
3217
0 50 1001 minus bb0 ()
1
EllipseStadium
(e) Δ119860Δ
Figure 3 Impact of applied geometrical ring models in absence of buckling ellipse (full line in ((a) (b) (d) and (e)) and subscript 119890 in (c))and stadium (dashed line in ((a) (b) (d) and (e)) and subscript 119904 in (c)) on normalised major axis 119886 and normalised area
119860 as a function ofpinching degree 1 minus
1198871198870 with the maximum difference indicated on each subfigure (ltsdot) (a) major axis 119886 and 1198860 = 05 (thin dash-dot line)(b) area
119860 and
1198600 = 1205874 (thin dash-dot line) (c) difference between stadium model (subscript 119890) and ellipse (subscript 119904) relative to thecircular value for major axis 119886 (with a maximum difference for pinching degree 52) and area
119860 (with a maximum difference for pinchingdegree 66) (d) sensitivity of major axis Δ119886 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 91) and(e) sensitivity of area Δ
119860 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 60) The maximum differenceis indicated as ltsdot
Mathematical Problems in Engineering 7
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
05
1
15
2
30
210
60
240
90
270
120
300
150
330
0180
EllipseStadium
EllipseStadium
(a) Ring shape 119903(120579) (left) and sensitivity Δ119903(120579)Δ (right) for 1 minus 0 = 50
0 50 1000
10
20
30
65
1
1 minus bb0 ()
EllipseStadium
gt1lt1
r(120579)Δb
maxΔ
(b) max(Δ119903(120579))Δ
0 50 1000
1
2
3
371Δ
r(120579)Δb
lt03
1 minus bb0 ()
EllipseStadium
(c) Δ119903(120579)Δ
Figure 4 Illustration of overall sensitivity Δ119903(120579) of elliptic (full line) and stadium (dashed line) ring shape 119903(120579) to a variation of the minoraxis Δ119887 as a function of pinching degree 1minus1198871198870 (a) example for pinching degree 50 (b) maximum value of Δ119903(120579) and (c) mean sensitivityΔ119903(120579)
globalmeasure of the ring shape to a variation of 119887Thereforethe maximum value of Δ119903(120579) and mean sensitivity Δ119903(120579) forthe entire ring as a function of pinching degree 1 minus
1198871198870
are plotted in Figures 4(b) and 4(c) respectively In contrastto the sensitivity of 119886 which decreases or remains constantfor increasing pinching degree as shown in Figure 3(d)both global measures indicate that the overall sensitivity ofthe shape increases with the pinching degree Whereas themean sensitivity increase remains limited (lt3) themaximumsensitivity increase is more important since it yields up togt5 times the variation imposed on Δ
119887 for both the ellipseand the stadium As was observed for the area 119860 and majoraxis 119886 the stadium ring model is more sensitive than theelliptic ring model to a variation of 119887 for severe constrictiondegrees (ge60) The observed maximum difference (gt1for pinching degrees gt65) is more of less averaged outsince the difference in mean sensitivity between both ringmodels remains small (lt03) regardless of the pinchingdegree
22 Buckling Ring Model Peanut In Section 21 two geo-metrical ring models with low computational cost were
considered Both models reduced to a single parameter ringmodel of the minor axis 119887 when applying the assumptionof conservation of the perimeter
119875 In this section a quasi-analytical elastic peanut ring models is outlined Analyticalelastic ring models are expected to become pertinent whenthe length of the parallel pinching bars (see Figure 1) reduces(119897119861 lt 2119886) so that the loading becomes axial and buckling islikely to occur [4 6]
221 Peanut Each quarter of the peanut ring with center 119900consists of two arc portions as schematised in Figure 5 Thepeanut ring is then expressed as a function of the radius 120588 ofthe outer arc (with origin 1199001) and 120585 twice the angle spannedby the inner arc (with origin 1199002) so that the peanut parameterset is (120588 120585) The peanut ring model is suitable to approximatenonbuckled (119887 ge 120588) and buckled (119887 lt 120588 second mode)elastic rings The assumption of conservation of perimeter ismade so that the normalized perimeter of the total peanutequals the perimeter of the circular tube
119875 = 120587 andconsequently the quarter peanut has a constant length 1205874The total curvature of each quarter peanut yields 1205872 Theminor axis 119887 le 05 and major axis 119886 ge 05 of the nonbuckled
8 Mathematical Problems in Engineering
1205852
b
ao2
o1
o
b ge Nonbuckled
(a)
o2 a
1205852
o1
b
o
Buckled b lt
(b)
Figure 5 Illustration of nonbuckled ((a) inner arc curvature is positive) and buckled ((b) inner arc curvature is negative) quarters of peanutrings with center 119900 and parameter set (120588 120585) Each ring consists of two arc portions outer arc (thick gray line) with origin 1199001 and radius 120588 andinner arc (thick black line) with origin 1199002 and angle 1205852 The major axis 119886 ge 05 and minor axis 119887 le 05 of the peanut are indicated For thenonbuckled (a) and buckled peanut (b) 119887 ge 120588 and 119887 lt 120588 hold respectively
and buckled peanuts are obtained from parameter set (120588 120585)as
non-buckled 119887 = 120587 (1 minus 2120588)
1 minus cos (1205852)2120585
+ 120588
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 120588
(18)
buckled 119887 = (
120587 (1 minus 2120588)
2120585
+ 120588)(cos 1205852
minus 1)
+ 120588 cos 1205852
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 2120588 sin 120585
2
+ 120588
(19)
The area
119860(120588 120585) of the total peanut is given as
119860 = 120587120588 minus 120587120588
2+
120587
2
4
(1 minus 2120588)
2119891 (120585)
with 119891 (120585) =
120585 minus sin 120585120585
2
(20)
Note that for 120585 = 0 the last term in (20) is omitted since119891(120585 = 0) = 0The bending energy of the quarter peanut yields
1
4
E =
120587 minus 120585
2120588
+
120585
2
120587 minus 2 (120587 minus 120585) 120588
(21)
with E(120588 120585) denoting the bending energy of the total peanutThe bending energy (21) is then used to express the Lagrange
functional L associated with the energy of deformation ofthe peanut ring as a function of (120588 120585) as well
1
BL =
120587 minus 120585
120588
+
120585
2
1205872 minus (120587 minus 120585) 120588
+ 120582(
D2minus
120587
2
16
(1 minus 2120588)
2(1 minus 120587119891 (120585)))
1
2120587
(22)
with isoperimetric difference of the quarter peanut D2 =
(1205874)
2minus 120587
1198604 (0 le
D) normalized pressure 120582 = 8PBdenoting the ratio of hydrostatic pressure P and flexuralrigiditymodulus of the ringB = Υ
119889
312(1minus]2)usingYoungrsquos
modulus Υ Poissonrsquos ratio ] and normalised wall thickness
119889For a given
119860 it is aimed to find the peanut definedby parameter set (120588 120585) which minimises the bending energyE When using the normalised pressure 120582 as a Lagrangemultiplier 120582 the constrained minimisation problem satisfiesthe Euler-Lagrange equations in terms of (120588 120585 120582)
0 =
120587120588 minus 1205872 minus 2120585120588
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588
2 1 minus 120587119891 (120585)
120587 minus 120585
0 =
minus2
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588119891
1015840(120585)
120588 =
2
120587
(
120587
4
minus
D
radic1 minus 120587119891 (120585)
)
(23)
where the prime denotes the derivative with respect to 120585 Thesystem (23) is then solved firstly by eliminating 120582(120588 120585) usingthe first two expressions of (23)
120582 =
8
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2 (24)
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
d
2(y + d) y = b0
2a0
2b0
P0
lB
(a) Circular 1198870 = 1198632
d2b
2a
P
lB
2(y + d) y ge b
(b) Pinched 119887 lt 1198632Figure 1 Illustration of a deformable ring (light grey) with constant wall thickness 119889 compressed between two parallel bars (dark grey) oflength 0 lt 119897119861 The internal central ring height yields twice the minor axis 2119887 and the internal central width yields twice the major axis 2119886 Theparallel bars are spaced by 2(119910 + 119889) with 119910 = 119887 for nonbuckled rings and 119910 gt 119887 for buckled rings (a) Circular (1198870 = 1198632 and 1198860 = 1198632) ringwith internal perimeter 1198750 = 120587119863 (internal diameter119863) and (b) elongated ring (119887 lt 1198632 and 119886 gt 1198632)
degree of the ring is expressed as 1minus 1198871198870 where the subscriptsdot0 refers to values associatedwith a circular ring For a circularring both the minor 1198870 and major 1198860 axis yield half theinternal diameter119863We consider a circular ring with internalperimeter 1198750 = 120587119863 which is compressed between twoparallel bars of length 0 lt 119897119861 as illustrated in Figure 1(a) It isassumed that the ring has constant wall thickness 119889 Forcingthe bars closer together (Figure 1(b)) deforms the ring so thatfor the ringrsquos minor axis 0 lt 119887 le 1198632 holds where 1198870 = 1198632
denotes the minor axis of the undeformed circular ring Thedistance between the pinching bars yields 0 lt 2(119910 + 119889) le
119863+2119889 Deforming the ring results in an elongated ring shapeso that the area 119860 enveloped by the ring reduces as the barsare forced closer together The deformed ring is assumed tobe symmetrical with respect to its center lines containing theminor and major axis
Parameterised geometrical ring models are presented inSection 2 for nonbuckled and buckled rings Next (Section 3)a simple data-driven procedure is proposed enabling to applythe ring models in order to obtain the geometry of an elastictube pinched between two parallel bars A concrete case studyfor 119897119861 ≫ 2119886 is discussed
2 Geometrical Ring Models
The internal ring shape is modeled using quasi-analyticalelastic ring models which are symmetrical with respect to thecenter lines The main input parameters are the minor axis119887 and internal diameter 119863 defining the perimeter 1198750 = 120587119863
of the unpinched circular ring Consequently the pinchingdegree 0 lt 1 minus 1198871198870 le 1 of the ring is assumed a knownquantity The considered ring shapes are invariant underhomothetic scaling Therefore in the following lengths arenormalised (sdot) by the internal diameter119863 of the circular ringso that 0 lt
119887 = 119887119863 le 05 1198750 = 120587 and
1198600 = 1205874 arequantities associated with a circle with unit diameter Notethat the pinching degree is not altered by the normalisationsince 1 minus 1198871198870 = 1 minus
119887
1198870 = 1 minus 2
119887 Furthermore sincethe wall thickness
119889 is assumed constant the shape of theexternal ring
119877(120579 1 minus 1198871198870) is derived from the shape of theinternal ring as 119877(120579 1minus1198871198870) = 119903(120579 1minus
119887
1198870)+ sgn(119903) sdot 119889 withsign function sgn and the ring expressed in polar coordinateswith normalised internal radius 119903 normalised external radius
119903 and angle 0 le 120579 le 2120587 Depending on the length ofthe parallel pinching bars 119897119861 compared to the length of themajor axis 119886 buckling of the elastic ring will occur 119897119861 ≫ 2119886
(Figure 1) or not 119897119861 ≪ 2119886 as the pinching degree 1 minus
119887
1198870 isincreased Therefore in Section 21 geometrical ring modelsare presented which does not allow to describe buckling InSection 22 a ring model is presented for which buckling ofthe elastic ring occurs when the pinching degree 1 minus
119887
1198870
increases
21 Geometrical Nonbuckling Ring Models Ellipse and Sta-dium The internal ring is pinched between long parallel barsso that 119897119861 ≫ 2119886 as depicted in Figure 1 The ring shape isapproximated as an ellipse or a stadium with rounded edgesas illustrated in Figure 2
211 Ellipse For given minor axis 119887 with 0 lt
119887 le 05 theelliptic ring shape is fully determined by considering a secondgeometrical parameter such as perimeter 119875 or major axis 119886
When the major axis 119886 and minor axis 119887 are known theelliptic ring shape in polar coordinates yields [12 13]
119903
119886119887(120579) =
radic
119887
2
1 minus 119890
2cos2 (120579)with 119890 =
radic
(1 minus (
119887
119886
)
2
)
(1)
where 120579 isin [0 2120587] and eccentricity 119890 le 1 The perimeter isthen obtained from the expression of Ramanujan [14]
119875
119886119887= 120587 [3 (119886 +
119887) minusradic(119886 + 3
119887) (3119886 +
119887)] (2)
and the area yields 119860119886119887
= 120587
119887119886with
119860 = 119860119863
2The sensitivityof the radius perimeter and area to a variation of the imposedparameters Δ119887 and Δ119886 yields
Δ119903
119886119887(120579) =
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119887
3cos2120579119886
3(1 minus 119890
2cos2 (120579))32
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ119886
+
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1 minus cos2120579(1 minus 119890
2cos2 (120579))32
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ
119887
Δ
119875
119886119887=
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus120587(
6119886 + 10
119887
2radic(119886 + 3
119887) (3119886 +
119887)
minus 3)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ119886
Mathematical Problems in Engineering 3
+
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus120587(
10119886 + 6
119887
2radic(119886 + 3
119887) (3119886 +
119887)
minus 3)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ
119887
Δ
119860
119886119887= 120587119886Δ
119887 + 120587
119887Δ119886
(3)
When the perimeter is used as a second model inputparameter instead of themajor axis 119886 we assume that its valueis conserved when the elastic ring deforms so that
119875
= 120587
holds for all pinching degrees 1 minus1198871198870 The elliptic ring shapein polar coordinates for 120579 isin [0 2120587] is again of the form (1) forwhich eccentricity 119890 is only a function of 119887 since using (2) itfollows that
119903
(120579) =
radic
119887
2
1 minus (1 minus
119887
2119886
2
) cos2 (120579)
with 119886
=
radic
minus20
119887
2+ 12
119887 + 3
6
minus
2
119887
3
+
1
2
(4)
The area
119860 yields again
119860
= 120587
119887119886
or
119860
= 120587
119887(
radic
minus20
119887
2+ 12
119887 + 3
6
minus
2
119887
3
+
1
2
) (5)
The sensitivity of the radius to a variation of the imposedparameter Δ119887 yields
Δ119903
(120579)
Δ
119887
=
1
119903
(120579)
minus
119887
2
119886
2
sdot
cos2 (120579)119903
3
(120579)
(1 +
119887
3119886
[
10
119887 minus 3
radic
minus20
119887
2+ 12
119887 + 3
+ 2])
(6)
where the last term expresses the sensitivity of the major axisto a variation of the imposed parameter Δ119887 since
Δ119886
Δ
119887
=
minus1
3
(
10
119887 minus 3
radic
minus20
119887
2+ 12
119887 + 3
+ 2) (7)
Finally the sensitivity of the area to a variation of theimposed parameter Δ119887 for a constant perimeter becomes
Δ
119860
Δ
119887
=
120587
2
minus
4120587
119887
3
+
120587 (minus20
119887
2+ 12
119887 + 3)
12
6
+
120587
119887 (minus20
119887
2+ 12
119887 + 3)
minus12
12
(8)
212 Stadium with Rounded Edges The stadium shapeillustrated in Figure 2 consists of two flat portions connectedby circular arcs with radius 119887 so that the minor axis of thestadium equals the minor axis of the ellipse The criticalangle 120593 isin [0 1205872] is defined by the critical distance
119871 =
119887 tan(120593) ge 0 which corresponds to half the extent of the flatportion in contact with the parallel pinching bars As for theellipse (Section 211) the stadium shape is fully determinedby considering the minor axis 119887 and a second geometricalparameter such as perimeter 119875 or major axis 119886
Firstly assume that the perimeter
119875 is conserved whenthe elastic rings deform so that 119875
= 120587 holds for all pinching
degrees 1minus1198871198870 In this case the stadium ring shape 119903(120579)withparameters (119887 119875 = 120587) in polar coordinates (119903 120579) is given as apiecewise function of 120579 isin [0 2120587]which depends explicitly onthe pinching degree 1 minus
119887
1198870 = 1 minus 2
119887
119903
(120579) =
119887
sin (120579) for120593 le 120579 le 120587 minus 120593
120587
4
sdot (1 minus 2
119887) sdot ((
16
120587
2sdot
119887
2
(1 minus 2
119887)
2minus sin2 (120579))
12
+ cos (120579)) for minus 120593 lt 120579 lt 120593
120587
4
sdot (1 minus 2
119887) sdot ((
16
120587
2sdot
119887
2
(1 minus 2
119887)
2minus sin2 (120579))
12
minus cos (120579)) for120587 minus 120593 lt 120579 lt 120587 + 120593
minus
119887
sin (120579) for120587 + 120593 le 120579 le 2120587 minus 120593
(9)
where critical angle 120593and critical distance 119871
are a function
of pinching degree 1 minus
119887
1198870 = 1 minus 2
119887
120593
= arctan( 4
120587
sdot
119887
1 minus 2
119887
)
119871
=
120587 (1 minus 2
119887)
4
(10)
The major axis 119886and area
119860
are then expressed by a
linear and respectively quadratic function of the imposedparameter 119887
119886
=
119887 +
120587 (1 minus 2
119887)
4
=
120587
4
+ (1 minus
120587
2
)
119887
119860
= 120587
119887
2+ 4120587
119887 (1 minus 2
119887) = 120587
119887 (1 minus 7
119887)
(11)
4 Mathematical Problems in Engineering
Ellipse (b P = 120587) or (a b)
b
a
(a)
L = btan(120593)
Stadium (b P = 120587) or (a b)
120593bb
a
b
(b)Figure 2 Overview geometrical nonbuckling ring models requiring two geometrical input parameters corresponding to a circular ring withperimeter
1198750 = 120587 and area
1198600 = 1205874 (a) elliptic cross-section with major axis 119886 and minor axis 119887 indicated and (b) stadium with roundededges with minor axis 119887 and critical angle 120593 indicated from which the critical distance 119871 =
119887 tan(120593) can be derived
Consequently the sensitivity of the major axis Δ119886and
area Δ
119860
to a variation of the imposed parameter Δ119887 results
in respectively a constant and a linear function of 119887
Δ119886
Δ
119887
= 1 minus
120587
2
Δ
119860
Δ
119887
= 2120587 (2 minus 7
119887)
(12)
whereas the sensitivity of the radius to a variation of theexperimental imposed parameter Δ119887 is again expressed by apiecewise function of 120579
Δ119903
(120579)
Δ
119887
=
1
sin (120579) for 120593 le 120579 le 120587 minus 120593
16
119887 minus 2120587
2(1 minus 2
119887) sin2 (120579)
4
radic
16
119887
2minus 120587
2(1 minus 2
119887)
2
sin2 (120579)minus
120587
2
cos (120579) for minus 120593 lt 120579 lt 120593
16
119887 minus 2120587
2(1 minus 2
119887) sin2 (120579)
4
radic
16
119887
2minus 120587
2(1 minus 2
119887)
2
sin2 (120579)+
120587
2
cos (120579) for 120587 minus 120593 lt 120579 lt 120587 + 120593
minus
1
sin (120579) for 120587 + 120593 le 120579 le 2120587 minus 120593
(13)
which 119903(9) depends explicitly on the pinching degree 1minus1198871198870
or 1 minus 2
119887Next the major axis 119886 instead of the perimeter
119875 is usedas a second known geometrical parameter in addition to the
minor axis 119887 The stadium ring shape 119903119886(120579) with parameters
(
119887 119886) in polar coordinates (119903 120579) is again a piecewise functionof 120579 isin [0 2120587]
119903
119886(120579) =
119887
sin (120579) for 120593 le 120579 le 120587 minus 120593
radic
119887
2minus 119886
2(1 minus
119887
2
119886
2) sin2 (120579) + (119886 minus
119887) cos (120579) for minus 120593 lt 120579 lt 120593
radic
119887
2minus 119886
2(1 minus
119887
2
119886
2) sin2 (120579) minus (119886 minus
119887) cos (120579) for 120587 minus 120593 lt 120579 lt 120587 + 120593
minus
119887
sin (120579) for 120587 + 120593 le 120579 le 2120587 minus 120593
(14)
Therefore 119903
119886(14) does not depend explicitly on the
pinching degree (1 minus 2
119887) as was the case for (9) but it doesdepend explicitly on the critical distance
119871 = 119886 minus
119887 and thestadium eccentricity 119890 =
radic
1 minus
119887
2119886
2 The perimeter
119875
119886and
Mathematical Problems in Engineering 5
area
119860
119886are respectively a linear and quadratic function of
119887 and depend linearly on 119886 as well
119875
119886= 2
119887 (120587 minus 2) + 4119886
119860
119886= 4119886
119887 +
119887
2(120587 minus 4)
(15)
Consequently the sensitivity of the perimeter Δ
119875
119886and
area Δ
119860
119886to a variation of the input parameters Δ119886 and
Δ
119887 is respectively constant and linearly dependent on
119887 and119886
Δ
119875
119886= 2120587Δ
119887 + 4Δ119886
Δ
119860
119886= 4
119887Δ119886 + (2 (120587 minus 4)
119887 + 4119886)Δ
119887
(16)
The sensitivity of the radius to a variation of the experi-mentally imposed parameter Δ119887 is a piecewise function of 120579
Δ119903
119886(120579) =
1
sin 120579 for 120593 le 120579 le 120587 minus 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119887cos2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
minus cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for minus 120593 lt 120579 lt 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119886sin2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
+ cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for 120587 minus 120593 lt 120579 lt 120587 + 120593
1
sin 120579 for 120587 + 120593 le 120579 le 2120587 minus 120593
(17)
which depends explicitly on the stadium eccentricity since(
119887
2minus 119886
2) = minus119890
2119886
2
213 Comparison of Ellipse and Stadium Ring Models withPerimeter 120587 The geometrical ellipse and stadium ring mod-els with the assumption of constant perimeter 119875 = 120587 are fullydetermined as a function of a single parameter We considerthe minor axis 119887 as a parameter since it is directly related tothe pinching degree 1minus1198871198870 and therefore a control parameterfor ring deformation regardless of the applied model Singleparameter models are in particular of interest to be used incombination with other analytical flow or acoustic modelswhich require a geometrical ring (or cross-section) parametersuch as major axis 119886 or enveloped area
119860 Therefore inFigure 3 we consider the impact of the applied geometricalring model in absence of buckling (ellipse or stadium) onthese parameters as a function of pinching degree 1 minus
1198871198870Following (11) the normalised major axis 119886 of the stadiumring model increases linearly (Figure 8(c)) as a function ofpinching degree 1minus1198871198870 whereas the normalised area
119860 of thestadium model decreases quadratically (Figure 8(d)) From(4) and (5) we deduce that the ring model results in the sametendencies for major axis 119886 (maximum difference betweenellipse and stadium for all pinching degrees lt01) and area
119860 (maximum difference between ellipse and stadium for allpinching degrees lt005) although the increase for 119886 is notlinear as is the case for the stadium model Major axis 119886
of the ellipse is longer than the value of stadium model sothat the area
119860 enveloped by the ellipse is smaller than thearea enveloped by the stadium model The relative differencebetween the stadium and ellipse model for major axis 119886 andarea
119860 is further illustrated in Figure 3(c) Where the models
are equivalent for both extreme pinching degrees associatedwith a circular cross section in absence of pinching (1 minus
1198871198870 = 0 and 119886 =
1198870 = 05) and a completely pinchedring (1 minus
1198871198870 = 100 and cross-section area
119860 = 0) themaximum difference (lt12) of major axis 119886 between theellipse and stadium model occurs for a pinching degree of52 whereas the maximum difference (lt7) between bothmodels for area
119860 occurs for a pinching degree of 66 Thesensitivity of major axis Δ119886 and area Δ
119860 to a variation ofminor axis Δ
119887 is illustrated in Figures 3(d) and 3(e) It isseen that Δ119886 le Δ
119887 so that for both the ellipse and stadium119886 varies less than
119887 (maximum difference for all pinchingdegrees between ellipse and stadium lt08) For the stadiumthe variation of 119886 is independent from the pinching degree 1minus
1198871198870 following (12) For the ellipse 119886 becomes less sensitive toa variation of 119887 as the pinching degree increases as expressedin (7) For small pinching degrees the impact of a variationof 119887 on the area
119860 is small (Δ
119860 le Δ
119887 for pinching degreesle17 for the ellipse and le32 for the stadium) As thepinching degree increases the impact of a variation of Δ119887 onthe area becomes more important In general the impact ofa variation of 119887 on the area is of the same order of magnitudefor the ellipse and the stadium (maximum difference forall pinching degrees lt07) Nevertheless the area using thestadium ring is less sensitive to a variation Δ
119887 than whenan ellipse is used for pinching degrees le60 for pinchingdegrees gt60 the opposite holds The ellipse and stadiumring shape 119903(120579) and associated sensitivity to a variation ofminor axis Δ119903(120579) are illustrated in Figure 4(a) for a pinchingdegree of 50 As expressed in (6) and in (13) the sensitivitydepends on the angle so that it is of interest to consider a
6 Mathematical Problems in Engineering
EllipseStadium
04
06
08
a(mdash
)
0 50 1001 minus bb0 ()
lt01
a0
(a) 119886
0
02
04
06
A(mdash
)
lt005
A0
EllipseStadium
0 50 1001 minus bb0 ()
(b) 119860
0
5
10
15
|middotsminusmiddote|middot 0
()
middot = Amiddot = a
lt7
52lt12
66
0 50 1001 minus bb0 ()
(c) |119886119904 minus 119886119890|1198860 and |119860119904minus 119860119890|1198600
0
05
1
ΔaΔ
b(mdash
)
lt08
91
0 50 1001 minus bb0 ()
EllipseStadium
(d) Δ119886Δ
60
lt07
0
2
4
ΔAΔ
b(mdash
)
3217
0 50 1001 minus bb0 ()
1
EllipseStadium
(e) Δ119860Δ
Figure 3 Impact of applied geometrical ring models in absence of buckling ellipse (full line in ((a) (b) (d) and (e)) and subscript 119890 in (c))and stadium (dashed line in ((a) (b) (d) and (e)) and subscript 119904 in (c)) on normalised major axis 119886 and normalised area
119860 as a function ofpinching degree 1 minus
1198871198870 with the maximum difference indicated on each subfigure (ltsdot) (a) major axis 119886 and 1198860 = 05 (thin dash-dot line)(b) area
119860 and
1198600 = 1205874 (thin dash-dot line) (c) difference between stadium model (subscript 119890) and ellipse (subscript 119904) relative to thecircular value for major axis 119886 (with a maximum difference for pinching degree 52) and area
119860 (with a maximum difference for pinchingdegree 66) (d) sensitivity of major axis Δ119886 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 91) and(e) sensitivity of area Δ
119860 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 60) The maximum differenceis indicated as ltsdot
Mathematical Problems in Engineering 7
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
05
1
15
2
30
210
60
240
90
270
120
300
150
330
0180
EllipseStadium
EllipseStadium
(a) Ring shape 119903(120579) (left) and sensitivity Δ119903(120579)Δ (right) for 1 minus 0 = 50
0 50 1000
10
20
30
65
1
1 minus bb0 ()
EllipseStadium
gt1lt1
r(120579)Δb
maxΔ
(b) max(Δ119903(120579))Δ
0 50 1000
1
2
3
371Δ
r(120579)Δb
lt03
1 minus bb0 ()
EllipseStadium
(c) Δ119903(120579)Δ
Figure 4 Illustration of overall sensitivity Δ119903(120579) of elliptic (full line) and stadium (dashed line) ring shape 119903(120579) to a variation of the minoraxis Δ119887 as a function of pinching degree 1minus1198871198870 (a) example for pinching degree 50 (b) maximum value of Δ119903(120579) and (c) mean sensitivityΔ119903(120579)
globalmeasure of the ring shape to a variation of 119887Thereforethe maximum value of Δ119903(120579) and mean sensitivity Δ119903(120579) forthe entire ring as a function of pinching degree 1 minus
1198871198870
are plotted in Figures 4(b) and 4(c) respectively In contrastto the sensitivity of 119886 which decreases or remains constantfor increasing pinching degree as shown in Figure 3(d)both global measures indicate that the overall sensitivity ofthe shape increases with the pinching degree Whereas themean sensitivity increase remains limited (lt3) themaximumsensitivity increase is more important since it yields up togt5 times the variation imposed on Δ
119887 for both the ellipseand the stadium As was observed for the area 119860 and majoraxis 119886 the stadium ring model is more sensitive than theelliptic ring model to a variation of 119887 for severe constrictiondegrees (ge60) The observed maximum difference (gt1for pinching degrees gt65) is more of less averaged outsince the difference in mean sensitivity between both ringmodels remains small (lt03) regardless of the pinchingdegree
22 Buckling Ring Model Peanut In Section 21 two geo-metrical ring models with low computational cost were
considered Both models reduced to a single parameter ringmodel of the minor axis 119887 when applying the assumptionof conservation of the perimeter
119875 In this section a quasi-analytical elastic peanut ring models is outlined Analyticalelastic ring models are expected to become pertinent whenthe length of the parallel pinching bars (see Figure 1) reduces(119897119861 lt 2119886) so that the loading becomes axial and buckling islikely to occur [4 6]
221 Peanut Each quarter of the peanut ring with center 119900consists of two arc portions as schematised in Figure 5 Thepeanut ring is then expressed as a function of the radius 120588 ofthe outer arc (with origin 1199001) and 120585 twice the angle spannedby the inner arc (with origin 1199002) so that the peanut parameterset is (120588 120585) The peanut ring model is suitable to approximatenonbuckled (119887 ge 120588) and buckled (119887 lt 120588 second mode)elastic rings The assumption of conservation of perimeter ismade so that the normalized perimeter of the total peanutequals the perimeter of the circular tube
119875 = 120587 andconsequently the quarter peanut has a constant length 1205874The total curvature of each quarter peanut yields 1205872 Theminor axis 119887 le 05 and major axis 119886 ge 05 of the nonbuckled
8 Mathematical Problems in Engineering
1205852
b
ao2
o1
o
b ge Nonbuckled
(a)
o2 a
1205852
o1
b
o
Buckled b lt
(b)
Figure 5 Illustration of nonbuckled ((a) inner arc curvature is positive) and buckled ((b) inner arc curvature is negative) quarters of peanutrings with center 119900 and parameter set (120588 120585) Each ring consists of two arc portions outer arc (thick gray line) with origin 1199001 and radius 120588 andinner arc (thick black line) with origin 1199002 and angle 1205852 The major axis 119886 ge 05 and minor axis 119887 le 05 of the peanut are indicated For thenonbuckled (a) and buckled peanut (b) 119887 ge 120588 and 119887 lt 120588 hold respectively
and buckled peanuts are obtained from parameter set (120588 120585)as
non-buckled 119887 = 120587 (1 minus 2120588)
1 minus cos (1205852)2120585
+ 120588
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 120588
(18)
buckled 119887 = (
120587 (1 minus 2120588)
2120585
+ 120588)(cos 1205852
minus 1)
+ 120588 cos 1205852
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 2120588 sin 120585
2
+ 120588
(19)
The area
119860(120588 120585) of the total peanut is given as
119860 = 120587120588 minus 120587120588
2+
120587
2
4
(1 minus 2120588)
2119891 (120585)
with 119891 (120585) =
120585 minus sin 120585120585
2
(20)
Note that for 120585 = 0 the last term in (20) is omitted since119891(120585 = 0) = 0The bending energy of the quarter peanut yields
1
4
E =
120587 minus 120585
2120588
+
120585
2
120587 minus 2 (120587 minus 120585) 120588
(21)
with E(120588 120585) denoting the bending energy of the total peanutThe bending energy (21) is then used to express the Lagrange
functional L associated with the energy of deformation ofthe peanut ring as a function of (120588 120585) as well
1
BL =
120587 minus 120585
120588
+
120585
2
1205872 minus (120587 minus 120585) 120588
+ 120582(
D2minus
120587
2
16
(1 minus 2120588)
2(1 minus 120587119891 (120585)))
1
2120587
(22)
with isoperimetric difference of the quarter peanut D2 =
(1205874)
2minus 120587
1198604 (0 le
D) normalized pressure 120582 = 8PBdenoting the ratio of hydrostatic pressure P and flexuralrigiditymodulus of the ringB = Υ
119889
312(1minus]2)usingYoungrsquos
modulus Υ Poissonrsquos ratio ] and normalised wall thickness
119889For a given
119860 it is aimed to find the peanut definedby parameter set (120588 120585) which minimises the bending energyE When using the normalised pressure 120582 as a Lagrangemultiplier 120582 the constrained minimisation problem satisfiesthe Euler-Lagrange equations in terms of (120588 120585 120582)
0 =
120587120588 minus 1205872 minus 2120585120588
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588
2 1 minus 120587119891 (120585)
120587 minus 120585
0 =
minus2
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588119891
1015840(120585)
120588 =
2
120587
(
120587
4
minus
D
radic1 minus 120587119891 (120585)
)
(23)
where the prime denotes the derivative with respect to 120585 Thesystem (23) is then solved firstly by eliminating 120582(120588 120585) usingthe first two expressions of (23)
120582 =
8
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2 (24)
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
+
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus120587(
10119886 + 6
119887
2radic(119886 + 3
119887) (3119886 +
119887)
minus 3)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
Δ
119887
Δ
119860
119886119887= 120587119886Δ
119887 + 120587
119887Δ119886
(3)
When the perimeter is used as a second model inputparameter instead of themajor axis 119886 we assume that its valueis conserved when the elastic ring deforms so that
119875
= 120587
holds for all pinching degrees 1 minus1198871198870 The elliptic ring shapein polar coordinates for 120579 isin [0 2120587] is again of the form (1) forwhich eccentricity 119890 is only a function of 119887 since using (2) itfollows that
119903
(120579) =
radic
119887
2
1 minus (1 minus
119887
2119886
2
) cos2 (120579)
with 119886
=
radic
minus20
119887
2+ 12
119887 + 3
6
minus
2
119887
3
+
1
2
(4)
The area
119860 yields again
119860
= 120587
119887119886
or
119860
= 120587
119887(
radic
minus20
119887
2+ 12
119887 + 3
6
minus
2
119887
3
+
1
2
) (5)
The sensitivity of the radius to a variation of the imposedparameter Δ119887 yields
Δ119903
(120579)
Δ
119887
=
1
119903
(120579)
minus
119887
2
119886
2
sdot
cos2 (120579)119903
3
(120579)
(1 +
119887
3119886
[
10
119887 minus 3
radic
minus20
119887
2+ 12
119887 + 3
+ 2])
(6)
where the last term expresses the sensitivity of the major axisto a variation of the imposed parameter Δ119887 since
Δ119886
Δ
119887
=
minus1
3
(
10
119887 minus 3
radic
minus20
119887
2+ 12
119887 + 3
+ 2) (7)
Finally the sensitivity of the area to a variation of theimposed parameter Δ119887 for a constant perimeter becomes
Δ
119860
Δ
119887
=
120587
2
minus
4120587
119887
3
+
120587 (minus20
119887
2+ 12
119887 + 3)
12
6
+
120587
119887 (minus20
119887
2+ 12
119887 + 3)
minus12
12
(8)
212 Stadium with Rounded Edges The stadium shapeillustrated in Figure 2 consists of two flat portions connectedby circular arcs with radius 119887 so that the minor axis of thestadium equals the minor axis of the ellipse The criticalangle 120593 isin [0 1205872] is defined by the critical distance
119871 =
119887 tan(120593) ge 0 which corresponds to half the extent of the flatportion in contact with the parallel pinching bars As for theellipse (Section 211) the stadium shape is fully determinedby considering the minor axis 119887 and a second geometricalparameter such as perimeter 119875 or major axis 119886
Firstly assume that the perimeter
119875 is conserved whenthe elastic rings deform so that 119875
= 120587 holds for all pinching
degrees 1minus1198871198870 In this case the stadium ring shape 119903(120579)withparameters (119887 119875 = 120587) in polar coordinates (119903 120579) is given as apiecewise function of 120579 isin [0 2120587]which depends explicitly onthe pinching degree 1 minus
119887
1198870 = 1 minus 2
119887
119903
(120579) =
119887
sin (120579) for120593 le 120579 le 120587 minus 120593
120587
4
sdot (1 minus 2
119887) sdot ((
16
120587
2sdot
119887
2
(1 minus 2
119887)
2minus sin2 (120579))
12
+ cos (120579)) for minus 120593 lt 120579 lt 120593
120587
4
sdot (1 minus 2
119887) sdot ((
16
120587
2sdot
119887
2
(1 minus 2
119887)
2minus sin2 (120579))
12
minus cos (120579)) for120587 minus 120593 lt 120579 lt 120587 + 120593
minus
119887
sin (120579) for120587 + 120593 le 120579 le 2120587 minus 120593
(9)
where critical angle 120593and critical distance 119871
are a function
of pinching degree 1 minus
119887
1198870 = 1 minus 2
119887
120593
= arctan( 4
120587
sdot
119887
1 minus 2
119887
)
119871
=
120587 (1 minus 2
119887)
4
(10)
The major axis 119886and area
119860
are then expressed by a
linear and respectively quadratic function of the imposedparameter 119887
119886
=
119887 +
120587 (1 minus 2
119887)
4
=
120587
4
+ (1 minus
120587
2
)
119887
119860
= 120587
119887
2+ 4120587
119887 (1 minus 2
119887) = 120587
119887 (1 minus 7
119887)
(11)
4 Mathematical Problems in Engineering
Ellipse (b P = 120587) or (a b)
b
a
(a)
L = btan(120593)
Stadium (b P = 120587) or (a b)
120593bb
a
b
(b)Figure 2 Overview geometrical nonbuckling ring models requiring two geometrical input parameters corresponding to a circular ring withperimeter
1198750 = 120587 and area
1198600 = 1205874 (a) elliptic cross-section with major axis 119886 and minor axis 119887 indicated and (b) stadium with roundededges with minor axis 119887 and critical angle 120593 indicated from which the critical distance 119871 =
119887 tan(120593) can be derived
Consequently the sensitivity of the major axis Δ119886and
area Δ
119860
to a variation of the imposed parameter Δ119887 results
in respectively a constant and a linear function of 119887
Δ119886
Δ
119887
= 1 minus
120587
2
Δ
119860
Δ
119887
= 2120587 (2 minus 7
119887)
(12)
whereas the sensitivity of the radius to a variation of theexperimental imposed parameter Δ119887 is again expressed by apiecewise function of 120579
Δ119903
(120579)
Δ
119887
=
1
sin (120579) for 120593 le 120579 le 120587 minus 120593
16
119887 minus 2120587
2(1 minus 2
119887) sin2 (120579)
4
radic
16
119887
2minus 120587
2(1 minus 2
119887)
2
sin2 (120579)minus
120587
2
cos (120579) for minus 120593 lt 120579 lt 120593
16
119887 minus 2120587
2(1 minus 2
119887) sin2 (120579)
4
radic
16
119887
2minus 120587
2(1 minus 2
119887)
2
sin2 (120579)+
120587
2
cos (120579) for 120587 minus 120593 lt 120579 lt 120587 + 120593
minus
1
sin (120579) for 120587 + 120593 le 120579 le 2120587 minus 120593
(13)
which 119903(9) depends explicitly on the pinching degree 1minus1198871198870
or 1 minus 2
119887Next the major axis 119886 instead of the perimeter
119875 is usedas a second known geometrical parameter in addition to the
minor axis 119887 The stadium ring shape 119903119886(120579) with parameters
(
119887 119886) in polar coordinates (119903 120579) is again a piecewise functionof 120579 isin [0 2120587]
119903
119886(120579) =
119887
sin (120579) for 120593 le 120579 le 120587 minus 120593
radic
119887
2minus 119886
2(1 minus
119887
2
119886
2) sin2 (120579) + (119886 minus
119887) cos (120579) for minus 120593 lt 120579 lt 120593
radic
119887
2minus 119886
2(1 minus
119887
2
119886
2) sin2 (120579) minus (119886 minus
119887) cos (120579) for 120587 minus 120593 lt 120579 lt 120587 + 120593
minus
119887
sin (120579) for 120587 + 120593 le 120579 le 2120587 minus 120593
(14)
Therefore 119903
119886(14) does not depend explicitly on the
pinching degree (1 minus 2
119887) as was the case for (9) but it doesdepend explicitly on the critical distance
119871 = 119886 minus
119887 and thestadium eccentricity 119890 =
radic
1 minus
119887
2119886
2 The perimeter
119875
119886and
Mathematical Problems in Engineering 5
area
119860
119886are respectively a linear and quadratic function of
119887 and depend linearly on 119886 as well
119875
119886= 2
119887 (120587 minus 2) + 4119886
119860
119886= 4119886
119887 +
119887
2(120587 minus 4)
(15)
Consequently the sensitivity of the perimeter Δ
119875
119886and
area Δ
119860
119886to a variation of the input parameters Δ119886 and
Δ
119887 is respectively constant and linearly dependent on
119887 and119886
Δ
119875
119886= 2120587Δ
119887 + 4Δ119886
Δ
119860
119886= 4
119887Δ119886 + (2 (120587 minus 4)
119887 + 4119886)Δ
119887
(16)
The sensitivity of the radius to a variation of the experi-mentally imposed parameter Δ119887 is a piecewise function of 120579
Δ119903
119886(120579) =
1
sin 120579 for 120593 le 120579 le 120587 minus 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119887cos2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
minus cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for minus 120593 lt 120579 lt 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119886sin2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
+ cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for 120587 minus 120593 lt 120579 lt 120587 + 120593
1
sin 120579 for 120587 + 120593 le 120579 le 2120587 minus 120593
(17)
which depends explicitly on the stadium eccentricity since(
119887
2minus 119886
2) = minus119890
2119886
2
213 Comparison of Ellipse and Stadium Ring Models withPerimeter 120587 The geometrical ellipse and stadium ring mod-els with the assumption of constant perimeter 119875 = 120587 are fullydetermined as a function of a single parameter We considerthe minor axis 119887 as a parameter since it is directly related tothe pinching degree 1minus1198871198870 and therefore a control parameterfor ring deformation regardless of the applied model Singleparameter models are in particular of interest to be used incombination with other analytical flow or acoustic modelswhich require a geometrical ring (or cross-section) parametersuch as major axis 119886 or enveloped area
119860 Therefore inFigure 3 we consider the impact of the applied geometricalring model in absence of buckling (ellipse or stadium) onthese parameters as a function of pinching degree 1 minus
1198871198870Following (11) the normalised major axis 119886 of the stadiumring model increases linearly (Figure 8(c)) as a function ofpinching degree 1minus1198871198870 whereas the normalised area
119860 of thestadium model decreases quadratically (Figure 8(d)) From(4) and (5) we deduce that the ring model results in the sametendencies for major axis 119886 (maximum difference betweenellipse and stadium for all pinching degrees lt01) and area
119860 (maximum difference between ellipse and stadium for allpinching degrees lt005) although the increase for 119886 is notlinear as is the case for the stadium model Major axis 119886
of the ellipse is longer than the value of stadium model sothat the area
119860 enveloped by the ellipse is smaller than thearea enveloped by the stadium model The relative differencebetween the stadium and ellipse model for major axis 119886 andarea
119860 is further illustrated in Figure 3(c) Where the models
are equivalent for both extreme pinching degrees associatedwith a circular cross section in absence of pinching (1 minus
1198871198870 = 0 and 119886 =
1198870 = 05) and a completely pinchedring (1 minus
1198871198870 = 100 and cross-section area
119860 = 0) themaximum difference (lt12) of major axis 119886 between theellipse and stadium model occurs for a pinching degree of52 whereas the maximum difference (lt7) between bothmodels for area
119860 occurs for a pinching degree of 66 Thesensitivity of major axis Δ119886 and area Δ
119860 to a variation ofminor axis Δ
119887 is illustrated in Figures 3(d) and 3(e) It isseen that Δ119886 le Δ
119887 so that for both the ellipse and stadium119886 varies less than
119887 (maximum difference for all pinchingdegrees between ellipse and stadium lt08) For the stadiumthe variation of 119886 is independent from the pinching degree 1minus
1198871198870 following (12) For the ellipse 119886 becomes less sensitive toa variation of 119887 as the pinching degree increases as expressedin (7) For small pinching degrees the impact of a variationof 119887 on the area
119860 is small (Δ
119860 le Δ
119887 for pinching degreesle17 for the ellipse and le32 for the stadium) As thepinching degree increases the impact of a variation of Δ119887 onthe area becomes more important In general the impact ofa variation of 119887 on the area is of the same order of magnitudefor the ellipse and the stadium (maximum difference forall pinching degrees lt07) Nevertheless the area using thestadium ring is less sensitive to a variation Δ
119887 than whenan ellipse is used for pinching degrees le60 for pinchingdegrees gt60 the opposite holds The ellipse and stadiumring shape 119903(120579) and associated sensitivity to a variation ofminor axis Δ119903(120579) are illustrated in Figure 4(a) for a pinchingdegree of 50 As expressed in (6) and in (13) the sensitivitydepends on the angle so that it is of interest to consider a
6 Mathematical Problems in Engineering
EllipseStadium
04
06
08
a(mdash
)
0 50 1001 minus bb0 ()
lt01
a0
(a) 119886
0
02
04
06
A(mdash
)
lt005
A0
EllipseStadium
0 50 1001 minus bb0 ()
(b) 119860
0
5
10
15
|middotsminusmiddote|middot 0
()
middot = Amiddot = a
lt7
52lt12
66
0 50 1001 minus bb0 ()
(c) |119886119904 minus 119886119890|1198860 and |119860119904minus 119860119890|1198600
0
05
1
ΔaΔ
b(mdash
)
lt08
91
0 50 1001 minus bb0 ()
EllipseStadium
(d) Δ119886Δ
60
lt07
0
2
4
ΔAΔ
b(mdash
)
3217
0 50 1001 minus bb0 ()
1
EllipseStadium
(e) Δ119860Δ
Figure 3 Impact of applied geometrical ring models in absence of buckling ellipse (full line in ((a) (b) (d) and (e)) and subscript 119890 in (c))and stadium (dashed line in ((a) (b) (d) and (e)) and subscript 119904 in (c)) on normalised major axis 119886 and normalised area
119860 as a function ofpinching degree 1 minus
1198871198870 with the maximum difference indicated on each subfigure (ltsdot) (a) major axis 119886 and 1198860 = 05 (thin dash-dot line)(b) area
119860 and
1198600 = 1205874 (thin dash-dot line) (c) difference between stadium model (subscript 119890) and ellipse (subscript 119904) relative to thecircular value for major axis 119886 (with a maximum difference for pinching degree 52) and area
119860 (with a maximum difference for pinchingdegree 66) (d) sensitivity of major axis Δ119886 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 91) and(e) sensitivity of area Δ
119860 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 60) The maximum differenceis indicated as ltsdot
Mathematical Problems in Engineering 7
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
05
1
15
2
30
210
60
240
90
270
120
300
150
330
0180
EllipseStadium
EllipseStadium
(a) Ring shape 119903(120579) (left) and sensitivity Δ119903(120579)Δ (right) for 1 minus 0 = 50
0 50 1000
10
20
30
65
1
1 minus bb0 ()
EllipseStadium
gt1lt1
r(120579)Δb
maxΔ
(b) max(Δ119903(120579))Δ
0 50 1000
1
2
3
371Δ
r(120579)Δb
lt03
1 minus bb0 ()
EllipseStadium
(c) Δ119903(120579)Δ
Figure 4 Illustration of overall sensitivity Δ119903(120579) of elliptic (full line) and stadium (dashed line) ring shape 119903(120579) to a variation of the minoraxis Δ119887 as a function of pinching degree 1minus1198871198870 (a) example for pinching degree 50 (b) maximum value of Δ119903(120579) and (c) mean sensitivityΔ119903(120579)
globalmeasure of the ring shape to a variation of 119887Thereforethe maximum value of Δ119903(120579) and mean sensitivity Δ119903(120579) forthe entire ring as a function of pinching degree 1 minus
1198871198870
are plotted in Figures 4(b) and 4(c) respectively In contrastto the sensitivity of 119886 which decreases or remains constantfor increasing pinching degree as shown in Figure 3(d)both global measures indicate that the overall sensitivity ofthe shape increases with the pinching degree Whereas themean sensitivity increase remains limited (lt3) themaximumsensitivity increase is more important since it yields up togt5 times the variation imposed on Δ
119887 for both the ellipseand the stadium As was observed for the area 119860 and majoraxis 119886 the stadium ring model is more sensitive than theelliptic ring model to a variation of 119887 for severe constrictiondegrees (ge60) The observed maximum difference (gt1for pinching degrees gt65) is more of less averaged outsince the difference in mean sensitivity between both ringmodels remains small (lt03) regardless of the pinchingdegree
22 Buckling Ring Model Peanut In Section 21 two geo-metrical ring models with low computational cost were
considered Both models reduced to a single parameter ringmodel of the minor axis 119887 when applying the assumptionof conservation of the perimeter
119875 In this section a quasi-analytical elastic peanut ring models is outlined Analyticalelastic ring models are expected to become pertinent whenthe length of the parallel pinching bars (see Figure 1) reduces(119897119861 lt 2119886) so that the loading becomes axial and buckling islikely to occur [4 6]
221 Peanut Each quarter of the peanut ring with center 119900consists of two arc portions as schematised in Figure 5 Thepeanut ring is then expressed as a function of the radius 120588 ofthe outer arc (with origin 1199001) and 120585 twice the angle spannedby the inner arc (with origin 1199002) so that the peanut parameterset is (120588 120585) The peanut ring model is suitable to approximatenonbuckled (119887 ge 120588) and buckled (119887 lt 120588 second mode)elastic rings The assumption of conservation of perimeter ismade so that the normalized perimeter of the total peanutequals the perimeter of the circular tube
119875 = 120587 andconsequently the quarter peanut has a constant length 1205874The total curvature of each quarter peanut yields 1205872 Theminor axis 119887 le 05 and major axis 119886 ge 05 of the nonbuckled
8 Mathematical Problems in Engineering
1205852
b
ao2
o1
o
b ge Nonbuckled
(a)
o2 a
1205852
o1
b
o
Buckled b lt
(b)
Figure 5 Illustration of nonbuckled ((a) inner arc curvature is positive) and buckled ((b) inner arc curvature is negative) quarters of peanutrings with center 119900 and parameter set (120588 120585) Each ring consists of two arc portions outer arc (thick gray line) with origin 1199001 and radius 120588 andinner arc (thick black line) with origin 1199002 and angle 1205852 The major axis 119886 ge 05 and minor axis 119887 le 05 of the peanut are indicated For thenonbuckled (a) and buckled peanut (b) 119887 ge 120588 and 119887 lt 120588 hold respectively
and buckled peanuts are obtained from parameter set (120588 120585)as
non-buckled 119887 = 120587 (1 minus 2120588)
1 minus cos (1205852)2120585
+ 120588
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 120588
(18)
buckled 119887 = (
120587 (1 minus 2120588)
2120585
+ 120588)(cos 1205852
minus 1)
+ 120588 cos 1205852
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 2120588 sin 120585
2
+ 120588
(19)
The area
119860(120588 120585) of the total peanut is given as
119860 = 120587120588 minus 120587120588
2+
120587
2
4
(1 minus 2120588)
2119891 (120585)
with 119891 (120585) =
120585 minus sin 120585120585
2
(20)
Note that for 120585 = 0 the last term in (20) is omitted since119891(120585 = 0) = 0The bending energy of the quarter peanut yields
1
4
E =
120587 minus 120585
2120588
+
120585
2
120587 minus 2 (120587 minus 120585) 120588
(21)
with E(120588 120585) denoting the bending energy of the total peanutThe bending energy (21) is then used to express the Lagrange
functional L associated with the energy of deformation ofthe peanut ring as a function of (120588 120585) as well
1
BL =
120587 minus 120585
120588
+
120585
2
1205872 minus (120587 minus 120585) 120588
+ 120582(
D2minus
120587
2
16
(1 minus 2120588)
2(1 minus 120587119891 (120585)))
1
2120587
(22)
with isoperimetric difference of the quarter peanut D2 =
(1205874)
2minus 120587
1198604 (0 le
D) normalized pressure 120582 = 8PBdenoting the ratio of hydrostatic pressure P and flexuralrigiditymodulus of the ringB = Υ
119889
312(1minus]2)usingYoungrsquos
modulus Υ Poissonrsquos ratio ] and normalised wall thickness
119889For a given
119860 it is aimed to find the peanut definedby parameter set (120588 120585) which minimises the bending energyE When using the normalised pressure 120582 as a Lagrangemultiplier 120582 the constrained minimisation problem satisfiesthe Euler-Lagrange equations in terms of (120588 120585 120582)
0 =
120587120588 minus 1205872 minus 2120585120588
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588
2 1 minus 120587119891 (120585)
120587 minus 120585
0 =
minus2
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588119891
1015840(120585)
120588 =
2
120587
(
120587
4
minus
D
radic1 minus 120587119891 (120585)
)
(23)
where the prime denotes the derivative with respect to 120585 Thesystem (23) is then solved firstly by eliminating 120582(120588 120585) usingthe first two expressions of (23)
120582 =
8
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2 (24)
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Ellipse (b P = 120587) or (a b)
b
a
(a)
L = btan(120593)
Stadium (b P = 120587) or (a b)
120593bb
a
b
(b)Figure 2 Overview geometrical nonbuckling ring models requiring two geometrical input parameters corresponding to a circular ring withperimeter
1198750 = 120587 and area
1198600 = 1205874 (a) elliptic cross-section with major axis 119886 and minor axis 119887 indicated and (b) stadium with roundededges with minor axis 119887 and critical angle 120593 indicated from which the critical distance 119871 =
119887 tan(120593) can be derived
Consequently the sensitivity of the major axis Δ119886and
area Δ
119860
to a variation of the imposed parameter Δ119887 results
in respectively a constant and a linear function of 119887
Δ119886
Δ
119887
= 1 minus
120587
2
Δ
119860
Δ
119887
= 2120587 (2 minus 7
119887)
(12)
whereas the sensitivity of the radius to a variation of theexperimental imposed parameter Δ119887 is again expressed by apiecewise function of 120579
Δ119903
(120579)
Δ
119887
=
1
sin (120579) for 120593 le 120579 le 120587 minus 120593
16
119887 minus 2120587
2(1 minus 2
119887) sin2 (120579)
4
radic
16
119887
2minus 120587
2(1 minus 2
119887)
2
sin2 (120579)minus
120587
2
cos (120579) for minus 120593 lt 120579 lt 120593
16
119887 minus 2120587
2(1 minus 2
119887) sin2 (120579)
4
radic
16
119887
2minus 120587
2(1 minus 2
119887)
2
sin2 (120579)+
120587
2
cos (120579) for 120587 minus 120593 lt 120579 lt 120587 + 120593
minus
1
sin (120579) for 120587 + 120593 le 120579 le 2120587 minus 120593
(13)
which 119903(9) depends explicitly on the pinching degree 1minus1198871198870
or 1 minus 2
119887Next the major axis 119886 instead of the perimeter
119875 is usedas a second known geometrical parameter in addition to the
minor axis 119887 The stadium ring shape 119903119886(120579) with parameters
(
119887 119886) in polar coordinates (119903 120579) is again a piecewise functionof 120579 isin [0 2120587]
119903
119886(120579) =
119887
sin (120579) for 120593 le 120579 le 120587 minus 120593
radic
119887
2minus 119886
2(1 minus
119887
2
119886
2) sin2 (120579) + (119886 minus
119887) cos (120579) for minus 120593 lt 120579 lt 120593
radic
119887
2minus 119886
2(1 minus
119887
2
119886
2) sin2 (120579) minus (119886 minus
119887) cos (120579) for 120587 minus 120593 lt 120579 lt 120587 + 120593
minus
119887
sin (120579) for 120587 + 120593 le 120579 le 2120587 minus 120593
(14)
Therefore 119903
119886(14) does not depend explicitly on the
pinching degree (1 minus 2
119887) as was the case for (9) but it doesdepend explicitly on the critical distance
119871 = 119886 minus
119887 and thestadium eccentricity 119890 =
radic
1 minus
119887
2119886
2 The perimeter
119875
119886and
Mathematical Problems in Engineering 5
area
119860
119886are respectively a linear and quadratic function of
119887 and depend linearly on 119886 as well
119875
119886= 2
119887 (120587 minus 2) + 4119886
119860
119886= 4119886
119887 +
119887
2(120587 minus 4)
(15)
Consequently the sensitivity of the perimeter Δ
119875
119886and
area Δ
119860
119886to a variation of the input parameters Δ119886 and
Δ
119887 is respectively constant and linearly dependent on
119887 and119886
Δ
119875
119886= 2120587Δ
119887 + 4Δ119886
Δ
119860
119886= 4
119887Δ119886 + (2 (120587 minus 4)
119887 + 4119886)Δ
119887
(16)
The sensitivity of the radius to a variation of the experi-mentally imposed parameter Δ119887 is a piecewise function of 120579
Δ119903
119886(120579) =
1
sin 120579 for 120593 le 120579 le 120587 minus 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119887cos2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
minus cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for minus 120593 lt 120579 lt 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119886sin2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
+ cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for 120587 minus 120593 lt 120579 lt 120587 + 120593
1
sin 120579 for 120587 + 120593 le 120579 le 2120587 minus 120593
(17)
which depends explicitly on the stadium eccentricity since(
119887
2minus 119886
2) = minus119890
2119886
2
213 Comparison of Ellipse and Stadium Ring Models withPerimeter 120587 The geometrical ellipse and stadium ring mod-els with the assumption of constant perimeter 119875 = 120587 are fullydetermined as a function of a single parameter We considerthe minor axis 119887 as a parameter since it is directly related tothe pinching degree 1minus1198871198870 and therefore a control parameterfor ring deformation regardless of the applied model Singleparameter models are in particular of interest to be used incombination with other analytical flow or acoustic modelswhich require a geometrical ring (or cross-section) parametersuch as major axis 119886 or enveloped area
119860 Therefore inFigure 3 we consider the impact of the applied geometricalring model in absence of buckling (ellipse or stadium) onthese parameters as a function of pinching degree 1 minus
1198871198870Following (11) the normalised major axis 119886 of the stadiumring model increases linearly (Figure 8(c)) as a function ofpinching degree 1minus1198871198870 whereas the normalised area
119860 of thestadium model decreases quadratically (Figure 8(d)) From(4) and (5) we deduce that the ring model results in the sametendencies for major axis 119886 (maximum difference betweenellipse and stadium for all pinching degrees lt01) and area
119860 (maximum difference between ellipse and stadium for allpinching degrees lt005) although the increase for 119886 is notlinear as is the case for the stadium model Major axis 119886
of the ellipse is longer than the value of stadium model sothat the area
119860 enveloped by the ellipse is smaller than thearea enveloped by the stadium model The relative differencebetween the stadium and ellipse model for major axis 119886 andarea
119860 is further illustrated in Figure 3(c) Where the models
are equivalent for both extreme pinching degrees associatedwith a circular cross section in absence of pinching (1 minus
1198871198870 = 0 and 119886 =
1198870 = 05) and a completely pinchedring (1 minus
1198871198870 = 100 and cross-section area
119860 = 0) themaximum difference (lt12) of major axis 119886 between theellipse and stadium model occurs for a pinching degree of52 whereas the maximum difference (lt7) between bothmodels for area
119860 occurs for a pinching degree of 66 Thesensitivity of major axis Δ119886 and area Δ
119860 to a variation ofminor axis Δ
119887 is illustrated in Figures 3(d) and 3(e) It isseen that Δ119886 le Δ
119887 so that for both the ellipse and stadium119886 varies less than
119887 (maximum difference for all pinchingdegrees between ellipse and stadium lt08) For the stadiumthe variation of 119886 is independent from the pinching degree 1minus
1198871198870 following (12) For the ellipse 119886 becomes less sensitive toa variation of 119887 as the pinching degree increases as expressedin (7) For small pinching degrees the impact of a variationof 119887 on the area
119860 is small (Δ
119860 le Δ
119887 for pinching degreesle17 for the ellipse and le32 for the stadium) As thepinching degree increases the impact of a variation of Δ119887 onthe area becomes more important In general the impact ofa variation of 119887 on the area is of the same order of magnitudefor the ellipse and the stadium (maximum difference forall pinching degrees lt07) Nevertheless the area using thestadium ring is less sensitive to a variation Δ
119887 than whenan ellipse is used for pinching degrees le60 for pinchingdegrees gt60 the opposite holds The ellipse and stadiumring shape 119903(120579) and associated sensitivity to a variation ofminor axis Δ119903(120579) are illustrated in Figure 4(a) for a pinchingdegree of 50 As expressed in (6) and in (13) the sensitivitydepends on the angle so that it is of interest to consider a
6 Mathematical Problems in Engineering
EllipseStadium
04
06
08
a(mdash
)
0 50 1001 minus bb0 ()
lt01
a0
(a) 119886
0
02
04
06
A(mdash
)
lt005
A0
EllipseStadium
0 50 1001 minus bb0 ()
(b) 119860
0
5
10
15
|middotsminusmiddote|middot 0
()
middot = Amiddot = a
lt7
52lt12
66
0 50 1001 minus bb0 ()
(c) |119886119904 minus 119886119890|1198860 and |119860119904minus 119860119890|1198600
0
05
1
ΔaΔ
b(mdash
)
lt08
91
0 50 1001 minus bb0 ()
EllipseStadium
(d) Δ119886Δ
60
lt07
0
2
4
ΔAΔ
b(mdash
)
3217
0 50 1001 minus bb0 ()
1
EllipseStadium
(e) Δ119860Δ
Figure 3 Impact of applied geometrical ring models in absence of buckling ellipse (full line in ((a) (b) (d) and (e)) and subscript 119890 in (c))and stadium (dashed line in ((a) (b) (d) and (e)) and subscript 119904 in (c)) on normalised major axis 119886 and normalised area
119860 as a function ofpinching degree 1 minus
1198871198870 with the maximum difference indicated on each subfigure (ltsdot) (a) major axis 119886 and 1198860 = 05 (thin dash-dot line)(b) area
119860 and
1198600 = 1205874 (thin dash-dot line) (c) difference between stadium model (subscript 119890) and ellipse (subscript 119904) relative to thecircular value for major axis 119886 (with a maximum difference for pinching degree 52) and area
119860 (with a maximum difference for pinchingdegree 66) (d) sensitivity of major axis Δ119886 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 91) and(e) sensitivity of area Δ
119860 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 60) The maximum differenceis indicated as ltsdot
Mathematical Problems in Engineering 7
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
05
1
15
2
30
210
60
240
90
270
120
300
150
330
0180
EllipseStadium
EllipseStadium
(a) Ring shape 119903(120579) (left) and sensitivity Δ119903(120579)Δ (right) for 1 minus 0 = 50
0 50 1000
10
20
30
65
1
1 minus bb0 ()
EllipseStadium
gt1lt1
r(120579)Δb
maxΔ
(b) max(Δ119903(120579))Δ
0 50 1000
1
2
3
371Δ
r(120579)Δb
lt03
1 minus bb0 ()
EllipseStadium
(c) Δ119903(120579)Δ
Figure 4 Illustration of overall sensitivity Δ119903(120579) of elliptic (full line) and stadium (dashed line) ring shape 119903(120579) to a variation of the minoraxis Δ119887 as a function of pinching degree 1minus1198871198870 (a) example for pinching degree 50 (b) maximum value of Δ119903(120579) and (c) mean sensitivityΔ119903(120579)
globalmeasure of the ring shape to a variation of 119887Thereforethe maximum value of Δ119903(120579) and mean sensitivity Δ119903(120579) forthe entire ring as a function of pinching degree 1 minus
1198871198870
are plotted in Figures 4(b) and 4(c) respectively In contrastto the sensitivity of 119886 which decreases or remains constantfor increasing pinching degree as shown in Figure 3(d)both global measures indicate that the overall sensitivity ofthe shape increases with the pinching degree Whereas themean sensitivity increase remains limited (lt3) themaximumsensitivity increase is more important since it yields up togt5 times the variation imposed on Δ
119887 for both the ellipseand the stadium As was observed for the area 119860 and majoraxis 119886 the stadium ring model is more sensitive than theelliptic ring model to a variation of 119887 for severe constrictiondegrees (ge60) The observed maximum difference (gt1for pinching degrees gt65) is more of less averaged outsince the difference in mean sensitivity between both ringmodels remains small (lt03) regardless of the pinchingdegree
22 Buckling Ring Model Peanut In Section 21 two geo-metrical ring models with low computational cost were
considered Both models reduced to a single parameter ringmodel of the minor axis 119887 when applying the assumptionof conservation of the perimeter
119875 In this section a quasi-analytical elastic peanut ring models is outlined Analyticalelastic ring models are expected to become pertinent whenthe length of the parallel pinching bars (see Figure 1) reduces(119897119861 lt 2119886) so that the loading becomes axial and buckling islikely to occur [4 6]
221 Peanut Each quarter of the peanut ring with center 119900consists of two arc portions as schematised in Figure 5 Thepeanut ring is then expressed as a function of the radius 120588 ofthe outer arc (with origin 1199001) and 120585 twice the angle spannedby the inner arc (with origin 1199002) so that the peanut parameterset is (120588 120585) The peanut ring model is suitable to approximatenonbuckled (119887 ge 120588) and buckled (119887 lt 120588 second mode)elastic rings The assumption of conservation of perimeter ismade so that the normalized perimeter of the total peanutequals the perimeter of the circular tube
119875 = 120587 andconsequently the quarter peanut has a constant length 1205874The total curvature of each quarter peanut yields 1205872 Theminor axis 119887 le 05 and major axis 119886 ge 05 of the nonbuckled
8 Mathematical Problems in Engineering
1205852
b
ao2
o1
o
b ge Nonbuckled
(a)
o2 a
1205852
o1
b
o
Buckled b lt
(b)
Figure 5 Illustration of nonbuckled ((a) inner arc curvature is positive) and buckled ((b) inner arc curvature is negative) quarters of peanutrings with center 119900 and parameter set (120588 120585) Each ring consists of two arc portions outer arc (thick gray line) with origin 1199001 and radius 120588 andinner arc (thick black line) with origin 1199002 and angle 1205852 The major axis 119886 ge 05 and minor axis 119887 le 05 of the peanut are indicated For thenonbuckled (a) and buckled peanut (b) 119887 ge 120588 and 119887 lt 120588 hold respectively
and buckled peanuts are obtained from parameter set (120588 120585)as
non-buckled 119887 = 120587 (1 minus 2120588)
1 minus cos (1205852)2120585
+ 120588
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 120588
(18)
buckled 119887 = (
120587 (1 minus 2120588)
2120585
+ 120588)(cos 1205852
minus 1)
+ 120588 cos 1205852
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 2120588 sin 120585
2
+ 120588
(19)
The area
119860(120588 120585) of the total peanut is given as
119860 = 120587120588 minus 120587120588
2+
120587
2
4
(1 minus 2120588)
2119891 (120585)
with 119891 (120585) =
120585 minus sin 120585120585
2
(20)
Note that for 120585 = 0 the last term in (20) is omitted since119891(120585 = 0) = 0The bending energy of the quarter peanut yields
1
4
E =
120587 minus 120585
2120588
+
120585
2
120587 minus 2 (120587 minus 120585) 120588
(21)
with E(120588 120585) denoting the bending energy of the total peanutThe bending energy (21) is then used to express the Lagrange
functional L associated with the energy of deformation ofthe peanut ring as a function of (120588 120585) as well
1
BL =
120587 minus 120585
120588
+
120585
2
1205872 minus (120587 minus 120585) 120588
+ 120582(
D2minus
120587
2
16
(1 minus 2120588)
2(1 minus 120587119891 (120585)))
1
2120587
(22)
with isoperimetric difference of the quarter peanut D2 =
(1205874)
2minus 120587
1198604 (0 le
D) normalized pressure 120582 = 8PBdenoting the ratio of hydrostatic pressure P and flexuralrigiditymodulus of the ringB = Υ
119889
312(1minus]2)usingYoungrsquos
modulus Υ Poissonrsquos ratio ] and normalised wall thickness
119889For a given
119860 it is aimed to find the peanut definedby parameter set (120588 120585) which minimises the bending energyE When using the normalised pressure 120582 as a Lagrangemultiplier 120582 the constrained minimisation problem satisfiesthe Euler-Lagrange equations in terms of (120588 120585 120582)
0 =
120587120588 minus 1205872 minus 2120585120588
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588
2 1 minus 120587119891 (120585)
120587 minus 120585
0 =
minus2
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588119891
1015840(120585)
120588 =
2
120587
(
120587
4
minus
D
radic1 minus 120587119891 (120585)
)
(23)
where the prime denotes the derivative with respect to 120585 Thesystem (23) is then solved firstly by eliminating 120582(120588 120585) usingthe first two expressions of (23)
120582 =
8
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2 (24)
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
area
119860
119886are respectively a linear and quadratic function of
119887 and depend linearly on 119886 as well
119875
119886= 2
119887 (120587 minus 2) + 4119886
119860
119886= 4119886
119887 +
119887
2(120587 minus 4)
(15)
Consequently the sensitivity of the perimeter Δ
119875
119886and
area Δ
119860
119886to a variation of the input parameters Δ119886 and
Δ
119887 is respectively constant and linearly dependent on
119887 and119886
Δ
119875
119886= 2120587Δ
119887 + 4Δ119886
Δ
119860
119886= 4
119887Δ119886 + (2 (120587 minus 4)
119887 + 4119886)Δ
119887
(16)
The sensitivity of the radius to a variation of the experi-mentally imposed parameter Δ119887 is a piecewise function of 120579
Δ119903
119886(120579) =
1
sin 120579 for 120593 le 120579 le 120587 minus 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119887cos2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
minus cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for minus 120593 lt 120579 lt 120593
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
119886sin2 (120579)
radic(
119887
2minus 119886
2) sin2 (120579) +
119887
2
+ cos (120579)
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
[Δ119886 + Δ
119887] for 120587 minus 120593 lt 120579 lt 120587 + 120593
1
sin 120579 for 120587 + 120593 le 120579 le 2120587 minus 120593
(17)
which depends explicitly on the stadium eccentricity since(
119887
2minus 119886
2) = minus119890
2119886
2
213 Comparison of Ellipse and Stadium Ring Models withPerimeter 120587 The geometrical ellipse and stadium ring mod-els with the assumption of constant perimeter 119875 = 120587 are fullydetermined as a function of a single parameter We considerthe minor axis 119887 as a parameter since it is directly related tothe pinching degree 1minus1198871198870 and therefore a control parameterfor ring deformation regardless of the applied model Singleparameter models are in particular of interest to be used incombination with other analytical flow or acoustic modelswhich require a geometrical ring (or cross-section) parametersuch as major axis 119886 or enveloped area
119860 Therefore inFigure 3 we consider the impact of the applied geometricalring model in absence of buckling (ellipse or stadium) onthese parameters as a function of pinching degree 1 minus
1198871198870Following (11) the normalised major axis 119886 of the stadiumring model increases linearly (Figure 8(c)) as a function ofpinching degree 1minus1198871198870 whereas the normalised area
119860 of thestadium model decreases quadratically (Figure 8(d)) From(4) and (5) we deduce that the ring model results in the sametendencies for major axis 119886 (maximum difference betweenellipse and stadium for all pinching degrees lt01) and area
119860 (maximum difference between ellipse and stadium for allpinching degrees lt005) although the increase for 119886 is notlinear as is the case for the stadium model Major axis 119886
of the ellipse is longer than the value of stadium model sothat the area
119860 enveloped by the ellipse is smaller than thearea enveloped by the stadium model The relative differencebetween the stadium and ellipse model for major axis 119886 andarea
119860 is further illustrated in Figure 3(c) Where the models
are equivalent for both extreme pinching degrees associatedwith a circular cross section in absence of pinching (1 minus
1198871198870 = 0 and 119886 =
1198870 = 05) and a completely pinchedring (1 minus
1198871198870 = 100 and cross-section area
119860 = 0) themaximum difference (lt12) of major axis 119886 between theellipse and stadium model occurs for a pinching degree of52 whereas the maximum difference (lt7) between bothmodels for area
119860 occurs for a pinching degree of 66 Thesensitivity of major axis Δ119886 and area Δ
119860 to a variation ofminor axis Δ
119887 is illustrated in Figures 3(d) and 3(e) It isseen that Δ119886 le Δ
119887 so that for both the ellipse and stadium119886 varies less than
119887 (maximum difference for all pinchingdegrees between ellipse and stadium lt08) For the stadiumthe variation of 119886 is independent from the pinching degree 1minus
1198871198870 following (12) For the ellipse 119886 becomes less sensitive toa variation of 119887 as the pinching degree increases as expressedin (7) For small pinching degrees the impact of a variationof 119887 on the area
119860 is small (Δ
119860 le Δ
119887 for pinching degreesle17 for the ellipse and le32 for the stadium) As thepinching degree increases the impact of a variation of Δ119887 onthe area becomes more important In general the impact ofa variation of 119887 on the area is of the same order of magnitudefor the ellipse and the stadium (maximum difference forall pinching degrees lt07) Nevertheless the area using thestadium ring is less sensitive to a variation Δ
119887 than whenan ellipse is used for pinching degrees le60 for pinchingdegrees gt60 the opposite holds The ellipse and stadiumring shape 119903(120579) and associated sensitivity to a variation ofminor axis Δ119903(120579) are illustrated in Figure 4(a) for a pinchingdegree of 50 As expressed in (6) and in (13) the sensitivitydepends on the angle so that it is of interest to consider a
6 Mathematical Problems in Engineering
EllipseStadium
04
06
08
a(mdash
)
0 50 1001 minus bb0 ()
lt01
a0
(a) 119886
0
02
04
06
A(mdash
)
lt005
A0
EllipseStadium
0 50 1001 minus bb0 ()
(b) 119860
0
5
10
15
|middotsminusmiddote|middot 0
()
middot = Amiddot = a
lt7
52lt12
66
0 50 1001 minus bb0 ()
(c) |119886119904 minus 119886119890|1198860 and |119860119904minus 119860119890|1198600
0
05
1
ΔaΔ
b(mdash
)
lt08
91
0 50 1001 minus bb0 ()
EllipseStadium
(d) Δ119886Δ
60
lt07
0
2
4
ΔAΔ
b(mdash
)
3217
0 50 1001 minus bb0 ()
1
EllipseStadium
(e) Δ119860Δ
Figure 3 Impact of applied geometrical ring models in absence of buckling ellipse (full line in ((a) (b) (d) and (e)) and subscript 119890 in (c))and stadium (dashed line in ((a) (b) (d) and (e)) and subscript 119904 in (c)) on normalised major axis 119886 and normalised area
119860 as a function ofpinching degree 1 minus
1198871198870 with the maximum difference indicated on each subfigure (ltsdot) (a) major axis 119886 and 1198860 = 05 (thin dash-dot line)(b) area
119860 and
1198600 = 1205874 (thin dash-dot line) (c) difference between stadium model (subscript 119890) and ellipse (subscript 119904) relative to thecircular value for major axis 119886 (with a maximum difference for pinching degree 52) and area
119860 (with a maximum difference for pinchingdegree 66) (d) sensitivity of major axis Δ119886 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 91) and(e) sensitivity of area Δ
119860 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 60) The maximum differenceis indicated as ltsdot
Mathematical Problems in Engineering 7
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
05
1
15
2
30
210
60
240
90
270
120
300
150
330
0180
EllipseStadium
EllipseStadium
(a) Ring shape 119903(120579) (left) and sensitivity Δ119903(120579)Δ (right) for 1 minus 0 = 50
0 50 1000
10
20
30
65
1
1 minus bb0 ()
EllipseStadium
gt1lt1
r(120579)Δb
maxΔ
(b) max(Δ119903(120579))Δ
0 50 1000
1
2
3
371Δ
r(120579)Δb
lt03
1 minus bb0 ()
EllipseStadium
(c) Δ119903(120579)Δ
Figure 4 Illustration of overall sensitivity Δ119903(120579) of elliptic (full line) and stadium (dashed line) ring shape 119903(120579) to a variation of the minoraxis Δ119887 as a function of pinching degree 1minus1198871198870 (a) example for pinching degree 50 (b) maximum value of Δ119903(120579) and (c) mean sensitivityΔ119903(120579)
globalmeasure of the ring shape to a variation of 119887Thereforethe maximum value of Δ119903(120579) and mean sensitivity Δ119903(120579) forthe entire ring as a function of pinching degree 1 minus
1198871198870
are plotted in Figures 4(b) and 4(c) respectively In contrastto the sensitivity of 119886 which decreases or remains constantfor increasing pinching degree as shown in Figure 3(d)both global measures indicate that the overall sensitivity ofthe shape increases with the pinching degree Whereas themean sensitivity increase remains limited (lt3) themaximumsensitivity increase is more important since it yields up togt5 times the variation imposed on Δ
119887 for both the ellipseand the stadium As was observed for the area 119860 and majoraxis 119886 the stadium ring model is more sensitive than theelliptic ring model to a variation of 119887 for severe constrictiondegrees (ge60) The observed maximum difference (gt1for pinching degrees gt65) is more of less averaged outsince the difference in mean sensitivity between both ringmodels remains small (lt03) regardless of the pinchingdegree
22 Buckling Ring Model Peanut In Section 21 two geo-metrical ring models with low computational cost were
considered Both models reduced to a single parameter ringmodel of the minor axis 119887 when applying the assumptionof conservation of the perimeter
119875 In this section a quasi-analytical elastic peanut ring models is outlined Analyticalelastic ring models are expected to become pertinent whenthe length of the parallel pinching bars (see Figure 1) reduces(119897119861 lt 2119886) so that the loading becomes axial and buckling islikely to occur [4 6]
221 Peanut Each quarter of the peanut ring with center 119900consists of two arc portions as schematised in Figure 5 Thepeanut ring is then expressed as a function of the radius 120588 ofthe outer arc (with origin 1199001) and 120585 twice the angle spannedby the inner arc (with origin 1199002) so that the peanut parameterset is (120588 120585) The peanut ring model is suitable to approximatenonbuckled (119887 ge 120588) and buckled (119887 lt 120588 second mode)elastic rings The assumption of conservation of perimeter ismade so that the normalized perimeter of the total peanutequals the perimeter of the circular tube
119875 = 120587 andconsequently the quarter peanut has a constant length 1205874The total curvature of each quarter peanut yields 1205872 Theminor axis 119887 le 05 and major axis 119886 ge 05 of the nonbuckled
8 Mathematical Problems in Engineering
1205852
b
ao2
o1
o
b ge Nonbuckled
(a)
o2 a
1205852
o1
b
o
Buckled b lt
(b)
Figure 5 Illustration of nonbuckled ((a) inner arc curvature is positive) and buckled ((b) inner arc curvature is negative) quarters of peanutrings with center 119900 and parameter set (120588 120585) Each ring consists of two arc portions outer arc (thick gray line) with origin 1199001 and radius 120588 andinner arc (thick black line) with origin 1199002 and angle 1205852 The major axis 119886 ge 05 and minor axis 119887 le 05 of the peanut are indicated For thenonbuckled (a) and buckled peanut (b) 119887 ge 120588 and 119887 lt 120588 hold respectively
and buckled peanuts are obtained from parameter set (120588 120585)as
non-buckled 119887 = 120587 (1 minus 2120588)
1 minus cos (1205852)2120585
+ 120588
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 120588
(18)
buckled 119887 = (
120587 (1 minus 2120588)
2120585
+ 120588)(cos 1205852
minus 1)
+ 120588 cos 1205852
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 2120588 sin 120585
2
+ 120588
(19)
The area
119860(120588 120585) of the total peanut is given as
119860 = 120587120588 minus 120587120588
2+
120587
2
4
(1 minus 2120588)
2119891 (120585)
with 119891 (120585) =
120585 minus sin 120585120585
2
(20)
Note that for 120585 = 0 the last term in (20) is omitted since119891(120585 = 0) = 0The bending energy of the quarter peanut yields
1
4
E =
120587 minus 120585
2120588
+
120585
2
120587 minus 2 (120587 minus 120585) 120588
(21)
with E(120588 120585) denoting the bending energy of the total peanutThe bending energy (21) is then used to express the Lagrange
functional L associated with the energy of deformation ofthe peanut ring as a function of (120588 120585) as well
1
BL =
120587 minus 120585
120588
+
120585
2
1205872 minus (120587 minus 120585) 120588
+ 120582(
D2minus
120587
2
16
(1 minus 2120588)
2(1 minus 120587119891 (120585)))
1
2120587
(22)
with isoperimetric difference of the quarter peanut D2 =
(1205874)
2minus 120587
1198604 (0 le
D) normalized pressure 120582 = 8PBdenoting the ratio of hydrostatic pressure P and flexuralrigiditymodulus of the ringB = Υ
119889
312(1minus]2)usingYoungrsquos
modulus Υ Poissonrsquos ratio ] and normalised wall thickness
119889For a given
119860 it is aimed to find the peanut definedby parameter set (120588 120585) which minimises the bending energyE When using the normalised pressure 120582 as a Lagrangemultiplier 120582 the constrained minimisation problem satisfiesthe Euler-Lagrange equations in terms of (120588 120585 120582)
0 =
120587120588 minus 1205872 minus 2120585120588
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588
2 1 minus 120587119891 (120585)
120587 minus 120585
0 =
minus2
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588119891
1015840(120585)
120588 =
2
120587
(
120587
4
minus
D
radic1 minus 120587119891 (120585)
)
(23)
where the prime denotes the derivative with respect to 120585 Thesystem (23) is then solved firstly by eliminating 120582(120588 120585) usingthe first two expressions of (23)
120582 =
8
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2 (24)
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
EllipseStadium
04
06
08
a(mdash
)
0 50 1001 minus bb0 ()
lt01
a0
(a) 119886
0
02
04
06
A(mdash
)
lt005
A0
EllipseStadium
0 50 1001 minus bb0 ()
(b) 119860
0
5
10
15
|middotsminusmiddote|middot 0
()
middot = Amiddot = a
lt7
52lt12
66
0 50 1001 minus bb0 ()
(c) |119886119904 minus 119886119890|1198860 and |119860119904minus 119860119890|1198600
0
05
1
ΔaΔ
b(mdash
)
lt08
91
0 50 1001 minus bb0 ()
EllipseStadium
(d) Δ119886Δ
60
lt07
0
2
4
ΔAΔ
b(mdash
)
3217
0 50 1001 minus bb0 ()
1
EllipseStadium
(e) Δ119860Δ
Figure 3 Impact of applied geometrical ring models in absence of buckling ellipse (full line in ((a) (b) (d) and (e)) and subscript 119890 in (c))and stadium (dashed line in ((a) (b) (d) and (e)) and subscript 119904 in (c)) on normalised major axis 119886 and normalised area
119860 as a function ofpinching degree 1 minus
1198871198870 with the maximum difference indicated on each subfigure (ltsdot) (a) major axis 119886 and 1198860 = 05 (thin dash-dot line)(b) area
119860 and
1198600 = 1205874 (thin dash-dot line) (c) difference between stadium model (subscript 119890) and ellipse (subscript 119904) relative to thecircular value for major axis 119886 (with a maximum difference for pinching degree 52) and area
119860 (with a maximum difference for pinchingdegree 66) (d) sensitivity of major axis Δ119886 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 91) and(e) sensitivity of area Δ
119860 to a variation of minor axis Δ119887 (switch of most sensitive model for pinching degree 60) The maximum differenceis indicated as ltsdot
Mathematical Problems in Engineering 7
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
05
1
15
2
30
210
60
240
90
270
120
300
150
330
0180
EllipseStadium
EllipseStadium
(a) Ring shape 119903(120579) (left) and sensitivity Δ119903(120579)Δ (right) for 1 minus 0 = 50
0 50 1000
10
20
30
65
1
1 minus bb0 ()
EllipseStadium
gt1lt1
r(120579)Δb
maxΔ
(b) max(Δ119903(120579))Δ
0 50 1000
1
2
3
371Δ
r(120579)Δb
lt03
1 minus bb0 ()
EllipseStadium
(c) Δ119903(120579)Δ
Figure 4 Illustration of overall sensitivity Δ119903(120579) of elliptic (full line) and stadium (dashed line) ring shape 119903(120579) to a variation of the minoraxis Δ119887 as a function of pinching degree 1minus1198871198870 (a) example for pinching degree 50 (b) maximum value of Δ119903(120579) and (c) mean sensitivityΔ119903(120579)
globalmeasure of the ring shape to a variation of 119887Thereforethe maximum value of Δ119903(120579) and mean sensitivity Δ119903(120579) forthe entire ring as a function of pinching degree 1 minus
1198871198870
are plotted in Figures 4(b) and 4(c) respectively In contrastto the sensitivity of 119886 which decreases or remains constantfor increasing pinching degree as shown in Figure 3(d)both global measures indicate that the overall sensitivity ofthe shape increases with the pinching degree Whereas themean sensitivity increase remains limited (lt3) themaximumsensitivity increase is more important since it yields up togt5 times the variation imposed on Δ
119887 for both the ellipseand the stadium As was observed for the area 119860 and majoraxis 119886 the stadium ring model is more sensitive than theelliptic ring model to a variation of 119887 for severe constrictiondegrees (ge60) The observed maximum difference (gt1for pinching degrees gt65) is more of less averaged outsince the difference in mean sensitivity between both ringmodels remains small (lt03) regardless of the pinchingdegree
22 Buckling Ring Model Peanut In Section 21 two geo-metrical ring models with low computational cost were
considered Both models reduced to a single parameter ringmodel of the minor axis 119887 when applying the assumptionof conservation of the perimeter
119875 In this section a quasi-analytical elastic peanut ring models is outlined Analyticalelastic ring models are expected to become pertinent whenthe length of the parallel pinching bars (see Figure 1) reduces(119897119861 lt 2119886) so that the loading becomes axial and buckling islikely to occur [4 6]
221 Peanut Each quarter of the peanut ring with center 119900consists of two arc portions as schematised in Figure 5 Thepeanut ring is then expressed as a function of the radius 120588 ofthe outer arc (with origin 1199001) and 120585 twice the angle spannedby the inner arc (with origin 1199002) so that the peanut parameterset is (120588 120585) The peanut ring model is suitable to approximatenonbuckled (119887 ge 120588) and buckled (119887 lt 120588 second mode)elastic rings The assumption of conservation of perimeter ismade so that the normalized perimeter of the total peanutequals the perimeter of the circular tube
119875 = 120587 andconsequently the quarter peanut has a constant length 1205874The total curvature of each quarter peanut yields 1205872 Theminor axis 119887 le 05 and major axis 119886 ge 05 of the nonbuckled
8 Mathematical Problems in Engineering
1205852
b
ao2
o1
o
b ge Nonbuckled
(a)
o2 a
1205852
o1
b
o
Buckled b lt
(b)
Figure 5 Illustration of nonbuckled ((a) inner arc curvature is positive) and buckled ((b) inner arc curvature is negative) quarters of peanutrings with center 119900 and parameter set (120588 120585) Each ring consists of two arc portions outer arc (thick gray line) with origin 1199001 and radius 120588 andinner arc (thick black line) with origin 1199002 and angle 1205852 The major axis 119886 ge 05 and minor axis 119887 le 05 of the peanut are indicated For thenonbuckled (a) and buckled peanut (b) 119887 ge 120588 and 119887 lt 120588 hold respectively
and buckled peanuts are obtained from parameter set (120588 120585)as
non-buckled 119887 = 120587 (1 minus 2120588)
1 minus cos (1205852)2120585
+ 120588
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 120588
(18)
buckled 119887 = (
120587 (1 minus 2120588)
2120585
+ 120588)(cos 1205852
minus 1)
+ 120588 cos 1205852
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 2120588 sin 120585
2
+ 120588
(19)
The area
119860(120588 120585) of the total peanut is given as
119860 = 120587120588 minus 120587120588
2+
120587
2
4
(1 minus 2120588)
2119891 (120585)
with 119891 (120585) =
120585 minus sin 120585120585
2
(20)
Note that for 120585 = 0 the last term in (20) is omitted since119891(120585 = 0) = 0The bending energy of the quarter peanut yields
1
4
E =
120587 minus 120585
2120588
+
120585
2
120587 minus 2 (120587 minus 120585) 120588
(21)
with E(120588 120585) denoting the bending energy of the total peanutThe bending energy (21) is then used to express the Lagrange
functional L associated with the energy of deformation ofthe peanut ring as a function of (120588 120585) as well
1
BL =
120587 minus 120585
120588
+
120585
2
1205872 minus (120587 minus 120585) 120588
+ 120582(
D2minus
120587
2
16
(1 minus 2120588)
2(1 minus 120587119891 (120585)))
1
2120587
(22)
with isoperimetric difference of the quarter peanut D2 =
(1205874)
2minus 120587
1198604 (0 le
D) normalized pressure 120582 = 8PBdenoting the ratio of hydrostatic pressure P and flexuralrigiditymodulus of the ringB = Υ
119889
312(1minus]2)usingYoungrsquos
modulus Υ Poissonrsquos ratio ] and normalised wall thickness
119889For a given
119860 it is aimed to find the peanut definedby parameter set (120588 120585) which minimises the bending energyE When using the normalised pressure 120582 as a Lagrangemultiplier 120582 the constrained minimisation problem satisfiesthe Euler-Lagrange equations in terms of (120588 120585 120582)
0 =
120587120588 minus 1205872 minus 2120585120588
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588
2 1 minus 120587119891 (120585)
120587 minus 120585
0 =
minus2
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588119891
1015840(120585)
120588 =
2
120587
(
120587
4
minus
D
radic1 minus 120587119891 (120585)
)
(23)
where the prime denotes the derivative with respect to 120585 Thesystem (23) is then solved firstly by eliminating 120582(120588 120585) usingthe first two expressions of (23)
120582 =
8
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2 (24)
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Complex AnalysisJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
05
1
15
2
30
210
60
240
90
270
120
300
150
330
0180
EllipseStadium
EllipseStadium
(a) Ring shape 119903(120579) (left) and sensitivity Δ119903(120579)Δ (right) for 1 minus 0 = 50
0 50 1000
10
20
30
65
1
1 minus bb0 ()
EllipseStadium
gt1lt1
r(120579)Δb
maxΔ
(b) max(Δ119903(120579))Δ
0 50 1000
1
2
3
371Δ
r(120579)Δb
lt03
1 minus bb0 ()
EllipseStadium
(c) Δ119903(120579)Δ
Figure 4 Illustration of overall sensitivity Δ119903(120579) of elliptic (full line) and stadium (dashed line) ring shape 119903(120579) to a variation of the minoraxis Δ119887 as a function of pinching degree 1minus1198871198870 (a) example for pinching degree 50 (b) maximum value of Δ119903(120579) and (c) mean sensitivityΔ119903(120579)
globalmeasure of the ring shape to a variation of 119887Thereforethe maximum value of Δ119903(120579) and mean sensitivity Δ119903(120579) forthe entire ring as a function of pinching degree 1 minus
1198871198870
are plotted in Figures 4(b) and 4(c) respectively In contrastto the sensitivity of 119886 which decreases or remains constantfor increasing pinching degree as shown in Figure 3(d)both global measures indicate that the overall sensitivity ofthe shape increases with the pinching degree Whereas themean sensitivity increase remains limited (lt3) themaximumsensitivity increase is more important since it yields up togt5 times the variation imposed on Δ
119887 for both the ellipseand the stadium As was observed for the area 119860 and majoraxis 119886 the stadium ring model is more sensitive than theelliptic ring model to a variation of 119887 for severe constrictiondegrees (ge60) The observed maximum difference (gt1for pinching degrees gt65) is more of less averaged outsince the difference in mean sensitivity between both ringmodels remains small (lt03) regardless of the pinchingdegree
22 Buckling Ring Model Peanut In Section 21 two geo-metrical ring models with low computational cost were
considered Both models reduced to a single parameter ringmodel of the minor axis 119887 when applying the assumptionof conservation of the perimeter
119875 In this section a quasi-analytical elastic peanut ring models is outlined Analyticalelastic ring models are expected to become pertinent whenthe length of the parallel pinching bars (see Figure 1) reduces(119897119861 lt 2119886) so that the loading becomes axial and buckling islikely to occur [4 6]
221 Peanut Each quarter of the peanut ring with center 119900consists of two arc portions as schematised in Figure 5 Thepeanut ring is then expressed as a function of the radius 120588 ofthe outer arc (with origin 1199001) and 120585 twice the angle spannedby the inner arc (with origin 1199002) so that the peanut parameterset is (120588 120585) The peanut ring model is suitable to approximatenonbuckled (119887 ge 120588) and buckled (119887 lt 120588 second mode)elastic rings The assumption of conservation of perimeter ismade so that the normalized perimeter of the total peanutequals the perimeter of the circular tube
119875 = 120587 andconsequently the quarter peanut has a constant length 1205874The total curvature of each quarter peanut yields 1205872 Theminor axis 119887 le 05 and major axis 119886 ge 05 of the nonbuckled
8 Mathematical Problems in Engineering
1205852
b
ao2
o1
o
b ge Nonbuckled
(a)
o2 a
1205852
o1
b
o
Buckled b lt
(b)
Figure 5 Illustration of nonbuckled ((a) inner arc curvature is positive) and buckled ((b) inner arc curvature is negative) quarters of peanutrings with center 119900 and parameter set (120588 120585) Each ring consists of two arc portions outer arc (thick gray line) with origin 1199001 and radius 120588 andinner arc (thick black line) with origin 1199002 and angle 1205852 The major axis 119886 ge 05 and minor axis 119887 le 05 of the peanut are indicated For thenonbuckled (a) and buckled peanut (b) 119887 ge 120588 and 119887 lt 120588 hold respectively
and buckled peanuts are obtained from parameter set (120588 120585)as
non-buckled 119887 = 120587 (1 minus 2120588)
1 minus cos (1205852)2120585
+ 120588
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 120588
(18)
buckled 119887 = (
120587 (1 minus 2120588)
2120585
+ 120588)(cos 1205852
minus 1)
+ 120588 cos 1205852
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 2120588 sin 120585
2
+ 120588
(19)
The area
119860(120588 120585) of the total peanut is given as
119860 = 120587120588 minus 120587120588
2+
120587
2
4
(1 minus 2120588)
2119891 (120585)
with 119891 (120585) =
120585 minus sin 120585120585
2
(20)
Note that for 120585 = 0 the last term in (20) is omitted since119891(120585 = 0) = 0The bending energy of the quarter peanut yields
1
4
E =
120587 minus 120585
2120588
+
120585
2
120587 minus 2 (120587 minus 120585) 120588
(21)
with E(120588 120585) denoting the bending energy of the total peanutThe bending energy (21) is then used to express the Lagrange
functional L associated with the energy of deformation ofthe peanut ring as a function of (120588 120585) as well
1
BL =
120587 minus 120585
120588
+
120585
2
1205872 minus (120587 minus 120585) 120588
+ 120582(
D2minus
120587
2
16
(1 minus 2120588)
2(1 minus 120587119891 (120585)))
1
2120587
(22)
with isoperimetric difference of the quarter peanut D2 =
(1205874)
2minus 120587
1198604 (0 le
D) normalized pressure 120582 = 8PBdenoting the ratio of hydrostatic pressure P and flexuralrigiditymodulus of the ringB = Υ
119889
312(1minus]2)usingYoungrsquos
modulus Υ Poissonrsquos ratio ] and normalised wall thickness
119889For a given
119860 it is aimed to find the peanut definedby parameter set (120588 120585) which minimises the bending energyE When using the normalised pressure 120582 as a Lagrangemultiplier 120582 the constrained minimisation problem satisfiesthe Euler-Lagrange equations in terms of (120588 120585 120582)
0 =
120587120588 minus 1205872 minus 2120585120588
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588
2 1 minus 120587119891 (120585)
120587 minus 120585
0 =
minus2
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588119891
1015840(120585)
120588 =
2
120587
(
120587
4
minus
D
radic1 minus 120587119891 (120585)
)
(23)
where the prime denotes the derivative with respect to 120585 Thesystem (23) is then solved firstly by eliminating 120582(120588 120585) usingthe first two expressions of (23)
120582 =
8
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2 (24)
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
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Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
1205852
b
ao2
o1
o
b ge Nonbuckled
(a)
o2 a
1205852
o1
b
o
Buckled b lt
(b)
Figure 5 Illustration of nonbuckled ((a) inner arc curvature is positive) and buckled ((b) inner arc curvature is negative) quarters of peanutrings with center 119900 and parameter set (120588 120585) Each ring consists of two arc portions outer arc (thick gray line) with origin 1199001 and radius 120588 andinner arc (thick black line) with origin 1199002 and angle 1205852 The major axis 119886 ge 05 and minor axis 119887 le 05 of the peanut are indicated For thenonbuckled (a) and buckled peanut (b) 119887 ge 120588 and 119887 lt 120588 hold respectively
and buckled peanuts are obtained from parameter set (120588 120585)as
non-buckled 119887 = 120587 (1 minus 2120588)
1 minus cos (1205852)2120585
+ 120588
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 120588
(18)
buckled 119887 = (
120587 (1 minus 2120588)
2120585
+ 120588)(cos 1205852
minus 1)
+ 120588 cos 1205852
119886 = 120587 (1 minus 2120588)
sin (1205852)2120585
+ 2120588 sin 120585
2
+ 120588
(19)
The area
119860(120588 120585) of the total peanut is given as
119860 = 120587120588 minus 120587120588
2+
120587
2
4
(1 minus 2120588)
2119891 (120585)
with 119891 (120585) =
120585 minus sin 120585120585
2
(20)
Note that for 120585 = 0 the last term in (20) is omitted since119891(120585 = 0) = 0The bending energy of the quarter peanut yields
1
4
E =
120587 minus 120585
2120588
+
120585
2
120587 minus 2 (120587 minus 120585) 120588
(21)
with E(120588 120585) denoting the bending energy of the total peanutThe bending energy (21) is then used to express the Lagrange
functional L associated with the energy of deformation ofthe peanut ring as a function of (120588 120585) as well
1
BL =
120587 minus 120585
120588
+
120585
2
1205872 minus (120587 minus 120585) 120588
+ 120582(
D2minus
120587
2
16
(1 minus 2120588)
2(1 minus 120587119891 (120585)))
1
2120587
(22)
with isoperimetric difference of the quarter peanut D2 =
(1205874)
2minus 120587
1198604 (0 le
D) normalized pressure 120582 = 8PBdenoting the ratio of hydrostatic pressure P and flexuralrigiditymodulus of the ringB = Υ
119889
312(1minus]2)usingYoungrsquos
modulus Υ Poissonrsquos ratio ] and normalised wall thickness
119889For a given
119860 it is aimed to find the peanut definedby parameter set (120588 120585) which minimises the bending energyE When using the normalised pressure 120582 as a Lagrangemultiplier 120582 the constrained minimisation problem satisfiesthe Euler-Lagrange equations in terms of (120588 120585 120582)
0 =
120587120588 minus 1205872 minus 2120585120588
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588
2 1 minus 120587119891 (120585)
120587 minus 120585
0 =
minus2
(1205872 minus (120587 minus 120585) 120588)
2+
1
4
120582120588119891
1015840(120585)
120588 =
2
120587
(
120587
4
minus
D
radic1 minus 120587119891 (120585)
)
(23)
where the prime denotes the derivative with respect to 120585 Thesystem (23) is then solved firstly by eliminating 120582(120588 120585) usingthe first two expressions of (23)
120582 =
8
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2 (24)
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
The resulting expression (24) is resubstituted in (23) tofind 120585 satisfying
0 =
2
120588119891
1015840(120585) (1205872 minus (120587 minus 120585) 120588)
2
+
(120587 minus 120585) (120587120588 minus 1205872 minus 2120585120588)
120588
2(1 minus 120587119891 (120585)) (1205872 minus (120587 minus 120585) 120588)
2
(25)
Equation (25) is solved by substituting 120588(120585) given bythe last equation of (23) so that a single equation for 120585 isobtained from which in turn the parameter 120588 is obtainedstraightforwardly
Once 120588 and 120585 are determined for given
119860 the polarequation of the corresponding peanut 119903120588120585(120579) with center 119900taken as the origin is directly given as a piecewise functiondescribing two connected arc portions with centers 1199001 and 1199002as depicted in Figure 5
1003816
1003816
1003816
1003816
1003816
119903120588120585 (120579)
1003816
1003816
1003816
1003816
1003816
=
120588 120579 isin [0
120587 minus 120585
2
] with 1199001 = (
(1205872 minus 120587120588)
120585
+ 2120588) sin 120585
2
(
(1205872 minus 120587120588)
120585
+ 120588) 120579 isin [
120587 minus 120585
2
120587
2
] with 1199002 = (
(1205872 minus 120587120588)
120585
+ 2120588) cos 1205852
(26)
Instead of imposing the normalised area
119860 the pinchingdegree 1 minus
1198871198870 can be used as a constrain parameterusing the expression of the minor axis (19) The relationshipbetween normalised area
119860 and pinching degree 1 minus
1198871198870
is depicted in Figure 6(a) It is seen that the peanut is fullypinched (1 minus
1198871198870 = 100) when the area
119860 yields about25 of
1198600 Pinching degrees greater than 100 result innonphysical peanut shapes since it corresponds to a negativevalue of the minor axis 119887 and are therefore not consideredfurther The onset of buckling occurs for pinching degreesgreater than 54 (or areas smaller than about 70 of
1198600) so that for those pinching degrees the value of theparameter 120588 is greater than the minor axis 119887 as illustrated inFigure 6(b)
The Euler-Lagrange equations (23) can be seen as amapping function 119865 (120588 120585 120582
119860) 997891rarr R3 for which the param-eter vector 119909 = (120588 120585 120582) is determined for each
119860 by solving119865(119909
119860) = 0 Consequently an expression for the modulusof deformation of the peanut 120583 = 119889120582119889119860 is computed byapplying the chain rule to find the derivatives 120597119909119894120597119860 andfurthermore using Cramerrsquos rule as [15]
120583 =
1
det (120597119865119895120597119909119894)
sdot
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
A119886 A119887 0
A119888 A119889 0
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 120587
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
(27)
with matrix
(
120597119865119895
120597119909119894
) =
[
[
[
[
[
[
[
[
[
[
A119886 A119887120588
2(1 minus 120587119891 (120585))
4 (120587 minus 120585)
A119888 A119889120588119891
1015840(120585)
4
120587
2
4
(1 minus 2120588) [1 minus 120587119891 (120585)]
120587
3
16
(1 minus 2120588)
2119891
1015840(120585) 0
]
]
]
]
]
]
]
]
]
]
(28)
where
A119886 =minus120587
22 + 120587
2120588 minus 3120587120588120585 + 2120588120585
2
(1205872 minus (120587 minus 120585) 120588)
3+
1
2
120582120588
1 minus 120587119891 (120585)
120587 minus 120585
A119887 =2120588
2120585
(1205872 minus (120587 minus 120585) 120588)
3
+
1
4
120582120588
2[
1 minus 120587119891 (120585)
(120587 minus 120585)
2minus
120587119891
1015840(120585)
120587 minus 120585
]
A119888 =minus4 (120587 minus 120585)
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582119891
1015840(120585)
A119889 =4120588
(1205872 minus (120587 minus 120585) 120588)
3+
1
4
120582120588119891
10158401015840(120585)
(29)
Figure 7(a) illustrates the increase in normalised pressure120582(120588 120585)when the normalised peanut area
119860 is decreased fromthe value of a circle with unit diameter (1198600) to completeclosure (119860 = 0) following (24) The resulting modulus of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
0 02 04 06 080
50
100
A0
A
1minusbb 0
() asymp 25 A0
(a) (119860)
0 50 1000
05
1
Nonbuckled Buckled
max
(b120588
)b 0
asymp54
b
1 minus bb0 ()
(b) max( 120588)0
Figure 6 Illustration of peanut parameters (a) pinching degree 1minus1198871198870 as a function of area
119860with
1198600 = 1205874 (dashed line) and (b)maximumvalue of minor axis 119887 and peanut parameter 120588 as a function of pinching degree 1 minus
119887
1198870 The shift from a nonbuckled to a buckled ring isindicated
200 400 600
0
05
1
A
120582
A0
(a) 119860(120582)
200 400 600
0
A0
120583
minus50
minus100
minus150
120582
(b) 120583(120582)
Figure 7 (a) Normalised pressure 120582 as a function of area
119860 The dashed line corresponds to
119860
0= 1205874 (b) Modulus of deformation 120583 as a
function of normalised pressure 120582 As a reference the value of the modulus for 1198600 = 1205874 is indicated (dashed line)
deformation 120583 as a function of normalised pressure 120582 isshown in Figure 7
222 Comparison of Peanut and Nonbuckling Ring Modelswith Perimeter 120587 Examples of ring shapes obtained usingthe elliptic stadium and peanut ring model for pinchingdegrees of 30 and 80 are illustrated in Figures 8(a) and8(b) respectivelyThe resulting rings are relatively similar for30 for which the peanut ring is nonbuckled whereas for80 the peanut ring buckles so that the difference betweenthe peanut shape and the other ring models becomes morepronounced The impact of the model choice is furtherquantified as a function of pinching degree 1minus1198871198870 for majoraxis 119886 (Figure 8(c)) and enveloped area
119860 (Figure 8(d)) It isseen that for the nonbuckled peanut shapes corresponding topinching degrees le54 the major axis and area of the peanutyields values in the range between the ones obtained withthe elliptic and stadium model Indeed the peanut modelresults in values of 119886 and
119860 similar to the elliptic model forlow pinching degrees (le15) and gradually evolves towards
values obtained with the stadium model as the pinchingdegree is increased from 15 to 54 Consequently forpinching degrees le54 the difference between the modeledring values for 119886 and
119860 increases with pinching degree dueto the increasing difference between the elliptic and stadiumring model which yields lt12 for 119886 and lt7 for
119860 (seeFigure 3(c)) For pinching degrees gt54 the peanut ring isbuckled compared to the nonbuckled rings obtained withthe elliptic and stadium model As a result the differencebetween modeled values of 119886 and
119860 obtained with thenonbuckling elliptic and stadium models reduces as thepinching degree increases whereas the difference between thenonbucklingmodels and the buckled peanut model increaseswith pinching degree up to 45 for 119886 and up to 25 for 119860
3 Application to an Elastic Tube with 119897119861 ≫ 2119886
It is aimed to evaluate if the two-dimensional ring modelsoutlined in the previous sections can be applied to model athree-dimensional elastic tube which is pinched somewhere
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(a) 119903(120579) for 1 minus 1198870 = 30
02
04
06
08
30
210
60
240
90
270
120
300
150
330
0180
(b) 119903(120579) for 1 minus 1198870 = 80
EllipseStadiumPeanut
0 50 10004
06
08
54
a(mdash
)
lt015
a0
1 minus bb0 ()
(c) 119886(1 minus 1198870)
0 50 1000
02
04
06
08
54A
(mdash)
A0
lt02
1 minus bb0 ()
EllipseStadiumPeanut
(d) 119860(1 minus 1198870)
Figure 8 Illustration of nonbuckling (ellipse stadium) and buckling (peanut) geometrical ring models with perimeter 120587 as a function ofpinching degree 1minus1198871198870 ((a) (b)) ring shapes 119903(120579) for pinching degrees of 30 and 80 and ((c) (d)) major axis 119886 and area
119860 As a reference1198860 = 05 and
1198600 = 1205874 are indicated In addition the threshold value of the pinching degree (asymp54) for the peanut between nonbuckling andbuckling is depicted (see Figure 6(b))
along the longitudinal direction using a simple data-drivenprocedure The parallel pinching bars have a length largerthan twice the major axis so that 119897119861 ≫ 2119886 holds regardlessof the pinching degree Consequently it is aimed to modeleach cross-section as a two-dimensional nonbuckled ringwith constant perimeter which is equal to the perimeter of theunpinched circular tube Therefore all previously discussedring models (ellipse stadium and peanut) can be appliedwhen one additional parameter that is local pinching degreeis known at each cross section The longitudinal variation ofthe pinching degree is expressed as follows 1 minus 119887(119909)1198870 sincethe minor axis 119887 depends on the longitudinal dimension 119909The tubersquos geometry the gathering of geometrical data and asimple data-driven procedure to apply the two-dimensional(2D) ring models to model a three-dimensional (3D) tubegeometry are outlined in Section 31 The comparison ofmodeled and measured geometrical tube features is thenpresented in Section 32
31 Extension from 2D Ring Models to 3D Tube Models AData-Driven Procedure Geometrical data are acquired fora cylindrical elastic tube of length 119897 = 184mm in thelongitudinal 119909-direction with wall thickness 3mm and theinternal diameter of the undeformed cylindrical tube yields25mm The tube is equipped with a pincer at longitudinal
position 119909119888 = 90mm Concretely the pincer consists oftwo parallel circular bars of diameter 64mm oriented alongthe 119910-axis which enforces a symmetrical deformation of thetubes cross-section shape with respect to its center-planeTheelastic tube with pincer is illustrated in Figures 9(a) and 9(b)
In order to obtain geometrical data the tube is mountedon a turning platform controlled by a step motor withaccuracy 18∘ At each rotational position a longitudinalprofile of the tube is quantified using a calibrated line laserscan [16 17] The turning platform with mounted tube andsingle longitudinal laser line is illustrated in Figure 9 At theposition of the pincer 119909119888 buckling is hindered due to thecondition 119897119861 ≫ 2119886 Consequently the distance between theparallel bars yields 2(119887119909
119888
+ 119889) and the minor axis 119887119909119888
is aknown quantity at the pincer position 119909119888 The pincher effortis expressed by the imposed pincher degree at 119909 = 119909119888 definedas 1minus119887119909
119888
1198870 with 1198870 = 125mmas before denoting the internalradius of the unpinched cylindrical tube
Concretely longitudinal profiles are assessed for 6 differ-ent pincher efforts 1 minus 119887119909
119888
1198870 in the range from 40 up to95 [16 17] In order to model the 3D tube using the 2D ringmodels the minor axis need to be estimated for each crosssection so that the required ring model parameter 1minus1198871199091198870 isa known quantityThis way the 2D ringmodels can be appliedto each cross section of the 3D tube resulting in an elliptic
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
xb0 + d
Pincerxc
a(x) + d
y
(a) Major axis 119886(119909) top view
x
bx119888 + d
b(x) + d
z
b0 + d
(b) Minor axis 119887(119909) side view
Laser line
Turning platform
(c) Laser line for scan
Figure 9 Illustration of deformable tube with internal diameter 2 times 119887
0= 25mm and wall thickness 119889 with pincer at longitudinal position
119909 = 119909119888 and pincer effort 1 minus 119887119909119888
1198870 (a) major axis 119886(119909) (119909119910-plane top view) (b) minor axis (119909119911-plane side view) and (c) laser line fromresulting in the longitudinal 119909 profile as a function of platform turning angle
stadium or peanut tube approximationTherefore firstly theminor (119887) and major (119886) axis are extracted as a functionof longitudinal position 0 le 119909 le 119897 from the measuredlongitudinal profiles Next the following parameters of themeasured longitudinal profiles are extracted the amplitudeat the pincer position 119909119888 of the major axis 119886119909
119888
ge 1198870 the halfwidth at half maximum amplitude of the major axis 120572119886 andof the minor axis 120572119887 The parameter values obtained fromthe 6 assessed pinching degrees are plotted in Figure 10(a)(119886119909119888
) and in Figure 10(b) (120572119886119887) Polynomial fits are applied tothe measured parameters in order to estimate the parametervalues for pinching degrees in the range from 40 up to95 Concretely a linear fit is used to estimate the maximumamplitude 119886119909
119888
as a function of pinching degree 1 minus 119887119909119888
1198870
(coefficient of determination 119877
2gt 099) A second order
polynomial fit is used to approximate the half width athalf maximum amplitude of the major axis 120572119886 (coefficientof determination 119877
2gt 072) and the half width at half
maximum amplitude of the minor axis 120572119887 (coefficient ofdetermination 119877
2gt 081) along the longitudinal direction
as a function of pinching degree in the range from 40up to 95 The polynomial parameter fits are shown inFigures 10(a) (119886119909
119888
) and 10(b) (120572119886119887)Using the polynomial parameter fits for 119886119909
119888
120572119886 and 120572119887
as a function of pinching effort 1 minus 119887119909119888
1198870 illustrated inFigure 10 the longitudinalminor 119887(119909) andmajor 119886(119909) axis aresubsequently approximated by symmetrical peak functionscentered around the position of the pincer 119909119888 and functionof the applied pincer effort 1 minus 119887119909
119888
1198870
119886(119909119888119886119909119888120572119886) (119909) = 1198870 + 1198870 sdot (
119886119909119888
1198870
minus 1)
sdot exp(minus ln 2 sdot (119909 minus 119909119888)
2
120572
2119886
)
with 119886119909119888
(119887119909119888
) and 120572119886 (119887119909119888
)
(30)
119887(119909119888119887119909119888120572119887) (119909) = 1198870 minus 1198870 sdot (1 minus
119887119909119888
1198870
)
sdot (
(119909 minus 119909119888)2
120572
2
119887
+ 1)
minus1
with 120572119887 (119887119909119888
)
(31)
An example of the measured and estimated 119886(119909) and 119887(119909)using the data-driven procedure to obtain the parametersneeded to apply (31) is illustrated in Figure 10(c) It is seen thatthe tube deforms symmetrically around the pincer positionover a total length of 8 times 1198870 which corresponds to 4 times theinternal diameter of the circular tube Following this simpleprocedure the considered 2D ring models can be applied toeach cross section of the tube for pinching efforts in the range40 lt 1 minus 119887119909
119888
1198870 lt 95 using (31) and the assumption of aconstant perimeter 21205871198870
32 Ring Model Selection Comparison of Modeled andMeasured Data Features The simple data-driven procedurepresented in Section 31 is applied to the tube for all 6experimentally assessed pinching efforts 1minus119887119909
119888
1198870 in order tocomment on the relevance of the geometrical ring models toestimate the tube geometry for different pinching degrees andthe potential impact of the model choice between the ellipticstadium and peanut ring models
We are particularly interested in validating the longitudi-nal major axis 119886(119909) since as argued in the introduction thisparameter impacts for instance physical phenomena relatedto wave propagation Figure 11(a) presents the modeled andmeasured values of 119886119909
119888
(1 minus 119887119909119888
1198870) for all of the consideredring models for pinching efforts in the range from 40 up to95The linear increase of 119886119909
119888
with pinching effort 1 minus 119887119909119888
1198870
characterizing the stadiummodel reflects the experimentallyobserved tendency In particular the rate of increase is wellrepresentedwhen using the stadiummodel whereas the offset
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
40 60 80 10012
13
14
15
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
(a) 119886119909119888
40 60 80 100
15
20
25
120572(m
m)
120572a120572b
1 minus bx119888 b0 ()
(b) 120572119886 120572119887
0 2 40
05
1
15
ab
0bb
0(mdash
)
ab
(x minus xc)b0 (mdash)minus2minus4
(c) 119886(119909) and 119887(119909) for 94
Figure 10 Fitting of symmetrical peak function parameters to experimentally measured (symbols) pincher effort 1 minus 119887119909119888
1198870 [] (a) linear fit(full line) to major axis maximum 119886119909
119888
(times) and (b) second order polynomial fits (full lines) to half width at half maximum for major axis 120572119886(times) and for minor axis 120572119887 (+) and (c) example of measured (symbols) and peak function approximation (full lines) major 119886(119909) (times) and minoraxis 119887(119909) (+) for pincher effort 1 minus 119887
119909119888
119887
0of 94
is overestimated so that 119886119909119888
is slightly (less than 5 of 1198870)overestimated for all pinching efforts The elliptic modelresults in a much more severe overestimation (up to 20 of1198870) for all pinching degrees so that solely on the estimationof 119886119909
119888
the elliptic model can be rejected The stadium modeldoes not exhibit the experimentally linear increase of 119886119909
119888
withpinching effort Nevertheless for pinching efforts smallerthan 70 the accuracy of estimated values of 119886119909
119888
is of thesame order ofmagnitude as obtained using the peanutmodelFor pinching efforts greater than 54 buckling starts to occurwhich results in a severe underestimation (up to 20 of 1198870)of 119886119909
119888
Consequently solely based on the estimation of 119886119909119888
it can be argued that the ring model pick order is stadiumpeanut and elliptic In general the discrepancy betweenthe modeled longitudinal 119909 profile and the measured profilealong the corresponding laser lines (such as the one depictedin Figure 9(c)) is less than 7 of 1198870 (or less than 4 of the
tubes diameter) when applying the stadium ring model toeach cross section with the local pinching degree computedusing the peak function approximation of 119887(119909) as expressedin (31)The discrepancy between themeasured and estimatedlongitudinal stadium profiles is illustrated in Figure 11(b) forpincher effort 1minus119887119909
119888
1198870 = 94using the local pinching degreeassociated with 119887(119909) plotted in Figure 10(c)
4 Conclusion
The minor axis of an elastic ring compressed between twoparallel bars is directly related to the imposed pinchingdegree Therefore homothetic quasi-analytical geometricalring models (ellipse stadium and peanut) are formulated fora circular ring with unit diameter as a function of pinchingdegree using the assumption of a constant perimeter All threering models reflect the same general tendencies for pinching
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Mathematical Problems in Engineering
0 20 40 60 80 1001
12
14
16
EllipseStadiumPeanut
a x119888b
0(mdash
)
1 minus bx119888 b0 ()
ax119888
(a) 119886119909119888
(1 minus 119887119909119888
1198870)1198870
0 2 4
1
12
14
16
aPeak functionStadium
ab
0(mdash
)
minus4 minus2
lt7
(x minus xc)b0 (mdash)
(b) 119886(119909)1198870Figure 11 (a) Modeled and measured 119886119909
119888
1198870 as a function of pinching effort 1 minus 119887119909119888
1198870 (b) Example of measured (symbols) and stadiummodel (dashed line) major axis 119886(119909)1198870 for pincher effort 1minus 119887119909
119888
1198870 = 94 of measured (times) and peak function approximation (full line) usingthe local pinching degree associated with 119887(119909) plotted in Figure 10(c)
degrees up to 54 Therefore the dynamics due to modelchoice for pinching degrees le 54 is limited (eg le7 forthe area and le12 for the major axis) and is determinedby the difference between an elliptic and stadium ring sincethe peanut ring evolves gradually from elliptic (le15) tostadium (at 54) For pinching degrees greater than 54 thepeanut ring starts to buckle As a consequence the dynamicsof geometrical features related to the model choice becomesmore important (eg le25 for the area and le45 for themajor axis)
Next the ring models are applied to each cross section ofan elastic circular tube compressed between two parallel barsfor different pinching efforts between 40 and 95 A simpledata-driven procedure is used to provide the local pinchingdegree required as an input parameter Overall the stadiumring model provides the best approximation of the tube witha characteristic error yielding less than 4 of the tubes diam-eter Moreover the experimentally observed linear increaseof the major axis with the pinching effort is an inherentproperty of the stadium model Consequently the case studyillustrates that the outlined quasi-analytical models such asthe stadium model can provide an accurate and computa-tional low cost approximation of the tubes geometry suitablefor applications such as geometry initialisation combinationwith other parameterised physicalmathematical models orexperimental setup design and validation purposes as afunction of pinching degree
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This work was partly supported by EU-FET Grant (EUNI-SON 308874) Daniel Manica and Guillaume Popovici areacknowledged for their contribution to the experiments
References
[1] P Kundu Fluid Mechanics Academic Press London UK 1990[2] A Pierce Acoustics An Introduction to Its Physical Principles
and Applications Acoustical Society of America New York NYUSA 1991
[3] S Houliara and S A Karamanos ldquoBuckling and post-bucklingof long pressurized elastic thin-walled tubes under in-planebendingrdquo International Journal of Non-Linear Mechanics vol41 no 4 pp 491ndash511 2006
[4] Y Zhu X Y Luo and R W Ogden ldquoAsymmetric bifurcationsof thick-walled circular cylindrical elastic tubes under axialloading and external pressurerdquo International Journal of Solidsand Structures vol 45 no 11-12 pp 3410ndash3429 2008
[5] A Goriely R Vandiver and M Destrade ldquoNonlinear Eulerbucklingrdquo Proceedings of the Royal Society A MathematicalPhysical amp Engineering Sciences vol 464 no 2099 pp 3003ndash3019 2008
[6] Y Zhu X Y Luo and R W Ogden ldquoNonlinear axisymmetricdeformations of an elastic tube under external pressurerdquo Euro-pean Journal of MechanicsmdashASolids vol 29 no 2 pp 216ndash2292010
[7] P A Djondjorov V M Vassilev and I M Mladenov ldquoAnalyticdescription and explicit parametrisation of the equilibriumshapes of elastic rings and tubes under uniform hydrostaticpressurerdquo International Journal of Mechanical Sciences vol 53no 5 pp 355ndash364 2011
[8] S K Kourkoulis C F Markides and E D Pasiou ldquoA combinedanalytic and experimental study of the displacement field in acircular ringrdquoMeccanica vol 50 no 2 pp 493ndash515 2015
[9] R von Mises ldquoDer kritische Aubendruck zulindrischer rohrerdquoVDI Zeitschrift vol 58 pp 750ndash755 1914
[10] A E H LoveTheMathematicalTheory of Elasticity CambridgeUniversity Press London UK 1927
[11] R Levien ldquoThe elastica a mathematical historyrdquo Tech RepEECS Department University of California Berkeley CalifUSA 2008
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 15
[12] M Abramovitz and I A Stegun Handbook of MathematicalFunctions US Government Printing Office Washington DCUSA 1972
[13] B D Gupta Mathematical Physics Vikas Publishing HouseNew Delhi India 2002
[14] S Ramanujan ldquoModular equations and approximations to 120587rdquoTheQuarterly Journal of Pure and Applied Mathematics vol 45no 1 pp 350ndash372 1914
[15] F Liu and A Treibergs ldquoOn the compression of elastic tubesrdquoTech Rep 2011 httpwwwmathutahedusimtreibergLT2006pdf
[16] D Manica ldquoRealisation calibration and implementation ofa laser scan for object reconstructionrdquo Tech Rep GrenobleUniversities Grenoble France 2013
[17] G Popovici ldquoModeling of deformed tube geometries fromreconstructed and measured datardquo Tech Rep Grenoble Uni-versities Grenoble France 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Elastic deformation of concrete - utcluj.ro2... · Elastic deformation of concrete. ... articles from the german standard DIN 1048-5:1991 [1] (which has concrete testing as the main](https://static.fdocuments.in/doc/165x107/5b6938767f8b9af23e8dc057/elastic-deformation-of-concrete-2-elastic-deformation-of-concrete-.jpg)
