Research Article Variational Principles for Buckling of ...Variational formulations were employed in...
Transcript of Research Article Variational Principles for Buckling of ...Variational formulations were employed in...
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Research ArticleVariational Principles for Buckling of MicrotubulesModeled as Nonlocal Orthotropic Shells
Sarp Adali
Discipline of Mechanical Engineering, University of KwaZulu-Natal, Durban 4041, South Africa
Correspondence should be addressed to Sarp Adali; [email protected]
Received 15 April 2014; Revised 19 June 2014; Accepted 20 June 2014; Published 5 August 2014
Academic Editor: Chung-Min Liao
Copyright Β© 2014 Sarp Adali. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A variational principle for microtubules subject to a buckling load is derived by semi-inverse method.The microtubule is modeledas an orthotropic shell with the constitutive equations based on nonlocal elastic theory and the effect of filament network taken intoaccount as an elastic surrounding. Microtubules can carry large compressive forces by virtue of the mechanical coupling betweenthe microtubules and the surrounding elastic filament network.The equations governing the buckling of the microtubule are givenby a system of three partial differential equations.The problem studied in the present work involves the derivation of the variationalformulation for microtubule buckling. The Rayleigh quotient for the buckling load as well as the natural and geometric boundaryconditions of the problem is obtained from this variational formulation. It is observed that the boundary conditions are coupledas a result of nonlocal formulation. It is noted that the analytic solution of the buckling problem for microtubules is usually adifficult task. The variational formulation of the problem provides the basis for a number of approximate and numerical methodsof solutions and furthermore variational principles can provide physical insight into the problem.
1. Introduction
Understanding the buckling characteristics of microtubulesis of practical and theoretical importance since they performa number of essential functions in living cells as discussedin [1β4]. In particular, they are the stiffest components ofcytoskeleton and are instrumental in maintaining the shapeof cells [1, 5].This is basically due to the fact thatmicrotubulesare able to support relatively large compressive loads as aresult of coupling to the surrounding matrix. Since thisfunction is of importance for cellmechanics and transmissionof forces, the study of the buckling behavior of microtubulesprovides useful information on their biological functions.Consequently the buckling of microtubules has been studiedemploying increasinglymore complicated continuummodelswhich are often used to simulate their mechanical behaviorand provide an effective tool to determine their load carryingcapacity under compressive loads. The present study facili-tates this investigation of the microtubule buckling problemby providing a variational setting which is the basis of anumber of numerical and approximate solution methods.
Buckling of microtubules occurs for a number of rea-sons such as cell contraction or constrained microtubulepolymerization at the cell periphery. To better understandthis phenomenon, an effective approach is to use continuummodels to represent a microtubule. These models includeEuler-Bernoulli beambyCivalek andDemir [6], Timoshenkobeam by Shi et al. [7], and cylindrical shells by Wang etal. [8] and Gu et al. [9]. The present study provides avariational formulation of the buckling ofmicrotubules usingan orthotropic shell model to represent their mechanicalbehavior. Variational principles form the basis of a numberof computational and approximate methods of solutionsuch as finite elements, Rayleigh-Ritz and Kantorovich. Inparticular Rayleigh quotient provides a useful expression toapproximate the buckling load directly. As such the resultspresented can be used to obtain the approximate solutionsfor the buckling of microtubules as well as the variationallycorrect boundary conditions which are derived using thevariational formulation of the problem.
Continuummodeling approach has been used effectivelyin other branches of biology and medicine [10], and their
Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2014, Article ID 591532, 9 pageshttp://dx.doi.org/10.1155/2014/591532
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2 Computational and Mathematical Methods in Medicine
accuracy can be improved by implementing nonlocal con-stitutive relations for micro- and nanoscale phenomenoninstead of classical local ones which relate the stress at agiven point to the strain at the same point. As such localtheories are of limited accuracy at the micro- and nanoscalesince they neglect the small scale effects which can besubstantial due to the atomic scale of the phenomenon.Recent examples of microtubule models based on the localelastic theory include [11β18] where Euler-Bernoulli andhigher order shear deformable beams and cylindrical shellsrepresented the microtubules. A review of the mechanicalmodeling of microtubules was given by Hawkins et al. [19]and a perspective on cell biomechanics by Ji and Bao [20].
In the present study the formulation is based on thenonlocal theory which accounts for the small scale effects andimproves the accuracy.The nonlocal theory was developed inthe early seventies by Eringen [21, 22] and recently appliedto micro- and nanoscale structures. Nonlocal continuummodels have been used in a number of studies to investigatethe bending and vibration behavior of microtubules usingnonlocal Euler-Bernoulli [6, 23] andTimoshenko beams [24].There have been few studies on the buckling of microtubulesbased on a nonlocal theory. Nonlocal Timoshenko beammodel was employed in [25β27] and nonlocal shell modelin [1]. In a series of studies, Shen [28β30] used nonlocalshear deformable shell theory to study the buckling andpostbuckling behavior of microtubules. Nonlocal problemsalso arise in other subject areas and have been studiedusing fractional calculus in a number studies [31, 32]. Themodels based on beam or isotropic shell theories neglect thedirectional dependence of the microtubule properties. Theaccuracy of a continuum model can be improved further byemploying an orthotropic shell theory to take this directionaldependence into account as discussed in [12, 33β35].
The objective of the present study is to derive a variationalprinciple and Rayleigh quotient for the buckling load as wellas the applicable natural and geometric boundary conditionsfor a microtubule subject to a compressive load. The partic-ular model used in the study is a nonlocal orthotropic shellunder a compressive load with the effect of filament networktaken into account as an elastic surrounding. Moreover thepressure force on the microtubule exerted by the viscouscytosol is calculated using the Stokes flow theory [1]. Theinclusion of these effects in the governing equations isimportant to model the phenomenon accurately since it isknown that the microtubules can carry large compressiveforces by virtue of the mechanical coupling between themicrotubules and the surrounding elastic filament networkas observed by Brangwynne et al. [36] and Das et al. [37].Moreover, the microtubules are surrounded by the viscouscytosol in addition to the soft elastic filament network and thebuckling causes the viscous flowof the cytosol [38].These twoprocesses result in an external stress field which improves thebuckling characteristics of microtubules.
Variational formulations were employed in a numberof studies involving microtubules. In particular, small scaleformulations for the linear vibrations of microtubules werederived using the energy expression in [39, 40] and fornonlinear vibrations in [41]. Previous studies on variational
principles involving nanoscale structures includemultiwalledcarbon nanotubes. In particular, variational principles werederived for nanotubes under buckling loads [42], for nan-otubes undergoing linear vibrations [43], and for nanotubesundergoing nonlinear vibrations [44] using the nonlocalEuler-Bernoulli beam theories. Variational principles werealso derived for nanotubes undergoing transverse vibrationsusing a nonlocal Timoshenko beam model in [45] and astrain-gradient cylindrical shell model in [46]. Apart fromproviding an insight into a physical problem, the variationalformulations are often employed in the approximate andnumerical solutions of the problems, in particular, in thepresence of complicated boundary conditions [47].Moreovernatural boundary conditions can be easily derived from thevariational formulation of the problem. In the present studyvariational formulation of a problem is derived by the semi-inverse method developed by He [48β51] which was appliedto several problems of mathematical physics to obtain thevariational formulations for problems formulated in termsof differential equations [52β55]. Recently the semi-inversemethod was applied to the heat conduction equation in [56]to obtain a constrained variational principle.The equivalenceof this formulation to the one obtained by He and Lee [57]has been shown in [58, 59]. A recent application of thesemi-inverse method involves the derivation of variationalprinciples for partial differential equations modeling watertransport in porous media [60].
2. Equations Based on Nonlocal Elastic Theory
In the present study the microtubule is modeled as anorthotropic cylindrical shell of length πΏ, radius π , andwall thickness β and surrounded by a viscoelastic medium(cytoplasm). It is subject to an axial compressive load π asshown in Figure 1.
The microtubule has the Youngβs moduli πΈ1and πΈ
2along
the axial and circumferential directions, shearmodulusπΊ andPoissonβs ratios π
1and π
2along the circumferential and axial
directions. Dimensionless longitudinal direction is denotedby π₯ = π/π and the circumferential direction by π as inFigure 1. For an orthotropic shell, the constitutive relationsbased on nonlocal elasticity theory are given by (see [1])
ππ₯β(ππ0)2
π 2β2ππ₯=
πΈ1
1 β π1π2
ππ₯+
πΈ2π2
1 β π1π2
ππ,
ππβ(ππ0)2
π 2β2ππ=
πΈ2
1 β π1π2
ππ+
πΈ1π1
1 β π1π2
ππ₯,
ππ₯πβ(ππ0)2
π 2β2ππ₯π= πΊππ₯π,
(1)
where β2 = (π2/ππ₯2) + (π2/ππ2) is the Laplace operator,ππ₯, ππ, and π
π₯πare stress components, and π
π₯, ππ, and π
π₯π
are the normal and shear strains. In (1), ππ0is the small
scale parameter reflecting the nanoscale of the phenomenonand has to be experimentally determined. The differentialequations governing the buckling of the microtubules are
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L
CytoplasmCytoplasm
NX
R
h
Cytoplasm
Microtubule
N
Microtubule
π
οΏ½ (x, π)
u(x, π)
w(x, π)
Figure 1: Microtubule under a compressive load and its surround-ing.
given in [1] based on the nonlocal constitutive relation (1).Thedifferential equation formulation of the problem is expressedas a system of partial differential equations given by
π·1(π’, V, π€) β‘ πΏ
1(π’) + π
1(V, π€) = 0,
π·2(π’, V, π€) β‘ πΏ
2(V) + π
2(π’, π€) = 0,
π·3(π’, V, π€) β‘ πΏ
3(π€) +π
3(π’, V) = 0,
(2)
where π’(π₯, π), V(π₯, π), and π€(π₯, π) are the displacement com-ponents in the axial, circumferential, and radial directions,respectively, as shown in Figure 1. The differential operatorsπΏπandπ
πare defined as
πΏ1(π’) = π
2(1 + π
2) π’ππ+ π’π₯π₯+π
πΎL (π’π₯π₯) , (3)
π1(V, π€) = (π
2+ π1) Vπ₯π+ π1π€π₯β π2π€π₯π₯π₯
+ π2π2π€π₯ππ, (4)
πΏ2(V) = π
2(1 + 3π
2) Vπ₯π₯+ π1Vππ+π
πΎL (Vπ₯π₯) , (5)
π2(π’, π€) = (π
2+ π1) π’π₯π+ π1π€πβ π2(3π2+ π1) π€π₯π₯π, (6)
πΏ3(π€) = β (1 + π
2) π1π€ β 2π
2π1π€ππ
β π2(π€π₯π₯π₯π₯
+ π1π€ππππ
+ (4π2+ 2π1) π€π₯π₯ππ
)
+ππ
πΎL (π€) +
π
πΎL (π€π₯π₯) +
π
πΎL (πππ) ,
(7)
π3(π’, V) = βπ
1π’π₯+ π2(π’π₯π₯π₯
β π2π’π₯ππ)
β π1Vπ+ π2(3π2+ π1) Vπ₯π₯π,
(8)
where the differential operator L(β ) is defined as L(β ) =π2β2β1, the subscriptsπ₯, πdenote differentiationwith respect
to that variable, and the dimensionless small scale parameterπ is given by π = ππ
0/π . The symbols in (3)β(8) are defined as
π1=πΈ1
πΈ2
,
π2=πΊ (1 β π
1π2)
πΈ1
,
π2=
β3
0
12π 2β,
πΎ =πΈ1β
(1 β π1π2),
(9)
where β0is the effective thickness for bending. The symbol
π appearing in (7) is the elastic constraint from the filamentsnetwork and is given by π = 2.7 πΈ
πwhere πΈ
πis the elastic
modulus of the surrounding viscoelastic medium.The radialpressure π
ππexerted by the motion of cytosol in (7) is
computed from the dynamic equations of cytosol given in [1].
3. Variational Formulation
Following the semi-inverse method, we construct a varia-tional trial-functional π(π’, V, π€) as follows:
π (π’, V, π€) = π1(π’) + π
2(V) + π
3(π€)
+ β«
2π
0
β«
π
0
πΉ (π’, V, π€) ππ₯ ππ,(10)
where π = πΏ/π , π₯ β [0, πΏ/π ] and the functionals π1(π’), π
2(V),
π3(π€) are given by
π1(π’) =
1
2β«
2π
0
β«
π
0
[ β π2(1 + π
2) π’2
πβ π’2
π₯
+π
πΎ(π2(π’2
π₯π₯+ π’2
π₯π) + π’2
π₯)] ππ₯ ππ,
π2(V) =
1
2β«
2π
0
β«
π
0
[ β π2(1 + 3π
2) V2π₯β π1V2π
+π
πΎ(π2(V2π₯π₯+ V2π₯π) + V2π₯)] ππ₯ ππ,
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4 Computational and Mathematical Methods in Medicine
π3(π€)
=1
2β«
2π
0
β«
π
0
[β (1 + π2) π1π€2+ 2π2π1π€2
π
β π2(π€2
π₯π₯+ π1π€2
ππ)
β (4π2+ 2π1) π€2
π₯π] ππ₯ ππ β β β
+1
2β«
2π
0
β«
π
0
[βππ
πΎ(π2(π€2
π₯+ π€2
π) + π€2) +
π
πΎ
Γ (π2(π€2
π₯π₯+ π€2
π₯π) + π€2
π₯)
+2π
πΎL (πππ) π€] ππ₯ ππ.
(11)
In (10), πΉ(π’, V, π€) is an unknown function to be determinedsuch that the Euler-Lagrange equations of the variationalfunctional (10) correspond to the differential equation (2).This establishes the direct relation between the variationalformulation and the governing equations in the sense thatdifferential equation (2) can be obtained from the derivedvariational principle using the Euler-Lagrange equations. Itis noted that the choice of the trial functionals defined by(10)-(11) is not unique. The review article by He [61] providesa systematic treatment on the use of semi-inverse methodfor the derivation of variational principles and the selectionof trial functionals as well as on variational methods for thesolution of linear and nonlinear problems.
It is noted that the Euler-Lagrange equations of thevariational functional π(π’, V, π€) are
πΏ1(π’) +
πΏπΉ
πΏπ’= 0,
πΏ2(V) +
πΏπΉ
πΏV= 0,
πΏ3(π€) +
πΏπΉ
πΏπ€= 0.
(12)
Thus the functionals π1(π’), π
2(V), and π
3(π€) are the varia-
tional functionals for the differential operators πΏ1(π’), πΏ
2(V),
and πΏ3(π€), respectively. In (12), the variational derivative
πΏπΉ/πΏπ’ is defined as
πΏπΉ
πΏπ’=ππΉ
ππ’βπ
ππ₯(ππΉ
ππ’π₯
) βπ
ππ(ππΉ
ππ’π
) +π2
ππ₯2(ππΉ
ππ’π₯π₯
)
+π2
ππ₯ππ(ππΉ
ππ’π₯π
) +π2
ππ2(ππΉ
ππ’ππ
)
βπ3
ππ₯3(
ππΉ
ππ’π₯π₯π₯
) β β β .
(13)
Comparing (12) with (2), we observe that the followingequations have to be satisfied for Euler-Lagrange equationsofπ(π’, V, π€) to represent the governing equation (2), namely,
πΏπΉ
πΏπ’= π1(V, π€)
= (π2+ π1) Vπ₯π+ π1π€π₯
β π2π€π₯π₯π₯
+ π2π2π€π₯ππ,
(14)
πΏπΉ
πΏV= π2(π’, π€) = (π
2+ π1) π’π₯π+ π1π€π
β π2(3π2+ π1) π€π₯π₯π,
(15)
πΏπΉ
πΏπ€= π3(π’, V)
= βπ1π’π₯+ π2(π’π₯π₯π₯
β π2π’π₯ππ)
β π1Vπ+ π2(3π2+ π1) Vπ₯π₯π.
(16)
The function πΉ(π’, V, π€) has to be determined such that(14)β(16) are satisfied. For this purpose we first determineπΉ(π’, V, π€) satisfying (14) to obtain
πΉ (π’, V, π€) = β (π2+ π1) Vπ₯π’π+ π1π€π₯π’ + π2π€π₯π₯π’π₯
+ π2π2π€π₯π’ππ+ Ξ¦ (V, π€) ,
(17)
where Ξ¦(V, π€) is an unknown function of V and π€. Next wecompute πΏπΉ/πΏV from (17), namely,
πΏπΉ
πΏV= (π2+ π1) π’π₯π+πΏΞ¦ (V, π€)
πΏV(18)
which should satisfy (15). From (15) and (18) it follows thatπΏΞ¦ (V, π€)
πΏV= π1π€πβ π2(3π2+ π1) π€π₯π₯π. (19)
The expression for Ξ¦(V, π€) satisfying (19) is determined as
Ξ¦ (V, π€) = βπ1π€Vπ+ π2(3π2+ π1) π€π₯π₯Vπ. (20)
Thus from (17) and (20), πΉ(π’, V, π€) is obtained as
πΉ (π’, V, π€) = β (π2+ π1) Vπ₯π’π+ π1π€π₯π’ + π2π€π₯π₯π’π₯
+ π2π2π€π₯π’ππβ π1π€Vπ
+ π2(3π2+ π1) π€π₯π₯Vπ.
(21)
Finally we note that ππΉ/ππ€ satisfies (16). Thus the functionπΉ(π’, V, π€) given by (21) satisfies the Euler-Lagrange equations(14)β(16) as required. Now the variational functional can beexpressed as
π (π’, V, π€) = π1(π’) + π
2(V) + π
3(π€) + π
4(π’, V, π€) , (22)
whereπ4(π’, V, π€)
= β«
2π
0
β«
π
0
πΉ (π’, V, π€) ππ₯ ππ
= β«
2π
0
β«
π
0
(β (π2+ π1) Vπ₯π’π+ π1π€π₯π’ + π2π€π₯π₯π’π₯
+ π2π2π€π₯π’ππβ π1π€Vπ
+π2(3π2+ π1) π€π₯π₯Vπ) ππ₯ ππ
(23)
and π1(π’), π
2(V), and π
3(π€) are given by (11). To verify
the validity of the variational formulation (22), one has toshow that the Euler-Lagrange equations of π(π’, V, π€) yieldthe governing (2). The fact that this is indeed the case can beshown easily.
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Computational and Mathematical Methods in Medicine 5
4. Rayleigh Quotient
The Raleigh quotient is obtained for the buckling load π bynoting that
π1(π’) = βπ
1(π’) + ππ (π’) ,
π2(V) = βπ
2(V) + ππ (V) ,
π3(π€) = βπ
3(π€) + ππ (π€) ,
(24)
where
π 1(π’) =
1
2β«
2π
0
β«
π
0
(π2(1 + π
2) π’2
π+ π’2
π₯) ππ₯ ππ,
π 2(V) =
1
2β«
2π
0
β«
π
0
(π2(1 + 3π
2) V2π₯+ π1V2π) ππ₯ ππ,
π 3(π€)
=1
2β«
2π
0
β«
π
0
((1 + π2) π1π€2β 2π2π1π€2
π
+ π2(π€2
π₯π₯+ π1π€2
ππ)
+ (4π2+ 2π1) π€2
π₯π) ππ₯ ππ β β β
+1
2β«
2π
0
β«
π
0
(ππ
πΎ(π2(π€2
π₯+ π€2
π) + π€2)
β2π
πΎL (πππ) π€) ππ₯ ππ,
π (π¦) =1
2πΎβ«
2π
0
β«
π
0
(π2(π¦2
π₯π₯+ π¦2
π₯π) + π¦2
π₯) ππ₯ ππ.
(25)
Thus from (22) and (24), it follows that
π (π’, V, π€) = β (π 1(π’) + π
2(V) + π
3(π’))
+ π4(π’, V, π€)
+ π (π (π’) + π (V) + π (π€))
(26)
and the Rayleigh quotient can be expressed as
π = minπ’,V,π€
π 1(π’) + π
2(V) + π
3(π€) β π
4(π’, V, π€)
π (π’) + π (V) + π (π€). (27)
5. Natural and GeometricBoundary Conditions
It is noted that the displacements are equal at the end pointsπ = 0 and π = 2π; that is,
π’ (π₯, 0) = π’ (π₯, 2π) ,
V (π₯, 0) = V (π₯, 2π) ,
π€ (π₯, 0) = π€ (π₯, 2π)
for π₯ β [0, π] .
(28)
The first variations of π(π’, V, π€) with respect to πΏπ’, πΏV, andπΏπ€, denoted by πΏ
π’π, πΏVπ, and πΏπ€π, respectively, can be
obtained by integration by parts and using (28). We firstobtain the variations of π
1(π’) and π
4(π’, V, π€) with respect to
πΏπ’ which are given by
πΏπ’π1(π’)
= β«
2π
0
β«
π
0
πΏ1(π’) πΏπ’ ππ₯ ππ + π΅
1(π’, πΏπ’, πΏπ’
π₯) ,
πΏπ’π4(π’, V, π€)
= β«
2π
0
β«
π
0
π1(V, π€) πΏπ’ ππ₯ ππ + π΅
2(π€, πΏπ’) ,
(29)
where
π΅1(π’, πΏπ’, πΏπ’
π₯)
= β«
2π
0
[(βπ’π₯βπ
πΎ(π2(π’π₯π₯π₯
+ π’π₯ππ) β π’π₯)) πΏπ’
+π
πΎπ2π’π₯π₯πΏπ’π₯]
π₯=π
π₯=0
ππ,
π΅2(π€, πΏπ’) = β«
2π
0
[π2π€π₯π₯πΏπ’]π₯=π
π₯=0ππ.
(30)
Similarly
πΏVπ2 (V) = β«2π
0
β«
π
0
πΏ2(V) πΏV ππ₯ ππ + π΅
3(V, πΏV, πΏV
π₯) ,
πΏVπ4 (π’, V, π€) = β«2π
0
β«
π
0
π2(π’, π€) πΏV ππ₯ ππ
+ π΅4(π’, πΏV) ,
(31)
where
π΅3(V, πΏV, πΏV
π₯)
= β«
2π
0
[(βπ2(1 + 3π
2) Vπ₯βπ
πΎ(π2(Vπ₯π₯π₯
+ Vπ₯ππ) β Vπ₯)) πΏV
+π
πΎπ2Vπ₯π₯πΏVπ₯]
π₯=π
π₯=0
ππ,
π΅4(π’, πΏV) = β«
2π
0
[β (π2+ π1) π’π]π₯=π
π₯=0πΏV ππ.
(32)
Finally we obtain πΏπ€π3(π€) and πΏ
π€π4(π’, V, π€), namely,
πΏπ€π3(π€)
= β«
2π
0
β«
π
0
πΏ3(π€) πΏπ€ππ₯ππ + π΅
5(π€, πΏπ€, πΏπ€
π₯)
+ π΅6(π€, πΏπ€, πΏπ€
π₯) ,
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6 Computational and Mathematical Methods in Medicine
πΏπ€π4(π’, V, π€)
= β«
2π
0
β«
π
0
π3(π’, V) πΏπ€ππ₯ππ + π΅
7(π’, V, πΏπ€, πΏπ€
π₯)
+ π΅7(π’, V, πΏπ€, πΏπ€
π₯) ,
(33)
where
π΅5(π€, πΏπ€, πΏπ€
π₯)
= β«
2π
0
[π2(π€π₯π₯π₯
+ (4π2+ 2π1) π€π₯ππ) πΏπ€
βπ2π€π₯π₯πΏπ€π₯]π₯=π
π₯=0ππ,
π΅6(π€, πΏπ€, πΏπ€
π₯)
= β«
2π
0
[(βππ
πΎπ2π€π₯βπ
πΎ(π2(π€π₯π₯π₯
+ π€π₯ππ) β π€π₯)) πΏπ€
+π
πΎπ2π€π₯π₯πΏπ€π₯]
π₯=π
π₯=0
ππ,
π΅7(π’, V, πΏπ€, πΏπ€
π₯)
= β«
2π
0
[(π1π’ + π2(π2π’ππβ π’π₯π₯) β π2(3π2+ π1) Vπ₯π) πΏπ€
+ (π2(3π2+ π1) Vπ+ π2π’π₯) πΏπ€π₯]π₯=π
π₯=0ππ.
(34)
Since the first variations of the functional π(π’, V, π€) are zero,that is,
πΏπ’π (π’, V, π€) = πΏVπ (π’, V, π€) = πΏπ€π (π’, V, π€) = 0 (35)
by the fundamental lemma of the calculus of variations, wehave
π΅1(π’, πΏπ’, πΏπ’
π₯) + π΅2(π€, πΏπ’) + π΅
3(V, πΏV, πΏV
π₯)
+ π΅4(π’, πΏV) + π΅
5(π€, πΏπ€, πΏπ€
π₯)
+ π΅6(π€, πΏπ€, πΏπ€
π₯) + π΅7(π’, V, πΏπ€, πΏπ€
π₯) = 0
(36)
which yields the boundary conditions. We first note that (36)can be written as
5
β
π=1
ππ= 0, (37)
where
π1(π’, π€, πΏπ’, πΏπ’
π₯)
= β«
2π
0
[(βπ’π₯βπ
πΎ(π2(π’π₯π₯π₯
+ π’π₯ππ) β π’π₯) + π2π€π₯π₯) πΏπ’
+π
πΎπ2π’π₯π₯πΏπ’π₯]
π₯=π
π₯=0
ππ,
π2(π’, V, π€, πΏV, πΏV
π₯)
= β«
2π
0
[( β (π2+ π1) π’πβ π2(1 + 3π
2) Vπ₯
βπ
πΎ(π2(Vπ₯π₯π₯
+ Vπ₯ππ) β Vπ₯)
βπ2(3π2+ π1) π€π₯π) πΏV]π₯=π
π₯=0
ππ
+ β«
2π
0
[π
πΎπ2Vπ₯π₯πΏVπ₯]
π₯=π
π₯=0
ππ,
π3(π€, πΏπ€)
= β«
2π
0
[(π2(π€π₯π₯π₯
+ (4π2+ 2π1) π€π₯ππ) β
ππ
πΎπ2π€π₯
βπ
πΎ(π2(π€π₯π₯π₯
+ π€π₯ππ) β π€π₯)) πΏπ€]
π₯=π
π₯=0
ππ,
π4(π’, V, πΏπ€)
= β«
2π
0
[(π1π’ + π2(π2π’ππβ π’π₯π₯)
βπ2(3π2+ π1) Vπ₯π) πΏπ€]
π₯=π
π₯=0ππ,
π5(π’, V, π€, πΏπ€
π₯)
= β«
2π
0
[(βπ2π€π₯π₯+π
πΎπ2π€π₯π₯+ π2(3π2+ π1) Vπ
+π2π’π₯) πΏπ€π₯]
π₯=π
π₯=0
ππ.
(38)
From (37)-(38), the natural and geometric boundary condi-tions are obtained at π₯ = 0 and π₯ = π as
π’π₯+π
πΎ(π2(π’π₯π₯π₯
+ π’π₯ππ) β π’π₯) β π2π€π₯π₯= 0 or π’ = 0,
π’π₯π₯= 0 or π’
π₯= 0,
(π2+ π1) π’π+ π2(1 + 3π
2) Vπ₯
+π
πΎ(π2(Vπ₯π₯π₯
+ Vπ₯ππ) β Vπ₯)
+ π2(3π2+ π1) π€π₯π= 0 or V = 0,
Vπ₯π₯= 0 or V
π₯= 0,
π2(π€π₯π₯π₯
+ (4π2+ 2π1) π€π₯ππ) β
ππ
πΎπ2π€π₯
βπ
πΎ(π2(π€π₯π₯π₯
+ π€π₯ππ) β π€π₯) β β β + π
1π’
+ π2(π2π’ππβ π’π₯π₯)
β π2(3π2+ π1) Vπ₯π= 0 or π€ = 0,
β π2π€π₯π₯+π
πΎπ2π€π₯π₯+ π2(3π2+ π1) Vπ
+ π2π’π₯= 0 or π€
π₯= 0.
(39)
-
Computational and Mathematical Methods in Medicine 7
6. Conclusions
The variational formulation for the buckling of a microtubulewas given using a nonlocal continuum theory wherebythe microtubule was modeled as an orthotropic shell. Thecontinuum model of the microtubule takes the effects ofthe surrounding filament network and the viscous cytosolinto account as well as its orthotropic properties. Methodsof calculus of variations were employed in the derivationof the variational formulation and in particular the semi-inverse approach was used to identify suitable variationalintegrals. The buckling load was expressed in the form ofa Rayleigh quotient which confirms that small scale effectslower the buckling load as has been observed in a numberof studies [1, 26, 28]. The natural and geometric boundaryconditions were derived using the formulations developed.The variational principles presented here form the basisof several approximate and numerical methods of solutionand facilitate the implementation of complicated boundaryconditions, in particular, the natural boundary conditions.
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
Theresearch reported in this paperwas supported by researchgrants from the University of KwaZulu-Natal (UKZN) andfrom National Research Foundation (NRF) of South Africa.The author gratefully acknowledges the support provided byUKZN and NRF.
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