Mechanics of Advanced omposite Structures
Transcript of Mechanics of Advanced omposite Structures
Mechanics of Advanced Composite Structures 8 (2021) 77-86
Semnan University
Mechanics of Advanced Composite
Structures
journal homepage: http://MACS.journals.semnan.ac.ir
*Corresponding author. Tel.: +91-9842647487 E-mail address: [email protected]
DOI: 10.22075/MACS.2020.19273.1230 Received 2019-12-09; Received in revised form 2020-05-27; Accepted 2020-06-19 Ā© 2021 Published by Semnan University Press. All rights reserved.
Assessment of Hydrostatic Stress and Thermo Piezoelectrici-ty in a Laminated Multilayered Rotating Hollow Cylinder
S. Mahesh a, R. Selvamani b*, F. Ebrahami c a Department of Mathematics, Kathir College of Engineering, Coimbatore, 641062, India.
b Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, 641114, India. c Department of Mechanical Engineering, Imam Khomeini International University Qazvin, Iran.
K E Y W O R D S A B S T R A C T
Initial hydrostatic stress
Thermoelasticity
Longitudinal waves
Bessel function
In this paper, we built a mathematical model to study the influence of the initial stress on
the propagation of waves in a hollow infinite multilayered composite cylinder. The elastic
cylinder assumed to be made of inner and outer thermo piezoelectric layer bonded togeth-
er with Linear Elastic Material with Voids (LEMV) layer. The model described by the equa-
tions of elasticity, the effect of the initial stress and the framework of linearized, three-
dimensional theory of thermo elasticity. The displacement components obtained by found-
ing the analytical solutions of the motionās equations. The frequency equations that include
the interaction between the composite hollow cylinders are obtained by the perfect-slip
boundary conditions using the Bessel function solutions. The numerical calculations car-
ried out for the material PZT-5A and the computed non-dimensional frequency against
various parameters are plotted as the dispersion curve by comparing LEMV with Carbon
Fiber Reinforced Polymer (CFRP). From the graph, it is clear that those are analyzed in the
presence of hydrostatic stress is compression and tension.
1. Introduction
Composite materials are generally utilized in en-gineering structures because of their pre-dominance over the basic materials in applica-tions requiring high quality and solidness in lightweight parts. Thusly, the portrayal of their mechanical conduct is taking imperative part in basic plan. Procedures that incite transversely isotropic flexible properties in them make most cylindrical parts, for example, poles, wires, cyl-inders, funnels and strands. Displaying the pro-liferation of waves in these parts is significant in different applications, including ultrasonic non-destructive assessment systems, progression of room explore and numerous others. Smart ma-terials are normally prestressed during the as-sembling procedure. As initial stresses are indi-visible in surface acoustic wave gadgets and res-onators, investigation of such impacts has been finished with various methodologies. A few crea-tors have considered wave engendering in pre-stressed piezoelectric structure.
Soniya Chaudhary et al. [1] derived secular equation of SH waves propagating in pre-stressed and rotating piezo-composite structure with imperfect interface. Abhinav Singhal et al. [2] analyticaly analysed interfacial imperfection study in pres-stressed rotating multiferroic cy-lindrical tube with wave vibration. Lotfy and El-Bary [3] discussed photothermal excitation for a semiconductor medium due to pulse heat flux and volumetric source of heat with thermal memory. Lotfy [4] discovered a novel model for photothermal excitation of variable thermal conductivity semiconductor elastic medium sub-jected to mechanical ramp type with two-temperature theory and magnetic feld. Soniya Chaudhary et al. [5] studied anatomy of flexo-electricity in micro plates with dielectrically highly/weakly and mechanically complaint in-terface. Ebrahimi et al. [6] discussed Magneto-electro-elastic analysis of piezoelectricāflexoelectric nanobeams rested on silica aerogel foundation. Abhinav Singhal et al. [7] investigate mechanics of 2D Elastic Stress Waves Propaga-
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tion Impacted by Concentrated Point Source Dis-turbance in Composite Material Bars. Lotfy [8] discover a novel model of photothermal diffu-sion (PTD) for polymer nano-composite semi-conducting of thin circular plate. Placidi and Hutter [9] studied thermodynamics of Polycrys-talline materials treated by the theory of mix-tures with continuous diversity. Altenbach and Eremeyev [10] discussed vibrtion analysis of non-linear 6-parameter prestressed shells. Sonal Nirwal et al. [11] analysised of different bounda-ry types on wave velocity in bedded piezo-structure with flexoelectric effect. Lot-fy and Wafaa Hassan [12] discovered the effect of rotation for two-temperature generalized thermoelasticity of two-dimensional under
thermal shock problem.Abhinav Singhal et al. [13] studied Liouville-Green approximation: an analytical approach to study the elastic waveās vibrations in composite structure of piezo mate-rial.Soniya Chaudhary and Abhinav Singhal [14], [15] discussed analytic model for Rayleigh wave propagation in piezoelectric layer overlaid or-thotropic substratum and surface wave propaga-tion in functionally graded piezoelectric materi-al. Abbas and Othman [16] generalized thermoe-lastic interacon in a fiber-reinforced anisotropic half-space under hydrostatic initial stress. Oth-man and Lotfy [17] discussed the influence of gravity on 2-D problem of two temperatures generalized thermoelastic medium with thermal relaxation. Abo-Dahab et al. [18] discovered ro-tation and magnetic field effect on surface waves propagation in an elastic layer lying overa gen-eralized thermoelastic diffusive half-space with imperfect boundary. Hobinyand and Abbas [19] studied analytical solution of magneto-thermoelastic interaction in a fiber-reinforced anisotropic material. Manoj Kumar Singh et al. [20] discussing approximation of surface wave velocity in smart composite structure using WentzelāKramersāBrillouin method. Othman, Abo-Dahab, and Lotfy [21] investigate gravita-tional effect and initial stress on generalized magneto-thermomicrostretch elastic solid for different theories. Abo-Dahab and Lotfy [22] designed two-temperature plane strain problem in a semiconducting medium under photo ther-mal theory. Abo-Dahab [23] discussed reflection of P and SV waves from stress-free surface elas-tic half-space under influence of magnetic field and hydrostatic initial stress without energy dissipation. Abhinav Singhal et al. [24], [25] studied Stresses produced due to moving load in a prestressed piezoelectric substrate and ap-proximation of surface wave frequency in piezo-composite structure. Alireza Mohammadi et.al [26] discover the results of influence of viscoe-lastic foundation on dynamic behaviour of the double walled cylindrical inhomogeneous micro
shell using MCST and with the aid of GDQM. Ebrahimi et.al [27] studied the impacts of ther-mal buckling and forced vibration characteris-tics of a porous GNP reinforced nanocomposite cylindrical shell. Habibi, Hashemabadi and Sa-farpour [28] discussed the vibration analysis of a high-speed rotating GPLRC nanostructure cou-pled with a piezoelectric actuator. MostafaHabi-bi et.al [29] studied stability analysis of an elec-trically cylindrical nanoshell reinforced with graphene nanoplatelets.MehranSafarpoura et.al [30] analysics frequency characteristics of FG-GPLRC viscoelastic thick annular plate with the aid of GDQM.
In this paper, we have built a mathematical model to study the effect of imposing thermal field on longitudinal wave propagation in a hol-low multilayered composite cylinder under the influence of initial hydrostatic stress. The cylin-der is made of tetragonal system material, such as PZT-5A. The displacement components are obtained by founding the analytical solutions of the motionās equations. After applying suitable boundary conditions, the frequency equation is presented as a determinant with elements con-taining Bessel functions. The numerical compu-tations obtained the roots of frequency equa-tions. The dispersion curve carried for various parameters.
2. Formulation of the problem and basic equations
Longitudinal wave propagation in a homoge-neous, transversely isotropic cylinder of tetrag-onal elastic material of inner and outer radius x and a subjected to an axial thermal and electric field is considered. The cylinder is treated as a perfect conductor and the regions inside and outside the elastic material is assumed to be vacuumed. The medium is assumed to be rotat-ing with uniform angular velocity . The dis-placement field, in cylindrical coordinates (š, š, š§) is given by
šš šš ,š
+ šššš§,š§
+ šā1(šššš)
+ š( Ī© Ć ( Ī© Ć )
+ 2( Ī© Ć ,š”)) = šš¢,š”š”
(1a)
šššš§ ,š
+ ššš§š§,š§
+šā1šššš§
+ š( Ī© Ć ( Ī© Ć )
+ 2( Ī© Ć ,š”)) = šš¤,š”š”
(1b)
The electric displacement equation
1
š
š
šš(šš·š
š) +šš·š
š§
šš§= 0 (1c)
The heat conduction equation
S. Mahesh, R. Selvamani, F. Ebrahami/ Mechanics of Advanced Composite Structures 8 (2021) 77-86
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š¾1(šš,šš + šā1šš
,š + šā2šš,šš) + š¾3šš
,š§š§
ā ššš£šš,š”
= š0
š
šš”[š½1(šš
šš + šššš)
+ š½3šš,š§š§ ā š3 š,š]
(1d)
The stress strain relations are given as fol-lows šš
šš = š11šššš + š12šš
šš + š13ššš§š§
ā š½1šš ā š31šš§š
(2)
ššš§š§ = š13šš
šš + š13šššš + š33šš
š§š§ ā š½3šš
ā š33šš§š
šššš§ = š44šš
šš§
š·š = š15šššš§ + ķ11šš
š
š·š§ = š31(šššš + šš
šš) + š33ššš§š§ + ķ33šš
š§
+ š3š
Where šššš , šš
šš , šššš§, šš
šš , ššš§š§ , šš
šš§ are the stress and šš
šš , ššš§š§ , šš
šš , šššš , šš
š§š , šššš§ are
the strain components, T is the temperature change about the equilibrium temperature š11, š12, š13, š33, š44, š66 are the five elastic con-stants,š½1, š½3 and š¾1, š¾3 respectively thermal ex-pansion coefficients and thermal conductivities along and perpendicular to the symmetry, Ļ is the mass density, šš£ is the specific heat capacity ,š3 is the pyroelectric effect.
The strains šššš related to the displacements
given by [32]
šššš = š¢š
,š šššš = šā1(š¢š +
š£š,š), šš
š§š§ = š¤š,š§
(3) šš
šš = š£šš ā šā1(š£š ā š¢š
,š),
ššš§š = š£š
,š§ + šā1š¤š,š , šš
šš§ = š¤š,š + š¢š
,š§
Substitution of the Eqs. (3) and (2) into Eqs. (1) results in the following three-dimensional equation of motion, heat and electric conduc-tion. We note that the first two equations under the influence hydrostatical stress become [31]:
š11(š¢š,šš + šā1š¢š
,š + šā2š¢š) + š13š¤š,šš§
+ š44(š¢š,š§š§ + š¤š
,šš§)
+ (š15 + š31)š,šš§š
ā š0(š¢š,šš + šā1š¢š
,š
+ šā2š¢š + š¢š,š§š§)
ā š½1šš,š + š( Ī©2š¢
+ 2Ī©š¤,š”) = šš¢,š”š”
(4a)
(š44 + š13)(š¢š,šš§ + šā1š¢š
,š§) + š33(š¤š,š§š§)
+ š44(š¤š,šš + šā1š¤š
,š)
+ š15(š,ššš + šā1š,š)
ā š0(š¤š,šš + šā1š¤š
,š
+ š¤š,š§š§) ā š½3šš
,š
+ š( Ī©2š¤ + 2Ī©š¢,š”)
= šš¤,š”š”
(4b)
šā1š15(š¤,ššš + š¢,šš§
š ) + ķ11š,šš + š13š¢,šš§
š
+ šā1š¢šš + š33š¤,š§š§
š
+ ķ33š,š§š + š3š = 0
(4c)
š11(šš,šš + šā1šš
,š) + š33šš,š§š§ ā š0ššš
,š”
= š0[š½1(š¢š,šš” + šā1š¢š
,š”)
+ š½3š¤š ,š§]
(4d)
Where ššš is heat conduction coefficient, š =šš¶š£
š0 , u and w are the displacements along r, z
direction, š is the mass density and t is the time. The solutions of Eq. (4) is considered in the
form š¢š = šš
,šexp š(šš§ + šš”)
(5)
š¤š = (š
ā) šš exp š(šš§ + šš”)
šš = (šš44
šš33
) šøšexp š(šš§ + šš”)
šš = (š44
š½3
) (šš
ā2)exp š(šš§ + šš”)
Where, š¢š , š¤š , šš , šš are displacement poten-tials, k is the wave number, p is the angular fre-
quency and š = āā1 . We introduce the non-
dimensional quantities š„ =š
š, ķ = šš, š = šš āaā
is the geometrical parameter of the composite
hollow cylinder. š11 =š11
š44ā , š13 =
š13š44
ā ,š33 =š33
š44ā , š66 =
š66š44
ā , =
š½1š½3
ā , š =(šš44)
12
š½32š0šĪ©
,
Substituting Eqs. (5) in Eqs. (4), we obtain: [(š11 ā š0)š»2 ā (1 ā š0)ķ2 + ķ2 + š2
+ ā¶]šš
ā [ķ(1 + š13)]šš ā ķ(31 + 15)šøš ā šš
= 0
(6a)
[ķ(1 + š13)ā2]šš + [(1 ā š0)ā2 ā (š33
ā š0)ķ2 + ķ2 + š2
+ ā¶]šš + (15ā2 ā ķ2)šøš
ā ķšš = 0
(6b)
ķ((31 + 15)ā2šš + (š15ā2 + ķ2)šš
ā (š¾112 ā2 + š¾33ķ2)šøš
ā šķšš = 0
(6c)
š»2šš + ķšš ā šķšøš
+ (šš¾1š»2 + šš¾3ķ2 ā )šš
= 0
(6d)
The above Eqs. [6] rearranged in the follow-ing form
||
(š11 ā š0)ā2 ā š 1 + š“1 āš“2 š“3 āš“4
š“2 (1 ā š0)ā2 ā š 2 + š“1 š152 ā2 + š“5 š“6
š“3ā2 š152 ā2 + š“5 š¾11
2 ā2 + š“7 āš“8
š“4ā2 š“6 āš“8 šš¾1ā2 + š“9
||
Ć (šš , šš , šøš , šš) = 0
(7)
Where
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š“1 = ķ2 + š2 + ā¶, š“2 = ķ(1 + š13), š“3 = ķ(31 + 15), š“4 = āš½, š“5 = ķ2, š“6 = ķ, š“7 =š¾33
2 ķ2 š“8 = āššķ , š“9 = š3ķ2 ā , š 1 =(1 ā š0)ķ2 , š 2 = (š33 ā š0)ķ2 .
Evaluating the determinant given in Eq. (7), we obtain a partial differential equation of the
form: (š“ā8 + šµā6 + š¶ā4 + š·ā2 + šø)(šš šš šøš šš)š
= 0 (8)
Factorizing the relation given in Eq. (8) into
biquadratic equation for(š¼ššš)
2, i=1, 2, 3, 4 the
solutions for the symmetric modes are obtained as
šš = ā[š“šš½š(š¼šš„) + šµšš¦š(š¼šš„)],
4
š=1
šš = ā ššš [š“šš½š(š¼šš„) + šµšš¦š(š¼šš„)],
4
š=1
šøš = ā ššš[š“šš½š(š¼šš„) + šµšš¦š(š¼šš„)],
4
š=1
šš = ā ššš[š“šš½š(š¼šš„) + šµšš¦š(š¼šš„)],
4
š=1
Here (š¼šššš„) > 0 , for (š = 1,2,3,4) are the
roots of algebraic equation
(š“(š¼ššš)
8+ šµ(š¼š
šš)6
+ š¶(š¼ššš)
4+
š·(š¼ššš)
2+ šø) (šš , šš , šøš , šš) = 0 (10)
The solutions corresponding to the root (š¼šš)2 = 0 not considered here, since š½š(0) is zero, except š = 0. The Bessel function š½š is used when the roots(š¼šš)2,(š = 1,2,3,4) are real or complex and the modified Bessel function š¼š is used when the roots (š¼šš)2, (š = 1,2,3,4) are imaginary.
The constants ššš , šš
š and ššš defined in Equa-
tion (10) can be calculated from the following equations [(š11 ā š0)ā2 ā (1 ā š0)ķ2 + ķ2 + š2 + ā¶]
ā ķ(1 + š13)ššš
ā ķ(31 + 15)ššš ā šš
š = 0
ķ(1 + š13)ā2 + [(1 ā š0)ā2 ā (š33 ā š0)ķ2 + ķ2
+ š2 + ā¶]ššš + (15ā2 ā ķ2)šš
š
ā ķššš = 0
ķ((31 + 15)ā2 + (15ā2 + ķ2)ššš
ā (š¾112 ā2 + š¾33ķ2)šš
š ā šķššš
= 0
ā2 + ķššš ā šķšš
š + (šš¾1ā2 + šš¾3ķ2 ā )ššš = 0
3. Equation of Motion for Linear Elastic Materials with Voids LEMV
The displacement equations of motion and equation of equilibrated inertia for an isotropic LEMV are ( š + 2š)(š¢,šš + šā1š¢,š ā šā2š¢) + šš¢,š§š§
+ (š + š)š¤,š§š§ + š½šø,š
= šš¢š”š”
(11a)
(š + š)(š¢,šš§ + šā1š¢,š§)
+ š(š¤,šš + šā1š¤,š)
+ (š + 2š)š¤,š§š§ + š½šø,š§
= šš¤,š”š”
(11b)
āš½(š¢,š + šā1š¢) ā š½š¤,š§ + š¼(šø,šš +
šā1šø,š + š,š§š§) ā šæššø,š”š” ā
ššø,š” ā ššø = 0
(11c)
The stress in the LEMV core materials are š,šš = (š + 2š)š¢,š + ššā1š¢ + šš¤,š§ + š½š
ššš§ = š(š¢,š” + š¤,š)
The solution for Eq. (11) is taken as š¢ = š,ššš„šš(šš§ + šš”)
(12) š¤ = (š
ā)ššš„šš(šš§ + šš”)
šø = (1
ā2) šø šš„šš(šš§ + šš”)
The above solution in (11) and dimensionless variables x and ķ, equation can be simplified as
|
(š + 2š)ā2 + š1 āš2 š3
š2ā2 ā2 + š4 š5
āš3ā2 š5 š¼ā2 + š6
| Ć
(š¢, š¤, šø) = 0
(13)
Where ā2=š2
šš„2 +1
š„
š
šš„
š1 =š
š1(šā)2 ā ķ2 , š2 = (š + )ķ ,š3 = ,
š4 =š
š1(šā)2 ā (š + )ķ2 ,š5 = ķ
š6 =š
š1(šā)2 ā ķ2 ā š(šā) ā š
The Eq. (13) can specified as, (ā6 + šā4 + šā2 + š )(š, š, šø) = 0 (14)
Thus, the solution of Eq. (14) is as follows,
š = ā[š“šš½0(š¼šš„) + šµšš¦0(š¼šš„)],
3
š=1
š = ā šš[š“šš½0(š¼šš„) + šµšš¦0(š¼šš„)],
3
š=1
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šø = ā šš[š“šš½0(š¼šš„) + šµšš¦0(š¼šš„)],
3
š=1
(š¼šš„)2
are the roots of the equation when re-
placing ā2= ā(š¼šš„)2
. The arbitrary constant šš
and šš are obtained from
š2ā2 + (š ā2 + š4)šš + š5šš = 0,
āš3ā2 + š5šš + (š¼ā2 + š6)šš = 0
By taking the void volume fraction E=0, and
the lameās constants as š = š12, š =š11āš12
2 in
the Eq. (11) we got the governing equation for CFRP core material.
4. Boundary conditions and fre-quency equations
The frequency equations can obtain for the following boundary condition
On the traction free inner and outer sur-face šš
šš = ššš§š = šøš = šš = 0 with š =
1,3 At the interface ššš
š = ššš; ššš§š =
ššš§; šøš = 0; šš = 0; š·š = 0. Substituting the above boundary condition,
we obtained as a 22Ć22 determinant equation
|(ššš)| = 0 , (š, š = 1,2,3, ā¦ .22) (15)
At š„ = š„0 Where š = 1,2,3,4
š1š = 2š66(š¼š
1
š„0
)š½1(š¼1šš„0)
ā [(š¼1šš)
2š11 + ķš13šš
š
+ 31ķššš + šš
š] š½0(š¼1ššš„0)
š2š = (ķ + šš1 + 15šš
1)(š¼š1) š½1(š¼1
šš„0)
š3š = šš1š½0(š¼1
šš„0)
š4š =āš
1
š„0
š½0(š¼1šš„0) ā (š¼š
1) š½1(š¼1šš„0)
In addition, the other nonzero elements š1,š+4, š2,š+4, š3,š+4 and š4,š+4are obtained by re-
placing š½0 by š½1and š0 byš1. At š„ = š„1
š5š = 2š66(š¼š
1
š„1
)š½1(š¼1šš„1)
ā [(š¼1šš)
2š11 + ķš13šš
š
+ 31ķššš + šš
š] š½0(š¼1ššš„1)
š5,š+8 = ā[2 (š¼š
š„1
) š½1(š¼š„1)
+ ā(š + )(š¼š)2
+ šš
ā šķšš š½0(š¼šš„1)
š6š = (ķ + šš1 + 15šš
1)(š¼š1) š½1(š¼1
ššš„1)
š6,š+8 = ā(ķ + šš)(š¼š)š½1(š¼šš„1)
š7š = (š¼šš)š½1(š¼š
šš„1)
š7,š+8 = ā(š¼š)š½1(š¼ššš„1)
š8š = šššš½0(š¼š
šš„1)
š8,š+8 = āšššš½0(š¼š
šš„1)
š9š = šššš½0(š¼š
šš„0)
š10š = šš(š¼š)š½1(š¼š1š„1)
š11š =šš
š
š„1
š½0(š¼ššš„1) ā (š¼š
š)š½1(š¼ššš„1)
and the other nonzero element at the inter-faces š„ = š„1 can be obtained on replacing š½0 by š½1 and š0 by š1 in the above elements. They are šš,š+4,šš,š+8,šš,š+11,šš,š+14, (š = 5,6,7,8) and
š9,š+4,š10,š+4,š11,š+4,. At the interface š„ = š„2, non-
zero elements along the following rows ššš , (š =
12,13, ā¦ ,18 and š = 8,9, ā¦ ,20) are obtained on replacing š„1 by š„2and superscript 1 by 2 in or-der. Similarly, at the outer surface š„ = š„3 , the nonzero elements ššš , (š = 19,20,21,22 and š =
14,15, , ā¦ ,22) can be had from the nonzero ele-ments of first four rows by assigning x3 for x0 and superscript 2 for 1. In the case of without voids in the interface region, the frequency equation obtained by taking šø = 0 in Eq. (11) which reduces to a 20 x 20 determinant equa-tion. The frequency equations derived above are valid for different inner and outer materials of 6mm class and arbitrary thickness of layers.
5. Numerical results and discussion
The frequency equation given in Eq. (15) is transcendental in nature with unknown fre-quency and wave number. The solutions of the frequency equations obtained numerically by fixing the wave number. The material chosen for the numerical calculation is PZT-5A. The materi-al properties of PZT-5A used for the numerical calculation given below: š¶11 = 13.9 Ć 1010 Nmā2; š¶12
= 7.78 Ć 1010 Nmā2; š¶13 = 7.43 Ć 1010 Nmā2; š¶33 = 11.5 Ć1010 Nmā2; š¶44 = 2.56 Ć 1010 Nmā2; š¶66 = 3.06 Ć1010 Nmā2; š½1 = 1.52 Ć 106 NKā1mā2; š0 = 298K; š½3 = 1.53 Ć 106 šš¾ā1šā2
, šš£ = 420Jkgā1Kā1 š3 = ā452 Ć 10ā6 CKā1mā2;š13 = ā6.98Cmā2
š¾1 = š¾3 = 1.5Wmā1Kā1; š33 = 13.8Cmā2
š15 = 13.4Cmā2 ;š = 7750Kgmā2;
ķ11 = 60.0 Ć 10ā10C2Nā1mā2; ķ33 = 5.47 Ć 10ā10C2Nā1mā2.
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The non-dimensional frequency versus the wave number and thickness h are plotted in Figs 2ā5, which delineate the impacts of the underly-ing hydrostatic stress on the longitudinal vibra-tions of a hollow circular cylinder for the benefit of turning parameter Ī©. From the Figs 2 and 3 there are a type of comparative changes gath-ered in non dimensional frequency against the wave number which differing in rotational speed Ī©. whereas,the non dimensional frequency ex-panding at the same time for the higher estima-tions of wave number to arrive at as far as pos-sible and again diminished in Fig: 1. From the Figs 4 and 5, there is a comparative change, which can be watched above all in non-dimensional frequency against the thickness which shifting in rotational speed Ī©.The study represents the conduct of turning at first focused on hollow cylinder. Not withstanding the idea of non-dimensional frequency against hydrostatic pressure (š0=-5, 0, 5Ć10^6) is watched.
Fig .1. Geometry of the problem
Fig. 2. Variation of frequency versus wave number against
hydrostatic stress p0 with Ī©=0.
Fig. 3. Variation of frequency versus wave number against
hydrostatic stress p0 with Ī©=0.4.
Fig. 4. Variation of frequency versus Thickness against hy-
drostatic stress p0 with Ī©=0
Fig. 5. Variation of frequency versus Thickness against hy-
drostatic stress p0 with Ī©=0.4
The non-dimensional frequency versus the wave number and thickness h is plotted in Figs 6ā9, which represent the impacts of the underly-ing hydrostatic stress on the longitudinal vibra-tions of a hollow circular cylinder for the estima-tion of Electric parameter E. From the Figs 6 and 7 there is a slight change happened in non-dimensional frequency against the wave number which shifting in Electric parameter E and the equivalent, which shows in hydrostatic pressure. Despite the fact that, the non dimensional fre-quency which stays consistent by expanding in wave number. From the Figs 8 and 9 it is seen that the essential piece of non dimensional fre-quency against the thickness which variety in Electric parameter E and furthermore a similar which shows in hydrostatic pressure. From this, it is seen that the conduct of electric parameter is focused on at hollow cylinder. The idea of non-dimensional frequency against hydrostatic pres-sure (š0=-5, 0, 5Ć10^6) consequently watched.
Fig. 6. Variation of frequency vs. wave number against hy-
drostatic stress p0 with šø = 0
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Fig. 7. Variation of frequency vs wave number against hy-drostatic stress p0 with šø = 0.5.
Fig. 8. Variation of frequency versus Thickness against hy-
drostatic stress p0 with šø = 0.
Fig. 9. Variation of frequency versus Thickness against hy-
drostatic stress p0 with šø = 0.5.
The non-dimensional frequency versus the thickness h against the with and without hydro-static stress are plotted in 3D Figs 10-13, which illustrate the effects of the initial hydrostatic stress on the longitudinal vibrations of a LEMV and CFRP layers of the hollow circular cylinder. From the Figs 10 and 11 compared that the LEMV and CFRP layers how to vary without hy-drostatic stress for increasing in thickness of the cylinder. From the Figs 12 and 13 compared that the LEMV and CFRP layers how to vary with hy-drostatic stress (š0 = 5 Ć 106) for increasing of thickness of the cylinder. In both cases a linear nature observed in LEMV layers against the in-fluences of with and without hydrostatic stress, but small deviations noted in CFRP layers against the influences of with and without hy-drostatic stress.
The 3D Figs: 14-15 shows the non-dimensional strain against the wave number and thickness h, which illustrate the effects of the initial hydrostatic stress on the longitudinal vi-brations of a hollow circular cylinder for the val-ue of thermal parameter š½.
Fig.10. Variation of nondimensional frequency versus thicness of cylinder with LEMV Layer and without
hydtrostatic stress.
Fig. 11. Variation of nondimensional frequency versus thickness of cylinder with CFRP layer and without hydrostat-
ic stress.
Fig. 12. Variation of nondimensional frequency versus thickness of the cylinder against hydrostatic stress in LEMV
layer.
6. Conclusions
The frequency equation for free axisymmet-ric vibration of the hollow circular cylinder with initial hydrostatic stress as core material is de-rived using three-dimensional linear theory of elasticity. Three displacement potential func-
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tions introduced to uncouple the equations of motion, electric and heat conduction. The fre-quency equation of the system consisting of ro-tating piezothermoelastic cylinder developed under the assumption of thermally insulated and electrically shorted free boundary conditions at the surface of the cylinder. The analytical equa-tions numerically studied through the MATLAB programming for axisymmetric modes of vibra-tion for PZT-5A material. The Dispersion curves shows the variation of the various physical pa-rameters of the hollow circular cylinders with a rotation speed, thermal and electrical impacts. The damping observed is not significant due to the rotating effect of the circular hollow cylinder and the presence hydrostatic stress. In addition, the damping observed significant due to the thermal and electric effect of the circular hollow cylinder. The scattering of the frequency in LEMV and CFRP layers are observed in the pres-ence of hydrostatic stress. The methods used in the present article are applicable to a wide range of problems in elasticity with different bonding layers.
Fig: 13. Variation of nondimensional frequency versus thickness of the cylinder against hydrostatic stress in CFRP
layer.
Fig.14. Variation of Strain versus wave number against hy-
drostatic stress p0 with thermal parameter š½.
Fig.15. Variation of Strain versus wave number against hy-
drostatic stress p0 without thermal parameter š½.
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