Document Type : Research Article
Authors
^{1} Department of Mathematics, Kathir College of Engineering,Coimbatore , 641062, India.
^{2} Department of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore, 641114, India.
^{3} Department of Mechanical Engineering, Imam Khomeini International University Qazvin, Iran
Abstract
Keywords
Assessment of Hydrostatic Stress and Thermo Piezoelectricity 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
KEYWORDS 

ABSTRACT 
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 together with Linear Elastic Material with Voids (LEMV) layer. The model described by the equations of elasticity, the effect of the initial stress and the framework of linearized, threedimensional theory of thermo elasticity. The displacement components obtained by founding the analytical solutions of the motion’s equations. The frequency equations that include the interaction between the composite hollow cylinders are obtained by the perfectslip boundary conditions using the Bessel function solutions. The numerical calculations carried out for the material PZT5A and the computed nondimensional 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.

Composite materials are generally utilized in engineering structures because of their predominance over the basic materials in applications 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, cylinders, funnels and strands. Displaying the proliferation of waves in these parts is significant in different applications, including ultrasonic nondestructive assessment systems, progression of room explore and numerous others. Smart materials are normally prestressed during the assembling procedure. As initial stresses are indivisible in surface acoustic wave gadgets and resonators, investigation of such impacts has been finished with various methodologies. A few creators have considered wave engendering in prestressed piezoelectric structure.
Soniya Chaudhary et al. [1] derived secular equation of SH waves propagating in prestressed and rotating piezocomposite structure with imperfect interface. Abhinav Singhal et al. [2] analyticaly analysed interfacial imperfection study in presstressed rotating multiferroic cylindrical tube with wave vibration. Lotfy and ElBary [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 subjected to mechanical ramp type with twotemperature theory and magnetic feld. Soniya Chaudhary et al. [5] studied anatomy of flexoelectricity in micro plates with dielectrically highly/weakly and mechanically complaint interface. Ebrahimi et al. [6] discussed Magnetoelectroelastic analysis of piezoelectric–flexoelectric nanobeams rested on silica aerogel foundation. Abhinav Singhal et al. [7] investigate mechanics of 2D Elastic Stress Waves Propagation Impacted by Concentrated Point Source Disturbance in Composite Material Bars. Lotfy [8] discover a novel model of photothermal diffusion (PTD) for polymer nanocomposite semiconducting of thin circular plate. Placidi and Hutter [9] studied thermodynamics of Polycrystalline materials treated by the theory of mixtures with continuous diversity. Altenbach and Eremeyev [10] discussed vibrtion analysis of nonlinear 6parameter prestressed shells. Sonal Nirwal et al. [11] analysised of different boundary types on wave velocity in bedded piezostructure with flexoelectric effect. Lotfy and Wafaa Hassan [12] discovered the effect of rotation for twotemperature generalized thermoelasticity of twodimensional under thermal shock problem.Abhinav Singhal et al. [13] studied LiouvilleGreen approximation: an analytical approach to study the elastic wave’s vibrations in composite structure of piezo material.Soniya Chaudhary and Abhinav Singhal [14], [15] discussed analytic model for Rayleigh wave propagation in piezoelectric layer overlaid orthotropic substratum and surface wave propagation in functionally graded piezoelectric material.Abbas and Othman [16] generalized thermoelastic interacon in a fiberreinforced anisotropic halfspace under hydrostatic initial stress. Othman and Lotfy [17] discussed the influence of gravity on 2D problem of two temperatures generalized thermoelastic medium with thermal relaxation. AboDahab et al. [18] discovered rotation and magnetic field effect on surface waves propagation in an elastic layer lying overa generalized thermoelastic diffusive halfspace with imperfect boundary. Hobinyand and Abbas [19] studied analytical solution of magnetothermoelastic interaction in a fiberreinforced anisotropic material. Manoj Kumar Singh et al. [20] discussing approximation of surface wave velocity in smart composite structure using Wentzel–Kramers–Brillouin method. Othman, AboDahab, and Lotfy [21] investigate gravitational effect and initial stress on generalized magnetothermomicrostretch elastic solid for different theories. AboDahab and Lotfy [22] designed twotemperature plane strain problem in a semiconducting medium under photo thermal theory. AboDahab [23] discussed reflection of P and SV waves from stressfree surface elastic halfspace 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 approximation of surface wave frequency in piezocomposite structure. Alireza Mohammadi et.al [26] discover the results of influence of viscoelastic 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 thermal buckling and forced vibration characteristics of a porous GNP reinforced nanocomposite cylindrical shell. Habibi, Hashemabadi and Safarpour [28] discussed the vibration analysis of a highspeed rotating GPLRC nanostructure coupled with a piezoelectric actuator. MostafaHabibi et.al [29] studied stability analysis of an electrically cylindrical nanoshell reinforced with graphene nanoplatelets.MehranSafarpoura et.al [30] analysics frequency characteristics of FGGPLRC 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 hollow multilayered composite cylinder under the influence of initial hydrostatic stress. The cylinder is made of tetragonal system material, such as PZT5A. The displacement components are obtained by founding the analytical solutions of the motion’s equations. After applying suitable boundary conditions, the frequency equation is presented as a determinant with elements containing Bessel functions. The numerical computations obtained the roots of frequency equations. The dispersion curve carried for various parameters.
Longitudinal wave propagation in a homogeneous, transversely isotropic cylinder of tetragonal 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 rotating with uniform angular velocity . The displacement field, in cylindrical coordinates is given by
(1a) 

(1b) 
The electric displacement equation

(1c) 
The heat conduction equation
(1d) 
The stress strain relations are given as follows
(2) 

Where , , are the stress and are the strain components, T is the temperature change about the equilibrium temperature are the five elastic constants, and respectively thermal expansion coefficients and thermal conductivities along and perpendicular to the symmetry, ρ is the mass density, is the specific heat capacity , is the pyroelectric effect.
The strains related to the displacements given by [32]

(3) 
Substitution of the Eqs. (3) and (2) into Eqs. (1) results in the following threedimensional equation of motion, heat and electric conduction. We note that the first two equations under the influence hydrostatical stress become [31]:
(4a) 

(4b) 

(4c) 

(4d) 
Where is heat conduction coefficient, , u and w are the displacements along r, z direction, is the mass density and t is the time.
The solutions of Eq. (4) is considered in the form
(5) 

Where, are displacement potentials, k is the wave number, p is the angular frequency and . We introduce the nondimensional quantities ‘a’ is the geometrical parameter of the composite hollow cylinder. , , , ,
Substituting Eqs. (5) in Eqs. (4), we obtain:
(6a) 

(6b) 

(6c) 

(6d) 
The above Eqs. [6] rearranged in the following form
(7) 
Where
, , , , , , , , . 
Evaluating the determinant given in Eq. (7), we obtain a partial differential equation of the form:
(8) 
Factorizing the relation given in Eq. (8) into biquadratic equation for , i=1, 2, 3, 4 the solutions for the symmetric modes are obtained as
Here for are the roots of algebraic equation
(10) 
The solutions corresponding to the root not considered here, since is zero, except . The Bessel function is used when the roots , are real or complex and the modified Bessel function is used when the roots are imaginary.
The constants and defined in Equation (10) can be calculated from the following equations
The displacement equations of motion and equation of equilibrated inertia for an isotropic LEMV are
(11a) 

(11b) 


(11c) 
The stress in the LEMV core materials are
The solution for Eq. (11) is taken as
(12) 

The above solution in (11) and dimensionless variables x and equation can be simplified as
(13) 
Where
, , ,

The Eq. (13) can specified as,
(14) 
Thus, the solution of Eq. (14) is as follows,
are the roots of the equation when replacing The arbitrary constant and are obtained from


By taking the void volume fraction E=0, and the lame’s constants as in the Eq. (11) we got the governing equation for CFRP core material.
The frequency equations can obtain for the following boundary condition
Substituting the above boundary condition, we obtained as a 22×22 determinant equation

(15) 
At Where
In addition, the other nonzero elements and are obtained by replacing by and by .
At








and the other nonzero element at the interfaces can be obtained on replacing by and by in the above elements. They are and . At the interface , nonzero elements along the following rows and are obtained on replacing by and superscript 1 by 2 in order. Similarly, at the outer surface , the nonzero elements and can be had from the nonzero elements 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 in Eq. (11) which reduces to a 20 x 20 determinant equation. The frequency equations derived above are valid for different inner and outer materials of 6mm class and arbitrary thickness of layers.
The frequency equation given in Eq. (15) is transcendental in nature with unknown frequency and wave number. The solutions of the frequency equations obtained numerically by fixing the wave number. The material chosen for the numerical calculation is PZT5A. The material properties of PZT5A used for the numerical calculation given below:
;
;
; ;
_{,}
_{ } ;
_{ }; ;_{ } ;
.
The nondimensional frequency versus the wave number and thickness h are plotted in Figs 2–5, which delineate the impacts of the underlying hydrostatic stress on the longitudinal vibrations of a hollow circular cylinder for the benefit of turning parameter Ω. From the Figs 2 and 3 there are a type of comparative changes gathered in non dimensional frequency against the wave number which differing in rotational speed Ω. whereas,the non dimensional frequency expanding at the same time for the higher estimations of wave number to arrive at as far as possible and again diminished in Fig: 1. From the Figs 4 and 5, there is a comparative change, which can be watched above all in nondimensional frequency against the thickness which shifting in rotational speed Ω.The study represents the conduct of turning at first focused on hollow cylinder. Not withstanding the idea of nondimensional frequency against hydrostatic pressure ( =5, 0, 5×10^6) is watched.
Fig .1. Geometry of the problem
Fig. 2. Variation of frequency versus wave number against hydrostatic stress p0 with Ω=0.
Fig. 3. Variation of frequency versus wave number against hydrostatic stress p0 with Ω=0.4.
Fig. 4. Variation of frequency versus Thickness against hydrostatic stress p0 with Ω=0
Fig. 5. Variation of frequency versus Thickness against hydrostatic stress p0 with Ω=0.4
The nondimensional frequency versus the wave number and thickness h is plotted in Figs 6–9, which represent the impacts of the underlying hydrostatic stress on the longitudinal vibrations of a hollow circular cylinder for the estimation of Electric parameter E. From the Figs 6 and 7 there is a slight change happened in nondimensional 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 frequency which stays consistent by expanding in wave number. From the Figs 8 and 9 it is seen that the essential piece of non dimensional frequency 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 nondimensional frequency against hydrostatic pressure ( =5, 0, 5×10^6) consequently watched.
Fig. 6. Variation of frequency vs. wave number against hydrostatic stress p0 with
Fig. 7. Variation of frequency vs wave number against hydrostatic stress p0 with .
Fig. 8. Variation of frequency versus Thickness against hydrostatic stress p0 with .
Fig. 9. Variation of frequency versus Thickness against hydrostatic stress p0 with .
The nondimensional frequency versus the thickness h against the with and without hydrostatic stress are plotted in 3D Figs 1013, 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 hydrostatic 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 hydrostatic stress ( for increasing of thickness of the cylinder. In both cases a linear nature observed in LEMV layers against the influences of with and without hydrostatic stress, but small deviations noted in CFRP layers against the influences of with and without hydrostatic stress.
The 3D Figs: 1415 shows the nondimensional strain against the wave number and thickness h, which illustrate the effects of the initial hydrostatic stress on the longitudinal vibrations of a hollow circular cylinder for the value of thermal parameter
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 hydrostatic stress.
Fig. 12. Variation of nondimensional frequency versus thickness of the cylinder against hydrostatic stress in LEMV layer.
The frequency equation for free axisymmetric vibration of the hollow circular cylinder with initial hydrostatic stress as core material is derived using threedimensional linear theory of elasticity. Three displacement potential functions introduced to uncouple the equations of motion, electric and heat conduction. The frequency equation of the system consisting of rotating piezothermoelastic cylinder developed under the assumption of thermally insulated and electrically shorted free boundary conditions at the surface of the cylinder. The analytical equations numerically studied through the MATLAB programming for axisymmetric modes of vibration for PZT5A material. The Dispersion curves shows the variation of the various physical parameters 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 presence 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 hydrostatic stress p0 with thermal parameter .
Fig.15. Variation of Strain versus wave number against hydrostatic stress p0 without thermal parameter .
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