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## Latin American Journal of Solids and Structures

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*On-line version* ISSN 1679-7825

### Lat. Am. j. solids struct. vol.11 no.14 Rio de Janeiro 2014

#### https://doi.org/10.1590/S1679-78252014001400006

**Modeling and analysis of waves in a heat conducting thermo-elastic plate of elliptical shape**

**R. Selvamani ^{I, *}; P. Ponnusamy^{II}**

^{I}Department of Mathematics, Karunya University, Coimbatore-641 114, Tamil Nadu, India

^{II}Departments of Mathematics, Govt Arts College, Coimbatore, 641 018, Tamil Nadu, India

**ABSTRACT**

Wave propagation in heat conducting thermo elastic plate of elliptical cross-section is studied using the Fourier expansion collocation method based on Suhubi's generalized theory. The equations of motion based on two-dimensional theory of elasticity is applied under the plane strain assumption of generalized thermo elastic plate of elliptical Â cross-sections composed of homogeneous isotropic material. The frequency equations are obtained by using the boundary conditions along outer and inner surface of elliptical cross-sectional plate using Fourier expansion collocation method. The computed non-dimensional frequency, velocity and quality factor are plotted in dispersion curves for longitudinal and flexural (symmetric and antisymmetric) modes of vibrations.

**Keywords: **Wave propagation in plate; vibration of thermal plate; Fourier collocation method; generalized thermo elastic plate; thermal relaxation times

**1 INTRODUCTION**

The plates of elliptical cross-sections are frequently used as structural components and their vibration characteristics are important for practical design. The propagation of waves in thermoelastic material has many applications in various fields of science and technology, namely, atomic physics, industrial engineering, thermal power plants, submarine structures, pressure vessel, aerospace, chemical pipes, and metallurgy. The importance of thermal stresses in causing structural damages and changes in functioning of the structure is well recognized whenever thermal stress environments are involved.

A method, for solving wave propagation in doubly connected arbitrary and polygonal cross-sectional plates and to find out the phase velocities in different modes of vibrations namely longitudinal, torsional and flexural, by constructing frequency equations was devised by Nagaya (1981a; 981b; 1983a; 1983b; 1983c). He formulated the Fourier expansion collocation method for this purpose, the same method is adopted in this problem. The generalized theory of thermo elasticity was developed by Lord and Shulman (1967) involving one relaxation time for isotropic homogeneous media, which is called the first generalization to the coupled theory of elasticity. These equations determine the finite speeds of propagation of heat and displacement distributions, the corresponding equations for an isotropic case were obtained by Dhaliwal and Sherief (1980). The second generalization to the coupled theory of elasticity is what is known as the theory of thermo elasticity, with two relaxation times or the theory of temperature-dependent thermoelectricity. A generalization of this inequality was proposed by Green and Laws (1972). Green and Lindsay (1972) obtained an explicit version of the constitutive equations. This theory contains two constants that act as relaxation times and modify not only the heat equations, but also all the equations of the coupled theory. The classical Fourier's law of heat conduction is not violated if the medium under consideration has a center of symmetry. Erbay and Suhubi (1986) studied the longitudinal wave propagation in a generalized thermoplastic infinite cylinder and obtained the dispersion relation for a constant surface temperature of the cylinder.

Sharma and Pathania (2005) investigated the generalized wave propagation in circumferential curved plates. Asymptotic of wave motion in transversely isotropic plates was analyzed by Sharma and Kumar (2012). Tso and Hansen (1995) have studied the wave propagation through cylinder/plate junctions. Heyliger and Ramirez (2000) analyzed the free vibration characteristics of laminated circular piezoelectric plates and disc by using a discrete-layer model of the weak form of the equations of periodic motion. Thermal deflection of an inverse thermo elastic problem in a thin isotropic circular plate was presented by Gaikward and Deshmukh (2005). Verma and Hasebe (2001) investigated the wave propagation in plates of general anisotropic media in generalized thermo elasticity. Later, Verma (2002) has presented the propagation of waves in layered anisotropic media in generalized thermo elasticity in an arbitrary layered plate. The free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates was studied by Chen et al. [19]. Kumar and Partap (2007) presented the free vibration of microstretch thermoelastic plate with one relaxation time. Ponnusamy and Selvamani (2012) have studied the dispersion analysis of generalized magneto-thermo elastic waves in a transversely isotropic cylindrical panel using the wave propagation approach. Later, Selvamani (2012) performed mathematical modeling and analysis for damping of generalized thermoelastic waves in a homogeneous isotropic plate.

In this paper, the in-plane vibration of heat conducting thermo elastic elliptical cross-sectional plate of homogeneous isotropic material is studied. The solutions to the equations of motion for an isotropic medium is obtained by using the two dimensional theory of elasticity. To satisfy the boundary conditions, the Fourier expansion collocation method is performed to the equations of the boundary conditions and the frequency equations are obtained for longitudinal and flexural (symmetric and antisymmetric) modes of vibrations. The computed non-dimensional frequency, velocity and quality factor are plotted in dispersion curves for longitudinal and flexural (symmetric and antisymmetric) modes of vibrations.

**2 FORMULATION OF THE PROBLEM**

We consider a homogeneous, isotropic, thermally conducting elastic plate of elliptical cross -sections with uniform temperature *T*_{0} in the undisturbed state initially. The system displacements and stresses are defined in polar coordinates *r* in θ an arbitrary point inside the plate and denote the displacements *u _{r}*Â in the direction of

*r*Â and

*u*

_{θ}Â in the tangential direction.

The two dimensional stress equations of motion, and strain –displacement relations and heat conduction equation in the absence of body force for a linearly elastic medium are

and

where σ_{rr}, σ_{θ}_{θ}, σ_{r}_{θ} are the stress components, *e*_{rr}, *e*_{θ}_{θ}, *e*_{r}_{θ}Â are the strain components, *T*Â is the temperature change about the equilibrium temperature *T*_{0},Â ρ is the mass density, *c _{v}*Â is the specific heat capacity,Â β is a coupling factor that couples the heat conduction and elastic field equations,

*K*is the thermal conductivity,Â η, τ is the thermal relaxation times,

*t*Â is the time,Â λ andÂ µÂ are Lame' constants.

The strain *e _{ij}*Â related to the displacements are given by

in which *u _{r}*Â and

*u*

_{θ}Â areÂ the displacement components along radial and circumferential directions, respectively. The comma in the subscripts denotes the partial differentiation with respect to the variables.

Substituting Eqs. (3) and (2) in Eq. (1), the following displacement equations of motions are obtained

**3 SOLUTIONS OF THE PROBLEM**

Eq. (4) is a coupled partial differential equation with two displacements and heat conduction components. To uncouple Eq.(4), we follow Mirsky (1964) by assuming the vibration and displacements along the axial direction z equal to zero. Hence assuming the solutions of Eq. (4) in the form

where ε_{n} = 1/2 for *n* = 0, ε_{n} = 1 for *n* __>__ 1, , ω is the frequency, ϕ_{n}(*r*, θ), ψ_{n}(*r*, θ), *T*_{n}(*r*, θ), (*r*, θ), (*r*, θ) and (*r*, θ) are the displacement potentials.

Introducing the dimensionless quantities such as , *x* = *r*/*a*, = (λ + 2µ)/ρ, α' = *c*_{1}*a*/*K*, Ω^{2} = ω^{2}*a*^{2}/, ε_{1} = *T*_{0}*a*β^{2}/(ρ^{2}*c _{v}c_{1}K*), ε

^{2}= τ/(

*c*), and ε

_{v}K_{3}=

*c*

_{1}η/

*a*and using Eq. (5) in Eq. (4), we obtain

where

and

where ∇^{2} ≡ ∂^{2}/∂*x*^{2} + *x*^{-1} ∂/∂*x* + *x*^{-2} ∂^{2}/∂θ^{2}

The parameters defined in Eq. (7) namely, ε_{1}Â couples the equations corresponding to the elastic wave propagation and the heat conduction which is called the coupling factor; theÂ coefficient ε_{2}, which is introduced by the theory of generalized thermo elasticity, may render the governing system of equations hyperbolic. The parameter ε_{3}Â is the coefficient of the term indicating the difference between empirical and thermodynamic temperatures.Â Solving the partial differential equation (6), the solutions for symmetric mode is obtained as

and the solution for the antisymmetric mode Â is obtained by replacing cos *n*θÂ by sin *n*θÂ in Eqs. (9a) and (9b), we get

where *J _{n}*Â is the Bessel function of first kind of order

*n*Â and

*Y*Â is the Bessel function of second kind of order

_{n}*n*. Solving Eq. (8), we obtain

for symmetric mode, and for the antisymmetric mode is obtained from Eq. (11a) by replacing sin *n*θÂ by cos *n*θ.

where (α_{3}*a*)^{2} = (2 + )Ω^{2} - ς^{2}. If, (α* _{i}a*)

^{2}< 0 (

*i*= 1, 2, 3), then the Bessel functions

*J*and

_{n}*Y*are replaced by the modified Bessel function

_{n}*I*and

_{n}*K*respectively.

_{n}

**4 BOUNDARY CONDITIONS AND FREQUENCY EQUATIONS**

In this problem, the vibration of thick arbitrary cross-sectional plate is considered. Since the boundary is irregular in shape, it is difficult to satisfy the boundary conditions along both outer and inner surface of the plate directly. Hence, the Fourier expansion collocation method is applied to satisfy the boundary conditions. For the plate, the normal stress , shearing stress and thermal field *T*' along the inner surface of the plate is equal to zero. Similarly, normal stress σ* _{xx}*, shearing stress σ

*and thermal field*

_{xy}*T*along the outer surface of the plate is equal to zero. Thus, the boundary conditions along the outer boundary of the plate are

and for the inner boundary, the boundary conditions are

where *x* is the coordinate normal to the boundary and *y* is the coordinate tangential to the boundary, σ* _{xx}*, are the normal stresses,σ

*, are the shearing stresses,*

_{xy}*T*,

*T*' are the thermal fields and ()

_{i}is the value at the

*i*-

*th*segment of the outer and inner boundary respectively.Â Since the cross-section of the plate is irregular, it is difficult to find the transformed expression of the stresses in the thick arbitrary cross-sectional plates because the coordinate and are vary with the angle θ. Therefore, the inner and outer boundary of the thick arbitrary cross-sections is divided into small segments such that the variations of the stresses are assumed to be constant. Assuming the angle γ

*, between the normal to the segment and the reference axis to be constant, the transformed expressions for the stresses are followed by Nagaya (1983b) as*

_{i}Substituting Eqs. (9)-(11)in Eqs. (12) and (13), the boundary conditions are transformed as follows:

for the inner surface and

for the outer surface, where

The coefficients for are given in the Appendix A.

Performing the Fourier series expansion to Eqs. (12) and (13) along the boundary, the boundary conditions along the inner and outer surfaces are expanded in the form of double Fourier series. When a plate is symmetric about more than one axis, the boundary conditions, in the case of symmetric mode can be written in the form of a matrix as given below:

where

Similarly, the matrix for the antisymmetric mode is obtained as

where

and where *j* = 1, 2, 3, 4, 5 and 6, *I* is the number of segments, is the coordinate at the inner boundary, *R _{i}* is the coordinate

*r*at the outer boundary and

*N*is the number of truncation of the Fourier series. For the nontrivial solution of the system of equations given in Eqs. (19) and (21), the determinant of the coefficient matrix must vanish and these determinants give the frequencies of symmetric and antisymmetric modes of vibrations respectively.

**4.1 Elliptic cross-sectional plate**

The geometry of elliptic cross-sectional ring shaped plate is shown in Figure 1.The geometrical relations of an elliptic ring shaped plate given by Nagaya (1981b) are used for numerical calculation and are given below:

for the outer surface and

for the inner surface, where *a*_{1} and *a*_{2} are the length of inner and outer semi major axis, and *b*_{1} and *b*_{2} are the length of semi minor axis of an elliptic cross-section. Also = (θ_{l} + θ_{l-1})/2 and *R _{i}* is the coordinate

*r*at the

*i*-

*th*boundary, γ

*is the angle between the reference axis and the normal to the segment. For in-plane vibration problem, there exist two types of vibrations; one of which is generated mainly by flexural motion and the other by longitudinal motion.*

_{i}In the present problem, there are three kinds of basic independent modes of wave propagation have been considered, namely, the longitudinal and two flexural (symmetric and antisymmetric) modes.

**5 NUMERICAL ANALYSIS**

The numerical analysis of the frequency equation is carried out for heat conducting thermo elastic elliptic cross-sectional plates. The material properties of copper at 42 K are taken approximately as Poisson ratio *v* = 0.3, density ρ = 8.96x10^{3} kg/m^{3}, the Young's modulus *E* = 2.139x10^{11} N/m^{2}, λ = 8.20x10^{11} kg/*m**s^{2}, µ = 4.20x10^{10} kg/*m**s^{2}, *c _{v}* = 9.1x10

^{-2}m

^{2}/K*s

^{2}, and

*K*= 113x10

^{-2}kg*m/K*s

^{2}. The thermal parameters such α', ε

_{1}, ε

_{2}, and ε

_{3}are chosen by the following arguments are given by Erbay and Suhubi (1986).

In the numerical calculation, the angle is taken as an independent variable and the coordinate *R _{i}* and are at the

*i*-

*th*segment of the boundary is expressed in terms of θ. Substituting

*R*, and the angle γ

_{i}_{i}, between the reference axis and the normal to the

*i*-

*th*boundary line, the integrations of the Fourier coefficients and can be expressed in terms of the angle θ. Using these coefficients in to equations (20) and (22), the frequencies are obtained for heat conducting thermo elastic elliptical cross-sectional plate.

**5.1 Longitudinal mode**

The geometrical relations for the elliptic cross-sections given in Eq. (23) are used directly for the numerical calculations, and three kinds of basic independent modes of wave propagation are studied.Â In case of the longitudinal mode of elliptical cross-section, the cross-section vibrates along the axis of the cylinder, so that the vibration and displacements in the cross-section is symmetrical about both major and minor axes. Hence, the frequency equation is obtained by choosing both terms of *n* and *m* as 0, 2, 4, 6, ... in Eq. (19) for the numerical calculations. Since the boundary of the elliptical cross-sections is irregular in shape, it is difficult to satisfy the boundary conditions along the curved surface, and hence Fourier expansion collocation method is applied. In this method, the curved surface, in the range θ = 0 and θ = π is divided into 20 segments, such that the distance between any two segments is negligible and the integrations is performed for each segment numerically by using the Gauss five point formula.The non-dimensional frequenciesÂ are computed for 0 < Ω __<__ 1.2 , using the secant method.

**5.2 Flexural mode**

In the case of flexural mode of elliptical cross-section, the vibration and displacements are antisymmetrical about the major axis and symmetrical about the minor axis. Hence, the frequency equations are obtained from Eq. (21) by choosing *n*, *m* = 1, 3, 5, ... . Two kinds of flexural (symmetric and antisymmetric) modes are considered. The computed non-dimensional frequencies are presented in the form of dispersion curves.

**5.3 Quality factor**

The ratio of the total elastic energy and the energy loss in one cycle of a material is defined as the quality factor. In engineering and physics, the quality factor of a material is a dimensionless parameter that compares the time constant for decay of an oscillating physical structure's amplitude to its oscillation period. Equivalently, it compares the frequency at which a structure oscillates to the rate at which it dissipates its energy. The quality factor is defined as

where *V* and *Q* represents the phase velocity and attenuation coefficient, respectively.

**5.4 Dispersion curve**

The variation of non-dimensional frequency versus dimensinonless wave number of longitudinalÂ modes ofÂ thermo-elastic circular cross sectional plate with and without thermal field for the two values of aspect ratio *a*_{1}/*b*_{1} = *a*_{2}/*b*_{2} = 0.5, 1.5 is shown in Figures 2 and 3 respectively. From Figure 2, it is observed that the dispersion linearly increase with respect to the wave number. For without thermal field, the relation between the wave number and frequency in both the cases of aspect ratios are alike, but for the thermal inclusion the behavior is oscillating throughout the full range of wave number. Figures 4 and 5 shows the dispersion characteristic of flexural (symmetric) modes of elliptic cross-sectional plate with and without thermal field.Â It shows that, the dispersion is common for both the plate with and without thermal inclusion, where the higher aspect ratio attains higher frequency compared with lower aspect ratio.

A graph is drawn between non-dimensional wave number versus dimensionless frequency of flexural (antisymmetric) modes of elliptic cross-sectional plate with and without thermal environment are shown in Figures 6 and 7 respectively. In both figures, it is observed that as the wave numberÂ increases, the non-dimensional frequencies also increases, also it could be inferred that, the dispersion characteristic are prominent only in lower orders with negligible variations in higher rang.Â From the figures, it is observed that the non-dimensionless frequency increases with respect to its wave number. It is also observed that, the cross over points between the modes of different aspect ratios of longitudinal and flexural symmetric and antisymmetric modes represents the transportation of energy between the two different vibration medium.

The variation of velocity versus dimensionless frequency of a flexural symmetric modes of elliptic cross-sectional plate with and without thermal environment are shown in Figures 8 and 9, respectively. From the Figures 8 and 9, it is observed that, for the increasing values of the aspect ratio modes are merges and oscilating through entire range of frequency. The merging of thermal modes and oscillation of point between the vibrational modes shows that, there is an energy transportation between the modes of vibrations by the effect of aspect ratio and anisotropic of the material.

Figures 10 and 11 reveals that the variation of quality factor with the dimensionless frequency for the flexural symmetric modes with and without thermal signal. The quality factor is quite high at lower range of frequency and starts to decay with increasing frequency. The quality factor profile are dispersive in trend forÂ flexural symmetrical modes with thermal signal than in flexural symmetrical modes withoutÂ thermal signal and experience oscillation in the wave number range 0.2 __<__ |ς| __<__ 0.6 for flexural symmetric thermal modes of vibration. The cross over points between the vibration modes represents the transfer of energy between the thermal modes.

**6 CONCLUSIONS**

In this paper, the wave propagation in a heat conducting thermo elastic plate of ellipticalÂ shaped is analyzed by satisfying the boundary conditions on the irregular boundary using the Fourier expansion collocation method and the frequency equation for the longitudinal and flexural (symmetric and antisymmetric) modes of vibrations are obtained. The computed dimensionless frequency, velocity and quality factor are plotted in graphs for longitudinal and flexural (symmetric and antisymmetric) modes with and without thermal signals. From the graphical representation, the effect of thermal energy with thermal relaxation times and the anisotropy of the material on the various considered wave characteristics are more significant and dominant in the flexural modes of elliptical cross sectional plate.

**References**

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Received 21.01.2014

In revised form 23.06.2014

Accepted 16.07.2014

Available online 17.08.2014

**Appendix A**

The expressions used in Eqs. (20) and (22) are given as follows: