On the propagation of Lamb waves in viscothermoelastic plates under fluid loadings

In this paper the propagation of plane and circular crested viscothermoelastic waves in a homogeneous isotropic, Kelvin-Voigt type viscoelastic thermally conducting, plate sandwiched between inviscid liquid layers is investigated in the context of classical and non-classical theories of thermoelasticity. The secular equations for the symmetric and skew-symmetric modes of plane and circular crested waves are derived in closed form and isolated mathematical conditions. It is noticed that the motion for both the plane and cylindrical waves in plates is governed by Rayleigh-Lamb-type secular equations. The secular equations for thin plate and short wave length waves are also obtained and discussed. The results in the absence of fluid loading, coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. The dispersion curves, attenuation profiles and specific loss in case of symmetric and skew-symmetric wave modes are also presented graphically for a polymethyl methacrylate material plate under fluid loadings. The effect of dissipation due to viscosity is noticed to be quite significant and clearly visible from various curves in the graphs.     

Thermoelastic Lamb waves in a transversely isotropic plate bordered with layers of inviscid liquid

Analysis for the propagation of thermoelastic waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides, is investigated in the context of coupled theory of thermoelasticity. Secular equations for homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and anti-symmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion of wave propagation and occurs along an in-plane axis of symmetry. The special cases, such as short wavelength waves and thin plate waves of the secular equations are also discussed. The secular equations for leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic plate of zinc material bordered with water. The dispersion curves for symmetric and anti-symmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement.     

Analysis of free vibrations for Rayleigh — Lamb waves in a microstretch thermoelastic plate with two relaxation times

The propagation of free vibrations in a microstretch thermoelastic homogeneous isotropic plate subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT) and Green and Lindsay (G—L) theories of thermoelasticity. The secular equations for the microstretch thermoelastic plate in closed form for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. At short wavelength limits, the secular equations for both modes in a stress-free thermally insulated and isothermal homogeneous isotropic microstretch thermoelastic plate reduce to the Rayleigh surface wave frequency equation. The results for symmetric and skew-symmetric wave modes are computed numerically and presented graphically. The theory and numerical computations are found to be in close agreement. Published in Inzhenerno-Fizicheskii Zhurnal, Vol. 82, No. 1, pp. 36–46, January–February, 2009.     

Circular crested waves in anisotropic thermoelastic plates bordered with inviscid liquid

The propagation of circularly crested waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides is investigated in the context of conventional coupled thermoelasticity, Lord-Shulman and Green-Lindsay theories of thermoelasticity. Secular equations for circular homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. The special cases such as short wavelength waves, thin plate waves and leaky Lamb waves of the secular equation are also deduced and discussed. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic circular plate of cobalt material bordered with water. The dispersion curves for symmetric and antisymmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The analytical and numerical results are found to be in close agreement.     

Elastodynamic Behavior of Waves in Thermo-Microstretch Elastic Plate Bordered with Layers of Inviscid Liquid

The elastodynamic behavior of waves in a thermo-microstretch elastic homogeneous isotropic plate bordered with layers of inviscid liquid on both sides subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of Lord and Shulman and Green and Lindsay theories of thermoelasticity. The mathematical model has been simplified by using the Helmholtz decomposition technique, and the frequency equations for different mechanical situations are obtained and discussed. The special cases such as short wavelength waves and regions of the secular equations are also discussed. Finally, the numerical solution is carried out for a magnesium crystal composite material plate bordered with water. The dispersion curves, attenuation coefficients, amplitudes of dilatation, microrotation, microstretch, and temperature distribution for the symmetric and skew-symmetric wave modes are presented graphically.     

Dispersion of Thermoelastic Waves in a Plate With and Without Energy Dissipation

In this paper, the dispersion and energy dissipation of thermoelastic plane harmonic waves in a thin plate bounded by insulated traction-free surfaces is studied on the basis of three generalized theories of thermoelasticity. The frequency equations corresponding to the symmetric and antisymmetric modes of vibration of the plate are obtained. Some limiting and particular cases of the frequency equations are then discussed. Results obtained in three theories of generalized thermoelasticity are compared. The results for the coupled thermoelasticity can be obtained as particular cases of the results by setting thermal relaxations times equal to zero. Numerical evaluations relating to the lower modes of the symmetric and antisymmetric waves are presented for an aluminum alloy plate.     

超急速传热过程中热弹性响应的解析分析

考虑超急速传热过程中诱发的热冲击效应,基于L-S广义热弹性理论,建立了温度突变加热条件下热弹性响应的控制方程组。借助于Laplace正逆变换,在适当简化的条件下推导了一维超急速传热问题热弹性响应的解析表达式。通过对温度场、位移场及应力场的解析求解,给出了超急速传热过程中热波和热弹性波在弹性体内的传递规律,并指出在超急速传热条件下,由于热波和热弹性波的相互叠加作用削弱了热作用产生的热冲击效应。     

On waves in a random micropolar conducting magneto-thermo-viscoelastic medium

Effects of random inhomogeneity on wave propagation in the interacting micropolar conducting magneto-generalized thermo-viscoelastic medium are studied. The couple stress theory relevant to micropolar solids is employed. The analysis is carried out under the smooth perturbation technique amenable to stochastic linear differential equations up to the second perturbation. The perturbing field has been assumed to be weakly conducting and weakly thermal. The generalized thermoelasticity has been used. Six different types of waves have been observed to propagate in the medium. The dispersion equations have been derived. Effects due to random variations of micropolar-elastic, conducting-electromagnetic and generalized thermo-visco-parameters have been computed. Effects of random heterogeneity of the conducting magnetic field are readily available up to first order perturbation. However effects of the generalized thermal field are discernible only in the domain of second order perturbation. Change of phase speed occurs on account of randomness. Attenuation coefficients for types of waves have been computed. A special type of generalized theromomechanical auto- and cross-correlation functions has been used to approximately measure effects of random variations of parameters. Uncoupled problem has been formulated for future investigations.     

On the propagation of waves in layered anisotropic media in generalized thermoelasticity

In view of the increased usage of anisotropic materials in the development of advanced engineering materials such as fibers and composite and other multilayered, propagation of thermoelastic waves in arbitrary anisotropic layered plate is investigated in the context of the generalized theory of thermoelasticity. Beginning with a formal analysis of waves in a heat-conducting N-layered plate of an arbitrary anisotropic media, the dispersion relations of thermoelastic waves are obtained by invoking continuity at the interface and boundary conditions on the surfaces of layered plate. The calculation is then carried forward for more specialized case of a monoclinic layered plate. The obtained solutions which can be used for material systems of higher symmetry (orthotropic, transversely isotropic, cubic, and isotropic) are contained implicitly in our analysis. The case of normal incidence is also considered separately. Some special cases have also been deduced and discussed. We also demonstrate that the particle motions for SH modes decouple from rest of the motion, and are not influenced by thermal variations if the propagation occurs along an in-plane axis of symmetry. The results of the strain energy distribution in generalized thermoelasticity are useful in determining the arrangements of the layer in thermal environment.     

Propagation of Waves Through Cylindrical Bore in a Cubic Micropolar Generalized Thermoelastic Medium

The article deals with the propagation of axial symmetric cylindrical surface waves in a cylindrical bore through a micropolar thermoelastic medium of infinite extent possessing cubic symmetry. The theories of generalized thermoelasticity developed by Lord and Shulman and Green and Lindsay are used to study the problem. The frequency equations, connecting the phase velocity with the wave number, radius of bore, and other material parameters for empty and liquid-filled bores are derived. Some special cases have been deduced. The numerical results obtained have been illustrated graphically to understand the behavior of the phase velocity and attenuation coefficient versus the wave number.     

A generalized variational model of magneto-thermo- elasticity for nonlinearly magnetized ferroelastic bodies

Based on a generalized variational principle of total energy functional, this paper presents a theoretical model to describe the magneto-thermo-elastic interaction of soft ferroelastic bodies with nonlinear magnetization under stationary thermal and magnetic fields. The energy functional of the magneto-thermo-elastic system is established by the summation of energy of sub-systems of nonlinearly magnetized magnetic field, thermal field, and mechanical deformation. By means of the manipulation of the mixed variational principle with independent variations of magnetic scalar potential, displacement vector, and temperature, all governing equations, which are nonlinear and coupling among magnetic, elastic and thermal fields, together with the expressions of magnetic forces are obtained from the variational approach. In order to valuate the obtained model, some existing models of the magneto-elasticity and the thermo-elasticity, which are validly demonstrated in literature, as special cases of the problem considered here are deduced out from the general case. Finally, an analytical analysis of magneto-thermo-elastic instability is conducted to a simply supported ferromagnetic rectangular thin plate under both a uniform distribution of temperature and a uniform transverse magnetic field by means of the linearized theory and the perturbation technique.     

Generalized thermoelastic waves in spherical curved plates without energy dissipation

In this article, the propagation of thermoelastic waves in orthotropic spherical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The theoretical formulation is based on the linear GN thermoelastic theory. The coupled wave equation and heat conduction equation expressed by the displacement and temperature are obtained. By the Legendre orthogonal polynomial series expansion approach, the coupled controlling equations are solved. The convergence of the method is demonstrated through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding purely elastic spherical plate are also shown to analyze the influence of thermoelasticity on elastic modes. The displacement, temperature and stress distributions of both elastic modes and thermal modes are calculated to show their differences. A thermoelastic spherical plate with a different ratio of radius to thickness is considered to show the influence of the ratio on the characteristics of thermoelastic waves.     

Generalized thermoelastic waves in functionally graded plates without energy dissipation

In this paper, a dynamic solution of the propagating thermoelastic waves in functionally graded material (FGM) plate subjected to stress-free, isothermal boundary conditions is presented in the context of the Green–Naghdi (GN) generalized thermoelastic theory. The FGM plate is composed of two orthotropic materials. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The coupled wave equation and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of the inhomogeneous thermoelastic plate and the corresponding pure elastic plate are compared to show the characteristics of thermal modes and the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions of elastic modes and thermal modes are shown to discuss their differences. A plate with a different gradient variation is calculated to illustrate the influence of the gradient field on the wave characteristics.     

Rayleigh-Lamb Waves in Transversely Isotropic Thermoelastic Diffusive Layer

Propagation of plane harmonic thermoelastic diffusive waves in a homogeneous, transversely isotropic, thin elastic layer of finite width is studied, in the context of the theory of coupled thermoelastic diffusion. According to the characteristic equation, three quasi-longitudinal waves, namely, quasi-elastodiffusive (QED) mode, quasi-mass diffusion (QMD) mode, and quasi-thermodiffusive (QTD) mode can propagate in addition to quasi-transverse waves (QSV) mode and the purely quasi-transverse motion (QSH) mode, which is not affected by thermal and diffusion vibrations, gets decoupled from the rest of the motion of wave propagation. The secular equations corresponding to the symmetric and skew symmetric modes of the layer are derived. The amplitudes of displacements, temperature change, and concentration for symmetric and skew symmetric modes of vibration of the layer are computed numerically. Anisotropy and diffusion effects on the phase velocity, attenuation coefficient, and amplitudes of displacements, temperature change, and concentration are presented graphically in order to illustrate and compare the results analytically. Some special cases of the frequency equation are also deduced and compared with the existing results.     

Rayleigh waves at the boundary surface of modified couple stress generalized thermoelastic with mass diffusion

In this problem, we have studied propagation of Rayleigh waves in an homogeneous isotropic modified couple stress generalized thermoelastic with mass diffusion solid half space in the context of Lord–Shulman (L-S), Green–Lindsay (G-L) theories of thermoelasticity. Secular equations are derived mathematically by using appropriate boundary conditions. The values of determinant of secular equation, Rayleigh wave velocity and attenuation coefficient with respect to angular velocity for different values of wave number and relaxation times in the absence and presence of mass diffusion, are computed numerically. The numerical simulated results are depicted graphically for copper material.     

Some considerations on generalized thermoelasticity

In this paper the analysis is based on the decoupled field equations of generalized thermoelasticity. These equations have been solved with the help of integral transforms. The dynamic behaviour of an elastic half space due to a thermal shock on the boundary is also discussed. Because the “second sound” effects are short lived, the small time approximations have been considered. The displacement is continuous and temperature is discontinuous on both the elastic as well as thermal wave fronts.     

Two-dimensional generalized thermal shock problem of a thick piezoelectric plate of infinite extent

The theory of generalized thermoelasticity, based on the theory of Green and Lindsay with two relaxation times, is used to deal with a thermoelastic–piezoelectric coupled two-dimensional thermal shock problem of a thick piezoelectric plate of infinite extent by means of the hybrid Laplace transform-finite element method. The generalized thermoelastic–piezoelectric coupled finite element equations are formulated. By using Laplace transform the equations are solved and the solutions of the temperature, displacement and electric potential are obtained in the Laplace transform domain. Then the numerical inversion is carried out to obtain the temperature, displacement and electric potential distributions in the physical domain. The distributions are represented graphically. From the distributions, it can be found the wave type heat propagation in the piezoelectric plate. The heat wavefront moves forward with a finite speed in the piezoelectric plate with the passage of time. This indicates that the generalized heat conduction mechanism is completely different from the classic Fourier’s in essence. In generalized thermoelasticity theory heat propagates as a wave with finite velocity instead of infinite velocity in media.     

Equations in Stresses for Two- and Three-Dimensional Dynamic Problems of Thermoelasticity in Spherical Coordinates

The basic system of differential equations for the six components of the dynamic stress tensor is deduced in a spherical coordinate system by using the equations of motion, Hooke's law, Cauchy relations, and Saint-Venant conditions of compatibility of strains. The obtained system of differential equations is reduced (without introducing additional potential functions) to a system of wave equations used for the successive evaluation of the first invariant and unknown components of the stress tensor. We also present the corresponding systems of wave equations for the axisymmetric, polar, and centrally symmetric problems of thermoelasticity.     

Thermo elastic waves with thermal relaxation in isotropic micropolar plate

In the present investigation, we have discussed about the features of waves in different modes of wave propagation in an infinitely long thermoelastic, isotropic micropolar plate, when the generalized theory of Lord–Shulman (L–S) is considered. A more general dispersion equation is obtained. The different analytic expressions in symmetric and anti-symmetric vibration for short as well as long waves are obtained in different regions of phase velocities. It is found that results agree with that of the existing results predicted by Sharma and Eringen in the context of various theories of classical as well as micropolar thermoelasticity.     

Rayleigh surface wave propagation in an orthotropic rotating magneto-thermoelastic medium subjected to gravity and initial stress

Abstract

The prime objective of the present article is to analyze the effects of rotation and initial stress on the propagation of Rayleigh surface waves in a homogeneous, orthotropic magneto-thermoelastic half space subjected to gravity field. The frequency equations in closed form are derived and the amplitude ratios of surface displacements, temperature change during the Rayleigh wave propagation on the surface of half space have been computed analytically. The highlights of this study are the effects of different parameters (rotation, magnetic field, initial stress, and gravity) on the velocity of Rayleigh waves. Variation in phase velocity of Rayleigh waves against a wave number is shown graphically. Some particular cases have been deduced. Also, the classical Rayleigh wave equation is obtained as a special case of the present study. Numerical example has been carried out and represented by the means of graphs. Impacts of various involved parameters appearing in the solutions are carefully analyzed. In fact, in the absence of various parameters, these equations are in agreement with the results for isotropic medium.