Thermodynamic aspects of transverse wave propagation in an incompressible hyperelastic solid

Summary Thermodynamic considerations in the propagation of finite amplitude plane shear waves in an incompressible isotropic hyperelastic solid are examined, in particular the validity of the assumption of zero thermal conductivity which results in the adiabatic approximation. Two problems are considered for a solid withdc/d|F|>0, wherec is the transverse wave speed andF is the shear component of the deformation gradient tensor for simple shear. These problems, which are used to illustrate thermodynamic effects, are the sudden application of a spatially uniform shearing stress to the free surface of an unstressed half space at rest and the sudden removal of the spatially uniform shearing stress on the surface of a half space at rest and in a state of simple shear.With 2 Figures     

An eigen theory of electro-magnetic acoustic waves in magnetoelectroelastic media

The Maxwell’s equations, coupled to the mechanical equations of equilibrium, and the mechanical equations of motion, coupled to the equations of static electric and magnetic field, are studied here based on the standard spaces of the physical presentation. The complete set of uncoupled electromagnetic wave equations and elastic wave equations in magnetoelectroelastic solids is deduced. The results show that the number of electromagnetic waves and elastic waves in magnetoelectroelastic solids is determined by both the subspaces of electromagnetically anisotropic media and ones of mechanically anisotropic media. Based on these laws, we discuss the propagation behavior of electromagnetic waves and elastic waves in the transverse isotropic magnetoelectroelastic material poled in x₃.     

Existence of longitudinal and transverse waves in anisotropic thermoelastic media

Four waves propagate in an anisotropic thermoelastic medium. The fastest among them is a quasi-longitudinal wave. The slowest of them is a thermal wave. The remaining two are called quasi-transverse waves. The prefix ‘quasi’ refers to their polarizations being nearly, but not exactly, parallel or perpendicular to the direction of propagation. The polarizations of these four waves are not mutually orthogonal. Hence, unlike anisotropic elastic media, the existence of a longitudinal wave may not imply the existence of a transverse wave, by default. The existence of a purely longitudinal wave in an anisotropic thermoelastic medium is ensured by the stationary characters of three expressions. These expressions involve components of phase direction with elastic (stiffness and coupling) and thermal coefficients of the thermoelastic medium. The existence of a purely transverse wave is ensured by the two equations restricting the choice of thermoelastic (stiffness and coupling) coefficients. The existence of longitudinal and transverse waves along the coordinate axes and in the coordinate planes are discussed for general anisotropy. The discussion is extended to orthotropic materials, and the existence of pure phases is explored along few specific phase directions.     

Comparative study of different nonconserving time integrators for wave propagation in hyperelastic waveguides

This paper addresses the formulation and numerical efficiency of various numerical models of different nonconserving time integrators for studying wave propagation in nonlinear hyperelastic waveguides. The study includes different nonlinear finite element formulations based on standard Galerkin finite element model, time domain spectral finite element model, Taylor–Galerkin finite element model, generalized Galerkin finite element model and frequency domain spectral finite element model. A comparative study on the computational efficiency of these different models is made using a hyperelastic rod model, and the optimal computational scheme is identified. The identified scheme is then used to study the propagation of transverse and longitudinal waves in a Timoshenko beam with Murnaghan material nonlinearity.     

Reflection and transmission of waves from a plane interface between two microstretch solid half-spaces

In this paper, we have investigated the wave propagation and their reflection and transmission from a plane interface between two different microstretch elastic solid half-spaces in perfect contact. It is shown that there exist five waves in a linear homogeneous isotropic microstretch elastic solid, one of them travel independently, while other waves are two sets of two coupled waves. It is also shown that these waves travel with different velocities, three of which disappear below a critical frequency. Amplitude ratios and energy ratios of various reflected and transmitted waves are presented when a set of coupled longitudinal waves and a set of coupled transverse waves is made incident. It is found that the amplitude ratios of reflected and transmitted waves are functions of angle of incidence, frequency and are affected by the elastic properties of the media. Some special cases have been reduced from the present formulation.     

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.     

Weak shock waves and shear bands in thermoelastic solids

The linear weak shock wave (acoustic wave) propagation and the existence of shear bands are examined in finitely deformed thermoelastic solids within the framework of the theory of singular surfaces. The jumps of certain field variables across the shock wave front are obtained by using Taylor series expansions of them. The propagation condition of shock waves in a thermoelastic solid is obtained by using the strain–energy function corresponding to Duhamel–Neumann expression. The propagation speeds of weak shock waves are determined for a particular state of deformation, that is, general dilation. The formation of shear bands and the magnitudes of critical stretches are obtained for the deformation states of uniaxial, biaxial extension and for uniform dilation.     

Finite amplitude waves from suddenly punched hole in hyperelastic sheet

Summary Wave propagation resulting from a suddenly punched circular hole in a thin hyperelastic sheet, subjected to finite deformation equibiaxial tension, is considered. It is assumed that longitudinal wave propagation takes place in an annular region and reflections from the outer fixed boundary and the edge of the hole are considered. A numerical scheme described in a previous paper is used to obtain results for sheets whose strain energy function is a particular case of the Mooney-Rivlin.     

Effects of an external magnetic field on thermo-acoustical waves in a linear isotropic thermo-elastic dielectric material

Summary Thermo-mechanico-electromagnetic coupled waves propagating in a linear isotropic thermo-elastic dielectric material are theoretically investigated, in case an external magnetic field is applied to the material. Here the constitutive equations derived from the Clausius-Duhem inequality and Vernotte's heat conduction law are adopted. There are three types of coupled waves: the predominantly electromagnetic wave, the predominantly mechanical transverse wave and the predominantly thermo-mechanical longitudinal wave. The first and second waves have no thermal coupling. The third wave has thermal coupling and its propagation velocity and attenuation constant are perturbed by the external magnetic field.     

General solution technique for transient thermoelasticity of transversely isotropic solids in cylindrical coordinates

Summary Goodier has proposed the thermoelastic potential function in order to analyze thermoelastic problems for isotropic solids. The thermoelastic problem can be reduced to the elastic problem by his technique. Elastic problems are in general analyzed by the generalized Boussinesq solutions and the Michell function. This paper discusses a new solution technique for thermoelastic problems of transversely isotropic solids in cylindrical coordinates. The present solution technique consists of five fundamental solutions which are developed from the Goodier's thermoelastic potential function, the generalized Boussinesq solutions and the Michell function. Considering the relations among the material constants of transverse isotropy, the present solution technique can be classified into two cases. One of them can be reduced to the three solution techniques above which are specifically for isotropic solids only. As an application of the present solution technique, a transient thermoelastic problem in a transversely isotropic cylinder with an external crack is analyzed.     

The simple tension problem at large volumetric strains computed from finite hyperelastic material laws

Summary In this paper, different formulations of finite isotropic hyperelastic material laws for compressible solids are considered. Material laws with an additive split of the hyperelastic strain energy function into isochoric parts and volumetric parts are often used in the numerical treatment of nearly incompressible solids. It will be shown that this formulation leads to unphysical results in the simple tension problem when we do not restrict ourselves to nearly incompressible materials.     

Reflection of Plane Waves in a Semiconducting Medium under Photothermal Theory

In this paper photothermal theory was used to study the reflection of waves at the surface of a semi-infinite semiconconducting medium. Using the harmonic wave method, the reflection coefficient ratios were obtained analytically under coupled thermoelastic theory and plasma theory for an incident CI wave, which is one coupled thermoelastic plasma wave and an incident rotational wave. The variations of the amplitude of reflection coefficient ratios with the angle of incidence are shown graphically for silicon. Effects of the thermal frequency, the thermoelastic coupling parameter, and the thermoelectric coupling parameter were given by numerical results. Also, the energy ratios for reflected waves were computed to check the numerical results.     

Finite amplitude spherically symmetric wave propagation in a compressible hyperelastic solid

Summary Some aspects of the wave propagation, resulting from the spherically symmetric expansion of a thick walled hyperelastic shell and the limiting case of expansion of a cavity in an unbounded medium, are investigated. It is assumed that the shell is isotropic and uniform in the natural reference state and its strain energy function is a particular compressible generalization of that for the neo-Hookean solid. The response of a compressible shell, due to a spatially uniform time dependent application of internal pressure, is compared with that for the neo-Hookean shell taken as a limiting case of the compressible shell. This is also done for an unbounded medium.A finite difference method which uses the relation along one of the families of characteristics is used to obtain numerical results. In order to implement this method the governing equations are expressed as a system of first order partial differential equations in conservation form.With 9 Figures     

Coupled thermoelasticity and second sound in finite length functionally graded thick hollow cylinders (without energy dissipation)

In this article, the coupled thermoelasticity behavior of functionally graded thick hollow cylinders is studied. The governing coupled thermoelasticity and the energy equations are solved for a finite length functionally graded cylinder subjected to thermal shock load. The coupled thermoelastic equations are considered based on Green–Naghdi theory. The mechanical properties of cylinder are graded across the thickness as a power law function of radius. The cylinder is assumed to be made of many isotropic sub-cylinders (layers) across the thickness. Functionally graded properties are created by suitable arrangement of layers and governing equations are expanded in longitudinal direction by means of trigonometric function expansion. The Galerkin Finite Element and Newmark Methods are used to analyze the cylinder. The dynamic behavior of temperature distribution, mechanical displacement and thermal stresses is obtained and discussed. The second sound and elastic wave propagation are determined for various kinds of variation in the mechanical properties. The comparison of present results with published data shows the excellent agreement.     

非对称周期结构中耦合波的传播特性

为了揭示周期结构中纵向波和弯曲波的耦合作用,设计了对称和非对称周期结构。考虑子结构中的纵向和弯曲耦合运动,利用导纳法和传递矩阵法,得到了周期单元的传递方程。由于结构中存在多种波的耦合作用,在求解周期单元的传播系数时将出现变态矩阵,采用波型分组法,求得了周期结构中多种波型的传播系数。推导了半无限长和有限长周期结构在纵向力、横向力和弯矩作用下的动态响应。数值计算结果表明,对称周期结构中纵向波和弯曲波的带隙结构相互独立;非对称周期结构中纵向波和弯曲波的耦合明显改变了两种波的带隙结构,只有在两种波阻带重叠的频段内结构上的振动响应才存在衰减。     

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.     

Acceleration waves through an isotropic mixture of two non-linear elastic solids

Summary In this paper we study the propagation of acceleration waves through an isotropic isothermal mixture of two non-linear elastic solids. After giving the constitutive equations of the mixture, we calculate the possible normal speeds of propagation. Then we state that, in general, it is possible to distinguish between longitudinal and transverse acceleration waves. Finally, we establish the evolution law of the discontinuities along the normal trajectories associated with the wave front.Work performed under the auspices of C.N.R. (GNFM) and supported by M.P.I. of Italy.     

Finite amplitude transverse waves in materials with memory

We consider the propagation of finite amplitude plane transverse waves in a class of homogeneous isotropic incompressible viscoelastic solids with memory. It is assumed that the Cauchy stress may be written as the sum of an elastic part and a dissipative viscoelastic part. The elastic part is of the form of the stress corresponding to a Mooney–Rivlin material, whereas the dissipative part depends not only on current but also on previous deformations. The body is first subjected to a homogeneous static deformation. It is seen that two finite amplitude transverse plane waves may propagate in every direction in the deformed body. It is also seen that finite amplitude circularly polarized waves may propagate along either n⁺ or n⁻, where n⁺, n⁻ are the normals to the planes of the central circular section of the ellipsoid x · B⁻¹x = 1. Here B is the left Cauchy–Green strain tensor corresponding to the finite static homogeneous deformation.     

Propagation of waves in magneto-thermo-viscoelastic solids subjected to a uniform magnetic field

The propagation of magneto-thermo-mechanical (MTM) plane waves in electrically and thermally conductive magneto-thermo-viscoelastic (MTVE) unbounded solids is investigated with account for the mutual effects of the magnetic, thermal and strain fields. Concerning the mutual and thermo-electric effects in isotropic solids the governing equations are first linearized. In the linearization, the material is assumed to be subjected to a uniform and primary magnetic field in any direction while the material undergoes infinitesimal deformations. It is shown that the governing equations at the intermediate state are fulfilled by the presumed MTM-fields. Furthermore, the dispersion relation which allows us to consider the entire frequency range, the effect of the magnetic field and some nondimensional material parameters is obtained. Therefore, several modes of MTM-waves arise depending upon the direction of the magnetic field such as the uncoupled magnetic and mechanical S-mwaves, the coupled S-wave, the modified mechanical $\text{P}$-and thermal waves, and the modified and coupled MTM-waves. It is seen that all modes of the wave are dispersive and dissipative due to the conductivity and the viscosity of the material. Then the phase velocities and the attenuation constants for the coupling modes are obtained, and some limiting values are discussed. From the expressions follow, in particular, the results for the elastic case, the propagation of mechanical waves in nonconductive materials.     

Thermoelastic plane waves in a rotating isotropic medium

Summary We discuss the theory of thermoelastic wave propagation in a rotating isotropic material. In any given direction there are four waves. In general, all of these waves are attenuated, and none of them is purely dilatational or transverse. Some earlier published results are found to be false.