Fractional-order generalized thermoelastic diffusion theory

The present work aims to establish a fractional-order generalized themoelastic diffusion theory for anisotropic and linearly thermoelastic diffusive media. To numerically handle the multi-physics problems expressed by a sequence of incomplete differential equations, particularly by a fractional equation, a generalized variational principle is obtained for the unified theory using a semi-inverse method. In numerical implementation, the dynamic response of a semi-infinite medium with one end subjected to a thermal shock and a chemical potential shock is investigated using the Laplace transform. Numerical results, i.e., non-dimensional temperature, chemical potential, and displacement, are presented graphically. The influence of the fractional order parameter on them is evaluated and discussed.     

On thermoelastic diffusion thin plate theory

The effect of diffusion on thermoelastic thin plates is investigated. The governing equations for thin thermoelastic diffusion plates under three different laws of heat and diffusion transmission are derived. By the C₀-semigroup theory, the well-posedness of the proposed equations is shown.     

A half-space problem in the theory of generalized thermoelastic diffusion

In this work we consider the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques.The temperature, displacement, stress and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.     

Rayleigh surface waves in the theory of thermoelastic materials with voids

In this paper we analyze the behavior of plane harmonic waves and Rayleigh waves in a linear thermoelastic material with voids. We take into account the damped effects of the thermal field upon the propagation waves. Consequently, the propagation condition is established in the form of an algebraic equation of 9th degree whose coefficients are complex numbers while the eigensolutions of the thermoelastodynamic with voids system are explicitly obtained in terms of the characteristic solutions. We show that the transverse waves are undamped in time and they are not influenced by the thermal and porous effects while the longitudinal waves are all damped in time and they are coupled with the thermal and porous effects. The related solution of the Rayleigh surface wave problem is expressed as a linear combination of the eigensolutions in concern. The secular equation is established in an implicit form and afterwards an explicit form is written for an isotropic and homogeneous thermoelastic with voids half-space. Furthermore, we use the numerical methods and computations to solve the secular equation for a specific material.     

Several results in the dynamic theory of thermoelastic Cosserat shells with voids

In this paper we investigate the boundary-initial-value problem of the dynamic linear theory for thermoelastic Cosserat shells with voids. We prove a reciprocity relation and derive a uniqueness theorem. Then, we study the continuous dependence of the solution on external body loads and heat supply and on initial data. A variational characterization of the solution is also established.     

Mechanical loads on a generalized thermoelastic medium with diffusion

The disturbance caused by the application of continuous mechanical source on the free surface of a homogeneous, isotropic elastic half space in the context of the theory of generalized thermoelastic diffusion with one relaxation time parameter is investigated in the Laplace-Fourier transform domain for a two dimensional problem using eigenvalue approach. The integral transforms are inverted by using a numerical technique. The expressions for displacement components, stresses, temperature field, concentration and chemical potential so obtained in the physical domain are computed numerically and illustrated graphically at different times, for copper like material. As a special case the effect of diffusion on various expressions has also been obtained analytically and depicted graphically.     

On the asymptotic spatial behaviour in the theory of mixtures of thermoelastic solids

This paper is concerned with the study of asymptotic spatial behaviour of solutions in a mixture consisting of two thermoelastic solids. A second-order differential inequality for an adequate volumetric measure and the maximum principle for solutions of the one-dimensional heat equation are used to establish a spatial decay estimate of solutions in an unbounded body occupied by the mixture. For a fixed time, the result in question proves that the mechanical and thermal effects are controlled by an exponential decay estimate in terms of the square of the distance from the support of the external given data. The decay constant depends only on the thermal constitutive coefficients of the mixture.     

A problem for an infinite elastic body with a spherical cavity in the theory of generalized thermoelastic diffusion

In this work, we study a one-dimensional problem in a generalized thermoelastic diffusion in infinite medium with a spherical cavity subjected to a time dependent thermal shock of its internal boundary which is assumed to be traction free. The chemical potential is also assumed to be a known function of time on the bounding cavity. Laplace transform techniques are used. The solution of the problem in the transformed domain is obtained by using a direct approach without the customary use of potential functions. By means of numerical Laplace inversion, the problem is solved in the physical domain. Numerical results predict finite speeds of propagation for thermoelastic and diffusive waves. To investigate the diffusions effects, a comparison is made with the results obtained in the thermoelastic problem.     

Two-dimensional interactions due to moving load in generalized thermoelastic solid with diffusion

The present paper is concerned with the investigation of disturbances in＇a homogeneous, isotropic elastic medium with generalized thermoelastic diffusion, when a moving source is acting along one of the co-ordinate axis on the boundary of the medium. Eigen value approach is applied to study the disturbance in Laplace-Fourier transform domain for a two dimensional problem. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using a numerical technique for the inversion of Laplace transform based on Fourier expansion techniques. These expressions are calculated numerically for a copper like material and depicted graphically. As special cases, the results in generalized thermoelastic and elastic media are obtained. Effect of presence of diffusion is analyzed theoretically and numerically.     

Slip/diffusion zone effects at rapidly-loaded cracks in thermoelastic solids

The effect of a zone of slip and diffusion mechanisms that form prior to the onset of plastic yield at the edge of a rapidly-loaded Mode I crack in a fully-coupled thermoelastic material is studied. The resulting initial/mixed boundary value problem is solved exactly in the transform space, despite the existence of poles and branch points that vary with the time transform variable.Crack edge stress singularity relaxation is assumed, and this condition leads to exact expressions for the time transforms of a zone characterization function and the temperature change induced ahead of the crack. Study of their inverses for long, i.e. O(1)s, times after loading shows that a slip/diffusion process that is consistent with rapid crack surface separation can indeed relax elastic stresses, but that high temperature increases may well have to await the development of full plasticity and fracture at the crack edge.     

Energy decay in thermoelastic diffusion theory with second sound and dissipative boundary

We study the energy decay of the solutions of a linear homogeneous anisotropic thermoelastic diffusion system with second sound and dissipative boundary of the form $$\mathbf{T}(x,t)n(x) = -\gamma_0v(x,t) -\int_0^\infty \lambda(s)v^t(x,s) ds. $$ This boundary condition well describes a material for which the domain outside the body consists in a material of viscoelastic type. Models of boundary conditions including a memory term which produces damping were proposed in Fabrizio and Morro (Arch. Ration. Mech. Anal. 136:359–381, 1996) in the context of Maxwell equations and in Propst and Prüss (J. Integral Equ. Appl. 8:99–123, 1996) for sound evolution in a compressible fluid. The thermal and diffusion disturbances are modeled by Cattaneo-Maxwell law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier’s law. The system of equations in this case is a coupling of three hyperbolic equations. By introducing a boundary free energy, we prove that, if the kernel λ exponentially decays in time then also the energy exponentially decays. Finally, we generalize the obtained results to the Gurtin-Pipkin model.     

Structural function theory of generalized variational principles for linear elastic materials with voids

In this paper,we have obtained generalized variational principles for linear elasticmaterials with voids from structural function theory.Correspondent relations betweenstructural functions and generalized variational principles are given.     

Three-dimensional vibration analysis of a thermoelastic cylindrical panel with voids

In this paper, based on three-dimensional linear generalized thermoelasticity, an exact analysis of free vibration of a simply supported homogeneous isotropic, thermally conducting, cylindrical panel with voids initially at uniform temperature and undeformed state has been presented. Three displacement potential functions are introduced for solving the equations of motion, heat conduction and volume fraction field. The purely transverse wave gets decoupled from rest of motion and is not affected by thermal and volume fraction (voids) fields. After expanding the displacement potentials, volume fraction and temperature functions with orthogonal series, the equations of the considered vibration problem are reduced to five-second order coupled ordinary differential equations whose formal solution can be expressed by using Bessel functions with complex arguments. The corresponding results for thermoelastic panel without voids, elastic panel with and without voids have been deduced as special cases from the present analysis. In order to illustrate the analytical results, the numerical solutions of various relations and equations have been obtained to compute the lowest frequency as function of different cylindrical panel parameters. The computer simulated results have been presented graphically.     

Propagation of plane waves at the imperfect boundary of elastic and electro-microstretch generalized thermoelastic solids

The problem of reflection and transmission of plane waves incident on the contact surface of an elastic solid and an electro-microstretch generalized thermoelastic solid is discussed. It is found that there exist five reflected waves, i.e., longitudinal displacement （LD） wave, thermal （T） wave, longitudinal microstretch （LM） wave and two coupled transverse displacement and microrotational （CD（I） and CD（II）） waves in the electro-microstretch generalized thermoelastic solid, and two transmitted waves, i.e., longitudinal （P） and transverse （SV） waves in the elastic solid. The amplitude ratios of different reflected and transmitted waves are obtained for an imperfect boundary and deduced for normal force stiffness, transverse force stiffness, and perfect bonding. The variations of amplitude ratios with incidence angles have been depicted graphically for the LD wave and the CD（I） wave. It is noticed that the amplitude ratios of reflected and transmitted waves are affected by the stiffness, electric field, stretch, and thermal properties of the media. Some particular interest cases have been deduced from the present investigations.     

On shock waves in a special class of thermoelastic solids

This paper describes a thermoelastic model for shock waves in uniaxial strain based on a subclass of the so-called materials of Mie–Grüneisen type. We compare the Hugoniot curve with the isotherms and isentropes for this model, and we construct the shock-wave solution to a simple impact problem.     

Propagation of Rayleigh waves on free surface of transversely isotropic generalized thermoelastic diffusion

The present paper is devoted to the study of Rayleigh wave propagation in a homogeneous, transversely isotropic, thermoelastic diffusive half-space, subject to stress free, thermally insulated/isothermal, and chemical potential boundary conditions in the context of the generalized thermoelastic diffusion theory. The Green-Lindsay（GL） theory is used in the study. In this theory, thermodiffusion and thermodiffusion mechanical relaxations are governed by four different time constants. Secular equations for surface wave propagation in the considered media are derived. Anisotropy and diffusion effects on the phase velocity, attenuation coefficient are graphically presented in order to present the analytical results and make comparison. Some special cases of frequency equations are derived from the present investigation.     

Fracture analysis of electromagnetic thermoelastic solids

This paper analyzes the problem of collinear permeable cracks in a magnetoelectroelastic solid subjected to uniform heat flow at infinity. The analysis is conducted according to the extended Stroh formalism. Concise expressions are given for the field intensity factors and the electric–magnetic field inside cracks. It is shown that all the field singularities are independent of the applied electric–magnetic loads, and the electric–magnetic field inside cracks is linearly variable with position along the crack line.     

Incremental response of thermoelastic solids with initial thermal and mechanical fields

The equations describing the incremental response of a thermoelastic solid under pre-existing mechanical and thermal fields are derived. The associated differential system is shown to be self-adjoint. This property in turn is used to establish the equivalence of linear static and dynamic stability criteria.Received: 10 November 2002, Accepted: 6 January 2003, Published online: 23 April 2003E. Soós: deceased     

On some basic principles in dynamic theory of elastic materials with voids

According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author^[1], some basic principles in dynamic theory of elastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of elastic materials with voids, but also to derive systematically the complementary functionals for the eight-field, six-field, four-field and two-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. The project supported by the Foundation of Zhongshan University Advanced Research Center