Thermoelastic diffusion with memory-dependent derivative

The constitutive equations are derived for the thermoelastic diffusion in anisotropic and isotropic solids, in the context of a new generalized thermoelasticity theory with two time delays and kernel functions. The coupled thermoelastic diffusion and the Lord–Shulman theories result from the given theory as particular cases. For anisotropic solid, the reciprocity theorem is proved; the convolutional variational principle is given and the uniqueness theorem based on the variational principle is proved.     

Fractional Order Theory of Thermoelastic Diffusion

In this work, a new theory of thermodiffusion in elastic solids is derived using the methodology of fractional calculus. The theories of coupled thermoelastic diffusion and of generalized thermoelastic diffusion problem with one relaxation time follow as limit cases. A variational theorem is then obtained for the governing equations. Finally, a uniqueness and reciprocity theorems for these equations are derived.     

KIRCHHOFF'S FORMULA FOR THERMOELASTIC SOLID

A generalization of Kirchhoff's theorem for the classical wave equation to the case of a central system of field equations of coupled thermoelasticity is established in this paper. The theorem asserts that a pair (Φ, θ), where Φ and θ denote the thermoelastic displacement potential and temperature, respectively, can be expressed by surface integrals over the boundary of a thermoelastic solid whose kernels have the form of an infinite series satisfying the wave-like and heat-like equations occurring in the decomposition theorem for the central system of equations ([3]).     

A domain of influence theorem for a natural stress–heat-flux disturbance in thermoelasticity of type-II

The aim of the present work is to establish the domain of influence theorem for a stress–heat-flux disturbance under Green and Naghdi thermoelasticity theory of type-II. We consider a mixed problem of natural type represented as stress–heat-flux disturbance in the context of Green–Naghdi thermoelasticity theory of general type. We establish a general energy identity for the problem. Then we establish the domain of influence theorem for natural stress–heat-flux disturbance in the context of Green–Naghdi model of thermoelasticity oftype-II. We prove that for a finite time t > 0, the pair of stress and the heat-flux field generates no disturbance outside a bounded domain. The domain of influence is shown to be dependent on the thermoelastic parameters.     

PLANE HARMONIC WAVES IN A MICROPERIODIC LAYERED THERMOELASTIC SOLID REVISITED

In previous work a one-dimensional averaged theory of thermoelastic waves in a microperiodic layered infinite solid was proposed in which an eighth-order-in-time partial differential equation involving a high intrinsic mechanical frequency Ω_M and a high intrinsic thermal frequency Ω_T is a central equation. Also, the existence of two harmonic thermoelastic waves of a given frequency ω propagating in a positive direction normal to the layering was previously established when Ω_M → ∞ and Ω_T < ∞, or Ω_M < ∞ and Ω_T → ∞. The existence of two harmonic thermoelastic waves of a given frequency ω propagating in the positive direction normal to the layering is proved when Ω_M < ∞ and Ω_T < ∞. Also, a closed form of the associated velocities and attenuation coefficients is obtained. Numerical results illustrating propagation of the two waves in a particular microperiodic layered thermoelastic solid are included.     

Generalized Theory of Thermoelastic Diffusion for Anisotropic Media

The equations of generalized thermoelastic diffusion, based on the theory of Lord and Shulman with one relaxation time, are given in anisotropic media. A variational principle for the governing equations is obtained. Then we show that the variational principle can be used to obtain a uniqueness theorem under suitable conditions. A reciprocity theorem for these equations is given. The obtained results are valid for some special cases that can be deduced from our generalized model.     

Qualitative Aspects in the Coupled Theory of Thermoelastic Diffusion

In this paper we derive some qualitative results of the coupled theory of thermoelastic diffusion for anisotropic media. We establish a reciprocity relation, which involves two thermoelastic diffusion processes at different instants. We show that this relation can be used to obtain reciprocity, uniqueness and continuous dependence theorems. The reciprocity theorem avoids both the use of the Laplace transform and the incorporation of initial conditions into the equations of motion. The uniqueness theorem is derived without the positive definiteness assumption on the elastic, conductivity and diffusion tensors. We prove also that the reciprocal relation leads to a continuous dependence theorem studied on external body loads. Finally, we prove the existence of a generalized solution by means of the semigroup of linear operators theory.     

SAINT-VENANT'S PRINCIPLE FOR A MICROPERIODIC COMPOSITE THERMOELASTIC SEMISPACE: THE DYNAMICAL REFINED AVERAGE THEORY

A Saint-Venant's principle associated with a one-dimensional dynamic coupled thermoelastic effective modulus theory (EMT) for a microperiodic layered semispace was presented in J. Thermal Stresses, vol. 23, pp. 1-14, 2000. It was shown there that a thermoelastic energy associated with a solution to an initial boundary value problem of the theory decays exponentially as a distance x from the thermomechanical load region goes to infinity and that its decay length depends on the time t, an effective velocity d c 1 * ¢ , an effective time unit d T* ¢ , and an effective thermoelastic coupling parameter d k * ¢ . In the present article the Saint-Venant's principle is extended to include a refined averaged theory (RAT) for a microperiodic layered thermoelastic semispace in which a microstructural length is taken into account (see [IFTR Report #25, pp. 1-158, 1995]). It is shown that for such an extended theory, a similar exponential decay estimate for a thermoelastic energy holds true. In the refined estimate the thermoelastic energy depends on a number of microstructural parameters while its decay length is independent of these parameters; and the decay length for small (large) times is comparable to that of a pure thermal (elastic) energy for a rigid (elastic) semispace for every time t > 0 .     

INERTIAL EFFECTS OF A TRANSIENT THERMOELASTIC PROBLEM IN A THICK PLATE WITH AN AXISYMMETRIC TEMPERATURE

This paper concerns an axisymmetric dynamic treatment of a three-dimensional thermoelastic problem in a thick plate that is exposed to the temperature of surrounding medium depending on the position and time. In the present investigation, the Laplace transform and the convolution theorem are used, and an exact solution that is valid for the whole time interval without approximation is obtained. The thermal stresses are compared with the corresponding quasi-static stresses and the corresponding one-dimensional stresses.     

Steady-state Green's functions for thermal stresses within rectangular region under point heat source

This article presents new steady-state Green's functions for displacements and thermal stresses for plane problem within a rectangular region. These results were derived on the basis of structural formulas for thermoelastic Green's functions which are expressed in terms of Green's functions for Poisson's equation. Structural formulas are formulated in a special theorem, which is proved using the author's developed harmonic integral representation method. Green's functions for thermal stresses within rectangle are obtained in the form of a sum of elementary functions and ordinary series. In the particular cases for a half-strip and strip, ordinary series vanish and Green's functions are presented by elementary functions. These concrete results for Green's functions and respective integration formulas for thermoelastic rectangle, half-strip and strip are presented in another theorem, which is proved on the basis of derived structural formulas. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle under a boundary constant temperature gradient also are derived. Analytical solutions were presented in the form of graphics. The fast convergence of the infinite series is demonstrated on a particular thermoelastic boundary value problem (BVP). The proposed technique of constructing thermal stresses Green's functions for a rectangle could be extended to many 3D BVPs for unbounded, semibounded, and bounded parallelepipeds.     

Laser-Generated Thermoelastic Waves in an Anisotropic Infinite Plate: Exact Analysis

In recent years, the study of thermoelastic waves generated by lasers has been undertaken by several researchers because the technique provides an efficient non-contact technique for generation and detection of ultrasonic waves. Laser-generated ultrasonic waves have diverse applications ranging from material characterization to defect characterization. Transient ultrasonic guided waves generated by a pulsed laser in anisotropic infinite plate are investigated in this article. An exact analytical method is adopted for this purpose. The governing equations and boundary conditions are first transformed from spatial-time domain into wavenumber-frequency domain using Fourier Transform. After solving these equations and satisfying the boundary conditions in the wavenumber-frequency domain, the Cauchy's residue theorem is used to get the response in the spatial domain and then the numerical integration is used to eventually obtain the response in time domain. Results for dispersion and transient guided waves in infinite silicon nitride (Si₃N₄) plates are presented. Numerical results show that pulsed laser excites mainly the lowest Lamb modes, namely, the lowest symmetric (S ₀) and antisymmetric (A₀) modes. They also show that the transient response is dominated by the antisymmetric mode A₀ which shows dispersion characteristic. This study provides a quantitative model for laser generated ultrasonic waves in an anisotropic plate and can be used for non-destructive evaluation.     

ANALYTICAL TREATMENT OF AXISYMMETRICAL THERMOELASTIC FIELD WITH KASSIR'S NONHOMOGENEOUS MATERIAL PROPERTIES AND ITS ADAPTATION TO BOUNDARY VALUE PROBLEM OF SLAB UNDER STEADY TEMPERATURE FIELD

The method of an analytical development of thermoelastic problems for a medium with Kassir's nonhomogeneous material properties is developed. For isothermal problems of such a nonhomogeneous body, an analytical method of development has already been proposed by M. K. Kassir under the assumption that the shear modulus of elasticity G is changed arbitrarily with the variable z of the axial coordinate according to the relation G(z) = G_oz^m. However, the analytical procedure for the thermoelastic field has not been established. In this article, introducing the thermoelastic displacement potential function, the analytical method of development for the axisymmetrical thermoelastic field is established. As an illustrative example, we consider the thermoelastic problem of a slab. Assuming that the shear modulus of elasticity G, the thermal conductivity λ, and the coefficient of linear thermal expansion α vary with the variable [zcirc] of the dimension-less axial coordinate according to the relation G([zcirc]) = G_o[zcirc]^m, λ([zcirc]) = λ₀[zcirc]^l, α([zcirc]) = α₀[zcirc]ⁿ, the axisymmetric temperature solution in a steady state for the slab is obtained and the associated thermal stress components are evaluated theoretically. Numerical calculations are carried out for several cases, taking into account the variety of the nonhomogeneous material properties. Numerical results are shown graphically.     

Solution in Elementary Functions to a BVP of Thermoelasticity: Green's Functions and Green's-Type Integral Formula for Thermal Stresses within a Half-Strip

This article presents new elementary Green's functions for displacements and stresses created by a unit heat source applied in an arbitrary interior point of a half-strip. We also obtain the corresponding new integration formulas of Green's and Poisson's types which directly determine the thermal stresses in the form of integrals of the products of internal distributed heat source, temperature, or heat flux prescribed on boundary and derived thermoelastic influence functions (kernels). All these results are presented in terms of elementary functions in the form of a theorem. Based on this theorem and on derived early by author general Green's type integral formula, we obtain a new solution to one particular boundary value problem of thermoelasticity for half-strip. The graphical presentation of thermal stresses created by a unit point heat source and of thermal stresses for one particular boundary value problem of thermoelasticity for half-strip is also included. The proposed method of constructing thermoelastic Green's functions and integration formulas are applicable not only for a half-strip but also for many other two- and three-dimensional canonical domains of Cartesian system of coordinates.     

Theory of Two-Temperature Thermoelasticity without Energy Dissipation

This paper deals with thermoelastic behavior without energy dissipation; it deals with linear theory of thermoelasticity. In particular, in this work, a new theory of generalized thermoelasticity has been constructed by taking into account two-temperature generalized thermoelasticity theory for a homogeneous and isotropic body without energy dissipation. The new theorem has been derived in the context of Green and Naghdi model of type II of linear thermoelasticity. Also, a general uniqueness theorem is proved for two-temperature generalized thermoelasticity without energy dissipation.     

QUASI-STATIC COUPLED PROBLEMS OF THERMOELASTICITY FOR CYLINDRICAL REGIONS

Abstract

The one-dimensional axisymmetric quasi-static coupled thermoelastic problem is investigated. The general solutions of its governing equations are obtained in the transform domain. The solutions in the real domain for the cases of an infinitely long solid cylinder and for an infinite medium with a cylindrical hole are also presented. The solution technique uses Laplace transform, and the inversion to the real domain is obtained by means of Cauchy's theorem of residues and the convolution theorem. Comparison with published results, when similar assumptions are made and the same boundary conditions are imposed, shows complete agreement.     

THERMOELASTIC PROPERTIES OF COMPOSITES CONTAINING ELLIPSOIDAL INHOMOGENEITIES

This article studies the thermal stresses and the effective thermoelastic properties of composites containing ellipsoidal inhomogeneities. The cluster scheme developed recently by A. Molinari and M. El Mouden in The Problem of Elastic Inclusions at Finite Concentration, Int. J. Solids Struct, vol. 33, pp. 3131 - 3150, 1996, for the case of elastic inclusions embedded in an isotropic elastic matrix, is generalized to the case of ellipsoidal thermoelastic inclusions embedded in an anisotropic thermoelastic matrix. The shape, spatial distribution, and orientation of the inhomogeneities are taken into account in our scheme. The theoretical results for a composite of S_iO₂ particles in a Kerimid matrix are in good agreement with experimental measurements.     

GENERALIZED THERMOELASTICITY: CLOSED-FORM SOLUTIONS

A study of the one-dimensional thermoelastic waves produced by an instantaneous plane source of heat in homogeneous isotropic infinite and semi-infinite bodies of the Green-Lindsay (G-L) type is presented. Closed-form Green's functions corresponding to the plane heat source are obtained using the decomposition theorem for a potential-temperature wave of the G-L theory. Qualitative analysis of the results is included.     

A Theory of Heat and Mass Transfer in Viscoelastic Solids with Microstructures

In the present work, the linear theory of micropolar thermo-viscoelasticity with mass diffusion is established. The constitutive relations are obtained for a mixture of diffusive masses in the bulk medium, and then the uniqueness theorem is proved using the Laplace transform and the positive definiteness assumptions on the initial case of the thermoelastic modulus. The reciprocity relation is established avoiding Laplace transform. Some consequences on the reciprocity relation are discussed. The variational theorem is proved. The integral representation is obtained for the model equations; hence, the Maysel's, Somigliana's and Green's formulae are derived. Finally, the mixed boundary value problem is considered and reduced to a system of four Fredholm-Volterra integral equations.     

Three-Dimensional Steady-State General Solution for Isotropic Thermoelastic Materials with Applications I: General Solutions

Three general solutions of the three-dimensional steady-state governing equations of isotropic thermoelastic materials are derived in this article. For this object, two displacement functions are first introduced to simplify the govering equation. Then, using the differential operator theory, three general solutions can be expressed in terms of two functions, one satisfies a harmonic equation and the other satisfies a six-order partial differential equation. By virtue of Almansi's theorem, three general solutions can be further transferred to two general solutions, which are expressed in terms of three harmonic functions. At last, one more relatively completed general solution expressed in four harmonic functions is obtained by superposing the two general solutions. The proposed general solutions are simple in form and hence they may bring more convenience to certain boundary problems. As two examples, the fundamental solutions for both a point heat source in the interior of infinite thermoelastic body and a point heat source on the surface of semi-infinite thermoelastic body are presented by virtue of the obtained general solutions.