2D fundamental solution for orthotropic pyroelectric media

Fundamental solutions play an important role in electroelastic analyses of piezoelectric media. However, most works available are on the topic of identical temperature. On the basis of the compact general solution of orthotropic pyroelectric media, which is expressed in harmonic functions, and employing the trial-and-error method, the 2D fundamental solution for a steady point heat source in the interior of an infinite pyroelectric plane or on the surface of a semi-infinite pyroelectric plane are presented by four newly induced harmonic functions. All components of the coupled field are expressed in terms of elementary functions and are convenient to use.     

Green's functions in orthotropic thermoelastic diffusion media

The aim of the present paper is to study the Green's function in orthotropic thermoelastic diffusion media. With this objective, firstly the two-dimensional general solution in orthotropic thermoelastic diffusion media is derived. On the basis of general solution, the Green's function for a steady point heat source in the interior of semi-infinite orthotropic thermoelastic diffusion material is constructed by four newly introduced harmonic functions. The components of displacement, stress, temperature distribution and mass concentration are expressed in terms of elementary functions. From the present investigation, a special case of interest is also deduced, to depict the effect of diffusion on components of stress and temperature distribution.     

The solution of steady-state axtsymmetric thermoelastic problems by means of complex variables

In this paper is introduced a method of solution of steady-state axisymmetrie thermoelastic problems by means of functions of complex variable. There the equations of the problem are obtained by a rotation of the plane state about an axis of symmetry or by a linear translation of the axisymmetrie state. The equations are utilized for the solution of steady-state thermoelastic problem in a sphere. The stresses in a sphere are given in cylindrical coordinates and coincide with the known solution.     

The general solution of three-dimensional problems in magnetoelectroelastic media

The general solution of three-dimensional problems in transversely isotropic magnetoelectroelastic media is obtained through five newly introduced potential functions. The displacements, electric potential, magnetic potential, stresses, electric displacements and magnetic inductions can all be expressed concisely in terms of the five potential functions, all of which are harmonic. The derived general solution is then applied to find the fundamental solution for a generalized dislocation and also to derive Green's functions for a half-space magnetoelectroelastic solid.     

Three‐dimensional fundamental solution for transversely isotropic piezothermoelastic material

Fundamental solutions play an important role in electroelastic analyses and numerical methods of piezoelectric material. However, most works available on this topic are on the case of identical temperature. We use the compact mono‐harmonic general solutions of transversely isotropic piezothermoelastic material to construct the three‐dimensional fundamental solution of a steady point heat source in an infinite piezothermoelastic material by four newly introduced mono‐harmonic functions. All components of coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results for cadmium selenide are given graphically by contours. Copyright © 2008 John Wiley & Sons, Ltd.     

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 linear steady thermoelastic problem for a strip with a collinear array of Griffith cracks parallel to its edges

The problem of determining the thermal stresses when a uniform heat flow in a thermoelastic strip is disturbed by a collinear array of cracks is discussed. The solution of the Duhamel-Neumann equations is posed in terms of harmonic functions, which leads to dual series relations whose solutions are known. Numerical results for the stress intensity factors at the crack tips are displayed in graphical form.     

General steady-state solution for thermo-poroelastic material

This study established a general steady-state solution in the z-convex domain (the domain boundary must have at most two points of intersection with any straight line parallel to the z-axis) for thermo-poroelastic materials. Two displacement functions to simplify the equations governing elasticity, pressure, and temperature fields into one Laplace equation and four eighth-order partial–differential governing equations are introduced. The general solutions of displacement, pressure, and temperature are derived in terms of five harmonic functions using the generalized Almansi’s theorem and considering equivalent substitution. The relationship between the Boussinesq–Galerkin general solutions and the general solution proposed in this paper is discussed without considering the changes in pore pressure and temperature to prove the completeness of the latter.     

Transient thermoelastic analysis for a multilayered hollow cylinder with piecewise power law nonhomogeneity due to asymmetric surface heating

This paper is concerned with the theoretical treatment of transient thermoelastic problems involving a multilayered hollow cylinder with piecewise power law nonhomogeneity due to the asymmetrical heating of its surfaces. The thermal and thermoelastic constants of each layer are expressed as power functions of the radial coordinate, and their values continue on the interfaces. The exact solution for the two-dimensional temperature change in a transient state is obtained using the Laplace transformation and separation-of-variables method. The exact solution for the thermoelastic response of a multilayered hollow cylinder under the state of generalized plane strain, where the strain is not bound, is obtained herein. Some numerical results for the temperature change and the stress distributions are presented in figures. The influence of the functional grading on the thermal stresses is investigated. Furthermore, the influence on the axial stress of the restraint condition in the axial direction is investigated.     

Thermal stresses in a viscoelastic trimaterial with a combination of a point heat source and a point heat sink

A general solution for a thermoviscoelastic trimaterial combined with a point heat source and a point heat sink is presented in this work. Based on the method of analytic continuation associated with the alternation technique, the solutions to the heat-conduction and thermoelastic problems for three dissimilar, sandwiched media are derived. A rapidly convergent series solution for both the temperature and stress field, expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties, while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking the inverse Laplace transform. A typical example concerning the interfacial stresses generated from a combined arrangement of a heat source and sink are discussed in detail. The corresponding thin-film problem is also discussed.     

Thermal stresses in a viscoelastic trimaterial

A general solution for a three dissimilar sandwiched medium subjected to a point heat source and a three-phase composite cylinder subjected to a remote uniform heat flow is presented in this work. The solutions to the heat conduction and thermoelastic problems are derived based on the method of analytic continuation associated with the alternation technique. A rapidly convergent series solution for both the temperature and the stress field, which is expressed in terms of an explicit general term of the corresponding homogeneous potential, is obtained in an elegant form. The hereditary integral in conjunction with the Kelvin–Maxwell model is applied to simulate the thermoviscoelastic properties while a thermorheologically simple material is considered. Based on the correspondence principle, the Laplace transformed thermoviscoelastic solution is directly determined from the corresponding thermoelastic one. The real-time solution can then be solved numerically by taking inverse Laplace transform. Two typical examples concerning interfacial stresses for plane-layered media and circularly cylindrical-layered media are discussed in detail.     

General solution of the coupled-wave equations of acousto-optics

The basic problem of the diffraction of an optical plane wave by an acoustic plane wave in an anisotropic homogeneous medium is considered. The acousto-optical interaction is considered indifferently of the isotropic or of the birefringent type. Coupled-wave equations are obtained rigorously and cast into an eigenvalue value problem. A general solution is obtained for the diffraction efficiency of diffracted orders, for any interaction length and diffraction regime. The theory includes the Bragg regime, the Raman-Nath regime, and all intermediate situations in the same formulation. The method of solution is both exact and computationally efficient. It is similar in character to the rigorous coupled-wave analysis of Moharam and Gaylord but differs by the choice of basis functions adapted to propagating rather than static gratings. Examples are given for acousto-optical interaction in paratellurite, TeO2.     

Three-dimensional steady-state general solution for transversely isotropic hygrothermopiezoelectric media and its application in fracture

The potential theory method is utilized to derive the steady-state, general solution for three-dimensional (3D) transversely isotropic, hygrothermopiezoelectric media in the present paper. Two displacement functions are introduced to simplify the governing equations. Employing the differential operator theory and superposition principle, all physical quantities can be expressed in terms of two functions, one satisfies a quasi-harmonic equation and the other satisfies a tenth-order partial differential equation. The obtained general solutions are in a very simple form and convenient to use in boundary value problems. As one example, the 3D fundamental solutions are presented for a steady point moisture source combined with a steady point heat source in the interior of an infinite, transversely isotropic, hygrothermopiezoelectric body. As another example, a flat crack embedded in an infinite, hygrothermopiezoelectric medium is investigated subjected to symmetric mechanical, electric, moisture and temperature loads on the crack faces. Specifically, for a penny-shaped crack under uniform combined loads, complete and exact solutions are given in terms of elementary functions, which serve as a benchmark for different kinds of numerical codes and approximate solutions.     

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.     

The biharmonic problem and progress in the development of analytical methods for the solution of boundary-value problems

A comparative analysis of harmonic and biharmonic boundary-value problems for 2D problems on a rectangle is given. Some common features of two types of problems are emphasized and special attention is given to the basic distinction between them. This distinction was thoroughly studied for the first time by L. N. G. Filon with respect to some plane problems in the theory of elasticity. The analysis permits to introduce an important aspect of the general solution of boundary-value problems. The procedure for solving the biharmonic problem involves both the method of homogenous solutions and the method of superposition. For some cases involving self-equilibrated loadings on one pair of sides of the rectangle, the complete solution, including calculation of the quantitative characteristics of the displacements and stresses, is given. The efficiency of the numerical implementation of the general solutions is shown. The analysis of the quantitative data allows to elucidate some main points of the Saint-Venant principle.     

Symmetric Edge Cracks in an Orthotropic Strip Under Normal Loading

The problem of two edge cracks of finite length, situated symmetrically in an orthotropic infinite strip of finite thickness 2 h, under normal point loading has been discussed. The displacements and stresses in plane strain conditions are expressed in terms of two harmonic functions. The problem is addressed by seeking the solution of a pair of simultaneous integral equations with Cauchy type singularities solved by finite Hilbert Transform technique. For large h, analytical expression for the stress intensity factor at the crack tip is obtained.     

Deformation due to Mechanical and Thermal Sources in a Thermoelastic Body with Voids under Axi-symmetric Distributions

The Laplace and Hankel transforms have been employed to find the general solution of a homogeneous, isotropic, thermoelastic half-space with voids for a plane axi-symmetric problem. The application of a thermoelastic half-space with voids subjected to a normal force and a thermal source acting at the origin has been considered to show the utility of the solution obtained. To obtain the solution in a physical form, a numerical inversion technique has been applied. The results in the form of displacements, stresses, temperature distribution, and change in volume fraction field are computed numerically and illustrated graphically for a magnesium crystal-like material to depict the effects of voids in the theory of coupled thermoelasticity (CT) and uncoupled thermoelasticity (UCT) for an insulated boundary and a temperature gradient boundary.     

耦合均载作用下的电磁热弹性简支圆板

耦合均载作用下的简支空心和实心圆板问题是一个经典问题,对于电磁热弹性材料尚无理论解。构造了5个含有待定常数的单调和函数,将其代入用单调和函数表示的横观各向同性电磁热弹性材料的通解,获得了表面力电磁热耦合均载作用下的简支空心圆板内耦合场的解,再将所得解代入边界条件获得确定待定常数的线性方程组。该解可以退化得到实心圆板对应问题的解。所得各解都是用初等函数表示,非常方便于工程应用。算例比较了在相同热力载荷作用下,具有相同物理常数的热弹性空心圆板、压电热弹性空心圆板和电磁热弹性空心圆板内的弹性场。     

The edge dislocation inside an elliptical inclusion

The plane elasticity problem of an edge dislocation located within an elliptical inclusion in an unbounded matrix is considered. A general solution to this problem is obtained in terms of complex potential functions and all coefficients in the series representation of these potential functions are explicitly obtained. Convergence of these series is thus assured. Two specific cases of this general solution are considered in detail. The first case considers a very long, thin inclusion corresponding physically to the geometry of typical crazes in glassy polymers. the second case considers the almost circular inclusion representative of a number of crystal defects and imperfections. For the second case, expressions for the resultant force on the dislocation agree with results previously obtained for the limiting circular inclusions.     

Thin elastic inclusions in a transversally isotropic material

We propose a model of thin elastic inclusions that reduces the problem to the treatment of plane surfaces of discontinuity of displacements in a transtropic space. By representing the solution of the equilibrium equations in terms of harmonic functions and applying the technique of integral Fourier transformation, the problem is reduced to the solution of a system of two-dimensional singular integral equations relative to displacement jumps of the discontinuity surfaces. For illustration we consider an ellipsoidal inclusion in a homogeneous uniaxial tension field for which exact solutions of the integral equations are found and expressions for the stresses within the inclusion and in its neighborhood are obtained in explicit form.Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 27, No. 2, pp. 80–85, March–April, 1991.