On some crack problems in anisotropic thermoelasticity

Some basic equations recently derived by Clements are used to consider crack problems in anisotropic thermoelasticity. The problems concern a single crack in an anisotropic material in which the displacement and stress are independent of one Cartesian coordinate. No symmetry elements of the material are assumed and the temperature, displacement and stress fields are determined for an arbitrary distribution of temperature or heat flux over the crack faces.     

Wave propagation in dual-phase-lag anisotropic thermoelasticity

The present paper is concerned with the propagation of plane waves in a transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The governing equations are solved in x–z plane to show the existence of three plane waves. Reflection of these plane waves from thermally insulated as well as isothermal stress-free surfaces is studied to obtain a system of three non-homogeneous equations in reflection coefficients of reflected waves. For numerical computations of speeds and reflection coefficients, a particular material is modeled as transversely isotropic dual-phase-lag generalized thermoelastic solid half-space. The speeds of plane waves are computed numerically for a certain range of the angle of propagation and are shown graphically against the angle of propagation for the cases of dual-phase-lag (DPL) thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity. Reflection coefficients of various reflected plane waves are computed numerically for thermally insulated as well as isothermal cases and are shown graphically against the angle of incidence for the cases of DPL thermoelasticity, coupled thermoelasticity and Lord–Shulman generalized thermoelasticity.     

Nonlinear Eulerian thermoelasticity for anisotropic crystals

A complete continuum thermoelastic theory for large deformation of crystals of arbitrary symmetry is developed. The theory incorporates as a fundamental state variable in the thermodynamic potentials what is termed an Eulerian strain tensor (in material coordinates) constructed from the inverse of the deformation gradient. Thermodynamic identities and relationships among Eulerian and the usual Lagrangian material coefficients are derived, significantly extending previous literature that focused on materials with cubic or hexagonal symmetry and hydrostatic loading conditions. Analytical solutions for homogeneous deformations of ideal cubic crystals are studied over a prescribed range of elastic coefficients; stress states and intrinsic stability measures are compared. For realistic coefficients, Eulerian theory is shown to predict more physically realistic behavior than Lagrangian theory under large compression and shear. Analytical solutions for shock compression of anisotropic single crystals are derived for internal energy functions quartic in Lagrangian or Eulerian strain and linear in entropy; results are analyzed for quartz, sapphire, and diamond. When elastic constants of up to order four are included, both Lagrangian and Eulerian theories are capable of matching Hugoniot data. When only the second-order elastic constant is known, an alternative theory incorporating a mixed Eulerian–Lagrangian strain tensor provides a reasonable approximation of experimental data.     

An explicit-form solution to the plane elasticity and thermoelasticity problems for anisotropic and inhomogeneous solids

This paper presents an analytical approach to solve the plane elasticity and thermoelasticity problems for inhomogeneous, orthotropic planes, half-planes, and strips. Solution of the problems is reduced to the governing Volterra integral equation formulated for the key function and accompanied by the corresponding integral conditions. By making use of the resolvent-kernel technique, the governing equation is solved and the solution to the original problem is presented in explicit form.     

A boundary element method for anisotropic coupled thermoelasticity

Summary A boundary element formulation is presented for the solution of the equations of fully coupled thermoelasticity for materials of arbitrary degree of anisotropy. By employing the fundamental solutions of anisotropic elastostatics and stationary heat conduction, a system of equations with time-independent matrices is obtained. Since the fundamental solutions are uncoupled and time-independent, a domain integral remains in the representation formula which contains the time-dependence as well as the thermoelastic coupling. This domain integral is transformed to the boundary by means of the dual reciprocity method. By taking this approach, the use of dynamic fundamental solutions is avoided, which enables an efficient calculation of system matrices. In addition, the solution of transient processes as well as, free and forced vibration analysis becomes straightforward and can be carried out with standard time-stepping schemes and eigensystem solvers. Another important advantage of the present formulation is its versatility, since it includes a number of simplified thermoelastic theories, viz. the theory of thermal stresses, coupled and uncoupled quasi-static thermoelasticity, and stationary thermoelasticity. The accuracy of the new thermoelastic boundary element method is demonstrated by a number of example problems. Support by the Deutsche Forschungsgemeinschaft (DFG) of the Graduate Collegium Modelling and discretization methods for continua and fluids (GKKS) at the University of Stuttgart is gratefully acknowledged.     

Numerical-analytical solution of plane problems in thermoelasticity

The scaled-boundary method is used as a numerical-analytical method to solve problems of thermoelasticity. As an example, the stress intensity factor for a heated thin circular orthotropic disk with an internal crack is evaluated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 12, pp. 122–126, December 2007.     

Solution of three-dimensional static problems of thermoelasticity

Zaporozh Institute of Mechanical Engineering. Translated from Prikladnaya Mekhanika, Vol. 30, No. 3, pp. 25–34, March, 1994.     

On the final boundary value problems in linear thermoelasticity

In the present study we derive some uniqueness criteria for solutions of the Cauchy problem for the standard equations of dynamical linear thermoelasticity backward in time. We use Lagrange-Brun identities combined with some differential inequalities in order to show that the final boundary value problem associated with the linear thermoelasticity backward in time has at most one solution in appropriate classes of displacement-temperature fields. The uniqueness results are obtained under the assumptions that the density mass and the specific heat are strictly positive and the conductivity tensor is positive definite.     

Some non-classical thermoelasticity problems for a rectangular parallelepiped

Meccanica - In the Cartesian system of coordinates, thermoelastic equilibrium of an isotropic homogeneous rectangular parallelepiped is considered. On the lateral faces of a parallelepiped either...     

Closed solutions of boundary-value problems of coupled thermoelasticity

Coupled equations of thermoelasticity take into account the effect of nonuniform heating on the medium deformation and that of the dilatation rate on the temperature distribution. As a rule, the coupling coefficients are small and it is assumed, sometimes without proper justification, that the effect of the dilatation rate on the heat conduction process can be neglected. The aim of the present paper is to construct analytical solutions of some model boundary-value problems for a thermoelastic bounded body and to determine the body characteristic dimensions and the medium thermomechanical moduli forwhich it is necessary to take into account that the temperature and displacement fields are coupled. We consider some models constructed on the basis of the Fourier heat conduction law and the generalized Cattaneo-Jeffreys law in which the heat flux inertia is taken into account. The solution is constructed as an expansion in a biorthogonal system of eigenfunctions of the nonself-adjoint operator pencil generated by the coupled equations of motion and heat conduction. For the model problem, we choose a special class of boundary conditions that allows us to exactly determine the pencil eigenvalues.     

Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity

In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a quasi-longitudinal Primary (qP) wave and a quasi-transverse Secondary (qS) wave; and (2) a mainly thermal (qT) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.