Analytical Solution of Coupled Thermoelastic Axisymmetric Transient Waves in a Transversely Isotropic Half-Space

Abstract

A half-space containing transversely isotropic thermoelastic material with a depth-wise axis of material symmetry is considered to be under the effects of axisymmetric transient surface thermal and forced excitations. With the use of a new scalar potential function, the coupled equations of motion and energy equation are uncoupled, and the governing equation for the potential function, is solved with the use of Hankel and Laplace integral transforms. As a result, the displacements and temperature are represented in the form of improper double integrals. The solutions are also investigated in detail for surface traction and thermal pulses varying with time as Heaviside step function. It is also shown that the derived solutions degenerate to the results given in the literature for isotropic materials. Some numerical evaluations for displacement and temperature functions for two different transversely isotropic materials with different degree of anisotropy are presented to portray the dependency of response on the thermal properties as well as the degree of anisotropy of the medium.

Transient response of a thermoelastic half‐space to mechanical and thermal buried sources

“…It is emphasized that in this case, where m is zero, the equations of motion and the energy equation are decoupled from each other (see Eq. (1)) and as a result, they could be solved independently (see Raoofian et al [50]).…”

“…Contrary to Hankel transforms, the numerical inversion of a Laplace transform results in an illposed problem, where no single algorithm exists in the literature to be applicable for all transformed functions (Davis and Martin [10]; Duffy [14]; Narayanan and Beskos [38]), and thus it is the most problematic case in the process of numerical computations in this paper. Here, among the aforementioned algorithms for the numerical inversion of Laplace transform, the multi-precision, which is very fast and efficient is adopted (see Raoofian et al [50]). This algorithm may be expressed as (Abate and Valko [1])…”

“…They have used one-sided and double bi-lateral Laplace transforms and derived a representation for vertical displacement by asymptotic expansion for short and long time spans. Recently, with the use of representations given by Eskandari-Ghadi et al [18], Raoofian et al [50] have analytically investigated the propagation of a transient thermoelastic wave in an axisymmetric transversely isotropic half-space under the effects of surface mechanical and thermal loads. Also, Hayaty et al [24] and [25], with the use of the aforementioned potential functions, have studied the time harmonic motion of thermoelastic axisymmetric isotropic and transversely isotropic halfspaces under the surface tractions and prescribed temperature changes.…”

Performance comparison of numerical inversion methods for Laplace and Hankel integral transforms in engineering problems

“…The joint Hankel-Laplace transform is a common tool for three-dimensional transient wave propagation in both elastic and thermoelastic boundary value problems, specially in cylindrical coordinate system [26,27,50,58,49]. Sometimes, by separation of two integral transforms that happens with the use of the so-called Cagniard-De Hoop method [3], it is possible to invert one of the integral transform in analytical form.…”

“…The interested reader may find a wider bibliography in Belman [6] and Cohen [7]. In a wide range of engineering problems, Laplace transform is used in conjunction with Hankel integral transform [20,21,29,44,48,50], where the Hankel integral transform of order m of function f(r), r e (0, 1), is defined as [53] f m ðnÞ ¼…”