A Mode-I Crack Problem for an Infinite Space in Generalized Thermoelasticity

Abstract

In this work, we solve a dynamical problem of an infinite space with a finite linear crack inside the medium. The Fourier and Laplace transform techniques are used. The problem is reduced to the solution of a system of four dual integral equations. The solution of these equations is shown to be equivalent to the solution of a Fredholm integral equation of the first kind. This integral equation is solved numerically using the method of regularization. The inverse Laplace transforms are obtained numerically using a method based on Fourier expansion techniques. Numerical values for the temperature, stress, displacement, and the stress intensity factor are obtained and represented graphically.During the second half of the 20th century, nonisothermal problems of the theory of elasticity became increasingly important. This is due mainly to their many applications in widely diverse fields. First, the high velocities of modern aircraft give rise to aerodynamic heating, which produces intense thermal stresses, reducing the strength of the aircraft structure. Second, in the nuclear field, the extremely high temperatures and temperature gradients originating inside nuclear reactors influence their design and operations [1]. Danilovskaya [2] was the first to solve an actual problem in the theory of elasticity with nonuniform heat. The problem was in the context of what became known as the theory of uncoupled thermoelasticity. In this theory, the temperature is governed by a parabolic partial differential equation, which does not contain any elastic terms. It was not much later that many attempts were made to remedy the shortcomings of this theory.Biot [3] formulated the theory of coupled thermoelasticity to eliminate the paradox inherent in the classical uncoupled theory that elastic changes have no effect on the temperature. The heat equations for both theories, however, are of the

A Green-Naghdi Model in a 2D Problem of a Mode I Crack in an Isotropic Thermoelastic Plate

“…The material constants are given as following [8] Before going to the analysis grid independence audit was carried out. The quadrilateral, eight-node isoparametric element is used for temperature and displacement components.…”

“…[7]. Sherief and ElMaghraby [8,9] studied mode I crack problems using the method of regularization. Prasad et al [10] applied the method of regularization in a two dimensional thermoelastic problem of a mode I crack under Green and Naghdi type III model.…”

A Thick Plate Problem Under the Action of a Body Force in Generalized Thermoelasticity

“…The uniqueness of a solution for this theory was proved under different conditions by Ignaczak [4,5] and by Sherief [6]. Sherief and El-Maghraby [7,8] have solved two crack problems. Mallik and Kanoria [9] have formulated a generalized thermoelasticity application for a penny-shaped crack analysis.…”

A Generalized Thermoelasticity Problem for a Half-Space with Heat Sources and Body Forces

“…The fundamental solution for this theory was obtained by Sherief [9] and by Sherief and Anwar [10]. Sherief and El-Maghraby [11] have solved a problem for a penny-shaped crack in an infinite thermoelastic solid and have obtained the solution for a Mode-I crack problem in an infinite space in [12]. Sherief and Helmy have solved a two-dimensional problem in [13].…”