Annular crack in a thermoelastic half-space

Abstract

The purpose of this article is to analytically investigate the three-dimensional thermoelastic fields in a semi-infinite medium weakened by an annular crack. The crack faces are exposed to a prescribed temperature load and the external surface of the medium is kept at the reference temperature. The problem is formulated as a three-part mixed boundary value problem and is treated through Goodier’s thermoelastic potential and the Boussinesq harmonic functions. The Hankel transform method is employed to convert this problem into triple integral equations and a system of coupled ones. Using some integral relations and the Gegenbauer addition formula, the set of triple integral equations is ultimately reduced to an infinite system of linear algebraic equations. Closed-form formulas for various quantities of physical interest are derived and expressed in terms of the solution of the obtained infinite algebraic system. Moreover, the combined mechanical and thermal loading conditions are also considered. Numerical results for thermoelastic fields and mixed-mode $I-I\mathrm{}I$ thermal stress intensity factors are shown graphically to analyze their dependence on the radii and depth of the crack. Results for the thermoelastic problem of an infinite medium containing an annular crack are also obtained as a special case of this study.

Keywords:

• Annular crack
• heat conduction
• mixed boundary value problem
• thermal stress intensity factor
• thermoelastic deformation
• triple integral equations

Acknowledgment

The Laboratoire de Génie Mécanique et Développement (LGMD) of Ecole Nationale Polytechnique (ENP) is gratefully acknowledged for the resources and support.