General steady-state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application

This paper presents a set of 3D general solutions for thermoporoelastic media for the steady-state problem. By introducing two displacement functions, the equations governing the elastic, pressure and temperature fields are simplified. The operator theory and superposition principle are then employed to express all the physical quantities in terms of two functions, one of which satisfies a quasi–Laplace equation and the other satisfies a differential equation of the eighth order. The generalized Almansiʹs theorem is used to derive the displacements, pressure and temperature in terms of five quasi-harmonic functions for various cases of material characteristic roots. To show its practical significance, an infinite medium containing a penny-shaped crack subjected to mechanical, pressure and temperature loads on the crack surface is given as an example. A potential theory method is employed to solve the problem. One integro-differential equation and two integral equations are derived, which bear the same structures to those reported in literature. For a penny-shaped crack subjected to uniformly distributed loads, exact and complete solutions in terms of elementary functions are obtained, which can serve as a benchmark for various kinds of numerical codes and approximate solutions.