Boundary Integral Method in the Theory of Bone Porothermoelasticity

Abstract

The concept of porous media is used in many areas of applied science(e.g., biology, biophysics, biomechanics) and engineering. The doubleporosity model would consider the bone fluid pressures in the vascularporosity and lacunar-canalicular porosity. A porothermoelastic approachfor double porosity materials combines the theory of heat conduction withporoelastic constitutive equations, coupling the temperature field with thestresses and the pore and fissure fluid pressures.This paper concerns with the quasi-static coupled linear theory of boneporothermoelasticity for materials with double porosity and some basic resultsof the classical theory of thermoelasticity are generalized. The systemof equations of this theory is based on the equilibrium equations, conservationof fluid mass, the effective stress concept, Darcy’s law for materialwith double porosity and Fourier law of heat conduction. The fundamentalsolution of the system of governing equations is constructed by means ofelementary functions and its basic properties are established. The Green’sformulas in the considered theory are obtained. The formulas of Somiglianatype integral representations of regular vector and regular (classical) solutionsare presented. The uniqueness theorems for classical solutions of theinternal and external boundary value problems are proved. The singlelayer,double-layer and volume potentials are constructed and their basicproperties are established. Finally, the existence theorems for classical solutionsof the boundary value problems are proved by means of the boundaryintegral method and the theory of singular integral equations.