Abstract This article is concerned with the coupled linear quasi-static theory of thermoporoelasticity under local thermal equilibrium. The system of equations of this theory is based on the constitutive equations, Darcy’s law of the flow of a fluid through a porous medium, Fourie’s law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of general governing equations is expressed in terms of the displacement vector field, the changes of the volume fraction of pores, the fluid pressure in pore network and temperature. The fundamental solution of the system of quasi-static equations in the considered theory is constructed and its basic properties are established. On the basis of Green’s formulas the uniqueness theorems for classical solutions of the internal and external boundary value problems (BVPs) are proved. Then, the surface and volume potentials are presented and their basic properties are given. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.

中文翻译：

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热孔弹性耦合线性理论中的准静态问题

摘要 本文研究了局部热平衡下热孔弹性的耦合线性准静态理论。该理论的方程组基于本构方程、流体通过多孔介质流动的达西定律、傅里叶热传导定律、平衡方程、流体质量守恒和热传递。一般控制方程组用位移矢量场、孔隙体积分数的变化、孔隙网络中的流体压力和温度来表示。构造了所考虑理论中拟静态方程组的基本解，并建立了其基本性质。在格林公式的基础上，证明了内外边值问题（BVP）经典解的唯一性定理。然后，给出了表面势和体积势，并给出了它们的基本性质。最后，利用势法（边界积分方程法）和奇异积分方程理论证明了BVP经典解的存在定理。