Steady vibration problems in the coupled linear theory of porous elastic solids

Abstract

This paper concerns the coupled linear theory of elasticity for isotropic porous materials. In this theory the coupled phenomena of the concepts of Darcy’s law and the volume fraction is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions, and its basic properties are presented. The radiation conditions are established and Green’s identities are obtained. The uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single layer and double layer) and volume potentials are constructed and the basic properties of these potentials 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.

Boundary Value Problems of Thermoelasticity for Porous Sphere and for A Space with Spherical Cavity

“…If supposing that h and j h are known functions, taking into account the representations (8), to define the displacement vector u from Eq. , ) 1 ( 1 we get the following non-homogeneous equation , ) 1 ( grad…”

“…The generalization of the theory of elasticity and thermoelasticity for materials with double voids belongs to Ieşan and Quintanilla (2014) [7]. In (2020) [8] Svanadze considered the coupled linear model of porous elastic solids by combining the following three variables: the displacement vector field, the volume fraction of pores and the change of fluid pressure in pore network. In this work the basic internal and external BVPs of steady vibrations are investigated, the uniqueness and the existence theorems are proved by means of the potential method and the theory of singular integral equations.…”

“…The quasi-static BVPs of the theories of elasticity and thermoelasticity for porous materials are studied by Mikelashvili (2020Mikelashvili ( ,2021 [10,11]. These works [8][9][10][11] represents a first step in the coupled linear theory of elasticity and thermoelasticity of porous materials.…”

A Reduced-Order Extrapolation (ROE) Method for Solution Coefficient Vectors in the Mixed Finite Element (MFE) Method for the Two-Dimensional (2D) Fourth-Order Hyperbolic Equation

“…y, 0) = u 0 (x, y), at t = 0 and in , u t (x, y, 0) = u 1 (x, y), at t = 0 and in , (1) where is an interconnected domian with bounded boundary ∂ , = ∪∂ , J = [0, T ], T is the final moment, f (x, y, t) is the given sufficiently smooth source function, g 1 (x, y, t), g 2 (x, y, t), u 0 (x, y) and u 1 (x, y) are given sufficiently smooth boundary functions and initial functions respectively.…”

“…The 2D fourth-order hyperbolic equation is an important partial differential equation describing vibration or wave phenomena, which has important application value in aerospace, petroleum exploration, urban construction, crustal sounding and so on [1], [2], [3]. For practical problems, due to the complexity of the physical problem itself and the solution region, it is often difficult to obtain the exact solution of the problem (1). In the last few years, many scholars have deeply studied the numerical solution of the problem (1) and put forward a variety of numerical calculation methods.…”