Explicit solution of the Dirichlet boundary value problem of elasticity for porous infinite strip
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Explicit solution of the Dirichlet boundary value problem of elasticity for porous infinite strip Lamara Bitsadze
Abstract. The object of the present paper is to consider the Dirichlet boundary value problem of the coupled linear quasistatic theory of elasticity for porous isotropic elastic infinite strip. The general representations of a regular solutions of a system of considered equations for a homogeneous isotropic medium are constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. Using the Fourier method, the Dirichlet BVP is solved effectively (in quadratures) for the infinite strip. Mathematics Subject Classification. 74G05, 74F10, 74F99. Keywords. Elastic porous materials, Explicit solution, Infinite strip.
1. Introduction Many materials such as rocks, sand, soil, etc., which occur on and below the surface of the earth, are known as porous materials and have applications in many fields of engineering, such as the petroleum industry, material science and biology. The theory of porous materials is used for investigated various types of geological and biological materials for which classical theory of elasticity is not adequate. In most of naturally or manufactured solids is not completely filled. In nearly every body, there are empty interspaces, which are called pores through which the liquid or gas may flow. For example, the human skin has a larger number of pores, bone tissue could be assumed to be transversely isotropic and most closely describes mechanical anisotropy of bone and cancellous bone is considered as a porous material. The foundations of the theory of elastic materials with voids were first proposed by Cowin and Nunziato [1,2]. They investigated the linear and nonlinear theories of elastic materials with voids. In these theories, the independent variables are displacement vector field and the change of volume fraction of pores. Elastic materials which contain a multi-porous structure have a multitude of applications in real life. The history of development of porous body mechanics, the main results and the sphere of their application are set forth in detail in the monographs [3–6] (see references therein). The generalization of the theory of elasticity and thermoelasticity for materials with double void pores belongs to Ie¸san and Quintanilla [7]. In [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 pressure of the fluid. The basic internal and external BVPs of steady vibrations are investigated, Green’s formulas are obtained, the uniqueness and the existence theorems are proved by means of the potential method and the theory of singular integral equations. The coupled linear quasi-static theory of elasticity for porous materials is considered in [9]. The fundamental solution is constructed, and its basic properties are established. The uniqueness and existence theorems of the
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