Plane Waves and Boundary Value Problems in the Theory of Elasticity for Solids with Double Porosity
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Plane Waves and Boundary Value Problems in the Theory of Elasticity for Solids with Double Porosity Merab Svanadze
Received: 14 December 2011 / Accepted: 8 February 2012 / Published online: 23 May 2012 © Springer Science+Business Media B.V. 2012
Abstract This paper concerns with the dynamical theory of elasticity for solids with double porosity. This theory unifies the earlier proposed quasi-static model of Aifantis of consolidation with double porosity. The basic properties of plane waves are established. The radiation conditions of regular vectors are given. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness theorems are proved. The basic properties of elastopotentials are given. The existence of regular (classical) solution of the external BVP by means of the potential method (boundary integral method) and the theory of singular integral equations are proved. Keywords Double porosity · Plane waves · Boundary value problems · Steady vibrations · Boundary integral method · Singular integral equations
1 Introduction Porous materials play an important role in many branches of engineering, e.g., the petroleum industry, chemical engineering, geomechanics, and, in recent years, biomechanics. The construction and the intensive investigation of the theories of continua with microstructures arise by the wide use of porous materials into engineering and technology. The general 3D theory of consolidation for materials with single porosity was formulated by Biot [9]. The model for consolidation requires the quasi-static assumption that the equations of motion are replaced by the corresponding equilibrium equations. One important generalization of Biot’s theory of poroelasticity that has been studied extensively started with the works by Barenblatt et al. [4]. In this paper the double porosity model was first proposed to express fluid flow in hydrocarbon reservoirs and aquifers. The quasi-static theory of elasticity for materials with double porosity in the framework of mixture theory to model the flow and deformation behavior of porous media M. Svanadze () Institute for Fundamental and Interdisciplinary Mathematics Research, Ilia State University, K. Cholokashvili Ave., 3/5, 0162, Tbilisi, Georgia e-mail: [email protected]
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characterized by two coexisting degree of porosity was presented by Aifantis and his coworkers [7, 14, 24]. The Aifantis’ theory unifies the earlier proposed models of Barenblatt’s for porous media with double porosity and Biot’s for porous media with single porosity. The fundamental solution of the system of equations of steady vibrations in the Aifantis’ quasistatic theory of elasticity for solids with double porosity was constructed using elementary functions and its basic properties were established by Svanadze [23]. Based on ideas similar to Biot’s theory, Berryman and Wang [5, 6] established the phenomenological equations of the quasi-static theory for double porosity media and presented the method to determine the
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