Method of potentials in elastostatics of solids with double porosity D. Ies ßan ⇑ Octav Mayer Institute of Mathematics, Romanian Academy, 700506 Ias ßi, Romania article info Article history: Received 12 November 2013 Received in revised form 10 February 2014 Accepted 14 April 2014 Available online xxxx Keywords: Porous elastic materials Boundary-value problems Fundamental solutions Potentials of single-layer and double-layer Existence and uniqueness theorems abstract This paper is concerned with the basic boundary-value problems in the equilibrium theory of elastic materials with a double porosity structure. First, a counterpart of the Boussinesq–Somigliana–Galerkin solution in the classical elastostatics is presented. Then, the fundamental solutions of the field equations are established. The potentials of single- layer and double-layer are used to reduce the boundary-value problems to singular integral equations. Existence and uniqueness results are presented. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction The linear theory of elastic materials with a double porosity structure has been investigated in many papers (see, e.g., Berryman & Wang, 2000; Cowin, 1999; Straughan, 2013; Svanadze, 2012, and references therein). The intended applications of the theory are to mechanics of geological materials and to mechanics of bone. The theory is governed by a system of five partial differential equations with five unknown functions: three components of the displacement vector field, the pressure associated with the pores, and a pressure associated with fissures. It is important to note that in the equilibrium theory the fluid pressures do not depend on the displacement vector. The theory of continua with single porosity structure has been extensively studied (see, e.g., De Boer (2000), Rajagopal (2007), Straughan (2008), and references therein). Nunziato and Cowin (1979) established a theory for the behavior of porous bodies in which the skeletal or matrix materials are elastic and the interstices are void of material. This theory has been studied in many papers (see, e.g., Bedford & Drumheller, 1983; Ciarletta & Straughan, 2007; Cowin & Nunziato, 1983; Cowin & Puri, 1983; Munoz-Rivera & Quintanilla, 2008, and references therein). In this paper we consider a linear theory of elastic solids with a double porosity structure which is derived from the theory of Nunziato and Cowin (1979). In contrast with the classical theory of elastic materials with double porosity, in the case of equilibrium the porosity structure depends on the displacement vector field. We study the basic equations of the equilibrium theory for homogenenous and isotropic solids. First, we present a counterpart of the Boussinesq– Somigliana–Galerkin solution in the classical elastostatics. This result is used to establish the fundamental solutions of the field equations. Then, representations of Somigliana type for the displacement and volume fraction fields are presented. We use the method of potentials (Kupradze, Gegelia, Bashelishvili, & Burchuladze, 1979) to study the basic boundary-value http://dx.doi.org/10.1016/j.ijengsci.2014.04.011 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Tel.: +40 0232310444. E-mail address: iesan@uaic.ro International Journal of Engineering Science xxx (2014) xxx–xxx Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci Please cite this article in press as: Ies ßan, D. Method of potentials in elastostatics of solids with double porosity. International Journal of Engi- neering Science (2014), http://dx.doi.org/10.1016/j.ijengsci.2014.04.011