European Journal of Mechanics A/Solids 23 (2004) 957–973 Potential method in the linear theory of swelling porous elastic soils C. Gale¸ s ∗ Faculty of Mathematics, University of Ia¸ si, 6600 Ia¸ si, Romania Received 26 January 2004; accepted 16 July 2004 Available online 9 September 2004 Abstract This paper deals with the isothermal linear theory of swelling porous elastic soils in the case of fluid saturation. Internal and external boundary value problems of steady vibrations are investigated using the potential method. The uniqueness and existence theorems of classical solutions of the aforementioned problems are proved.  2004 Elsevier SAS. All rights reserved. Keywords: Mixtures; Vibrations; Potentials 1. Introduction Eringen (1994) pointed out the importance of the theory of mixtures (see the review articles by Bowen (1976), Atkin and Craine (1976a, 1976b), and Bedford and Drumheller (1983) for a presentation of the work on the subject) to the applied field of swelling. In this connection Eringen (1994) has developed a continuum theory of swelling porous elastic soils as a continuum theory of mixture consisting of three components: an elastic solid, a viscous fluid and a gas. The intended applications of the theory are in the field of swelling, oil exploration, slurries and consolidation problems. The theory is relevant to problems in the oil exploration industry, since oil is viscous and is usually accompanied by gas in underground rocks, porous solid in slurries and muddy river beds. Consolidation problems in the building industry, earthquake problems, swelling of plants and living tissues and a plethora of other problems fall into domain of mixture theory considered by Eringen (1994). In the context of Eringen’s mixture theory, some results concerning existence, uniqueness, continuous dependence, insta- bility, structural stability and spatial behavior of the solutions in the case of fluid and/or gas saturation have been studied in several recent articles by Chiri¸ tˇ a (2003), Quintanilla (2002a, 2002b) and Gale¸ s (2002a, 2002b, 2003, 2004). Also, the anti- plane shear deformations of swelling porous elastic soils forward and backward in time have been considered by Bofill and Quintanilla (2003a, 2003b). In the present paper, using the classical potential method (see Kupradze et al., 1968), we investigate some internal and external boundary value problems of steady vibration, in the case that solid matrix is saturated with fluid. To this end, we consider the fundamental solutions of the steady vibration, established by Gale¸ s (2004), and study some of their properties. Then, in view of these results, we establish some representation formulas which are used to introduce the single layer and the double layer potentials. Making use of their properties, we reduce the boundary value problems to two dimensional singular * Tel.: +40-232-201226 E-mail address: cgales@uaic.ro (C. Gale¸ s). 0997-7538/$ – see front matter  2004 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2004.07.003