Complex variable BEM for thermo- and poroelasticity Vadim Koshelev a , Ahmad Ghassemi b, * a Institute for Problems of Mechanical Engineering, Russian Academy of Sciences, St-Petersburg 199178, Russia b University of North Dakota, Room 330A, Leonard Halll, Grand Forks 58202, USA Received 5 June 2003; revised 11 July 2003; accepted 26 August 2003 Abstract A stationary thermoelastic (poroelastic) boundary element method is suggested based on the Complex Variable Hypersingular Boundary Integral Equation. The method is developed for heterogeneous blocky media. Various conditions on the contacts between the blocks are considered, namely discontinuity of temperature (pore pressure) or discontinuity of heat (fluid) flux. The problem of the basic integrals calculation is discussed and numerical examples are presented to demonstrate the potential of the method. q 2003 Elsevier Ltd. All rights reserved. Keywords: Thermoelasticity; Poroelasticity; Boundary element method; Complex variables 1. Introduction Many problems of mechanics of materials, in general and geomechanics in particular, deal with the stress and displacement calculations for heterogeneous media consist- ing of blocks, grains and inclusions. These problems often involve thermal and pore pressure effects. Heterogeneity of both mechanical and thermal (or poromechanical) charac- teristics necessitates numerical analysis of the problem of interest. The Boundary Element Method (BEM) based on the Hypersingular Boundary Integral Equations (BIE) has shown to be the most effective technique for treatment of problems with complicated contact conditions. In this work, the Complex Variable (CV) Hypersingular BIE [1,2] is utilized. The equation is formulated in the terms of direct values of integrals. An alternative treatment without involving the direct value of a finite part integral can be found in other works [3,4] that use the Hadamard type integral which is a derivative of Cauchy type integral. The equation under consideration contains displacements discontinuities and tractions on the boundaries of the blocks. Thus, there is no need to use additional relations to calculate the characteristics of the contacts interaction as in the case of Singular Equations. The CV-BEM presented herein has been developed by considering additional terms that depend on the given values of a potential (temperature or pore pressure) and its normal derivative (heat or fluid flux) at the boundary of the domain under consideration. These values may be found using the CV-BEM for potential problems [5]. 2. Integral identities of direct BEM Below, we deal with BIE and the direct formulation BEM. Their derivation is based on the reciprocity theorem of work. The formulation of the thermoelastic version follows [6] ð S ðs i u p i 2 s p i u i Þds þ ð V ðc i u p i 2 c p i u i Þdn þ ð V gðT 1 p kk 2 T p 1 kk Þdn ¼ 0; ð1Þ where a repeated index implies a sum; and ðc i ; s i ; u i ; T Þ and ðc p i ; s p i ; u p i ; T p Þ characterize two independent states of body force, traction, displacement and temperature for the thermoelastic body V having a surface S: The coefficient g is a constant from the constitutive equation s ij ¼ C ijkl 1 kl 2 gd ij T : ð2Þ For the poroelastic case g is defined as Biot’s effective stress coefficient [7,8]. The common method to arrive at the integral identities of the direct method is to take the actual state under consideration as the first state mentioned above; and consider the second state to be generated by a unit body 0955-7997/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2003.08.006 Engineering Analysis with Boundary Elements 28 (2004) 825–832 www.elsevier.com/locate/enganabound * Corresponding author. Tel.: þ 1701-777-3213; fax: þ 1701-777-4449. E-mail address: ahmad_Ghassemi@mail.und.nodak.edu (A.Ghassemi).