In the present paper, an up-scaling thermo-mechanical approach is presented for rock masses considered as anisotropic composite materials. A mathematical framework is proposed to estimate overall thermo-mechanical properties based on local geometrical considerations using the Mori-Tanaka scheme. In addition, a homogenization-based effective thermal conductivity formulation is proposed for composite materials subjected to periodic micro-scale heat fluxes and governed by Fourier's heat law and the steady state balance equations. The particular case of isotropic material is also presented for completeness.

The response of a Representative Elementary Volume (REV) to external mechanical loading is modeled using finite element numerical simulation in order to conclude the effective mechanical properties and compare them with the results of analytical calculations. The porosity of the REV is varied by changing the number or the radii of the embedded inclusions. As a first result, the two homogenization techniques are compared with reference to proximity to the numerical results. Moreover, it was noticed that the curves presenting the variation of effective bulk modulus with respect to the number of inclusions for different radii are convergent. Finally, the inclusions number and radius in the RVE which give close numerical and analytical results could then be determined.

1. Introduction

The constitutive laws of rock materials are usually established using phenomenological approaches based on results of tests on specimens that are assumed to be homogeneous. The response of materials to loading is thus crucial.

Despite their effectiveness and wide use, such approaches are limited since the responses of natural materials can be scale-dependent. Each constituent shows different material properties and/or material orientations. Therefore, the behavior of local entities needs to be studied to estimate the overall behavior. Consequently, the phenomenological approaches of materials' characterization can be insufficient. This leads to a technique that links macro-mechanical properties to micro-structural considerations known as up-scaling or homogenization. Homogenization techniques assume that scales are separated, and an intermediate level is introduced: The Representative Volume Element "RVE". The choice of this volume, its dimensions and the number of inclusions is important. Indeed, it must be smaller enough than the size of the whole structure and bigger enough than the size of the heterogeneities. It is used to get an approximation of stress and strain fields under appropriate boundary conditions. In the work of [1], the following definition of RVE is proposed: "It is the smallest material volume element of the composite for which the usual spatially constant (overall modulus) macro-scale constitutive representation is a sufficiently accurate model to represent the mean constitutive response". It is also called DRVE, deterministic representative volume element in [2]. Several researches were made in order to find the correct RVE which gives the best mechanical and thermal results. For instance, a numerical study of the size of the (RVE) for both the heat flow and linear elasticity problems is presented in [3]. Another numerical approach to evaluate the effective properties consisted on working on the sequence of microstructure volumes with prescribed sizes smaller than the DRVE [4–7].

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