Numerical Homogenization of Elastic Behavior of Fractured Rock Masses and Micro-cracked Materials by FEM

A review of recent works on numerical homogenisation of fractured rock masses gives witness of difficulties in defining rigorously and determining numerically the macroscopic strain and stress that are to be used in upscaling relations. Moreover, for finite-size domains, relations between macroscopic strain and stress are dependant on boundary conditions and this constitutes an additional difficulty for upscaling. Due to these difficulties the methods used for numerical upscaling remains somehow empirical and lead to non-accurate results, for instance to non-symmetric elastic tensors. In this paper, some fundamental results on homogenization of finite-size heterogeneous domains established by Huet (1988) are extended to discontinuous; fractured or micro-cracked, elastic materials. Boundary conditions leading to lower and upper bounds of compliance tensors for finite-size domains are rigorously defined and, in context of Finite Elements, numerical methods for determining these tensors are given. It is shown that even for a finite-size domain the elastic tensor obtained numerically will be rigorously symmetric if correctly defined and calculated.