A micromechanics-based strain gradient damage model for fracture prediction of brittle materials – Part I: Homogenization methodology and constitutive relations

In this paper, we first describe a homogenization methodology with the aim of establishing strain gradient constitutive relations for heterogeneous materials. The methodology presented in this work includes two main steps. The first one is the construction of the average strain-energy density for a well-chosen RVE by using a homogenization technique. The second one is the transformation of the obtained average strain-energy density to that for the continuum. An important characteristic of this method is its self-consistency with respect to the choice of the RVE: the strain gradient constitutive law built by using the present method is independent of the size and the form of the RVE. In the frame of this homogenization procedure, we have constructed a strain gradient constitutive relation for a two-dimensional elastic material with many microcracks by adopting the self-consistent scheme. It was shown that the effective behavior of cracked solids depends not only on the crack density but also on the average crack size with which the strain gradient is associated. The proposed constitutive relation provides a starting point for the development of an evolution law of damage including strain gradient effect, which will be presented in the second part of this work.