Effects of the orientational distribution of cracks in isotropic solids

The paper deals with the elastic characterisation of microcracked solids: we analyse dispersions of cracks with arbitrary non-random orientational distributions. Particular cases of angular distributions are given by cracks all oriented in a given direction or cracks uniformly random oriented in the space. A unified theory covers all the orientational distributions between the random and the parallel ones. The micromechanical averaging inside the composite material is carried out by means of explicit results which allows us to obtain closed-form expressions for the macroscopic or equivalent elastic moduli of the overall material. The analysis has been performed in two-dimensional (2D) elasticity (plane stress and plane strain) with slit like cracks and in three-dimensional (3D) elasticity with planar circular cracks. The elastic behaviour of the microcracked solid depends upon the density of cracks and upon their orientational distribution. In particular, this study allows us to state that in two-dimensions the elastic behaviour of such a microcracked material is completely defined by one order parameter, which depends on the given angular distribution while the elastic characterisation in three-dimensions depends on two order parameters. The particular cases of isotropic orientations of cracks (both in 2D and in 3D) have been generalised to higher values of the cracks density by means of the method of the iterated homogenisation, which leads to some differential equations. Their solutions show that the equivalent elastic moduli depend exponentially on the cracks density.