An efficient homogenization method for elastic media with multiple cracks

An efficient numerical method for calculation of the effective stiffness tensor of 3D-elastic media with multiple interacting cracks is developed. A representative volume element (RVE) of the medium with a finite number of cracks is embedded into the background homogeneous medium. The presence of cracks outside of the RVE is accounted by placing the RVE into an effective external stress field. This field does not coincide with the external stress field applied to the cracked medium, and for construction of this field, a version of a self-consistent effective field method is proposed. The problem is reduced to a system of 2D-integral equations for the crack opening vectors inside the RVE. Discretization of these equations is performed by Gaussian approximation functions centered at a set of nodes uniformly distributed on crack surfaces. For such functions, construction of the matrix of the discretized problem is reduced to calculation of five standard 1-D integrals that can be tabulated. As a result, numerical integration is not necessary, and the matrix elements are calculated fast. Examples of various arrangements of planar cracks in isotropic elastic media are considered. The results are compared with numerical solutions and experimental data available in the literature.