On the use of effective properties for the fracture analysis of microstructured materials

This paper addresses some fundamental theoretical and numerical issues concerning the application of effective properties for the failure analysis of microstructured materials, with a focus on functionally graded materials. An edge-cracked square region is considered which is taken to be on the size of a representative volume element in a larger structural system. The region is made functionally graded by utilizing a non-homogeneous probability density function (PDF) governing the spatial distribution of the constituents. The non-homogeneity of the PDF exists only in the direction of the crack plane. Based upon the type of microstructure considered in the region, an accurate homogenization method is employed to obtain effective properties. For functional forms of the PDF, energy release rates and stress intensity factors (SIFs) are then calculated using both effective properties based on pointwise homogenization and realizations of specific microstructures generated from the PDF. For all calculations, we employ the eXtended finite element method, alleviating the need to remesh the domain between different microstructural realizations. Enrichment strategies are employed by the method to capture the singular stress fields near the crack tip as well as the discontinuous strain fields at fiber–matrix interfaces, even with relatively coarse meshes. SIFs are calculated using a domain form of the J-integral that does not exhibit any domain dependence. The results show excellent agreement between the implicit and the mean of the explicit trials for the energy release rates and strain energy densities. The results also seem to indicate that the implicit SIFs predict the mean SIF of the explicit trials for an arbitrary initial crack geometry. Furthermore, the results indicate that a good approximation of the SIF at specific crack tips may be estimated by re-interpreting the implicit results according to knowledge of the distinct crack-tip location in the microstructure. Quasi-static crack growth studies are performed to investigate the probability of finding crack tips within either of the composite phases.