Multiscale analysis of mixed-mode fracture and effective traction-separation relations for composite materials

A multiscale framework for the analysis of fracture is developed in order to determine the effective (homogenized) strength and fracture energy of a composite material based on the constituent's material properties and microstructural arrangement. The method is able to deal with general (mixed-mode) applied strains without a priori knowledge of the orientation of the cracks. Cracks occurring in a microscopic volume element are modeled as sharp interfaces governed by microscale traction-separation relations, including interfaces between material phases to account for possible microscale debonding. Periodic boundary conditions are used in the microscopic volume element, including periodicity that allows cracks to transverse the boundaries of the volume element at arbitrary orientations. A kinematical analysis is presented for the proper interpretation of a periodic microscopic crack as an equivalent macroscopic periodic crack in a single effective orientation. It is shown that the equivalent crack is unaffected by the presence of parallel periodic replicas, hence providing the required information of a single localized macroscopic crack. A strain decomposition in the microscopic volume element is used to separate the contributions from the crack and the surrounding bulk material. Similarly, the (global) Hill-Mandel condition for the volume element is separated into a bulk-averaged condition and a crack-averaged condition. Further, it is shown that, though the global Hill-Mandel condition can be satisfied a priori using periodic boundary conditions, the crack-based condition can be used to actually determine the effective traction of an equivalent macroscopic crack.