Cracks interacting with particles or fibers in composite materials

Micromechanics analysis of damage in heterogeneous media and composites cannot ignore the interactions among cracks as well as between cracks and inclusions or voids. Previous investigators can to this conclusion upon finding that states of distributed (diffuse) cracking (damage) cannot be mathematically represented merely as crack systems in a homogeneous medium, even though stable states with distributed damage have been experimentally observed in heterogeneous materials such as concrete. This paper presents a method for modeling interactions between a crack and many inclusions. Based on the Duhamel-Neuman analogy, the effect of the inclusions is equivalent to unbalanced forces acting on the contour of each inclusion in an infinite homogeneous solid. The problem is solved by superposition; it is decomposed into several standard problems of elasticity for which well-known solutions are available. The problem is finally reduced to a system of linear algebraic equations similar to those obtained by Kachanov for a system of interacting cracks without inclusions. The calculated estimates of the stress intensity factors differ from some known exact solutions by less than 10% provided the cracks or the inclusions are not very close to each other. Approximately, the problem can be treated as crack propagation in an equivalent homogeneous macroscopic continuum for which the apparent fracture toughness increases or decreases as a function of the crack length. Such variations are calculated for staggered inclusions. They are analogous to /R-curves in nonlinear fracture mechanics. They depend on the volume fraction of the inclusions, their spatial distribution and the difference between the elastic properties of the inclusions and the matrix. Large variations (of the order of 100%) are found depending on the location of the crack and its propagation direction with respect to the inclusions.