Modeling micro-cracking and failure in short fiber-reinforced composites

Composites made of reinforcing short fibers embedded into brittle matrices, like, e.g., fiber-reinforced concretes, exhibit enhanced strength and ductility properties. Their failure process induced by tensile loadings involves hardening and softening stages as a result of matrix multiple micro-cracking, due to stress bridging of fibers across matrix micro-cracks, and strain localization phenomena. In the present paper, a variational model is proposed for the description of the intriguing failure mechanisms observed in short fibre-reinforced composites subjected to tensile loadings. The key modeling idea is to schematize the composite as a mixture of two phases, a brittle phase, representative of the matrix, and a ductile phase, accounting for the fibers reinforcement, which are coupled by elastic bonds. Different modeling levels of increasing complexity are proposed, ranging from a simplified one-dimensional analytical model to a three-dimensional variational model. Within the variational formulation, specific damage and plastic energies are assigned to the two phases, incorporating non-local gradient terms, and governing equations and evolution laws for the internal variables, as yield inequalities, consistency conditions and normality rules, are deduced from minimum principles. Parameters calibration is discussed as well as the importance of three internal lengths incorporated into the model. Moreover, the variational structure of the problem allow for a straightforward finite element implementation based on an incremental energy minimization algorithm and several aspects of the response are highlighted by means of numerical examples.