A study of stability and bifurcation in micro-cracked periodic elastic composites including self-contact

A novel analysis of microscopic instability and bifurcation phenomena is here developed in the context of homogenization theory of finitely deformed periodic elastic composites containing micro-cracks in unilateral self-contact. Original stability and non-bifurcation conditions together with their interrelations are determined for a composite microstructure driven along quasi-static equilibrium paths by a macroscopic deformation, taking into account the effects of micro-cracks and of frictionless self-contact between crack faces. Innovative upper and lower bounds to primary instability and bifurcation loads are also formulated, by introducing linear comparison problems corresponding to bonding or free to penetrate incremental kinematical conditions at the current crack contact interface. These theoretical developments, obtained in full generality for incrementally linear materials and also useful for non-periodic RVEs, are then numerically illustrated by using a finite element approach with reference to a 2D hyperelastic model of a continuously fiber reinforced composite with interface debonding. Numerical results confirm the validity of the proposed formulation and show the notable influence of self-contact and debonding phenomena on instability and bifurcation loads, together with size dependence effects and consequent loss of periodicity during the unstable regime.