Finite strain HFGMC analysis of damage evolution in nonlinear periodic composite materials

This work studies the evolution of damage in periodic composites with hyperelastic constituents prone to mechanical degradation under sufficient loading. The micromechanical problem is solved for quasistatic far-field loading for plane-strain conditions, using the finite strain high-fidelity general method of cells (FSHFGMC) approach to discretize the conservation equations. Damage is treated as degradation of material cohesion, modeled by a material conservation law with a stress-dependent damage-source (sink) term. The two-way coupled formulation with the internal variable representing damage is reminiscent of the phase-field approach to gradual cracks growth, albeit with a mechanistically derived governing equation, and with important theoretical differences in consequences. The HFGMC approach consists in enforcing equilibrium in each phase (in the cell-average sense) by stress linearization, using instantaneous tangent moduli, and subsequent iterative enforcement of continuity conditions, a formulation arguably natural for composite materials. The inherent stiffness of the underlying differential equations is treated by use of a predictor-corrector scheme. Various examples are solved, including those of porous material developing cracks close to the cavity, for various sizes and shapes of the cavity, damage in a two-phase composite of both periodic and random structure, etc. The proposed methodology is physically tractable and numerically robust and allows various generalizations.