Statistical damage theory of 2D lattices: Energetics and physical foundations of damage parameter

The paper presents an in-depth analysis of two-dimensional disordered lattices of statistical damage mechanics for the study of quasi-brittle materials. The strain energy variation in correspondence to damage formation is thoroughly examined and all the different contributions to the net energy changes are identified and analyzed separately. We demonstrate that the introduction of a new defect in the microstructure produces a perturbation of the microscopic random fields according to a principle of maximum energy dissipation. A redistribution parameter η is introduced to measure the load redistribution capability of the microstructure. This parameter can be estimated from simulation data of detailed models. This energetic framework sets the stage for the investigation of the statistical foundations of the damage parameter as well as the damage localization. Logical statistical arguments are developed to derive two analytical models (a maximum field model and a mean field one) for the estimate of the damage parameter via a bottom-up approach that relates the applied load to the microstructural disorder. Simulation data provided input to the statistical models as well as the means of validation. Simulated tensile tests of honeycomb lattices with mechanical disorder demonstrate that long-range interactions amongst sets of microcracks with different orientations play a fundamental role already in damage nucleation as well as in the homogeneous-heterogeneous transition. A functional "hierarchy of sets" of grain boundaries, based on their orientation in relation to the applied load, seems to emerge from this study. Results put in evidence the ability of discrete models of capturing seamlessly the damage anisotropy. The ideas exposed inhere should be useful to develop a full rational model for disordered lattices and, later, to extend the approach to discrete models with solid elements. The findings suggest that statistical damage mechanics might aid in the quest of reliable and physically sound constitutive relations of damage, even in synergy with micromechanics.