Approaches to description of damageable lattices dynamics

This paper discusses approaches to a theory of micro- and mesoscale behavior of damageable structures based on a concept of structure with reserved capacities. We suggest methods of analysis of nonmonotonic structural response using geometric characteristics of lattices. The proper lattice geometry provides reserved capacities and guarantees the nonmonotonic response that lead to damage spread, controllable waves of damage, and increase of lattice resistivity. Damage is viewed as cascades of phase transitions that include stiffness decrease at a given equilibrium, elimination of a part of a structure, and transition between equilibria. Criteria for the transitions are established. In lattices, damage propagation is described by a random walk model. Simulations confirm that without a proper randomization, the numerical predicted evolution of damage is symmetric and unrealistic. The paper discusses a continuum limit of lattice dynamics by introducing a lattice damage tensor (LDT) and a damage evolution: A system of thermodynamic equations for decaying material that couples elastic variables, temperature, and damage tensor.