Sweeping process approach to stress analysis in elastoplastic lattice spring models with applications to network materials

Disordered network materials abound in both nature and synthetic situations while rigorous analysis of their nonlinear mechanical behaviors remains challenging. The purpose of this paper is to connect the mathematical framework of the sweeping process originally proposed by Moreau to the generic class of lattice spring models that incorporate plasticity. We derive the equations of quasistatic evolution of an elastic-perfectly plastic lattice and relate them to concepts from rigidity theory and structural mechanics. Then we explicitly construct a sweeping process and provide numerical schemes to find the evolution of stresses in the model. In particular, we develop a highly efficient "leapfrog" computational framework that allows us to rigorously track the progression of plastic events in the system based on the sweeping process theory. The utility of our framework is demonstrated by analyzing the elastoplastic stresses in a novel class of disordered network materials exhibiting the property of hyperuniformity, in which the (normalized) infinite-wavelength density fluctuations associated with the distribution of network nodes are completely suppressed. We find enhanced mechanical properties such as increasing stiffness, yield strength, and tensile strength as the degree of hyperuniformity of the material system increases. Our results have implications for optimal network material design and our event-based framework can be readily generalized to nonlinear stress analysis of other heterogeneous material systems.