Homogenization of rate-independent elastoplastic spring network models with non-local random fields

Abstract

We investigate the time-evolution of elastoplastic materials reinforced by randomly distributed long-range interactions. Starting from a rate-independent system on a discrete spring lattice that combines local linearized elasticity, gradient-regularized plasticity and stochastic non-local links modeling stiff fibers, we establish a discrete-to-continuum limit in the energetic formulation. We prove that as the lattice spacing tends to zero, an evolutionary solution of the discrete system converges to the unique energetic solution of a continuum limit problem. The limiting continuum model couples classical elastoplasticity with a non-local energy featuring fractional-order interactions that capture the homogenized influence of random long-range reinforcements. These results extend previous static homogenization studies by rigorously treating path-dependent dissipation and showing existence, uniqueness and Lipschitz continuity of the evolving solutions. The work therefore provides a mathematical foundation for simulating time-dependent mechanical response of fiber-reinforced composites with random architecture.