On the Stability of Moreau’s Sweeping Process With Applications to Networks of Elastoplastic Springs

The dissertation is devoted to the stability properties of Moreau’s sweeping processes and develops an analytic framework to study those stability properties in the context of onedimensional networks of elastoplastic springs governed by quasistatic evolution under periodic loading. It is known that trajectories of a sweeping process converge to a periodic regime, but the limiting periodic trajectory may not be unique. We study the properties of the set of periodic trajectories, which we call attractor, and provide conditions for the uniqueness of the limiting trajectory. As an application, we rigorously define the problem of quasistatic evolution of networks of elastoplastic springs and convert it into a Moreau’s sweeping process with a moving polyhedron. The study of attractors allows us to spot a class of sweeping processes and closed-form estimates on eligible loadings where the limiting elastic stress does not depend on the initial condition and therefore all the trajectories approach the same T-periodic solution. This fact represents a discrete analogue of the Frederick-Armstrong theorem. We also investigate the structural stability of the attractor with respect to the change of parameters of a network of springs. We offer specific examples for the presented theoretical results.