Stabilization of the response of cyclically loaded lattice spring models with plasticity
This paper develops an analytic framework to design both stress-controlled and displacement-controlled T -periodic loadings which make the quasistatic evolution of a one-dimensional network of elastoplastic springs converging to a unique periodic regime. The solution of such an evolution problem is a function t->(e(t),p(t)), where ei(t) and pi(t) are the elastic and plastic deformations of spring i, that satisfies the initial condition (e(t0),p(t0)). After we rigorously convert the problem into a Moreau sweeping process with a moving polyhedron C(t) in a vector space E of dimension d, it becomes natural to expect (based on a result by Krejci) that the elastic component t->e(t) always converges to a T-periodic function. The achievement of the present paper is in spotting a class of loadings where the Krejci's limit doesn't depend on the initial condition (e(t0),p(t0)) and so all the trajectories approach the same T-periodic regime. The proposed class of sweeping processes is the one for which the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent. We further link this geometric condition to mechanical properties of the given network of springs. In this way we obtain an analogue of the Frederick-Armstrong theorem from continuum mechanics.