Stabilization of the response of cyclically loaded lattice spring models with plasticity

TL;DR

This paper presents an analytic framework for designing loadings that ensure the convergence of elastoplastic spring networks to a unique periodic response, extending the understanding of stability in such systems.

Contribution

It identifies a class of loadings under which the elastoplastic response converges to a unique periodic regime, independent of initial conditions, based on geometric conditions of the system.

Findings

Normal vectors of polyhedron facets are linearly independent under certain load conditions.

Convergence to a periodic regime occurs when the number of displacement loadings is two less than the number of nodes.

Large stress-controlled loads ensure the geometric condition for convergence.

Abstract

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 e_i(t) and p_i(t) are the elastic and plastic deformations of spring i, defined on [t0,\infty) by 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 solution t->(e(t),p(t)) always converges to a T-periodic function. The achievement of this 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. We discover that the normal vectors of any d different facets of the moving polyhedron C(t) are linearly independent, if the number of displacement-controlled loadings is two less the number of nodes of the given network of springs and when the magnitude of the stress-controlled loading is sufficiently large (but admissible). The result can be viewed as an analogue of the high-gain control method for elastoplastic systems. In continuum theory of plasticity, the respective result is known as Frederick-Armstrong theorem.