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.
Networks of elastoplastic springs (elastoplastic systems) have been linked to differential equations with polyhedral constraints in the pioneering paper by Moreau (1974). Periodic loading of an elastoplastic system, therefore, corresponds to a periodic motion of the polyhedral constraint. According to Krejci (1996), every solution of a sweeping process with a periodically moving constraint asymptotically converges to a periodic orbit. Understanding whether such an asymptotic periodic orbit is unique or there can be an entire family of asymptotic periodic orbits (that form a periodic attractor) has been an open problem since then. Since suitable small perturbation of a polyhedral constraint seems to be always capable to destroy a potential family of periodic orbits, it is expected that none of potential periodic attractor is structurally stable. In the present paper we give a simple example to prove that even though the periodic attractor (of nonstationary periodic solutions) can be destroyed by little perturbation of the moving constraint, the periodic attractor resists perturbations of the physical parameters of the mechanical model (i.e. the parameters of the network of elastoplastic springs).
We offer a finite-time stability result for Moreau sweeping processes on the plane with periodically moving polyhedron. The result is used to establish the convergence of stress evolution of a simple network of elastoplastic springs to a unique cyclic response in just one cycle of the external displacement-controlled cyclic loading. The paper concludes with an example showing that smoothing the vertices of the polyhedron makes finite-time stability impossible.
This paper develops an analytic framework to design both stresscontrolled 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 [t 0 , ∞) by the initial condition (e(t 0 ), p(t 0 )). 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(t 0 ), p(t 0 )) and so all the trajectories approach the same T -periodic regime. The proposed class of sweeping processes is the one for which the normals 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 normals 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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.