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) is the elastic elongation and pi(t) is the relaxed length of spring i, defined on [t0, ∞) 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 elastic component t↦e(t) always converges to a T-periodic function as t →∞. 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 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

Exponential stability of positive semigroups in Banach spaces

The paper establishes a link between the stability of the semigroup e^(−Γ+M)t and the spectral radius of Γ^(−1)M in ordered Banach spaces. On the one hand our result allows utilizing simple estimates for the eigenvalues of −Γ +M in order to provide general conditions for the convergence of the successive approximation scheme for semilinear operator equations. On the other hand, this paper helps examining the stability of the semigroup e^(−Γ+M)t for those classes of matrices −Γand M, which lead to observable expressions for Γ(−1)M, e.g. when M is a coupling applied to disjoint systems representing Γ. The novelty of the paper is in the development of an infinite-dimensional framework, where an absolute value function induced by a cone is introduced and a way to deal with the lack of global continuity of eigenvalues is presente

Formation of a nontrivial finite-time stable attractor in a class of polyhedral sweeping processes with periodic input

We consider a differential inclusion known as a polyhedral sweeping process. The general sweeping process was introduced by J.-J. Moreau as a modeling framework for quasistatic deformations of elastoplastic bodies, and a polyhedral sweeping process is typically used to model stresses in a network of elastoplastic springs. Krejčí’s theorem states that a sweeping process with periodic input has a global attractor which consists of periodic solutions, and all such periodic solutions follow the same trajectory up to a parallel translation. We show that in the case of polyhedral sweeping process with periodic input the attractor has to be a convex polyhedron χ of a fixed shape. We provide examples of elastoplastic spring models leading to structurally stable situations where χ is a one- or two- dimensional polyhedron. In general, an attractor of a polyhedral sweeping process may be either exponentially stable or finite-time stable and the main result of the paper consists of sufficient conditions for finite-time stability of the attractor, with upper estimates for the settling time. The results have implications for the shakedown theory

One-period stability analysis of polygonal sweeping processes with application to an elastoplastic model

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