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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 ) is the elastic elongation and p i ( t ) is the relaxed length 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 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 ( 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. 

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.