## Stabilization of the response of cyclically loaded lattice spring models with plasticity.(English)Zbl 07369299

Summary: 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 $$t \mapsto (e(t), p(t))$$, where $$e_i(t)$$ is the elastic elongation and $$p_i(t)$$ is the relaxed length of spring $$i$$, defined on $$[t_0, \infty )$$ 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 P. Krejčí [Hysteresis, convexity and dissipation in hyperbolic equations. Tokyo: Gakkotosho (1996; Zbl 1187.35003)]) that the elastic component $$t \mapsto e(t)$$ always converges to a $$T$$-periodic function as $$t \rightarrow \infty$$. The achievement of this paper is in spotting a class of loadings where the Krejci’s limit does not 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.

### MSC:

 74H55 Stability of dynamical problems in solid mechanics 74M05 Control, switches and devices (“smart materials”) in solid mechanics 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) 93C15 Control/observation systems governed by ordinary differential equations

Zbl 1187.35003
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