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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

Citations:

Zbl 1187.35003
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