Vol. 8, No. 1, 2020

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Second-order work criterion and divergence criterion: a full equivalence for kinematically constrained systems

Jean Lerbet, Noël Challamel, François Nicot and Felix Darve

Vol. 8 (2020), No. 1, 1–28
Abstract

This paper presents stability results for rate-independent mechanical systems, associated with general tangent stiffness matrices including symmetric and nonsymmetric ones. Conservative and nonconservative as well as associate and nonassociate elastoplastic systems are concerned by such a theoretical study. Hill’s stability criterion, also called the second-order work criterion, is here revisited in terms of kinematically constrained systems. For piecewise rate-independent mechanical systems (which may cover inelastic and elastic evolution processes), such a criterion is also a divergence Lyapunov stability criterion for any kinematic autonomous constraints. This result is here extended for systems with nonsymmetric tangent matrices. By virtue of a new type of variational formulation on the possible kinematic constraints, and thanks to the concept of kinematical structural stability (KISS), both criteria, Hill’s stability criterion and the divergence stability criterion under kinematic constraints, are shown to be equivalent.

Keywords
Hill's criterion, kinematic structural stability, nonconservative systems, nonassociate materials
Physics and Astronomy Classification Scheme 2010
Primary: 46.32.+x
Mathematical Subject Classification 2010
Primary: 34D30, 37C20
Milestones
Received: 4 July 2019
Revised: 6 November 2019
Accepted: 7 January 2020
Published: 11 May 2020

Communicated by Angelo Luongo
Authors
Jean Lerbet
Université Paris-Saclay
CNRS
Université d’Evry
Laboratoire de Mathématiques et Modélisation d’Evry
Évry
France
Noël Challamel
Institut de Recherche Dupuy de Lôme
UMR CNRS 6027
Université de Bretagne Sud
Lorient
France
François Nicot
INRAE
Université Grenoble Alpes
Unité de Recherche ETNA
Domaine Universitaire
Saint-Martin-d’Hères
France
Felix Darve
Laboratoire 3SR
UMR CNRS 5521
Université Grenoble Alpes
Institut Polytechnique de Grenoble
Grenoble
France