European Journal of Mechanics A/Solids 24 (2005) 572–592 Study of a rheological model with a friction term and a cubic term: deterministic and stochastic cases Claude-Henri Lamarque a,∗ , Frédéric Bernardin a,b , Jérôme Bastien c a URA 1652 CNRS, Département Génie Civil et Bâtiment, Laboratoire Géomatériaux, École Nationale des Travaux Publics de l’Etat, rue Maurice Audin, 69518 Vaulx-en-Velin cedex, France b UMR 5585 CNRS, MAPLY, Laboratoire de mathématiques appliquées de Lyon, Université Claude Bernard Lyon I, 69622 Villeurbanne cedex, France c Laboratoire Mécatronique 3M, Equipe d’accueil A 3318, Université de Technologie de Belfort-Montbéliard, 90010 Belfort cedex, France Received 2 April 2004; accepted 4 May 2005 Available online 28 June 2005 Abstract First, results on existence of solutions and their numerical approximations are given for the studied models. Some results of the identification of hysteretic discrete mechanical systems with damping submitted to deterministic or stochastic forcing are given. The identification is obtained thanks to hysteresis cycles which are convex or nonconvex.  2005 Elsevier SAS. All rights reserved. Keywords: Identification; Discrete mechanical systems; Friction; Damping; Stochastic excitation 1. Introduction In previous works, rheological models including friction terms have been studied: first, well/ill-posed problems have been addressed in the spirit of Coulomb type models (Schatzman et al., 1999). The models with one degree of freedom or n degrees of freedom based on combinations in parallel or series of elementary constitutive elements (linear springs, dashpots, Saint-Venant elements) have been considered (Bastien et al., 2000). These models were described by using differential inclusions. Existence and uniqueness results based on known mathematical results of Brézis (1973) have been established and special implicit Euler numerical scheme of order 1 has been presented. Extensions to a one mass model involving an infinite number of degrees of freedom via infinite number of Saint-Venant elements have been considered from the point of view of mechanical modelling, mathematics, and numerics (Bastien et al., 2002). In paper Bastien et al. (2000), identification process from the knowledge of half of an hysteretic cycle has been given, for systems without damping. These works consider only linear springs: they provide the simplest Lipschitz continuous functions from the mathematical point of view. Clearly they do not take into account damping in the identification process. In these papers the external forces have been assumed to be deterministic ones. The aim of this paper is to extend these results for modelling and identification to cases where damping, nonlinear springs and stochastic excitation are taken into account. For the sake of simplicity, extension is presented for one degree of freedom systems. * Corresponding author. E-mail address: lamarque@entpe.fr (C.-H. Lamarque). 0997-7538/$ – see front matter  2005 Elsevier SAS. All rights reserved. doi:10.1016/j.euromechsol.2005.05.001