Finite-time stability of elastoplastic lattice spring systems under simultaneous displacement-controlled and stress-controlled loadings

According to Moreau (C.I.M.E. 1973), plastically deforming springs of lattice spring systems with a time-dependent displacement-controlled loading correspond to the attractor of a differential inclusion with a moving constraint of the form $ C(t) = C+c(t) $, where $ C $ is a polytope and $ c(t) $ is a time-dependent vector. Finite-time stability of differential inclusions of this type is established in Gudoshnikov et al. [SIAM J. Control Optim. 60 (2022)]. The work by Moreau also implies that accounting for a stress-controlled loading no longer allows to split $ C(t) $ as $ C+c(t) $. In the present paper we show that if we are interested in attractivity of a particular vertex of $ C(t) $, then $ C(t) $ can again be viewed as $ C+c(t) $ for a specially constructed $ c(t) $ (which depends on the vertex of interest), so that the technique of Gudoshnikov et al. can be used to obtain a criterion for finite-time stability of the vertex. The criterion obtained is illustrated with a benchmark example where we discover a drastic increase of diversity of possible combinations of plastically deforming springs when stress-controlled loading is introduced on top of displacement-controlled loading compared to the case where displacement-controlled loading is the only forcing.