Well-posedness of a one-dimensional nonlinear kinematic hardening model

Abstract

We investigate the quasistatic evolution of a one-dimensional elastoplastic body at small strains. The model includes general nonlinear kinematic hardening but no nonlocal compactifying term. Correspondingly, the free energy of the medium is local but nonquadratic. We prove that the quasistatic evolution problem admits a unique strong solution.

A note about hardening-free viscoelastic models in Maxwellian-type rheologies at large strains

“…The definition of weak solutions follows directly from (13). It can be recovered by formally testing ( 14) by smooth functions and use Green formulas (see ( 2)) together with the surface Green formula in (3), the boundary conditions in (16), and multiple integration by parts in time, keeping in account the initial conditions in (17). Altogether, we arrive at the following definition.…”

“…Within the mathematical purview, there is a general agreement that the rigorous analysis of large-strain inelastic time-evolving phenomena requires higher-order regularization of inelastic strains [7, 9–15]. Existence theories without gradient regularization are available only in one space dimension [16], at an incremental level [17–19], or under stringent modeling restrictions [6, 17]. In the engineering literature, conversely, gradient theories at large strains are seldom considered (see [1–3, 20, 21]) and the existence of solutions is not in focus.…”

A note about hardening-free viscoelastic models in Maxwellian-type rheologies at large strains

“…Within the mathematical purview, there is a general agreement that the rigorous analysis of large-strain inelastic time-evolving phenomena requires higher-order regularizations of the inelastic strains [8,13,24,29,[32][33][34]44]. Existence theories without gradient regularization are available only in one space dimension [27], at the incremental level [30,31,47], or under stringent modeling restrictions [20,30]. In the engineering literature, on the other hand, gradient theories at large strains are seldom considered, see [3,10,37], [18,Ch.…”