Finite thermoelastoplasticity and creep under small elastic strains

Abstract

A mathematical model for an elastoplastic continuum subject to large strains is presented. The inelastic response is modeled within the frame of rate-dependent gradient plasticity for nonsimple materials. Heat diffuses through the continuum by the Fourier law in the actual deformed configuration. Inertia makes the nonlinear problem hyperbolic. The modelling assumption of small elastic Green-Lagrange strains is combined in a thermodynamically consistent way with the possibly large displacements and large plastic strain. The model is amenable to a rigorous mathematical analysis. The existence of suitably defined weak solutions and a convergence result for Galerkin approximations is proved.

Quasistatic hypoplasticity at large strains Eulerian

“…that φ(det F p ) not only blows up to ∞ if det F p → 0+ but is it very large if det F p = 1; cf. [10,26,36,49] for a Lagrangian formulation. One advantage of the rate-formulation in Section 3 below will be that we can involve the isochoric constraint det F p = 1 into the model exactly and thus will be able to eliminate φ from the model completely.…”

“…A highly applicable assertion was originally devised for situations when F = ∇y with y ∈ W 2,p (Ω; R d ) but actually it holds in more general situations, as used also in [26,36,37,49]: Lemma 4.1 (T.J. Healey and S. Krömer [20]). Let κ > rd/(r−d) for some r > d. Then, for any C < +∞, there is ǫ > 0 such that, for any F ∈ W 1,r (Ω; R d×d ) with det F > 0 a.e.…”

“…Any reference configuration (i.e. the Lagrangian approach as in [10,26,32,36,49]) is thus eliminated from the formulation of the problem. This is very natural especially for materials where such a reference configuration cannot be identified naturally, as e.g.…”

Fatigue and phase transition in an oscillating elastoplastic beam

“…We remark that our viewpoint is different from the contributions already present in the literature: on one hand we focus on the dynamics (compared, for instance, with [18,26,30] that go into the direction of rateindependent damage processes in nonlinear elasticity and/or thermo-elasto-plasticity, see also [8,28,29]) and also because the approach used in other papers, that interpret damage processes as a kind of phase transition in the material (see for instance [1][2][3]27]), is based on the idea that damage processes are driven by large deformations. Moreover, the multiyield character emerging from our bending problem does not seem to have been taken into consideration before, neither in the multidimensional setting: see for instance [5,7,21,24].…”

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].…”