Numerical approximation of incremental infinitesimal gradient plasticity

Abstract

We investigate a representative model of continuum infinitesimal gradient plasticity. The formulation is an extension of classical rate‐independent infinitesimal plasticity based on the additive decomposition of the symmetric strain tensor into elastic and plastic parts. It is assumed that dislocation processes contribute to the storage of energy in the material whereby the curl of the plastic distortion appears in the thermodynamic potential and leads to an additional nonlocal backstress tensor. The formulation is cast into a numerical framework by a saddle point approximation of the corresponding minimization problem in each incremental loading step. This allows one to reformulate the (nonlocal) dissipation inequality to a point‐wise flow rule and yields a solution scheme, which is a direct extension of the standard approach in classical plasticity. Our numerical results show the regularizing effects of the additional physically motivated terms. Copyright © 2008 John Wiley & Sons, Ltd.

Torsion in Strain-Gradient Plasticity: Energetic Scale Effects

“…skw , the so-called plastic spin, enters the theory [20] (see also [26,27,29] for a related gradient plasticity model dealing with both infinitesimal and finite strains). As pointed out in [21], the ensuing flow rule is then much more complicated; not surprisingly, well-posedness has not been established for such model, unless one includes appropriate hardening terms [10], or restricts attention to particular symmetries [6].…”

A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing

“…We finish with numerical experiments (in Section 7) for the evaluation of the condition number of the parallel multigrid preconditioner in dependence on the block size and on damping factors in the smoothing scheme. This parallel programming model is designed for maximal flexibility, so that it can easily be applied to many other applications: it is already successfully applied to a geomechanical problem [32] via an element-based interface to an engineering model, to a problem in biomechanics [35], to a triphasic porous media model [33], to infinitesimal plasticity [30,31], to Cosserat plasticity [24], to nonlocal plasticity [25], and to periodic eigenvalue computations of photonic crystals [20].…”

A Primal-Dual Finite Element Approximation for a Nonlocal Model in Plasticity

“…More precisely, we study a subclass of gradient plasticity, where the corresponding elasto-plastic energy includes only the curl of the plastic strain, see, e.g., [7,8,9,13,14,26,27] for the analytical, mechanical, and physical properties of such models. In [17,18] it is shown that a representative nonlocal model of such type can also be transformed into a variational inequality.The nite element analysis for variational inequalities in plasticity with kinematic hardening yields a priori estimates involving the best approximation error and additional nonconformity terms, see [1,3]. Here, we extend these results to our non-local formulation and work with curl-conforming nite elements for the plastic variable.…”

“…More precisely, we study a subclass of gradient plasticity, where the corresponding elasto-plastic energy includes only the curl of the plastic strain, see, e.g., [7,8,9,13,14,26,27] for the analytical, mechanical, and physical properties of such models. In [17,18] it is shown that a representative nonlocal model of such type can also be transformed into a variational inequality.…”

“…The higher regularity of β implies an additional regularity on p, i.e., curl curl p ∈ L 2 (Ω, R 3×3 ) and β = H 0 p + L 0 curl curl p, see also [7,17,18], and thus the standard innitesimal model of kinematic hardening is recovered in the limit L 0 tends to zero. 3.…”