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Incompatibility and a simple gradient theory of plasticity. (English) Zbl 1030.74015

The author develops a nonlinear formulations in the setting of small strains. The displacement gradient is decomposed additively into elastic and plastic parts which are not compatible, i.e. they cannot be represented as a gradient of a continuous differentiable vector field. In the extension of the local \(J_{2}\)-flow theory, the measure of incompatibility is chosen to be the skewsymmetric part of the gradient of plastic strain tensor. A nonlocal hardening yield conditon of von Mises type describes the plastic behavior of material. The nonlocal hardeness is taken to be dependent on effective plastic strain and on the second invariant of incompatibility. The proposed hardening function generates an increase in the hardening rate due to incompatibility. As an example, the author examines the torsion of a thin cylindrical wire, and for an appropriate set of material parameters the predictions of the model reasonably reproduce known in the literature results. In the small strain formulation for crystal plasticity, the plastic part of displacement gradient is described by slip mechanism, and the measure of incompatibility is taken to be Nye’s dislocation density. The hardening activation Schmid condition involves a rate of hardening with instantaneous slip-system matrix dependent on the slip and on the Nye’s dislocation density. As an application, a single slip problem is considered in two dimensions. The author concludes that the hardening relations adopted in the local formulation account resonably for the effect of incompatibility on overall elastic plastic response.

MSC:

74C20 Large-strain, rate-dependent theories of plasticity
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
74S05 Finite element methods applied to problems in solid mechanics
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