A fourth-order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor

Abstract

In this paper we derive a novel fourth-order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner’s incompatibility tensor $\mathrm{inc}\left({\epsilon }_p\right):=\mathrm{Curl}\left[{\left(\mathrm{Curl}{\epsilon }_p\right)}^{\mathrm{T}}\right]$ , where ${\epsilon }_p$ is the symmetric plastic strain tensor. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution that is quadratic in the tensor $\mathrm{inc}\left({\epsilon }_p\right)$ and it contains isotropic hardening based on the rate of the plastic strain tensor ${\stackrel{·}{\epsilon }}_p$ . We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric plastic distortion $\stackrel{·}{p}$ to their symmetric counterpart ${\stackrel{·}{\epsilon }}_p=\mathrm{sym}\stackrel{·}{p}$ . In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings.