A fourth order gauge-invariant gradient plasticity model for polycrystals based on Kr\"oner's incompatibility tensor

In this paper we derive a novel fourth order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kr\"oner's incompatibility tensor $inc(\epsilon_p):= Curl[(Curl \epsilon_p)^T]$, where $\epsilon_p=sym p$ is the symmetric infinitesimal plastic strain tensor and $p$ is the (non-symmetric) infinitesimal plastic distortion. 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 which is quadratic in the tensor $inc(\epsilon_p)$ and it contains isotropic hardening based on the rate of the symmetric infinitesimal plastic strain tensor $\dot{\epsilon_p}$. 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 infinitesimal plastic distortion $\dot{p}$ to their symmetric counterpart $\dot{\epsilon_p}$. 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.