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 inc(ε p ) := Curl [(Curl ε p ) T ], where ε 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 which is quadratic in the tensor inc(ε p ) and it contains isotropic hardening based on the rate of the plastic strain tensorε 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 distortionṗ to their symmetric counterpartε p = symṗ. 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. Appendix 322 Aifantis writes in [4, p. 218]: "... In conformity with established results -that the plastic strain rateεp is a state variable, rather than the strain εp itself."3 The plastic spin in the finite deformation flow theory of plasticity is defined as the skew-symmetric part of the so-called plastic distortion rate i.e., Wp := skew(ḞpF −1 p ) , while its counter-part in the small strain theory is simply skew(ṗ) , where p is the non-symmetric infinitesimal plastic distortion.

A canonical rate-independent model of geometrically linear isotropic gradient plasticity with isotropic hardening and plastic spin accounting for the Burgers vector

“…Our development clearly shows that claims such as in [66] are unfounded and seem to indicate that there are different notions involved of what isotropy precisely means. This subject is also discussed further in [32].…”

“…• it contains only properly defined state-variables ( [113,32]). In this context, notice that, as mentioned in De Wit [24, p.1478]: ".…”

“…the plastic strain [sym p] is not a state quantity, i.e., it cannot be determined from the [current] state of the body." Through a proper definition of infinitesimal state-variables, this will be clearly presented in [32].…”

“…transformations of the reference system, i.e., in the linearized context it is invariant under ∇u(x) −→ ∇u(x) + ∇ϑ p (x) ∀ϑ p ∈ C 2 (R 3 , R 3 ) p(x) −→ p(x) + ∇ϑ p (x) , which is also known as translational T(3)-gauge invariance ( [68,69,70,32]);…”

Nečas-Lions lemma revisited: An $L^p$-version of the generalized Korn inequality for incompatible tensor fields

“…where f ∈ L 2 (Ω, R 3 ) describes the body force, see e.g. [33,34,35,36,37,52,71,79,80,81,87]. Here, sym e 2 represents the elastic energy, sym P 2 induces linear hardening and Curl P is known as the dislocation density tensor.…”