Eulerian finite-deformation extension of Gurtin's distortion gradient plasticity: formulation, H(curl) finite element implementation, and applications

Andrea Panteghini, Lorenzo Bardella

Department of Civil, Environmental, Architectural Engineering and Mathematics, Università degli Studi di Brescia, Via Branze, 43, 25123, Brescia, Italy

M.B. Rubin

Faculty of Mechanical Engineering, Technion-Israel Institute of Technology, 32000 Haifa, Israel

Abstract

This work proposes an Eulerian finite-deformation formulation for phenomenological higher-order distortion gradient plasticity (DGP) as a basis to model the size-dependent behavior of metals when geometrically necessary dislocations (GNDs) play a role. Here, the adjective phenomenological refers to disregarding the crystal lattice and DGP recalls that the theory constitutively prescribes the plastic spin. The developed DGP extends to finite deformations the pivotal theory of Gurtin (2004, "A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin" J. Mech. Phys. Solids 52, 2545-2568), which relies on the Nye-Kröner dislocation density tensor. In this work, the extension of such a tensor to finite deformations is denoted as the Nye-Kröner-like tensor, alpha, to emphasize the alternatives discussed in the literature for its definition. The main pillars of the proposed Eulerian theory are two evolution equations: one for alpha and one for the elastic distortional deformation tensor. The source term in the evolution equation for alpha is the curl, operating in the current configuration, of the plastic rate tensor, which, for isochoric plastic flow, is a deviatoric non-symmetric tensor. Also, this evolution equation extends that based on the Jaumann rate by linearly combining the total spin rate and the plastic spin rate; such a combination is governed by a model parameter whose effect is assessed by resorting to numerical results. The evolution equation for the elastic distortional deformation tensor is discussed on the basis of different choices for the second-order tensor governing the contribution due to the rate of plasticity. This choice is important because it has an impact both on the mechanical response and on the complexity of the numerical time-integration. Additionally, the results of the theory are discussed in the light of two different constitutive choices for the nonlinear measure of the deviatoric elastic strain, one of them being Hencky's logarithmic strain. The higher-order contribution in the free-energy density is simply assumed to be quadratic in alpha, which is adequate for the purposes of this investigation, aiming to present the essential features of the proposed finite-deformation framework; in future studies, more sophisticated models will be combined with the theory here developed to predict actual size effects caused by GNDs. The exclusive use of alpha as a higher-order primal field requires a H(curl) finite element (FE) formulation, which is consistent with the weakest continuity requirements on the plastic rate tensor. A fully implicit FE algorithm is provided for plane strain. Its performance and results are evaluated on the basis of two large-deformation benchmark problems: the shearing of a strip under monotonic and cyclic loadings, allowing verification against analytical estimates, and the necking of a plate, highlighting the role of the internal length in controlling strain localization and in governing size-dependent response. Overall, it is established that the proposed evolution equation for alpha yields results that are free from spurious oscillations. When rotations are important in the boundary value problem at hand, the obtained results suggest that the spin rate governing the evolution equation for alpha should be the elastic spin rate. For small elastic deformations, which is typical of metals, the predicted mechanical response is essentially insensitive to the adopted nonlinear elastic strain measure, so that computational efficiency may guide this choice. The necking problem shows that the notion of size dependence should compare the material length to the size of the region in which strain gradients are distributed (e.g. localized regions) and not to structural dimensions.

Author Keywords: small-scale metal plasticity; strengthening and mechanisms; Nye-Kröner tensor; Finite deformation; Eulerian formulation; H(curl) finite element