Theory and numerics of geometrically non-linear gradient plasticity

Abstract

This work presents the theory and the numerics of a thermodynamically consistent formulation of geometrically non-linear gradient plasticity. Due to the lack of the classical local continuum formulation to produce physically meaningful and numerically converging results within localization computations, a thermodynamically motivated gradient plasticity formulation is envisioned. Especially within the framework of crystal plasticity we resort to physically motivated arguments in terms of geometrically necessary dislocations densities that imply the incorporation of higher gradients. In a first simplified approach presented here we adopt the gradient of the internal hardening variable as a provision for geometrically necessary dislocations. We start from a thermodynamic formulation within a geometrically non-linear setting including the additional contribution of the gradient of the internal history variable. This introduces e.g. the vectorial hardening flux and the quasi-nonlocal drag stress. At the numerical side, besides the balance of linear momentum, the algorithmic consistency condition has to be solved in weak form. Thereby, the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non-linear programming. Finally, some demonstrative numerical examples complement the presentation.

Introduction

The objectives of the present work are the theory and in particular the appropriate numerics of a thermodynamically consistent geometrically non-linear gradient plasticity formulation. In particular we emphasize the physical motivation of higher gradients in terms of geometrically necessary dislocation densities. To this end we develop a general continuum mechanical framework of a dislocation based gradient model within multiplicative plasticity. In a simplified model of a single crystal with only one active slip system we later adopt the internal hardening variable as a substitute for the geometrically necessary dislocation density. We then derive and apply the appropriate numerics for the proposed prototype model.

Recently much attention has been focused on the physical motivation of the incorporation of higher gradients in plasticity. In addition to the meanwhile well-established regularizing effect w.r.t. discretization dependencies in standard numerical computations of localized plastic deformations in softening materials, higher gradients have been motivated by dislocations in crystal plasticity, see e.g. [1] and for an alternative framework [2]. In particular, higher gradients can be physically justified for single and polycrystalline material like metals, if one considers the dislocation density and incompatibility, respectively, see e.g. [3]. An alternative recent approach by Gurtin [4] deals as well with geometrically necessary dislocations, which are incorporated in a gradient theory of single crystal plasticity.

Several experimental evidences for microstructural interaction w.r.t. plastic deformation phenomena can be found in the literature, e.g. the Hall–Petch effect [5], [6], dislocation related hardening effects, see [7] or grain size effects in the works by Ashby [8]. In particular special dedication to various profound aspects of dislocations w.r.t. crystallographic characteristics can be found in an article by Amelinckx [9]. Furthermore torsion tests on copper wires were investigated by Fleck et al. [10], thereby scale effects motivate a particular strain gradient plasticity formulation as advocated by Fleck and Hutchinson [11].

Whenever the influence of material substructure cannot be neglected multifield theories enter the stage, see e.g. [12] and for a recent contribution [13]. Thereby one has to distinguish on the one hand theories that introduce extra observable fields resulting in couple stresses satisfying additional balance equations, e.g. the micropolar theory of the Cosserat brothers [14]. On the other hand, the introduction of internal variables and their gradients reflects the microstructural response. For a comprehensive study of the thermodynamics with internal variables we refer to Maugin [15]. Moreover in order to include nonlocal effects, apart from the here advocated particular gradient theory nonlocal integral models are a possibility, see e.g. [16].

For gradient continua, a variety of numerical strategies, different from the one proposed in this work, were investigated e.g. by Sluys et al. [17], Pamin [18], de Borst and Pamin [19], Peerlings et al. [20], [21], Mikkelsen [22], Steinmann [23], Comi [24], Chambon et al. [25], Svedberg and Runesson [26], Svedberg [27] and Nedjar [28].

For the local continuum description an early attempt to set up a mixed finite element formulation was provided by Pinsky [29], whereby a plastic strain-like variable is discretized in addition to the displacement field. An alternative proposal based on a complementary mixed finite element formulation is due to Simo et al. [30], wherein the flow rule is enforced in a weak sense at the element level. Likewise, a two-field finite element formulation for elasticity coupled to damage was proposed by Florez-Lopez et al. [31].

Our own recent attempts to the numerics of gradient plasticity at small strains are documented in [32].