A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations

Abstract

We propose a deformation theory of strain gradient crystal plasticity that accounts for the density of geometrically necessary dislocations by including, as an independent kinematic variable, Nye's dislocation density tensor [1953. Acta Metallurgica 1, 153–162]. This is accomplished in the same fashion as proposed by Gurtin and co-workers (see, for instance, Gurtin and Needleman [2005. J. Mech. Phys. Solids 53, 1–31]) in the context of a flow theory of crystal plasticity, by introducing the so-called defect energy. Moreover, in order to better describe the strengthening accompanied by diminishing size, we propose that the classical part of the plastic potential may be dependent on both the plastic slip vector and its gradient; for single crystals, this also makes it easier to deal with the “higher-order” boundary conditions. We develop both the kinematic formulation and its static dual and apply the theory to the simple shear of a constrained strip (example already exploited in Shu et al. [2001. J. Mech. Phys. Solids 49, 1361–1395], Bittencourt et al. [2003. J. Mech. Phys. Solids 51, 281–310], Niordson and Hutchinson [2003. Euro J. Mech. Phys. Solids 22, 771–778], Evers et al. [2004. J. Mech. Phys. Solids 52, 2379–2401], and Anand et al. [2005. J. Mech. Phys. Solids 53, 1789–1826]) to investigate what sort of behaviour the new model predicts. The availability of the total potential energy functional and its static dual allows us to easily solve this simple boundary value problem by resorting to the Ritz method.

Introduction

As is well-known nowadays, metallic components, when subjected to inhomogeneous plastic flow, exhibit a size dependent behaviour, with smaller being stronger (the experimental evidence is well documented, e.g., in Stelmashenko et al., 1993; Fleck et al., 1994; Stölken and Evans, 1998). The size range of engineering relevance for this effect spans from a few hundreds of nanometers to a few tens of micrometers. As argued by Nye (1953) and Ashby (1970), the size effect in this range of length scales is mainly due to the interaction between statistically stored dislocations (SSDs) and (density of) geometrically necessary dislocations (GNDs), the former growing with the plastic strain, the latter being induced by the plastic strain gradients.

Standard (i.e., local) plasticity theories are unsuitable to describe the size effect since they do not involve any intrinsic material length scale. In order to overcome this problem, such theories have been extended in various ways.

Perhaps, the most exploited and successful approach to the continuum modelling of the size effect through strain gradients consists of properly adding to standard constitutive laws a (tensorial) variable involving the gradients of some plastic strain measure; we call these models phenomenological. This strategy may require the introduction of proper stress measures conjugated to the chosen kinematic variables, and related “higher-order” boundary conditions (see, e.g., Fleck and Hutchinson, 1997, Fleck and Hutchinson, 2001; Gurtin and Needleman, 2005), but this is not always the case (see, e.g., the flow theory of plasticity of Acharya and Bassani (2000), where strain gradients enter the instantaneous hardening moduli only, thus preserving the classical mathematical framework of standard plasticity). However, it is quite clear (see, e.g., Shu et al., 2001; Niordson and Hutchinson, 2003) that the continuum modelling of phenomena related to the size effect such as the development of boundary layers in constrained plastic flow requires a theory involving “higher-order” boundary conditions. An approach closer to the dislocation mechanics consists of making use of evolutive relations for the dislocation densities (which may even account for the different geometry inherent to edge and screw dislocations) for incorporation into the continuum model (Harder, 1999, Evers et al., 2004, Arsenlis et al., 2004).

For simple problems, the predictions of some continuum models have been compared with those obtained from discrete dislocation simulations (e.g., Van der Giessen and Needleman, 1995, Cleveringa et al., 1999, Bittencourt et al., 2003, Yefimov et al., 2004).

Another source of discrimination among existing theories is then the fact that some richer continuum models can account for the different role of GNDs and SSDs. The importance of such a modelling has been documented by Ashby (1970), Sun et al. (2000), and many researchers involved in the discussion in “Scripta Materialia” 48, 2003. Among the models which enjoy such a desirable feature let us mention those of Acharya and Bassani (2000), Evers et al. (2004), Arsenlis et al. (2004), Gurtin and Needleman (2005), and Han et al. (2005).

In particular, the phenomenological models able to properly include the peculiar effect of GNDs are usually those involving Nye's dislocation density tensor (Nye, 1953), which accounts for the part of the deformation which would lead to lattice incompatibility if there were no GNDs. Here, we focus on this approach.

GNDs are also responsible for long-range interaction effects leading, in flow theory of plasticity models, to a kinematic hardening (resulting in the Bauschinger effect upon cyclic loading).

We propose a deformation theory of strain gradient crystal plasticity that accounts for GNDs by including, as an independent kinematic variable, Nye's tensor. This is accomplished, as proposed by Gurtin and co-workers (see, e.g., Gurtin, 2002; Gurtin and Needleman, 2005) in the context of the flow theory of crystal plasticity, by the introduction of a defect energy which, in our holonomic context, can be seen as an extension either of the plastic potential or of the free energy, the latter being Gurtin's viewpoint.

Moreover, we introduce a power law plastic potential also dependent on the gradient of the plastic slip vector. Whereas this further extension requires the definition of one more stress variable, it is characterised by two noticeable features: (i) it allows a better description of the strengthening accompanied by diminishing size and (ii) with respect to the theory developed by Gurtin and Needleman (2005), it permits one to prescribe, as kinematic “higher-order” boundary conditions, all the plastic slips, independent of the glides available; this might be useful in the Finite Element implementation of the theory, even though, for what concerns the modelling of polycrystals, it might cause relevant complications in dealing with the (dis)continuity conditions over internal grain boundaries that are any less defective than impermeable to dislocations.

Drawbacks and advantages of deformation theory models are well appreciated (see, e.g., Budiansky (1959), and, for the gradient case Fleck and Hutchinson (2001) and Fleck and Willis (2004)). Whereas they allow the description of the material behaviour under monotonic loading conditions only, they are much easier to deal with, with respect to their rate form counterparts, from both analytical and numerical viewpoints, and in many cases provide very accurate approximations. In fact, the availability of the minimum principles for the total potential energy and its dual will allow us to easily solve simple boundary value problems and, in the near future, it shall be exploited in order to homogenise the overall plastic properties of polycrystals, in the same fashion as Fleck and Willis (2004) did for non-crystalline strain gradient composites.