Geometrically necessary dislocations and strain-gradient plasticity: a few critical issues
Introduction
One of the earliest mentions of a hardening effect due to the plastic accommodation of elastic strain gradients by dislocations is found in the book by Friedel [1]. With reference to the bending of a crystal to curvature c (and with μ and b being respectively the shear modulus and the modulus of the Burgers vector), Friedel wrote “… the minimum dislocation density necessary to produce the deformation is given by ρ=c/b […]. This density […] introduces short-range stresses on a scale comparable with the average distance ℓ between dislocations […]. One expects therefore a parabolic law σ≈σ0+(μ/2π)(bc)1/2. […] Similar but more elaborate equations can be given in the same way for any type of macroscopic distortion which is not a uniform shear”. These are of course geometrically necessary dislocations (GNDs) in the sense of Ashby [2], taken to induce increased hardening to the crystal in addition to that caused by statistically stored dislocations.
In pure fcc crystals, the resolved flow stress τ is then given by the well-known relationship:where α is a constant coefficient, ρs is the density of statistically stored dislocations and ρg the density of GNDs. With ρg proportional to the strain gradients, where Eq. (1) applies, GND-governed scale-dependent hardening is recognizable through a linear dependence of the square of the flow stress on the inverse of the distance d characteristic of the deformation problem at hand (or, if ρs≪ρg, a linear dependence of τ on ). Such a dependence has indeed been found in several important cases, including the response to microhardness indentations of fcc crystals, the large-strain flow stress of precipitate or dispersion hardened metals, and the yield and flow stress of particle reinforced metals [4], [5], [6].
GND models have attracted much attention because, in simple cases, the density of dislocations needed to relax a given strain gradient can be calculated assuming essentially static dislocation arrays dictated by local equilibrium. In a way, this is reminiscent of the low energy dislocation structure models [3], or of the models developed to explain the formation of arrays of misfit dislocations in epitaxial layers.
The resulting possibility of modeling scale effects in plastic deformation has motivated the proposal of several continuum plasticity theories which incorporate a dual dependence of the plastic flow stress on strain and strain gradients. The assumptions underlying these theories are variably based on the theory of GNDs, which is used to propose length scales that serve to quantify in the continuum the contribution of strain gradients to hardening. Our purpose in this short contribution is to discuss, from a materials science perspective, a few critical issues in connection with the dislocation phenomena that underlie the continuum strain gradient plasticity (SGP) approach.
Section snippets
Hardening by geometrically necessary dislocations
How valid is Eq. (1)? In pure fcc metals, it is quite robust. The flow stress is then mostly governed by short-range attractive intersections of non-coplanar dislocations. As was shown by early theoretical studies [7], [8], and confirmed by experimental investigations [9], this leads to Eq. (1) with α=0.3±0.1. This “forest” hardening should be distinguished from “dipolar” or Taylor hardening which stems from long-range dislocation interactions. Then, the same scaling law is recovered but the
Length scale dependence
As mentioned in the previous section, the observed dependence of hardening on length scale predicted by Eq. (1) has, indeed, been observed in several important cases. This provides strong justification for the use of GND theory towards quantification of scale-dependent plasticity; however, as is well known, this dependence is not universal. In particular, the initial yield stress of undeformed materials, which at a fine scale of structures is also scale-dependent, generally cannot be explained
Specific nature of geometrically necessary dislocation patterns
As pointed out by Gil Sevillano: “GND arrays are not univocally determined” [17]. This is well illustrated by the variety of dislocation configurations that exist in a system as basic as a crystal deforming by single slip around a hard spherical particle [5], [18].
As an illustration of the importance of this point, consider the SGP model proposed by Gao et al. [19]. A salient feature of this model is its clear concern for basing the SGP equations on specific GND arrays, which it describes in
Gradient plasticity and size effects
Several difficulties arise when formulating continuum models for the evolution of dislocation densities inside a crystal. We discuss here the homogenization process that transforms a discrete dislocation density into a continuum one, within a purely dislocational framework and without making reference to GNDs. In addition we show that the modeling of size effects of dislocational origin does not necessarily involves the consideration of GNDs.
Models describing the coupled evolution of
Concluding remarks
GND models can efficiently be used to describe size effects in plasticity, but within some limits, however. These limits are set by the static and equilibrium characters of the GND framework, by kinetic effects such as lattice friction and by yield phenomena also, which may exhibit different, non-GND related, scaling properties at micron-scale distances. Other limitations stem from the importance of GND array specifics or, in other words, by the potential lack of universality of characteristic
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