A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations

doi:10.1016/S0022-5096(03)00094-2

Journal of the Mechanics and Physics of Solids

Volume 52, Issue 2, February 2004, Pages 279-300

https://doi.org/10.1016/S0022-5096(03)00094-2 Get rights and content

Abstract

A two-dimensional nonlocal version of continuum crystal plasticity theory is proposed, which is based on a statistical-mechanics description of the collective behavior of dislocations coupled to standard small-strain crystal continuum kinematics for single slip. It involves a set of transport equations for the total dislocation density field and for the net-Burgers vector density field, which include a slip system back stress associated to the gradient of the net-Burgers vector density. The theory is applied to the problem of shearing of a two-dimensional composite material with elastic reinforcements in a crystalline matrix. The results are compared to those of discrete dislocation simulations of the same problem. The continuum theory is shown to be able to pick up the distinct dependence on the size of the reinforcing particles for one of the morphologies being studied. Also, its predictions are consistent with the discrete dislocation results during unloading, showing a pronounced Bauschinger effect. None of these features are captured by standard local plasticity theories.

Introduction

Crystal plasticity theories have become popular and successful models for the anisotropic plastic deformation of single crystals. They have a hybrid, discrete/continuum, nature in the sense that they adopt a continuum description of the plastic flow by averaging over dislocations, but account for the discreteness of the available slip systems. Constitutive descriptions of the flow strengths and the hardening matrix have been given on purely phenomenological grounds, by e.g. Asaro (1983), but also on the basis of dislocation models, e.g. by Kocks et al. (1975).

Irrespective of the precise formulation, conventional continuum plasticity theory predicts that the plastic response is size independent. There is a considerable, and growing, body of experimental evidence, however, that shows that the response is in fact size dependent at length scales of the order of tens of microns and smaller, e.g. Fleck et al. (1994), Ma and Clarke (1995) and Stölken and Evans (1998). Various so-called nonlocal plasticity theories have been proposed that incorporate a size dependence, e.g. Aifantis (1984), Fleck and Hutchinson (1997), Acharya and Bassani (2000), Gurtin 2000, Gurtin 2002, but they differ strongly in origin and mathematical structure. Although dislocation-based arguments have sometimes been used as a motivation, the theories mentioned above are phenomenological and have not been quantitatively derived from considerations of the behavior of dislocations. Therefore, the material length scale that enters in such theories needs to be fitted to experimental results (see, e.g., Fleck et al., 1994; Fleck and Hutchinson, 1997) or results of numerical discrete dislocation simulations, e.g. Bassani et al. (2001), Bittencourt et al. (2003).

This paper is concerned with a new nonlocal plasticity theory that combines a standard crystal plasticity model with a two-dimensional statistical-mechanics description of the collective behavior of dislocations due to Groma (1997) and to Groma and Balogh (1999). The resulting theory contains a length scale through a set of coupled transport equations for two dislocation density fields: one is the total dislocation density and the other is a net-Burgers vector density. After a summary of the theory for single slip, we proceed to numerical implementation of the theory and to a comparison with direct simulations of discrete dislocation plasticity in a model composite material based on the work of Cleveringa 1997, Cleveringa 1998, Cleveringa 1999a. Similar comparisons have been carried out by Bassani et al. (2001) and Bittencourt et al. (2003) with the nonlocal theories of Acharya and Bassani (2000) and Gurtin (2002), respectively.

Section snippets

Statistical-mechanics description for single slip

Let us consider N parallel edge dislocations positioned at the points rⁱ, i=1,…,N, in $R^{2}$ . In single slip, the Burgers vector of the ith dislocation is bⁱ=±b where b is parallel to the slip direction s, i.e. b=bs. With the commonly accepted assumption of over-damped dislocation motion, the glide velocity of the ith dislocation in the slip direction s is given by vⁱ=B⁻¹Fⁱ in terms of the dislocation drag coefficient B and the glide component of the Peach-Koehler force, Fⁱ. This can be further

Incorporation into crystal plasticity theory

The above description of the dislocation structure in terms of the density fields ρ and κ is incorporated into the well-known framework of single crystal continuum plasticity (see, e.g., Asaro, 1983). Confining attention to small displacement gradients, the total strain rate $ε ̇$ in such a constitutive model is decomposed as $ε ̇ = ε ̇^{e} + ε ̇^{p}$ in terms of the elastic strain rate $ε ̇^{e}$ and the plastic strain rate $ε ̇^{p}$ which, for single slip, is expressed in terms of the slip rate $γ ̇$ on the slip system as $ε ̇^{p} = 12 γ$

Problem formulation

As a first application of the theory we consider the deformation of a two-dimensional model material containing rectangular particles arranged in a doubly periodic hexagonal packing, as illustrated in Fig. 1. This is the same problem as analyzed using discrete dislocation dynamics by Cleveringa 1997, Cleveringa 1998, Cleveringa 1999a, and is used to check the quality of the present nonlocal theory in reproducing the size-dependent results. Two reinforcement morphologies are analyzed which have

Summary of discrete dislocation results

In this section, we briefly summarize the results of discrete dislocations simulations of the problem at hand. The results to be presented are close to those obtained by Cleveringa 1997, Cleveringa 1998, Cleveringa 1999a but differ in the fact that here we do not assume an initial distribution of dislocation obstacles inside the matrix. There are no dislocations present initially, and dislocation sources are assumed to be distributed randomly in the matrix with a uniform density of ρ_n=61.2L⁻²

Nonlocal crystal plasticity results

The discrete dislocation simulations discussed above will now be compared with the calculations based on the nonlocal continuum plasticity theory put forward in 2 Statistical-mechanics description for single slip, 3 Incorporation into crystal plasticity theory. As in the discrete calculations, we start out from a stress free and dislocation free state, ρ(r,t₀)=κ(r,t₀)=0. The results to be presented have been obtained using the same material parameters, whenever possible, as in the discrete

Conclusion

We have formulated a non-local continuum crystal plasticity theory for single slip that involves standard continuum kinematics and two state variable fields: the dislocation density and the net-Burgers vector density. These densities are governed by two coupled evolution equations that describe their balance during drag-controlled dislocation glide, and which are derived from a statistical-mechanics treatment of an ensemble of gliding dislocations. The conservation law for the dislocation

Acknowledgements

This research was carried out under project number MS97006 in the framework of the Strategic Research Program of the Netherlands Institute for Metals Research in the Netherlands (www.nimr.nl). IG is grateful for support by the OTKA program of the Hungarian Academy of Sciences under contract number T 030791.

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A comparison of a statistical-mechanics based plasticity model with discrete dislocation plasticity calculations

Abstract

Introduction

Section snippets

Statistical-mechanics description for single slip

Incorporation into crystal plasticity theory

Problem formulation

Summary of discrete dislocation results

Nonlocal crystal plasticity results

Conclusion

Acknowledgements

J. Mech. Phys. Solids

Adv. Appl. Mech.

Int. J. Solids Struct.

J. Mech. Phys. Solids

Acta Mater.

Int. J. Plasticity

Adv. Appl. Mech.

Acta Metall. Mater.

J. Mech. Phys. Solids

J. Mech. Phys. Solids

J. Mech. Phys. Solids

Acta Mater.

Scr. Materialia

On the microstructural origin of certain inelastic models

J. Eng. Mater. Technol.