Thermodynamic coarsening of dislocation mechanics and the size-dependent continuum crystal plasticity

Abstract

Starting from the standard coarsening of dislocation kinematics, we derive the size-dependent continuum crystal plasticity by systematic thermodynamic coarsening of dislocation mechanics.

First, we observe that the energies computed from different kinematic descriptions are different. Then, we consider systems without boundary dissipation (relaxation) and derive the continuum approximation for the free energy of elastic–plastic crystals. The key elements are: the two-dimensional nature of dislocation pile-ups at interfaces, the localized nature of the coarsening error in energy, and, the orthogonal decomposition theorem for compatible and incompatible elastic strain fields. Once the energy landscape is defined, the boundary dissipation is estimated from the height of energy barriers.

The characteristic lengths are the average slip plane spacing for each slip system. They may evolve through the double-cross slip mechanism. The theory features the slip-dependent interface free energy and interface dissipation for penetrable interfaces.

The main constitutive parameters are derived from elasticity. The exception is the dependence of interface energy on slip plane orientation, which is determined from numerical results.

The theory requires no higher order boundary conditions. The only novel boundary conditions are kinematic, involving slip relaxation on the two sides of an interface.

Introduction

Classical crystal plasticity is size-invariant, and incapable of incorporating boundary conditions for slip. Yet, numerous experiments¹ indicate pronounced size dependence, once the experimental length scale is below about 100 μm. Moreover, the need to impose boundary conditions on slip at an interface between two crystals is evident (Sun et al., 2000).

The physical mechanism for the observed size-dependence has been elucidated by Ashby (1970): In small volumes the motion of dislocations is constrained, resulting in pile-ups of dislocations with identical Burgers vector—the geometrically necessary dislocations. Following Ashby's physical argument, a number of heuristic mathematical formulations have been proposed during the last two decades.² The theories include higher order kinematic gradients, which, in turn, require characteristic length(s) (Mindlin, 1964; Aifantis, 1987), and inevitably produce qualitative prediction of size effect. The higher order gradients are typically related to Nye's (1953) dislocation density tensor. While this choice is intuitive, it is by no means unique. Moreover, the characteristic lengths are typically phenomenological, without clear physical meaning that can be connected to dislocation mechanics.

More recently, the focus has been shifted to interfaces³, with particular emphasis on dissipative models. However, analyses are mostly on the continuum level; starting from phenomenological assumptions, mathematical consequences are explored.

Attempts to fit the predictions of a theory to either experiments⁴ or simulations⁵ indicate a need for more rigorous approach, i.e., a derivation of a continuum theory by systematically coarsening dislocation mechanics.

Advances in statistical analysis of collective dislocation behavior⁶ have resulted in statistical continuum dislocation dynamics formulations⁷ (SCDD). The SCDD formulations emphasize dynamics of dislocation motion, while representing dislocations as continuum density fields. The key ingredient stems from the attempt to resolve the fundamental problem of discrete dislocation dynamics: long-range interactions. This is achieved by computing a local approximation to the driving force (the Peach–Koehler force). The basis for this approximation is the assumption that dislocation–dislocation correlation function is short-range (Groma, 1997). There are several important differences between these formulations and the present approach.

• In this paper, we take a point of view that size effects in plasticity are the result of dislocation pile-ups against the boundaries, and consider highly correlated stacked dislocation pile-ups.

• In our formulation, the long-range forces are the part of the continuum problem (Kroner, 1958), while the additional free energy is the error caused by kinematic coarsening (Roy et al., 2007; Geers et al., 2009). We show that this error is localized. This leads to natural treatment of all possible boundary conditions that can be applied to a continuum crystal plasticity problem.

• We aim to develop a more economical (and inevitably, less accurate) quasi-static formulation, in which the number of degrees of freedom will be comparable to the one in the standard crystal plasticity. In contrast to SCDD formulations, we will not attempt to follow the dynamics of dislocation fields, but concentrate on constrained thermodynamics. Moreover, we will assume that the phenomenological treatment of statistical hardening in standard crystal plasticity is sufficiently accurate for our purposes, and focus on the thermodynamic differences that arise from ordering of dislocations in pile-ups against the boundaries.

It should be stated at the outset, that the complexities of dislocation mechanics (complex geometry and long-range interactions) all but preclude a mathematically rigorous approach. Nevertheless, while some simplifying assumptions are necessary, the errors should be, in principle, quantifiable. Moreover, the requirement that all new quantities, fields and constitutive parameters, have a clear physical meaning at the level of dislocation mechanics is not contradictory to quantifiable assumptions.

The purely kinematic coarsening has been accomplished half century ago (Nye, 1953; Kroner, 1958). In this work we focus on thermodynamic coarsening. In particular, we aim to accurately express the free energy of the elastic–plastic crystals in terms of continuum fields, but on the basis of the accurate free energy, as computed using the discrete dislocation model.

The dissipation is, in general, a more difficult problem. Elementary dissipative events occur at the atomic length at time scales, so that the discrete dislocation model is of limited use. Nevertheless, dissipated energy can often be estimated if the energy landscape is known. The typical event consists of a peak energy configuration, followed by rapid relaxation, and the dissipated energy is approximated as the difference between the two energy levels.

The paper is organized as follows. We summarize different kinematic descriptions in Section 2. Then, in Section 3, we briefly review some relevant elements of the standard crystal plasticity and Kroner's (1958) continuum theory of dislocations.

In Section 4, we first note that the energies computed from actual and coarsened kinematics must be different, as illustrated earlier by Roy et al. (2007). Then, we define the framework in which the correction to the energy, i.e., the error resulting from kinematic coarsening, can be introduced into the framework of crystal plasticity. The orthogonal decomposition theorem for compatible and incompatible strain fields guarantees that this correction is orthogonal to the compatible strain, and therefore additive. It is a novel result, possibly with broader implications.

In Section 5, we consider several dislocation mechanics problems with simple geometries and thus amenable to analysis. The key observation is that the relevant configurations are the stacked pile-ups in parallel slip planes, and those—much like crack tips, have predominantly two-dimensional nature. We conclude that for slowly varying slip gradients, the coarsening error in energy density can be represented as a local function of slip gradients. Sharp boundary layers, resulting from pile-ups of dislocations, violate the slow variation condition, and are best modeled by a slip-dependent surface energy. The characteristic lengths that emerge from the analysis—one for each activated slip system, are the average spacings of slip planes in each slip system.

In Section 6, we formulate the continuum theory for single slip, and in Section 7, we generalize the results for multiple slip problems.

Consistent with our view that a continuum expression for the free energy is the primary goal from which the dissipation model can be deduced, we considered first the interfaces impenetrable to dislocations. The penetrable interfaces, interface relaxation, and the accompanying dissipation are discussed in Section 8. The summary and discussion are given in Section 9.

At the outset, we limit our analysis to: linearized kinematics theory with additive decomposition of elastic and plastic deformation, elastic isotropy, and isothermal deformation.