A model of crystal plasticity containing a natural length scale

Abstract

A crystal plasticity model is developed which is embedded with a natural length scale. The model is developed within the framework of Coleman-Gurtin thermodynamics of internal state variables. The normal multiplicative decomposition of the deformation gradient utilized in crystal plasticity is expanded to add an extra degree of freedom. The internal state variables introduced in the free energy include the elastic strain associated with the statistically stored dislocations, and the curvature, which is derived from the continuum theory of dislocations. In this theory, the curvature is the curl of the elastic rotation associated with the polar decomposition of the elastic deformation gradient. This leads to an internal stress field that results from the presence of geometrically necessary dislocations and possesses an inherent length scale, in addition to the normal mechanical resistance that is proportional to the square root of the statistically stored dislocations. A crystal plasticity model incorporating these internal stresses is implemented into a FEM code. The results of the models prediction of the formation of misoriented cells, whose size is determined by the length scale of the model, as well as the prediction of the gradients of misorientation at the interface of the two cells is compared with bi-crystal experiments on Aluminum. During the compression of these bi-crystals, the dislocation patterns that develop encompass local lattice rotations or cells of misorientation that have a hierarchical of length scales. Therefore, the kinematics of the model are generalized to include an additional rotational degree of freedom, and additional tensor state variables are derived from the kinematics. This model has many similarities with the multipolar dislocation/disclination theory of Eringen and Claus and may be viewed as a finite deformation extension of that theory. The relationship of this model, which now has multiple length scales, to existing deformation theory type of strain gradient models is then discussed, and the length scales are related to actual physical mechanisms.

Introduction

There has been a resurgence of interest in regularizing or adding a mathematical length scale to continuum models of deformation [1]. The motivation for this comes from several sources. From the point of view of numerical solutions, it is well-established that in the post-bifurcation regime of a solution (initiation of either strain localization or damage associated with material softening), the system of differential equations changes and the problem is ill posed [2], [3], [4], [5]. In codes modeling hyperbolic systems, the differential equations become elliptic in the post-bifurcation regime, but the boundary conditions are still prescribed for hyperbolic systems. Similarly, in static codes, elliptic systems transform into hyperbolic systems but the associated boundary conditions are still prescribed for the original elliptic system. This problem can be resolved by regularizing or adding a mathematical length scale to the continuum, either in the form of spatial gradients in the constitutive model or by some numerical construction. Another motivation results from the attempt to solve boundary value problems at extremely small length scales. Recent experimental studies have revealed that material properties change as a function of size [6], [7], [8]. For example, in mechanical tests on small specimens, when the specimen dimensions reaches a critically small size, the yield strength begins to increase sharply with further decrease in specimen dimension. This dimension is generally smaller than the reasonable applicability of local continuum theory, but still much larger than that required for tractable solutions of the problem with atomistic methods. This problem can also be resolved by the aforementioned introduction of a length scale using spatial gradients. And finally, as the demands of design require the incorporation of more of the underlying physics, it is becoming important to develop a means to bridge the length scales from the macroscopic continuum to the atomistic levels. One approach to achieve this is generalization or the addition of more kinematic degrees of freedom to the continuum. This approach results in the development of a crystal plasticity model with a physical length scale. Briefly, as a crystalline material is deformed, the crystals begin to rotate to align themselves with the imposed loading. At large enough strain, geometrical necessary dislocations (GNDs) or geometrically necessary boundaries (GNBs) are introduced as an energetically favorable alternative to the formation of gaps, overlaps and holes [9]. In the description of the material as a continuum field, these GNDs lead to incompatibilities resulting in internal stresses that are described by higher order spatial gradients of the deformation. Hence, the resulting model of the crystal includes a natural, physical length scale.