Dislocation mechanics-based constitutive equations
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6/30/04
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Dislocation Mechanics–Based Constitutive Equations FRANK J. ZERILLI A review of constitutive models based on the mechanics of dislocation motion is presented, with focus on the models of Zerilli and Armstrong and the critical influence of Armstrong on their development. The models were intended to be as simple as possible while still reproducing the behavior of real metals. The key feature of these models is their basis in the thermal activation theory propounded by Eyring in the 1930’s. The motion of dislocations is governed by thermal activation over potential barriers produced by obstacles, which may be the crystal lattice itself or other dislocations or defects. Typically, in bcc metals, the dislocation-lattice interaction is predominant, while in fcc metals, the dislocation-dislocation interaction is the most significant. When the dislocation-lattice interaction is predominant, the yield stress is temperature and strain rate sensitive, with essentially athermal strain hardening. When the dislocation-dislocation interaction is predominant, the yield stress is athermal, with a large temperature and rate sensitive strain hardening. In both cases, a significant part of the athermal stress is accounted for by grain size effects, and, in some materials, by the effects of deformation twinning. In addition, some simple strain hardening models are described, starting from a differential equation describing creation and annihilation of mobile dislocations. Finally, an application of thermal activation theory to polymeric materials is described.
I. INTRODUCTION
IN the 1970s and 1980s, as well as in more recent years, there has been a great deal of interest in constitutive relations that could describe material behavior sufficiently well enough to produce accurate predictions of deformation and fracture when used in large scale computer simulations. An important advance was made by Johnson and Cook,[1] who successfully described cylinder impact (Taylor) test results for a variety of materials using the Lagrangian materialdynamics code EPIC-2, which they had developed. They employed a temperature and strain rate-dependent constitutive relationship relating the von Mises yield stress to the von Mises effective strain: # m [1] s (A Bn )(1 C ln )(1 T * ) # where is the strain rate, T* is the ratio (T Troom)/(Tmelt Troom), and T is the absolute temperature. The terms A, B, n, C, and m are material constants determined from limited straining tests done in tension or torsion. In this equation, strain hardening, strain rate hardening, and thermal softening are taken into account, but it turns out that the variation of thermal softening with strain rate is not reproduced well for real materials. Since 1934, when Taylor,[2] Orowan,[3] and Polany,[4] trying to understand slip in crystals, independently proposed that the presence of imperfections, in particular, edge dislocations, is able to account for the discrepancy between the large theoretical shear strength and the much lower obse
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