Discrete Dislocation Plasticity
Plastic deformation of metals governs several length scales, Fig. 1. At the large length scale of the macroscopic world, plastic deformation is conveniently described by a phenomenological continuum theory. When zooming in, one will first start to observe
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CISM COURSES AND LECTURES
Series Editors: The Rectors Manuel Garcia Velarde - Madrid Jean Salenrl are instantaneous concentration tensors that must be evaluated by some appropriate micromechanical model on the basis of the instantaneous constituent behavior. The overall instantaneous stiffness tensor E; can then be obtained from equivalents of eqn. (2.9) such that it is a continuum tangent operator with da = E;d.s. Incremental approaches are not subject to limitations in terms of the (mechanical) load cases they can handle and they can be extended to thermal loading (Pettermann, 1997). They have, however, typically shown a marked tendency towards being excessively
A Short Introduction to Continuum Micromechanics
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stiff in the post-yield regime for matrix dominated deformation modes. The reason for this behavior is thought to be an accumulation of errors over the incremental procedure, so that recent algorithmic developments have been specifically aimed at this problem area (Doghri and Ouaar, 2003). The intraphase variations of the stress and strain fields within the assumption of phase-wise uniformity tend to have a considerable influence on the predicted behavior. A typical case in point are the phase averaged von Mises equivalent stresses, which govern the onset of yielding in many plasticity models and which may be severely underestimated if they are evaluated from the phase averaged stress components. Improvements in this respect have been obtained by evaluating the phase averages of the von Mises equivalent stresses on the basis of energy considerations (Qiu and Weng, 1992) and by evaluating higher statistical moments of the microfields, see Buryachenko (1996) and Ju and Sun (2001). Such refinements have become an integral part of "modified secant models" (Ponte Castaneda and Suquet, 1998) but have seen little use in incremental approaches. A specific type of geometrical nonlinearity can occur in porous elastoplastic materials, where the geometries of the voids may evolve in the course of loading histories. Problems of this type can be studied by Hashin-Shtrikman-models in combination with appropriate void shape and void orientation evolution conditions, see e.g. Kailasam et al. (2000).
Bounds for the Nonlinear Behavior. Analogs to the Hill bounds for nonlinear inhomogeneous materials were introduced by Bishop and Hill (1951). For polycrystals the nonlinear equivalents to Voigt and Reuss expressions are usually referred to as Taylor and Sachs bounds, respectively. In analogy to mean field estimates for elastoplastic material behavior, nonlinear bounds are typically obtained by evaluating a sequence of linear bounds. An important development was the derivation by Ponte Castaneda (1992) of a variational principle that allows upper bounds on the effective nonlinear behavior of inhomogeneous materials to be generated on the basis of upper bounds for the elastic response. Essentially, the variational principle guarantees the best choice for the "comparison composite" at any given loading state. The Ponte C
