Mechanism-based strain gradient crystal plasticity
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Mechanism-based strain gradient crystal plasticity Chung-Souk Han , Huajian Gao , Yonggang Huang† , and William D. Nix †
Max Planck Institute for Metals Research, Heisenbergstr. 3, D-70569 Stuttgart, Germany Dept. of Mater. Sci. and Engng., Stanford University, Stanford, CA 94305, USA Dept. of Mech. and Indust. Engng., University of Illinois, Urbana, IL 61801, USA
ABSTRACT To model size dependent plastic deformation at micron and submicron length scales the theory of mechanism-based strain gradient plasticity (MSG) was developed. The MSG approach incorporates the concept of geometrically necessary dislocations into continuum plastic constitutive laws via Taylor hardening relation. This concept is extended here to develop a mechanism-based strain gradient theory for crystal plasticity (MSG-CP) based on the notions of dislocation density tensor and resolved density force corresponding to the Peach-Koehler force in dislocation theory. An effective density of geometrically necessary dislocations is defined on the basis of resolved density force for specific slip systems and is incorporated into the plastic constitutive laws via Taylor relation. INTRODUCTION Size effects in plasticity at micron and submicron scales has been modeled by strain gradients by various authors, e.g. [3]. Analyzing micro-indentation experiments Nix & Gao [11] found that the measured indentation hardness strongly suggests a linear dependence between the square of plastic flow stress and strain gradient. This has led to the development of mechanism-based strain gradient (MSG) plasticity [4]. The core idea of MSG has been to incorporate the concept of geometrically necessary dislocations into the plastic formulation √ by the Taylor relation τ = αµb ρ, where µ denotes the shear modulus, b the Burgers vector length, α is an empirical coefficient usually taken to be 0.2-0.5, and ρ the dislocation density. Most of the theories on strain gradient plasticity have been developed for isotropic and polycrystalline materials. Crystal plasticity models, however, offer a higher resolution of plastic deformation at sizes comparable to single grains, which are often on the order of microns where significant size effects have been observed. The anisotropy of size effects has for example been observed in micro-indentation experiments of silver crystals in the (100) and (110) orientations [8]. In order to obtain a better resolution a MSG-type crystal plasticity model is presented. Notation. To set the stage, the notation has to be provided. The decomposition of the deformation gradient is adopted as, F = Fe Fp , where the lattice rotation is contained in Fe and the plastic slip is included in Fp . Further expressions in the intermediate configuration B˜ are given in Table 1 (see [6] for details). The scalar product used therein is defined by M · N = ij Mij Nij . DISLOCATION DENSITY TENSOR Conventional crystal plasticity assumes that the lattice remains essentially undistorted in the deformation process. Lattice distortions can be described with the dislocati
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