13th International Conference on Fracture June 16-21, 2013, Beijing, China Finite Deformation Modeling of Crystalline Defects in Hyper-elastic Material Akihiro Nakatani1,∗, Mitsuhiro Akita1 1 Department of Adaptive Machine Systems, Osaka University, Suita, Osaka 565-0871, Japan ∗ Corresponding author: nakatani@ams.eng.osaka-u.ac.jp Abstract An updated Lagrangian formulation is established to deal with the finite deformation of hyper-elastic body including the discontinuous slips due to the nucleation of lattice defects. The principle of virtual power is modified to treat the discontinuous deformation measured by initial lattice and a computational model is established by a penalty method with Lagrange multiplier. A few examples are shown for the application of geometrically non-linear deformation of hyper-elastic body including some dislocation structures and they are compared with the solutions of linear elasticity. The results have showed that the proposed method can be applicable to the analysis of kink deformation which had been observed experimentally in long-period stacking ordered magnesium alloy. Keywords Computational Mechanics, Finite Deformation, Hyper-elasticity, Driving Force 1. Introduction Magnesium alloys containing long-period stacking order (LPSO) phase have attracted a great deal of attention. The origins of both high strength and high ductility of these alloys are thought to be related to the grain refinement of matrix phase and kink deformation band of LPSO phase. Collective behaviors of dislocations on basal slip planes are important for the formation of kink deformation which consists of unique arrangement of microscopic lattice defects. Theoretical studies of the relationship between lattice defect theory and generalized continuum theory have been carried out since early time, but they are still developping[4, 5, 6, 7]. There are quite a few literature of the deformation analysis using discrete dislocation plasticity and some disclination dynamics; e.g. deformation mechanism and mechanical properties[1], Interface crack[2], grain boundary[3], and so on. However, there are few studies of formulation based on finite deformation except for Refs.[8, 9, 10] In this study, the boundary value problem based on discrete defect theory is formulated to study the kink deformation using finite deformation theory. The essential mechanism of kink deformation is related to instabilities like buckling or localization of deformation. It will be important to consider the boundary value problem including lattice defects such as dislocations and disclinations. 2. Formulation Some relative discontinuous displacement on planes ∂Γis considered in a hyper-elastic bodyΩ. Current coordinates for deformed positionxis referred to the reference coordinates Xfor initial position, as shown in Fig.1. The displacement uis defined byu=x−X. The tensors F, v, andL=∂v/∂X -1-
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