Finite Element Studies of Homogeneous and Heterogeneous Dislocation Nucleation based on the Rice-Peierls Framework
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Finite Element Studies of Homogeneous and Heterogeneous Dislocation Nucleation based on the Rice-Peierls Framework T.L. Li,1 J.H. Lee,2 Y.F. Gao,1,3 1
Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996, U.S.A. 2 Division for Research Reactor, Korea Atomic Energy Research Institute, Daejeon 305-353, Republic of Korea 3 Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A. ABSTRACT The study of dislocation nucleation has gained increasing attentions recently primarily due to the advancement of small scale mechanical testing methods. Based on the classic Rice model of dislocation nucleation from a crack tip in which the dislocation core is modeled by a continuous slip field, a nonlinear finite element method can be formulated with the interplanar potential as the input, and the development of interplanar slip field can be solved from the resulting boundary value problems. The effects of geometric boundary conditions, loading patterns, etc. can be conveniently determined, as opposed to the time consuming molecular simulations. To validate the method, we compare the simulations results of homogeneous dislocation nucleation and heterogeneous dislocation nucleation from a two-dimensional crack tip to the literature results. As proposed by Rice and Beltz (J. Mech. Phys. Solids, 1994), the activation energy for dislocation nucleation from a three-dimensional crack tip depends on the finite thickness in the direction parallel to the crack tip, which has been successfully reproduced in the finite element simulation results reported here. INTRODUCTION With the rapid development of micro- and nano-scale material structures and small scale mechanical testing methods, the study of dislocation nucleation has gained increasing attentions recently. Examples include the homogeneous dislocation nucleation under indentation (thus leading to the pop-in behavior on the load-displacement curves) [1-5] and heterogeneous dislocation nucleation from sharp features in strained nano-electronics [6-10]. A dislocation is usually modeled either by the Volterra model [11-13], which treats the dislocation as a mathematical discontinuity, or by the Peierls-Nabarro model, which treats the dislocation core as a continuous slip field [14-18]. Based on the diffused-core model, dislocation nucleation from stress concentration sites such as a crack tip is viewed as a gradual development of the interplanar slip field until an instability is reached [14,15]. The relative slip between two adjacent layers of the slip plane, Δα , and the shear stress on the slip plane, τ α , are related through τ α = ∂Φ ∂Δα , where Φ ( Δα ) is a periodic interplanar
potential and is also denoted as the γ surface, and α = 1, 2 are the two slip directions on the slip plane [14,15]. The total potential energy Π is
1 n ⋅ σ ⋅ ΔdS − ∫ n ⋅ σ elastic ⋅ ΔdS , (1) ∫ S S S 2 where n denotes the slip plane normal, σ is the self stress due to a non-uniform Δ when the applied load is zero, and σ
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