13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- crack tip based on the results obtained by our atomistic calculations for alpha iron, to evaluate the many-body effects of dislocations around a stress singularity field of a crack tip. 2. Methods of Dislocation Dynamics Analysis In this study, we employ the analysis model as shown in Figure 1. In this model, one-dimensional edge dislocations are emitted from a mode II crack tip. The crack length is set as 2a and the dislocation source S is located at some finite distance of xS from a crack tip. In this model, xS is taken as 10−9 m, which corresponds to the order of the dislocation core, thus this dislocation source is regarded to be located at the crack tip from the viewpoint of continuum mechanics. Figure 1. The model of dislocation emission and motion under the local stress field around crack tip In this study, dislocation dynamics method which has been performed by A. T. Yokobori, Jr. et al. is adopted. Since the detailed analysis methods were reported in the previous papers [6-7], here we show the outline of the analysis method. As the stress distribution near a crack tip τa(x, t), the stress singularity near a crack tip was taken into account under a constant rate of stress application as follows: ( ) x a t x t a τ τ = & , , (1) where τ& is the increasing rate of stress application, t is time and x is the distance from the crack tip. The relationship between effective stress exerted on each dislocations moving along x direction and dislocation velocity are written for each individual dislocation in a coplanar array by the following equation: ( ) ( )b Q E t x v i i i eff , dis d d τ = = Δ , (2) where, Q( ΔEdis) is the frequency of dislocation motion with 1b, ΔEdis is the free energy barrier for dislocation motion and is a function of effective stress τeff,i and temperature T. b is the length of Burger’s vector. The frequency of dislocation motion is determined by atomistic simulation as described later. Here, the effective stress exerted on the i th dislocation is written as follows: ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + + + − = ∑ ≠ = n i j j i j i j i i i i x x x x x x A x a t 1 * eff , 1 1 2 1 τ τ & , (3) where, A* = μb/2 π1− ν ( ), μ is the shear modulus, ν is the Poisson’s ratio, x i is the distance of the i th dislocation from the crack tip. The first term is the macroscopic stress field around a crack tip by applied stress, the second and third terms are the image force of dislocations by the free boundary of
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