13th International Conference on Fracture June 16-21, 2013, Beijing, China 2.2. Internal Discontinuous Deformation The relative slip of bτ(t) on a slip plane ∂Γis described by bτ(t)MA =X+ A −X−A, on ∂Γ, (7) where MA is the tangential vector along slip direction. X+ andX− are the upper and lower reference coordinates on the slip plane at time t, respectively. b means the amount of unit discontinuous slip such as the norm of Burgers’s vector. A monotonically increasing function τ(t) is varied from 0 to 1; τ(0) = 0 and τ(tend) = 1, where tend is the finish time. A linearized assumption of slip amount from t =t(i) to t =t(i) +∆t gives bτ(t(i) +∆t) =bτ(t (i)) +b˙τ∆t, (8) where ˙τ means the temporal rate of τ. From Eq.(7), bτ(t(i))MA =(F− 1)+ Aix +i −(F− 1)− Aix−i , (9) and bτ(t(i) +∆t)MA =((F− 1)+ Ai +(F− 1)+• Ai ∆t)(x +i + ˙x +i ∆t) −((F−1)− Ai +(F− 1)−• Ai ∆t)(x−i + ˙x−i ∆t). (10) Then, a rate type constraint condition is described by b˙τMA−(F− 1)+ Akl + kix +i −(F− 1)− Akl−kix−i −((F− 1)+ Aiv +i −(F− 1)− Aiv−i ) =0, (11) and b˙τMA−(F− 1)+ AkL+ kBX+ B −(F− 1)− AkL−kBX−B −((F− 1)+ Aiv +i −(F− 1)− Aiv−i ) =0, (12) in current coordinates system and in reference coordinates system, respectively. For the sake of simplicity in this study, we use the approximation as following b˙τ ¯FiAMA −(v +i −v−i ) =0, where ¯FiA =F+iA +F−iA. (13) There are several approached for the computational setup. For example, the extended finite element formulation can be used as well as some cohesive zone model. In this study, we use the double nodes on the slip plane. A Lagrange multiplier λL is introduced to take into account the constraint condition (13), and the principle of virtual power (6) is modified as ∫Ω δdTCddv +∫ Ω δlTTldv +∫ ∂Γ λL (b˙τ ¯FM−(v + −v−)) · (δv+ −δv−)ds =∫ Ω ρ˙b· δvdv +∫ ∂Ω ˙t∗ · δvda. (14) -3-
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