13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. The energy release rate for boundary cracking This section describes the method we used to derive the strain energy release rate associated with sliding contact crack initiation from the crack-free boundary. The energy release rate related to a boundary movement has been investigated by Eshelby [6], Sih [7], Budiansky and Rice [8,9]. Consider a three-dimensional (3D) elastostatic boundary problem with material contained within the surface boundary S+s (Fig. 1), where the portion s of the boundary is traction-free, and the external loading is imposed only on S. Without changing the boundary conditions on S, impose a continuously varying sequence of static solutions, related to the displacements u, given by a time-like parameter t. Details of the procedure can be consulted in references [9], and here, only the result of energy release rate per unit time, ∂Π/∂t, is given, i.e. , (1) where denotes the ‘velocity’ of the points on s and is the current outward normal to s. In the case of two-dimensional deformation fields as shown in Fig. 2, relevant to the present problem, the energy release rate remains of the same form as Eq. (1). Figure 1. Two-dimensional deformation fields and integration path. Let , which corresponds to two components of unit boundary movement, so that , , where is the angle between boundary movement and x1. Let be the unit tensor inward normal to boundary s, which means that the boundary s moves inward, and let all points on boundary s move in the same direction. Thus, the energy release rate for the boundary movement is given by , (2) Figure 2. Boundary movement or cracking as the notch-like boundary becomes a crack. ∫ = ∂ ∂ s i i wvmds t Π iv im Δ Δ/ i i iv e= = α cos 1 = e α sin 2 = e α Δ i in m−= α α Δ Π sin cos 2 1 J wends eJ J G i i s i i + = = = ∂ ∂ −= ∫ 0→s s m v n sin x1 x2 n α S
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