13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 5). From this figure we see that dR = ±Rdφ (5) where R and φ are polar coordinates. This equation gives two orthogonal families of slip lines R(φ) = Ce±ϕ; R| φ=0 = R* = C, R(φ) = R*e ±ϕ (6) where C is the integration constant, R* – the radius of the hole. Taking R* and W* = 2 2 πγR*, (7) as the length and energy reference scales, respectively, pass on to dimensionless variables r, λ, l, w by the formulas * 2 2 * , * , * , * R W W W w R L l R R R r πγ = = = Λ λ= = (8) R = R*r, Λ = R*λ, L = R*l, W = W*w (9) Then (6) takes the form r(φ) = e±ϕ, and choosing to be definite one of the branches (families) with the sign +, we obtain the dimensionless radial coordinate of the "petal"-wedge end rn rn = r(φn) = |(2)| = r(π/n) = e π/n, n = π/lnr n (10) Writing now the expression for the differential of arc length in polar coordinates, expressing φ in R using (5) and integrating over the entire crack length, we find the length of one crack Λ, λ and the total dimensionless length of all the cracks l ( ) ( ) ( ) ( ) 2 1 2 * 1 * 2 2 (5) 0 1 * 2 2 0 − λ = − = = = = ϕ + Λ= = ∫ ∫ ∫ ∫ π ϕ= ϕ= π ϕ= ϕ= n n n r R R n r dr R r dR R dR Rd dS n n (11) ( ) n n n r r n r l n ln 1 2 22 1 (10) 22 − − = = π = λ= (12) For the dimensionless cracking energy with (8), (3), (11) we have ( ) ( ) ( ) ( ) ( ) n n n r r R n R r R n W W w ln 1 10 2 2 * 2 2 * 1 11 2 2 * 2 3 , 8 * − = = πγ − γ = = πγ γΛ = = = 13) A plot of the dimensionless cracking energy w on rn is represented by the upper curve (almost straight) in Fig. 6. It is the function defined for rn ≥ 1, monotonically increasing, concave upwards and equal to 1 for rn = 1.
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