13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) ( ) 2 3 2 / 6 cos / sin u n n Eh B γ π π = (7) By computing the derivative ∂W/∂L and by equating it with zero, we obtain 2 1 3 = L B (8) ( ) ( ) ( ) 3 1 2 3 2 3 / 3 cos / sin 2 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ γ π π = = u n n Eh L B (9) Now we substitute A, L, and B/L3 computed by Eq. (6), Eq. (9) and Eq. (8) into Eq. (5) and obtain ( ) ( ) 1/3 3 / cos / sin ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ π π = n n W Cn (10) 2 2 1/3 3 2 3 ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ γ = γ Eh u C h (11) Since C is independent of n, it is convenient to divide W by πC and consider the inverse function of πC/W. In Eq. (10), we pass from the discrete variable n to the continuous variable x by the formulas π/n → x, n > 2, 0 < x < π/2 (12) and from Eq. (10) we obtain the following expression for the cube of this new function ( ) ( ) ( ) ( ) sin cos / sin / cos ( ) 3 3 3 3 x x x x x n n n n W n C ≡Ω → → π → π π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛π ⎟⎟ = ⎠ ⎞ ⎜⎜ ⎝ ⎛ π (13) We replace the minimization of the function W(n) by the maximization of the function Ω(x) with respect to x. One can readily show that this function has a single maximum at x0 ≈ 0.84. Since, according to Eq. (12), the discrete variable n and the continuous variable x are related as x ↔ π/n, it follows that the extreme value of n is one of the two integers nearest to π/x0 ≈ π/0.84 ≈ 3.74. By checking the minimum of the function ( ) ( )n n n C W n w n / cos / sin ( ) ( ) 3 3 3 π π ⎥ = ⎦ ⎤ ⎢ ⎣ ⎡ ≡ for n = 3, 4, we obtain 4 (3), (4) min = < w w n (14) 3. The Case of a Wedge (n–1 Cracks and n Sectors) We assume that the plate has the shape of a wedge with opening angle Φ, 0 ≤ Φ ≤ 2π, and the point indenter acts at the vertex of this wedge. Then the appearance of n-1 cracks in this plate corresponds to the formation of n sectors with opening angle Φ/n. In problems on an icebreaker in ice fields, the case Φ = 2π corresponds to the case of an icebreaker in the mouth of the channel crushed by it ([1], p. 72), and Φ = π corresponds to the case of an icebreaker coming over the ice field edge or a slant smooth support. Operating similar to paragraph 2, we obtain in this case instead of Eq. (10), Eq. (13)
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