13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- [ ] [ ] /(2 ) cos /(2 ) 1 sin 1 ( , ) ( , ) 3 2 3 3 n n n n C W n w n Φ Φ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥ = − ⎦ ⎤ ⎢ ⎣ ⎡ Φ Φ ≡ Φ Φ (15) ( ) ( ) x x x x x sin /2 cos ( , ) 2 3 Φ − Ω Φ = Φ (16) and replace the minimization of WΦ(n, Φ) by the maximization of ΩΦ(x, Φ) with respect to x. Then, for the complete plane (plate) with half-infinite cut Φ = 2π, Eq. (16) acquires the form ( ) ( ) ( )2 2 3 ( ) sin cos ( ,2 ) x x x x x x x π− Ω = π− Ω π = Φ (17) for which the relation of the type Eq. (14) remains valid, wΦ(4, 2π) < wΦ(3, 2π). Thus, for Φ = 2π the function WΦ(n, 2π) of energy expenditures in crack formation attains its minimum for nmin = 4 as well. For a plate-half-plane, Φ = π, and Eq. (16) becomes ( ) ( ) ( )2 2 3 /2 ( ) sin /2 cos ( , ) x x x x x x x π − Ω = π − Ω π = Φ (18) Here the function WΦ(n,π) attains its minimum at nmin = 2. Thus, as the opening angle Φ of the loaded wedge decreases, the number n of sectors minimizing the total energy expenditures necessary for their formation decreases from n = 4 for Φ = 2π to n = 2 for Φ = π. The natural question arises: How does nmin vary as Φ varies from 0 to 2π; in particular, for what values of the wedge opening angle Φ does nmin vary from n = 2 to n = 3 (Φ2→3) and from n = 3 to n = 4 (Φ3→4)? To answer this question, it suffices to compute wΦ(n,Φ) for Φ varying from π to 2π by Eq. (15) for n = 2, 3, 4. In Fig. 2, we present the graph of the dependence of ln[wΦ(n,Φ)] on the wedge opening angle Φ. The points of intersection of wΦ(2,Φ) with wΦ(3,Φ) and of wΦ(3,Φ) with wΦ(4,Φ) give precisely the values of the wedge opening angles Φ at which the number nmin of the formed sectors (or cracks) is changed, Φ2→3 ≈ 4.43 and Φ3→4 ≈ 5.94. The sectors with maximum opening angle (near Φ2→3/2 ≈ 2.21 rad) are formed for Φ close to Φ2→3 ≈ 4.43. Figure 2
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