13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- number of arising cracks (and sectors), L is the length of arising cracks, h is the plate thickness, γ is the effective surface energy of fracture, and U is the bending energy of each of the arising triangular sectors-beams. Write out the expression for the elastic bending energy U of one sector Π = 2 2u U (2) where Π is the bending compliance of the sector, u – the indenter vertical displacement (descent). Figure 1 Treating the sector as a cantilever beam triangular in plan (i.e., a cantilever of variable width) working in bending (Fig. 1), we write out the expression for its compliance in the form ([3], Table 18) ( ) ( ) sin /2 3 cos /2 3 3 2 ϕ ϕ Π= = Eh L Q u (3) where Q is the force acting at the end of each beam, L is the length of the lateral surface of the sector (equal to the length of the arising cracks), E is the Young modulus, h is the plate thickness, and φ is the central angle of the sector. In the case of formation of n equal cracks in a solid plate, φ = 2π/n. Taking into account this relationship and substituting successively of Eq. (3) in Eq. (2) and then in Eq. (1) we obtain for the function of total energy expenditures ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ π π = γ+ 2 3 2 3 / 6 cos / sin u n L n Eh W n Lh (4) 2. Minimization of the Expression for the Energy Expenditure. The Case of a Solid Plate Let minimize the obtained expression for W with respect to the crack length L and their number n. We rewrite Eq. (4) as ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = + 3 1 L B W AL (5) A = nhγ (6)
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