13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- bending of the arising sectors) be minimal over all n and L. This minimization with respect to L implies the condition that the energy required to form the cracks is equal to the doubled energy used to bend the arising sectors (Eq. 5, Eq. 8) and the “optimal” crack length L is expressed in terms of the thickness and the plate physical characteristics by a power law (Eq. 9). As a result, it follows from these relations that the total energy W is proportional to the crack formation energy whose expression contains all the above parameters only as factors raised to various powers and which then disappear in the process of optimization. Since we only take into account the strain of the plate central part cut by cracks, the solution does not contain the plate fixation conditions in any way. The character of variation in the number of arising sectors n with varying wedge opening angle Φ may also be explained qualitatively. The function W of total energy expenditures is the sum of the crack formation energy nLhγ and the energy nU of elastic bending of sectors-beams. For small Φ < Φ2→3, the arising sectors are narrow, and their total elastic energy weakly decreases as n increases, but the crack formation energy always increases linearly in n. Therefore, the minimum of W is realized for the minimum feasible value n = 2 at which the energy is minimal. For large Φ and small n, the elastic energy U is very sensitive to variations in n (moreover, as Φ → 2π, in the framework of the accepted scheme, the value n = 2 is associated with U → ∞). As a result, the minimum point moves upwards, first, towards n = 3 for Φ = Φ2→3 and then towards n = 4 for Φ = Φ3→4. In this case, Un = 2, Φ → 2π → ∞, which conceptually reflects the fact of a sharp increase in the rigidity of the arising sectors and hence in the accumulated elastic energy and formally shows that the beam model cannot be used. As follows from the results in Sec. 4, the computed length of the arising cracks can be comparable with the general dimensions of the plate Lp (for example, for typical window glass). This means that, on the one hand, there is a natural upper limit for possible values of lengths of the arising cracks, and on the other hand, it is necessary to take into account the plate dimensions and the corresponding boundary conditions. Consider one purely kinematic consequence of the boundedness of possible crack lengths. As the crack length L = Lp is attained in the energy balance Eq. (4), the further increase in W in the left-hand side can be counterbalanced in the right-hand side for fixed L = Lp only by an increase in n. For a small excess over the calculated L > Lp, the energy excess is small and obviously can be radiated as elastic vibrations and waves (which is not detected by the proposed model). But, starting from a certain moment, the accumulated energy becomes sufficient for the formation of a picture with five rather than four symmetric cracks, and then with six, etc. Then, in general, the number n of arising cracks is always determined as the integral part of the solution of an equation of the form Eq. (4) with respect to n for Lp and given values of W (or u*) and the other quantities contained in it. Thus, for a sufficiently small imperfection (high strength) of the plate, which permits accumulating a large amount of elastic energy, the finiteness of its dimensions may result in an increase in the number of cracks arising in it. In the case of nonsymmetrical conditions of the plate support (when the lengths of the arising cracks are limited only in several directions), the symmetry of the crack formation picture is generally violated. For example, consider a plate in the form of a long strip clamped along the long sides. Let us trace the evolution of the crack formation picture as the accumulated elastic energy and, respectively, the lengths of the arising cracks increase. As long as these lengths are much less than the characteristic dimensions of the plate, the picture remains symmetric (for simplicity, we assume that the cracks are oriented as in Fig. 4). But for sufficiently large cracks such that L/21/2 ≅ b/2 (where b is the plate
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