13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 4. Example After the number n of cracks formed in fracture is found, one can use the Eq. (9) to estimate the order of the lengths of these cracks L, for example, in the case of a solid plate made of window glass. But here we encounter another difficulty. By Eq. (9), the length of the formed cracks is determined by the value of the critical deflection u*. In the framework of this model, nothing can be said about u*, since we do not specify any local fracture criterion and do not study the stress field distribution. But if for some u* the plate is destroyed according to the above model, then in this plate there arise four symmetric cracks of length determined by Eq. (9) with n = 4. This implies an interesting observation. Suppose that an artificial stress concentrator, a conic cave (countersinking) is placed on the lower part of the plate under the indenter. Then different u* are realized depending on the dimensions (depth and opening angle) of this cave, and, respectively, systems of cracks of different L will be formed. In a similar way, in the case of symmetric extension of the strip edges (Fig. 3) by u under the action of loads applied on a small part of dimension d > δ (δ is the unknown dimension of the defect in the material), we assume that the plate is mechanically isotropic, in strength and in imperfection, and we do not precisely know what defects are contained in the plate. But since the crack-like defects perpendicular to the load direction are most dangerous, the fracture occurs for different displacements u* of the force application points depending on the maximum initial dimension δ of such defects. The lengths of the arising cracks L are different and correspond to the energy U accumulated at this time. Figure 3 To obtain estimates of the length L of the arising cracks by Eq. (9) in the framework of the proposed model, it is necessary to introduce some reasonable values of the critical deflection u* and some actual values of the glass mechanical characteristics E, h, and γ. The effective surface fracture energy γ can be expressed in terms of the crack growth resistance KIC by the Irwin formula E KIC ) (1 2 2 −μ γ = (19) where μ is the Poisson ratio. For glass, we set E = 6×1010 N/m2, μ = 0.3 ([4], p. 116), and KIC = 0.8 kg/mm3/2 = 0.8⋅10 N/(10−3 m)3/2 = 8×104.5 N/m3/2 ([5], p. 620); the typical values of the glass thickness h are h ≅ (1 ÷ 10) mm = (10-3 ÷ 10-2) m; for u*, we take several values proportional to h by the formula u* = αh, where α = 1; 10−1; 10−2; 10−3 . By substituting γ expressed by Eq. (19) into Eq. (9) and by taking n = 4, we see that for such parameter values the lengths of the cracks arising in glass can be of the order of several centimeters already for α = 10-3. For the thickness h = 4⋅10-3 m typical of window glass, the relative deflections α = 10-3, 10-2 imply the values L ≅ 0.1 m and L ≅ 0.5 m, respectively, for L.
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