13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- ( ) 2 2 2 2 1 ln , 1 1 ln 1 : n n n n n n e r r Ap r r Ap r r W W = + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − = (23) Tangency condition we obtain by differentiating (23) n n n n n n n n n e r r r r Ap r Ap r r r r dr dW dr dW 2 1 ln , 2 ln 1 ln : 2 3 2 2 ⎟⎟= ⎠ ⎞ ⎜⎜ ⎝ ⎛ + + = − − = (24) Dividing (231) to (241), we obtain governing equations for rn 3 2 2 2 1 1 ln 1 ln ln 1 n n n n n n n n r r r r r r r r ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − − − (25) 2 1 1 2 1 ln + = − + + = n n n n r r r r (26) Equation (26) for rn is transcendental and needs numerical solution, but it can be seen that its root is close to e, therefore, representing rn in (26) as rn = e(1 + ε), expanding in powers of ε and holding the first order, one can obtain simple approximate estimates for ε, rn and n ( ) ( ) [ ] 4.2 0.75 3.14 1 ln 1 1 ln 0.25, ≈ ≈ +ε π ≈ +ε π = = +ε = π = ε≈− e r e r n n n It is seen, that the value for n obtained is a fractional number. For the model assumed it means that the solution will be one of the two integers closest to the fractional value found (ie, either 4 or 5), which gives a lower value for the elastic energy. Those integer solutions n appear with changing We(p) on increasing p, which, as is clear from Fig. 6, leads to the splitting of the solution found above into two solutions (corresponding to one and the same p), one of which, as can be seen from the graphs in Fig. 6, crawls down (rn1 ↓), and the other – up (rn2 ↑). By (13) w(rn) = (rn – 1)/lnrn is a monotonically increasing function (↑), ie rn1 <rn2 ⇒ w(rn1) <w(rn2); vice versa, n(rn) decreases monotonically (by (10): n = π /lnrn ↓). The lower elastic energy corresponds to the lower value of rn and, accordingly, to the larger value of n. Consequently, nmin = 5. Here, with the growth of p the total stored elastic energy in the body increases, but cracking consumes less energy due to the fact that though the number n of cracks increases, but because of the reduction in rn their total length l in (12) becomes less.
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