13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- Going from P and Wf with the aid of scales of pressure E and energy ( ) 2* ** 1 ER W ≡π +μ (17) respectively, to dimensionless p and wf by formulas f f f f P pE W W w W W w E P p ** , , ** , = = = = (18) we obtain for the dimensionless energy wf ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = − 2 2 1 1 ( , ) n n f r w p r p (19) The formula (19) determines for rn ≥ 1 monotonically increasing function, concave upwards, equal to 0 for rn = 1 and getting onto a horizontal asymptote at rn → ∞ . The graphs of this function for two values of p are shown in Fig. 6 by two lower curves . 5. Calculation of the index n For small internal pressures p shear energy plot wf(rn) lies below the cracking energy plot w(rn). The pressure arising, the in-tube stored elastic energy increases monotonically, the plot wf(rn) goes higher and at some moment it touches the graph w(rn) for some rn. The moment of contact will be the first moment when the elastic energy be equal to the energy required for the formation of an appropriate system of cracks (see similar arguments in Mohr theory [25, § 61, p. 300-306]). Touching specifies two conditions (equality of functions and equality of their derivatives) to determine two unknowns: the pressure and the thickness of the elastic layer, which gives its elastic energy for cracking. From the condition of equality of functions, by substituting into (1) the expressions (7), (13), (17), (19), we obtain W = W*w = Wf = W**wf (20) ( ) ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − γ +μ = − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − =π +μ − πγ 2 2 2 2 2 1 1 2 2 * 1 ln 1 , 1 * 1 1 ln 1 2 2 * n n n n n n r p ER r r r Ep R r r R (21) or, setting the dimensionless constant ( ) γ +μ = 2 2 * 1 ER A (22)
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