13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Fig. 6. 4. The elastic energy Now we write the expression for the right hand side of (1) – elastic energy We, stored in the near-well layer, cut through with the cracks, which goes to the formation of cracks W W W RdR nR fs f e π = = ∫ 2 1 where Wf is shear energy, Wfs (s = specific) – specific shear energy, and the integration is over the ring R ∈ [1, Rn]. According to [25, p. 284, formula (7.28)], the expression for the specific shear energy can be written as ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ]2 1 2 2 2 2 1 2 1 3 2 3 2 2 2 1 6 1 6 1 σ −σ +σ +σ +μ = σ−σ +σ−σ +σ−σ = +μ = E E Wfs (14) where μ is Poisson's ratio, E – Young's modulus, σ1,2,3 – principal stresses. Substituting for the right side of (14) the well-known Lame solution for the very thick cylinder with inner radius R*, being under internal pressure P [25, p. 338-339, formula (9.21)] 2 2 1,2 * r P R R P ⎟ =± ⎠ ⎞ ⎜ ⎝ ⎛ σ =± (15) we obtain for the specific Wfs and entire Wf shear energy in the ring (layer) r ∈ [1, rn] ( )( ) ( ) ( ) ( ) ( ) ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ − ⎠ ⎞ ⎜ ⎝ ⎛ π = = =π +μ +μ = +μ + + = +μ = ∫ 2 2 2 1 4 2 4 2 4 2 2 1 1 * * 1 2 1 1 2 1 1 6 1 n R f fs r E P ER RdR R R r r P E W r r P r E P E W n (16)
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