13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- 1 2 2 2 0 0 1 2 ( ) 3 3 D m uN M u N A Narcsinh Y D M u ρ σ σ λ ⎡ ⎤ + ⎛ ⎞ Π , = − + − ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ . (15) with the notations introduced in (19), (27) and (28). For the sake of completeness, sections 3.3 and 3.4 respectively determine the contribution (18) of the shell mΩ (volume integral in (10)) and the contribution (26) of the interface I (surface integral in (10)) which has led to (15). Section 4 will consider the minimization w.r.t. parameter A. 3.3 Contribution of the shell to dissipation For a given value of parameter A, the contribution of the matrix to the macroscopic dissipation reads 0 1 ( ) D EQ m m v A d d dV σ λ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ Ω , = − Π |Ω| ∫ . (16) In order to obtain an analytical expression of ( ) D m A, Π , the approximation introduced in [4] is applied. Let ( ) S r denote the sphere of radius r. As a consequence of the Cauchy-Schwarz inequality, it is readily seen that 2 2 ( ) ( ) 4 EQ EQ S r S r d dS r d dS π ≤ ∫ ∫ . (17) We observe that the average ( ) 3 1 r r S r e e − ⊗ of 3 1 r r e e − ⊗ over the orientations of re on the sphere ( ) S r is equal to 0. Then, using (8) and (17) in (16), ( ) D m A, Π reads 1 6 2 2 2 2 0 6 2 2 2 0 1 4 2 2 ( ) 4( ) 3 3(1 ) D e m i r m e d m v r r D A A r D A dr d dV r uN M u N Narcsinh A M u ρ πσ λ α σ ρ λ Ω , = + − + − Π Ω Ω ⎡ ⎤ + ⎛ ⎞ = − − − ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ ∫ ∫ . (18) with 2 2 2 2 2 2 2 4( ) 3 d m A D M N D A α = + , = − . (19) 3.4. Inclusion-matrix interface Unlike the classical Gurson’s ’hollow sphere model’, the model proposed in this paper substitutes a rigid core for the void in the center of the thick-walled sphere. Therefore, owing to null velocity ( 0 Ov = ) in the rigid core, a velocity discontinuity tales place at the core boundary: ( ) ( ) a a A A r O A r r r v v e v v e = − = . 3.4.1. Surface density of dissipation The velocity field Av being discontinuous across the surface I (rigid core boundary), its gradient and the associated strain rate are to be defined in the sense of the distribution theory:
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