13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ( ) 1 { } 2 d d I A A n n v v δ = + ⊗ + ⊗ . (20) where {}d is the standard expression of the strain rate corresponding to its smooth part, Iδ is the Dirac distribution which support is the surface of discontinuity I and r n e= is the unit normal to this surface. The surface density of dissipation ( )Avπ contributed by the velocity jump Av is therefore related to the support function F π according to (see e.g. [10]): ( ) ( ) d F I Avπ π = . (21) where dI is defined as ( ) 1 2 dI A A n n v v = ⊗ + ⊗ . (22) Recalling (6) and (22), the strain rate dI associated with the velocity jump can be obtained and written as ( ) 1 1 1 ( (1 ) ) ( ) ( ) 2 d D D I i m d d r r r r r r r D A e e e e e e ρ ρ ⎛ ⎞ = + − ⊗ + ⋅ ⊗ + ⊗ ⋅ ⎜ ⎟ ⎝ ⎠ . (23) Eventually, the surface density of dissipation is derived from the combination of (21) and (3) 0 2 ( ) with tr 3 d d d EQ EQ I I I I I I I v v A d d d H d vπ σ λ = − = : : ; = . (24) 3.4.2. Contribution of the interface to dissipation Recalling (16), the macroscopic dissipation related to the part of inclusion-matrix interface depending also on the scalar A can be written as 0 1 ( ) D EQ i I I I v r r A d d dS σ λ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ = , = − Π Ω∫ . (25) Again, the integration of I eq d is approximated by the upper bound (17) which yields 2 0 4 2 ( ) 3 D d d i i I I I r r r A Y X Y H dS π σ λ = , = − ; = : : Π Ω ∫ . (26) Using (23), Y takes the form 2 2 15 P Q Y α + = . (27) with [ ]2 2 2 2 2 51 6 45(1 2 ) (1 ) 2 d m Q D P D A α ρ α ρ ⎛ ⎞ = + , = + − − ⎜ ⎟ ⎝ ⎠ . (28) In turn, observing that the average D i r d r r r e e = ⋅ ⋅ of D i r d r r r e e = ⋅ ⋅ over the orientations of re on the sphere I is equal to 0, it is readily seen that ( ) ( ) 1 3 (1 ) a I v m S r X d dS D A ρ = = − − Ω∫ . (29)
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