13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- ( ) 3 3 1 ( ) 1 3 d D e A d m r r r A D A e e r = + + − − ⊗ . (7) From (7), with 2 d d D D dD = : , the expression of EQ d in (3) can be written as ( ) 6 2 3 2 2 2 6 3 2 4( ) 2 4( ) 1 3 3 3 D EQ e d m e m d r r r D D A r d A D A e e r r α − = + − + + : − ⊗ . (8) The velocity is 0 Ov = in the rigid core. Note that the condition 0 Av = on the boundary i r r = cannot be fulfilled by the velocity field defined in (6). This implies that the dissipation associated with a discontinuity of velocity must be considered at the boundary I ( i r r = ) (section 3.4). 3.2 Macroscopic support function Defining the macroscopic strength domain hom G as the set of admissible macroscopic stress statesΣ, the macroscopic support function reads ( ) sup( ) D D hom hom G Π = Σ: ,Σ∈ . Considering the set K of k.a. velocity fields withD, ( )D hom Π is characterized as [5]: ( ) 1 ( ) inf ( ) ( ) v D d F m hom A A I K dV v dS π π ∈ Ω Π = + |Ω| ∫ ∫ . (9) where 3 4 3 erπ |Ω|= / . In the surface integral, v denotes the velocity discontinuity at the core boundary I and ( )vπ represents the associated surface density of dissipation. In the line of reasoning of Gurson approach, ( )D hom Π is approximated by the minimal dissipation obtained among the velocity fields Av defined in (6): ( ) 1 ( ) inf ( ) ( ) D d F m hom A A I A R dV v dS π π ∈ Ω Π = + |Ω| ∫ ∫ . (10) For further use, let us introduce the following notation: 1 1 ( ) ( ) ( ) ( ) d D D F m m I A A I dV A dS A v π π Ω = , ; = , Π Π |Ω| |Ω| ∫ ∫ . (11) Accordingly: ( ) inf ( ) D D hom A R A ∈ Π = Π , . (12) ( ) ( ) ( ) D D D m I A A A Π , = , + , Π Π . (13) Once ( )D hom Π is determined, the macroscopic admissible stress states on the boundary hom G∂ are derived according to: ( )D D hom ∂Π Σ= ∂ . (14) The stress state of (14) lies on the boundary of hom G at the location where the normal is parallel to D. The overall dissipation of (12) proves to read in the following form:
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