13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 4. Macroscopic criterion The macroscopic support function can be determined by minimizing the sum ( ) ( ) ( ) D D D m I A A A Π , = , + , Π Π with respect to the parameter A. Accordingly, the boundary of hom G is determined according to (14)[7]: ( ) ( ) with 0 D D D A A A ∂Π , ∂Π , Σ= , = ∂ ∂ . (30) 5. Comparison with the result obtained by a variational approach Predictions according to (30) of the macroscopic criterion derived in the framework of the cinematic approach of limit analysis are now compared with the result obtained by the variational approach [11]. For the derivation of their criterion, these authors consider a variational approach in the two homogenization steps. Their criterion reads: 2 hom 2 2 2 2 3 3 2 3 ( ) 1 2(1 ) 0 (1 ) 2 3 2 d m m f f f F f T f h f h T f ρ + + ⎛ ⎞ Σ, , =ΘΣ+ − Σ+ − Σ− − = ⎜ ⎟ + ⎝ ⎠ . (31) with ( ) 2 2 2 2 1 2 3 3 2 3 2 4 12 9 6 13 6 1 1 f f T T T f T f ρ ρ + / − − − − + − Θ= + . (32) Applying the parameters f=0.25 and T=0.525, the comparison between the results predicted by the two different methods is shown in Fig.1 As it can be seen in Fig.1, the analytical estimate (31) obtained by the variational approach and the prediction from (30) based on the ’rigid core sphere model’ show a very good agreement for purely isotropic stress states, both in traction and compression. It is noteworthy that the strength under purely isotropic stress seems surprisingly almost unaffected by the volume fraction ρ of the rigid core. Although the shapes of the yield surfaces predicted by the two methods are similar, the strength predicted by limit analysis always overestimates that predicted by the variational method. In particular, as far as the strength under pure shear loading is concerned, the difference becomes very important when the volume fraction of the rigid core ρis larger. In order to gain a deeper understanding of the effect of the parameter ρ, the isotropic strength will now be compared with the exact solution predicted by the so-called static approach (sections 6). On the other hand, we note that the strength predicted by limit analysis (upper bounds) overestimates that predicted by the variational method under shear loading.(We focus on the strength proprieties under isotropic loading. The strength under shear loading has not been discussed in the present paper.)
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