13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- However, we observe that m ±Σ remain in the neighborhood of 2m ±Σ . We therefore propose to approximate the functions D+ andD− by series expansions in the neighborhood of 2m +Σ and 2m −Σ . In the case of of series expansion to the second order, The analytical solutions at order 2 read: ( )(1 ) 1 m c c η η ρ κ η ρ ± − Σ = + − − + . (45) with the following parameters for isotropic compression or traction: 2 2 1 1 1 3 2 1 3 2 or 2 6( 1) 2 6(1 ) m m m m f f f f hT f f hT c c η η κ κ ⎧ ⎧ − + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − + ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ =Σ =Σ + + = = − − =Σ =Σ . (46) 7. An approximate analytical macroscopic criterion We seek an approximation of the criterion by an ellipse in the ( ) m d Σ ,Σ plane. The ellipse intersects the mΣ -axis at the points ( 0) m −Σ , and ( 0) m +Σ , . The center of the ellipse is located at the middle of these points. We still have to determine the ordinate of the center of the ellipse which corresponds to the strength under pure shear ( 0 mD = ). With the condition 0 mD = , it is readily seen that the value of the parameter A which minimizes ( ) D A Π , is A=0. With Dm=0 and A=0, closed-form expressions of the mean stress and the maximum shear stress can be obtained from (30): 0 2 17 4 (1 ) 3 15 m c d c α λ σ ρ ρ α , , ⎡ + ⎤ Σ =− ; Σ = ⎢ − + ⎥ ⎣ ⎦ . (47) where 0 , , λ σ α have been defined in (4) and (5). The subscript c recalls that this point is the center of the ellipse. Now let us try to approximate the criterion given in parametric form (obtained by means of cinematic approach) found in section 4 by an analytical elliptic criterion. Analytical expressions of the strength have been established at particular stress states, namely under isotropic loading and under pure shear loading (with m λ Σ =− ). Recalling (45) (46) for the expressions of ,m m + − Σ Σ in the case of a second order expansion, together with (47) for the expression of d c, Σ , the macroscopic criterion can be approached by the following analytical elliptic function: 2 2 1 2 m d L d c λ , ⎛ ⎞ ⎛ ⎞ Σ − Σ + = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Σ / Σ ⎝ ⎠ ⎝ ⎠ . (48) with L m m + − Σ =Σ −Σ and ( ) 2 m m λ + − = Σ +Σ / . Recall that , , m m d c + − , Σ Σ Σ are quantified in (45),(47) by (42),(43),(44); and 0 , , λ σ α are given in (4),(5). According to the comparison between the predictions of the analytical macroscopic criterion, (48), and the parametric criterion predicted by (30). We found that the comparison shows an excellent accuracy of (48).
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