13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- µ23! x3 x2 x1 µ23 >0 µ23 <0 Flexure crack µ13! x3 x2 x1 µ13 >0 µ13 <0 Flexure crack (a) (b) Figure 2. The direction of bending and the microcracking for (a) Mode I and (b) Mode II cracks. The corresponding bond is broken independently of the direction of microcrack (flexure crack) propagation, which is controlled by the sign of the corresponding moment stress. 3. Small-scale Cosserat continuum We consider here the case of isotropic particulate material with internal rotations. The corresponding Cosserat continuum is defined by the following equilibrium and constitutive equations in the co-ordinate frame (x1, x2, x3) (e.g. [34]) σji, j =0, µji, j +εijk σjk =0, i, j =1,2,3 σji = µ+α ( ) γji + µ−α ( ) γij + λγkk µji = γ+ε ( ) κji + γ−ε ( ) κij + βκkk (2) where σij and µij are stress and moment stress, εijk is the alternating tensor. Here we use the deformation measures - the strain and curvature-twist tensors γji =ui, j − εkji ϕk , κji = ϕi, j (3) where ui and ϕi are independent displacement and rotation vectors respectively and index (,j) denotes differentiation with respect to xj, µ, α, γ, ε, λ, β are the Cosserat elastic moduli. According to [29-31] the main term in the asymptotics lm H→0 can formally be obtained from the equations of the Cosserat continuum with constrained rotations (the couple stress continuum),
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