13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- results for the cases analysed here. Eq. (5) is used after each failure event to calculate the evolution of the moment with damage. In addition, the maximal component is monitored. This is the largest connected cracked surface. 3. Results and discussion Figure 3 shows the results of the evolution of the damage parameters defined by Eq. (4) as functions of the moment of crack population defined by Eq. (5). Recall that a damage parameter, based on the relative change of the Young’s modulus equals the damage parameters based on the individual stress components, deviatoric and hydrostatic, when the material remains macroscopically isotropic. In this case the same damage parameter describes the relative change of the shear modulus. The results for the cases of uniaxial extension, unconfined (a) and confined (d), show equality of the four damage parameters (approximate in case 4). This suggests that microscopic isotropy is maintained during damage evolution and the results reproduce very closely the linear relation predicted by the theory and given by Eq. (1). Interestingly, an estimate for the slope of the linear function from the figures is about 1.5, which is very close to the value of β reported in relation to Eq. (1). Figure 3. Damage parameters relation to crack population moment. In the cases of plane stress (b) and plain strain (c), however, the development of damage is radically different, illustrating the development of damage-induced anisotropy. In this case the damage parameter Di represents the relative reduction of the longitudinal shear modulus in direction Xi. Note that this is not the shear modulus relating shear stress to shear strain. In both plane cases, the evolution of D1 suggest that the system undergoes transition into negative longitudinal shear resistance, quite more pronounces in the plane strain case (c), while the shear resistance in direction X2 increases from its initial value. This behaviour may seem unusual, but it is not impossible for anisotropic materials. The bounds for Poisson’s ratios in such materials calculated in [20, 21] allow for negative longitudinal shear moduli with the values recorded here. The results merely show that extreme anisotropy has been developed in the material with the evolution of micro-crack population under the two plane cases. The development of the hydrostatic damage is also affected in these cases, as it cannot be described as a linear function of the cracked area moment.
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