13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- To understand what causes the anisotropy in the plane cases the structure of the crack population needs to be studied in more detail. This requires a single damage parameter; an appropriate choice is the relative reduction of the strain energy density in the system, D = 1 – W / W(0), which is found to be approximately equal to the damage parameter defined via the relative reduction of the hydrostatic stress in all cases, see Fig. 4(a). The development of the maximal graph component, i.e. the main crack, with damage is shown in Fig. 4(b) with the ratio between the area of the maximal component, Am, to the total cracked area, A. It is clear that the main crack becomes dominant very early in the development of damage (at damage less than 1%) and its relative area grows nearly exponentially for all cases. It seems therefore sufficient to examine the structure of the maximal component as the damage develops. Figure 4. Hydrostatic damage (a) and relative area of main crack (b) development with damage defined as relative reduction of strain energy density. Figure 5 shows the development of the maximal component area, split into the areas of surfaces normal to the three principal axes, A1, A2, A3, and the surfaces formed on octahedral planes, A4. All areas are normalised with the total areas of the corresponding boundaries in the cellular structure. In the cases of uniaxial extension, unconfined (a) and confined (d), the development of the main crack involves creation of surfaces normal to the applied load and on octahedral planes. Although there is a difference between the two cases in the rates of creation of normal and octahedral surfaces, the overall balance results in isotropic damage, see Fig. 3(a),(d). Figure 5. Structure of maximal graph component with damage.
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