For stress-strain curves with prepeak inelasticity, the energy dissipation consists of two terms. One corresponds to the energy dissipated in the prepeak range, which is proportional to the volume, and the second corresponds to the energy dissipated after peak which, in the cases of localization in a single band, is proportional to the volume of the band hcA, where A is the area of the main surface of the crack band. Therefore, the problem is identical to that for the cohesive crack with bulk dissipation (Section 7.1.6).

An analysis analogous to that for the cohesive cracks was performed by f. dices and Planas (1989): for a uniform bar in tension (Fig. 8.3.5a), with the stress-strain curve shown in Fig. 8.3.5b, the material follows initially the path OP up to the peak. Then the bifurcation occurs and the material outside the crack band follows the unloading path PBB’ while the material in the crack band follows the path PAA’. Therefore, the total work of fracture is

WF = A(L — hc) area{OPBB’O) + Ahc area(OPAA’O) .-

AL area {OPBB’O) + Ahc яка.{ВВ’АЛ’В) (8.3.14)

The area OPBB’O (lightly shaded in the figure) is the energy supplied to a unit volume when it is loaded up to the peak and then unloaded; we represent it by 7ц. The area BB’AA’B (darker shading in the figure) represents the extra energy supply required to break a unit volume of material in the crack band. Therefore, we may write the foregoing equation as

WF = AI/yu + AhcjF (8.3.15)

Now, identifying the second term as the surface energy dissipation, we can apply (8.3.3) and get an

expression identical to (7.1.19). From this point on, the analysis is identical to that for the cohesive cracks with bulk dissipation, which leads to a dependence of the mean fracture energy given by (7.1.22). This can obviously be recast in terms of the properties of the stress-strain curve using (8.3.3):