# Band Models with Bulk Dissipation

The crackirtg in reality does not begin upon reaching the tensile strength, but earlier, and so the diagram of stress vs. strain should have the form shown in Fig. 8.3.3a. As discussed in Section 7.1.6 for cohesive crack models, such behavior can be incorporated into the computational models with relative concep­tual simplicity, but it considerably complicates the numerical treatment and experimental interpretation. Moreover, at least for concrete, neglecting the prepeak nonlinearity is generally acceptable for practical engineering use, and so the elastic-softening models we previously discussed are those most used. This seems to be clearly established (Planas, Elices and Guinea 1992) when there is one main crack, i. e., sharp localization occurs, because then the large postpeak strains dominate over the prepeak deformation. However, for situations where the localization is not sharp, the prepeak nonlinearity may play a dominant role, and its inclusion might be necessary. This may be the case in the prelocalization stages when there is reinforcement or when the stress field has a high gradient (as in the case for shrinkage stresses).

In our discussion of cohesive crack models, the inclusion of bulk dissipation (prepeak inelasticity) required defining an inelastic constitutive equation for the bulk in addition to the softening curve for the cohesive crack. One of the appealing features of the crack band model is that such a dichotomy is not necessary. Indeed, it is enough to define a single curvilinear stress-strain curve such as that shown in Fig. 8.3.3a. Then we can split the strain into the elastic and inelastic or fracturing part and use the curve of stress vs. inelastic strain as shown in Fig. 8.3.3b. For example, Ba/.ant and Client (1985a) proposed

the following-power-exponential curve:

a -= ф{є1) = CE{Ve~be!" (8.3.9)

where C , p, b, and q are constants, Fig. 8.3.3c shows the appearance of this curve forp = 1/3, q = 0.55 as derived by Bazant and Chern (1985a) by lining of various experimental data. Note that the curve has been nondimensionalized so that its peak and area are equal to one.

Alternatively, the complete stress-strain curve can be given in the form a — ф(є). This is equivalent to giving the a(s-f) curve in parametric form as: a — ф(є)

ES = Є 1;ф(є)

Among the formulas of this kind we have the power-exponential form (Bazant 1985a)

a = Еее~ы"

where E is the elastic modulus and b, and q are constants. Another expression is  Ее

1-foe-f beq

which was introduced by Saenz (1964) for compression strain softening, and in which a, b and q arc constants.