Elastic-Softening Crack Band Models
As for cohesive cracks, the prepeak stress-strain relation can be nonlinear, but for many purposes it is enough to assume a linear behavior up to the peak followed by softening (Section 8.2,2). Then, the stress-elongation curve is given by (8.2.4), for arbitrary h. In Baz. ant’s approach, the width of the band cannot be less than a certain characteristic value hc. Thus, substituting h — hc in (8.2.4), we get an expression that is formally identical to the corresponding expression for the cohesive crack if we identify hcgf with the cohesive crack opening displacement w.
Thus, the stress-elongation curve for the band model and for the cohesive model will coincide if we relate the softening curve of stress vs. fracturing strain ф(е/) to the softening curve of stress vs. crack opening of the cohesive crack, i. e.,
4>(ef) = /(«’) = f(hcsf) or f(w) = <j>(w/hr) (8.3.2)
where f(w) is the equation of the softening curve for the cohesive crack model. Therefore, there is a unique relationship between the crack band model and the cohesive crack model, at least for the simple elastic-softening case that we are analyzing. The correspondence is illustrated in Fig. 8.3.1 which shows the softening curve for the cohesive crack (Fig. 8.3.1a) and the corresponding stress-strain curve for the crack band (Fig. 8.3.1b). Also shown is the correspondence for the initial linear approximation to the curve, the horizontal intercept of which satisfies e = W/hc. It follows that a linear approximation for the softening of crack bands will be a good approximation in the same circumstances as it was for the cohesive crack model, principally for peak loads of not too large specimens, if notched, but any size specimens, if unnotched). This explains why the use of linear softening was very successful in the work of Bazant and Oh (1983a).
Figure 8.3.1 Correspondence between the softening curve of the cohesive crack model (a), and the stress-strain curye of the crack band model (b).
The correspondence is obviously maintained for the specific fracture energy Gf – Indeed, from (8.2.6) it follows that the energy required to form a complete crack (or a fully softened band) is
Gp — – Д – = hc 7p (8.3.3)
From this, the characteristic size £ck can be easily obtained in terms of the properties of the crack band model as
where є і is the horizontal intercept of the initial tangent (Fig. 8.3.1b).
A parameter of interest in numerical calculations using the crack band model is the softening modulus Et for the linear approximation (Fig. 8.3.1b). It is a simple matter to show that
The correspondence between the two models can further be systematized by defining a characteristic strain ec/,’and a reduced fracturing strain є* as
heft ft – r e!
£ch – = —- and ef ———————– (8.3.7)
Gf If ech
With this, the nondimensiona! expression for the softening function is identical to that for the cohesive model, with the obvious change <-> tt>.,i. e.:
о = f[i!) (8.3.8)
Therefore, all the softening curves discussed in the previous chapter can be directly implemented in the crack band model. The only difference between the results for one and other model is in the strain and displacement distribution. Figs. 8.3.2a and b show the comparison of the axial displacement distribution in a bar for a cohesive crack and a crack band model. Figs. 8.3.2c and d show the corresponding strain distributions. Obviously, the difference is nil for engineering purposes if hc L. This is almost invariably true in practical situations because, as we will see later in more detail, hc is of the order of a few maximum aggregate sizes (Baz. ant and Oil 1983a).
Figure 8.3.3 (a) Curvilinear stress-strain curve, (b) Curvilinear stress-fracturing strain curve, (c) Plot of the curve defined in Eq. (8.3.9).