Nonlocalizedl Fracture: Third Parameter

As we recall from Section 7.1.3, if the shape of the tensile softening curve is fixed, then the cohesive crack (fictitious crack) model is defined by two material parameters, Gf and f[. The crack band model, on the other hand, is defined by three material parameters, Gf, f[, and hc. For the fictitious crack model, too, a third material parameter with dimensions of length, namely ich, has been defined (see Section 7.1.3); however, this is a derived parameter, not an independent one, while hc is an independent material parameter. Why does the crack band model, in its simplest form, have one more material parameter?

In answer to this question, we must first recall from Section 8.3.6 that, for localized fracture, the effect of the value of hc on the results is almost negligible, provided, of course, that the softening part of the stress-strain diagram is adjusted so as to always yield the same fracture energy Gf for any value of


Л-л cohesive cracks («>) and yS – crack band with h —»0 ,

– c чі/ —



crack band yS’ s’

with h =21/9


-1 1 | r-r



. . 1-і-.


– ^"20 ^cohesive cracks


in 1 with unadherent’ : ^ Ireinforcement



/-•-ib – reinforcement only :

2 4 6 8 10 12

(E/f ) A1./L




Nonlocalizedl Fracture: Third Parameter

Figure 8.7.1 (a) Reinforced concrete bar and definition oflinear cohesive crack model and the corresponding band model, (b) Resulting stress-mean strain curves for several possibilities (full lines are for adherent bar, dashed lines for unbonded bar).

hc. Therefore, in the case of isolated fracture, the crack band model has, in effect, only two material parameters, G f and //.just like the fictitious crack model.

The value of crack band width, hc, however, does make a difference in the case of nonlocalized fracture, that is, when densely spaced parallel cracks can form. Such situations, in which the strain softening state is stable against localization (in the macroscopic sense), can arise in various situations; for example, when there is sufficient reinforcement that can stabilize distributed cracking against localization (this occurs when the reinforcement is so strong that the tangential stiffness matrix of the composite of steel and cracked concrete is positive definite even though that of cracked concrete alone is not). Another possibility is the parallel cracks caused by drying shrinkage, which may be stabilized (against localization into isolated fractures) by the intact concrete in front of the cracks, due to shear stiffness of the material. The same situation can arise in bending, if the beam is sufficiently reinforced.

From these examples it transpires that the physical significance of hc is not really the width of the actual cracking zone at the fracture front but the minimum possible spacing of parallel cohesive cracks, each of which is equivalent to one crack band. Since adjacent crack bands cannot overlap (the material cannot be cracked twice), the distance between the symmetry lines of the adjacent crack bands is at least


Now, is it necessary that the minimum possible spacing of parallel cracks be a material fracture pa­rameter? In the early analysis of the problem it seemed, based on some examples, that it was so. For example, Bazant (1985b, 1986a) discussed the problem of a reinforced concrete bar loaded in centric tension, see Fig. 8.7.1, where the reinforcing bar represents five percent of the cross section area. In that case, smeared cracking is stable against localization. Bazant’s (1985b, 1986a) interpretation was that the cohesive cracks could form at any spacing, s, and as far as the fictitious crack model is concerned, these cracks could be arbitrarily close. He concluded that the number of cracks per unit length can approach infinity while each crack can have a finite opening width w. But this would mean that, according to the fictitious crack model, the energy dissipated by the cracking of concrete in the bar could approach infinity – a paradoxical result. On the other hand, if there is such a condition as the minimum spacing s, then, of course, the energy dissipated by the cracking in the bar is bounded, even according to the computations based on the fictitious crack model.

However, Bazant’s theoretical example can be reinterpreted in different terms, as done by Planas and Elices (1993b). For these authors, the cracks can be infinitely close while having a vanishingly small crack opening. This implies that no appreciable softening takes place, and thus the concrete deforms at <r = //. Therefore, the heavily reinforced concrete behaves in an elastic-perfectly plastic fashion. Moreover, this solution, with infinitely many cracks, is consistent with Bazant’s simplified analysis, which assumed that the cross-sections of the bar remained plane during the stretching process and that full bond existed between the bar and the concrete. However, it is easily shown that if two ctacks form at any finite spacing

Nonlocalizedl Fracture: Third Parameter

Figure 8.7.2 Strain distribution across the fracture process zone: (a) actual distribution; (b) cohesive crack ap­proximation; (c) crack band model approximation; (d) nonlocal approximation; (e) finite element approximation (adapted from report of ACI Committee 446 1992). (f) Tortuous crack path.

s, the subsequent infinitesimal loading step causes the tensile strength to be exceeded at the middle point between them. Therefore, a third crack must form and we have cracks spaced at s/2. Repeating the reasoning ad infinitum it turns out that if the bar is bonded and if we assume plane cross sections to remain plane, then the only solution consistent with the cohesive crack model is that cohesive cracks form infinitely close to each other. However, we know by experiment that, at the end, a collection of discrete cracks appear, even for strongly reinforced bars. The key point in the explanation of this effect is that, upon localization, the sections cease to be plane, an effect that cannot be caught by the simple classical analysis.

It is worth to note that the solution based on the assumption of Planas and Flices docs converge to the crack band solution for hc —> 0, as shown in Fig. 8.7.1 by the full lines. The dashed lines correspond to localized crack solutions valid only if bond is neglected, in which case the reinforcement is not interacting with concrete except at the ends of the bar, and then the bifurcation analysis given in the first section indicates that both the cohesive crack model and the crack band model predict that a single crack will occur.

Therefore, the simplified analysis of this problem seems to show that, in accordance with Planas and Elices (1993b) hypothesis, it is still possible to use the cohesive crack model for fully distributed cracking in conjunction with an associated elastic-perfectly plastic Rankinc model. The problem still remains, however, of determining when the fracture will localize. In their work on shrinkage, Planas and lilices made some special assumptions for the localization point, but pointed out that the actual localization must be determined by bifurcation analysis, which could be based on the principle of minimum second-order work (as done in the simple case of the series coupling model in Section 8.1.2 and justified thermodynamically in Bazant and Cedolin 1991, Sec. 10.1).