Relation to Micromechanics of Fracture
The normal microstrains across the fracture process zone may be distributed roughly as shown in Fig. 8.7.2a. The discrete fictitious crack model simplifies this random strain distribution as a Dirac delta function, Fig. 8.7.2b. The crack band model simplifies it as a rectangular strain distribution, Fig. 8.7.2c. The nonlocal continuum model, which wc will discuss in Chapter 13, describes this strain distribution as a smooth bell-shaped profile across the crack band, as shown in Fig. 8.7.2d (cf. Bazant and Pijaudier-Cabot 1988). The finite element approximation to the nonlocal continuum simplifies the strain distribution in the form of several steps as in Fig. 8.7.2e. Now, which representation is more correct?
Among the simple distributions, i. e., those for the fictitious cohesive crack and the crack band (Figs. 8.7.2b-c) neither one is better or worse, as an approximation to the true distribution in Fig. 8.7.2a. Efforts have been made to physically observe the microcracks and strains throughout the fracture process zone. In optical microscopic observations, distributed cracking has not been seen in concrete (although it has been clearly observed in ceramics). The explanation might be that it is difficult to distinguish new very fine microcracks from the pre-existing ones, or that the microcracks on the fringes of the fracture process zone have extremely small openings while being extremely numerous and thus still contributing significantly to the overall relative displacement across the width of the fracture process zone.
With regard to the optical observations, it must be noted that fracture in concrete is normally highly tortuous, meandering to each side of the fracture axis by a distance approximately equal to the maximum aggregate size (Fig. 8.7.20- Now, even if all the microcracking is concentrated on a line, but this line is highly tortuous, the fracture is represented by a straight line crack no better than by a crack band of width of about tvvo aggregate sizes. So, even if cohesive cracks are a reality for concrete, one still cannot claim that a straight fictitious crack is a more realistic model than a crack band.
It is also pertinent to mention that measurements of the localizations of the acoustic emission during the fracture process in concrete (Labuz, Shah and Dowding 1985; Maji and Shah 1988) indicate, despite considerable scatter, that the locations of the emission sources are spread over a relatively wide band in the frontal region of fracture. This tends to support the crack band model. On the other hand, various measurements of strains on the surface, for example by interferometry (Cedolin, DeiPoli and Iori 1983, 1987) or by laser holography (Miller, Shah and Bjclkhagen 1988), indicate that very high strains arc concentrated within a very narrow zone at the front of fracture. This might be better modeled by a cohesive crack than a crack band. It should be noted, though, that the fracture strains might be localized at the surface of concrete specimens to a greater extent than in the invisible interior, due to the wall effect as well as other effects.
In view of the foregoing three arguments, there seems to be no compelling reason for rejecting cither the crack band model or the cohesive (fictitious) crack model. The choice seems to be a matter of convenience of analysis. When the fracture shape is known in advance, both formulations appear to be about equally convenient. However, if the shape of the fracture is unknown in advance, the crack band model might be more convenient.