Constitutive Relation and Gradient Approximation
As is clear from the foregoing exposition, the constitutive relation is defined only locally. It yields the inelastic stress increment Д5′(|)(х), illustrated by segment 32 shown in Mg. 13.3.2. In the previous nonlocal formulations, by contrast, the nonlocaljnelastic strain, stress, or damage was part of the constitutive relation. This caused conceptual difficulties as well as continuity problems with formulating the unloading criterion. Furthermore, in the case of nonlocal plasticity, this may also cause difficulties with the consistency condition for the subsequent loading surfaces.
Here these difficulties do not arise, because the nonlocal spatial integral is separate from the constitutive relation. Thus the unloading criterion can, and must, be defined strictly locally. If plasticity is used to define the local stress-strain relation, the consistency condition of plasticity is also local.
In principle, the nonlocal model based on crack interaction can be applied to any constitutive model for strain-softening, for example, parallel smeared cracking, isotropic damage theory, plasticity with yield limit degradation, plastic-fracturing theory, and endochronic theory. But to fully realize the potential of this approach, a more realistic model, such as the microplane model, appears more appropriate and has already been applied by Ozboll and Bazant (1996). This will be discussed and documented in the next chapter.
Recently there has been much interest in limiting localization of cracking by means of the so-called gradient models. These models can be looked at as approximations of the nonlocal integral-type models, and can be obtained by expanding the nonlocal integral in Taylor series (Bazant 1984b); see Section 13.1.3. Unlike the present model, there have been only scant and vague attempts at physical justifications for the gradient models, especially for aggregate-matrix composites such as concrete. It seems that the physical justification for the gradient models of such materials must come indirectly, through the integral-type model. However, if that is the case, the present results signal a problem, if the spatial integral in (13.3.9) were expanded into Taylor series and truncated, the long-range decay of the type r~2 or r~3 could not be preserved. Yet it seems that this decay is important for microcrack systems. If so, then the gradient approximations are physically unjustified.