Crack Band Models and Smeared Cracking
Modeling of fracture by discrete line cracks, which has been discussed in the preceding chapters, is not the only viable approach. Another approach, which has gained wide popularity in finite element analysis of concrete structures (Meyer and Okamura, fids., 1986) and is used almost exclusively in design practice, is to represent fracture in a smeared manner. In this approach, introduced by Rashid (1968), infinitely many parallel cracks of infinitely small opening are imagined to be continuously distributed (smeared) over the finite element. This can be conveniently modeled by reducing the material stiffness and strength in the direction normal to the cracks after the peak strength of the material has been reached. Such changes of the stiffness matrix are relatively easy to implement in a finite element code, and, hence, the appeal of smeared cracking. The evolution of the cracking process down to full fracture implies strain softening, a term which describes the postpeak gradual decline of stress at increasing strain.
The term evolved from the terminology of plasticity where work hardening describes the gradual increase of yield stress resulting in a rising stress-strain diagram of a slope that is positive but smaller than the elastic slope. After it was realized that the hardening is not merely a function of the plastic work, a scalar, but depends on all the components of the strain tensor, the term strain hardening has been adopted. From the viewpoint of plasticity, the postpeak decline of stress may be regarded as a gradual decrease of the yield limit, i. c., softening. This phenomenon again is not just a function of work (in which case we could speak of work softening) but of all the strain components; hence, strain softening.
The smeared cracking (with strain softening), however, leads to certain theoretical difficulties which were initially unknown or unappreciated. They consist of the so-called localization instabilities and spurious mesh sensitivity of finite element calculations. After years of controversies and polemics, it has now been generally accepted that these difficulties can be adequately tackled by supplementing the material model with some mathematical condition that prevents localization of smeared cracking into arbitrarily small regions. The simplest way to attain this goal is the crack band model, which is the object of this chapter.
Since it is essentia] to understand why fracture cannot be consistently and objectively described just by postulating a stress-strain curve with softening and nothing else, we first analyze in this chapter the strain localization in systems displaying softening. We start with the series coupling of discrete elements (Section 8.1), which serves as the starting point for the analysis of the localization of strain in a softening bar (Section 8.2). From this, it follows that some kind of localization limiter must be associated with the softening stress-strain curve to get meaningful results. Next, we analyze the basic issues in the crack band model, in the simplest uniaxial approximation (Section 8.3). Then we deal with the underlying stress-strain relations with softening, first in the simple uniaxial version (Section 8.4) and then in full three-dimensional analysis (Section 8.5). After this, we discuss the triaxia! features of the crack band models and smeared cracking, with emphasis placed on the numerical issues (Section 8.6). A comparison of the crack band and cohesive crack approaches closes the chapter (Section 8.7).