Continuum-Based Theories

The foregoing description is, at least for concrete, only half of the story: the half dealing with researchers interested in discovering when and how a preexisting crack-like flaw or defect would grow. The other half deals with structural engineers wanting to describe the crack formation and growth from an initially flaw-free structure (in a macroscopic sense). The first finite element approaches to that problem consisted in reducing to zero the stiffness of the elements in which the tensile strength was reached (Rashid 1968). Later, more sophisticated models were used with progressive failure of the elements (progressive softening) and, starting with the work of Kachanov (1958), there was a great proliferation of continuum damage mechanics models with internal variables describing softening.

However, even though some results were very promising, it later became apparent that numerical analysis using these continuum models with softening yiclded’results strongly dependent on the size of the elements of the finite element mesh (see the next section for details). To overcome this difficulty while keeping the continuum mechanics formulation —which seems more convenient for structural analysis— Bazant developed the crack band model in which the crack was simulated by a fracture band of a fixed thickness (a material property) and the strain was uniformly distributed across the band (Bazant 1976, 1982; Bazant and Cedolin 1979, 1980; Bazant and Oh 1983a; Rots et al. 1985). This approximation, analyzed in depth in Chapter 8, was initially rivalling Hillerborg’s model, but it soon became apparent that they were numerically equivalent (Elices and Planas 1989).

Since the 1980s, a great effort, initiated by Bazant (1984b) with the imbricate continuum, was devoted to develop softening continuum models that can give a consistent general description of fracture processes without further particular hypotheses regarding when and how the fracture starts and develops. In the nonlocal continuum approach, discussed in Chapter 13, the nonlinear response al a point is governed not only by the evolution of the strain at that point but also by the evolution of the strains at other points in the neighborhood of that point. These models, which probably constitute the most general approach to fracture, evolved from the early nonlocal elastic continua (Eringen 1965, 1966; Kroner 1967) to nonlocal continua in which the nonlocal variables are internal irreversible variables such as damage or inelastic strain (Pijaudier-Cabot and Bazant 1987; Bazant and Lin 1988a, b). Iligher-order continuum models, in which the response at a point depends on the strain tensor and on higher order gradients (which include Cosserat continua) are related to the nonlocal model and are also intended to handle fracture in a continuum framework (e. g., de Borst and Miihlhaus 1991). However, the numerical difficulties associated with using generalized continuum models make these models available for practical use to only a few research groups. Moreover, sound theoretical analysis concerning convergence and uniqueness is still lacking, which keeps these models somewhat provisional. Nevertheless, the generalizing power of these models is undeniable and they can provide a firm basis to extend some simpler and well accepted models. It has been recently shown, for example, that the cohesive crack models arise as rigorous solutions of a certain class of nonlocal models (Planas, Elices and Guinea 1993).