For some metals and other materials it was later noted that the cohesive zone should exhibit a gradual rather than sudden stress drop, characterized by a softening relation of the normal stress a across the crack vs. the crack-opening displacement Sc (Fig. 3.2d); see e.g. Knauss (1974), Kfouri and Miller (1974), and Wnuk (1974). For concrete, this type of model was proposed by Hillerborg, Modeer and Petersson (1976) under the name of fictitious crack model; see also Petersson (1980a, 1981), Hillerborg (1983, 1985a, 1985b). (The term “fictitious” refers to the fact that the portion of a crack which transmits tensile stress cannot be a continuous crack with full separation of the surfaces; the real crack ends at the point where the stress is reduced to zero; Fig. 3.2d.) In this model, which has been widely applied in finite element analysis of concrete fracture, the material fracture properties are defined by the softening stress-displacement relation:
= f ( 6 c) (3.1)
(see Fig. 3.2d), where ay is the stress in the direction normal to the crack. The area under the curve represents the fracture energy of the material, i.e.:
Jroo1 <Jyd6c (3.2) 0
The crack begins to open when the stress at the tip reaches the tensile strength limit / / . If the shape of the softening curve is fixed, then the fracture properties are completely characterized by two parameters: / / and Gf. The precise shape of the softening stressdisplacement diagram has a considerable influence on the calculation results. In various works this shape is considered triangular or bilinear, in which case the stress is reduced to zero at a finite displacement So. An exponential shape has also been used. When the softening zone is unloaded and reloaded (Fig. 3.2d), the behavior is as sketched in Fig. 3.2d.