## F Kj Kj Kjjj KC

Several functional forms of equation 5.1 have been proposed in the literature. In LEFM models, a sudden release of stress on the surface is assumed with the extension of the crack. Most investigators adopt the discrete crack propagation finite element model (DCPM) with the LEFM constitutive models. Techniques to apply the LEFM criteria in smeared crack propagation finite element model have also been reported in the literature. Pekau et al. (1991) have proposed a numerical procedure to apply the linear elastic fracture mechanics criterion in discrete crack propagation analysis with boundary element model for the dams.

The question of whether the fracture process in concrete can take place at a localized point has been a subject of intense debate for quite long time. In reality, the fracture process zone (FPZ) must have some finite size (Figure5.2-b). It is argued that the LEFM could be applied if the FPZ is much smaller than the dimension of the structure under consideration. Very large concrete structures like dams are usually cited as the possible candidates for application of LEFM models. However, no rational experimental evidence has ever been put forward appraising the extent of FPZ in dam concrete. The disregard of nonlinear behaviour in the FPZ is an assumption of unknown consequences in influencing the global response of the structure. It seems

Figure 5.2: Fracture process zone (FPZ); (a) LEFM; (b) NLFM

appropriate to consider the nonlinear behaviour in the FPZ if the localization of the crack profile is a primary objective of the finite element analysis. In a gravity dam, a relatively stiff structure, crack opening displacements may be very small, which means that a long fracture process zone may exist (Dungar et al. 1991). Hence the argument of small fracture process zone in comparison to the thickness of the structure, usually cited to apply LEFM models, may not be true even for concrete gravity dams. The choice between LEFM and NLFM models could also be influenced by the strain rate under consideration.

The primary characteristic of nonlinear fracture mechanics (NLFM) is the recognition of the strain softening behaviour of concrete in the FPZ. Two apparently different models have been proposed in the literature considering only the model I nonlinear fracture propagation in concrete. The most referenced work is due to Hillerborg et al. (1976). They characterized the existence of FPZ as a fictitious crack lying ahead of the real crack tip (figure5.3-a). The behaviour of concrete in the FPZ was represented by a diminishing stress, a, versus crack opening displacement (COD), S, relationship; the tensile resistance being ceased at a critical COD value, S, (Figure 5.3-b). The area under the a — S curve represents the energy, Gf, dissipated during fracture Process on unit area:

Gf is a material property and often referred to as fracture energy or specific fracture energy. The special feature of the Hillerborg's fictitious crack model is the dissipation of energy over a discrete line crack. The basic nature of the model has made its extensive applications possible in discrete crack propagation analysis.

The LEFM models, in the context of discrete crack propagation, have also been modified to take account of the FPZ through an effective crack length calculated as the true crack length plus a portion of the fictitious crack. In some recent studies the key assumption of Hillerborg's model, that the tensile stress at the tip of fictitious crack is equal to the tensile strength of concrete has been modified using the concept of singular stress distribution at the fictitious crack tip.

Bazant and Oh (1983) came forward with the argument that the energy dissipation cannot take place in a diminishing volume, it must involve a finite

Figure 5.3: Nonlinear fracture mechanics models: (a,b) fictitious crack model,(c,d) crack band model volume of the material. The fracture process is thus assumed to propagate as a blunt front (Figure5.3-c). The width of the blunt crack (or the band of micro-cracks), wc, is assumed to be a material property in this model. The strain softening behaviour of concrete in the FPZ is represented by a stress-strain relationship (Figure5.3-d) and the fracture energy, Gf, is given by r £f

The inherent characteristic of the proposed crack band model is the smeared nature of crack distribution over a band width, wc, The crack band width, wc, however, has not been determined by any direct experimental investigation. Bazant and Oh (1983), in the original presentation, attempted to establish the value of wc by fitting their proposed model in a series of stress-strain and fracture energy data collected from the literature. The values of wc in the range of single aggregate dimension to six times that size gave equally good results. The assumption of fracture energy dissipation over a certain band area of material characteristic dimension, thus, seems to be a numerical speculation.

An important aspect of the nonlinear fracture mechanics models is the shape of the softening branch. Various proposals have been made in the literature about the form of a — S and a — £ softening relationships. A bilinear relationship is usually applied to interpret the experimental observations (Briihwiler 1990; Nomura et al. 1991). The precise shape of the softening diagram has been reported to have considerable influence on the numerical results (ACI 1991). There has been an unrelenting debate as to which softening model, the fictitious crack model or the crack band model, should be used. The smeared nature of the crack band model is a tempting feature for application in finite element analysis when the direction and location of crack propagation are not known a priori. Application of the crack band model, in its original form, to smeared fracture propagation analysis requires the size of finite elements to be limited to wc. The very small value of wc, according to the definition presented by Bazant and Oh (1983), renders any practical finite element analysis of large concrete gravity dams too expensive.

Special finite element techniques have been proposed to ease this limit on element size, where the fracture process effects are smeared over a zone of the finite element and the average stress-strain relationship is adjusted to conserve the fracture energy. A linear strain softening relationship has been

Figure 5.4: (a) average stress-strain curve for smeared crack element; (b) characteristic dimension, /c = /i,/2; (c) characteristic dimension. lc= v7/7/77

assumed in most of these models. The area under the average stress-strain curve for a finite element undergoing fracture process is adjusted such that the dissipated fracture energy, Gf, for unit area of crack extension remains independent of the element characteristic dimension, lc, (Figure5.4-a) . A favored approach in the nonlinear smeared fracture models (NSFM) is to adjust the slope of the softening branch, assuming that the softening process initiates when the tensile stress reaches the material tensile strength. The strain-softening modulus, Et, in figure5.4-a can be derived as:

vi le if the tensile strength, at, the elastic modulus, E, and the strain fracture energy, Gf, are known for the material, the strain softening modulus for the particular element size, lc, can be determined from equation 5.2. For a special case of , lc = wc (wc is the width of crack band), the nonlinear smeared fracture model reduces to the crack band model of Bazant and Oh (1983). Unlike the crack band width, wc, the characteristic dimension, lc, is a geometric property of the element. For cracks parallel to a side of square finite elements, as in Figure5.4-b, the characteristic dimension, lc, equals the element dimension across the crack plane (lc =li, l2,l3, etc.) For oblique crack propagation, lc can be taken as the square root of the element area under consideration (Figure5.4-c).

The softening modulus, Et, given by equation 5.2, becomes stiffer for increasing value of lc up to a certain limit, beyond which an unrealistic snap-back appears in the tensile stress strain relationship of concrete. In the limit case, the softening constitutive model degenerates to the traditional elas-tobrittle failure criterion, dissipating the stored elastic strain energy instantly upon reaching the tensile strength of material. The maximum finite element size, denoted by l0, that can be modelled with strain-softening behaviour is determined from equation 5.2:

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