For typical dam concrete properties of E = 30 000 MPa, Gf = 0.2 N/mm, and vt = 2.0MPa, the limiting value is l0 < 3.0m. This limit on maximum dimension, which is much higher than the material characteristic dimension, wc, of Bazant and Oh (1983), also has often been considered stringent for large scale finite element analysis at reasonable cost.

Figure 5.5: Nonlinear fracture mechanics in smeared crack propagation model

To circumvent this limit on the size of finite elements, and at the same time respect the principle of conservation of energy, one proposition is to reduce the fracture initiation stress, a0, with increasing finite element size and assume an elasto-brittle failure criterion for element sizes greater than /0 (Figure5.5-a). This is the so-called size reduced strength (SRS) criterion proposed by Bazant (1984) and Bazant and Cedolin (1979, 1983). This size reduced strength criterion (or in other words the elastic fracture criterion) can be criticized for two reasons: (i) the size independent critical COD value, Sf, of no tensile resistance, implied by the conservation of fracture energy in the nonlinear fracture model, is violated (Figure5.5-b), and (ii) when applied with a strength based crack initiation criterion, the principle of fracture energy conservation is likely to be violated in the interior element as well as in the exterior element (figure5.5-c). An important numerical side effect of the elasto-brittle SRS failure criterion is the generation of spurious shock waves in the finite element model. El-Aidi and Hall (1989a) encountered such numerical difficulties in the nonlinear seismic analysis of concrete gravity dams. The alternate proposition of Bazant (1985) is to take the softening modulus, Et, as a material property and reduce the fracture initiation stress with increasing size of element (Figure5.5-d) for conserving the fracture energy. This approach may allow numerically stable algorithm as opposed to the elasto-brittle model, but it seems to be based on a weaker theoretical consideration, and the questions raised for the elasto-brittle criterion are still present. Moreover, taking Et, as a material property has not been justified from experimental investigations. The limit on the maximum size of finite elements thus appears to be a requirement to ensure a reliable application of the nonlinear fracture mechanics criteria in smeared crack propagation analysis.

Several other models with some variations of the crack band model have also been proposed in the literature. Gajer and Dux (1990) decomposed the finite element strain increment in the following form:

where the incremental strains aee and aecr correspond to the uncracked concrete and the crack band respectively, and a is the averaging factor defined as the ratio between the crack band area and the gross element area. For decomposition of strain in the fracturing direction, equation 5.3 essentially represents the nonlinear smeared fracture model described earlier. Gajer and Dux (1990) apparently ignored the directional property of cracking by applying the scalar factor, a, in the global finite element direction. An enhanced form of the crack band model is the so-called nonlocal cracking model (Bazant and Lin 1988). In that approach, the crack strain at an integration point is smeared over the neighboring points of the finite element mesh. A crude form of the nonlocal model can be considered to be the nonlinear smeared fracture model, described earlier, where the crack strain is averaged over a characteristic dimension of single element.

Figure 5.6: Closing and reopening of partially formed cracks

Application of nonlinear fracture mechanics models in dynamic analysis requires the definition of unloading and reloading behaviours during the fracture process. Very few studies have been reported in the literature in this respect. Bazant and Gambarova (1984) proposed a hypothetical nonlinear stress-strain relationship for closing and reopening behaviours of partially open cracks as depicted in Figure 5.6-a. Numerical simulation of such nonlinear model is very complex. de Borst and Nauta (1985) applied the linear tangent softening modulus (line 1-2 in Figure5.6-b) to characterize the FPZ behaviour of increasing strain with decreasing stress and a secant formulation (line 2-0 in Figure5.6-b) to represent the closing of partially open cracks. Gambarova and Valente (1990) have applied an assumption of sudden stress release when the closing of partially open cracks is detected at any instant of the fracture process (Figure5.6-c). Dahlblom and Ottosen (1990) proposed the following relationship for closing and reopening behaviours of partially fractured concrete:

^max where A is the ratio between the residual strain upon closing of cracks and the strain of open cracks (Figure5.6-d). It appears that the techniques applied by de Borst and Nauta (1985) and Gambarova and Valente (1990) are subsets of this generalized model with A = 0.0 and 1.0 respectively. The physical phenomenon of crack closing and reopening taking place before the complete fracture of the material is, however, yet to be investigated rationally.

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