Triaxial Nonlocal Smeared Cracking Models
The nonlocal concept can, in principle, be applied to any inelastic constitutive model. It has been applied to the smeared cracking model described in Chapter 8. There are two variants of this model, both of which have been studied. One variant is the cracking of fixed direction (Section 8.5.3) in which the damage w, which is used to modify the compliance matrix, is considered to be a function of tire normal strain enn in the direction normal to the cracks. The nonlocal generalization is obtained by consideiing the nonlocal damage w to be the same function of the averaged strain s„ in the direction normal to the cracks (for details, see Bazant and Lin 1988a).
Another variant is the rotating crack model, for which the local formulation was presented in Section 8.5.6. Again, the nonlocal generalization is obtained by replacing the dependence of the normal compliance Cpi on the local principal strain by an identical dependence calculated as a function of the nonlocal principal strain є,. Of course, when the cracks do not rotate, the first and second variant coincide. When they rotate, the second variant seems to be closer to reality.
The model was used by Bazant and Lin (1988a) to simulate three-point-bend fracture specimens, and particularly the size effect. Fig. 13.2.1 shows the finite element meshes for three specimens sizes in the proportions 1:2:4. Fig. 13.2.2 shows a comparison of the nonlocal finite clement analysis with test results. The strain-softening law has been considered in two forms: exponential (dashed) and linear (dash-dot). The calculations are compared to the test results of Bazant and Pfeiffer (1987) and to the optimum fit of these results with the size effect law, Bq (1.4.10). The results demonstrate that the nonlocal model eliminates mesh sensitivity (because the ratio of the element size to specimen size is very different for the three specimens). They also demonstrate that the transitional size effect is well described by the nonlocal model. The width of the fracture process zone is, in these calculations, found to be roughly 2.7 £, where l — characteristic length, in agreement with the calculations of Bazant and Pijaudier-Cabot (1988).
Fig. 13.2.3 shows finite element calculations on unnotched beams with deliberately slanted meshes. These calculations show that the nonlocal model in which the characteristic length is sufficiently larger than the element size is free of directional mesh bias. The cracking band can propagate in any direction, without bias, to the mesh lines or the diagonal directions.