Triaxial Nonlocal Models with Yield Limit Degradation
The plasticity models can also be adapted to nonlocal analysis of distributed damage. To this end, plastic hardening is replaced by softening, which means, for example, that the plastic hardening modulus H becomes negative, as illustrated by the negative slope in Fig. 13.2.4 for a Mohr-Coulomb yield surface model. If this is done, of course, Drucker’s stability postulate for plasticity ceases to be satisfied, but this is not fundamentally incorrect (see Chapters 10 and 13 in Bazant and Ccdolin 1991) because this postulate cannot be expected to apply in the case of damage. The nonlocal concept is intioduccd into the model
Figure 13.2.2 Size effect plot comparing Ihc lest results of Bazant and Pfeiffer (1987) to the size effect law as well as to finite element results of the nonlocal smeared cracking model for linear and exponential softenings (after Bazant and Lin 1988a).
Figure 13.2.3 Strain localization zones at three loading stages for a mesh aligned with the crack path (left) and for a skew mesh (right) (from Bazant and Lin 1988a).
by replacing the plastic strain increment, as soon as it is calculated, by its spatial average and using this average in the constitutive relation.
A debatable feature of this formulation is the fact that Prager’s continuity condition of plasticity (consistency condition) is satisfied by the local rather than the nonlocal plastic strain increments, which means that the constitutive law is local and the nonlocality is introduced as separate adaptation. This approach appears to be in line with the conclusions of the analysis of crack interactions (Bazant 1994b) which will be explained later. Some theorists (e. g., de Borst) have insisted that the continuity relation must be satisfied by the nonlocal strains, which, however, would cause a tremendous complication of the model because the continuity condition would become an integral equation over the entire structure. Such a complexity would defeat the advantages of the nonlocal approach. It is true, however, that if Prager’s continuity condition is not satisfied by nonlocal strains, there is no precisely defined nonlocal constitutive law. Theoretically, this is a weak point of this type of formulation.
Fig. 13.2.5 shows an example (Bazant and Lin 1988b) of a rectangular panel solved by meshes of three different refinements. The local plasticity solution with a degrading yield limit gives the response in Fig. 13.2.5b and the nonlocal model gives the responses shown in Fig. 13.2.5c.
This model has also been applied to the analysis of failure of a tunnel excavation in grouting soil; see Fig. 13.2.6, which shows meshes of four different refinements and the boundaries of the strain softening zones obtained by the four meshes. Note again that the nonlocal approach is basically free of mesh sensitivity.
Figure 13.2.6 Ba%ant and bin’s (1988b) finite element analysis of a tunnel excavation in a grouted soil with a degrading yield limit: (a) finite element meshes; (b) boundaries of the softening zone at full tunnel excavation obtained for the four meshes shown in (a); (c) exaggerated deformation at full excavation. (Adapted from Bazant and Lin 1988b.)