Simple Numerical Issues
Strict application of the crack band model, as formulated by Bazant, with hc equal to a material constant, would require a finite element mesh in which the cracking band has exactly a width hc. Thus, if the crack-band location is not known in advance, all the finite elements would have to be of width hc as depicted in Fig. 8.3.6a for the uniaxial case. This is unpractical and, fortunately, unnecessary. The fundamental reason is that hc does not enter explicitly the essential macroscopic parameters, of which the most important is Gp = hc7/.’. Therefore, if finite elements larger than hc need to be used, it is possible to keep the essential response if we preserve the fracture energy. To do so, an adequate approximation is to distribute uniformly the fracturing strain over the element and rescale the softening part of the stress-strain curve to keep Gp constant. The resulting stress-strain curve will depend on the element size, and must be scaled so that
where is the size of the element and 7^ the density of fracture energy to be used for this element.
For models of the clastic-softening type with stress-strain curves such as the one shown in Fig. 8.3.1b, the scaling is easy: just multiply c? by the factor in the preceding formula, i. e.,
Note that only the fracturing part of the strain must be scaled. The result of the scaling is shown in Figs. 8.3.6c-e for the simple linear softening. Note also that if the size of the element is too large, as deliberately shown in Fig. 8.3.6e, the resulting softening branch for the stress-strain curve of the element will show a snapback. In these cases, the finite element will become unstable at the snapback point, and the stress will drop suddenly to zero. Then the energy dissipated cannot be made equal to Gp, since all the elastic energy in the element is released. Therefore, either the element must be kept small so that no snapback would occur, or the curve must be modified to preserve the energy dissipation. This problem will be addressed in Section 8.6 in a wider three-dimensional framework.
The matters are a bit more complicated if the stress-strain curve has a prepcak nonlinearity as in Fig. 8.3.7a. In that case, the strain in the hardening and unloading branches must not be scaled; only the
(a) (b) (c)
strain contribution to the surface component of energy must. Therefore, the strain on (he softening branch has to be scaled so that
£S(C) -£“ = ^~y(£S -£“) (8.3.19)
where es and єи are the strains on the softening branch and on the unloading branch, respectively, for the same level of stress (Fig. 8.3.5a-c). This can be rewritten as
Thus, as shown in Fig. 8.3.7b-c, only the part of the softening curve on the right of the unloading branch is to be scaled. This may substantially complicate the use of otherwise simple stress-strain curves (i. e., with simple, beautiful expressions). Note, again, that snapback may occur in this case, too, if the element is too large, as shown in Fig. 8.3.7c.
In the foregoing it is implicitly assumed that the fracture will localize in a single element, since we showed that this is the solution in the preceding twosections. However, if a homogeneous case, such as the tensioned bar (or, in three dimensions, a pure bend beam), is numerically analyzed, and if the elements are given exactly the same properties, a normal finite element code will not catch the bifurcation. The reason is that the program will search for a solution by extrapolating from the previous step, and thus all elements will go through the peak into the softening branch simultaneously. And they will stay there! To avoid sophisticated bifurcation analysis (which is more elegant and more robust, but much more complicated), a simple expedient may be used: put imperfections into the material. Then either one element selected at random is taken to be a few percent weaker than the rest, or the strength of each element is assigned at random using a narrow strength distribution function. This is necessary only for structures with a nominally homogeneous distribution of elastic stresses and strains (laboratory specimens, typically), in most structures the elastic fields have stress concentrations which trigger localization without the need for introducing imperfections. (However, in some situations, the danger remains that the loading step is too large for the imperfections assumed to trigger localization. Theoretically, without bifurcation analysis, one is not sure, in general, that a localization has not been missed.)
We have addressed here two basic aspects of the numerical computation: the stress-strain curves to use, and the way to trigger the localization. This, of course, does not exhaust the discussion on the numerical models, but the other important aspects are fundamentally three-dimensional and are discussed later in Section 8.6, after presenting the three-dimensional softening models.