Multi-Directional Fixed Cracking
A difficult question with the foregoing formulation is the orientation of cracking. The practice which has been and is still typical of most large finite element codes is to set the crack direction to be normal to the maximum principal stress at the moment the tensile strength (or the tensile yield surface) is reached (Fig. 8.5.2a). During the subsequent loading process, the direction of the maximum principal stress can rotate. At the moment the cracks begin to form, there is, by definition, no shear stress on the cracking planes. However, due to keeping the cracking orientation fixed and assuming shear interlocking of the opposite crack faces, shear stresses can arise later if the principal stress direction rotates (Fig. 8.5.2b). It was for this reason that the diagonal compliances or stiffnesses for shear had to be included in F. qs. (8.5.17) and (8.5.23).
When the principal stress direction rotates, it is possible that the tensile strength // is reached again in another direction that is inclined with regard to the normal of the originally formed cracks. In that case, it is assumed that a second system of smeared cracks forms in the direction normal to the current principal stress (Fig. S.5.2c). This system is inclined at some general angle Act with regard to the orientation of the primary cracks. The cracking strain due to the formation of secondary cracks is then superposed on the original cracking strain, which means that another fracturing strain tensor (є( ® ii)s is added to the right-hand side of Eq. (8.5.5). The orientation of the secondary cracks is also kept fixed even when the principal stress directions subsequently rotate during the loading process. Thus, it may happen that the tensile strength is again reached in a third direction (Fig. 8.5.2d), in which case tertiary smeared cracks begin to form and the corresponding cracking strain needs to be again superimposed in Eq. (8.5.5).
The laws governing the secondary and tertiary cracking strains may be assumed to be the same as
Figure 8.5.3 Multi-crack system with fixed angular separation.
for the primary cracking, although some formulations allow for interaction between the various crack systems. The formulation of multiple cracking with fixed directions, which has been worked out in the greatest generality perhaps by de Borst (1986), can obviously get quite complicated when some crack systems open and cause others to close. Special computational strategies must then be devised to follow the possible bifurcation paths.
In the method just described, the secondary (and tertiary) cracks can have arbitrary orientations with regard to the primary ones. In this manner, cracks of many directions can form. In that case, it may be more convenient to assume that the cracks may form only in certain specified spatial orientations which are uniformly distributed among all spatial directions (Fig. 8.5.3). Such an assumption is also more realistic because it prevents the angle between intersecting cracks from being too small (say 10°, which is unlikely to occur). This approach, in which, again, the cracking strains from all the assumed discrete crack orientations arc superimposed, makes it possible to describe the fact that a principal stress of a certain direction may cause microcracking with various intensities at various orientations (this is captured more systematically by the context of the microplane approach to be discussed in Chapter 14; see Carol and Prat 1990, and Carol and Bazant 1997).