Fracture of Arbitrary Direction
Finite element modeling of fracture is easy and accurate only if the fracture runs along the mesh lines. If the fracture path is known in advance, cither from experience or some preliminary calculations, then it is possible to design the mesh to accommodate the fracture path as a smooth path along the mesh lines. In general practical problems, however, the fracture path is not known. It is one of the unknowns to be found by analysis. In such general situations, which need to be tackled in general purpose finite element programs, there are basically two possibilities to proceed: either to modify the finite element mesh each time the fracture advances, or to keep a fixed mesh and allow the fracture to have a rugged boundary and zig-zag shape (Fig. 8.7.3c). The second possibility is not possible with the fictitious crack model, since it would cause serious problems with interlocking in the case of shear. On the other hand, the first possibility, that is, remeshing, exists both for the discrete fictitious crack and for the crack band, although in practice it has so far been used apparently only for the cohesive crack approach. The automatic remeshing (Fig. 8.7.3a, b) at crack advance is not simple to program; however, various research groups have nevertheless succeeded in developing finite element programs which do just that (sec Section 7.2.3). So far, however, the remeshing approach has not gained a wide popularity, due to the complexity of remeshing.
Although remeshing has not yet been used in conjunction with the crack band modeling of fracture, one must realize that this is a possibility which would be no more complex than remeshing for the cohesive crack. The algorithm for remeshing as developed by Ingraffca and co-workers (Section 7.2.3) could, no doubt, be easily adapted for remeshing in the case of crack bands (Fig. 8.7.3b).
As normally perceived, however, one of the advantages of the crack band approach is that fracture of arbitrary direction can be represented without any remeshing. The next clement that undergoes cracking is decided on the basis of either the strength criterion (for the tensile sudden stress drop) Or the stress-strain relation with strain softening. The zig-zag band is normally found to propagate roughly in the direction of previous cracking, however, it is possible under certain situations (for example a strip of concentrated shear stress) to obtain propagation of the band of cracked elements in a direction that is inclined to the direction of cracking in the elements. This represents shear fracture or mixed mode fracture (e. g., Bazant and Pfeiffer 1986).
However, as discussed in Section 8.6.2 and Section 8.6.3, there are certain errors associated with a zig-zag crack band. Due to the inevitable development of shear stresses on the planes parallel to the. overall direction of the zig-zag band, there is some degree of interlocking, i. e., an increased resistance to shear, larger than that obtained with a smooth crack (cohesive crack) or a smooth band with remeshing (Fig. 8.7.3a, b). Although, to a large extent, the errors arc tolerable compared to other errors involved in the analysis of fracture, remedies are needed to obtain accurate results.
The problem can be partially alleviated by using a square mesh in which each square is subdivided
Figure 8.7.3 Description of fracture path inclined with respect to initial mesh lines: (a) cohesive crack with remeshing; (b) crack band with remeshing; (c) zig-zag crack band in a square mesh; (d) mesh allowing better representation of inclined fracture.
into four triangular elements. In this case, there are not only horizontal and vertical mesh lines, but also mesh lines at 45° inclinations (Fig. 8.7.3d). This kind of mesh, which allows approximating an arbitrary fracture propagation direction more closely, should always be used with the crack band model when the propagation direction is unknown.
A better remedy is to employ a nonlocal approach, to be discussed in the next chapter. This, however, brings the penalty that there must be several finite elements across the width of the crack band, which in turn necessitates either a more refined mesh in the fracture zone or an artificial increase of the width of the crack band (with the corresponding modification of the postpeak stress-strain relation). Probably the simplest solution is to use a standard mesh (of the type shown in Fig. 8.7.3d) to get the approximate crack path, and then remesh to fit the mesh lines to the crack path, as indicated in Fig. 8.7.3b, and recalculate.