Page 261 - The Combined Finite-Discrete Element Method

244 TRANSITION FROM CONTINUA TO DISCONTINUA

propagation is to a large extent influenced by the topology of the finite element mesh
in the vicinity of the crack tip. In other words, the result of analysis will be sensitive
to the topology of the finite element mesh employed. The fracture pattern is therefore a
function of both problem parameters (overall geometry, applied load, material properties)
and model parameters (mesh orientation, mesh size).
Sensitivity to mesh size and orientation is due to the singularity of the stress field at
the crack tip. The influence of such singularity can be illustrated on the example of the
following function:
1
σ = √ (7.26)
x

being approximated by constant value finite elements, as shown in Figure 7.12.
The relative error of such an approximation can be estimated by
1 1
√ − √ - .
(n − 1)h nh n − 1
ε = 100% = 1 − 100% (7.27)
1 n
√
(n − 1)h
It is evident that the relative error of approximation does not decrease with the decreasing
element size. This means that, for a zero length plastic zone, no mesh refinement would
increase the ability of the model to predict the fracture pattern. In other words, fracture
models (often found in the literature) based on the sudden release of stress (acoustic
release) cannot be objective, regardless of the finite element mesh or other type of grid
employed. In contrast, the combined single and smeared crack model is based on the
assumption of a finite plastic zone. Thus, very fine meshes in conjunction with the com-
bined single and smeared crack model should result in accurate prediction of both the
critical load and fracture pattern.
The problem is that such fine meshes are in many practical applications simply not
affordable. Small scale problems such as rock crushers, the fracture of smaller size blocks,
and so on, are already affordable on modern day computers. Coarser meshes would
also make large scale problems affordable in terms of CPU time (for instance, blasting

s

h h h h x
nh

Figure 7.12 A singular stress field approximated by constant stress finite elements.