Page 260 - The Combined Finite-Discrete Element Method
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DISCRETE CRACK MODEL 243
s
s
Figure 7.11 A crack tip approximation using a very coarse mesh – thick line indicates the
failure pattern.
ignored by a weak formulation of the boundary value problem, and the failure load
appears to be approximately
σ
= 1 (7.25)
f t
where σ and f t are the critical load and maximum bonding stress, respectively. The
critical load predicted is wrong by an order of magnitude, for the simple reason that no
plastic zone or stress concentration has been taken into account due to the element size
being greater than the size of the plastic zone. The direction of the crack propagation is
mostly influenced by the global elastic stress field, and does not depend much upon either
the crack presence or element orientation.
For a very fine mesh, the element size is in general only a fraction of the size of the
plastic zone. Thus the plastic zone itself spreads through a relatively large number of
finite elements, and stress and strain fields within the plastic zone are well represented
by the finite element approximation. In addition, by the Sant-Venant principle the stress
field towards the end of the plastic zone is not influenced much by the small change in
the shape of the crack walls corresponding to the finite element size (which is assumed
to be only a fraction of the plastic zone size). In other words, for a very fine mesh the
critical load, together with the general direction of crack propagation, should be accurately
predicted due to the insignificant effect of element orientation on the stress field within
the plastic zone.
For intermediate element sizes, the critical load is overestimated. The stress field in the
vicinity of the crack tip is, to a large extent, influenced by the size, shape and orientation
of the finite elements in the vicinity of the crack tip. Because crack propagation is locally
driven (i.e. the direction of the crack propagation is determined on the basis of the stress
and strain fields in the vicinity of the crack tip), it follows that the direction of crack
