Page 258 - The Combined Finite-Discrete Element Method

DISCRETE CRACK MODEL 241

where the variable D is given by


0,
 if δ ≤ δ t

1, if δ> δ c

D = (7.13)
 δ − δ t
 , otherwise

δ c − δ t
while the parameters a, b and c are obtained from experimental stress displacement curves
by curve fitting. Note that for any value of these parameters, the above heuristic formula
results in a bonding stress of f t for D = 0 and a bonding stress equal to zero for D = 1.
The tangent at the stress displacement curve at D = 0 is horizontal. Thus, parameters a,
b and c control the slope of the curve at D = 1 and the shape of the curve (curvature) at
D = 0 together, with the inflection point.
In the discrete crack model, it is assumed that the crack walls coincide with the finite
element edges. Thus initially the total number of nodes for each of the finite element
meshes (every single discrete element is associated with its separate finite element mesh)
is doubled, and nodes are held together through a penalty function method. Thus the
separation δ t is a function of the penalty term p employed. In the limit no separation of
adjacent edges takes place before stress f t is reached, i.e.

lim δ t = 0 (7.14)
p→∞
With increasing separation δ> δ t the bonding stress decreases, and at separation δ> δ c
it is zero and the crack is assumed to propagate.
In finite element discretisation of the governing equations, only approximate stress
and strain fields close to the crack tip are obtained. With the bonding stress model as
described above, the stress and strain fields close to the crack tip are influenced by the
magnitude and distribution of the bonding stress close to the crack tip. For the bonding
stress to have a significant effect on stress distribution results, it is necessary that the size
of finite elements close to the crack tip be smaller than the actual size of the plastic zone.
The coarser mesh results in bonding stress in all elements close to the crack walls being
reduced to zero, except for the few elements adjoining the crack tip. The propagation of
the crack is therefore influenced by the orientation of those elements close to the crack
tip. The coarse finite element mesh does not accurately represent the stress field in the
proximity of the crack tip, and as a result, the stress field obtained is influenced by the
mesh topology close to the crack tip. i.e. the de-bonding and separation of crack walls
occurs on an element-by-element basis.
One way to avoid this problem is to have an element size close to the crack tip much
smaller than the size of the plastic zone. The approximate length of the plastic zone
for a plane stress mode I loaded crack can be approximated from Muskhelishvili’s
exact solution for a crack loaded in mode I. For an infinite body under plane stress
conditions, Muskhelishvili’s solution gives at points y = 0 normal stress σ y and crack
opening displacement δ as a function of the coordinate x:

1 4a #
and δ = σ 1 − (x/a) 2 (7.15)
σ y = σ #
1 − (a/x) 2 E