Chalmers Finite Element Center - FEniCS Project

10 PER HEINTZ
6. Numerical examples
In this section we present three numerical examples that demonstrate the crack propagation
algorithm, implemented as described in the previous sections. First, we consider
a simple model problem of a single edged crack for which we compute and compare the
energy release rate with the analytical value. Secondly, a weak and a stiff inclusion problem
are considered. The final example demonstrates a more complicated loading case, taken
from the litterature, where a simply supported beam is subjected to a pointload.
In all examples, the parameter α is chosen such that the normal tractions are evaluated
from the uncracked element ahead of the cracktip. The parameter δ must be chosen high
enough to ensure stability of the method. In all examples δ was set to δ = C · (2µ + 3λ)
where C depends on the approximation order of the basis functions. For linear triangles
C = 2 was sufficient in all examples. If the material properties are different between the
last cracked element and the element ahead of the cracktip, we simply use the highest value
obtained from the two different materials. For an in depth discussion and mathematical
proofs concerning the choice of the parameter δ see [6], [8] and references therein. The
elastic properties are given in terms of Youngs modulus E and poissons ratio ν and we use
the relations
Eν
(6.1)
λ =
(1 + ν)(1 − 2ν)
Eν
(6.2)
λ =
1 − ν 2
E
(6.3)
µ =
2(1 + ν)
where (6.1) and (6.2) are for plane strain and plain stress problems respectivily.
In example 2 and 3 we use a simple mesh refinement algorithm to increase the accuracy
of the computed kink angle. The elements that are marked with indicator 1, see Figure
7, are subdivided into four new triangles in two steps. Thus the original elements with
indicator 1 are subdivided into 16 new triangles. Due to the properties of the refinement
algorithm, the refinement is diffused through the grid such that no hanging nodes and no
sharp corners are obtained.
6.1. Single edged crack. Consider the single edged crack in Figure 9 subjected to a
modus I load with n · σ · n = σ 0 = 1. The body is in plane stress and the elastic
parameters are E = 1.0 Pa and ν = 0.3. The dimensions of the body are W = 1.0 m and
h = 2.0 m. The stress intensity factor K I , representing the strength of the singularity at
the cracktip, is now obtained from
(6.4)
K I = √ πa · f(a,W),
where f(a,W) is a geometry factor. The relation between the stress intensity and the
energy release rate is
(6.5)
J = K2 I
E