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25th
ICAF Symposium – Rotterdam, 27–29 May 2009
ADVANCES IN CRACK GROWTH
MODELLING OF 3D AIRCRAFT
STRUCTURES
S.C.Mellings1, J.M.W.Baynham
1, R.A.Adey
1
1 BEASY, Southampton, UK
Abstract: This work is focussed on the 3D simulation of crack
growth in metal structures which are exposed to complex multi-
axial loadings, as regularly occur in aircraft. Simulation of these
cracks under the applied loading can be used by the design
engineers to investigate changes in the design, loading or
materials.
Given an initial crack, the aims of the simulation are to identify
the stress intensity factors occurring at any stage of the loading
cycle, and to predict the time taken for the crack to grow through
the structure. This means that vulnerability to fast crack growth is
assessed, and that fatigue growth calculations of growth direction
and distance are performed and accumulated. The result is that
crack life predictions are made of the crack size variation with
number of cycles.
The use of multi-scale techniques allows sub-models to be created
from any part of the model. This enables large-scale FE models to
be used for a detailed BE analysis. The paper describes the crack
growth in one such sub-model using a variety of initiated crack
shapes and positions, and for a range of different loading regimes.
INTRODUCTION The analysis of cracks in airframe structures is a critical part of the design process.
In this paper various crack growth analyses are presented in which a crack is
grown within a single model, a machined stiffened panel. In the analyses various
crack growth scenarios are presented.
S.C.Mellings, J.M.W.Baynham
2
Initially in this paper, details of the analysis methodology are presented, followed
by descriptions of the model creation, from the initial FE model, and the automatic
crack growth process. The paper then goes on to apply the described method to a
number of crack growth scenarios for the given model. These examples present
cracks initiated at different locations and under a variety of loading regimes.
BOUNDARY ELEMENT ANALYSIS The numerical analysis presented in this paper is based on the Boundary Element
Method (BEM) rather than the ubiquitous Finite Element Method (FEM).
The BEM has significant advantages for mathematical modelling of fracture
specifically that only the surfaces of the part need to be meshed. The crack, as a
surface of the model, is meshed as well, using a layer of elements on both the top
and the bottom surfaces of the crack. Figure 7 shows a view from the inside of a
structure, showing the elements on the crack surfaces and those on external
surfaces. The quarter-penny-shaped crack has a roughly quarter-circular crack
front.
The simplicity of the mesh required to represent behaviour of cracks is a major
advantage when using BEM for fracture analysis. The method allows very refined
meshing near the crack front without any difficulty at all. By contrast FE meshes
often suffer from the unwanted side-effect that the refinement tends to propagate
through the volume of the structure. The simplicity provided by the BEM can
especially be appreciated during the growth of a crack, since the only parts of the
model that are affected are the surface mesh on the crack and the mesh on the
immediately adjacent surfaces. In particular with BEM, it is not necessary to
recreate the mesh of the entire structure.
Additional advantages of the BEM are first that it is a mixed method which
calculates both displacement components and stress components directly, and
secondly that these values are calculated at positions on the surface of the
structure. Some other methods derive stresses by differentiation of displacement,
and often calculate results at “gauss points” inside the volume, thereafter
extrapolating to the surface. It is well known that differentiation tends to dilute
accuracy, and in pressure-vessels, the highest stresses are generally at the surfaces,
with sometimes very rapid variation near the surface, which means that
extrapolation to the surface can be very misleading.
The end result is that BEM provides a methodology which fits very conveniently
with fracture and crack growth simulation requirements, and makes it feasible to
perform automatic fatigue crack growth modelling.
Advances in crack growth modelling of 3D aircraft structures
3
In this paper a specific form of the BEM is applied, involving use of dual elements
on the crack. These allow the greatest simplicity of modelling, since the same
element geometry can be used to represent the behaviour on both sides of the
crack, hence the name “dual”. From a mathematical viewpoint, a node (at which
problem variables are defined) on one side of the crack is at the same position as
the node on the corresponding dual element on the other side of the crack. This
situation would ordinarily prevent a solution from being obtained, but the dual
BEM avoids this difficulty by using an influence equation (for the dual node)
which is derived in a different way [1-6].
The examples shown in this paper were modelled using the BEASY simulation
package, which additionally allows selection of an initial crack from a library, and
automatic insertion of the crack into a BEM model of an un-cracked structure.
The benefits of mathematical modelling are obtained when the method is applied
to situations which are not covered either by standards, previous experience,
reference solutions, or experiment.
CREATING BEM MODELS FROM FE GEOMETRY AND
RESULTS The methodology presented above describes how the boundary element analysis is
performed. However in most cases, the parts to be analysed have already been
modelled with finite elements, rather than boundary elements. The multi-scale
approach used involves extracting a sub-model from part of the larger scale FE
model.
In the case presented here, a finite element model of the part is available in
ABAQUS, as shown in Figure 1. The part is a section of a curved panel that is
stiffened by two ribs.
Figure 1 ABAQUS model of a stiffened panel
S.C.Mellings, J.M.W.Baynham
4
This part is modelled in ABAQUS using tetrahedral elements and 3 load cases are
modelled. One load case generates a uniform tension load in the panel, aligned
with the stiffener direction. The other two load cases generate non-uniform tension
loading along the panel edge, again with the loading aligned with the stiffener axis.
For the purposes of the crack growth analysis it is not necessary to analyse the
complete stiffener model, therefore the BEM model that will be generated will be a
sub-model of this part. The sub-model section used is shown in Figure 2.
Figure 2 Sub-model selection in ABAQUS
This group of finite elements is used to generate the BEM model using a
“skinning” process. In this process the selected elements are used to define the
extent of the BEM sub-model. The external surfaces of this region are then used to
define the BEM elements which are then loaded from the FE results. This process
is controlled using a JAVA based tool, the BEASY model generation wizard,
which extracts the required information from the ABAQUS result file and then
creates and loads the required boundary element model. The resultant BEM mesh
is shown in Figure 3.
Figure 3 Sub-model selection in ABAQUS
This tool allows for BEM sub-models to be created from finite element meshes,
generated using a variety of FE tools and the loading on the sub-model can be
Advances in crack growth modelling of 3D aircraft structures
5
generated from the realistic stress results computed in the FE analysis. This allows
for complex stress fields to be represented without the requirement to manually
interpolate the boundary conditions. In some cases, however, the FE mesh may not
be suitably refined in the area to be investigated. For example, if the FE model is
represented with linear elements, then facetted rather than true curved surfaces will
be generated. An example of this is shown in Figure 4. On the left is a bolt that has
been represented using linear elements in FE, this will give a very different local
stress field to the quadratic representation shown on the right, although the applied
loading to the model is the same. In addition, features may not be included in a
more global model that will be required in a more detailed crack growth, such as
fillets or inclusions in the model, all of which will affect the crack growth rate and
direction.
Figure 4 Linear and quadratic representation of a bolt
If an FE model is not directly suitable for a detailed crack growth study then the
model can be recreated directly as a BEASY model. However the loading from the
global FE model can still be used and provides valuable data on the stress fields
that are applied to the part.
Additionally, residual stress fields may be applied to the part and these can be
obtained by reading the FE stress results in the vicinity of the crack. These stresses
are then automatically applied to the crack faces to provide loading either opening
or closing the crack.
THE AUTOMATIC CRACK GROWTH PROCESS
S.C.Mellings, J.M.W.Baynham
6
In this paper the automatic crack growth process uses a toolkit for fracture analysis
in which a crack can be initiated into an existing, un-cracked, BEM mesh. In the
examples presented here the mesh used was the BEM sub-model created from the
larger scale FE model of the stiffened panel. An example of the initial mesh is
shown in Figure 3 where the mesh of boundary elements has been created, but the
crack has not yet been initiated in the part.
The simulation process continues with selection of the required initial crack shape
from a library of shapes. This is achieved within a GUI which leads the user
through the various necessary tasks, such as choosing the size of the initial crack
and its position and orientation within the component. Figure 5 shows an example
of the appearance of the GUI during selection of a crack shape. Other necessary
tasks include selection of a “growth distance” used during simulation of fatigue
growth, and selection of appropriate fatigue growth data and a crack growth rate
equation. Optionally, further settings can be adjusted to control mesh density,
number of J-integral calculations to be performed along each crack front and so on.
Having defined all the necessary data, the simulation process is started, the GUI
can be closed, and the steps required for fatigue crack growth then proceed
automatically.
The very first step involves placement of the required initial crack into the mesh of
the un-cracked component, and modification of the mesh on the crack and nearby
surfaces of the component so that it is suitable for the fracture calculation. The
mesh after completion of this step is as shown in Figure 6, with detail around the
crack shown in Figure 7.
Figure 5 Appearance of the GUI during selection of initial crack shape from the
crack library
Advances in crack growth modelling of 3D aircraft structures
7
Figure 6 Mesh of corner crack modelled into stiffener
Figure 7 Wireframe close-up view of the model with the initial crack
In the next step of the process, the new mesh is first used to compute stresses and
displacements, and then to determine the stress intensity factors, typically using a
J-integral approach.
Next, the fatigue behaviour is simulated, to “grow” the crack. This involves use of
the stress intensity factors and load variation details to select appropriate direction
and distance of growth. This calculation is made at multiple positions along the
crack front, and a new position of the crack front, for the grown crack, is thus
S.C.Mellings, J.M.W.Baynham
8
predicted. This crack growth allows for variable crack growth rates and directions
along the crack front, due to the predicted stress intensity factor values computed
in the analysis.
In the next step, the predicted new crack front position is used together with the old
crack, to define the surface of the new grown crack.
To complete the cycle, the final step involves placement of the new grown crack
into the mesh of the un-cracked component, and as before, modification of the
mesh on the crack and nearby surfaces of the component so that it is suitable for
the fracture calculation.
The process, shown diagrammatically in Figure 8, is continued until one of several
stopping criteria is met. These criteria include for example occurrence of critical
stress intensity factor, or the attainment of the required total growth distance.
Figure 8: Flowchart outlining the process of fatigue crack growth simulation
Using this process it is simple to change the crack that is initiated into the model.
Figure 9 show the same model that can be re-used with different crack shapes,
Advances in crack growth modelling of 3D aircraft structures
9
orientation, sizes or initiation positions. In this paper three different initial cracks,
in the same base model, will be examined.
Figure 9 Crack growth studies 1-3 using the BEM sub-model.
EXAMPLE 1: THROUGH CRACK IN A STIFFENER
For the initial study, a through crack will initiated into the stiffener itself. The
crack is grown using an initial cycling of the uniform tension load in the panel.
Figure 10 shows a cross section of the stiffener model with the through crack
added into the part.
Figure 10 Initial crack modelled into the stiffener
Example 2
Example 1
Example 3
S.C.Mellings, J.M.W.Baynham
10
The crack is now grown under the cyclic loading. Figure 11 and Figure 12 show
the crack after 8 and 10 crack growth increments respectively. As can be seen the
crack shape given is not pre-defined, rather it is computed directly from the
stresses in the model.
Figure 11 Crack grown though the stiffener
Figure 12 Crack growing into the base panel
As the crack continues to grow, under the loading used in this model, the crack
grows faster along the breakout edge and slower though the depth of the panel.
Advances in crack growth modelling of 3D aircraft structures
11
However it eventually breaks though the base panel and a transition can be made to
a crack that can grow in two directions along the base panel.
Figure 13 Crack just before transition to a through crack
Figure 14 Crack transitioned to a through crack
EXAMPLE 2: CORNER CRACK IN A STIFFENER The analysis presented above assumes that the crack starts as a through crack that
is modelled at the top of the stiffener. However, since the initial geometry does not
contain the initial crack and this is added in from a library of pre-defined cracks, it
is easy to start the analysis either with a different crack or with the crack in an
alternative location or orientation. In this second analysis a crack is grown from a
corner crack in the stiffener to identify the grown crack shape.
S.C.Mellings, J.M.W.Baynham
12
Figure 15 Initial corner crack added to the model
Initially the crack growth proceeds as a through crack, however when it reaches the
corner of the stiffener it transitions to become a “through” crack as seen in Figure
16. This initially has the same circular crack front, as shown in Figure 15, however
as the crack grows through the stiffener this quickly transitions to become a
through crack which is very similar to that grown from the initial through crack.
Figure 16 Corner crack transition to a through crack and then growing into base
panel
EXAMPLE 3: CRACK IN THE PANEL The cracks grown above show a crack growing from the stiffener into the panel.
Alternatively the crack may initiate in the panel itself and grow though into the
Advances in crack growth modelling of 3D aircraft structures
13
stiffener. In this analysis a crack is initiated as a though crack starting at the edge
of the sub-model. This analysis assumes that there is a similar crack growing into
the remainder of the part that has not been modelled. Note that in this analysis the
model used in the previous studies has been re-used and only the selected crack
and initiation points needed to be changed.
Figure 17 Initial crack in panel
Figure 18 Crack grown into fillet
Figure 19 Crack grown into thickened section
EXAMPLE 4: FATIGUE ANALYSIS WITH MULTI-AXIAL
LOADING The example presented above show various examples of uni-axial fatigue crack
growth in the part. Each stage of the analysis yields a set of stress intensity factors
at each mesh point along the crack front. This is used to grow the crack to the next
increment, as shown in the previous examples, but will also allow a fatigue life to
be computed. During the model build phase, it was noted that three load cases were
S.C.Mellings, J.M.W.Baynham
14
included in this model; a uniform tension and two non-uniform load cases. In the
analysis above, the fatigue load applied was a cyclic load of the uniform tensile
load case from full to zero loading.
Using the crack shapes from example 1, investigations into effects of changes of
loading can be performed. Three fatigue cases will be studied looking at different
combinations of the non-uniform loading using three different load spectrum files.
In the first case the loads will act 180 degrees out of phase, in the second they will
act in phase and in the last case they will be 90 degrees out of phase.
5.0000
10.0000
15.0000
20.0000
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
Cycles
Cra
ck S
ize
180 deg out of phase In Phase 90 deg out of phase Figure 20 Comparison of crack growth rates
As can be seen, the crack growth rate when the two loads are applied 180 degrees
out of phase gives a longer crack life than when they either apply in phase or 90
degrees out of phase.
These fatigue life results were evaluated without re-growing the crack. This
situation is typical of forensic studies where crack shapes may be known from
photographic data. The way in which the multi-axial loads are combined and
cycled may however affect the crack shape, and so now we will re-grow the crack
to investigate these effects.
The crack has therefore been re-grown in each case for a short distance to compare
the crack growth results. In each case the resultant grown crack for the 90 degree
and 180 degree out of phase load cases are show in the pictures below.
Advances in crack growth modelling of 3D aircraft structures
15
Figure 21 Crack growth with loading 90 degrees out of phase
Figure 22 Crack growth with loading 180 degrees out of phase
The analysis shows very similar grown crack shapes in the planar directions but, as
can be seen, the case with the loads applied further out of phase generates a higher
“twist” in the resultant crack.
Using this data to re-compute the fatigue life we see that the fatigue life, shown in
Figure 23, gives a similar behaviour to that estimated from the planar analysis,
shown in Figure 20, but the fatigue calculation is now higher and generates a faster
crack growth for the larger crack sizes. This is because the twist in the crack causes
it to be aligned normal to the maximum principal stress direction. This results in
higher computed SIF values and therefore a reduced life computation. This
S.C.Mellings, J.M.W.Baynham
16
removes some of the uncertainty in the results resulting in a more realistic (and
more conservative) life.
5.0000
10.0000
15.0000
20.0000
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
Cycles
Cra
ck
Siz
e
In phase 90 deg out of phase 180 deg out of phase Figure 23 Comparison of crack growth rates with full analysis
EXAMPLE 5: CONTACT BETWEEN THE CRACK FACES
In the models, examined tensile loading was applied and the crack growth was
dominated by mode 1 behaviour. However in many realistic cases, the cyclic
loading may, at one extreme, introduce compression across the crack face. If the
resulting contact is not taken into account, the calculated stress intensity factors
will not be realistic. For this example an alternative load case has been applied to
this model, in which the loading creates a compressive load across the crack faces.
In a standard BEM analysis, the crack faces are assumed to be load free and
therefore in a compressive field the crack faces are allowed to interfere. This
causes a negative stress intensity factor in the mode 1 direction.
Advances in crack growth modelling of 3D aircraft structures
17
-4.0000E+02
-3.0000E+02
-2.0000E+02
-1.0000E+02
0.0000E+00
1.0000E+02
2.0000E+02
3.0000E+02
0.0000 0.2000 0.4000 0.6000 0.8000 1.0000
Local position along crack front
SIF
K1
K2
K3
Figure 24 SIF values computed from a compressive loading
In reality however, the crack faces would not interfere but would transfer loading.
This is modelled with contact applied to the crack faces. This allows for the mode
2 and 3 results to be accurately computed.
-4.0000E+02
-3.0000E+02
-2.0000E+02
-1.0000E+02
0.0000E+00
1.0000E+02
2.0000E+02
3.0000E+02
0.0000 0.2000 0.4000 0.6000 0.8000 1.0000
Local position along crack front
SIF
K1
K2
K3
Figure 25 SIF values computed with contact on crack faces
The application of contact on crack faces can best be seen with a simple example.
In this model an angled though edge crack has been modelled into a plate, which is
subjected to a compressive loading.
S.C.Mellings, J.M.W.Baynham
18
Figure 26 Initial crack in plate (front & back surfaces removed)
At the first increment the contact load generates a sliding along the crack faces
which generates a shear loading in the part. This causes the crack to turn as shown
in Figure 27.
Figure 27 Crack after first growth step
As can be seen this now generates a tensile (opening) load on the crack face which
leads to SIF values that are dominated by mode 1. The crack continues to grow in
this direction until a “pit” is generated.
Advances in crack growth modelling of 3D aircraft structures
19
Figure 28 Pit generated after 5 crack growth steps
CONCLUSION A methodology has been presented in this paper which can be applied to crack
growth in real components. This process takes into account the real stress fields
which occur in practise, including effects of section thickness change, use of the
real component and crack geometry, and multi-axial loading. The method has been
applied in a multi-scale fashion, in which sub-models have been extracted from
larger FE models.
It has been shown that the methodology predicts how a crack will grow without
being constrained by standard reference solutions. By extension, it can be
concluded that this technique allows investigations to be performed into the grown
crack shapes and enables effects of design changes to be studied in a realistic
manner.
Although not discussed, the technique of applying loading to the crack faces can be
extended to allow effects of residual stresses to be studied. This is necessary for the
study of effects on crack growth of different surface treatments or different
methods of welding. Additionally, multi-cracks can be analysed together in the
same part and the interference of the cracks will affect the crack growth paths.
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Southampton, UK, 2008.
S.C.Mellings, J.M.W.Baynham
20
[2] Portela, A.; Aliabadi M.H; Rooke, D.P., “The Dual Boundary Element Method:
Efficient Implementation for Cracked Problems”, International Journal for
Numerical Methods in Engineering, Vol 32, pp 1269-1287, (1992).
[3] Mi Y; Aliabadi M.H., “Three-dimensional crack growth simulation using
BEM”, Computers & Structures, Vol. 52, No. 5, pp 871-878, (1994).
[4] Prasad, N.N.V.; Aliabadi M.H; Rooke, D.P., “The Dual Boundary Element
Method for Thermoelastic Crack Problems”, International Journal of Fracture, Vol
66, pp 255-272, (1994).
[5] Dell’Erba D.N.; Aliabadi M.H; Rooke, D.P., “BEM analysis of fracture
problems in three-dimensional thermoelasticity using J-Integral”, International
Journal of Solids and Structures, Vol. 38, pp. pp.4609-4630, (2001).
[6] Neves A; Niku S.M., Baynham J.M.W., Adey, R.A., “Automatic 3D crack
growth using BEASY”, Proceedings of 19th Boundary Element Method
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