advances in crack growth modelling of 3d aircraft … · 2012-08-28 · advances in crack growth...

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


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


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


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


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


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,


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


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.


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.


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


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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


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.


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


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.


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-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.


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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.


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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.

REFERENCES [1] BEASY User Guide, Computational Mechanics BEASY Ltd, Ashurst,

Southampton, UK, 2008.


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[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

Conference, Computational Mechanics Publications, Southampton, pp 819-827,

1997.

advances in crack growth modelling of 3d aircraft … · 2012-08-28 · advances in crack growth...

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