T. Siegmund, Purdue 1

Numerical Simulation of Fatigue Crack Growth: Cohesive Zone

Models vs. XFEM

Thomas SiegmundPurdue University

T. Siegmund, Purdue

Funding

• JOINT CENTER OF EXCELLENCE for ADVANCED MATERIALS, FAA Cooperative Agreement 04-C-AM-PU.

• Damage Tolerance and Durability of Adhesively Bonded Composite Structures

• Technical Monitor: Curt Davies• Cost share: Purdue University• Additional Funding: Catterpillar Inc.

T. Siegmund, Purdue

Research Accomplishements

• Compare cohesive zone model approach to XFEM.

• Implement fatigue crack growth in XFEM• Study contact fatigue failure

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XFEM-ABAQUS

• eXtended Finite Element Method • Extension of conventional FEM based on

the concept of partition of unity;

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[ ]1

( ) ( )

shape function conventional nodal dispalcement enriched nodal displacements

N

I I II

I

I

I

N x H x

N=

= +∑u u a

ua

XFEM-ABAQUS

T. Siegmund, Purdue

Traction-Sparation Law

σ σ

φ φΔ

Δ

Δ⎛ ⎞ ⎡ ⎤ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎢ ⎥ Δ⎣ ⎦ ⎝ ⎠⎝ ⎠

⎛ ⎞⎜ ⎟ =⎜ ⎟⎝ ⎠

= − = −

Δ=

−∫ 0

,0 0

,0 0

,0 ,0

max,0 max,0

,0 ,0

0

00

max , 1

(1 ) , (1 )

m

m

n n

t t

n t

n n t t

eff mm

el

T KT K

T T

T D T T D T

T dD

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Example: CZM vs. XFEM

• Plate with a central hole• Remote displacements• CZ elements along several radial lines• XFEM enrichment throughout

T. Siegmund, Purdue

Damage Accumulation Rule► Damage accumulation starts if a deformation measure, accumulated or

current, is greater than a critical magnitude.► There exists an endurance limit which is a stress level below which cyclic

loading can proceed infinitely without failure.► The increment of damage is related to the increment of absolute value of

deformation as weighted by the ratio of current load level relative to strength.

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Damage Accumulation Rules

σf…..cohesive endurance limitδΣ….cyclic cohesive length

δ σ∑

Δ ⎡ ⎤= − =⎢ ⎥

⎣ ⎦∫

max

: ,m effc f c c

TCZM dD C D dD

ε σε σ∑

⎡ ⎤= − =⎢ ⎥

⎣ ⎦∫

1 1

max

: ,p pc f c cXFEM dD C D dD

σf…..cohesive endurance limitεΣ cyclic reference strain

δεΣ

Σ

= 0

0

KE

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Fatigue Damage Model

( )

0

0

max,0 0

max,0 0

Cyclic damage variable:

ˆEffective tractions: (1 ) (1 )

ˆ ˆSince: ,

(1 ),

Current Cohesive Proper

damagedC

nn

C C

n n

n n C

AD

AF TT

A D D

T T

T T D

σ δ

σ δ

=

= =− −

=

= −⎡ ⎤⎣ ⎦

( )

max max,0

max 0 max,0 0

ties: (1 ) , (1 )

C

n C

DD

σ σφ χσ δ χσ δ

= −

= = −

C C C n t n tNeed to define : D = D D ,Δ ,Δ , T , T ...

T. Siegmund, Purdue

Monotonic Loading

0.0E+00

2.0E+06

4.0E+06

6.0E+06

8.0E+06

1.0E+07

1.2E+07

1.4E+07

0 0.002 0.004 0.006 0.008

E22

Σ22

[Pa]

CZM

XFEMELASTIC

T. Siegmund, Purdue

Cyclic Loading: Damage Evolution in Mode I

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

Normalized Time [ ]

Cyc

lic D

amag

e [ ]

the first three elements of the row of elements emerging from the hole at . ---- 1st element CZM (both integration points shown), - -- 1st element XFEM, ---- 2nd element CZM (lagging integration point shown), - - -2nd element XFEM, ---- 3rd element CZM (lagging integration point shown), - - -3rd element XFEM.

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Cyclic Loading: Mixed Mode

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Contact Fatigue

• Multiaxial Fatigue Criteria:

{ },max

,max

Critical Plane ApproachE.g.: Findley Criterion:

max

shear stress amplitude on a planemax. normal stress on that plane

a f n f

a

n

τ α σ β

τσ

+ =

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Contact Fatigue

• Multiaxial Fatigue Criteria:

{ },max

,max

Critical Plane ApproachE.g.: Findley Criterion:

max

shear stress amplitude on a planemax. normal stress on that plane

a f n f

a

n

τ α σ β

τσ

+ =

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Example Result

Provides location of crack initiation but not:-- number of cycles to failure-- not applicable to variable amplitude loading

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CZM Approach

• Cohesive Zone Model LE

LE

LE

LE

LE

LE

LE

LELE

LE

LE

LE

CE

CE

CE

CE

CE

CE

CECE

CE

CE

CE

Mesh generator available“Tie” CZ mesh to main model

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Redefine: Effective Traction

( )2

20.643

00.64

3 0

n t

n

n t

n

T T T

if T

T T T

if T

⎛ ⎞= × +⎜ ⎟⎝ ⎠

>

= × +

<

( ) ( )2 2n tT T Tη β= × + ×( ) ( )2 2

n tT T T= +

Common Fatigue Crack Growth Contact Fatigue

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Pmax = 16800 N at Failure

Contact Radius

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Damage Evolution

Subsurface Crack Initiation Site

Contact Radius

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Damage Evolution Pmax = 2800 N

Subsurface Crack Initiation Site

Contact Radius

T. Siegmund, Purdue

Comparison

0

0.05

0.1

0.15

0.2

0.25

0 5000 10000 15000 20000Load

Dam

age

Acc

umul

atio

n R

ate

SubsurfaceSurface

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Summary

• CZM vs. XFEM:– Provide close results if correctly calibrated– CZM is mesh dependent, XFEM less

• Contact Fatigue:– Multifacet CZM– Effective traction– Variable amplitude loading or tilted geometry