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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
T. Siegmund, Purdue 4
XFEM-ABAQUS
• eXtended Finite Element Method • Extension of conventional FEM based on
the concept of partition of unity;
T. Siegmund, Purdue 5
[ ]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
T. Siegmund, Purdue
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.
T. Siegmund, Purdue 9
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
T. Siegmund, Purdue 10
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.
T. Siegmund, Purdue
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
+ =
T. Siegmund, Purdue 15
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
+ =
T. Siegmund, Purdue 16
Example Result
Provides location of crack initiation but not:-- number of cycles to failure-- not applicable to variable amplitude loading
T. Siegmund, Purdue 17
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
T. Siegmund, Purdue 18
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
T. Siegmund, Purdue 23
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
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