Albert Turon, Josep CostaAMADE. Universitat de Girona
Pedro P. CamanhoDEMEGI. Universidade do Porto
Carlos G. DávilaNASA Langley Research Center
Simulation of delamination under high cycle fatigue in composite materials
COMPTEST200610th-12th April. Universidade do Porto
COMPTEST2006 10th-12th April. Universidade do Porto
Girona
COMPTEST2006 10th-12th April. Universidade do Porto
Introduction
• Delamination: Interlaminar crack formation and/or propagation
Approaches to the study of delamination:
(1) Direct application of Fracture Mechanics Delamination propagationVirtual Crack Closure Technique (VCCT), J integral …
(2) Damage Mechanics Initiation and propagation of delaminationCohesive Zone Model approach, based on the Dugdale-Barenblatt concept: A
cohesive damage zone -or softening plasticity- is developed ahead of the crack tip
• There are numerical tools to analyze initiation or propagation of delamination under quasi-static loading, but “not” under cycling load.
COMPTEST2006 10th-12th April. Universidade do Porto
• Quasi - static model
• High cycle fatigue
• Damage evolution law
• Cycle jump strategy
• Results
• Conclusions
OUTLINE
COMPTEST2006 10th-12th April. Universidade do Porto
Quasi static model
COMPTEST2006 10th-12th April. Universidade do Porto
Damage Mechanics models
Constitutive equations model the constitutive behaviour of the cohesive zone.
τInitiation criteria
Propagation criteria 21 3 4 5P
P
Δ0 ΔF Δ
1
2
4 5
3K
(1-d)K
0
τ0
Gc
COMPTEST2006 10th-12th April. Universidade do Porto
( ) ( )λλ
λλ
∂∂=
∂∂=
t.tt.. Grr,Frd
( )
( ) ( )( )0ft
0tft
tt
ss
0t
rrrG
rGd
ts0max,rmaxr
ΔΔΔΔ
λ
−−=
=
≤≤⎭⎬⎫
⎩⎨⎧=
( ) ( ) 0, ; 0, ; ..
=≤≥ tttt rFrrFr λλ0
• Kuhn-Tucker conditions for loading/unloading/neutral load conditions
• Evolution of internal variables
Initiation
Propagation
d = 0.9d = 0.5d = 0.1
�3
�shear
Damage evolution under quasi-static loading
COMPTEST2006 10th-12th April. Universidade do Porto
Implemented using Decohesion Elements
• Zero-thickness elements placed at the interfaces of Solid Elements
• Simulate the cohesive forces of the interface
In different element technologies such as
• Elements with Embedded Interfaces
Finite element implementation
COMPTEST2006 10th-12th April. Universidade do Porto
0
100
200
300
400
500
600
700
800
0 2 4 6 8 10 12
Displacement [mm]
Load
[N]
ENF
MMB (GII/GT=50%)
MMB (GII/GT=20%)
Experimental
Numerical
MMB (GII/GT=80%)
DCB
Simulation results
COMPTEST2006 10th-12th April. Universidade do Porto
Simulation results (II)
x
25º
25º-25º
-25º
90º90º
25 mm
0.7
92
mm
z
yF
F
COMPTEST2006 10th-12th April. Universidade do Porto
High cycle fatigue
COMPTEST2006 10th-12th April. Universidade do Porto
Fatigue loading
�max
�min
�
�
(1-d )k0
1
2
t
u1
2
1
2
t
u
• Low cycle fatigue Cycle by cycle analyses
• High cycle fatigue
• Damage evolution with the number of cycles
• Cycle jump strategy
cyclicstatic ddd +=
COMPTEST2006 10th-12th April. Universidade do Porto
• … a damage evolution law as a function of the number of cycles isestablished a priori, Peerling’s law, for example:
→ The parameters of the law (C, λ, β) have to be adjusted calibrating the whole numerical model with experimental results.
• In this presentation: The evolution of the damage variable was derived by linking Fracture Mechanics and Damage Mechanics to relate damage evolution to crack growth rates.
Damage evolution with the number of cycles
βλ
ΔΔ⎟⎠⎞
⎜⎝⎛
=∂∂
a
CeN
dd
COMPTEST2006 10th-12th April. Universidade do Porto
The evolution of the damage variable is related with the evolution of the crack surface:
Different approaches:
(1) Damage Mechanics
(2) Fracture Mechanics
Damage evolution with the number of cycles (II)
NA
AN ∂∂
∂∂=
∂∂ d
d
dd
AAdd =
A1
A=
∂∂
d
d
cGAA Ξ=d
Ξ∂∂=
∂∂ dd
d AG
Ac
Δ
τ
Ξ
(1-d)K
Δ0 Δf
COMPTEST2006 10th-12th April. Universidade do Porto
…the crack growth rate equals to the sum of the damaged surface growth rate in the cohesive zone:
…the area of the cohesive zone can be computed using Rice’s model:
Damage evolution with the number of cycles (III)
NA
AN ∂∂
∂∂=
∂∂ d
d
dd
NA
AA
NA
NA CZ
Ae
e
CZ∂∂≈
∂∂=
∂∂ ∑
∈
dd
NA
AA
NA
CZ ∂∂=
∂∂ d
( )23
329
oCZGEbA
τπ=
COMPTEST2006 10th-12th April. Universidade do Porto
→ Experimental characterization
Crack growth rate
log(dA/dN)
log(ΔG)
Region IIIRegion IIRegion I
ΔGth
Gc
1m
m
cGGC
NA
⎟⎠
⎞⎜⎝
⎛=
∂∂ Δ
NA
AA
NA
CZ ∂∂=
∂∂ d
COMPTEST2006 10th-12th April. Universidade do Porto
ΔG is computed from the constitutive equation
Crack growth rate (II)
Δ
τ
ΔG
ΔmaxΔminΔ0 Δf
Δ
τ
Gmax
Δmax
Δ
τ
Gmin
Δmin
m
cGGC
NA
⎟⎠
⎞⎜⎝
⎛=
∂∂ Δ
1
2
t
u
COMPTEST2006 10th-12th April. Universidade do Porto
Different approaches:
(1) Damage Mechanics
(2) Fracture Mechanics
Summary of damage evolution under cyclic loading
NA
AN ∂∂
∂∂=
∂∂ d
d
dd
AAdd =
cGAA Ξ=d
NA
AN CZ ∂∂=
∂∂ 1d
NA
AG
N CZ
c
∂∂
Ξ∂∂=
∂∂ dd
cyclicstatic ddd +=
Experimental
Constitutive model
COMPTEST2006 10th-12th April. Universidade do Porto
Determination of cycle jump ΔNi
→ Fixed→ Variable
Integration of the constitutive equation
Cycle jump strategy
i
i
i1i NN
Δ∂∂+=+ ddd
maxdd ΔΔ ≤∂∂ i
iN
N
t
u ΔNi-1 ΔNi ΔNi+1
COMPTEST2006 10th-12th April. Universidade do Porto
Two elements connected by only one decohesion element:
Results
COMPTEST2006 10th-12th April. Universidade do Porto
Crack growth velocity under mode I loading:
Results (II)
COMPTEST2006 10th-12th April. Universidade do Porto
• A thermodynamically consistent Interface Damage Model has been formulated.
• The Interface Damage Model has been modified to simulate high cycle fatigue loading.
• The evolution of the damage variable was derived by linking Fracture Mechanics and Damage Mechanics.
• The model reproduce test data without the need of additional parameters that are typically used in other fatigue growth models.
Concluding remarks