Post on 27-Jun-2020
Computational & Multiscale
Mechanics of Materials CM3www.ltas-cm3.ulg.ac.be
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece
Prediction of intra- and inter-laminar failure of laminates using
non-local damage-enhanced mean-field homogenization
Ling Wu (CM3), F. Sket (IMDEA), L. Adam (e-Xstream), I. Doghri (UCL), Ludovic Noels. (CM3)
Contributors: J.M. Molina (IMDEA), A. Makradi (Tudor)
STOMMMAC The research has been funded by the Walloon Region under the agreement no 1410246-STOMMMAC (CT-INT 2013-03-
28) in the context of M-ERA.NET Joint Call 2014.
SIMUCOMP The research has been funded by the Walloon Region under the agreement no 1017232 (CT-EUC 2010-10-12) in the
context of the ERA-NET +, Matera + framework.
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 2
• Introduction
– Failure of composite laminates
– Multi-scale modelling
– Mean-Field-Homogenization (MFH)
• Micro-scale modelling
– Incremental-Secant MFH
– Damage-enhanced incremental-secant MFH
• Multi-scale method for the failure analysis of composite
laminates
– Intra-laminar failure: Non-local damage-enhanced mean-field-
homogenization
– Inter-laminar failure: Hybrid DG/cohesive zone model
– Experimental validation
Content
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 3
Failure of composite laminates
• Difficulties
– Different involved mechanisms at
different scales
• Inter-laminar failure
• Intra-laminar failure
– Direct finite element simulation
On Micro-scale volume
Not possible at structural scale
DelaminationMatrix rupture
Pull out
Bridging
Fiber rupture
Debonding
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 4
Failure of composite laminates
• Difficulties
– Different involved mechanisms at
different scales
• Inter-laminar failure
• Intra-laminar failure
– Direct finite element simulation is not
possible at structural scale
– Continuum damage models do not
represent accurately the intra-laminar
failure
• Damage propagation direction is not in
agreement with experiments
DelaminationMatrix rupture
Pull out
Bridging
Fiber rupture
Debonding
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 5
Failure of composite laminates
• Difficulties
– Different involved mechanisms at
different scales
• Inter-laminar failure
• Intra-laminar failure
– Direct finite element simulation is not
possible at structural scale
– Continuum damage models do not
represent accurately the intra-laminar
failure
• Damage propagation direction is not in
agreement with experiments
• Solution:
– Embed damage model in a multi-scale
formulation
– For computational efficiency: use of
mean-field-homogenization
– For macro cracks: using hybrid
DG/Cohesive zone model
DelaminationMatrix rupture
Pull out
Bridging
Fiber rupture
Debonding
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 6
Multi-scale modelling
• Multi-scale modelling
– One way: homogenization
– 2 problems are solved
(concurrently)
• The macro-scale problem
• The meso-scale problem (on
a meso-scale Volume
Element)
• Length-scales separation
Lmacro>>LVE>>Lmicro
BVP
Macro-scale
Material
response
Extraction of a meso-
scale Volume Element
For accuracy: Size of the meso-
scale volume element smaller than
the characteristic length of the
macro-scale loading
To be statistically representative:
Size of the meso-scale volume
element larger than the
characteristic length of the micro-
structure
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 7
• Mean-Field-Homogenization
– Macro-scale
• FE model
• At one integration point e is know, s is sought
– Transition
• Downscaling: e is used as input of the MFH model
• Upscaling: s is the output of the MFH model
– Micro-scale
• Semi-analytical model
• Predict composite meso-scale response
• From components material models
Multi-scale modelling
wI
w0
ε σε
σ
Mori and Tanaka 73, Hill 65, Ponte Castañeda 91, Suquet
95, Doghri et al 03, Lahellec et al. 11, Brassart et al. 12, …
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 8
• Key principles
– Based on the averaging of the fields
– Meso-response
• From the volume ratios ( )
• One more equation required
– Difficulty: find the adequate relations
Mean-Field-Homogenization
V
VaV
a d)(1
X
1I0 vv
II00I0I0
σσσσσσ vvvv ww
II00I0I0
εεεεεε vvvv ww
II εσ f
00 εσ f
0I : εBε e
?eB
0I : εBε e
wI
w0
matrix
inclusions
composite ?
σ
ε
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 9
• Key principles (2)
– Linear materials
• Materials behaviours
• Mori-Tanaka assumption
• Use Eshelby tensor
with
Mean-Field-Homogenization
wI
w0
matrix
inclusions
composite ?
σ
ε
1
01
1
0 )](::[ CCCSIBe
0I0I :,,I εCCBε e
0εε
III : εCσ
000 : εCσ
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 10
• Key principles (2)
– Linear materials
• Materials behaviours
• Mori-Tanaka assumption
• Use Eshelby tensor
with
– Non-linear materials
• Define a Linear Comparison Composite (LCC)
• Common approach: incremental tangent
Mean-Field-Homogenization
0
algalg
I :,,II0
εCCBε e
inclusions
composite
σ
ε
algIC
alg0C
matrix: 0σ
Iε 0εε
wI
w0
matrix
inclusions
composite ?
σ
ε
1
01
1
0 )](::[ CCCSIBe
0I0I :,,I εCCBε e
0εε
III : εCσ
000 : εCσ
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 11
Content
• Micro-scale modelling
– Incremental-Secant Mean-Field-Homogenization (MFH)
– Damage-enhanced incremental-secant MFH
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 12
Incremental-secant mean-field-homogenization
• Material model
– Elasto-plastic material
• Stress tensor
• Yield surface
• Plastic flow &
• Linearization
0, eq pRpf Ysσσ
)(: plel εεCσ
Nε p pl
σN
f
σ
ε
elC
plε
εCσ :alg
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 13
• New incremental-secant approach
– Perform a virtual elastic unloading from
previous solution
• Composite material unloaded to reach the
stress-free state
• Residual stress in components
New Linear Comparison Composite (LCC)
Incremental-secant mean-field-homogenization
inclusions
composite
σ
ε
matrix
unload
Iε unload
0εunloadε
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 14
• New incremental-secant approach
– Perform a virtual elastic unloading from
previous solution
• Composite material unloaded to reach the
stress-free state
• Residual stress in components
New Linear Comparison Composite (LCC)
– Apply MFH from unloaded state
• New strain increments (>0)
• Use of secant operators
Incremental-secant mean-field-homogenization
inclusions
composite
σ
ε
matrix
unload
Iε unload
0εunloadε
r
0
SrSrr
I :,,II0
εCCBε e
unload
I/0I/0
r
I/0 εεε inclusions
composite
σ
ε
matrix
r
Iε rε r
0ε
SrIC
Sr0C
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 15
• New incremental-secant approach (2)
– Equations summary
• Inputs
– Internal variables at last increment
– Residual tensor after virtual unloading
– from FE resolution
• Solve iteratively the system
• With the stress tensors
Incremental-secant mean-field-homogenization
unload
II
r
I εεε
unload
00
r
0 εεε
r
0
Sr
I
Sr
0
r
I :,,I εCCBε e
r
0
Sr
0
res
00 : εCσσ
r
I
Sr
I
res
II : εCσσ
II00 σσσ vv
ε
r
II
r
00
unloadr εεεεε vv
inclusions
composite
σ
ε
matrix
r
Iε rε r
0ε
SrIC
Sr0C
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 16
• Zero-incremental-secant method
– Continuous fibres
• 55 % volume fraction
• Elastic
– Elasto-plastic matrix
– For inclusions with high hardening (elastic)
• Model is too stiff
Incremental-secant mean-field-homogenization
0
2
4
6
8
10
12
0 0.002 0.004
s/s
Y0
e
Longitudinal tension
FE (Jansson, 1992)
C0Sr
0
0.5
1
1.5
2
2.5
3
0 0.003 0.006 0.009
s/s
Y0
e
Transverse loading
FE (Jansson,1992)
C0Sr
inclusions
composite
σ
ε
matrix
r
Iε rε r
0ε
SrIC
Sr0C
0, eq pRpf Ysσσ
is underestimatedeqσ
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 17
• Zero-incremental-secant method (2)
– Continuous fibres
• 55 % volume fraction
• Elastic
– Elasto-plastic matrix
– Secant model in the matrix
• Modified if negative residual stress
Incremental-secant mean-field-homogenization
0
2
4
6
8
10
12
0 0.002 0.004
s/s
Y0
e
Longitudinal tension
FE (Jansson, 1992)
C0Sr
C0S0
0
0.5
1
1.5
2
2.5
3
0 0.003 0.006 0.009
s/s
Y0
e
Transverse loading
FE (Jansson,1992)C0
Sr
C0S0
inclusions
composite
σ
ε
matrix
r
Iε rε r
0ε
SrIC
Sr0C
inclusions
composite
σ
ε
matrix
r
Iε rε r
0ε
SrIC
S00C
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 18
• Verification of the method
– Spherical inclusions
• 17 % volume fraction
• Elastic
– Elastic-perfectly-plastic matrix
– Non-proportional loading
Incremental-secant mean-field-homogenization
.000
.020
.040
.060
.080
.100
0 10 20 30 40
e
t
e13 e23
e332e112e22
-70
-45
-20
5
30
0 10 20 30 40
s13
[Mpa
]
t
FE (Lahellec et al., 2013)MFH, incr. tg.MFH, var. (Lahellec et al., 2013)MFH, incr. sec.
-80
-40
0
40
80
120
0 10 20 30 40
s33
[Mp
a]
t
FE (Lahellec et al., 2013)
MFH, incr. tg.
MFH, var. (Lahellec et al., 2013)
MFH, incr. sec.
FFT
FFT
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 19
Non-local damage-enhanced MFH
• Material models
– Elasto-plastic material
• Stress tensor
• Yield surface
• Plastic flow &
• Linearization
0, eq pRpf Ysσσ
)(: plel εεCσ
Nε p pl
σN
f
σ
ε
elC
plε
εCσ :alg
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 20
Non-local damage-enhanced MFH
• Material models
– Elasto-plastic material
• Stress tensor
• Yield surface
• Plastic flow &
• Linearization
– Local damage model
• Apparent-effective stress tensors
• Plastic flow in the effective stress space
• Damage evolution
0, eq pRpf Ysσσ
)(: plel εεCσ
Nε p pl
σN
f
σ
ε
elC
plε
σ
ε
elC
plε
σ̂
σσ ˆ1 D
el1 CD
εCσ :alg
σσ ˆ1 D
),( pFD D ε
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 21
Non-local damage-enhanced MFH
• Material models
– Elasto-plastic material
• Stress tensor
• Yield surface
• Plastic flow &
• Linearization
– Local damage model
• Apparent-effective stress tensors
• Plastic flow in the effective stress space
• Damage evolution
– Non-Local damage model
• Damage evolution
• Anisotropic governing equation
• Linearization
0, eq pRpf Ysσσ
)(: plel εεCσ
Nε p pl
σN
f
σ
ε
elC
plε
σ
ε
elC
plε
σ̂
σσ ˆ1 D
el1 CD
εCσ :alg
σσ ˆ1 D
),( pFD D ε
)~,( pFD D ε
ppp ~~gc
pp
FFD DD ~
~ˆ:ˆ1 alg
σε
εσCσ
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 22
• Equations summary: zero-approach
– For soft matrix response
• Remove residual stress in matrix
• Avoid adding spurious internal energy
– Solve iteratively the system
– With the stress tensors
Non-local damage-enhanced MFH
unload
II
r
I εεε
unload
00
r
0 εεε
r
0
Sr
I
0S
0
r
I :,)1(,I εCCBε De
matrix:
inclusions
composite
σ
ε
matrix: 0σ̂
0σ
r
Iε rε r
0ε
SrIC
S00C
r
0
S0
00 :)1( εCσ D
r
I
Sr
I
res
II : εCσσ
II00 σσσ vv
)r(
II
)r(
00
)r( εεε vv
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 23
• Mesh-size effect
– Fictitious composite
• 30%-UD fibres
• Elasto-plastic matrix with damage
– Notched ply
Non-local damage-enhanced MFH
0
50
100
150
200
250
300
350
400
0 0.2 0.4 0.6 0.8 1 1.2
Fo
rce [
N/m
m]
Displacement [mm]
Mesh size: 0.43 mm
Mesh size: 0.3 mm
Mesh size: 0.15 mm
Mesh size: 0.1 mm
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 25
• Multi-scale method for the failure analysis of composite
laminates
– Intra-laminar failure: Non-local damage-enhanced mean-field-
homogenization
– Inter-laminar failure: Hybrid DG/cohesive zone model
– Experimental validation
Content
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 26
• Weak formulation of a composite laminate
– Strong form
for each homogenized ply Ω𝑖
for the matrix phase
– Boundary conditions
– Macro-scale finite-element discretization
Intra-laminar failure: Non-local damage-enhanced MFH
0fσ T
aa
pNp p~~~
Tnσ
ppp ~~gc
0~g pcn
aa
uN uu
ppppup
puuu
d
d
~
intext
~~~
~
~ FF
FF
p
u
KK
KK
i
ii
i
ii
1LL 1
i
i
X
X
1
2
3
5
4
i
iww
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 27
• 60%-UD Carbon-fibres reinforced epoxy
– Carbon fibres
• Use of transverse isotropic elastic material
– Elasto-plastic matrix with damage
• Use manufacturer Young’s modulus
• Use manufacturer strength
• Delamination: Double Cantilever Beam
– Critical energy realise rates
• GIC = 600 J/m2
• GIIC = 1200 J/m2 (assumption)
Experimental validation
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 29
• [45o4 / -45o
4]S- open hole laminate
– Tensile test on several coupons
– Propagation of the damaged zones in agreement with the fibre direction
Experimental validation
40 220
300
4.68±0.0539.60±0.35 O13
-45o-ply 45o-ply
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 30
• [45o4 / -45o
4]S- open hole laminate (2)
– Predicted delamination zones in agreement with experiments
– Tensile stress within 15 %
Experimental validation
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 32
• [90o / 45o / -45o / 90o / 0o]S- open hole laminate (2)
– Propagation of the damaged zones in agreement with the fibre direction
Experimental validation
90o-ply (out) 45o-ply -45o-ply 90o-ply (in) 0o-ply
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 33
• [90o / 45o / -45o / 90o / 0o]S- open hole laminate (3)
– Predicted delamination zones in agreement with experiments
Experimental validation
90o (out) / 45o 45o / -45o -45o / 90o (in) 90o (in) / 0o
CM3 ECCOMAS Congress, 5 - 10 JUNE 2016 Crete Island, Greece - 35
• Multi-scale method for the failure analysis of composite laminates
– Damage-enhanced MFH
– Non-local implicit formulation
– Hybrid DG/CZM for delamination
• Experimental validation
– Open-hole laminates
– Different stacking sequences
Conclusions