computational & multiscale cm3 mechanics of materials · cm3 eccomas congress, 5 - 10 june 2016...

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


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



• Difficulties


different scales



– 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


Pull out

Bridging

Fiber rupture

Debonding



• Difficulties


different scales



– 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


Pull out

Bridging

Fiber rupture

Debonding


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


• 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, …


• 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


• Key principles (2)

– Linear materials

• Materials behaviours

• Mori-Tanaka assumption

• Use Eshelby tensor

with


wI

w0

matrix

inclusions

composite ?

σ

ε

1

01

1

0 )](::[ CCCSIBe

0I0I :,,I εCCBε e

0εε

III : εCσ

000 : εCσ


• 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


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


Content

• Micro-scale modelling

– Incremental-Secant Mean-Field-Homogenization (MFH)

– Damage-enhanced incremental-secant MFH


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


• 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)


inclusions

composite

σ

ε

matrix

unload

Iε unload

0εunloadε


• 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


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


• 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


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


• Zero-incremental-secant method

– Continuous fibres

• 55 % volume fraction

• Elastic

– Elasto-plastic matrix

– For inclusions with high hardening (elastic)

• Model is too stiff


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


• Zero-incremental-secant method (2)

– Continuous fibres


• Elastic

– Elasto-plastic matrix

– Secant model in the matrix

• Modified if negative residual stress


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


• Verification of the method

– Spherical inclusions


• Elastic

– Elastic-perfectly-plastic matrix

– Non-proportional loading


.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


Non-local damage-enhanced MFH

• Material models


• Stress tensor

• Yield surface

• Plastic flow &

• Linearization

0, eq pRpf Ysσσ

)(: plel εεCσ

Nε p pl

σN

f

σ

ε

elC

plε

εCσ :alg



• Material models


• 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



• Material models


• Stress tensor

• Yield surface

• Plastic flow &

• Linearization

– Local damage model

• Apparent-effective stress tensors

• Plastic flow in the effective stress space


– Non-Local damage model


• 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


• 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


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


• Mesh-size effect

– Fictitious composite

• 30%-UD fibres

• Elasto-plastic matrix with damage

– Notched ply


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


• 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


• 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


• 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


• [45o4 / -45o

4]S- open hole laminate

– Tensile test on several coupons

– Propagation of the damaged zones in agreement with the fibre direction


40 220

300

4.68±0.0539.60±0.35 O13

-45o-ply 45o-ply


• [45o4 / -45o

4]S- open hole laminate (2)

– Predicted delamination zones in agreement with experiments

– Tensile stress within 15 %



• [90o / 45o / -45o / 90o / 0o]S- open hole laminate (2)

– Propagation of the damaged zones in agreement with the fibre direction


90o-ply (out) 45o-ply -45o-ply 90o-ply (in) 0o-ply


• [90o / 45o / -45o / 90o / 0o]S- open hole laminate (3)

– Predicted delamination zones in agreement with experiments


90o (out) / 45o 45o / -45o -45o / 90o (in) 90o (in) / 0o


• 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

computational & multiscale cm3 mechanics of materials · cm3 eccomas congress, 5 - 10 june 2016...

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