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Computational & Multiscale Mechanics of Materials

CM3

www.ltas-cm3.ulg.ac.be

CM3

EMMC15 7 - 9 September 2016, Brussels, Belgium

Simulations of composite laminates inter- and intra-laminar failure using on a non-local mean-field damage-enhanced multi-scale method

Ling Wu (CM3), L. Adam (e-Xstream), B. Bidaine (e-Xstream), Ludovic Noels. (CM3) Experiments: F. Sket (IMDEA), J.M. Molina (IMDEA), A. Makradi (List)

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

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

• Introduction of uncertainties

– As a random field

Content

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

Delamination Matrix rupture Pull out Bridging Fiber rupture Debonding

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

Delamination Matrix rupture Pull out Bridging Fiber rupture Debonding

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

Delamination Matrix rupture Pull out Bridging Fiber rupture Debonding

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

σ

ε 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

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

V a V a d ) ( 1 X 1

I

  v v

I I I

I

σ σ σ σ σ σ v v v v     

w w I I I

I

ε ε ε ε ε ε v v v v     

w w

 

I I

ε σ f 

 

ε σ f 

I

: ε B ε

e

 ?

e

B

I

: ε B ε

e

 wI w0

matrix inclusions composite

?

σ ε

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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 1 1

)] ( : : [

 

   C C C S I Be

 

I I

: , , I ε C C B ε

e

 ε ε 

 I I I

:ε C σ  : ε C σ 

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

 

alg alg I

: , , I

I

ε C C B ε   

e

inclusions composite

σ ε

alg

I

C

alg

C

matrix:

σ

I

ε  ε  ε  wI w0

matrix inclusions composite

?

σ ε

1 1 1

)] ( : : [

 

   C C C S I Be

 

I I

: , , I ε C C B ε

e

 ε ε 

 I I I

:ε C σ  : ε C σ 

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 11

Content

• Micro-scale modelling

– Incremental-Secant Mean-Field-Homogenization (MFH) – Damage-enhanced incremental-secant MFH

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 12

Incremental-secant mean-field-homogenization

• Material model

– Elasto-plastic material

• Stress tensor
• Yield surface
• Plastic flow

&

• Linearization

   

,

eq

    p R p f

Y

s σ σ ) ( :

pl el

ε ε C σ   N ε p   

pl

σ N    f σ ε

el

C

pl

ε ε C σ   :

alg



CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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 I

ε 

unload

ε 

unload

ε 

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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 I

ε 

unload

ε 

unload

ε 

 

r Sr Sr r I

: , , I

I

ε C C B ε   

e unload I/0 I/0 r I/0

ε ε ε     

inclusions composite

σ ε

matrix

r I

ε 

r

ε 

r

ε 

Sr

I

C

Sr

C

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

2 4 6 8 10 12 0.002 0.004

s/sY0 e

Longitudinal tension

FE (Jansson, 1992) C0

Sr

0.5 1 1.5 2 2.5 3 0.003 0.006 0.009

s/sY0

e

Transverse loading

FE (Jansson, 1992) C0

Sr

inclusions composite

σ ε

matrix

r I

ε 

r

ε 

r

ε 

Sr

I

C

Sr

C

   

,

eq

    p R p f

Y

s σ σ is underestimated

eq

σ

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

2 4 6 8 10 12 0.002 0.004

s/sY0 e

Longitudinal tension

FE (Jansson, 1992) C0

Sr

C0

S0

0.5 1 1.5 2 2.5 3 0.003 0.006 0.009

s/sY0

e

Transverse loading

FE (Jansson, 1992) C0

Sr

C0

S0

inclusions composite

σ ε

matrix

r I

ε 

r

ε 

r

ε 

Sr

I

C

Sr

C

inclusions composite

σ ε

matrix

r I

ε 

r

ε 

r

ε 

Sr

I

C

S0

C

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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 10 20 30 40

e t

e13 e23 e332e112e22

• 70
• 45
• 20

5 30 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

40 80 120 10 20 30 40

s33 [Mpa] t

FE (Lahellec et al., 2013) MFH, incr. tg. MFH, var. (Lahellec et al., 2013) MFH, incr. sec. FFT FFT

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 19

Non-local damage-enhanced MFH

• Material models

– Elasto-plastic material

• Stress tensor
• Yield surface
• Plastic flow

&

• Linearization

   

,

eq

    p R p f

Y

s σ σ ) ( :

pl el

ε ε C σ   N ε p   

pl

σ N    f σ ε

el

C

pl

ε ε C σ   :

alg



CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

   

,

eq

    p R p f

Y

s σ σ ) ( :

pl el

ε ε C σ   N ε p   

pl

σ N    f σ ε

el

C

pl

ε σ ε

el

C

pl

ε σ ˆ

 σ

σ ˆ 1 D  

 

el

1 C D  ε C σ   :

alg



 σ

σ ˆ 1 D   ) , ( p F D

D

   ε

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

   

,

eq

    p R p f

Y

s σ σ ) ( :

pl el

ε ε C σ   N ε p   

pl

σ N    f σ ε

el

C

pl

ε σ ε

el

C

pl

ε σ ˆ

 σ

σ ˆ 1 D  

 

el

1 C D  ε C σ   :

alg



 σ

σ ˆ 1 D   ) , ( p F D

D

   ε ) ~ , ( p F D

D

   ε

 

p p p       ~ ~

g

c

 

p p F F D

D D

~ ~ ˆ : ˆ 1

alg

                  σ ε ε σ C σ

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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 I I r I

ε ε ε     

unload r

ε ε ε     

 

r Sr I S r I

: , ) 1 ( , I ε C C B ε     D

e

matrix: inclusions composite

σ ε

matrix:

ˆ σ σ

r I

ε 

r

ε 

r

ε 

Sr

I

C

S0

C

r S0

: ) 1 ( ε C σ    D

r I Sr I res I I

: ε C σ σ   

I I

σ σ σ v v  

) r ( I I ) r ( ) r (

ε ε ε      v v

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 24
• Mesh-size effect

– Fictitious composite

• 30%-UD fibres
• Elasto-plastic matrix with damage

– Notched ply

Non-local damage-enhanced MFH

50 100 150 200 250 300 350 400 0.2 0.4 0.6 0.8 1 1.2 Force [N/mm] Displacement [mm]

Mesh size: 0.43 mm Mesh size: 0.3 mm Mesh size: 0.15 mm Mesh size: 0.1 mm

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 26
• Weak formulation of a composite laminate

– Strong form

for each homogenized ply Ω𝑗 for the matrix phase

– Boundary conditions – Macro-scale finite-element discretization

Intra-laminar failure: Non-local damage-enhanced MFH

f σ    

T a a p

N p p ~ ~

~

 T n σ  

 

p p p       ~ ~

g

c

 

~

g

    p c n

a a u

N u u                        

p p p p u p p u u u

d d

~ int ext ~ ~ ~ ~

~ F F F F p u K K K K

 

 



i i i i

i i

1 L L

1



     





i i

  

X X

1



2



3



5



4





i i

w w 

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 29
• [45o

4 / -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±0.05 39.60±0.35 O13

• 45o-ply

45o-ply

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 30
• [45o

4 / -45o 4]S- open hole laminate (2)

– Predicted delamination zones in agreement with experiments – Tensile stress within 15 %

Experimental validation

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 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

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 34
• [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

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 36
• [45o / -45o]S laminate under uniform tension

– No hole to trigger localization – Material defects trigger localization

Introduction of uncertainties

Uniform volume fraction 5% variation in volume fraction

GFRP [45/-45]

https://m.youtube.com/watch?v=PotKTduzTxg

CM3

EMMC15 - 7 - 9 September 2016, Brussels, Belgium

• 37
• 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

• Introduction of material uncertainties in the model

– First simulations

Conclusions