MFH for composite materials and computational ...A350 wing lower cover CM3 April 2014 - Northwestern...

Computational & Multiscale

Mechanics of Materials CM3 www.ltas-cm3.ulg.ac.be

CM3 April 2014 Northwestern Polytechnical University- Xi'an, P.R.China

Muti-scale methods with strain-softening: damage-enhanced

MFH for composite materials and computational

homogenization for cellular materials with micro-buckling

L. Noels, G. Becker, V.-D. Nguyen,

L. Wu , L. Adam (x-Stream), I. Doghri (UCL)

Non-local damage mean-field-homogenization

FE2

homogenization of

cellular structures

CM3 April 2014 - Northwestern Polytechnical University- Xi'an, P.R.China - 2

One-scale modelling

• Numerical model

– Resolution of the model equations using a

numerical approximation

• Finite element model

– One of the most popular numerical method

– Continuum mechanics equations integrated

in an average way on a tessellation

– The material law is known

• Metal at large scale

• Composites in the linear regime

• …

and the model can be solved solely at the macro-

scale

σ

ε

elC

plε


Multi-scale modelling: Why?

• Materials in aeronautics

– More and more engineered

– Multi-scale in nature

A350 wing lower cover



• Materials in aeronautics (2)

– More and more engineered

– Multi-scale in nature

A350 XWB wing trailing edge



• Limitations of one-scale models

– Physics at the micro-scale is too complex to

be modelled by a simple material law at the

macro-scale

• Engineered materials

• Multi-physics/scale problems

• ….

• See next slides

– Lack of information of the micro-scale state

during macro-scale deformations

• Required to predict failure

• …..

– Effect of the micro-structure on the macro-

structure response

• Fibres distribution …

• …

• Solution: multi-scale models

Delamination

Matrix rupture

Pull out

Bridging

Fiber rupture

Debonding


Content

• Introduction

– Multi-scale modelling: How?

– Strain softening issues

• Non-local damage-enhanced mean-field-homogenization

• Computational homogenization for cellular materials

• Other researches

– DG-based fracture mechanics: blast, fragmentation, …

• Conclusions


Multi-scale modelling: How?

• Principle

– 2 problems are solved concurrently

• The macro-scale problem

• The micro-scale problem (Representative Volume Element)

– Scale transitions coupling the two scales

• Downscaling: transfer of macro-scale quantities (e.g. strain) to the micro-scale to determine

the equilibrium state of the Boundary Value Problem

• Upscaling: constitutive law (e.g. stress) for the macro-scale problem is determined from the

micro-scale problem resolution

Assumptions:

Lmacro>>LRVE>>Lmicro

BVP

Macroscale

Material

response

Extraction of a RVE


• Computational technique: FE2

– Macro-scale

• FE model

• At one integration point e is know, s is sought


ε ?σ



– Macro-scale

• FE model


– Micro-scale

• Usual 3D finite elements

• Periodic boundary conditions


?σε



– Macro-scale

• FE model


– Transition

• Downscaling: e is used to define the BCs

• Upscaling: s is known from the reaction forces

– Micro-scale




σεε

σ



– Macro-scale

• FE model


– Transition

• Downscaling: e is used to define the BCs

• Upscaling: s is known from the reaction forces

– Micro-scale



– Advantages

• Accuracy

• Generality

– Drawback

• Computational time


Ghosh S et al. 95, Kouznetsova et al. 2002, Geers et al. 2010, …

σεε

σ


• Mean-Field Homogenization

– Macro-scale

• FE model



ε ?σ



– Macro-scale

• FE model


– Micro-scale

• Semi-analytical model

• Predict composite meso-scale response

• From components material models


wI

w0

ε ?σ



– Macro-scale

• FE model


– Transition

• Downscaling: e is used as input of the MFH model

• Upscaling: s is the output of the MFH model

– Micro-scale





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, …



– Macro-scale

• FE model


– Transition

• Downscaling: e is used as input of the MFH model

• Upscaling: s is the output of the MFH model

– Micro-scale




– Advantages

• Computationally efficient

• Easy to integrate in a FE code (material model)

– Drawbacks

• Difficult to formulate in an accurate way

– Geometry complexity

– Material behaviours complexity


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, …


Strain softening of the microscopic response

• Finite element solutions for strain softening problems suffer from:

– The loss the uniqueness and strain localization

– Mesh dependence

• Requires a non-local formulation of the macro-scale problem

The numerical results change with the size of

mesh and direction of mesh Homogenous unique solution

Lose of uniqueness

Strain localized

The numerical results change without

convergence


Multi-scale simulations with strain softening

• Two cases considered

– Composite materials

• Mean-field homogenization

• Non-local damage formulation

– Honeycomb structures

• Computational homogenization

• Second-order FE2

• Micro-buckling




Non-local damage-enhanced mean-field-homogenization

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.

L. Wu (ULg), L. Noels (ULg), L. Adam (e-Xstream), I. Doghri (UCL)


• Semi analytical Mean-Field Homogenization

– Based on the averaging of the fields

– Meso-response

• From the volume ratios ( )

• One more equation required

– Difficulty: find the adequate relations


V

VaV

a d)(1

X

1I0 vv

II00I0I0

σσσσσσ vvvv ww

II00I0I0

εεεεεε vvvv ww

II εσ f

00 εσ f

0I : εBεe

wI

w0

matrix

inclusions

composite ?

σ

ε

? eB

0I : εBεe


• Mean-Field Homogenization for different materials

– Linear materials

• Materials behaviours

• Mori-Tanaka assumption

• Use Eshelby tensor

with

– Non-linear materials

• Define a Linear Comparison Composite

• 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



• Material models

– Elasto-plastic material

• Stress tensor

• Yield surface

• Plastic flow &

• Linearization

0, eq pRpf Ysσσ

)(: plelεεCσ

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

)(: plelεεCσ

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



• Finite element solutions for strain softening problems suffer from:

– The loss the uniqueness and strain localization

– Mesh dependence

• Implicit non-local approach [Peerlings et al 96, Geers et al 97, …]

– A state variable is replaced by a non-local value reflecting the interaction between

neighboring material points

– Use Green functions as weight w(y; x)

New degrees of freedom

The numerical results change with the size of

mesh and direction of mesh Homogenous unique solution

Lose of uniqueness

Strain localized

The numerical results change without

convergence

C

d);()(1

)(~

CV

VwaV

a xyyx

aaca ~~ 2



• 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

)(: plelεεCσ

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


• Limitation of the incremental tangent method

– Fictitious composite

• 50%-UD fibres

• Elasto-plastic matrix with damage

– Due to the incremental formalism, stress in fibres cannot decrease during loading


No fibres unloading

-150

-100

-50

0

50

100

150

200

0 0.05 0.1

s [

MP

a]

e

Periodical Cell

MFH, incr. tg

-150

-100

-50

0

50

100

150

0 0.05 0.1

s0 [

MP

a]

e

Periodical Cell

MFH, incr. tg.

-200

-150

-100

-50

0

50

100

150

200

250

0 0.05 0.1s

I [M

Pa]

e

Periodical Cell

MFH, incr. tg


• Problem

– We want the fibres to get unloaded during

the matrix damaging process

• For the incremental-tangent approach

• To unload the fibres ( ) with such

approach would require

• We cannot use the incremental tangent MFH

– We need to define the LCC from another

stress state


0I ε

0algI C

0

algalg

I :,)1(,II0

εCCBε De

matrix:

inclusions

composite

σ

ε

algIC

alg0C

matrix: 0σ̂

0σ

Iε 0εε


• Idea

– 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


matrix:

inclusions

composite

σ

ε

matrix: 0σ̂

0σ

unload

Iε unload

0εunloadε


• Idea

– 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

• Apply MFH from unloaded state

– New strain increments (>0)

– Use of secant operators

– Possibility of have unloading


r

0

SrSrr

I :,)1(,II0

εCCBε De

unload

I/0I/0

r

I/0 εεε

0r

I ε

0I ε

matrix:

inclusions

composite

σ

ε

matrix: 0σ̂

0σ

r

Iε rε r

0ε

SrIC

Sr0C

matrix:

inclusions

composite

σ

ε

matrix: 0σ̂

0σ

unload

Iε unload

0εunloadε


• Zero-incremental-secant method

– Continuous fibres

• 55 % volume fraction

• Elastic

– Elasto-plastic matrix (no damage)

– For inclusions with high hardening (elastic)

• Model is too stiff


inclusions

composite

σ

ε

matrix: 0σ

r

Iε rε r

0ε

SrIC

Sr0C

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) C0

Sr


• Zero-incremental-secant method (2)

– Continuous fibres

• 55 % volume fraction

• Elastic

– Elasto-plastic matrix (no damage)

– Secant model in the matrix

• Modified if stiffer inclusions (negative residual stress)


inclusions

composite

σ

ε

matrix: 0σ

r

Iε rε r

0ε

SrIC

Sr0C

inclusions

composite

σ

ε

matrix: 0σ

r

Iε rε r

0ε

SrIC

S00C

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

S0

C0Sr


• New results for damage

– Fictitious composite

• 50%-UD fibres

– Elasto-plastic matrix with damage


-150

-100

-50

0

50

100

150

200

0 0.05 0.1

s [

MP

a]

e

Periodical Cell

MFH, incr. tg

MFH, incr. sec.

Fibres unloading


• Weak formulation

– Strong form

for the homogenized composite material

for the matrix phase

– Boundary conditions

– Finite-element discretization


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


• Laminate plate with hole

– Carbon-fibres reinforced epoxy

• 60%-UD fibres


– [-452/452]S staking sequence


40 220

300

4.68±0.05

39.60±0.35 O13


• Laminate plate with hole (2)

– Carbon-fibres reinforced epoxy

• 60%-UD fibres


– [-452/452]S staking sequence





Computational homogenization for cellular materials


• Challenges

– Micro-structure

• Not perfect with non periodic mesh

How to constrain the periodic boundary

conditions?

Computational homogenization for foamed materials


• Challenges

– Micro-structure



conditions?

• Thin components

• Experiences micro-buckling

How to capture the instability?



• Challenges

– Micro-structure



conditions?

• Thin components



– Transition

• Homogenized tangent not always elliptic

• Localization bands

How can we recover the solution unicity

at the macro-scale?



• Challenges

– Micro-structure



conditions?

• Thin components



– Transition

• Homogenized tangent not always elliptic


How can we recover the solution unicity

at the macro-scale?

– Macro-scale


How to remain computationally efficient




• Recover solution unicity: second-order FE2

– Macro-scale

• High-order Strain-Gradient formulation

• Partitioned mesh (//)

– Transition

• Gauss points on different processors

• Each Gauss point is associated to

one mesh and one solver

– Micro-scale

• Usual continuum


𝐏 𝑿 ⋅ 𝛁0- 𝐐 𝑿 : (𝛁0 ⊗𝛁0)=0

𝐏 , 𝐐

𝜕𝐏

𝜕𝐅 ,

𝜕𝐏

𝜕(𝐅 ⊗𝛁),

𝜕𝐐

𝜕𝐅 ,

𝜕𝐐

𝜕(𝐅 ⊗𝛁)

𝐅 , 𝐅 ⊗ 𝛁

𝐏 𝑿 ⋅ 𝛁0=0


• Micro-scale periodic boundary conditions

– Defined from the fluctuation field

– Stated on opposite RVE sizes

– Can be achieved by constraining opposite nodes

• Foamed materials

– Usually random meshes

– Important voids on the boundaries

• Honeycomb structures

– Not periodic due to the imperfections


𝒘 = 𝒖 − 𝐅 − 𝐈 ⋅ 𝑿 +𝟏

𝟐𝐅 ⊗ 𝛁𝟎 : (𝑿⊗ 𝑿)

𝒘 𝑿+ = 𝒘 𝑿−

𝒘 𝑿 𝒅𝝏𝑽 = 𝟎𝝏𝑽−


• Micro-scale periodic boundary conditions (2)

– New interpolant method

– Use of Lagrange, cubic spline .. interpolations

– Fits for

• Arbitrary meshes

• Important voids on the RVE sides

– Results in new constraints in terms of the boundary and control nodes displacements


𝒘 𝑿− = N 𝑿 𝒘𝑘

𝑘

𝒘 𝑿+ = N 𝑿 𝒘𝑘

𝑘

N 𝑿 𝒘𝑘

𝑘

𝒅𝝏𝑽 = 𝟎𝝏𝑽−

𝑪 𝒖 𝑏 − 𝒈 𝐅 , 𝐅 ⊗ 𝛁𝟎 =0


• Discontinuous Galerkin (DG) implementation of the second order continuum

• Finite-element discretization

• Same discontinuous polynomial approximations for the

• Test functions h and

• Trial functions

• Definition of operators

on the interface trace:

• Jump operator:

• Mean operator:

• Continuity is weakly enforced, such that the method

• Is consistent

• Is stable

• Has the optimal convergence rate


(a-1)+ (a)+

x

(a+1)-

Fie

ld

(a+1)+ (a)- (a-1)- ∙ =∙+ − ∙−

∙ =·+ + ∙−

2



𝑎 𝒖 , 𝛿𝒖 = 𝑎bulk 𝒖 , 𝛿𝒖 + 𝑎PI 𝒖 , 𝛿𝒖 + 𝑎QI 𝒖 , 𝛿𝒖 = 𝑏(𝛿𝒖 )

𝐏 𝑿 ⋅ 𝛁0- 𝐐 𝑿 : (𝛁0 ⊗𝛁0)=0

• Second-order FE2 method

– Macro-scale second order continuum

– Requires C1 shape functions on the mesh

– The C1 can be weakly enforced using the DG method






• Usual volume terms


𝑎bulk 𝒖 , 𝛿𝒖 = 𝐏 𝒖 : 𝛿𝒖 ⊗ 𝛁𝟎 + 𝐐 𝑿 ⋮ (𝛿𝒖 ⊗ 𝛁0 ⊗𝛁0) 𝑑𝑉𝑉


𝐏 𝑿 ⋅ 𝛁0- 𝐐 𝑿 : (𝛁0 ⊗𝛁0)=0



𝑎PI 𝒖 , 𝛿𝒖 =

𝛿𝒖 ⋅ 𝐏 − 𝐐 ∙ 𝛁0 ⋅ 𝑵 + 𝒖 ⋅ 𝐏 𝛿𝒖 − 𝐐 𝛿𝒖 ∙ 𝛁0 ⋅ 𝑵 +

𝒖 ⊗𝑵 :𝛽𝑃ℎ𝑠

𝐂𝟎 : 𝛿𝒖 ⊗𝑵 𝑑𝑉

𝜕𝐼𝑉





• Weak enforcement of the C0

– Continuity

– Consistency

– Stability

between the finite elements

– Allows efficient parallelization as elements are disjoint


𝐏 𝑿 ⋅ 𝛁0- 𝐐 𝑿 : (𝛁0 ⊗𝛁0)=0






• Weak enforcement of the C1

– Continuity

– Consistency

– Stability

between the finite elements

– Allows efficient parallelization as elements are disjoint


𝑎QI 𝒖 , 𝛿𝒖 =

𝛿𝒖 ⊗ 𝛁𝟎 ⋅ 𝐐 ⋅ 𝑵 + 𝒖 ⊗𝛁𝟎 ⋅ 𝐐 𝛿𝒖 ⋅ 𝑵 +

𝒖 ⊗𝛁𝟎 ⊗𝑵 :𝛽𝑃ℎ𝑠

𝐉𝟎 : 𝛿𝒖 ⊗ 𝛁𝟎 ⊗𝑵 𝑑𝑉

𝜕𝐼𝑉


𝐏 𝑿 ⋅ 𝛁0- 𝐐 𝑿 : (𝛁0 ⊗𝛁0)=0


• Capturing instabilities

– Macro-scale: localization bands

• Path following method on the applied loading

• Arc-length constraint on the load increment


𝑎 𝒖 , 𝛿𝒖 = 𝜇 𝑏(𝛿𝒖 )

ℎ Δ𝒖 , Δ𝜇 =Δ𝒖 ⋅ Δ𝒖

𝑋 02 + Δ𝜇 2 − Δ𝐿2 = 0


• Capturing instabilities

– Macro-scale: localization bands

• Path following method on the applied loading


– Micro-scale

• Path following method on the applied boundary conditions



𝑎 𝒖 , 𝛿𝒖 = 𝜇 𝑏(𝛿𝒖 )

𝐏 , 𝐐

𝜕𝐏

𝜕𝐅 ,

𝜕𝐏

𝜕(𝐅 ⊗𝛁),

𝜕𝐐

𝜕𝐅 ,

𝜕𝐐

𝜕(𝐅 ⊗𝛁)

𝐅 , 𝐅 ⊗ 𝛁

ℎ Δ𝒖 , Δ𝜇 =Δ𝒖 ⋅ Δ𝒖

𝑋 02 + Δ𝜇 2 − Δ𝐿2 = 0

𝑪 𝒖 𝑏 − 𝒈 𝐅 , 𝐅 ⊗ 𝛁𝟎 =0

𝐅 = 𝐅 0 + 𝜇 Δ𝐅

𝐅 ⊗ 𝛁𝟎 = 𝐅 ⊗ 𝛁𝟎 0 + 𝜇 Δ 𝐅 ⊗ 𝛁𝟎

ℎ Δ𝒖, Δ𝜇 =Δ𝒖 ⋅ Δ𝒖

𝑋02 + Δ𝜇2 − Δ𝑙2 = 0


• Compression of an hexagonal honeycomb


• Comparison of different solutions

Full direct simulation Multiscale with different macro-meshes



• Compression of an hexagonal honeycomb (2)

– Captures of the softening onset

– Captures the softening response

– No macro-mesh size effect





Ludovic Noels, G. Becker,

L. Homsi, V. Lucas, S. Mulay,

V.-D. Nguyen, V. Péron-Lührs,

V.-H. Truong, F. Wan, L. Wu

QC method for grain-boundary sliding

DG-based fracture framework

Stiction failure in a MEMS sensor

g

t

DG-based fracture

framework

SVE size effect on meso-scale properties


• Examples

DG-based fracture mechanics: blast, fragmentation, …


• Non-local damage-enhanced mean-field-homogenization

– MFH with damage model for the matrix material

– Non-local implicit formulation

– Can capture the strain softening

– More in

• 10.1016/j.ijsolstr.2013.07.022

• 10.1016/j.ijplas.2013.06.006

• 10.1016/j.cma.2012.04.011

• 10.1007/978-1-4614-4553-1_13

• Computational homogenization for foamed materials

– Second-order FE2 method

– Micro-buckling propagation

– General way of enforcing PBC

– More in

• 10.1016/j.cma.2013.03.024

• 10.1016/j.commatsci.2011.10.017

• 10.1016/j.ijsolstr.2014.02.029

• Other research fields

• Open-source software

– Implemented in GMSH

• http://geuz.org/gmsh/

Conclusions

http://geuz.org/gmsh/




谢谢

Conclusions

MFH for composite materials and computational ...A350 wing lower cover CM3 April 2014 - Northwestern...

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