CM3 Mechanics of Materials www.ltas-cm3.ulg.ac.be Multi-scale - - PowerPoint PPT Presentation

▶
cm3
SMART_READER_LITE
LIVE PREVIEW

CM3 Mechanics of Materials www.ltas-cm3.ulg.ac.be Multi-scale - - PowerPoint PPT Presentation

Mar 08, 2023 294 likes •699 views

Computational & Multiscale CM3 Mechanics of Materials www.ltas-cm3.ulg.ac.be Multi-scale modelling Multiscale models Ludovic Noels for composite failure G. Becker, S. Mulay, V.-D. Nguyen, V. Pron-Lhrs, L. Wu FE2 computations of

slide-1
SLIDE 1

Computational & Multiscale Mechanics of Materials

CM3

www.ltas-cm3.ulg.ac.be

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science

Multi-scale modelling

Ludovic Noels

  • G. Becker, S. Mulay, V.-D. Nguyen,
  • V. Péron-Lührs, L. Wu

Multiscale models for composite failure QC method for grain-boundary sliding Stiction failure in a MEMS sensor (picture Sandia National

Laboratories )

FE2 computations of foamed structures DG-based fracture framework

slide-2
SLIDE 2

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 2

Content

  • Introduction

– Multi-scale modelling: Why? – Multi-scale modelling: How?

  • Mean-Field-Homogenization with non-local damage
  • Conclusions
slide-3
SLIDE 3

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 3

Multi-scale modelling: Why?

  • 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

  • Grain-size effect in metals
  • …
  • Solution: multi-scale models
slide-4
SLIDE 4

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 4

Multi-scale modelling: Why?

  • Examples of multi-scale problems

– Different physics at the different scales – Stiction (adhesion of MEMS) Stiction failure in a MEMS sensor ( Jeremy A.Walraven Sandia National

  • Laboratories. Albuquerque, NM USA)

Continuum solid mechanics Van der Waals/capillary/Hertz forces at the asperity level Statistical representation of rough surfaces

n

F

a 2

c 2

z

l g t s

slide-5
SLIDE 5

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 5

Multi-scale modelling: Why?

  • Examples of multi-scale problems (2)

– Continuum solid mechanics at the different scales – Non-linear response of [-452/452]S composites

slide-6
SLIDE 6

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 6

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

slide-7
SLIDE 7

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 7

  • Computational technique: FE2

– Macro-scale

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

Multi-scale modelling: How?

ε ? σ

slide-8
SLIDE 8

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 8

  • Computational technique: FE2

– Macro-scale

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

– Micro-scale

  • Usual 3D finite elements
  • Periodic boundary conditions

Multi-scale modelling: How?

? σ ε

slide-9
SLIDE 9

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 9

  • Computational technique: FE2

– Macro-scale

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

– Transition

  • Downscaling: e is used to define the BCs
  • Upscaling: s is known from the reaction forces

– Micro-scale

  • Usual 3D finite elements
  • Periodic boundary conditions

Multi-scale modelling: How?

σ ε ε σ  

slide-10
SLIDE 10

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 10

  • Computational technique: FE2

– Macro-scale

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

– Transition

  • Downscaling: e is used to define the BCs
  • Upscaling: s is known from the reaction forces

– Micro-scale

  • Usual 3D finite elements
  • Periodic boundary conditions

– Advantages

  • Accuracy
  • Generality

– Drawback

  • Computational time

Multi-scale modelling: How?

ε σ ε σ  

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

slide-11
SLIDE 11

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 11

  • Mean-Field Homogenization

– Macro-scale

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

Multi-scale modelling: How?

ε ? σ

slide-12
SLIDE 12

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 12

  • Mean-Field Homogenization

– Macro-scale

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

– Micro-scale

  • Semi-analytical model
  • Predict composite meso-scale response
  • From components material models

Multi-scale modelling: How?

wI w0 ε ? σ

slide-13
SLIDE 13

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 13

  • 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: How?

wI w0 ε ? σ ε σ ε σ  

slide-14
SLIDE 14

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 14

  • 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

– 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

Multi-scale modelling: How?

wI w0 ε ? σ 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, …

slide-15
SLIDE 15

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 15

  • 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

Multi-scale modelling: How?

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

 wI w0

matrix inclusions composite

?

σ ε ?

e

B

I

:ε B ε

e

slide-16
SLIDE 16

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 16

  • Mean-Field Homogenization for linear materials

– System of equations

  • From the volume ratios ( )
  • Assume linear behaviours
  • Relation between average strains

– Single inclusion problem from Eshelby tensor

  • with
  • Results from a phase transformation analysis

– Multiple inclusions problem

  • Mori-Tanaka assumption

Multi-scale modelling: How?

1

I

  v v

I I

σ σ σ v v  

I I

ε ε ε v v  

I I I

: ε C σ  : ε C σ  wI w0

matrix inclusions composite

?

σ ε

 e

e

: ) , , (

I I

C C H ε I

1 1 1

)] ( : : [

 

   C C C S I H e S

 

I I

: , , I ε C C B ε

e

 ε ε 

 e e

H B 

I

:ε B ε

e

slide-17
SLIDE 17

Computational & Multiscale Mechanics of Materials

CM3

www.ltas-cm3.ulg.ac.be

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science

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)
slide-18
SLIDE 18

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 18

Non-local damage mean-field-homogenization

  • 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 ) ( ~

C V

V w a V a x y y x a a c a    ~ ~

2

slide-19
SLIDE 19

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 19

Non-local damage mean-field-homogenization

  • 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

slide-20
SLIDE 20

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 20

Non-local damage mean-field-homogenization

  • 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

slide-21
SLIDE 21

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 21

Non-local damage mean-field-homogenization

  • 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

slide-22
SLIDE 22

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 22

  • Incremental-tangent model with damage in the matrix

– From the volume ratios ( ) – Non-linear phases behaviours

  • Elasto-plastic inclusions
  • Non-local damaged matrix
  • Composite

Non-local damage mean-field-homogenization

1

I

  v v

I I

σ σ σ v v  

I 1

ε ε ε v v  

I alg I

:

I

ε C σ   

 

p p F F D

D D

~ ~ ˆ : ˆ 1

alg

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

 

p p F F D

D D

~ ~ ˆ : ˆ : 1 :

alg I alg I I

                   σ ε ε σ ε C ε C σ

 

alg alg I

: , ) 1 ( , I

I

ε C C B ε     D

e

Mori-Tanaka on one loading interval: wI w0

matrix: inclusions composite

σ ε

alg

I

C

alg

C

matrix:

ˆ σ σ

I

ε  ε  ε 

slide-23
SLIDE 23

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 23

  • Finite-element implementation

– Strong form

for the homogenized composite material for the matrix phase

– Boundary conditions – Finite-element discretization

Non-local damage mean-field-homogenization

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 u u p p u uu

d d

~ int ext ~ ~ ~

~ F F F F p u K K K K

slide-24
SLIDE 24

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 24

  • DNS vs. FE/MFH

– Fictitious composite

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

Non-local damage mean-field-homogenization

  • 150
  • 100
  • 50

50 100 150 200 0.05 0.1

Stress (MPa) Strain

Random Cells Periodical Cell MFH

slide-25
SLIDE 25

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 25

  • Mesh-size effect

– Fictitious composite

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

– Notched ply

Non-local damage mean-field-homogenization

50 100 150 200 250 300 0.5 1

Force (N/mm) Displacement (mm)

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

slide-26
SLIDE 26

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 26

  • Laminate: calibration

– Carbon-fibres reinforced epoxy

  • 60%-UD fibres
  • Elasto-plastic matrix with damage

– [-452/452]S staking sequence

Non-local damage mean-field-homogenization

10 20 30 40 50 60 70 80 90 0.2 0.4 0.6

Stress (MPa) Strain (%)

Experimental Loading Experimental Unloading Numerical Loading Numerical Unloading

50 160 260 N x ei 25.3±0.2

slide-27
SLIDE 27

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 27

  • Laminate plate with hole

– Carbon-fibres reinforced epoxy

  • 60%-UD fibres
  • Elasto-plastic matrix with damage

– [-452/452]S staking sequence

Non-local damage mean-field-homogenization

40 220 300 4.68±0.05 39.60±0.35 O13

slide-28
SLIDE 28

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 28

  • Laminate plate with hole (2)

– Carbon-fibres reinforced epoxy

  • 60%-UD fibres
  • Elasto-plastic matrix with damage

– [-452/452]S staking sequence

Non-local damage mean-field-homogenization

slide-29
SLIDE 29

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 29

  • Limitation of the method

– Fictitious composite

  • 30%-UD fibres

– Less accurate during softening for high fibres-volume-ratios

Non-local damage mean-field-homogenization

  • 150
  • 100
  • 50

50 100 150 200 0.05 0.1

Stress (MPa) Strain

Random Cells Periodical Cell MFH

  • 50%-UD fibres
  • 150
  • 100
  • 50

50 100 150 200 0.05 0.1

Stress (MPa) Strain

Random Cells Periodical Cell MFH

slide-30
SLIDE 30

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 30

  • Limitation of the method (2)

– Fictitious composite

  • 50%-UD fibres
  • Analyse phases behaviours

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

Non-local damage mean-field-homogenization

No fibres unloading

  • 150
  • 100
  • 50

50 100 150 200 0.05 0.1

s [MPa] e

Periodical Cell MFH, incr. tg

  • 150
  • 100
  • 50

50 100 150 0.05 0.1

s0 [MPa] e

Periodical Cell MFH, incr. tg.

  • 200
  • 150
  • 100
  • 50

50 100 150 200 250 0.05 0.1

sI [MPa] e

Periodical Cell MFH, incr. tg

slide-31
SLIDE 31

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 31

  • 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

Non-local damage mean-field-homogenization

I 

ε

alg

I

 C

 

alg alg I

: , ) 1 ( , I

I

ε C C B ε     D

e

matrix: inclusions composite

σ ε

alg

I

C

alg

C

matrix:

ˆ σ σ

I

ε  ε  ε 

slide-32
SLIDE 32

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 32

  • 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

Non-local damage mean-field-homogenization

matrix: inclusions composite

σ ε

matrix:

ˆ σ σ

unload I

ε 

unload

ε 

unload

ε 

slide-33
SLIDE 33

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 33

  • 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

Non-local damage mean-field-homogenization

 

r Sr Sr r I

: , ) 1 ( , I

I

ε C C B ε     D

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

ε ε ε     

r I 

ε

I 

ε

matrix: inclusions composite

σ ε

matrix:

ˆ σ σ

unload I

ε 

unload

ε 

unload

ε 

matrix: inclusions composite

σ ε

matrix:

ˆ σ σ

r I

ε 

r

ε 

r

ε 

Sr

I

C

Sr

C

slide-34
SLIDE 34

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 34

  • Zero-incremental-secant method

– Continuous fibres

  • 55 % volume fraction
  • Elastic

– Elasto-plastic matrix – For inclusions with high hardening (elastic)

  • Model is too stiff

Non-local damage mean-field-homogenization

inclusions composite

σ ε

matrix:

σ

r I

ε 

r

ε 

r

ε 

Sr

I

C

Sr

C

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

slide-35
SLIDE 35

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 35

  • Zero-incremental-secant method

– Continuous fibres

  • 55 % volume fraction
  • Elastic

– Elasto-plastic matrix – Secant model in the matrix

  • Modified if stiffer inclusions (negative residual stress)

Non-local damage mean-field-homogenization

inclusions composite

σ ε

matrix:

σ

r I

ε 

r

ε 

r

ε 

Sr

I

C

Sr

C

inclusions composite

σ ε

matrix: 0

σ

r I

ε 

r

ε 

r

ε 

Sr

I

C

S0

C

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

S0

C0

Sr

slide-36
SLIDE 36

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 36

  • Verification of the method

– Spherical inclusions

  • 17 % volume fraction
  • Elastic

– Elastic-perfectly-plastic matrix – Non-radial loading

Non-local damage mean-field-homogenization

0.00 0.02 0.04 0.06 0.08 0.10 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.

slide-37
SLIDE 37

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 37

  • New results for damage

– Fictitious composite

  • 50%-UD fibres
  • Analyse phases behaviours
  • 200
  • 150
  • 100
  • 50

50 100 150 200 250 0.05 0.1

sI [MPa] e

Periodical Cell MFH, incr. tg MFH, incr. sec.

Non-local damage mean-field-homogenization

  • 150
  • 100
  • 50

50 100 150 200 0.05 0.1

s [MPa] e

Periodical Cell MFH, incr. tg MFH, incr. sec.

  • 150
  • 100
  • 50

50 100 150 0.05 0.1

s0 [MPa] e

Periodical Cell MFH, incr. tg. MF, incr. sec. Fibres unloading

slide-38
SLIDE 38

CM3

2013 - Effective Properties of Materials: Perspectives from Mathematics and Engineering Science - 38

  • Multi-scale methods

– Allows considering

  • Micro-structure geometry
  • Non-linear behaviours of the micro-constituents

– Rely on different techniques

  • Computational
  • MFH
  • …

– Accuracy depends

  • On the model
  • On the micro-structure complexity
  • …
  • Non-local damage-enhanced MFH

– Good description of the meso-scale response – Can be used to study coupons problems

Conclusions