Computational & Multiscale CM3 Mechanics of Materials · Computational & Multiscale Mechanics of Materials CM3 CM3 18-21 December 2019 - APCOM2019 –Taipei, Taiwan A stochastic Mean-Field-Homogenization-based

Computational & Multiscale

Mechanics of Materials CM3www.ltas-cm3.ulg.ac.be

CM3 18-21 December 2019 - APCOM2019 – Taipei, Taiwan

A stochastic Mean-Field-Homogenization-based micro-

mechanical model of unidirectional composites

𝑥

𝑦

SVE 𝑥, 𝑦

SVE 𝑥′, 𝑦

SVE 𝑥′, 𝑦′

t

𝑙SVE

L. Wu, J. M. Calleja, V.-D. Nguyen, L. Noels

The research has been funded by the Walloon Region under the agreement no.7911-VISCOS in the context of the 21st

SKYWIN call and no 1410246-STOMMMAC (CT-INT 2013-03-28) in the context of the M-ERA.NET Joint Call 2014. SEM

images by Major Zoltan, Nghia Chnug Chi, JKU, Austria

.

CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 2

• Multi-scale modelling

– 2 problems are solved

concurrently

• The macro-scale problem

• The meso-scale problem

(on a meso-scale Volume

Element)

– Length-scales separation

Objectives

P, σ, q, … F, Ɛ, T, 𝛁T, …

BVP

Macro-scale

Material

response

Extraction of a meso-

scale Volume Element

Lmacro>>LVE>>Lmicro

• Failure of composite materials:• Statistical representativity is lost

• Main objective: To develop an efficient integrated stochastic multiscale approach

to predict failure of composites

BVP on a meso-scale Volume Element

Direct FE simulation vs. Semi-analytical method

- All the details of SVE - General information e.g.

volume fraction of inclusions,

Lmacro>>LVE~Lmicro


• Proposed methodology

– To develop a stochastic Mean Field Homogenization method able to predict the

probabilistic distribution of material response at an intermediate scale from micro-

structural constituents characterization

Methodology

𝜔 =∪𝑖 𝜔𝑖

wI

w0

Stochastic

Homogenization

Stochastic

MF-ROM

SVE

realizations

Micro-structure

stochastic model

strain

stress

d [mm]

PDF

0 1 2 3 4

4

3

2

1

0

rf [mm]

PDF

2.4 2.8 3.2 3.6

4

3

2

1

0


• Uncertainty quantification of micro-structure & micro-structure generator

– 2000x and 3000x SEM images, fibers detection

– Basic geometric information

– Distributions Random variables:

• 𝑝𝑅 𝑟 ,

• 𝑝𝑑1st 𝑑 ,

• 𝑝𝜗1st 𝜃 ,

• 𝑝Δ𝑑 𝑑 with Δ𝑑 = 𝑑2nd − 𝑑1st

• 𝑝Δ𝜗 𝜃 with Δ𝜗 = 𝜗2nd − 𝜗1st

* L. Wu, C.N. Chung, Z. Major, L. Adam, L. Noels, From SEM images to elastic responses: A stochastic multiscale analysis

of UD fiber reinforced composites, Compos. Struct. (ISSN: 0263-8223) 189 (2018a) 206–227

Experimental measurements

𝑅0

𝑅1

𝜗1st

𝑑1st∆𝜗

𝜗2nd𝑑2nd

𝑅2


• Numerical micro-structures are generated by a fiber additive process

– Arbitrary size

– Arbitrary number

– Possibility to generate non-homogenous distributions

Micro-structure stochastic model


• Window technique

– Extraction of Stochastic Volume Elements

• 𝑙SVE = 25 𝜇𝑚

• Correlation

– For each SVE

• Extract apparent homogenized material tensor ℂM

• Consistent boundary conditions:

– Periodic (PBC)

– Minimum kinematics (SUBC)

– Kinematic (KUBC)

Stochastic homogenization on the SVEs

𝑥

𝑦

SVE 𝑥, 𝑦

SVE 𝑥′, 𝑦

SVE 𝑥′, 𝑦′

t

𝑙SVE𝜺M =1

𝑉 𝜔 𝜔

𝜺m𝑑𝜔

𝝈M =1

𝑉 𝜔 𝜔

𝝈m𝑑𝜔

ℂM =𝜕𝝈M

𝜕𝒖M ⊗𝛁M

𝑅𝐫𝐬 𝝉 =𝔼 𝑟 𝒙 − 𝔼 𝑟 𝑠 𝒙 + 𝝉 − 𝔼 𝑠

𝔼 𝑟 − 𝔼 𝑟2

𝔼 𝑠 − 𝔼 𝑠2


• Mean-Field-Homogenization (MFH)

– Linear composites

– We use Mori-Tanaka assumption for 𝔹𝜀 I, ℂ0 , ℂI

• Stochastic MFH

– How to define randomness?

Stochastic Mean-Field Homogenization

𝛆M = 𝛆 = 𝑣0𝛆0 + 𝑣I𝛆I

𝛆I = 𝔹𝜀 I, ℂ0 , ℂI : 𝛆0

𝛔M = 𝛔 = 𝑣0𝛔0 + 𝑣I𝛔I

inclusions

composite

matrix

𝛆I

𝛔

𝛆

ℂ0

𝛆 = 𝛆M𝛆0

ℂI

ℂM = ℂM I, ℂ0 , ℂI , 𝑣I


wI

w0

L. Wu, V.-D. Nguyen, L. Adam, L. Noels, An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites (2019)


• Mean-Field-Homogenization (MFH)


• Consider an equivalent system

– For each SVE realization 𝑖:

ℂM and 𝜈I known

– Anisotropy from ℂM𝑖

𝜃 is evaluated

– Fiber behavior uniform

ℂI for one SVE

– Remaining optimization problem:



𝛆I = 𝔹𝜀 I, ℂ0 , ℂI : 𝛆0


inclusions

composite

matrix

𝛆I

𝛔

𝛆

ℂ0

𝛆 = 𝛆M𝛆0

ℂI

Defined as

random variables

ℂM ≃ ℂM( I, ℂ0 , ℂI , 𝑣I, 𝜃)

ℂ0ℂI

ℂ0

ℂI

𝜃

𝑎

𝑏

Equivalent

inclusion

min𝑎

𝑏, 𝐸0 , 𝜈0

ℂM − ℂM(𝑎

𝑏, 𝐸0 , 𝜈0 ; 𝑣I, 𝜃, ℂI )

ℂM = ℂM I, ℂ0 , ℂI , 𝑣I


wI

w0


• Inverse stochastic identification

– Comparison of homogenized

properties from SVE realizations

and stochastic MFH


ℂM ≃ ℂM( I, ℂ0 , ℂI , 𝑣I, 𝜃)

ℂ0ℂI

ℂ0

ℂI

𝜃

𝑎

𝑏

Equivalent

inclusion

* L. Wu, C. Nghia Chung, Z. Major, L. Adam, L. Noels, A micro-mechanics-based inverse study for stochastic

order reduction of elastic UD-fiber reinforced composites analyzes, Internat. J. Numer. Methods Engrg. (ISSN:

0263-8223) 115 (2018b) 1430–1456


• Non-linear Mean-Field-homogenization


– Non-linear composites

Non-linear stochastic Mean-Field Homogenization


𝛆I = 𝔹𝜀 I, ℂ0 , ℂI : 𝛆0


inclusions

composite

matrix

𝛆I

𝛔

𝛆

ℂ0

𝛆 = 𝛆M𝛆0

ℂI

inclusions

composite

matrix

𝚫𝛆I

𝛔

𝛆𝚫𝛆M 𝚫𝛆0

𝚫𝛆M = Δ𝛆 = 𝑣0Δ𝛆0 + 𝑣IΔ𝛆I

𝚫𝛆I = 𝔹𝜀 I, ℂ0LCC, ℂ𝐼

LCC : 𝚫𝛆0


Define a linear

comparison

composite material


• Incremental-secant Mean-Field-homogenization

– Virtual elastic unloading from previous state

• Composite material unloaded to reach the stress-

free state

• Residual stress in components


inclusions

composite

matrix

𝚫𝛆Iunload

𝛔

𝛆

𝚫𝛆Munload

𝚫𝛆0unload

ℂIel

ℂ0el


• Incremental-secant Mean-Field-homogenization



free state


– Define Linear Comparison Composite

• From unloaded state

• Incremental-secant loading

• Incremental secant operator


𝚫𝛆M𝐫 = Δ𝛆 = 𝑣0Δ𝛆0

𝐫 + 𝑣IΔ𝛆I𝐫

𝚫𝛆I𝐫 = 𝔹𝜀 I, ℂ0

S, ℂIS : 𝚫𝛆0

𝐫


inclusions

composite

matrix

𝚫𝛆Iunload

𝛔

𝛆

𝚫𝛆Munload

𝚫𝛆0unload

ℂIel

ℂ0el

𝚫𝛆I/0𝐫 = Δ𝛆I/0 + 𝚫𝛆I/0

unload

𝚫𝛔M = ℂMS I, ℂ0

S, ℂIS, 𝑣I : 𝚫𝛆M

𝐫 ℂ0S

inclusions

composite

matrix

𝚫𝛆I𝐫

𝛔

𝛆𝚫𝛆M

𝐫

𝚫𝛆0𝐫

ℂIS

ℂMS


• Non-linear inverse identification

– First step from elastic response


ℂ0ℂI

ℂ0el

ℂIel

𝜃

𝑎

𝑏

Equivalent

inclusion

ℂMel ≃ ℂM

el( I, ℂ0el, ℂI

el, 𝑣I, 𝜃)

inclusions

composite

matrix

𝚫𝛆Iunload

𝛔

𝛆

𝚫𝛆Munload

𝚫𝛆0unload

ℂIel

ℂ0el



– First step from elastic response

– Second step from the LCC

• New optimization problem

• Extract the equivalent hardening 𝑅 𝑝0 from the

incremental secant tensor


ℂ0S ≃ ℂ0

S( 𝑅 𝑝0 ; ℂ0el)

ℂ0ℂI

ℂ0S

ℂIS

𝜃

𝑎

𝑏

Equivalent

inclusion

ℂMel ≃ ℂM

el( I, ℂ0el, ℂI

el, 𝑣I, 𝜃)

ℂ0S ≃ ℂ0


ℂ0S

inclusions

composite

matrix

𝚫𝛆I𝐫

𝛔

𝛆𝚫𝛆M

𝐫

𝚫𝛆0𝐫

ℂIS

ℂMS

𝚫𝛔M ≃ ℂMS I, ℂ0

S, ℂIS, 𝑣I, 𝜃 : 𝚫𝛆M

𝐫

𝑝0

𝑅 𝑝0

inMPa

Extracted from SVEs

Identified hardening

Input matrix law



– Comparison SVE vs. MFH


ℂ0S ≃ ℂ0


ℂ0ℂI

ℂ0S

ℂIS

𝜃

𝑎

𝑏

Equivalent

inclusion

ℂMel ≃ ℂM

el( I, ℂ0el, ℂI

el, 𝑣I, 𝜃)


• Damage-enhanced Mean-Field-homogenization



free state



inclusions

composite

matrix

𝚫𝛆Iunload

𝛔

𝛆𝚫𝛆M

unload𝚫𝛆0

unload

ℂIel

(1 − 𝐷0)ℂ0el

effective

matrix


• Damage-enhanced inverse identification

– Elastic unloading

• Identify damage evolution 𝐷0

– Reloading with the LCC

•

• Extract the equivalent hardening 𝑅 𝑝0 & damage

evolution D0 𝑝0 from incremental secant tensor:


1 − 𝐷0 ℂ0el

ℂIel

1 − 𝐷0 ℂ0el

ℂIel

𝜃

𝑎

𝑏

Equivalent

inclusion

1 − D0 ℂ0𝑆 ≃ 1 − D0 𝑝0 ℂ0

𝑆( 𝑅 𝑝0 ; ℂ0el)

inclusions

composite

matrix

𝚫𝛆Ir

𝛔

𝛆𝚫𝛆M

r𝚫𝛆0

r

ℂIS

(1 − 𝐷0)ℂ0S

effective

matrix

ℂMS

𝚫𝛔M = ℂMS I, 1 − 𝐷0 ℂ0

S, ℂIS, 𝑣I : 𝚫𝛆M

𝐫

ℂMel(𝐷) ≃ ℂM

el( I, 1 − 𝐷0 ℂ0el, ℂI

el, 𝑣I, 𝜃)

inclusions

composite

matrix

𝚫𝛆Iunload

𝛔

𝛆𝚫𝛆M

unload𝚫𝛆0

unload

ℂIel

(1 − 𝐷0)ℂ0el

effective

matrixℂMel(𝐷)


• Damage-enhanced inverse identification

– Similar process as for elasto-plasticity


1 − 𝐷0 ℂ0ℂIel

1 − 𝐷0 ℂ0S

ℂIel

𝜃

𝑎

𝑏

Equivalent

inclusion

1 − D0 ℂ0𝑆 ≃ 1 − D0 𝑝0 ℂ0

𝑆( 𝑅 𝑝0 ; ℂ0el)

* L. Wu, V.-D. Nguyen, L. Adam, L. Noels, An inverse micro-mechanical analysis toward the stochastic homogenization of

nonlinear random composites, Comp. Meth. in App. Mech. and Engineering (ISSN: 0045-7825) 348, (2019) 97-138


• Generation of random field

– Inverse identification vs. diffusion map –based generator [Soize, Ghanem 2016]

Damage model Parameters

Use of stochastic Mean-Field Homogenization



• One single ply loading realization

– Random field and finite elements discretization

– Non-uniform homogenized stress distributions

– Creates damage localization



• Ply loading realizations

– Preliminary results (softening part not implemented)


Micro-structural model of fiber-reinforced highly crosslinked epoxy

• UD Composites with RTM6 epoxy matrix

– Identified matrix material behaviour

• Hyperelastic viscoelastic-viscoplastic constitutive model enhanced by a multi-mechanism

nonlocal damage model

• Hardening:

• Softening:

• Failure: &

– 2D simulations of 25 x 25 μm 40% volume fraction composite SVEs

𝑅 𝑝 = σ𝑦 + ℎ (1 − 𝑒−ℎ𝑒𝑥𝑝 𝑝)

𝐷𝑠 = (𝐻𝑠 𝑝 − 𝑝𝑖𝑛𝑖𝑡α)

Φ𝑓 = γ − 𝑎 exp −𝑏𝑇 − 𝑐 = 0 𝐷𝑓 = 𝐻𝑓 (χ𝑓)ζ𝑓(1 − 𝐷𝑓)

ζ𝑑 χ𝑓

0

0.5

1𝐷𝑓


• Apparent response

– Independent Stochastic Volume Elements

• 𝑙SVE = 25 𝜇𝑚

– Stochastic homogenization

• Extract apparent responses

– 3 stages

• Linear response

• (Damage-enhanced) elasto-plasticity

• Failure (loss of size objectivity)

Stochastic homogenization on the SVEs

𝜺M =1

𝑉 𝜔 𝜔

𝜺m𝑑𝜔

𝝈M =1

𝑉 𝜔 𝜔

𝝈m𝑑𝜔

ℂM =𝜕𝝈M

𝜕𝒖M ⊗𝛁M

Stochastic Homogenization

SVE realizations

Linear Non-linear Failure


• Non-linear SVE simulations

Mean-Field Homogenization with failure

0.0408

0.0780

0.0654

0.1019

Gc [N/mm]


• Extracted failure characteristics

Mean-Field Homogenization with failure


• Stochastic micro-structures

– Geometrical features from statistical measurements

– Micro-structure geometry generator

– Experimentally calibrated/validated epoxy model with length scale effect

• Inverse MFH identification

– MFH is used as a micro-mechanics based model

– Parameters identified from SVE simulations

– Localization behaviour identified using objective fields

• Stochastic Finite elements

– Stochastic MFH is used as material law

– Random fields (MFH parameters) generated using data-driven approach

– First ply simulations

Conclusions


Thank you for your attention!

Special thanks to:

VISCOS


• T. Mori, K. Tanaka, Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall.

21 (5) (1973) 571–574

• L. Wu, L. Noels, L. Adam, I. Doghri, A combined incremental–secant mean–field homogenization scheme with per–phase

residual strains for elasto–plastic composites, Int. J. Plast. 51 (2013a) 80–102,

http://dx.doi.org/10.1016/j.ijplas.2013.06.006

• L. Wu, L. Noels, L. Adam, I. Doghri, An implicit-gradient-enhanced incremental-secant mean-field homogenization scheme

for elasto-plastic composites with damage, Int. J. Solids Struct. (ISSN: 0020-7683) 50 (24) (2013b) 3843–3860,

http://dx.doi.org/10.1016/j.ijsolstr.2013.07.022.

• L. Wu, L. Adam, I. Doghri, L. Noels, An incremental-secant mean-field homogenization method with second statistical

moments for elasto-visco-plastic composite materials, Mech. Mater. (ISSN: 0167-6636) 114 (2017) 180–200,

http://dx.doi.org/10.1016/j.mechmat.2017.08.006.

• V.-D.Nguyen F.Lani T.Pardoen X.P.Morelle L.Noels, A large strain hyperelastic viscoelastic-viscoplastic-damage

constitutive model based on a multi-mechanism non-local damage continuum for amorphous glassy polymers. Int. Journal

of Sol. and Struct., (ISSN: 0020-7683) 96 (2016) 192-216, https://doi.org/10.1016/j.ijsolstr.2016.06.008

• V.-D. Nguyen, L. Wu, L. Noels, A micro-mechanical model of reinforced polymer failure with length scale effects and

predictive capabilities. Validation on carbon fiber reinforced high-crosslinked RTM6 epoxy resin , Mech. Mater. (ISSN:

0167-6636) 133 (2019) 193-213, https://dx.doi.org/10.1016/j.mechmat.2019.02.017

References

Documents

Computational & Multiscale CM3 Mechanics of Materials · Computational & Multiscale Mechanics of Materials CM3 CM3 18-21 December 2019 - APCOM2019 –Taipei, Taiwan A stochastic Mean-Field-Homogenization-based