computational & multiscale cm3 mechanics of materials · 2019-10-07 · computational &...

Computational & Multiscale

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

CM3 1-4 October 2019 - CMCS 2019 – Glasgow, UK

An inverse Mean-Field-Homogenization-based micro-mechanical

model for stochastic multiscale simulations 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

CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 3

• 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

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


• Non-linear responses of composites

– RVE size is too large

– Structural response suffers from scatter

– Requires the use of smaller VEs

• Main objective

– To develop an integrated stochastic multiscale approach to predict

failure of composites

Objectives

Lmacro>>LVE>~Lmicro

For accuracy: Size of the meso-

scale volume element smaller than

the characteristic length of the

macro-scale loading

Meso-scale volume element no

longer statistically representative:

Stochastic Volume Elements


• 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

Methodology


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


• 2000x and 3000x SEM images

• Fibers detection

Experimental measurements


• Basic geometric information of fibers' cross sections

– Fiber radius distribution 𝑝𝑅 𝑟

• Basic spatial information of fibers

– The distribution of the nearest-neighbor net distance

function 𝑝𝑑1st 𝑑

– The distribution of the orientation of the undirected line

connecting the center points of a fiber to its nearest

neighbor 𝑝𝜗1st 𝜃

– The distribution of the difference between the net

distance to the second and the first nearest-neighbor

𝑝Δ𝑑 𝑑 with Δ𝑑 = 𝑑2nd − 𝑑1st

– The distribution of the second nearest-neighbor’s

location referring to the first nearest-neighbor 𝑝Δ𝜗 𝜃

with Δ𝜗 = 𝜗2nd − 𝜗1st

Micro-structure stochastic model

𝑅0

𝑅1

𝜗1st

𝑑1st∆𝜗

𝜗2nd𝑑2nd

𝑅2


• Histograms of random micro-structures’ descriptors


𝑅0

𝑅1

𝜗1st

𝑑1st∆𝜗

𝜗2nd𝑑2nd

𝑅2


• Dependency of the four random variables 𝑑1st, ∆𝑑, 𝜗1st, ∆𝜗

• Correlation matrix

• Distances correlation matrix

𝑑1st and ∆𝑑 are dependent

they will have to be generated

from their empirical copula


𝑑1st ∆𝑑 𝜗1st ∆𝜗

𝑑1st 1.0 0.27 0.04 0.08

∆𝑑 1.0 0.05 0.06

𝜗1st 1.0 0.05

∆𝜗 1.0

𝑑1st ∆𝑑 𝜗1st ∆𝜗

𝑑1st 1.0 0.21 0.01 0.02

∆𝑑 1.0 0.002 -0.005

𝜗1st 1.0 0.02

∆𝜗 1.0

𝑅0

𝑅1

𝜗1st

𝑑1st∆𝜗

𝜗2nd𝑑2nd

𝑅2


• 𝑑1st and ∆𝑑 should be generated using their empirical copula


∆𝑑

SEM sample Generated sample

𝑑1st

∆𝑑

∆𝑑

𝑑1st

Directly from

copula generator

Statistic result from

generated SVE

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



• SVEs definition

Methodology


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


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

– To be used in micro-structural analysis

• Behaviour in composite is different

• Introduce a length-scale effect

• Resin model implementation:

– Requires non-local form [Bažant 1988]

• Introduction of characteristic length 𝑙𝑐

• Weighted average: 𝑍 𝒙 = 𝑉𝐶𝑊 𝒚;𝒙, 𝑙𝑐 𝑍 𝐲 𝑑𝒚

– Implicit form [Peerlings et al. 1998]

• New degrees of freedom: 𝑍

• New Helmholtz-type equations: 𝑍 − 𝑙𝑐2 Δ 𝑍 = 𝑍

– Has a length scale effect

Local: no mesh

convergence

𝜎

𝜀


• Resin model

– Material changes represented via internal variables

– Constitutive law 𝐏 𝐅; 𝒁 𝑡′

• Internal variables 𝒁 𝑡′

• Multi-damage strategy

– Non-local damage evolution laws

• Saturation law

• Failure law

0 0.2 0.4 0.6 0.80

50

100

150

200

-True Strain

-Tru

e S

tress [

MP

a]

Kinematic hardening

Both hardening

& saturated damage

Isotropic hardening

& failure damage

𝐏 = 𝟏 − 𝐷𝑠 𝟏 − 𝐷𝑓 𝐏

𝐷𝑠 = 𝐷𝑠 𝐷𝑠, 𝐅 𝑡 , 𝜒𝑠 𝑡 ; 𝑍 𝜏 , 𝜏 ∈ [0 𝑡] 𝜒𝑠

𝛾𝑠 − 𝑙𝑠2 Δ 𝛾𝑠 = 𝛾𝑠

𝜒𝑠 𝑡 = max𝜏

𝛾𝑠 𝜏

𝐷𝑓 = 𝐷𝑓 𝐷𝑓, 𝐅 𝑡 , 𝜒𝑓 𝑡 ; 𝑍 𝜏 , 𝜏 ∈ [0 𝑡] 𝜒𝑓

𝛾𝑓 − 𝑙𝑓2 Δ 𝛾𝑓 = 𝛾𝑓

𝜒𝑓 𝑡 = max𝜏

𝛾𝑓 𝜏




• Resin model: saturated softening

– Saturated damage evolution

– Several hardening/softening combinations• Requires composite material tests

0 0.2 0.4 0.6 0.80

50

100

150

200

-True Strain

-Tru

e S

tress [M

Pa]

𝐷𝑠∞ = 0.2𝐷𝑠∞ = 0.31

Experiment

𝐷𝑠∞ = 0.51𝐷𝑠∞ = 0.62

𝐷𝑠 = 𝐻𝑠 𝜒𝑠 − 𝜒𝑠0𝜁𝑠 𝐷𝑠∞ − 𝐷𝑠 𝜒𝑠

𝛾𝑠 − 𝑙𝑠2 Δ 𝛾𝑠 = 𝛾

𝜒𝑠 = max𝜏

𝜒𝑠0, 𝛾𝑠 𝜏

Pure resin uniaxial

tests



• Resin model: saturated softening

– Several hardening/softening combinations• Requires composite material tests

– Length scale effect

– Non-local BC at fibre interface

0 0.2 0.4 0.6 0.80

50

100

150

200

-True Strain

-Tru

e S

tress [M

Pa]

𝐷𝑠∞ = 0.2𝐷𝑠∞ = 0.31

Experiment

𝐷𝑠∞ = 0.51𝐷𝑠∞ = 0.62

Pure resin uniaxial

tests

0 0.02 0.04 0.06 0.080

60

120

180

240

-True Strain

-Tru

e S

tre

ss [M

Pa

]

𝐷𝑠∞ = 0.2𝐷𝑠∞ = 0.31

Experiment

𝐷𝑠∞ = 0.51𝐷𝑠∞ = 0.62

0

0.25

0.5𝐷𝑠

𝑙𝑠 = 3μm 1 −𝐷𝑠𝐷𝑠∞

𝛾𝑠 = 0

Composites


• Resin model: failure softening

– Failure surface

– Damage evolution

– Affect ductility

– Calibration

• From localization simulation

• Recover the epoxy 𝐺𝑐


𝐷𝑓 = 𝐻𝑓 𝜒𝑓𝜁𝑓

1 − 𝐷𝑓−𝜁𝑑

𝜒𝑓

𝛾𝑓 − 𝑙𝑓2 Δ 𝛾𝑓 = 𝛾𝑓

𝜒𝑓 = max𝜏

𝛾𝑓 𝜏

𝑙𝑓 = 3 μm 𝛻0 𝛾𝑓 ⋅ 𝐍 = 0

𝜙𝑓 = 𝛾 − 𝑎 exp −𝑏tr 𝝉

3 𝜏𝑒𝑞− 𝑐

𝜙𝑓 − 𝑟 ≤ 0; 𝑟 ≥ 0; and 𝑟 𝜙𝑓 − 𝑟 = 0

𝛾𝑓 = 𝑟

-True Strain

-Tru

e S

tress

𝐻𝑓

𝜒𝑓

𝐷𝑓

𝑅

Total failure

Unloading path

Loading path

Localization

onset

Dissipation onset

𝜎

𝒟

𝒟loc

𝒟end

𝐺𝑐 =𝒟end − 𝒟loc

𝐴0



• UD Composites with RTM6 epoxy matrix

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

0

0.5

1𝐷𝑓


• Extraction of apparent responses

Methodology


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


• 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


• Apparent elastic properties: distribution & Correlation

– For large enough RVE (𝑙SVE > 10 𝜇𝑚)


𝑙SVE = 25 𝜇𝑚

Auto/cross correlation vanishes at 𝜏 = 𝑙SVE

Distributions get closer to normal

𝑙SVE = 25 𝜇𝑚

Apparent properties of different SVEs

are independent

However the distributions depend on

• 𝑙SVE• The boundary conditions


• 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)


𝜺M =1

𝑉 𝜔 𝜔

𝜺m𝑑𝜔

𝝈M =1

𝑉 𝜔 𝜔

𝝈m𝑑𝜔

ℂM =𝜕𝝈M

𝜕𝒖M ⊗𝛁M

Stochastic Homogenization

SVE realizations

Linear Non-linear Failure


• Stochastic Mean-Field Homogenization-based model

– First stage: linear elasticity

Methodology


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

Linear


• 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



– Second stage: damage-enhanced elasto-plasticity

Methodology


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

Non-linear


• 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



• View to damage Mean-Field-homogenization

– Incremental forms

• Strain increments in the same direction

– Because of the damaging process, the fiber

phase is elastically unloaded during matrix

softening

• Solution

– We need to define the LCC from another state

inclusions

composite

matrix:

Δ𝛆I Δ 𝛆 Δ𝛆0

𝛔

𝛆

ℂ0alg

ℂIalg

matrix

inclusions

𝛔

𝛆Δ𝛆I Δ𝛆0

𝚫𝛆𝐈 = 𝔹𝜀 I, ℂ0alg, ℂI

alg: 𝚫𝛆𝟎


• 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


• Non-linear inverse identification

– 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

S( 𝑅 𝑝0 ; ℂ0el)

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


• Damage-enhanced Mean-Field-homogenization



free state



inclusions

composite

matrix

𝚫𝛆Iunload

𝛔

𝛆𝚫𝛆M

unload𝚫𝛆0

unload

ℂIel

(1 − 𝐷0)ℂ0el

effective

matrix


• Damage-enhanced Mean-Field-homogenization



free state


– Define Linear Comparison Composite

• From elastic state

• Incremental-secant loading

• Incremental secant operator


𝚫𝛆M𝐫 = Δ𝛆 = 𝑣0Δ𝛆0

𝐫 + 𝑣IΔ𝛆I𝐫

𝚫𝛆I𝐫 = 𝔹𝜀 I, 1 − 𝐷0 ℂ0

S, ℂIS : 𝚫𝛆0

𝐫


𝚫𝛆I/0𝐫 = Δ𝛆I/0 + 𝚫𝛆I/0

unload

inclusions

composite

matrix

𝚫𝛆Iunload

𝛔

𝛆𝚫𝛆M

unload𝚫𝛆0

unload

ℂIel

(1 − 𝐷0)ℂ0el

effective

matrix

𝚫𝛔M = ℂMS I, 1 − 𝐷0 ℂ0


𝐫

inclusions

composite

matrix

𝚫𝛆Ir

𝛔

𝛆𝚫𝛆M

r𝚫𝛆0

r

ℂIS

(1 − 𝐷0)ℂ0S

effective

matrix

ℂMS


• Damage-enhanced inverse identification

– Second step: elastic unloading

• Identify damage evolution 𝐷0


1 − 𝐷0 ℂ0el

ℂIel

1 − 𝐷0 ℂ0el

ℂIel

𝜃

𝑎

𝑏

Equivalent

inclusion

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

– Second step: elastic unloading

• Identify damage evolution 𝐷0

– Third step from 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


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)




– Third stage: Failure

Methodology


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

Failure


Mean-Field Homogenization with failure

• Need to recover size objectivity

– Strength is objective

– Critical energy release rate is objective

0

0.5

1𝐷𝑓

Total failure

Unloading paths

Loading path

Localization

onset

Dissipation onset

𝜎

𝒟

𝒟loc

𝒟end


𝐴0

𝜎𝑐

𝜎𝑐


• Non-linear SVE simulations


0.0408

0.0780

0.0654

0.1019

Gc [N/mm]



• Damage-enhanced mean-field homogenization

– Requires non-local form for the matrix part of homogenized

behavior

• Following implicit form [Peerlings et al. 1998]

– Apparent matrix plastic strain 𝑝0

– Non-local apparent matrix plastic strain 𝑝0

– Matrix damage evolution

• New Helmholtz-type equations:

–

– Definition of non-local length 𝑙𝑐

ℂ0SD 𝑝0

ℂIS

𝜃

𝑎

𝑏

)Δ𝐷0 = 𝐹𝐷(𝚫𝛆𝟎, Δ 𝑝0

𝑝0 − 𝑙𝑐2𝛻 ⋅ 𝛻 𝑝0 = 𝑝0

)Δ𝐷0 = 𝐹𝐷(𝚫𝛆𝟎, Δ𝑝0



• Damage-enhanced mean-field homogenization

– Requires non-local form for the matrix homogenized behavior

• Matrix damage evolution

• New Helmholtz-type equations:

–

– Definition of non-local length 𝑙𝑐

• Evaluation of non-local length

– To recover the energy release rate of SVEs

ℂ0SD 𝑝0

ℂIS

𝜃

𝑎

𝑏)Δ𝐷0 = 𝐹𝐷(𝚫𝛆𝟎, Δ 𝑝0

𝑝0 − 𝑙𝑐2𝛻 ⋅ 𝛻 𝑝0 = 𝑝0

𝑅 ≫ 𝑙𝑐

𝐿 ≫ 𝑙𝑐

Non-local MFH

material law

Total failure Loading path

Localization

onset

Dissipation onset

𝜎

𝒟

𝒟loc

𝒟end


𝐴0

𝑙𝑐 (mm) MFH 𝐺𝑐 [N mm-1] SVE 𝐺𝑐 [N mm-1]

20 1.008

0.10355 0.363

2 0.0922



• More convenient to fix non-local length

– For macro-scale Stochastic FEM

– To increase its length

• Modify the damage evolution to recover 𝐺𝑐– Before localization onset: identified evolution

– After localization onset:

•

• Iterate on α and 𝛽

)Δ𝐷0 = 𝐹𝐷(𝚫𝛆𝟎, Δ 𝑝0 𝑅 ≫ 𝑙𝑐

𝐿 ≫ 𝑙𝑐

Non-local MFH

material lawΔ𝐷0 = α( 𝑝0 + Δ 𝑝0 − 𝑝0onset)

β Δ 𝑝0

Total failure Loading path

Localization

onset

Dissipation onset

𝜎

𝒟

𝒟loc

𝒟end


𝐴0

Method 𝐺𝑐 [N mm-1]

SVE 0.0945

MFH 0.0932


• Use Stochastic Mean-Field Homogenization as constitutive law

Methodology


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


Use of stochastic Mean-Field Homogenization

• Stochastic simulations require two discretizations

– Random vector field discretization

• Of MFH-model parameters

• Stochastic MFH-model parameters identified

• Potentially more cells than SVE realizations

parameters need to be generated

– Finite element discretization

• Finer than random field grid


• Generation of random field

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




• Ply loading realizations

– Preliminary results (softening part not implemented)


• 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:

STOMMMAC project


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References

computational & multiscale cm3 mechanics of materials · 2019-10-07 · computational &...

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