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Computational & Multiscale CM3 Mechanics of Materials · 2019-10-07 · Computational & Multiscale...
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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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 4
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 5
• 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]
0 1 2 3 4
4
3
2
1
0
rf [mm]
2.4 2.8 3.2 3.6
4
3
2
1
0
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 6
• Uncertainty quantification of micro-structure & micro-structure generator
Methodology
𝜔 =∪𝑖 𝜔𝑖
wI
w0
Stochastic
Homogenization
Stochastic
MF-ROM
SVE
realizations
Micro-structure
stochastic model
strain
stress
d [mm]
0 1 2 3 4
4
3
2
1
0
rf [mm]
2.4 2.8 3.2 3.6
4
3
2
1
0
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 7
• 2000x and 3000x SEM images
• Fibers detection
Experimental measurements
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 8
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 9
• Histograms of random micro-structures’ descriptors
Micro-structure stochastic model
𝑅0
𝑅1
𝜗1st
𝑑1st∆𝜗
𝜗2nd𝑑2nd
𝑅2
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 10
• 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
Micro-structure stochastic model
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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 11
• 𝑑1st and ∆𝑑 should be generated using their empirical copula
Micro-structure stochastic model
∆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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 12
• Numerical micro-structures are generated by a fiber additive process
– Arbitrary size
– Arbitrary number
– Possibility to generate non-homogenous distributions
Micro-structure stochastic model
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 19
• SVEs definition
Methodology
𝜔 =∪𝑖 𝜔𝑖
wI
w0
Stochastic
Homogenization
Stochastic
MF-ROM
SVE
realizations
Micro-structure
stochastic model
strain
stress
d [mm]
0 1 2 3 4
4
3
2
1
0
rf [mm]
2.4 2.8 3.2 3.6
4
3
2
1
0
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 20
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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 22
• 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
Micro-structural model of fiber-reinforced highly crosslinked epoxy
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 23
Micro-structural model of fiber-reinforced highly crosslinked epoxy
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 24
Micro-structural model of fiber-reinforced highly crosslinked epoxy
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 25
• Resin model: failure softening
– Failure surface
– Damage evolution
– Affect ductility
– Calibration
• From localization simulation
• Recover the epoxy 𝐺𝑐
Micro-structural model of fiber-reinforced highly crosslinked epoxy
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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 27
Micro-structural model of fiber-reinforced highly crosslinked epoxy
• UD Composites with RTM6 epoxy matrix
– 2D simulations of 25 x 25 μm 40% volume fraction composite SVEs
0
0.5
1𝐷𝑓
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 28
• Extraction of apparent responses
Methodology
𝜔 =∪𝑖 𝜔𝑖
wI
w0
Stochastic
Homogenization
Stochastic
MF-ROM
SVE
realizations
Micro-structure
stochastic model
strain
stress
d [mm]
0 1 2 3 4
4
3
2
1
0
rf [mm]
2.4 2.8 3.2 3.6
4
3
2
1
0
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 29
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 30
• Apparent elastic properties: distribution & Correlation
– For large enough RVE (𝑙SVE > 10 𝜇𝑚)
Stochastic homogenization on the SVEs
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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 41
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 42
• Stochastic Mean-Field Homogenization-based model
– First stage: linear elasticity
Methodology
𝜔 =∪𝑖 𝜔𝑖
wI
w0
Stochastic
Homogenization
Stochastic
MF-ROM
SVE
realizations
Micro-structure
stochastic model
strain
stress
d [mm]
0 1 2 3 4
4
3
2
1
0
rf [mm]
2.4 2.8 3.2 3.6
4
3
2
1
0
Linear
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 43
• 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
𝜔 =∪𝑖 𝜔𝑖
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)
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 44
• Mean-Field-Homogenization (MFH)
– Linear composites
• 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:
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
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
𝜔 =∪𝑖 𝜔𝑖
wI
w0
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 45
• Inverse stochastic identification
– Comparison of homogenized
properties from SVE realizations
and stochastic MFH
Stochastic Mean-Field Homogenization
ℂM ≃ ℂM( I, ℂ0 , ℂI , 𝑣I, 𝜃)
ℂ0ℂI
ℂ0
ℂI
𝜃
𝑎
𝑏
Equivalent
inclusion
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 49
• Stochastic Mean-Field Homogenization-based model
– Second stage: damage-enhanced elasto-plasticity
Methodology
𝜔 =∪𝑖 𝜔𝑖
wI
w0
Stochastic
Homogenization
Stochastic
MF-ROM
SVE
realizations
Micro-structure
stochastic model
strain
stress
d [mm]
0 1 2 3 4
4
3
2
1
0
rf [mm]
2.4 2.8 3.2 3.6
4
3
2
1
0
Non-linear
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 50
• Non-linear Mean-Field-homogenization
– Linear composites
– Non-linear composites
Non-linear 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
inclusions
composite
matrix
𝚫𝛆I
𝛔
𝛆𝚫𝛆M 𝚫𝛆0
𝚫𝛆M = Δ𝛆 = 𝑣0Δ𝛆0 + 𝑣IΔ𝛆I
𝚫𝛆I = 𝔹𝜀 I, ℂ0LCC, ℂ𝐼
LCC : 𝚫𝛆0
𝛔M = 𝛔 = 𝑣0𝛔0 + 𝑣I𝛔I
Define a linear
comparison
composite material
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 51
Non-linear stochastic Mean-Field Homogenization
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 52
• Incremental-secant Mean-Field-homogenization
– Virtual elastic unloading from previous state
• Composite material unloaded to reach the stress-
free state
• Residual stress in components
Non-linear stochastic Mean-Field Homogenization
inclusions
composite
matrix
𝚫𝛆Iunload
𝛔
𝛆
𝚫𝛆Munload
𝚫𝛆0unload
ℂIel
ℂ0el
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 53
• Incremental-secant Mean-Field-homogenization
– Virtual elastic unloading from previous state
• Composite material unloaded to reach the stress-
free state
• Residual stress in components
– Define Linear Comparison Composite
• From unloaded state
• Incremental-secant loading
• Incremental secant operator
Non-linear stochastic Mean-Field Homogenization
𝚫𝛆M𝐫 = Δ𝛆 = 𝑣0Δ𝛆0
𝐫 + 𝑣IΔ𝛆I𝐫
𝚫𝛆I𝐫 = 𝔹𝜀 I, ℂ0
S, ℂIS : 𝚫𝛆0
𝐫
𝛔M = 𝛔 = 𝑣0𝛔0 + 𝑣I𝛔I
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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 54
• Non-linear inverse identification
– First step from elastic response
Non-linear stochastic Mean-Field Homogenization
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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 55
• 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
Non-linear stochastic Mean-Field Homogenization
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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 57
• Damage-enhanced Mean-Field-homogenization
– Virtual elastic unloading from previous state
• Composite material unloaded to reach the stress-
free state
• Residual stress in components
Non-linear stochastic Mean-Field Homogenization
inclusions
composite
matrix
𝚫𝛆Iunload
𝛔
𝛆𝚫𝛆M
unload𝚫𝛆0
unload
ℂIel
(1 − 𝐷0)ℂ0el
effective
matrix
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 58
• Damage-enhanced Mean-Field-homogenization
– Virtual elastic unloading from previous state
• Composite material unloaded to reach the stress-
free state
• Residual stress in components
– Define Linear Comparison Composite
• From elastic state
• Incremental-secant loading
• Incremental secant operator
Non-linear stochastic Mean-Field Homogenization
𝚫𝛆M𝐫 = Δ𝛆 = 𝑣0Δ𝛆0
𝐫 + 𝑣IΔ𝛆I𝐫
𝚫𝛆I𝐫 = 𝔹𝜀 I, 1 − 𝐷0 ℂ0
S, ℂIS : 𝚫𝛆0
𝐫
𝛔M = 𝛔 = 𝑣0𝛔0 + 𝑣I𝛔I
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
S, ℂIS, 𝑣I : 𝚫𝛆M
𝐫
inclusions
composite
matrix
𝚫𝛆Ir
𝛔
𝛆𝚫𝛆M
r𝚫𝛆0
r
ℂIS
(1 − 𝐷0)ℂ0S
effective
matrix
ℂMS
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 60
• Damage-enhanced inverse identification
– Second step: elastic unloading
• Identify damage evolution 𝐷0
Non-linear stochastic Mean-Field Homogenization
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)
L. Wu, V.-D. Nguyen, L. Adam, L. Noels, An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites (2019)
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 61
• 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:
Non-linear stochastic Mean-Field Homogenization
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)
L. Wu, V.-D. Nguyen, L. Adam, L. Noels, An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites (2019)
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 63
• Stochastic Mean-Field Homogenization-based model
– Third stage: Failure
Methodology
𝜔 =∪𝑖 𝜔𝑖
wI
w0
Stochastic
Homogenization
Stochastic
MF-ROM
SVE
realizations
Micro-structure
stochastic model
strain
stress
d [mm]
0 1 2 3 4
4
3
2
1
0
rf [mm]
2.4 2.8 3.2 3.6
4
3
2
1
0
Failure
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 64
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
𝐺𝑐 =𝒟end − 𝒟loc
𝐴0
𝜎𝑐
𝜎𝑐
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 65
• Non-linear SVE simulations
Mean-Field Homogenization with failure
0.0408
0.0780
0.0654
0.1019
Gc [N/mm]
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 66
Mean-Field Homogenization with failure
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 67
Mean-Field Homogenization with failure
• 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
𝐺𝑐 =𝒟end − 𝒟loc
𝐴0
𝑙𝑐 (mm) MFH 𝐺𝑐 [N mm-1] SVE 𝐺𝑐 [N mm-1]
20 1.008
0.10355 0.363
2 0.0922
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 68
Mean-Field Homogenization with failure
• 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
𝐺𝑐 =𝒟end − 𝒟loc
𝐴0
Method 𝐺𝑐 [N mm-1]
SVE 0.0945
MFH 0.0932
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 69
• Use Stochastic Mean-Field Homogenization as constitutive law
Methodology
𝜔 =∪𝑖 𝜔𝑖
wI
w0
Stochastic
Homogenization
Stochastic
MF-ROM
SVE
realizations
Micro-structure
stochastic model
strain
stress
d [mm]
0 1 2 3 4
4
3
2
1
0
rf [mm]
2.4 2.8 3.2 3.6
4
3
2
1
0
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 70
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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 71
• Generation of random field
– Inverse identification vs. diffusion map –based generator [Soize, Ghanem 2016]
Use of stochastic Mean-Field Homogenization
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 73
Use of stochastic Mean-Field Homogenization
• Ply loading realizations
– Preliminary results (softening part not implemented)
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 74
• 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
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 75
Thank you for your attention!
Special thanks to:
STOMMMAC project
CM3 1-4 October 2019 Computational Modeling of Complex Materials across the Scales (CMCS) 76
• 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,
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• L. Wu, L. Noels, L. Adam, I. Doghri, A combined incremental–secant mean–field homogenization scheme with per–phase
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• L. Wu, L. Noels, L. Adam, I. Doghri, An implicit-gradient-enhanced incremental-secant mean-field homogenization scheme
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