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
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 3
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 4
• 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
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 5
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 6
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 7
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 8
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 9
• 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
* 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
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 10
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 11
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 12
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 13
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 14
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 15
• Non-linear inverse identification
– Comparison SVE vs. MFH
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)
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 16
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 17
• 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:
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)
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 18
• Damage-enhanced inverse identification
– Similar process as for elasto-plasticity
Non-linear stochastic Mean-Field Homogenization
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
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 19
• Generation of random field
– Inverse identification vs. diffusion map –based generator [Soize, Ghanem 2016]
Damage model Parameters
Use of stochastic Mean-Field Homogenization
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 20
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
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 21
Use of stochastic Mean-Field Homogenization
• Ply loading realizations
– Preliminary results (softening part not implemented)
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 22
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
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 23
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 24
• Non-linear SVE simulations
Mean-Field Homogenization with failure
0.0408
0.0780
0.0654
0.1019
Gc [N/mm]
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 25
• Extracted failure characteristics
Mean-Field Homogenization with failure
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 26
• 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 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 27
Thank you for your attention!
Special thanks to:
VISCOS
CM3 18-21 December 2019 Asian Pacific Congress on Computational Mechanics (APCOM) 28
• 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