A stochastic Mean-Field-Homogenization-based micro-mechanical model of unidirectional composites failure

Computational & Multiscale

Mechanics of Materials

CM3

www.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

•

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

L

_macro

>>L

_VE

>>L

_micro

• 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,

•

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 𝜔_𝑖

w

_I

w

₀ 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 r_f[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:

• 𝑝_𝑅 𝑟 , • 𝑝_𝑑1st 𝑑 , • 𝑝_𝜗1st 𝜃 ,

• 𝑝_Δ𝑑 𝑑 with Δ𝑑 = 𝑑_2nd− 𝑑_1st • 𝑝_Δ𝜗 𝜃 with Δ𝜗 = 𝜗_2nd− 𝜗_1st

* 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 𝜗_1st 𝑑_1st ∆𝜗 𝜗_2nd 𝑑_2nd 𝑅₂

•

Numerical micro-structures are generated by a fiber additive process

– Arbitrary size

– Arbitrary number

– Possibility to generate non-homogenous distributions

•

Window technique

– Extraction of Stochastic Volume Elements

• 𝑙_SVE = 25 𝜇𝑚 • Correlation

– For each SVE

• Extract apparent homogenized material tensor ℂ_M

• 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 𝑙SVE 𝜺_M = 1 𝑉 𝜔 _𝜔𝜺m𝑑𝜔 𝝈_M = 1 𝑉 𝜔 _𝜔𝝈m𝑑𝜔 ℂM = 𝜕𝝈_M 𝜕𝒖_M ⊗ 𝛁_M 𝑅𝐫𝐬 𝝉 = 𝔼 𝑟 𝒙 − 𝔼 𝑟 𝑠 𝒙 + 𝝉 − 𝔼 𝑠 𝔼 𝑟 − 𝔼 𝑟 2 𝔼 𝑠 − 𝔼 𝑠 2

•

Mean-Field-Homogenization (MFH)

– Linear composites

– We use Mori-Tanaka assumption for 𝔹𝜀 _{I, ℂ} 0 , ℂI

•

Stochastic MFH

– How to define randomness?

Stochastic Mean-Field Homogenization

𝛆_M = 𝛆 = 𝑣₀𝛆₀ + 𝑣_I𝛆_I 𝛆_I = 𝔹𝜀 I, ℂ₀ , ℂ_I : 𝛆₀ 𝛔_M = 𝛔 = 𝑣₀𝛔₀ + 𝑣_I𝛔_I inclusions composite matrix 𝛆_I 𝛔 𝛆 ℂ0 𝛆 = 𝛆_M 𝛆0 ℂ_I ℂ_M = ℂ_M I, ℂ₀ , ℂ_I , 𝑣_I 𝜔 =∪_𝑖 𝜔_𝑖 w_I w₀

•

Mean-Field-Homogenization (MFH)

– Linear composites

•

Consider an equivalent system

– For each SVE realization 𝑖:

ℂ_M and 𝜈_I known – Anisotropy from ℂ_M𝑖

𝜃 is evaluated – Fiber behavior uniform

ℂ_I for one SVE

– Remaining optimization problem:

Stochastic Mean-Field Homogenization

𝛆_M = 𝛆 = 𝑣₀𝛆₀ + 𝑣_I𝛆_I 𝛆_I = 𝔹𝜀 I, ℂ₀ , ℂ_I : 𝛆₀ 𝛔_M = 𝛔 = 𝑣₀𝛔₀ + 𝑣_I𝛔_I inclusions composite matrix 𝛆_I 𝛔 𝛆 ℂ0 𝛆 = 𝛆_M 𝛆0 ℂ_I Defined as random variables ℂ_M ≃ ℂ_M( I, ℂ₀ , ℂ_I , 𝑣_I, 𝜃) ℂ₀ ℂ_I ℂ₀ ℂ_I 𝜃 𝑎 𝑏 Equivalent inclusion min 𝑎 𝑏, 𝐸0 , 𝜈0 ℂ_M − ℂ_M(𝑎 𝑏, 𝐸0 , 𝜈0 ; 𝑣I, 𝜃, ℂI ) ℂ_M = ℂ_M I, ℂ₀ , ℂ_I , 𝑣_I 𝜔 =∪_𝑖 𝜔_𝑖 w_I w₀

•

Inverse stochastic identification

– Comparison of homogenized

properties from SVE realizations and stochastic MFH

Stochastic Mean-Field Homogenization

ℂ

_M

≃

ℂ

_M

( I,

ℂ

₀

,

ℂ

_I

, 𝑣

_I

, 𝜃)

ℂ

₀

ℂ

_I

ℂ

₀

ℂ

_I

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

– Linear composites

– Non-linear composites

Non-linear stochastic Mean-Field Homogenization

𝛆_M = 𝛆 = 𝑣₀𝛆₀ + 𝑣_I𝛆_I 𝛆_I = 𝔹𝜀 I, ℂ₀ , ℂ_I : 𝛆₀ 𝛔_M = 𝛔 = 𝑣₀𝛔₀ + 𝑣_I𝛔_I inclusions composite matrix 𝛆_I 𝛔 𝛆 ℂ0 𝛆 = 𝛆_M 𝛆0 ℂ_I inclusions composite matrix 𝚫𝛆_I 𝛔 𝛆 𝚫𝛆_M 𝚫𝛆₀ 𝚫𝛆_M = Δ𝛆 = 𝑣₀Δ𝛆₀ + 𝑣_IΔ𝛆_I 𝚫𝛆_I = 𝔹𝜀 I, ℂ₀LCC, ℂ_𝐼LCC : 𝚫𝛆₀ 𝛔M = 𝛔 = 𝑣0𝛔0 + 𝑣I𝛔I 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

Non-linear stochastic Mean-Field Homogenization

inclusions composite matrix 𝚫𝛆_Iunload 𝛔 𝛆 𝚫𝛆_Munload 𝚫𝛆₀unload ℂ_Iel ℂ₀el

•

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

𝚫𝛆_M𝐫 = Δ𝛆 = 𝑣₀Δ𝛆₀𝐫 + 𝑣_IΔ𝛆_I𝐫 𝚫𝛆_I𝐫 = 𝔹𝜀 I, ℂ₀S, ℂ_IS : 𝚫𝛆₀𝐫 𝛔_M = 𝛔 = 𝑣₀𝛔₀ + 𝑣_I𝛔_I inclusions composite matrix 𝚫𝛆_Iunload 𝛔 𝛆 𝚫𝛆_Munload 𝚫𝛆₀unload ℂ_Iel ℂ₀el

𝚫𝛆_I/0𝐫 = Δ𝛆_I/0 + 𝚫𝛆_I/0unload

𝚫𝛔

_M

= ℂ

_MS

I, ℂ

₀S

, ℂ

_IS

, 𝑣

_I

: 𝚫𝛆

_M𝐫 ℂ0S inclusions composite matrix 𝚫𝛆_I𝐫 𝛔 𝛆 𝚫𝛆_M𝐫 𝚫𝛆₀𝐫 ℂ_IS ℂ_MS

•

Non-linear inverse identification

– First step from elastic response

Non-linear stochastic Mean-Field Homogenization

ℂ₀ ℂ_I ℂ₀el ℂ_Iel 𝜃 𝑎 𝑏 Equivalent inclusion ℂ_Mel ≃ ℂ_Mel( I, ℂ₀el, ℂ_Iel, 𝑣_I, 𝜃) inclusions composite matrix 𝚫𝛆_Iunload 𝛔 𝛆 𝚫𝛆_Munload 𝚫𝛆₀unload ℂ_Iel 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

Non-linear stochastic Mean-Field Homogenization

ℂ₀S ≃ ℂ₀S( 𝑅 𝑝0 ; ℂ0el) ℂ₀ ℂ_I ℂ₀S ℂ_IS 𝜃 𝑎 𝑏 Equivalent inclusion ℂ_Mel ≃ ℂ_Mel( I, ℂ₀el, ℂ_Iel, 𝑣_I, 𝜃) ℂ₀S ≃ ℂ₀S( 𝑅 𝑝₀ ; ℂ₀el) ℂ₀S inclusions composite matrix 𝚫𝛆_I𝐫 𝛔 𝛆 𝚫𝛆_M𝐫 𝚫𝛆₀𝐫 ℂIS ℂMS 𝚫𝛔_M ≃ ℂ_MS I, ℂS₀, ℂ_IS, 𝑣_I, 𝜃 : 𝚫𝛆_M𝐫 𝑝₀ 𝑅 𝑝 0 in MPa

Extracted from SVEs Identified hardening Input matrix law

•

Non-linear inverse identification

– Comparison SVE vs. MFH

Non-linear stochastic Mean-Field Homogenization

ℂ₀S ≃ ℂ₀S( 𝑅 𝑝₀ ; ℂ₀el) ℂ₀ ℂ_I ℂ₀S ℂ_IS 𝜃 𝑎 𝑏 Equivalent inclusion ℂ_Mel ≃ ℂ_Mel( I, ℂ₀el, ℂ_Iel, 𝑣_I, 𝜃)

•

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 𝚫𝛆_Iunload 𝛔 𝛆 𝚫𝛆_Munload_𝚫𝛆 0 unload ℂ_Iel (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:

Non-linear stochastic Mean-Field Homogenization

1 − 𝐷₀ ℂ₀el ℂ_Iel 1 − 𝐷₀ ℂ₀el ℂ_Iel 𝜃 𝑎 𝑏 Equivalent inclusion 1 − D₀ ℂ₀𝑆 ≃ 1 − D₀ 𝑝₀ ℂ₀𝑆( 𝑅 𝑝₀ ; ℂ₀el) inclusions composite matrix 𝚫𝛆_Ir 𝛔 𝛆 𝚫𝛆_Mr _𝚫𝛆 0 r ℂ_IS (1 − 𝐷₀)ℂ0S effective matrix ℂ_MS 𝚫𝛔_M = ℂ_MS I, 1 − 𝐷₀ ℂS₀, ℂ_IS, 𝑣_I : 𝚫𝛆_M𝐫 ℂ_Mel(𝐷) ≃ ℂ_Mel( I, 1 − 𝐷₀ ℂ₀el, ℂ_Iel, 𝑣_I, 𝜃) inclusions composite matrix 𝚫𝛆_Iunload 𝛔 𝛆 𝚫𝛆_Munload_𝚫𝛆 0 unload ℂ_Iel (1 − 𝐷₀)ℂ0el effective matrix ℂ_Mel_(𝐷)

•

Damage-enhanced inverse identification

– Similar process as for elasto-plasticity

Non-linear stochastic Mean-Field Homogenization

1 − 𝐷₀ ℂ₀ ℂ_Iel 1 − 𝐷₀ ℂ₀S ℂ_Iel 𝜃 𝑎 𝑏 Equivalent inclusion 1 − D₀ ℂ₀𝑆 ≃ 1 − D₀ 𝑝₀ ℂ₀𝑆( 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

Use of stochastic Mean-Field Homogenization

•

Ply loading realizations

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 = σ_𝑦+ ℎ (1 − 𝑒−ℎ𝑒𝑥𝑝𝑝₎ 𝐷_𝑠 = (𝐻_𝑠 𝑝 − 𝑝_{𝑖𝑛𝑖𝑡} α) Φ_𝑓 = γ − 𝑎 exp −𝑏𝑇 − 𝑐 = 0 𝐷_𝑓 = 𝐻_𝑓 (χ_𝑓)ζ𝑓_{(1 − 𝐷} 𝑓)ζ𝑑 χ𝑓

0

0.5 1

𝐷

_𝑓

•

Apparent response

– Independent Stochastic Volume Elements

• 𝑙_SVE = 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

𝜺_M = 1 𝑉 𝜔 _𝜔𝜺m𝑑𝜔 𝝈_M = 1 𝑉 𝜔 _𝜔𝝈m𝑑𝜔 ℂ_M = 𝜕𝝈M 𝜕𝒖_M ⊗ 𝛁_M Stochastic Homogenization SVE realizations

•

Non-linear SVE simulations

Mean-Field Homogenization with failure

0.0408

0.0780

0.0654

0.1019

•

Extracted failure characteristics

•

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

Thank you for your attention!

Special thanks to:

• 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