A stochastic Mean Field Homogenization model of Unidirectional composite materials

A stochastic Mean Field Homogenization model of

Unidirectional composite materials

Wu Ling, Noels Ludovic

𝑦 SVE 𝑥, 𝑦 SVE 𝑥′, 𝑦 SVE 𝑥′, 𝑦′ t 𝑙

– One method: homogenization

– 2 problems are solved (concurrently)

• The macro-scale problem

• The meso-scale problem (on a meso-scale Volume Element)

BVP

Macro-scale

Material response

Extraction of a meso-scale Volume Element

•

Material uncertainties affect structural behaviors

E_matrix Probability E_Fibers Probability Fiber orientation (a) Probability w₀ w_I a Composite stiffness Probability Probabilistic homogenization Loading Homogenized material properties distribution Probability Stochastic structural analysis UQ

•

Illustration assuming a regular stacking

– 60%-UD fibers

– Damage-enhanced matrix behavior

0 20 40 60 80 100 120

s

[M pa ] 0 degree 15 degree 30 degree 45 degree 60 degree 75 degree 90 degree Effect of the loading direction q for d_min = 0.5 µm 0 20 40 60 80 100 120 s [Mpa ] dmin=0.2 dmin=0.35 dmin=0.5 dmin=0.65 dmin=0.8 ϴ d_min Effect of the distance d_min for q = 30o

•

Proposed methodology:

𝜔 =∪_𝑖 𝜔_𝑖 w_I w₀ Stochastic Homogenization SVE size Average strength SVE size Variance of strength Stochastic reduced model Experimental measurements SVE realisations ∆𝑑 Copula (𝑑_1st, ∆𝑑) 𝑑_1st Micro-structure stochastic model

•

2000x and 3000x SEM images

•

Basic geometric information of fibers' cross sections

– Fiber radius distribution 𝑝_𝑅 𝑟

•

Basic spatial information of fibers

– The distribution of the nearest-neighbor net distance function 𝑝_𝑑1st 𝑑

– The distribution of the orientation of the undirected line connecting the center points of a fiber to its nearest neighbor 𝑝_𝜗1st 𝜃

– The distribution of the difference between the net distance to the second and the first nearest-neighbor 𝑝_Δ𝑑 𝑑 with Δ𝑑 = 𝑑_2nd − 𝑑_1st

– The distribution of the second nearest-neighbor’s location referring to the first nearest-neighbor 𝑝 𝜃

𝑅₀ 𝑅₁ 𝜗_1st 𝑑_1st ∆𝜗 𝜗_2nd 𝑑_2nd 𝑅₂

•

Histograms of random micro-structures’ descriptors

𝑅₀ 𝑅1 𝜗_1st 𝑑_1st ∆𝜗 𝜗_2nd 𝑑_2nd 𝑅₂

•

Dependency of the four random variables

𝑑

_1st

, ∆𝑑, 𝜗

_1st

, ∆𝜗

•

Correlation matrix

•

Distances correlation matrix

𝑑_1st and ∆𝑑 are dependent

they will have to be generated from their empirical copula

𝑑_1st ∆𝑑 𝜗_1st ∆𝜗 𝑑_1st 1.0 0.27 0.04 0.08 ∆𝑑 1.0 0.05 0.06 𝜗_1st 1.0 0.05 𝑑_1st ∆𝑑 𝜗_1st ∆𝜗 𝑑_1st 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 𝜗_1st 𝑑_1st ∆𝜗 𝜗2nd 𝑑2nd 𝑅₂

•

𝑑

_1st

and ∆𝑑 should be generated using their empirical copula

∆𝑑

SEM sample Generated sample

𝑑

_1st

∆𝑑

∆𝑑

Directly from copula generator

Statistic result from generated SVE 𝑅₁ 𝑑_1st ∆𝜗 𝜗_2nd 𝑑_2nd 𝑅₂

•

The numerical

micro-structure is generated

by

a

fiber

additive

process

1) Define 𝑁 seeds with first and second neighbors distances 𝑅_𝑘 𝑑_1st𝑘 𝑑_2nd𝑘 𝑅𝑘+1 𝑑_1st𝑘+1 𝑑_2nd𝑘+1 Seed 𝑘 Seed 𝑘 + 1

𝑅₁ 𝑑1st1 𝑑2nd₁ 𝑅₀ 𝑑_1st0 𝑑_2nd0 𝑅0 𝑑_1st0 𝑑_2nd0 Seed 𝑘 Seed 𝑘 + 1 𝜗_1st

•

The numerical

micro-structure is generated

by

a

fiber

additive

process

1) Define 𝑁 seeds with first and second neighbors distances 2) Generate first

neighbor with its own first and second neighbors distances

𝑅₁ 𝑑1st1 𝑑2nd₁ 𝑅₀ 𝑑_1st0 𝑑_2nd0 𝑅₀ 𝑑_1st0 𝑑_2nd0 Seed 𝑘 Seed 𝑘 + 1 𝜗_1st 𝑅₂ 𝑑_1st2 𝑑_2nd2 Δ𝜗

•

The numerical

micro-structure is generated

by

a

fiber

additive

process

1) Define 𝑁 seeds with first and second neighbors distances 2) Generate first

neighbor with its own first and second neighbors distances 3) Generate second

neighbor with its own first and second neighbors distances

•

The numerical

micro-structure is generated

by

a

fiber

additive

process

1) Define 𝑁 seeds with first and second neighbors distances 2) Generate first

neighbor with its own first and second neighbors distances 3) Generate second

neighbor with its own first and second neighbors distances 4) Change seeds & then

change central fiber of the seeds 𝑑_1st0 𝑅₀ 𝑑_2nd0 𝑅₀ 𝑑_1st0 𝑑_2nd0 Seed 𝑘 Seed 𝑘 + 1 𝑅_𝑖𝑘 𝑑_1st 𝑖𝑘 𝑑_2nd 𝑖𝑘 𝑅𝑖_𝑘+1 𝑑_1st 𝑖𝑘+1 𝑑_2nd 𝑖𝑘+1 𝑅₁ 𝑑1st₁ 𝑑_2nd1 𝜗_1st

•

The numerical micro-structure is generated by a fiber additive process

– The effect of the initial number of seeds N and

– The effect of the maximum regenerating times n_max after rejecting a fiber due to overlap

SEM: Average V_fof 103 windows;

•

Comparisons of fibers spatial information

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

•

Stochastic homogenization

– Extraction of Stochastic Volume Elements

• 2 sizes considered: 𝑙SVE = 10 𝜇𝑚 & 𝑙SVE = 25 𝜇𝑚

• Window technique to capture correlation

– For each SVE

• Extract apparent homogenized material tensor ℂ_M

• Consistent boundary conditions:

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

•

Apparent properties

𝑙SVE = 10 𝜇𝑚 𝑙SVE = 25 𝜇𝑚

Increasing 𝑙

_SVE

When 𝑙_SVE increases

• Average values for different BCs get closer (to PBC one) • Distributions narrow

•

When 𝑙

_SVE

increases: marginal distributions of random properties closer to normal

– l_SVE = 10 µm

•

Correlation

𝑙_SVE = 10 𝜇𝑚 𝑙SVE = 25 𝜇𝑚

Increasing 𝑙

_SVE

(1) Auto/cross correlation vanishes at 𝜏 = 𝑙_SVE

(2) When 𝑙_SVE increases, distributions get closer to normal (1)+(2) Apparent properties are independent random variables

•

Quid larger SVEs?

– Computational cost affordable in linear elasticity

– Computational cost non affordable in failure analyzes

•

How to deduce the stochastic content of larger SVEs?

– Take advantages of the fact that the apparent tensors can be considered as random variables

𝑙

𝐿

BS

•

Quid larger SVEs?

– Computational cost affordable in linear elasticity

– Computational cost non affordable in failure analyzes

•

How to deduce the stochastic content of larger SVEs?

– Take advantages of the fact that the apparent tensors can be considered as random variables

𝑙 𝐿 BS VE

Level I Level II

…

…

•

Quid larger SVEs?

– Computational cost affordable in linear elasticity

– Computational cost non affordable in failure analyzes

•

How to deduce the stochastic content of larger SVEs?

– Take advantages of the fact that the apparent tensors can be considered as random variables

– Accuracy depends on Small/Large SVE sizes 𝑙SSVE

𝐿

BS

VE

•

Numerical verification of 2-step homogenization

– Direct homogenization of larger SVE (BSVE) realizations – 2-step homogenization using BSVE subdivisions

Level I Level II

…

•

Stochastic model of the anisotropic elasticity tensor

– Extract (uncorrelated) tensor realizations ℂ_M𝑖

– Represent each realization ℂ_M𝑖 _{by a vector 𝓥 of 9 (dependant) 𝓥}(𝒓) _variables

– Generate random vectors 𝓥 using the Copula method

•

Stochastic model of the anisotropic elasticity tensor

– Extract (uncorrelated) tensor realizations ℂ_M𝑖

– Represent each realization ℂ_M𝑖 _{by a vector 𝓥 of 9 (dependant) 𝓥}(𝒓) _variables

– Generate random vectors 𝓥 using the Copula method

•

Simulations require two discretizations

– Random vector discretization

– Finite element discretization

•

Ply loading realizations

– Non-uniform homogenized stress distributions

– Different realizations yield different solutions

𝜎_𝑀𝑥𝑥 [Mpa]

43 53 63

𝜎_𝑀𝑥𝑥 [Mpa]

•

Mean-Field-homogenization (MFH)

– Linear composites

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

•

Stochastic MFH

– How to define random vectors 𝓥_MT of I, ℂ₀ , ℂ_I , 𝑣_I ? 𝛆_M = 𝛆 = 𝑣₀𝛆₀ + 𝑣_I𝛆_I 𝛆_I = 𝐁𝜀 I, ℂ₀ , ℂ_I : 𝛆₀ 𝛔_M = 𝛔 = 𝑣₀𝛔₀ + 𝑣_I𝛔_I composite matrix 𝛆_I 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 𝛆_M = 𝛆 = 𝑣₀𝛆₀ + 𝑣_I𝛆_I 𝛆_I = 𝐁𝜀 I, ℂ₀ , ℂ_I : 𝛆₀ 𝛔_M = 𝛔 = 𝑣₀𝛔₀ + 𝑣_I𝛔_I composite matrix 𝛆_I 𝛆 ℂ0 𝛆 = 𝛆_M 𝛆0 ℂ_I Defined as random fields ℂ_M ≃ ℂ_M( I, ℂ₀ , ℂ_I , 𝑣_I, 𝜃) ℂ₀ ℂ_I ℂ₀ ℂ_I 𝜃 𝑎 𝑏 Equivalent inclusion 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 M M 0 I I

ℂ

₀

ℂ

_I

ℂ

₀

ℂ

_I

𝜃

𝑎

𝑏

Equivalent inclusion

•

Comparison Random fields vs. Stochastic elastic MFH

Random anisotropic material tensor

Stochastic MFH

𝜎_𝑥𝑥 [Mpa] 0 33 66 𝜎_𝑥𝑥 [Mpa] 0 33 66 𝜎eq[Mpa] 0 30 60 𝜎eq[Mpa] 0 30 60

•

Stochastic MFH model:

– Homogenized properties ℂ_M(𝑎 𝑏, 𝐸0 , 𝜈0 𝑣I, 𝜃; ℂI ) – Random vectors 𝓥_MT • Realizations v_MT = 𝑎 𝑏, 𝐸0 , 𝜈0 𝑣I, 𝜃

• Characterized by the distance correlation matrix • Generator using the copula method

ℂ_M( 𝑏, 𝐸0 , 𝜈0 𝑣I, 𝜃; ℂI ) ℂ₀ ℂ_I 𝜃 𝑎 𝑏 𝒗_𝐈 𝜃 𝒂 𝒃 𝑬_𝟎 𝝂_𝟎 𝒗_𝐈 1.0 0.015 0.114 0.523 0.499 𝜃 1.0 0.092 0.016 0.014 𝒂 𝒃 1.0 0.080 0.076 𝑬_𝟎 1.0 0.661 𝝂

•

Stochastic simulations

– 2 discretization: Random field 𝓥_MT & Stochastic finite-elements

•

Non-linear Mean-Field-homogenization

– Linear composites – Non-linear composites 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 𝛆 ℂ0 𝛆 = 𝛆_M 𝛆0 ℂ_I inclusions composite matrix 𝛔 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

composite matrix 𝚫𝛆_Iunload 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

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

𝚫𝛆_I/0𝐫 = Δ𝛆_I/0 + 𝚫𝛆_I/0unload

inclusions composite matrix 𝛔 ℂ_IS ℂ_MS

•

Non-linear inverse identification

– First step from elastic response

ℂ₀ ℂ_I ℂ₀el ℂ_Iel 𝜃 𝑎 𝑏 Equivalent inclusion ℂ_Mel ≃ ℂ_Mel( I, ℂ₀el, ℂ_Iel, 𝑣_I, 𝜃) composite matrix 𝚫𝛆_Iunload 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

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 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

•

Non-linear inverse identification

– Comparison SVE vs. MFH ℂ₀S ≃ ℂ₀S( 𝑅 𝑝₀ ; ℂ₀el) ℂ₀ ℂ_I ℂ₀S ℂ_IS 𝜃 𝑎 𝑏 Equivalent inclusion M M 0 I I

•

Damage-enhanced Mean-Field-homogenization

– Virtual elastic unloading from previous state

• Composite material unloaded to reach the stress-free state

• Residual stress in components

composite matrix 𝚫𝛆_Iunload 𝛆 𝚫𝛆_Munload_𝚫𝛆 0 unload ℂ_Iel (1 − 𝐷₀)ℂ0el effective matrix

•

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

𝚫𝛆_M𝐫 = Δ𝛆 = 𝑣₀Δ𝛆₀𝐫 + 𝑣_IΔ𝛆_I𝐫

𝚫𝛆_I𝐫 = 𝐁𝜀 I, 1 − 𝐷₀ ℂ₀S, ℂ_IS : 𝚫𝛆₀𝐫 𝛔_M = 𝛔 = 𝑣₀𝛔₀ + 𝑣_I𝛔_I

𝚫𝛆_I/0𝐫 = Δ𝛆_I/0 + 𝚫𝛆_I/0unload

composite matrix 𝚫𝛆_Iunload 𝛆 𝚫𝛆_Munload_𝚫𝛆 0 unload ℂ_Iel (1 − 𝐷₀)ℂ0el effective matrix inclusions composite matrix 𝛔 ℂ_IS effective matrix ℂ_MS

•

Damage-enhanced inverse identification

– First step from elastic response

• Before damage occurs composite

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

•

Damage-enhanced inverse identification

– Second step: elastic unloading

• Identify damage evolution 𝐷₀

1 − 𝐷₀ ℂ₀el ℂ_Iel 1 − 𝐷₀ ℂ₀el ℂ_Iel 𝜃 𝑎 𝑏 Equivalent inclusion ℂ_Mel(𝐷) ≃ ℂ_Mel( I, 1 − 𝐷₀ ℂ₀el, ℂ_Iel, 𝑣_I, 𝜃) composite matrix 𝚫𝛆_Iunload 𝛆 𝚫𝛆_Munload_𝚫𝛆 0 unload ℂ_Iel (1 − 𝐷₀)ℂ0el effective matrix ℂ_Mel_(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 ℂ_Iel 1 − 𝐷₀ ℂ₀el ℂ_Iel 𝜃 𝑎 𝑏 Equivalent inclusion inclusions composite matrix 𝛔 ℂ_IS effective matrix ℂMS 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) composite matrix 𝚫𝛆_Iunload 𝛆 𝚫𝛆_Munload_𝚫𝛆 0 unload ℂ_Iel (1 − 𝐷₀)ℂ0el effective matrix ℂ_Mel_(𝐷)

•

Damage-enhanced inverse identification

– Comparison SVE vs. MFH _ℂ 1 − 𝐷₀ ℂ₀ I el 1 − 𝐷₀ ℂ₀S ℂ_Iel 𝜃 𝑎 𝑏 Equivalent inclusion 1 − D₀ ℂ₀𝑆 ≃ 1 − D₀ 𝑝₀ ℂ₀𝑆( 𝑅 𝑝₀ ; ℂ₀el)

•

Stochastic generator based on SEM measurements of unidirectional

fibers-reinforced composites

•

Computational homogenization on SVEs

•

Two-step computational homogenization for Big SVEs

•

Definition of a Stochastic MFH method