Stochastic study of unidirectional composites: from micro-structural statistical information to multi-scale analyzes

Mechanics of Materials

www.ltas-cm3.ulg.ac.be

Stochastic study of unidirectional composites: from

micro-structural statistical information to multi-scale analyzes

The research has been funded by the Walloon Region under the agreement no 1410246-STOMMMAC (CT-INT

2013-Wu Ling, Bidaine Benoit, Major Zoltan,

– One way: 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 Failure load Probability Stochastic structural analysis UQ

•

Illustration assuming a regular stacking

– 60%-UD fibers

– Damage-enhanced matrix behavior

•

Question: what does happen for a realistic fibre stacking?

0 20 40 60 80 100 120 0 0.01

s

[M pa ]

e

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 0 0.01 s [Mpa ]

e

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_Δ𝜗 𝜃 with Δ𝜗 = 𝜗_2nd − 𝜗_1st 𝑅₀ 𝑅₁ 𝜗_1st 𝑑_1st Δ𝜗 𝜗_2nd 𝑑_2nd 𝑅₂

•

Histograms of random micro-structures’ descriptors

𝑅₀ 𝑅1 𝜗_1st 𝑑_1st Δ𝜗 𝜗_2nd 𝑑_2nd 𝑅₂

•

Dependency of the four random variables

𝑑

_1st

, Δ𝑑, 𝜗

_1st

, Δ𝜗

•

Correlation matrix

•

Distances correlation matrix

𝑑_1st and Δ𝑑 are dependent they will be generated from their empirical copula 𝑑_1st Δ𝑑 𝜗_1st Δ𝜗 𝑑_1st 1.0 0.27 0.04 0.08 Δ𝑑 1.0 0.05 0.06 𝜗_1st 1.0 0.05 Δ𝜗 1.0 𝑑_1st Δ𝑑 𝜗_1st Δ𝜗 𝑑_1st 1.0 0.21 0.01 0.02 Δ𝑑 1.0 0.002 -0.005 𝜗1st 1.0 0.02 Δ𝜗 1.0 𝑅₀ 𝑅₁ 𝜗_1st 𝑑_1st Δ𝜗 𝜗2nd 𝑑2nd 𝑅₂

•

𝑑

_1st

and ∆𝑑 should be generated using their empirical copula

∆𝑑

SEM sample Generated sample

𝑑

_1st

∆𝑑

∆𝑑

𝑑

Directly from copula generator

Statistic result from generated SVE 𝑅₀ 𝑅₁ 𝜗_1st 𝑑_1st Δ𝜗 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 Δ𝜗 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: – Periodic (PBC)

– Minimum kinematics (SUBC) – Kinematic (KUBC) 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 However the distribution depend on

•

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_SSVE

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_SSVE 𝐿 BS VE

Level I Level II

Computational homogenization

…

…

•

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

Computational homogenization

…

•

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] 44 55 66 𝜎_𝑀𝑥𝑥 [Mpa] 43 55 67 𝜎_𝑀𝑥𝑥 [Mpa] 43 55 67

•

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 – Fibre behaviour uniform

ℂ_I for one SVE

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 fields

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 ℂ_M = ℂ_M I, ℂ₀ , ℂ_I , 𝑣_I 𝜔 =∪_𝑖 𝜔_𝑖 w_I w₀ 𝑎

•

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

•

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

(

𝑎

𝑏

,

𝐸

0

, 𝜈

0

𝑣

I

, 𝜃;

ℂ

I

)

ℂ

₀

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_𝟎

•

Stochastic simulations

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

• Stochastic generator based on SEM measurements

of unidirectional fibre reinforced composites

• Computational homogenization on SVEs

• Two-step computational homogenization for Big

SVEs

• Definition of a Stochastic MFH method