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IntroductionFE2 method

Asymptotic Numerical MethodTechniques of Fourier series with slowly varying coefficients

A Novel-FE2 Method Based Fourier MacroscopicModel for Instability Phenomena of Long Fiber

Reinforced Composites

Y. Hui1,2,3, R. Xu1, H. Hu1, G. Giunta2, S. Belouettar2

1School of Civil Engineering, Wuhan University, China2Luxembourg Institute of Science and Technology, Luxembourg

3Politecnico di Torino, Italy

ICCS 19§§§Porto, Portugal Hui Yanchuan



Compressive failure of long-fiber reinforced composites

Drapier et al., IJSS, 2007.




Research Methods:

1 FE2 Method

2 Asymptotic Numerical Method

3 Techniques of Fourier series with slowly varying coefficients

1 Kousnetzova et al., IJNME, 2002.

2 Damil et al., JMPS, 2010.

3 Cochelin et al., IJNME, 1994.

4 Liu et al., IJSS, 2012.




Finite Element Square Method

1 Capture the microscopic information compared to the classicalhomogenization technique.

2 Save lots of DOFs compared to the full microscopic model.3 Still heavy because each Gauss point needs a RVE whose mesh should be

fine enough to describe the local details, e.g., micro-buckling.4 Difficult to pilot the non-linear calculations when there exist lots of

bifurcation curves around the useful one, often requires special

imperfection.

Fourier-related envelope model

1 Transform the fast oscillation unknown fields to slowly variable fields.2 Improve computational efficiency.3 Make it easy to pilot the non-linear analysis.

Asymptotic Numerical Method

1 Inverse one stiffness matrix in one non-linear step.2 Very efficient in the case of strong non-linear calculations.




Exchange Information Between Two Scales

Research Methods:

1 FE2 Method







Nezamabadi S., Potier-Ferry M., Zahrouni H., Yvonnet J., 2015. Compos. Struct.





Macroscopic

∫Ω

P : δF dΩ =

∫∂Ωf

f · δu dΓ

S = C : γ

P = F · S

γ = 12 (F

t · F− I)

F = I +∇u

+ B.C

Microscopic

∫ω

P : δF dω = 0

S = C(r) : γ

P = F · S

γ = 12 (Ft · F− I)

F = I +∇u

u+−u− = (F−I)·(X+−X−) on ∂ω





Marco to Microu = A : F

F = A,X : F

Micro to Marco

P = F · S = Hr : F = Hr : A,X : F = L : F

P =1

ω

∫ωP(X)dω =

1

ω

∫ωL : Fdω = L : F





u can be divided into four parts for 2D problems

u = F(11)

u(11) + F(12)

u(12) + F(21)

u(21) + F(22)

u(22)

u can be given by four kinds of boundary conditionsµL (u(ij), δu) = 0, in ω.

u(ij)+ − u(ij)− = X(ij)+ −X(ij)−, on ∂ω.

X(11) =

[1 0

0 0

](X+ −X−)

X(21) =

[0 0

1 0

](X+ −X−)

X(12) =

[0 1

0 0

](X+ −X−)

X(22) =

[0 0

0 1

](X+ −X−)





bb

b b

+

+4

1 2

3b

b−

−

X+

X−

bb

X+X+

X+X−

bb

b b

+

+4

1 2

3b

b−

−

X+

X−

bb

X+X+

X−

X+X+

X+

X(11)

=

[1 0

0 0

](X

+ − X−) X

(12)=

[0 1

0 0

](X

+ − X−)

bb

b b

+

+

1 2

3b

b−

X+

X−

bb

X+X+

X+X−

−−

−

4

bb

b b

+

+4

1 2

3b

b−

−

X+

X−

b b

X+X+

X+X−

X(21)

=

[0 0

1 0

](X

+ − X−) X

(22)=

[0 0

0 1

](X

+ − X−)




Model based on Multiscale-ANM

Research Methods:

1 FE2 Method







Solver for Nonlinear ProblemAdapt the asymptotic numerical method (ANM) to solve the

nonlinear equations.

Strengths: 1. Every step of iteration to solve one inversion

2. Step length self adaption

Uj+1 = Uj + apUp

λj+1 = λj + apλpp ∈ [1, n]

Cochelin B, Damil N, Potier-Ferry M.2007.





FE2method, DOF: 177012

Fully meshed model, DOF: 910810

0 10 20 30 40 50 60 700

100

200

300

400

500

600

700

Displacement

Load

pro

port

iona

lity

fact

or

λ

Multiscale−ANM

Fully meshed model





⇓





Micro-buckling of long fiber reinforced composites




Model based on Multiscale-Fourier

Research Methods:

1 FE2 Method







AimBuild a model coupling the mean field and the envelope of the

oscillation.

Damil et al., JMPS, 2010





PrincipleFourier series with slowly varying coefficients

U(x) =

+∞∑j=−∞

Uj(x)ejiqx

U(x) including the displacement vector, its gradient, strain

and stress tensors, is expanded in the form of a Fourier series. The

wave number q is defined as q = πLq0, in which, L is the length of

the RVE (fiber expansion axis), and q0 is a chosen wave length

(q0 ∈ N, q0 ≥ 2). The Uj(X) represents the envelope for the

jth(j = 0, 1...) order harmonic, vary more slowly than the harmonic

function over a period [x, x+ 2πq ], here we keep the order j up to 1.





Virtual work and constitutive law of the RVE

δPint=∫ L0

+1∑j=−1

δγ−jSjdx

=∫ L0 δγ0S0+2δγ1RS1R+2δγ1IS1Idx

[D]−1S0=γ0=[H]θ0+ 12[A0]θ0+[A1R]θ1R+[A1I ]θ1I

[D]−1S1R=γ1R=[H]θ1R+[A0]θ1R

[D]−1S1I=γ1I=[H]θ1I+[A0]θ1I

For the microscopic scale, the resolution of the equations below is

performed by using Asymptotic Numerical Method:∫ω

tδγgenSgendω = λ∫∂ω

tδuFγgen = [Hgen]θgen+ 1

2 [A(θgen)]θgenSgen = [Dgen]γgen





Global-local coupling instability of sandwich composite beam, 2D fourier

based model (558 DOFs) VS 2D complete model (7390 DOFs)

Transversal displacement at (x=l/2, z=(hc+hf)/2) (m)0 0.005 0.01 0.015 0.02 0.025 0.03

Load

F (

N/m

)0

1000

2000

3000

4000

5000

6000

7000

8000 2D fourier related model (558 DOF)2D complete model (7390 DOF)





Microscopic Bifurcation Curve (RVE)

Macroscopic Nonlinear Response(RVE: 198 DOFs VS 640 DOFs)





Conclusions:

1 Macroscopic compressive failure comes from the microscopic

fiber buckling.

2 Multiscale model based on FE2 theory meet a good agreement

with the results of commercial finite element codes, which

shows that accurate results can be obtained with a reduced

computational cost.

3 Multiscale model based on FE2 theory and Fourier slowly

varying technique can detect the bifucation of the microscale.




AcknowledgementsThis work has been carried out within the FULLCOMP project

funded by the European Union’s Horizon’s 2020 research and innovation

programme under grant agreement No 642121.

Many thanks to all of you for kind attention!


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