asymptotic numerical method for multiscale buckling...

Asymptotic Numerical Method Asymptotic Numerical Method for Multiscale Buckling Analysis for Multiscale Buckling Analysis

of Heterogeneous Materialsof Heterogeneous Materials

Saeid NEZAMABADISaeid NEZAMABADI11, Hamid ZAHROUNI, Hamid ZAHROUNI11, Julien YVONNET , Julien YVONNET 22, , Michel POTIERMichel POTIER--FERRYFERRY11

¹¹ UniversitUniversitéé Paul VerlainePaul Verlaine-- Metz, Laboratoire de Physique et MMetz, Laboratoire de Physique et Méécanique des Matcanique des Matéériaux, FRE CNRS 3236riaux, FRE CNRS 3236²² UniversitUniversitéé ParisParis--Est, Laboratoire ModEst, Laboratoire Modéélisation et Simulation Multi lisation et Simulation Multi ÉÉchelle, FRE CNRS 3160chelle, FRE CNRS 3160

Workshop of Asymptotic methods, mechanics and other applications

August 31st and September 1st 2009 ENS CachanBretagne.

01/09/20092

Plan

¢ Introductionl Numerical homogenization of the nonlinear problemsl Using homogenization to deal with instabilitiesl Our homogenization approach

¢ Micro-macro problem formulation¢ ANM for the nonlinear constitutive laws¢ Applications:

l Bending of a rectangular beaml Buckling of a shallow archl Elastic microbuckling of a heterogeneous materiall Plastic microbuckling of a fiber reinforced composite

¢ Conclusions et Perspectives

01/09/20093

IntroductionNumerical homogenization of the

nonlinear problemsq The fully meshed model: the whole structure is meshed, including all heterogeneities;

Ladevèze et al. (2001).q The multiscale finite element method (FE2): This technique simply correlates the

micro structural response with the overall macroscopic behaviour, irrespective of the deformation level or the material behaviour; Smit 1998, Terada 2001, Ghosh 2001, Feyel 2000-2003, Yvonnet 2007.

01/09/20094

The instability phenomena in the context of multiscale homogenization analysis of heterogeneous materials are of structural (macro) and/or material (micro) types.

Abeyaratne and Triantafyllidis (1984) studied the macroscopic instability of a finitely strained periodic porous elastomer with a strictly rank-one convex matrix by homogenizing the tangent moduli.

Miehe et al. (2002): Investigation of the properties and the interaction of two coupled minimization principles: the principle of minimum potential energy of the macrostructure and the principle of minimum average energy of the microstructure.

IntroductionUsing homogenization to deal with

instabilities

01/09/20095

IntroductionOur homogenization approach

We combine the multilevel finite element method (FE2) and the asymptotic numerical method (ANM).

The principle of the ANM is to build up the solution branch in the form of power series that are truncated at rather large orders.

Introducing such expansion in the nonlinear problems at both macro and micro scales, it results a series of linear problems.

The superposition principle is applied on each linear problem.

The advantages with the ANM:n The localization tensor is constructed for each asymptotic

step;n High order approximation;

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IntroductionOur homogenization approach

n Only one decomposition of the tangent matrix per step;n Large asymptotic steps;n No iteration;n Estimate a posteriori the length of each step of calculation in

contrary to the method of Newton-Raphson.

üReduction of calculation time;üFollowing the complicated responses.

01/09/20097

Micro-macro problem formulation

Ω=∇ dans0.P

Ω∂=Ω∂Λ=

u

t

suruusurFN

*

*.P

γCS :)(r=

Ω=∇ dans0.P

∫∫ ΩΩΩ

Ω=Ω

Ω= dd FFPP 11

Boundary conditions in the microscopic scale

01/09/20098


( )−+−+−+

Ω∂

Ω∂

ΩΩ

−−=−⇒=

−=⇒=

=Γ⊗Ω

Γ⊗Ω

+=

Ω∇Ω

+ΩΩ

=

∇+=

+=

∫

∫

∫∫

XXuuwwLCXuwLC

dNw

dNw

dwdX

wwXx

)(:2..)(0:1..

01

1

11

IFIF

FF

FF

FFF

Periodic condition

Linear deformation

Boundary conditions in the microscopic scale

01/09/20099

Macro problem formulation

∑=

Λ+

Λ=

Λ

N

pp

p

pp

ua

u

aaau

10

0

0

)()()(

PPP

The weak form associated with the macroscopic problem is written as:

The ANM: Asymptotic expansion:

Collecting terms with the same powers of ”a” induces a sequence of linear problems defined as follows:

Ω∂=

ΓΛ=Ω ∫∫ Ω∂Ω

u

t

suruu

duFdt

*

* .: δδ FP

NpduFdt

ppt ...1.: * =∀ΓΛ=Ω ∫∫ Ω∂Ω

δδ FP

01/09/200910

Micro problem formulation

The microscopic problem is written as following:

( )

( ) Ω∂−−=−+

Ω

+∇=+=

==

=Ω

−+−+

Ω∫

surXXuuLC

dans

u

dr

t

)(:..

:0:

21

)(

IF

IFIFFγ

SFPγCS

FP

t

δ

01/09/200911


( )

( ) Ω∂−=−+

Ω

∇=+=

+==

=Ω

−+−+

Ω∫

surXXuuLC

dans

u

dr

t

111

11

011021

1

01101

1)(

1

1

:..

:0:

F

FFFFFγ

SFSFPγCS

FP

tt

δ

Perturbation :

Expanding the microscopic problem gives:

At order 1:

( ) Ω∂−=−+

Ω

∇=

++=

++=

=

=Ω

−+−+

=−

=−

Ω

∑

∑

∫

surXXuuLC

dans

u

d

ppp

pp

p

rrprppp

p

rrprppp

pr

p

pt

F

F

FFFFFFγ

SFSFSFP

γCSFP

ttt

:..

:0:

1002

1

100

)(

δ

At order p:

01/09/200912


( )

Ω∂−=−+Ω=

−+−+ surXXuuLCdansuu

111

1

:..0),(

FδtL

At order 1:

( )

Ω∂−=−+Ω=

−+−+ surXXuuLCdansuuu

ppp

nlpp

F:..)(),( δδ FLt

At order p:

Substitution:

Using the substitution technique leads to:

01/09/200913


pF

Ω+++= dansuFuFuFuFu )22(221

)21(211

)12(121

)11(1111

~~~~

Localization: The linear microscopic problems are solved partially because is unknown. Hence, at order 1, u1 can be expressed as a linear combination of the solution modes obtained by imposing independently the homogeneous deformation fields through the boundary conditions (e.g. 2D):

where ữ(ij) are the solutions of the following problem:

Ω∂−=−+

Ω=−+−+ surXXuuLC

dansuuijijijij

ij

)()()()(

)(

~~:..0),~( δtL

01/09/200914

nlppp uu += FA:

11 : FA=uTherefore, at order 1:

At order p, we can obtain in the same manner:


where A is a third-order tensor. The microscopic deformation gradient tensor, F, is written as:

1,11,1 : FAF XX uu ==∇=11 F=∇u Localization tensor

nlXppXp u ,, : += FAF

where upnl is obtained:

Ω= dansuuu nlp

nlp )(),( δδ FLt

Hence, we have:

01/09/200915


nlppp PFLP += :

The effective stresses can then be obtained at each order p by the coupling relation between micro and macro:

∫∫∫ ΩΩ∂ΩΩ−ΓΛ=Ω dduFd nl

p

t

ppt

t

FPFLF δδδ :.:: *

Finally, the expansion of macro problem is rewritten:

is the macroscopic constitutive tensor.L

∫∫ Ω∂ΩΓΛ=Ω

t

duFd ppt δδ .: *FP

Therefore, at order p:nlppp PFLP += :

01/09/200916

ANM for the nonlinear constitutive laws

pp γCS :=Linear constitutive law:

In the context of ANM, the expansion of the constitutive law can be generally written as following:

respptp SγCS += :Nonlinear constitutive law:

Here, two behaviour of materials are considered:

ØHyperelasticity

ØPlasticity

01/09/200917


Isotropic hyperelastic materials:

where C is the symmetric right Cauchy-Green strain tensor.

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Hence, we have:

The second Piola-Kirchhoff stress tensor can be obtained as following:

where:

01/09/200919


The difficult terms to treat in the context of the ANM are : C-1 , J and Ln J.

Three models of the isotropic compressible hyperelastic materials are considered in this work (for the incompressible case: J = 1):

Mooney-Rivlin model:

Neo-Hookean model:

(extension of Hook’s law)

Gent model:

01/09/200920


Treatment of the difficulties:

01/09/200921


Mooney-Rivlin model:

Neo-Hookean model:

(extension of Hook’s law)

Gent model:

respptp SγCS += :

01/09/200922


Plasticity:

ØDeformation theory:

Loi de comportement non linéaire Ramberg-Osgood:

d

n

y

eqd Pυ)(υ)(E SS

ISγ1

23211

−

+−−+=

σα

Pour plus d’information, voir Thèse de Zahrouni (1998).

ØIncremental theory:

Assidi et al. (2009)

01/09/200923

Applications

Four examples:

¢ Bending of a rectangular beam¢ Buckling of a shallow arch¢ Elastic microbuckling of a heterogeneous material¢ Plastic microbuckling of a fiber reinforced composite

In all examples: ANM: N = 15 δ = 10 -6Poisson ratio ν = 0,3

01/09/200924

Bending of a rectangular beam

We have tested:ØDifferent rigidity ratios;ØDifferent volume fractions;ØDifferent inclusion shapes.

01/09/200925

Circular inclusion


01/09/200926

Circular inclusion and rigidity ratio of s E2/E1= 10

Point BPoint A


01/09/200927

Circular inclusion, E2/E1=10


01/09/200928

Buckling of a shallow arch

01/09/200929

Buckling of a shallow arch

E2/E1=10

01/09/200930

Elastic microbuckling of a heterogeneous material

Elementary Cell (EFiber /EMatrix=1000)

Different RVE can be constructed from the elementary cell.

01/09/200931


01/09/200932

Point A Point B


01/09/200933

22VER11VER


01/09/200934

Point A Point B


01/09/200935

Plastic microbuckling of a fiber reinforced composite

Drapier et al. (1997), (1998) et (2001)

01/09/200936


01/09/200937

Deformed shapes of the microstructure at IP I


01/09/200938

Conclusions

Combining ANM and FE² permits to:

l Deal efficiently with the instabilities which may occur on both the micro and macro levels in the heterogeneous materials;

l Apply superposition principle on linear problems obtained by perturbation to construct a localization tensor;

l Large asymptotic step thankful of high order approximation;

l Save computational time compared to the classical Newton-Raphson method.

01/09/200939

Perspectives

¢ Reduction of computational time (Parallel computational, POD, …);

¢ Second order homogenization:

wXXXx

dtt

++=

ΩΩ

=+ ∫Ω

GF

FPGQFP

21

:1: δδδ

01/09/200940

Second order homogenization

γCS :)(r=

Ω=∇ dans0.P

∫∫

∫∫

ΩΩ

ΩΩ

ΩΩ

=ΩΩ

=

ΩΩ

=ΩΩ

=

dd

dd

GGQQ

FFPP

11

11

∫

∫Ω∂

Ω

Γ+Λ

=Ω+

t

duF

dt

t

Fr

GQFP

δδ

δδ

:.

:*

Kousenetzova et al. 2002 et 2004

01/09/200941

Thank you for your attentionThank you for your attentionS. Nezamabadi, J. Yvonnet, H. Zahrouni and M. Potier-Ferry (2009)A multilevel computational strategy for microscopic and macroscopic instabilities, Computer Methods in Applied Mechanics and Engineering, 198 : 2099-2110.S. Nezamabadi , H. Zahrouni, J. Yvonnet and M. Potier-Ferry (2009) A multiscale finite element approach for buckling analysis of elastoplastic long fiber composites, International Journal for Multiscale Computational Engineering, to appear.

Workshop of Asymptotic methods, mechanics and other applications

August 31st and September 1st 2009 ENS CachanBretagne.

asymptotic numerical method for multiscale buckling...

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