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Influence of local wrinkling on membrane behaviour: A newapproach by the technique of slowly variable Fourier coefficients

Noureddine Damil a, Michel Potier-Ferry b,

a Laboratoire de Calcul Scientifique en Mecanique, Facultedes Sciences Ben MSik, Universite Hassan II Mohammedia - Casablanca, Sidi Othman,

Casablanca, Moroccob Laboratoire de Physique et Mecanique des Materiaux, FRE CNRS 3236, UniversitePaul Verlaine de Metz, Ile du Saulcy, 57045 Metz, France

a r t i c l e i n f o

Article history:

Received 18 July 2009

Received in revised form

21 January 2010

Accepted 3 April 2010

Keywords:

Buckling

LandauGinzburg equation

Bifurcation

Slowly variable Fourier coefficient

Pattern formation

a b s t r a c t

In this paper, a new technique using slowly variable Fourier coefficients and the

asymptotic LandauGinzburg approach are re-discussed and compared. The aim is to

define simple macroscopic models describing the influence of local wrinkling on

membrane behaviour. This question is analyzed by considering the simple example of a

beam resting on a non-linear Winkler foundation.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Very thin sheets do not support high compressive stresses because the appearance of local buckling reduces strain and

stress in the compressed zones. Often the modal wavelength is small as compared with the size of the structure. Such

instabilities occur during the process of thin metal sheets by rolling where the plastic deformation in the bite induces

compressive residual stresses. These stresses generate sheet wrinkling as depicted inFig. 1, which releases the compressive

stresses and can affect the rolling process.

The aim of this research is to define macroscopic models which couples 2D linear elasticity with equations governing

the evolution of buckles. The present paper is limited to the study of the well known example of a beam resting on a non-

linear Winkler foundation.

Such local instabilities can be modelized by bifurcation analysis according to the LandauGinzburg theory (Wesfreidand Zaleski, 1984). This famous LandauGinzburg equation follows from an asymptotic double scale analysis. At the local

level, one accounts for the periodic nature of the buckles, while the slow variations of the envelope are described at the

macroscopic scale.

A slightly different approach has been proposed recently where the nearly periodic fields are represented by Fourier

series with slowly varying coefficients. This leads to macroscopic models that are generalized continua with the

macroscopic stress defined by Fourier coefficients of the microscopic stress ( Damil and Potier-Ferry, 2006). In this sense,

this technique is similar to a homogenization theory, where a double scale analysis permits to deduce macroscopic

generalized continua from microscopic classical ones (Forest and Sab, 1998; Kouznetsova et al., 2004).

Contents lists available atScienceDirect

journal homepage: www.elsevier.com/locate/jmps

Journal of the Mechanics and Physics of Solids

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0022-5096/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.jmps.2010.04.002

Corresponding author.

E-mail addresses: [email protected] (N. Damil), [email protected] (M. Potier-Ferry).

Journal of the Mechanics and Physics of Solids 58 (2010) 11391153
http://-/?-http://www.elsevier.com/locate/jmpshttp://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.jmps.2010.04.002mailto:[email protected]:[email protected]:[email protected]:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.jmps.2010.04.002http://www.elsevier.com/locate/jmpshttp://-/?-

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In this paper, the asymptotic approach and the technique using slowly variable Fourier coefficients will be re-discussed

and compared, with the aim to define simple macroscopic models that describe the influence of local buckling on the

global response of the beam.

These two scale methods are generic and could be applied in all cases of instabilities with spatially periodic patterns.

For instance, the asymptotic approach has been used for the RayleighBenard convection ( Newell and Whitehead, 1969;

Segel, 1969), for several fluid instabilities (Wesfreid and Zaleski, 1984;Cross and Hohenberg, 1993), for the buckling of a

beam on foundation that is revisited in the present paper (Amazigo et al., 1970; Pomeau and Zaleski, 1981; Potier-Ferry,

1983), for plate buckling (Pomeau, 1981; Damil and Potier-Ferry, 1986; Boucif et al., 1991 ) and cylindrical shell buckling(Amazigo and Fraser, 1971;Abdelmoula et al., 1992). In the same way, such cellular instabilities appear in the buckling of

carbon nanotubes (Ru, 2001;He et al., 2005) and in the buckling of thin elastic film bound to compliant substrate ( Chen

and Hutchinson, 2004;Audoly and Boudaoud, 2008;Wang et al., 2008).

The second approach, that is based on Fourier series, is able to account for the coupling between local and global

buckling as in the buckling of stiffened plate (Sridharan and Zeggane, 2001) or of sandwich structures (Leotoing et al.,

2002) or to account for the influence of wrinkles on the behaviour of membranes (Wong and Pellegrino, 2006). There are

many papers about the computation of very thin membranes with many applications like inflated airbag, life jackets or

fabric tension structures. According toRossi et al. (2005), two different approaches have been proposed to account for the

influence of wrinkling within a membrane model. The first family (Roddeman et al., 1987a, 1987b;Lu et al., 2001) changes

the deformation gradient while the second one introduces a modified constitutive law (Tabarrok and Qin, 1992;Liu et al.,

2001;Diaby et al., 2006;Trouffard et al., in press). In the two approaches, the new membrane behaviour tries to drop the

compressive stresses. These two approaches will be re-discussed shortly within the new framework of slowly variable

Fourier coefficients.Another class of application is the buckling of metal sheets under the influence of residual stress (Fischer et al., 2000),

which is of interest in the modelisation of industrial processes as web handling or rolling (Jacques et al., 2007;Abdelkhalek

et al., 2009). In the case of rolling, a modified membrane model has been proposed by Counhaye (2000),

that is more or less similar to the membrane wrinkling models of Roddeman et al. (1987a) and Tabarrok and

Qin (1992). This approach has been proved to be useful to predict flatness defects (Abdelkhalek et al., 2008, 2009). Last, let

us mention fibre microbuckling in long fibre composites materials (Kyriakides et al., 1995; Waas and

Schultheisz, 1996; Drapier et al., 2001) that is a plastic instability that permits to explain compressive failure of these

materials.

In this paper, the LandauGinzburg asymptotic method and the double scale Fourier series approach are applied

to the classical model of beam on foundation, with a view to represent the influence of local buckling on the membrane

behaviour. In general, this leads to a coupling of the membrane model with envelope equations, which are more

or less similar to the LandauGinzburg equation. Finally, we discuss the numerical implementation, the

boundary conditions for the envelope equations and the influence of non-linear boundary layers on the membranebehaviour.

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Fig. 1. Wrinkling patterns at bite exit.

N. Damil, M. Potier-Ferry / J. Mech. Phys. Solids 58 (2010) 113911531140

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The two spatial variablesxandXare supposed to be independent, which leads to the following rule to account for rapid

variations with x and for slow variations with X:

d

dx @

@x ffiffiffiep @

@X 10

When the expansion rules (9) and (10) are applied to (8), we get the following equations at the orders

Offiffiffie

p,Oe,Oe3=2:

Lv1 EI@4

v1

@x4 lc@

2

v1

@x2 cv1 0 11

Lv2 4EI @@X

@3v1

@x3 2lc @

@X

@v1

@x c2v12 12

Lv3 4EI @@X

@3v2

@x3 2lc @

@X

@v2

@x 6EI @

2

@X2@2v1

@x2 lc@

2v1

@X2 @

2v1

@x2 2c2v1v2c3v13 13

The solution of (11) is a modulated oscillation, the complex amplitude V1(X) of which is unknown at this stage. In the

same way, the solution of (12) depends on the square of this amplitude:

v1x,X V1XexpiqxV1Xexpiqx av2x,X V0XV2Xexp2iqxV2Xexp2iqx b

V0X 2c2

c V1X 2, V2X c2

9cV1X2

c

8>>>>>:

14

Note that the two first terms in (12) vanish because of (14a) and (7) and this implies that the right hand side of the

second equation (12) is proportional to the square ofv1. It is not necessary to solve the third equation (13). With account of

(14), the right hand side of (13) involves three harmonics. The solution v(3) is unbounded with respect to the rapid variable

x, if there is a term proportional to the main harmonic in this right hand side. To avoid this, one drops the coefficient of this

first harmonic, which leads to a differential equation for the complex amplitude, that is the famous LandauGinzburg

equation:

4EId2V1dX2

V1V19V192 3c3

q2 38c

22

cq2

0 15

From a multiscale point of view, the LandauGinzburg equation is a macroscopic model because it depends on the slow

variableX. On the contrary, Eqs. (11)(13) describe the microscopic evolutions.

This LandauGinzburg equation is generic for the bifurcation modes breaking the symmetry x-

x, as established

in Iooss et al. (1989) and Damil and Potier-Ferry (1992). The same envelope equation has been obtained in a lot of

physical problems, for instance in fluid mechanics. However, there is a drawback: it is valid only in a neighbor-

hood of the bifurcation point. In this paper, we present another approach, which can remain valid away from the bifurcation.

The aim of the paper is to deduce a macroscopic model that is able to couple the evolutions of local and global buckling

patterns. Of course, this coupling cannot be deduced from the asymptotic approach that assumes small perturbations close

to a specific state.

3. Multiscale Fourier analysis for macroscopic modelling of local instabilities

3.1. New macroscopic models

We study phenomena such that the response of the system is the sum of a slowly varying mean field and a fluctuation

that is nearly periodic in one spatial direction. As shown in Fig. 2, at least two slowly varying functions are needed to modelthese phenomena.

In this part, a general method is presented to deduce the equations satisfied by these slowly varying fields. All the

unknowns of the models U= {u(x), v(x), n(x), m(x)} are sought in the form of Fourier series, whose coefficients vary slower

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Mean field +amplitude

Mean field

Fig. 2. At least two macroscopic fields are necessary to describe a nearly periodic response: the mean field and the amplitude of the fluctuation.


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3.2. A model with five envelopes

As an example, we first express the model coupling global and local instabilities obtained by considering only five terms

in (21)(24) or in (25): U0AR4, U1AC

4, U2AC4, U1 U1 , and U2 U2 .

The macroscopic version of the principle of virtual work is

RL

0 n0dg0 n1dg1 n1dg1 n2dg2 n2dg2 m0dk0 m1dk1 m1dk1 m2dk2 m2dk2dx

RL0 f0du0 f1du1 f1du1 f2df2 f2du2 g0dv0 g1dv1 g1dv1 g2dv2 g2dv2dx 28The macroscopic constitutive equations are

n0ES

g0 du0

dx 1

2

dv0dx

2 d

dx iq

v1

2

ddx

2iq

v2

2

29

n1ES

g1 d

dx iq

u1

d

dxiq

v1

d

dx2iq

v2

dv0dx

d

dx iq

v1 30

n2ES

g2 d

dx2iq

u2

dv0dx

d

dx2iq

v2

1

2

d

dx iq

v1

231

m0

EI k0 d2v

0dx2 32

m1EI

k1 d

dx iq

2v1 33

m2EI

k2 d

dx2iq

2v2 34

g0 cv0 c2v20 2v1j j2 2v2j j2

c3v30 6v0 v1j j2 6v0 v 2j j2 2v1j j2v2 2v1j j2v2 v21v2 v1 2v2 35

g1 cv1 c22v0v1 2v1 v2c33v20v1 6v0v1 v2 3v1 v1j j2 6v1 v2j j2 36

g2 cv2 c22v0v2 v21c33v0v21 3v20v2 6v1j j2v2 3v2 v 2j j2 37The macroscopic version of the potential energy is

Puj,vj RL

0

ES

2

X2j 2

gjgj

0@

1A EI

2

X2j 2

kjkj

0@

1A

0@

1Adx RL0 X2

j 2f0u0 dx

Z L

0

c

2

X2j 2

vjvj

0@

1A c2

3

X2j1 2

XJM2j2 Jm2

vj1 vj2 vj1 j2

0@

1A c3

4

X2j1 2

X2j2 2

XJM3Jm

3

vj1 vj2 vj3 vj1 j2 j3

0@

1A

0@

1Adx 38

with

Jm2 Max2,2j1

JM2 Min2,2j1

Jm3 Max2,2j1j2

JM3 Min2,2j1j2Or more explicitly

Pu0,u1,u2,v0,v1,v2 RL

0 ES

2g20 29g19

2 29g292 EI

2k20 29k19

2 29k292dx

Z L

0f0u0dx

Z L0

c

2v20 29v19

2 29v292dx

Z L0

c23v30 6v09v192 6v09v292 3v21v2 3v1 2v2dx

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The approximate solution (41) has two advantages. First, the initial bifurcation curve can be described exactly in the

case of a uniform compression. Especially it will permit to recover the LandauGinzburg equation (15). Second, the second

harmonics can be dropped to deduce differential equations involving a single envelope.

4.2. A model with one complex envelope

We express in this section the model obtained by considering only three terms in the macroscopic potential energy:

U0AR4, U1AC4, and U1 U1 . The influence of the second harmonic is accounted in a simplified manner, as explained inSection 4.1. Moreover, with a view to build a relatively simple model in U0, U1, the influence of the second envelope is

limited to the main terms in the bifurcation analysis. More precisely, we drop the terms of order v0v41 and v

51 in the

potential energy or, equivalently, of orderv0v31andv

41in the variational equation. Furthermore, let us recall that it has been

assumed thatn1(x) =n2(x)=0,u1(x)= u2(x)=0. According to these principles, we establish simplified versions of the proposed

model.

Let us consider the variational equation (28), by disregarding the equations accounted in Section 4.1:Z L0

n0dg0 m0dk0 m1dk1 m1dk1g0dv0g1dv1g1dv1dx Z L

0f0du0 dx 42

According to the previously presented principles and by considering (41), the constitutive equations for the foundation

are simplified as

g0 cv0 c2v20 29v192

c3v30 6v09v192

43

g1 cv1 2c2v0v1 3c3v20v1 dv19v192 44

whered 3c32c22=9c. In the same way, the membrane constitutive equation is reduced to

n0ES

g0 du0

dx 1

2

dv0dx

2 d

dx iq

v1

2

45

By assembling (43)(45), one deduces that the variational equation (42) is the derivative of the following potential

energy:

Pu0,v0,v1 RL

0 ES

2g20

EI

2k20 2k1

2dxZ L

0f0u0 dx

Z L0

c2

v20 2v1j j2dxZ L

0

c23v30 6v0 v1j j2dx

Z L0

c34v40 12v20 v 1j j2 d2 v1j j

4

dx 46

4.3. A model with one real envelope

The aim is to build the simplest model that is able to couple membrane behaviour with the LandauGinzburg equation.

Let us start from the model with one complex envelope established in Section 4.2. One further assumes v0=0. The

remaining fields are the global membrane unknowns u0,n0and a local bending unknownv1. An additional simplification is

introduced, the bending envelope v1(x)= v(x) being assumed real. This means that only the amplitude of the fluctuation is

modulated, the phase being constant. With the notation u0(x)= u(x), the potential energy is reduced to

Pu,v Z L

0

ES

2g2 EI k

2 cv2 d2

v4fu dx 47g u0 v02 q2v2 48

9k92 v00q2v2 4q2v02 49A further simplification is done in the bending term

REI9k92 dxwith the objective to recover an equation similar to the

LandauGinzburg. One eliminates the second derivative of the envelope v(x). After integration by parts, this bending

energy is rewritten asZ L0

EIv00q2v2 4q2v02dx Z L

0

EIq4v2 6q2v02dx 50

Finally, the simplified potential energy is given by the formula:

Pu,v Z L

0

ES

2u0 v02

q2

v2 2 EI6q2v02 q4v2cv2 d2 v4fudx 51

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The differential equations of the system follow from the stationarity of the potential energy dP=0. This leads to

dn

dxf 0 52

n ESu0 v02 q2v2 53

d

dx 6EIq2

n

dv

dx EIq4 nq2 c vdv3 0 54

Hence Eqs. (52)(54) couple the global membrane behaviour with an equation similar to the LandauGinzburg. Indeed,

the third equation (54) that governs the local bending instability can be identified with the LandauGinzburg (15) if the

membrane force is equal to its value at the bifurcation point ( n= lc).

4.4. A classical membrane model accounting for the local instability

A last simplification is introduced in the previous model that couples the global membrane behaviour (53) with the

local instability equation (54). The idea is the same as in Section 4.1: one eliminates the envelope of the first harmonic from

a bifurcation analysis by neglecting the spatial variation of the envelope (all the spatial derivatives in (54) are dropped):

EIq4 nq2 cvdv3 0 ) lcnvdv3 0 55

In other words, the LandauGinzburg approach has been replaced by a Landau one. The bifurcation equation (55) hastwo solution branches

v 0v2 q

2lcnd

8>>>>>>>>>>>>:

58

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-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10

Fig. 3. Classical membrane model accounting for the local instability. The bilinear constitutive equation (58), for ES=1, lc=2, qc=1, andd =1.

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In this way, the elimination of wrinkling displacements,v(x) andv2(x), permits to recover a classical non-linear stress

strain law. In the formula (58) there is a slope discontinuity at the bifurcation state, more or less like in the approach of

Counhaye (2000)to predict the residual stresses generated by the rolling process. Similar non-linear membrane behaviour

is assumed in most of the modelisations of the wrinkling of very thin membranes but the slope change is generally located

at the beginning of the compression lc=0.

5. Few numerical solutions of some envelope equations

In this last part, the previously established macroscopic models are analyzed from theoretical and numerical

points of view. Let us recall that these models have been deduced from the assumption of slowly varying

Fourier coefficients. As usual in model reduction, it is not obvious that the reduced model applies correctly also near

the boundaries. Hence, one has to discuss carefully the boundary conditions that are consistent with the macroscopic

models. Furthermore, the established macroscopic models sometimes involve boundary layers. For instance, in the

transition from the initial model (47) to the LandauGinzburg equation (54), a fourth order derivative has been dropped,

that suggests the existence of solutions that are not slowly variable near the boundary. However, this LandauGinzburg

equation follows from an asymptotic analysis near the bifurcation point. Far from this critical zone the quantity

EIq4+nq2+ cis not small and the solution v(x) of LandauGinzburg equation (54) may be rapidly variable, especially close to

the boundary.

In this work, the amplitude equations were discretized by classical finite elements. Two kinds of numerical tests

will be presented. The first example is a long beam on a foundation subjected to a uniform compression n(x)=l. The

linear version of (22)(24), (26), (27) will be considered with the data EI=1, c=1 and with the choice of q=1 for the

wavenumber. We limit ourselves to a single harmonic j =1, which leads to the following eigenvalue problem satisfied by

the envelope:

d

dx i

4v1 l

d

dx i

2v1 v1 0 59

As (59) is similar to a beam bending equation, it is solved in the same way by Hermite polynomials but with twice the

number of degrees of freedom because the envelope v1 is a complex number.

The second example is the non-linear model (52)(54) with only one real amplitude. As these differential equations are

of the second order, the unknowns u(x), v(x) can be discretized by C0 finite elements. A 3-node element with quadratic

shape functions is chosen.

In these two cases, it has been assumed that the macroscopic unknowns are slowly variable, so that few macroscopic

elements are sufficient, independently of the wavelength of the local oscillations.

5.1. A linear eigenvalue analysis of a complex envelope equation

5.1.1. Theoretical remarks

Let us consider the linear buckling problem described by the macroscopic equation (59). The corresponding microscopic

model is the linearized version of (3c) and (3d). The unknowns are the compression force l0 and the associated buckling

modev(x):

d4v

dx4l0

d2v

dx2v 0 60

In this example, analytical solutions are known. Disregarding boundary conditions, the smallest eigenvalue is l0=2 and

the corresponding eigenmodes are v(x)= eix and v(x)= e ix. The critical wavenumber is q =1.

As this model is linear, one can verify that Eqs. (60) and (59) are equivalent, via the relation v(x) =v1(x)eix. Hence (59)has the same eigenvaluel0=2 and the corresponding modes are v1(x)=1, that is slowly varying, and v1(x)= e

2ix, that is

varying rapidly, thus, not consistent with the assumptions below. Therefore, the generalized continuum model has some

slowly varying solutions, as expected, but it has also other very oscillating solutions. Note that the rapid oscillations are

dropped by the asymptotic approach. Indeed, if the rules of the asymptotic approach are applied to Eq. (59), one recovers

the linearized LandauGinzburg equation, that is:

4d2v1dx2

l2v1 0 61

One remarks that the solutions of (61) are varying slowly if the parameter l is close to its critical value. Hence the

LandauGinzburg approach is able to select the adequate solutions, according to an asymptotic criterion. In what follows,

we shall try to drop the oscillating solutions by discretizing the model (59) by a coarse finite element mesh. In other words,

the asymptotic low-pass filter will be replaced by a numerical one.

Let us now describe the exact solutions of (60) and therefore of (59) in the case of a finite beam of length Land of clamped boundaries, i.e. v dv=dx 0 at x=0, L. The symmetric solutions of (59) are given by the following

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equation (x=xL/2):vx cosq1xcosq2L=2cosq2xcosq1L=2

q21 l0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffil0

24q

2 , q22

l0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffil0

24q

2

8>: 62

The first one can be rewritten as

vx cosq1xcosq2L=2cosq1L=2cosq1q2xsinq1xcosq1L=2sinq1q2x 63Close to the bifurcation, q1 1, q2 1, q1q2

ffiffiffiffiffiffiffiffiffiffiffiffil02

p Oe, so that the exact solution (63) satisfies the

assumptions of a Fourier series with slowly varying coefficients.

5.1.2. Numerical study of the eigenvalue problem

The boundary conditions to be associated with the macroscopic model can be established in the same manner as in the

asymptotic approach. One has to assume that the macroscopic model is more or less exact up to the boundary.

We study the linear beam buckling problem, described by Eq. (60), with a length L =100pand with clamped boundaryconditions. The beam length is large, in such a way that the instability mode has about 50 cells. A direct analysis of (60)

requires at least 400 Hermite finite elements. This direct analysis established that the first mode is a modulated oscillation,

the amplitude having a sinusoidal shape. Incidentally, the LandauGinzburg equation (61), when associated with a

Dirichlet boundary condition, predicts correctly this sinusoidal shape.

Let us consider clamped boundary conditions and assume that the fluctuationsRev1xeix vR1xcosxvI1xsinx 64

vanish at x =0, as well as its derivative. This implies that the real and imaginary parts of the envelope satisfy the two

following boundary conditions:

vR10 0,dvR10

dx vI10 0 65

If, in addition, the approximation of slowly varying amplitude is taken into account, it appears that the amplitudev1(x)

vanishes at the end, but the derivative does not.

In the first numerical experiment we now assume that v1(x) is zero at the ends, but not its derivative. As shown in Fig. 4,

the sinusoidal shape is recovered with a single macroscopic element. This implies that the macroscopic mesh is not related

with the microscopic wavelength and the boundary conditions of the macroscopic model v1(0)=v1(L)=0 have been settled

correctly.Let us try to apply now the macroscopic model with the same boundary conditions as in the initial model, i.e. require

thatv10 v1L dv1=dx0 dv1=dxL 0. In this case, the obtained mode does not have the expected sinusoidal shape(Fig. 5a) butFig. 5b shows that one comes near to this shape with 10 macroscopic elements. However, there are small

differences compared to the reference solution near the boundaries, because of the previously mentioned boundary layer

effects. Nevertheless, a correct solution cannot be obtained with a very fine mesh. For instance, with 100 macroscopic

elements, one gets very oscillating solutions as predicted in Section 5.1.1

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0 50 100 150 200 250 300 350

-200

-150

-100

-50

0

50

100

150

200

Fig. 4. Beam of length 100p. The envelope v1 is zero at the boundaries. The sinusoidal shape is recovered with only one macroscopic element.


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In other words, the best solutions are obtained by disregarding any condition on the derivative of the envelope, even

using the fourth order envelope equation (59). Nevertheless, more or less correct solutions are obtained if the number of

macroscopic elements is sufficiently large, to limit the effect of the boundary conditions on the slope and sufficiently small

to filter the very oscillating solutions.

5.2. A non-linear analysis coupling mean field and local buckling equations

The non-linear local buckling of a long beam is simulated in a case where the instability first appears in a small

region. The considered model is given by Eqs. (52)(54) that couple the macroscopic membrane variables u(x), n(x)

with the real amplitude of local buckles v(x). The data are EI=ES=c= d =1, L=100p and one chooses q=1. Two mainloadings will be applied to the beam: first a constant uniform force f(x) = 2/L and an increasing global shorteningu(L)= lL, u(0)= 0. With these boundary conditions, the pre-bifurcation membrane fields are solutions of (52)and (53) in the case v(x)=0:

nx 2xLl1, ux x

2

Ll1x 66

Thus, the maximal compressive membrane stress is located at the end x =0. One can expect that instability starts when

the maximal membrane stress reaches the same critical value as in the case of a uniform pre-stress. Hence the bifurcation

would occur forn(0)E2,lE1. The differential equations are discretized by three-node quadratic elements. Of course asmall number of elements are sufficient to solve the macroscopic model so we choose 10 elements. The classical NewtonRaphson algorithm with load control is used. To ensure a continuous transition from the fundamental to the bifurcating

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Fig. 5. Beam of length 100p. The envelopev1and its derivative are zero at the ends: (a) two macroscopic finite elements and (b) ten macroscopic finiteelements. The envelope has the true sinusoidal shape, with boundary effects.

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Fig. 6. The response curves vmax ,l of a beam of length 100pwith 10 macroscopic finite elements: (a) in the case of Neumann boundary conditions and(b) Dirichlet boundary conditions.


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branch, a uniform small transverse load is addedgpert= 2 103. Dirichlet or Neumann boundary conditions are prescribedto the envelope. The complete system of equations is summarized as follows:

dn

dx 2

L 0 a

n dudx

dvdx

2v2 b

ddx

6n dvdx

2 n vv3 lgpert c

u0 0, uL lL dv0 0, vL 0 or dv

dx0 0, dv

dxL 0 e

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

67

The response curvesvmax,l are plotted in Fig. 6a (respectively, 6b) in the case of Neumann and Dirichlet boundaryconditions, respectively. As established in Damil and Potier-Ferry (1986), these two kinds of macroscopic boundary

conditions correspond to a clamped or a supported beam in the starting model, respectively. First one notes that the

bifurcation occurs for the expected value lE1, in the two cases of boundary conditions. The two response curves aresimilar, the maximal deflection being slightly smaller in the Dirichlet case.

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Fig. 7. The evolution of the buckling patterns of a beam of length 100p, in the Neumann case. The deflection vxsinx is represented for 4 values of theload: (a)l =1.08, (b) l=1.2, (c)l =1.65, and (d) l=2.55.

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Fig. 8. The evolution of the buckling patterns of a beam of length 100p, in the Dirichlet case. The deflection vxsinxis represented for two values of theload: (a)l =1.08 and (b) l=1.65.


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To represent the evolution of the buckling patterns, the deflection vxsinxis represented for four values of the loadl= 1.08, 1.2, 1.65, and 2.55 in the Neumann case (seeFig. 7). Near the bifurcation, the wrinkling is localized in the region of

maximal membrane stress at x =0. Next the deflection increases and the instability pattern spreads in a large part of the

beam. In the Dirichlet case, the deflection is plotted for l=1.2 and 1.65 (seeFig. 8). Inside the domain these patterns are the

same as in Neumann case. They are different in a boundary layer within a width that corresponds to about 45 periods.

Using these data, the considered model leads to a consistent answer of the structure and it permits to account for the

differences between a clamped and a supported beam: one sees that this difference is located in a boundary layer.

6. Few comments about the validity range of the macroscopic models

The main restrictive condition to build the considered macroscopic models is the assumption of slowly varying

amplitudes. In other words the length scales of the oscillations and of the envelope must be clearly distinguished. This

limitation holds for the Fourier approach as well as for the asymptotic one.

The evolution of the non-linear modes close to boundaries is the first example of inadequacy of the macroscopic models

in some regions of the spatial domain. Indeed it is known that a reduced model or a homogenized model is not able to

represent accurately the exact solution near the boundary and that the boundary conditions are satisfied only in a mean

sense. To explain the differences between fine and reduced models one has to account for boundary layers: for instance,

such an analysis has been done by an asymptotic method in Daya et al. (2003)in the case of the vibration of large repetitive

structure. Such a boundary analysis could also be performed in a numerical framework by considering the fine model in

the boundary layers and by applying a bridging technique ( Ben Dhia, 1998) to match the solutions of the fine model (3) and

of the reduced model (52)(54). In brief, the macroscopic models can be seen as bulk models, whose validity near the

boundary can be discussed and sometimes corrected.

The localization of buckling patterns is another case, where the assumption of a slowly varying amplitude is no longer

valid anywhere. This problem of localization of buckling patterns has been addressed by many authors, for instance

Potier-Ferry (1983),Hunt et al. (1989),Lee and Waas (1996),Hunt et al. (2000)andComan et al. (2003). This localization

phenomenon appears generically for long domains and in the case of a softening non-linearity. Only the initial

post-buckling can be represented accurately by amplitude equations. In a case as the one ofFig. 8b ofHunt et al. (1989), the

amplitude is reduced by half in a simple period: clearly the assumption of two different length scales is no longer valid in

the region of localization. In other words, such a much localized behaviour cannot be represented by slowly varying

amplitude and another numerical technique involving bridging should be used as sketched above in the discussion of

boundary effects.

As established in Damil and Potier-Ferry (2006), one can recover the results of the asymptotic LandauGinzburg

approach from the present envelope equation (54) or (67c), if the pre-buckling load is uniform and close to the bifurcation

value:n(x)Elc= 2. The converse result is not true and the non-linear example of Section 5.2 cannot be analyzed by theasymptotic approach. The main difference is the account of the influence of the wrinkling on the membrane behaviour (see

(67b)). Moreover, the evolution of the amplitude that is governed by (67c) is coupled non-linearly with the membrane

equations (67a) and (67b). Clearly, this strong coupling cannot be accounted by the asymptotic approach of Section 2,

especially away from the bifurcation point as in the case ofFig. 7d.

7. Conclusion

A two scale approach to predict quasi-periodic instabilities has been presented, that led to a generalized continuum. The

so defined macroscopic stresses are Fourier coefficients of the microscopic stress. Clearly, this approach is interesting when

the modal wavelength is small with respect to the macroscopic wavelength. It seems much more manageable than the

LandauGinzburg methods, especially using discretization. The choice of the boundary conditions for the macroscopic

model is not an obvious task. We have not discussed here the number of harmonics to be accounted for but it was foundthat at least five ones 0, 7q, 72q are required to recover the LandauGinzburg equation.

References

Abdelkhalek, S., Montmitonnet, P., Legrand, N., Buessler, P., 2008. Manifested flatness predictions in thin strip cold rolling. International Journal ofMaterial Forming Suppl. 1, 339342.

Abdelkhalek, S., Zahrouni, H., Potier-Ferry, M., Montmitonnet, P., Legrand, N., Buessler, P., 2009. Mode lisation numerique du flambage de plaques minceset applications au laminage. Mecanique et Industries 10, 305309.

Abdelmoula, R., Damil, N., Potier-Ferry, M., 1992. Influence of distributed and localized imperfections on the buckling of cylindrical shells. InternationalJournal of Solids and Structures 29, 125.

Amazigo, J.C., Budiansky, B., Carrier, G.F., 1970. Asymptotic analyses of the buckling of imperfect columns on non-linear elastic foundations. InternationalJournal of Solids and Structures 6, 13411356.

Amazigo, J.C., Fraser, W.B., 1971. Buckling under external pressure of cylindrical shells with dimple shaped initial imperfection. International Journal ofSolids and Structures 7, 883900.

Audoly, B., Boudaoud, A., 2008. Buckling of a stiff film bound to a compliant substrate

Part II: a global scenario for the formation of herringbone pattern.Journal of the Mechanics and Physics of Solids 56, 24222443.

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