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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).
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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
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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.
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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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-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 50 100 150 200 250 300 350
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
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.
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
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.
N. Damil, M. Potier-Ferry / J. Mech. Phys. Solids 58 (2010) 113911531150
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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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0.1
0.05
0
0.05
0.1
0.15
0.2
0 50 100 150 200 250 300 3500.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0 50 100 150 200 250 300 350
0.80.6
0.4
0.2
0
0.2
0.4
0.6
0.8
0 50 100 150 200 250 300 350
1.5
1
0.5
0
0.5
1
1.5
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.
0 50 100 150 200 250 300 3500.25
0.2
0.15
0.10.05
0
0.05
0.1
0.15
0.2
0.25
0 50 100 150 200 250 300 350
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
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.
N. Damil, M. Potier-Ferry / J. Mech. Phys. Solids 58 (2010) 11391153 1151
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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
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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.
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