### The method of Fourier series with slowly variable coefficients

#### A new multiple scale approach: the method of Fourier series with slowly variable coefficients

We present here the methodology that will be used to deduce macroscopic nonlinear models of membrane wrinkling. For simplicity, this first discussion is limited to one-dimensional case (space variable *X ∈ IR*) and to a one-dimensional beam model of Von Karman type that was studied in numerous papers [38–41, 44–49]:

\frac{\mathit{dn}}{\mathit{dx}}+f=0

(5-a)

\frac{n}{\mathit{ES}}=\gamma =\frac{\mathit{du}}{\mathit{dx}}+\frac{1}{2}{\left(\frac{\mathit{dv}}{\mathit{dx}}\right)}^{2}

(5-b)

\mathit{EI}\frac{{d}^{4}v}{d{x}^{4}}-\frac{d}{\mathit{dx}}\left(n\frac{\mathit{dv}}{\mathit{dx}}\right)+\mathit{cv}+{c}_{3}{v}^{3}=0.

(5-c)

Let us suppose that the instability wavenumber *Q* is known. Within this method, the unknown field U\left(X\right)=\left(\begin{array}{ccccc}\hfill u\left(X\right)\hfill & \hfill v\left(X\right)\hfill & \hfill n\left(X\right)\hfill & \hfill K\left(X\right)\hfill & \hfill \gamma \hfill \end{array}\left(X\right)\right), whose components are axial displacement, transverse displacement, resultant stress, curvature and membrane strains, is written in the following form:

U\left(X\right)={\displaystyle \sum _{m=-\infty}^{+\infty}{U}_{m}\left(X\right)exp\left(\mathit{imQX}\right)}

(6)

The new macroscopic unknown fields *U*_{
m
}(*X*) vary slowly on a single period \left[X,X+\frac{2\pi}{Q}\right] of the pattern oscillation. As pictured in Figure 2, at least two functions *U*_{
0
} *(X)* and *U*_{1} *(X)* are necessary to describe nearly periodic patterns: *U*_{
0
}*(X)* can be identified to the mean value and *U*_{1 }*(X)* represents the envelope or the amplitude of the spatial oscillations.

The first envelope *U*_{
0
}*(X)* is real-valued while the next ones are complex. In the whole paper, we limit ourselves to two envelopes *U*_{
0
}*(X)* and *U*_{1 }*(X)*. Most of the time, we shall assume that they are real.

The derivation operators are calculated exactly according to the following rule:

\phantom{\rule{0.25em}{0ex}}{\left(\frac{\mathrm{da}}{\mathrm{dX}}\right)}_{m}=\frac{d{a}_{m}}{\mathit{dX}}+\mathit{miQ}{a}_{m}=\left(\frac{d}{\mathit{dX}}+\mathit{miQ}\right){a}_{m}

(7)

\phantom{\rule{0.25em}{0ex}}{\left(\frac{{\mathrm{d}}^{2}\mathrm{a}}{{\mathrm{dX}}^{2}}\right)}_{m}=\frac{{d}^{2}{a}_{m}}{d{X}^{2}}-{m}^{2}{Q}^{2}{a}_{m}+2\mathit{imQ}\frac{d{a}_{m}}{\mathit{dX}}.

(8)

It was established [38, 40] that it is necessary to keep some spatial derivatives of the envelopes, despite of the assumption of slowly varying envelopes *d/dX < <Q*. This will be re-discussed in the membrane case. At first all the derivatives are kept as in (7) (8), but some ones can be dropped later.

#### Application to the membrane constitutive law

The application of the Fourier method to simple nonlinear equation is straightforward and it can follow from a simple identification of Fourier coefficients. For instance let us consider the 1-D membrane constitutive law (5-b) that relates the membrane stress *n*, the membrane strain γ and the displacement *(u,v)*. From (5-b), one can deduce a macroscopic constitutive law for the m-th Fourier envelope:

\frac{{n}_{m}}{\mathit{ES}}={\gamma}_{m}=\left(\frac{d}{\mathit{dX}}+\mathit{imQ}\right){u}_{m}+\frac{1}{2}{\displaystyle \sum _{k=-\infty}^{+\infty}\left(\frac{d}{\mathit{dX}}+\mathit{ikQ}\right)}{v}_{k}\left(\frac{d}{\mathit{dX}}+i\left(m-k\right)Q\right){v}_{m-k}.

(9)

Especially the constitutive law for the mean stress *n*_{
0
}*(X)* will be very useful. In the case of two real envelopes *(u*_{
0
}*, v*_{
0
}*, n*_{
0
}*, γ*_{
0
}*, K*_{
0
}*)* and *(u*_{
1
}*, v*_{
1
}*, n*_{
1
}*, γ*_{
1
}*, K*_{
1
}*)*, it can be expressed as:

\frac{{n}_{0}}{\mathit{ES}}={\gamma}_{0}=\frac{d{u}_{0}}{\mathit{dX}}+\frac{1}{2}{\left(\frac{d{v}_{0}}{\mathit{dX}}\right)}^{2}+{\left(\frac{d{v}_{1}}{\mathit{dX}}\right)}^{2}+{Q}^{2}{v}_{1}^{2}.

(10)

Let us mention the last two terms are positive, what means that the membrane stretches out when it wrinkles.

#### Energetic approach

There is another manner to derive the full macroscopic model [38, 40] by starting from the potential energy. In the elastic beam problem (5), the initial potential is given by:

P\left(u,v\right)=\frac{1}{2}{\displaystyle \int \left(\mathit{EI}\phantom{\rule{0.25em}{0ex}}{K}^{2}+\mathit{ES}\phantom{\rule{0.25em}{0ex}}{\gamma}^{2}+c{v}^{2}+\frac{{c}_{3}{v}^{4}}{2}\right)}\phantom{\rule{0.25em}{0ex}}\mathit{dX},\phantom{\rule{2.5em}{0ex}}K=\frac{{d}^{2}v}{d{X}^{2}}.

(11)

As a consequence of the assumptions of slowly varying envelopes, these envelopes are assumed to be constant on each period so that only terms corresponding to the harmonic zero of *K*^{2}*,γ*^{2}*,v*^{2} and *v*^{4}contribute to the approximated values of the potential energy. According to Parseval formula, the potential energy of the macroscopic model can be written as (the contribution of *v*^{4} is omitted here for simplicity, full reduced models can be found in [40]):

\begin{array}{l}P\left({u}_{j},{v}_{j}\right)=\frac{1}{2}{\displaystyle \int \left(\mathit{EI}\phantom{\rule{0.5em}{0ex}}\left(\phantom{\rule{0.25em}{0ex}}{K}_{0}^{2}+2{\displaystyle \sum _{j=1}^{\infty}{\left|{K}_{j}\right|}^{2}}\right)+\mathit{ES}\phantom{\rule{0.25em}{0ex}}\left(\phantom{\rule{0.25em}{0ex}}{\gamma}_{0}^{2}+2{\displaystyle \sum _{j=1}^{\infty}{\left|{\gamma}_{j}\right|}^{2}}\right)+c\left(\phantom{\rule{0.25em}{0ex}}{v}_{0}^{2}+2{\displaystyle \sum _{j=1}^{\infty}{\left|{v}_{j}\right|}^{2}}\right)+\dots \right)}\phantom{\rule{0.25em}{0ex}}\mathit{dX}\\ \phantom{\rule{3.75em}{0ex}}\approx \frac{1}{2}{\displaystyle \int \left(\mathit{EI}\phantom{\rule{0.5em}{0ex}}\left(\phantom{\rule{0.25em}{0ex}}{K}_{0}^{2}+2{\left|{K}_{1}\right|}^{2}\right)+\mathit{ES}\phantom{\rule{0.25em}{0ex}}\left(\phantom{\rule{0.25em}{0ex}}{\gamma}_{0}^{2}+2{\left|{\gamma}_{1}\right|}^{2}\right)+c\phantom{\rule{0.25em}{0ex}}\left(\phantom{\rule{0.25em}{0ex}}{v}_{0}^{2}+2{\left|{v}_{1}\right|}^{2}\right)+\dots \right)\phantom{\rule{0.25em}{0ex}}}\mathit{dX}.\end{array}

(12)

Further simplifications can be introduced in this model if one looks only at local bending instabilities from one-dimensional elastic states. In this respect, we can drop the mean deflection *v*_{
0
}*(X)* and the envelope of the axial displacement *u*_{
1
}*(X)*. In this framework, the curvature and membrane strains of the wrinkled beam are approximated by (see (7) (9) and suppress the imaginary terms):

\left\{\begin{array}{cc}\hfill {K}_{0}=0\hfill & \hfill {\left|{K}_{1}\right|}^{2}={\left(\frac{{d}^{2}{v}_{1}}{d{X}^{2}}-{Q}^{2}{v}_{1}\right)}^{2}+4{\left(\frac{d{v}_{1}}{\mathit{dX}}\right)}^{2}\hfill \\ \hfill {\gamma}_{0}=\frac{d{u}_{0}}{\mathit{dX}}+{\left(\frac{d{v}_{1}}{\mathit{dX}}\right)}^{2}+{Q}^{2}{v}_{1}^{2}\hfill & \hfill {\gamma}_{1}=0.\hfill \end{array}\right.

(13)

A last approximation can be introduced by disregarding \frac{{d}^{2}{v}_{1}}{d{X}^{2}} in the potential energy. This latter approximation has been done in previous works [40] and it is justified by the slow spatial variations of the envelope. Finally, the potential energy can be approximated in the following form:

P\left({u}_{0},{v}_{1}\right)={\displaystyle \int \left(6\mathit{EI}{Q}^{2}{\left(\frac{d{v}_{1}}{\mathit{dX}}\right)}^{2}\phantom{\rule{0.5em}{0ex}}+\mathit{EI}{Q}^{4}{v}_{1}^{2}+\frac{\mathit{ES}}{2}\phantom{\rule{0.25em}{0ex}}{\left(\frac{d{u}_{0}}{\mathit{dX}}+{\left(\frac{d{v}_{1}}{\mathit{dX}}\right)}^{2}+{Q}^{2}{\mathrm{v}}_{1}^{2}\right)}^{2}+c{v}_{1}^{2}+3{\mathrm{c}}_{3}{v}_{1}^{4}/2\right)}\phantom{\rule{0.25em}{0ex}}\mathit{dX}

(14)

Thus the extrema of the macroscopic energy (14) are solutions of the following system:

#### Comments

Hence the membrane law (15-b) accounts for the wrinkling oscillation *v*_{
1
}*(X)* that is governed by a sort of bifurcation Equation (15-c) looking like a Ginzburg-Landau equation. The full model is a nonlinear system coupling membrane behavior and evolution of wrinkles. It has been established [38, 40] that the Fourier approach generalizes the asymptotic Ginzburg- Landau method, but it is not limited to the neighborhood of the bifurcation.

This modeling by few Fourier envelopes can be applied with several levels of approximation. The system (15) is the simplest possible with an internal length. It can still be simplified by neglecting the derivatives of *v*_{1} in (15-c): this leads to a nonlinear relation between membrane stress and strain, see [40] § 4.4. This pure membrane model will be extended in the 2D case in the following. It is also possible to include more harmonics, see [40] § 3.2, with obviously the cost of a greater complexity. The model with a complex envelope *v*_{1}*(X)* has been evaluated in [62]: it permits for instance a better account of the phase in the bulk and improves the response near the boundary.

The previous approach can be considered as a multi-scale method or a computational homogenization technique. The account of the local behavior is very simplified by the assumption of a harmonic local variation and it is described only by the instability wavenumber *Q*. If the Equation (15-c) is reduced to its linear version without modulation of the envelope, ones recovers the classical approach (2) (3) that gives the wavenumber at the first bifurcation by minimizing the critical load. When solving a nonlinear macroscopic problem such as (15), the wavenumber *Q* has to be prescribed and this could be considered as a weak point of the present macroscopic approach. Nevertheless it is known that, in a cellular bifurcation problem, many solutions can exist [45, 46, 63], each one being characterized by its wavelength. Furthermore a model with a complex envelope permits to predict a wavelength slightly deviating from the one a priori prescribed [62]. Higher order harmonics could also be accounted: for instance, it was established in [40] that a rough account of the second harmonic is sometimes necessary to recover a consistent post-bifurcation behavior. But this will be not necessary in the case of membrane wrinkling.

In the next parts, the interaction membrane-wrinkling will be modelised in a bi-dimensional case within a framework similar to (14) (15), the starting model being the Föppl-Von Karman equations.

### Föppl-Von Karman plate equations as a microscopic model

The main objective of the paper is to obtain nonlinear membrane models where the wrinkling is described by a bifurcation equation deduced from the initial plate model. The method of Fourier series with slowly variable coefficients is applied to deduce the sought model from the well known Föppl-Von Karman equations for elastic isotropic plates that will be considered here as the reference model:

\left\{\begin{array}{c}\hfill D{\text{\Delta}}^{2}w-\mathit{div}\left(\mathbf{N}\mathbf{.}\nabla w\right)=0\hfill \\ \hfill \mathbf{N}={\mathbf{L}}^{\mathbf{m}}\mathbf{.}\mathbf{\gamma}\hfill \\ \hfill 2\mathbf{\gamma}=\nabla \mathbf{u}+{}^{t}\nabla \mathbf{u}+\nabla w\otimes \nabla w\hfill \\ \hfill \mathit{div}\mathbf{N}=0\hfill \end{array}\right.

(16)

where **u** = (u,v) ∈ IR^{2} is the in-plane displacement, w is the deflection, **N** and **γ** are the membrane stress and strain. With the vectorial notations (**N**→^{t}(N_{X}N_{Y}N_{XY}), γ→^{t}(γ_{X}γ_{Y}2γ_{XY})), the membrane elasticity tensor is represented by the matrix \frac{\mathit{Eh}}{1-{\nu}^{2}}\left[\begin{array}{ccc}\hfill 1\hfill & \hfill \nu \hfill & \hfill 0\hfill \\ \hfill \nu \hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1-\nu}{2}\hfill \end{array}\right]. The corresponding energy E can be split into a membrane part *E*_{
mem
} and a bending part *E*_{
ben
}, as follows:

\left\{\begin{array}{c}\hfill E\left(\mathbf{u},w\right)={E}_{\mathit{ben}}\left(w\right)+{E}_{\mathit{mem}}\left(\mathbf{u},w\right)\hfill \\ \hfill 2{E}_{\mathit{ben}}\left(w\right)=D{\displaystyle \int \int \left({\left(\mathit{\Delta w}\right)}^{2}-2\left(1-\nu \right)\left(\frac{{\partial}^{2}w}{\partial {X}^{2}}\frac{{\partial}^{2}w}{\partial {Y}^{2}}-{\left(\frac{{\partial}^{2}w}{\partial X\partial Y}\right)}^{2}\right)\right)\mathit{d\omega}}\hfill \\ \hfill 2{E}_{\mathit{mem}}\left(\mathbf{u},w\right)={\displaystyle \int \int {}^{\mathbf{t}}\mathbf{\gamma}\mathbf{.}{\mathbf{L}}^{m}\mathbf{.}\mathbf{\gamma}\mathit{dw}=\frac{\mathit{Eh}}{1-{\nu}^{2}}{\displaystyle \int \int \left({\gamma}_{X}^{2}+{\gamma}_{Y}^{2}+2\left(1-\nu \right){\gamma}_{\mathit{XY}}^{2}+2\nu {\gamma}_{X}{\gamma}_{Y}\right)\mathit{d\omega}.}}\hfill \end{array}\right.

(17)

### Macroscopic modeling of membrane strain

As explained previously, the unknown fields U*(X,Y)* are expressed in terms of two harmonics: the mean field U_{0}*(X,Y)* and the first order harmonicsr, {\mathbf{U}}_{1}\left(X,Y\right){e}^{\mathit{iQX}}, {\overline{\mathbf{U}}}_{1}\left(X,Y\right){e}^{-\mathit{iQX}}. The second harmonic should be taken into account to recover the results of the asymptotic Ginzburg-Landau bifurcation approach, see [40]. Nevertheless the second harmonic does not contribute to the membrane energy in the present case, because the rapid one-dimensional oscillations *e*^{iQX} are inextensional so that **N**_{2} = 0, *w*_{2} = 0. Hence the second harmonic does not influence the simplest macroscopic models. For simplicity, the details of this calculation are omitted.

A unique direction *OX* for the wrinkling oscillations is chosen in the whole domain. Of course this assumption is a bit restrictive and should be removed in the future. A true multi-scale approach should include two levels of modelisation as in the FE2 method [64], but with a basic cell that is not a priori known [65], what should require a rather intricate management. Thus a more realistic goal is to first discuss the multi-scale approach with a given wrinkling wavelength and a given direction of the wrinkles.

The derivation rules (7) (8) can be extended easily in a bi-dimensional framework. For instance, the first Fourier coefficient of the gradient and the 0^{th} order coefficient of the strain (i.e. its mean value on a period) are given by:

\left\{{\left(\nabla w\right)}_{1}\right\}=\left\{\begin{array}{c}\hfill \frac{\partial {w}_{1}}{\partial X}+\mathit{iQ}{w}_{1}\hfill \\ \hfill \frac{\partial {w}_{1}}{\partial Y}\hfill \end{array}\right\}

(18)

\begin{array}{rl}\left\{{\gamma}_{0}\right\}& =\left\{\begin{array}{c}\hfill {{\gamma}_{X}}_{0}\hfill \\ \hfill {{\gamma}_{Y}}_{0}\hfill \\ \hfill 2{{\gamma}_{\mathit{XY}}}_{0}\hfill \end{array}\right\}=\left\{{\gamma}^{\mathit{FK}}\right\}+\left\{{\gamma}^{\mathit{wr}}\right\}\\ \left\{{\gamma}^{\mathit{FK}}\right\}& =\left\{\begin{array}{c}\hfill \frac{\partial {u}_{0}}{\partial X}+\frac{1}{2}{\left(\frac{\partial {w}_{0}}{\partial X}\right)}^{2}\hfill \\ \hfill \frac{\partial {v}_{0}}{\partial Y}+\frac{1}{2}{\left(\frac{\partial {w}_{0}}{\partial Y}\right)}^{2}\hfill \\ \hfill \frac{\partial {u}_{0}}{\partial Y}+\frac{\partial {v}_{0}}{\partial X}+\frac{\partial {w}_{0}}{\partial X}\frac{\partial {w}_{0}}{\partial Y}\hfill \end{array}\right\}\end{array}

(19)

\left\{{\gamma}^{\mathit{wr}}\right\}=\left\{\begin{array}{c}\hfill {\left|\frac{\partial {w}_{1}}{\partial X}+\mathit{iQ}{w}_{1}\right|}^{2}\hfill \\ \hfill {\left|\frac{\partial {w}_{1}}{\partial Y}\right|}^{2}\hfill \\ \hfill \left(\frac{\partial {w}_{1}}{\partial X}+\mathit{iQ}{w}_{1}\right)\frac{\partial {\overline{w}}_{1}}{\partial Y}+\left(\frac{\partial {\overline{w}}_{1}}{\partial X}-\mathit{iQ}{\overline{w}}_{1}\right)\frac{\partial {w}_{1}}{\partial Y}\hfill \end{array}\right\}

(20)

As in formula (10), the strain is split into a classical part *γ*^{FK}that has the same form as the original Föppl-Von Karman model (16) and a wrinkling part *γ*^{wr} which depends only on the envelope of the deflection *w*_{1}.

Let us now turn to a simplified version of the displacement–strain law (19) (20), in the same spirit as for the 1D model (15). First the displacement field is reduced to a membrane mean displacement and to a bending wrinkling, i.e. **u**_{1} = 0, *w*_{0} = 0. This means that we account for the influence of the local buckling on the membrane behavior, but not for the coupling between local and global buckling as in [66, 67]. Second the deflection envelope *w*_{1}(*X*, *Y*) is assumed to be real, which disregards the phase modulation of the wrinkling pattern. Thus the envelope of the displacement has only three components **u**_{0} = (*u*_{0}, *v*_{0}) and *w*_{1} that will be rewritten for simplicity as \left(u,v,w\right)\stackrel{\mathit{def}}{=}\left({u}_{0},{v}_{0},{w}_{1}\right). The simplified version of the strain field becomes:

\left\{\gamma \right\}\stackrel{\mathit{def}}{=}\left\{{\gamma}_{0}\right\}=\left\{\u03f5\left(\mathbf{u}\right)\right\}+\left\{{\gamma}^{\mathit{wr}}\right\}

(21)

\left\{\u03f5\left(\mathbf{u}\right)\right\}=\left\{\begin{array}{c}\hfill \frac{\partial u}{\partial X}\hfill \\ \hfill \frac{\partial v}{\partial Y}\hfill \\ \hfill \frac{\partial u}{\partial Y}+\frac{\partial v}{\partial X}\hfill \end{array}\right\},\phantom{\rule{3em}{0ex}}\left\{{\gamma}^{\mathit{wr}}\right\}=\left\{\begin{array}{c}\hfill {\left(\frac{\partial w}{\partial X}\right)}^{2}+{Q}^{2}{w}^{2}\hfill \\ \hfill {\left(\frac{\partial w}{\partial Y}\right)}^{2}\hfill \\ \hfill 2\frac{\partial w}{\partial X}\frac{\partial w}{\partial Y}\hfill \end{array}\right\}

(22)

The simplified membrane strain formula (21) (22) is quite similar to that of the initial Von Karman model. It is split, first in a linear part *ϵ*(**u**) that is the symmetric part of the mean displacement gradient corresponding to the pure membrane linear strain, second in a nonlinear part *γ*^{wr}(*w*) more or less equivalent to wrinkling stain of [23]. The main difference with the initial Föppl-Von Karman strain (16) is the extension *Q*^{2}*w*^{2} in the direction of the wrinkles. This wrinkling strain is always positive and this corresponds to a stretching. In the case of a compressive membrane strain, this wrinkling term leads to a decrease of the true strain.

As for the 1D model (14), we limit ourselves to the 0^{th} order harmonic to compute the reduced membrane energy:

2{\mathrm{E}}_{\mathit{mem}}\left(\mathbf{u},w\right)=\frac{\mathit{Eh}}{1-{\nu}^{2}}{\displaystyle \int \int \left\{\begin{array}{l}{\left(\frac{\partial u}{\partial X}+{\left(\frac{\partial w}{\partial X}\right)}^{2}+{Q}^{2}{w}^{2}\right)}^{2}+{\left(\frac{\partial v}{\partial Y}+{\left(\frac{\partial w}{\partial Y}\right)}^{2}\right)}^{2}+\\ 2\left(1-\nu \right){\left(\frac{1}{2}\left(\frac{\partial u}{\partial X}+\frac{\partial v}{\partial Y}\right)+\frac{\partial w}{\partial X}\frac{\partial w}{\partial Y}\right)}^{2}+\\ 2\nu \left(\frac{\partial u}{\partial X}+{\left(\frac{\partial w}{\partial X}\right)}^{2}+{Q}^{2}{w}^{2}\right)\left(\frac{\partial v}{\partial Y}+{\left(\frac{\partial w}{\partial Y}\right)}^{2}\right)\end{array}\right\}}\mathit{d\omega}.

(23)

### Macroscopic bending energy

We saw two possible ways to get a reduced-order model via the technique of Fourier series with slowly variable coefficients: either identify the Fourier coefficient as in (9), or simplify the energy by keeping only the 0th order term in the energy, as in (12). Here we shall prefer the second approach which permits to provide a formulation easier to be managed for the numerical discretisation. The computation of the energy is based on the fact that only the 0th order harmonic *φ*_{0} of a function *φ* has a non zero mean value:

{\displaystyle \underset{\omega}{\int \int}\phi}\mathit{d\omega}={\displaystyle \underset{\omega}{\int \int}{\phi}_{0}}\mathit{d\omega}.

(24)

The identity (24) is applied to the two terms of the bending energy in the same framework as in (15) (21) (22), i.e. **u**_{1} = (*u*_{1}, *v*_{1}) = (0, 0), *w*_{0} = 0, *w*_{1} ∈ *IR*:

\phi ={\left(\mathit{\Delta w}\right)}^{2}-2\left(1-\nu \right)\left[\frac{{\partial}^{2}w}{\partial {X}^{2}}\frac{{\partial}^{2}w}{\partial {Y}^{2}}-{\left(\frac{{\partial}^{2}w}{\partial X\partial Y}\right)}^{2}\right]={\phi}^{A}-2\left(1-\nu \right){\phi}^{B}

\begin{array}{ll}{\phi}_{0}^{A}& ={\left(\mathit{\Delta w}\right)}^{2}={\displaystyle \sum _{n=-\infty}^{+\infty}{\left(\mathit{\Delta w}\right)}_{n}{\left(\mathit{\Delta w}\right)}_{-n}=}2{\left(\mathit{\Delta w}\right)}_{1}{\left(\mathit{\Delta w}\right)}_{-1}=2{\left|{\left(\mathit{\Delta w}\right)}_{1}\right|}^{2}\\ =2{\left|\Delta {w}_{1}-{Q}^{2}{w}_{1}+2\mathit{iQ}\frac{\partial {w}_{1}}{\partial X}\right|}^{2}.\end{array}

Because of the assumption of a real envelope w = w_{1}, the first term of the bending energy is obtained:

{\phi}_{0}^{A}=2{\left(\mathit{\Delta w}-{Q}^{2}w\right)}^{2}+8{Q}^{2}{\left(\frac{\partial w}{\partial X}\right)}^{2}.

(25)

The second term of the bending energy {\phi}_{0}^{B} is computed in the same way:

{\phi}_{0}^{B}=2\left(\frac{{\partial}^{2}w}{\partial {X}^{2}}-{Q}^{2}w\right)\frac{{\partial}^{2}w}{\partial {Y}^{2}}-2{\left(\frac{{\partial}^{2}w}{\partial X\partial Y}\right)}^{2}-2{Q}^{2}{\left(\frac{\partial w}{\partial Y}\right)}^{2}.

(26)

To be consistent with the approach in the previous 1D case, the derivatives of order three or four in the differential equations are neglected. This has been justified in [39]: indeed spurious oscillations can appear in the response of the macroscopic model if these high order derivatives are kept. Finally the bending energy in the simplified framework is reduced as:

{\mathrm{E}}_{\mathit{ben}}\left(w\right)=D{\displaystyle \int \int \left\{{Q}^{4}{w}^{2}-2{Q}^{2}\mathit{w\Delta w}+4{Q}^{2}{\left(\frac{\partial w}{\partial X}\right)}^{2}+2\left(1-{\nu}^{2}\right){Q}^{2}\left[w\frac{{\partial}^{2}w}{\partial {Y}^{2}}+{\left(\frac{\partial w}{\partial Y}\right)}^{2}\right]\right\}}\mathit{d\omega}.

(27)

### Three macroscopic membrane wrinkling models

In this section, one establishes the Partial Differential Equations of the macroscopic models associated with the strain energies previously presented. In fact, we have not defined a closed model, but a family of models that can depend on the number of harmonics and various other assumptions, such as the assumption of a real wrinkling envelope or the hypothesis w_{0} = 0, which means no bending before wrinkling. Three cases will be considered. First the model presented in [33] couples a linear membrane behavior with a real envelope governed by a sort of Ginzburg-Landau equation. It corresponds to (15) in the 1D case and can be considered as a reference model because it is simple and able to account for the influence of wrinkling on membrane behavior. Next a pure membrane model will be derived by neglecting the dependence with respect to the derivatives of the envelopes. Last we shortly describe a model with a complex envelope, as the one studied in [62] in the 1D case that is a relatively simple improvement of the reference model.

#### A reference macroscopic membrane model

The first proposed macroscopic membrane model follows from the minimum of the potential energy. In the absence of body forces, the sum of the membrane energy (23) and of the bending energy (27) is stationary at equilibrium. Let us recall that these energies associate 0th order harmonic for membrane quantities and a real first order harmonics for the deflection, as in the 1D model (15). In other words, it permits to couple a spatially modulated wrinkling with a linear membrane model:

\delta {\mathrm{E}}_{\mathit{ben}}+\delta {\mathrm{E}}_{\mathit{mem}}=0,

for any virtual displacement that is zero at the boundary. This gives:

\delta {E}_{\mathrm{ben}}+{\displaystyle \underset{\omega}{\iint}\mathbf{N}:\delta {\mathbf{\gamma}}^{\mathit{wr}}\mathit{d\omega}}=0

(28)

{\displaystyle \underset{\omega}{\iint}\mathbf{N}:\delta \mathbf{\u03f5}\mathit{d\omega}}=0.

(29)

After straightforward calculations, one obtains the partial differential equations of the macroscopic problems in the following form, the wrinkling membrane strain *γ*^{wr}(*w*) being given in (22):

\mathit{div}\mathbf{N}=0

(30)

\mathbf{N}={\mathbf{L}}^{m}:\left[\mathbf{\u03f5}\left(u\right)+{\mathbf{\gamma}}^{\mathit{wr}}\left(w\right)\right]

(31)

-6D{Q}^{2}\frac{{\partial}^{2}w}{\partial {X}^{2}}-2D{Q}^{2}\frac{{\partial}^{2}w}{\partial {Y}^{2}}+\left(D{Q}^{4}+{N}_{X}{Q}^{2}\right)w-\mathit{div}\left(\mathbf{N}.\mathbf{\nabla}w\right)=0.

(32)

The nonlinear model (30) (31) (32) couples nonlinear membrane equations with a bifurcation Equation (32) satisfied by the envelope of wrinkling pattern. It extends the previous analysis of the 1D case that couples a beam membrane with a one-dimensional Ginzburg-Landau equation. Hence the bifurcation Equation (32) is a sort of bi-dimensional Ginzburg-Landau equation, but it differs from the amplitude equation of Segel-Newell-Whitehead [42, 43] who consider cases where the pre-bifurcation state is invariant under rotation. We shall see that finite element discretisation of (30) (31) (31) is straightforward. Two analytical solutions will be also presented in the section “Two analytical solutions for clamped rectangular membranes”.

Within the nonlinear model (30) (31) (32), one recovers two ideas of classical macroscopic membrane models. First the splitting between membrane and wrinkling strain of Roddeman theory [23] has been deduced from Föppl-Von Karman Equation, see (31), without any phenomenological assumption. Then the final bifurcation Equation (32) includes an internal length, which permits to retrieve the multi-scale instability analysis of Part 2. Finally this model can be qualitatively compared with the models of Banerjee et al. [29, 30], where an internal length is introduced via Cosserat theory. Body forces and boundary forces can be introduced easily by the same procedure. This requires that these forces can also be put in the form of Fourier series with slowly variable coefficients. For instance if these forces vary slowly at the scale of the wrinkles, this leads to a classical body force in the membrane Equation (30), as well as in the corresponding boundary conditions. In the same way, transverse forces can also be accounted for within the more complete model entitled “A more sophisticated macroscopic membrane model with a complex envelope”.

#### A pure membrane model

The reference membrane model (30) (31) (32) is not a pure membrane model because the Equation (32) includes spatial derivatives of the envelope of the wrinkles. It is natural to try to recover a pure membrane model, where the kinematic unknown is only the in-plane displacement, as suggested in [40], § 4.4. By dropping all the spatial derivatives in (32), one gets a bifurcation equation (*N*_{
X
} + *DQ*^{2})*w* = 0 that can be transformed in a perturbed bifurcation equation as:

\left({N}_{X}+D{Q}^{2}\right)w=\delta

(33)

From (33), one can obtain the deflection as a function of one component of the membrane stress. In (33), *δ* is a small perturbation parameter that transforms the perfect bifurcation equation into a perturbed bifurcation, what is more convenient for numerical path following calculations. If one simplifies the wrinkling strain (22) γ^{wr}(w) = Q^{2}w^{2}e_{x} ⊗ e_{x} and if one combines (31) and (33), one can drop the deflection and deduce a nonlinear relation between membrane strain **ϵ**(**u**) and membrane stress **N**. With account of the balance of membrane forces, the obtained full model is restricted to:

\left\{\begin{array}{l}\mathit{div}\mathbf{N}=0\phantom{\rule{13.25em}{0ex}}\hfill \\ \mathbf{\u03f5}\left(u\right)+\frac{{Q}^{2}{\delta}^{2}}{{\left({N}_{X}+D{Q}^{2}\right)}^{2}}{\mathbf{e}}_{X}\otimes {\mathbf{e}}_{X}={\left({\mathbf{L}}^{m}\right)}^{-1}:\mathbf{N}.\hfill \end{array}\right.

(34)

The model (34) is consistent with the pure membrane theories of the literature, see for instance [23], because *N*_{
X
} cannot be lower than the wrinkling stress − *DQ*^{2}.Generally, this wrinkling threshold is approximated by zero so that the membrane does not admit compressive stresses. In sections “Two analytical solutions for clamped rectangular membranes” and “Two numerical solutions”, this model will be compared with our reference model (30) (31) (32).

#### A more sophisticated macroscopic membrane model with a complex envelope

More sophisticated models can be introduced in a rather easy way. For instance models with five envelopes (harmonics 0, ±1, ±2) have been presented in [39, 40], starting models being respectively 2D hyperelasticity and the beam model of section The method of Fourier series with slowly variable coefficients. The reference model of section A reference macroscopic membrane model can be improved at least in two ways. First one can reintroduce the mean deflection *w*_{0} to account for a coupling between local wrinkling and global buckling, as in [66]: in this manner, one will associate an envelope equation with the full Föppl-Von Karman plate equations. Moreover, this envelope can be a complex one for a better account of the phase field and of the boundary behavior [62]. To keep a relatively simple model, a more questionable hypothesis will be done: the membrane behavior will be described only by the 0^{th} order harmonic, i.e. **u**_{1} = 0, **N**_{1} = 0, **γ**_{1} = 0. Hence the remaining unknown fields will be:

\mathbf{u}\stackrel{\mathit{def}}{=}{\mathbf{u}}_{0}\mathbf{,}\phantom{\rule{0.5em}{0ex}}\mathbf{N}\stackrel{\mathit{def}}{=}{\mathbf{N}}_{0}\mathbf{,}\mathbf{\gamma}\stackrel{\mathit{def}}{=}{\mathbf{\gamma}}_{0}\mathbf{,}{w}_{0}\in \mathit{IR},{w}_{1}\in C,

(35)

i.e. the same variables as in the basic Föppl-Von Karman equations completed by the complex envelope *w*_{1} of the wrinkles. This leads to the following system:

\mathit{div}\mathbf{N}=0

(36)

\mathbf{N}={\mathbf{L}}^{m}:\left[\mathbf{\u03f5}\left(u\right)+\frac{1}{2}\nabla {w}_{0}\otimes \nabla {w}_{0}+{\mathbf{\gamma}}^{\mathit{wr}}\left({w}_{1}\right)\right]

(37)

D{\Delta}^{2}{w}_{0}-\mathit{div}\left(\mathbf{N}\mathbf{.}\nabla {w}_{0}\right)=0

(38)

-6D{Q}^{2}\frac{{\partial}^{2}{w}_{1}}{\partial {X}^{2}}-2D{Q}^{2}\frac{{\partial}^{2}{w}_{1}}{\partial {Y}^{2}}+\left(D{Q}^{4}+{N}_{X}{Q}^{2}\right){w}_{1}-\mathit{iQ}\left(\left(\mathbf{N}.{e}_{X}\right).\mathbf{\nabla}{w}_{1}+\mathit{div}\left({w}_{1}\mathbf{N}.{e}_{X}\right)\right)-\mathit{div}\left(\mathbf{N}.\mathbf{\nabla}{w}_{1}\right)=0.

(39)

This system is only presented as an example of the multi-scale procedure and it will not be discussed further in this paper.