Membrane wrinkling revisited from a multi-scale point of view

This paper deals with macroscopic modeling of very thin shells, also called membranes in the scientific literature and in everyday life. Membranes have increasing application fields such as spacecraft structures (antenna, telescope lenses, Gossamer structures…), civil structures, life jackets, electronics, biological tissues, textile composites [1–6]. The appearance of wrinkles is a major feature in the mechanical behavior of membrane, due to their vanishing bending stiffness.

Membrane wrinkling is a multi-scale phenomenon, but it is generally described by one-scale models that can be “microscopic” or “macroscopic”. Microscopic models are simply based on elastic shell theory; see for instance [7–11]. Nowadays many commercial finite element codes permit to carry out such nonlinear shell computations. Shell analyses are able to describe the details of the membrane response: size, wavelength, orientation of the wrinkles, instability threshold, etc., but this leads to large scale numerical models that are moreover very difficult to be controlled in cases with a large number of wrinkles and therefore with a large number of equilibrium solutions. Indeed wrinkling can be seen as a small wavelength buckling phenomenon and, in such a case, there are several neighbor bifurcation modes and it is well known that this leads to compound response curves involving secondary bifurcations, see for instance [12, 13] among an extensive literature. Even if the complexity of wrinkling patterns requires numerical treatments, one can mention several papers by Coman et al. [14–16] that were able to find analytically wrinkling modes in cases with inhomogeneous pre-bifurcation stresses.

The second group of numerical descriptions are pure membrane models that account for the effect of wrinkling on membrane behaviour. The bending stiffness is neglected and the compressive stresses are dropped: this yields nonlinear constitutive laws that are of unilateral type, the tensile behaviour being about linear and the compressive states being forbidden. This has led to many numerical studies; see for instance [17–22]. In most of these papers, the constitutive law relates simply membrane stress and strain and it distinguishes three states: slack, wrinkled and taut. Some authors prefer the method of Roddeman [23, 24] that splits the deformation gradient into consistent membrane part and wrinkling part [25–28]. The partial differential equations deduced from these membrane models are not elliptic (or hyperbolic in the dynamical case) in the presence of a non-positive principal stress so that this problem is mathematically ill-posed. A well-posed problem can be obtained if the macroscopic model includes an internal length, for instance within Cosserat theory [29, 30]. However this regularisation may be not necessary for an explicit dynamic computation. All these membrane models have a phenomenological character, the only mention to full shell models being the vanishing of the membrane compressive stress. A variant has been introduced by [31] and used for buckling problems of sheets under residual stresses generated by manufacturing processes [32].

The aim of this paper is to revisit membrane modelling from a multi-scale point of view. It will be clearly established that a consistent wrinkling model depends both on microscopic quantities (wavelength, orientation of the wrinkles…) and macroscopic data like boundary conditions, size and shape of the structure… In this respect, a new multi-scale membrane model was briefly presented [33] that couple a nonlinear 2D membrane model with an amplitude equation governing the evolution of wrinkles. Amplitude equations to describe spatial pattern formation follow generally from the asymptotic bifurcation analyses of Ginzburg-Landau type [34–37]. Here a slightly different method will be applied, where the nearly periodic fields are represented by Fourier series with slowly varying coefficients [38–41]. In other terms, we use a multi-scale modelling method whose result is a generalised continuum including an internal length and where the macroscopic stresses are Fourier coefficients of the microscopic stress.

In this paper, the new membrane model of [33] is discussed in detail. Some additional variants, several analytical and numerical solutions and the corresponding numerical schemes will be presented. The new membrane models including wrinkling will be deduced from Föppl-Von Karman plate theory. These analyses will be carried out by applying standard techniques of multi-scale bifurcation analyses: asymptotic multi-scale method for the linear bifurcation analysis and multi-scale Fourier approach leading to nonlinear models valid away from the wrinkling threshold. Note that, contrarily to most of macroscopic membrane models, the present models are deduced from the shell equations without any phenomenological assumptions. The bending stiffness effects are included not only to define the wrinkling wavelength, but also to predict the macroscopic evolution of the buckling pattern.

The paper is organized as follows. In section “About the origin of short wavelength instabilities”, the origin of the wrinkling instability is briefly recalled, in order to underline its multi-scale origin and the generic character of this instability mechanism. section “Multi-scale nonlinear models for membrane wrinkling” is devoted to deducing new nonlinear wrinkling models from Föppl-Von Karman plate theory. Then two analytical solutions of these models will be presented (section “Two analytical solutions for clamped rectangular membranes”), the first one for defining the bifurcation features (bifurcation stress, wavelength and shape of the envelope) and the second one for a post-bifurcation analysis characterizing the evolution of the size of the wrinkles as a function of the applied load. The numerical analysis is presented in section “Numerical implementation” (techniques of implementation) and section Two numerical solutions where two examples of plate wrinkling will be solved. Finally, in section “Linear wrinkling analysis revisited by a double scale asymptotic analysis”, a more classical asymptotic multi-scale analysis will be performed in a special case and the results will be compared with those arising from the multi-scale Fourier approach.

Spatially periodic instability patterns are encountered in a lot of physical problems. The instability often occurs in domains whose dimension is smaller than the others, typically in rectangles or parallelepipeds with a small aspect ratio. The instability wavelength can be of the same order as the smallest dimension of the domain as in Rayleigh-Bénard convection [42, 43].

A physical example of such instability is the symmetric microbuckling mode of long fiber composite materials that are represented by an infinite stack of hard and soft layers (fiber, matrix), see Figure 1 and Rosen [51].

In this problem, the coefficients of (1) are related to Young moduli and thickness’s of the layers ($A\cong E_F{h}_F^3$, B ≅ E _M /h _M ), the microbuckling wavelength is related to the width of the fiber and of the matrix by the following formula: $\ell \cong h_F^{3/4}{h}_M^{1/4}{\left(E_F/E_M\right)}^{1/4}$. Quite similar wrinkling phenomena with a small wavelength can be found in studies about thin films coupled with compliant substrate, see for instance [52–55], or in the buckling of sheets due to residual stresses and forming processes [56, 57]. In this second class of instability problems, the wavelength is larger than the microscopic lengths h _F , h _M because of a large stiffness ratio. Here is the origin of the multi-scale behavior: the characteristic distance in the width is smaller than the longitudinal wavelength and moreover these longitudinal waves are generally modulated on larger distances.

The instability modes are in the form V(Y)cos(QX+φ) and their instability wavelength 2π/Q is generally small with respect to the characteristic length in the transverse direction. The Equations (1) and (4) can be related by assuming that the coefficient B is an eigenvalue of the differential operator − C∂²/∂Y², or equivalently $B=C/L_Y^2$, where L _Y is a macroscopic characteristic length in the transverse direction.

In the problem of a plane membrane of thickness h submitted to a tensile stress σ _Y in the transverse direction, A≅Eh ³ , C = σ _Y h and σ _X  = − λ/h is a compressive stress in the X-direction so that the wrinkling wavelength is ${\ell }_{\mathrm{x}}\cong {\left(E/{\sigma }_Y\right)}^{1/4}{h}^{1/2}{L}_Y^{1/2}$[3]. Hence this wavelength is much smaller than the transverse length L _Y because a membrane is very thin (h < <L_Y) and it decreases when the tensile stress increases. From Equation (3), one can deduce the critical compressive stress $\left|{\sigma }_X\right|\cong \sqrt{E{\sigma }_Y}h/L_Y$ that is much smaller than the prescribed tensile stress σ_y if this stress σ_y is not too small. It is worth to mention that the wrinkling stress depends both on the microscopic length h and on the macroscopic transverse one L_Y.

Note that the Equation (4) seems to be more or less generic. It has also been established in the antisymmetric microbuckling of long fiber composite [51, 58], with A≅E _F h_F³, C≅E _M h _F (this antisymmetrical mode occurs when the thickness of the fiber and of the matrix are of the same order h _M ≅h _F ). Hence the instability wavelength is ${\ell }_x\cong {\left(\frac{E_F}{E_M}\right)}^{1/4}{h}_F^{1/2}{L}_Y^{1/2}$ where the transverse length L _Y can be associated with the composite plate thickness or to the ply thickness [58]. Note that the nonlinear versions of these fiber microbuckling models permit to predict the compression failure of long fiber composite materials [59–61].

Hence a wrinkling mode follows from two stiffness’s: first the bending stiffness A, that is very small for a thin membrane or a fiber in a composite material, second a transverse stiffness C that can be purely elastic in the antisymmetrical microbuckling or the so called geometric stiffness due to the tensile stress in the wrinkling of a membrane. Clearly, all these instability problems involve several length scales. That is why we propose several multi-scale approaches to analyze the behavior of membranes in the presence of wrinkling.

A family of nonlinear membrane models is presented that is based on a multi-scale approach. Unlike macroscopic models of the literature, there is no phenomenological assumption and the models are completely deduced from the full plate model. The deduction method relies on the concept of Fourier series with slowly varying coefficients [38, 40], whose principle is to work with envelopes of the spatial oscillations. These envelopes are solutions of systems of nonlinear partial differential equations established in the next sections “Multi-scale nonlinear models for membrane wrinkling”. These equations can be solved, sometimes analytically and in general numerically, as sketched in the section “numerical implementation”.

In this Section, the reference macroscopic model (22) (30) (31) (32) is studied by seeking closed-form solutions in the case of clamped rectangular membranes: ω = [0, L _X ] × [0, L _Y ] as pictured in Figure 3. The membrane is submitted to a large uniform tensile stress N _Y  = hσ _Y  > 0 and to a small uniform compressive stress loading N _X  = hσ _X  = − λ < 0. If the plate is clamped, it is known [40] that the envelope w vanishes on the boundary. So the corresponding boundary conditions will be

$\left\{\begin{array}{c}\hfill \mathbf{N}\mathbf{.}{\mathbf{e}}_X=-\lambda {\mathbf{e}}_X\mathrm{on}\mathrm{the}\mathrm{sides}X=0,X=L_X\hfill \\ \hfill \mathbf{N}\mathbf{.}{\mathbf{e}}_Y=N_Y{\mathbf{e}}_Y\mathrm{on}\mathrm{the}\mathrm{sides}Y=0,Y=L_Y\hfill \\ \hfill w\left(X,Y\right)=0\mathrm{on}\partial \omega .\hfill \end{array}\right.$

(40)

Classically, we start with a “linear stability analysis” that is quite simple due to the stress state depending linearly on the applied force. This will establish unambiguously the multi-scale character of membrane wrinkling. Next a classical nonlinear bifurcation analysis will be done, in the same manner as for a classical post-buckling computation [68, 69]. This will provide a relation between the applied compressive stress and the size of the wrinkles.

An analytical solution for wrinkling initiation

The linearised version of the envelope Equation (32) is rewritten as

$-6D{Q}^2\frac{{\partial }^{2}w}{\partial X^2}-\left(2D{Q}^2+h{\sigma }_Y\right)\frac{{\partial }^{2}w}{\partial Y^2}+D{Q}^{4}w=h\left|{\sigma }_X\right|\left(Q^{2}w-\frac{{\partial }^{2}w}{\partial Y^2}\right).$

(41)

Since the envelope vanishes at the boundary, the smallest eigenfunction of (41) is in the form: w(X, Y) = sin(πX/L _x )sin(πY/L _Y ), as shown in Figure 4. This leads to a classical relation between compressive stress and wavenumber.

$h\left|{\sigma }_X\right|\left(Q\right)=\frac{6D{Q}^2\frac{{\pi }^2}{L_X^2}+\left(2D{Q}^2+h{\sigma }_Y\right)\frac{{\pi }^2}{L_Y^2}+D{Q}^4}{Q^2+\frac{{\pi }^2}{L_X^2}}$

(42)

Classically, the instability wavenumber Q is chosen by minimizing the stress as a function of the wavenumber. For simplicity, we take into account the orders of magnitude 1 < < QL _X ,  2DQ²/h ≈ |σ _X | < < σ _Y to simplify (42) in the following manner:

$\left|{\sigma }_X\right|\left(Q\right)=\frac{{\sigma }_Y{\pi }^2}{Q^2{L}_Y^2}+\frac{D{Q}^2}{t}$

(43)

The minimum of the latter yields values of the wavenumber and of the critical compressive stress that are consistent with the results of the literature [2, 7]:

$Q^{\mathit{wr}}=\sqrt[4]{12{\pi }^2\left(1-{\nu }^2\right)}\frac{1}{\sqrt{h{L}_Y}}\sqrt[4]{\frac{{\sigma }_Y}{E}}\approx 3.2\frac{1}{\sqrt{h{L}_Y}}\sqrt[4]{\frac{{\sigma }_Y}{E}}{\ell }^{\mathit{wr}}=\sqrt{h{L}_Y}\sqrt[4]{\frac{E}{{\sigma }_Y}}\left|{\sigma }_X^{\mathit{wr}}\right|=\frac{\pi }{\sqrt{3\left(1-{\nu }^2\right)}}\sqrt{E{\sigma }_Y}\frac{h}{L_Y}$

(44)

This simple calculation brings out the multiple scale character of the wrinkling phenomenon: indeed, the wrinkling threshold depends on the wavelength that is a microscopic quantity, but this wavelength depends of the width of the plate that is a macroscopic length. Thus a full wrinkling analysis has to associate micro and macro scales.

Recent experimental results [9] have established that the wavelength increases when the thickness h increases and decreases when the tension σ _Y increases. The scaling law $l\approx \sqrt{h}{\sigma }_Y^{-\frac{1}{4}}$ seems qualitatively consistent with experimental results.

The wrinkling stress can also be written in terms of the wavenumber by N _X  = − 2DQ², to be compared with the bifurcation load N _X  = − DQ² of the pure membrane model (34), but of course the pure membrane model is not able to predict the wavenumber.

Initial post-bifurcation analysis

Now we try to connect the applied loads and the size of the wrinkles. As in any plate buckling, the deflection is proportional to the square root of the difference between the current load and its critical value. This post-bifurcation analysis will be done by starting from the new reference membrane model (22) (30) (31) (32) in order to illustrate its ability to characterize the evolution of the wrinkles beyond the instability threshold.

The membrane model (22) (30) (31) (32) is a macroscopic one already simplified by a multi-scale analysis. Therefore there is no need to come back to a multiple scale bifurcation analysis of Ginzburg-Landau type. A standard post-bifurcation analysis as in [68, 69] will be sufficient to predict the amplitude of wrinkles.

According to [69], the solution of the symmetric bifurcation problem (22) (30) (31) (32) can be solved by seeking the unknowns and the compression load as Taylor series with respect to a scalar bifurcation parameter “a ”:

$\left\{\begin{array}{c}\hfill \mathbf{u}\left(X,Y\right)\hfill \\ \hfill \mathbf{N}\left(X,Y\right)\hfill \\ \hfill w\left(X,Y\right)\hfill \\ \hfill \lambda \hfill \end{array}\right\}=\left\{\begin{array}{c}\hfill {\mathbf{u}}^{\left(0\right)}\hfill \\ \hfill {\mathbf{N}}^{\left(0\right)}\hfill \\ \hfill 0\hfill \\ \hfill {\lambda }^{\left(0\right)}\hfill \end{array}\right\}+a\left\{\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill w^{\left(1\right)}\hfill \\ \hfill 0\hfill \end{array}\right\}+a^2\left\{\begin{array}{c}\hfill {\mathbf{u}}^{\left(2\right)}\hfill \\ \hfill {\mathbf{N}}^{\left(2\right)}\hfill \\ \hfill 0\hfill \\ \hfill {\lambda }^{\left(2\right)}\hfill \end{array}\right\}+a^3\left\{\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill w^{\left(3\right)}\hfill \\ \hfill 0\hfill \end{array}\right\}+\dots$

(45)

To avoid any ambiguity with Fourier expansion (U_i), the terms of the Taylor expansions have been denoted with a superscript U⁽ⁱ⁾. This straightforward bifurcation analysis leads to Partial Differential Equations at each order.

At order 0, the solution is a membrane state:

${\mathbf{N}}^{\left(0\right)}\left(X,Y\right)=-{\lambda }^{\left(0\right)}{\mathbf{e}}_X\otimes {\mathbf{e}}_X+N_y{\mathbf{e}}_Y\otimes {\mathbf{e}}_Y.$

(46)

By introducing the linear operator

$\mathrm{L}\left(w\right)=-6D{Q}^2\frac{{\partial }^{2}w}{\partial X^2}-2D{Q}^2\frac{{\partial }^{2}w}{\partial Y^2}+D{Q}^{4}w-N_Y\frac{{\partial }^{2}w}{\partial Y^2}.$

(47)

the PDE’s at orders 1, 2, 3 are:

$\left\{\begin{array}{l}\mathrm{L}\left(w^{\left(1\right)}\right)-{\lambda }^{\left(0\right)}{Q}^2{w}^{\left(1\right)}+{\lambda }^{\left(0\right)}\frac{{\partial }^2{w}^{\left(1\right)}}{\partial X^2}=0\\ w^{\left(1\right)}=0\mathrm{on}\partial \omega \end{array}\right.$

(48)

Of course the first order Equation (48) corresponds exactly to the bifurcation Equation (41) that has been solved previously. It is not necessary to re-discuss this point. The smallest buckling mode is

$w^{\left(1\right)}\left(X,Y\right)=sin\left(\frac{\mathit{\pi X}}{L_X}\right)sin\left(\frac{\mathit{\pi Y}}{L_Y}\right).$

(51)

Let us solve the second order problem (49). The equilibrium Equation (49-a) can be solved by introducing a stress function:

$N_X^{\left(2\right)}=\frac{{\partial }^2\phi }{\partial Y^2},N_Y^{\left(2\right)}=\frac{{\partial }^2\phi }{\partial X^2},N_{\mathit{XY}}^{\left(2\right)}=-\frac{{\partial }^2\phi }{\partial X\partial Y}.$

(52)

The membrane constitutive law (49-b) can be detailed as:

$\left\{\begin{array}{l}\frac{\partial u^{\left(2\right)}}{\partial X}+{\left(\frac{\partial w^{\left(1\right)}}{\partial X}\right)}^2+Q^2{\left(w^{\left(1\right)}\right)}^2=\frac{1}{\mathit{Eh}}\left[\frac{{\partial }^2\phi }{\partial Y^2}-\nu \frac{{\partial }^2\phi }{\partial X^2}\right]\\ \frac{\partial v^{\left(2\right)}}{\partial Y}+{\left(\frac{\partial w^{\left(1\right)}}{\partial Y}\right)}^2=\frac{1}{\mathit{Eh}}\left[\frac{{\partial }^2\phi }{\partial X^2}-\nu \frac{{\partial }^2\phi }{\partial Y^2}\right]\\ \frac{\partial u^{\left(2\right)}}{\partial Y}+\frac{\partial v^{\left(2\right)}}{\partial X}+2\frac{\partial w^{\left(1\right)}}{\partial X}\frac{\partial w^{\left(1\right)}}{\partial Y}=-2\frac{1+\nu }{\mathit{Eh}}\frac{{\partial }^2\phi }{\partial X\partial Y}.\end{array}\right.$

(53)

The in-plane displacement u⁽²⁾,v⁽²⁾ can be eliminated from the combination $\frac{{\partial }^2}{\partial Y^2}\left(53-a\right)+\frac{{\partial }^2}{\partial X^2}\left(53-b\right)-\frac{{\partial }^2}{\partial X\partial Y}\left(53-c\right)$. The resulting equation is:

$\frac{1}{\mathit{Eh}}{\Delta }^2\phi =2\left\{{\left(\frac{{\partial }^2{w}^{\left(1\right)}}{\partial X\partial Y}\right)}^2-\frac{{\partial }^2{w}^{\left(1\right)}}{\partial X^2}\frac{{\partial }^2{w}^{\left(1\right)}}{\partial Y^2}\right\}+2Q^2\left\{{\left(\frac{\partial w^{\left(1\right)}}{\partial Y}\right)}^2+w^{\left(1\right)}\frac{{\partial }^2{w}^{\left(1\right)}}{\partial Y^2}\right\}.$

(54)

It looks like the in-plane equation of the starting Föppl-Von Karman model, with a new second term in the r.h.s. due to the rapid oscillations. This second term is of the order $O\left(\frac{1}{{\ell }^2{L}^2}\right)$ while the first term of the r.h.s. involves only the macroscopic length and is $O\left(\frac{1}{L^4}\right)$. In what follows, this first term will be neglected. To account for the second order term λ⁽²⁾ in (52-c), one can introduce a new stress function ψ(X, Y) by:

$\phi \left(X,Y\right)=-{\lambda }^{\left(2\right)}\frac{Y^2}{2}+\psi \left(X,Y\right),$

so that ψ(X, Y) is solution of the following boundary value problem:

The Dirichlet boundary conditions in (55) are deduced from (49-c) (49-d) by integration along the boundary, by a quite well known calculation when dealing with a stress function.

Because of the boundary conditions, it seems difficult to get a closed form solution of (55). We propose to build an approximate equation by using a Galerkin approximation. Let us transform the domain ω = [0, L _X ] × [0, L _Y ] into a reference domain ω _ref  = [−1, + 1] × [−1, + 1] by:

$\begin{array}{l}X=\frac{1+r}{2}{L}_X,\mathrm{Y}=\frac{1+s}{2}{L}_Y\\ \Rightarrow w^{\left(1\right)}\left(r,s\right)=cos\left(\frac{\mathit{\pi r}}{2}\right)cos\left(\frac{\mathit{\pi s}}{2}\right).\end{array}$

Let us consider a shape function that satisfies the boundary conditions (55-b) (55-c) (55-d), for instance

$W\left(r,s\right)={\left(1-r^2\right)}^2{\left(1-s^2\right)}^2.$

(56)

The PDE (55-a) can be rewritten as (with $\alpha =\frac{L_Y}{L_X}$):

${\left({\alpha }^2\frac{{\partial }^2}{\partial r^2}+\frac{{\partial }^2}{\partial s^2}\right)}^2\psi ={\pi }^2\mathit{Eh}{Q}^2\frac{L_Y^2}{16}\left(1-cos\pi \left(1+r\right)\right)cos\pi \left(1+s\right).$

Its Galerkin approximated solution is:

$\begin{array}{l}\psi ={\pi }^2\mathit{Eh}{Q}^2\frac{L_Y^2}{16}\frac{I_2}{I_1}W\left(r,s\right)\\ I_1={\displaystyle \underset{{\omega }_{\mathit{ref}}}{\int \int }\left[{\left({\alpha }^2\frac{{\partial }^2}{\partial r^2}+\frac{{\partial }^2}{\partial s^2}\right)}^{2}W\right]\mathit{Wdrds}}\\ I_2={\displaystyle \underset{{\omega }_{\mathit{ref}}}{\int \int }\left[1-cos\pi \left(1+r\right)\right]cos\pi \left(1+s\right)\mathit{Wdrds}.}\end{array}$

(57)

The corresponding axial stress field is:

$N_X^{\left(2\right)}=-{\lambda }^{\left(2\right)}+\frac{{\pi }^2}{4}\mathit{Eh}{Q}^2\frac{I_2}{I_1}\frac{{\partial }^{2}W}{\partial s^2}.$

(58)

Finally we take into account the third order Equation (50-a), but we only need to write an orthogonality condition between the mode w⁽¹⁾ and the r.h.s. of (50-a):

${\displaystyle \underset{{\omega }_{\mathit{ref}}}{\int \int }\left\{N_X^{\left(2\right)}{Q}^2{\left(w^{\left(1\right)}\right)}^2-\mathit{div}\left[{\mathbf{N}}^{\left(2\right)}.\nabla w^{\left(1\right)}\right]w^{\left(1\right)}\right\}}\mathit{d\omega }=0.$

(59)

For simplicity we disregard the second term of (59) that is of order $O\left(\frac{1}{L^2}\right)$ while the first one is of order $O\left(Q^2\right)=O\left(\frac{1}{{\ell }^2}\right)$ (ℓ is the small instability wavelength), which leads to a simpler solvability condition:

${\displaystyle \underset{{\omega }_{\mathit{ref}}}{\int \int }{N}_X^{\left(2\right)}{\left(w^{\left(1\right)}\right)}^2}\mathit{d\omega }=0.$

This leads to the value of the second order load λ⁽²⁾, that is written in terms of four adimensional integrals I₁, I₂, I₃, I₄

${\lambda }^{\left(2\right)}=\frac{{\pi }^2}{4}\mathit{Eh}{Q}^2\frac{I_2{I}_4}{I_1\left(\alpha \right)I_3}$

(60)

$\begin{array}{l}{I}_3={\displaystyle \underset{{\omega }_{\mathit{ref}}}{\int \int }{\left(w^{\left(1\right)}\right)}^2\mathit{drds}}=1\\ I_4={\displaystyle \underset{{\omega }_{\mathit{ref}}}{\int \int }\frac{{\partial }^{2}W}{\partial s^2}{\left(w^{\left(1\right)}\right)}^2\mathit{drds}}.\end{array}$

This identity (60) permits to evaluate the increment of the load due to the wrinkles. It is convenient to compare this load to the approximated critical load λ⁽⁰⁾ = 2DQ²:

$\frac{\lambda }{{\lambda }^{\left(0\right)}}=1+\frac{{\lambda }^{\left(2\right)}}{{\lambda }^{\left(0\right)}}{a}^2=1+\frac{3}{2}{\pi }^2\left(1-{\nu }^2\right)\frac{I_2{I}_4}{I_1\left(\alpha \right)I_3}{\left(\frac{a}{h}\right)}^2.$

(61)

Curiously, this ratio depends rather little on the macroscopic lengths L _X , L _Y and the wrinkling wavelength ℓ, except via the aspect ratio $\alpha =\frac{L_Y}{L_X}$. Obviously the critical load depends strongly on ℓ or on L _Y by

$\frac{{\lambda }^{\left(0\right)}}{\mathit{Eh}}\approx {\left(\frac{h}{l}\right)}^2\approx \frac{h}{L_Y}.$

This means that for a given aspect ratio $\alpha =\frac{L_Y}{L_X}$, there is an universal law connecting the ratio “amplitude of wrinkles over the thickness” to the ratio “loading over critical wrinkling load”, but clearly the critical wrinkling load depends on the traction load. Finally by noting;

$C\left(\alpha \right)=\sqrt{\frac{2}{3{\pi }^2\left(1-{\nu }^2\right)}\frac{I_1\left(\alpha \right)I_3}{I_2{I}_4}},$

and by accounting for (44), we obtain the law connecting the wrinkling amplitude as a function of the compressive stress:

$\frac{a}{h}=C\left(\alpha \right)\sqrt{\frac{\left|{\sigma }_X\right|}{\sqrt{E{\sigma }_Y}}\frac{L_Y}{h}\sqrt{\frac{3\left(1-{\nu }^2\right)}{\pi }}-1}.$

(62)

Note that this analytical formula comes from a structural analysis. Hence the results depend on the domain via the coefficient $\alpha =\frac{L_Y}{L_X}$ and on the boundary conditions. The formula (62) is illustrated by numerical values in the case of a square membrane in Table 1 and Figure 4, for four values of the tensile force.

Let us underline that the previous bifurcation analysis depends strongly on the spatial derivatives in the envelope Equations (32) and therefore on the boundary conditions satisfied by the envelope w. For instance, if one considers a simply supported membrane, the macroscopic boundary conditions are of Newman type ($\frac{\partial w}{\partial n}=0$) and the mode is w⁽¹⁾ = 1 so that the bifurcation analysis from (32) should imply a bifurcation branch with a constant load, as in straight beam buckling. In other terms, the stable postbuckling response obtained in (62) is strongly related to the spatial evolution of the wrinkling mode. The non-linearity of this response is not negligible, as shown by the last column of Table 1 and Figure 5, which means that a significant wrinkling amplitude requires a significant increasing of the compression.

N Y (N/mm)	${\mathbf{N}}_{\mathbf{X}}^{\mathit{wr}}$ (N/mm)	$\frac{{\mathbf{L}}_{\mathbf{X}}}{{\mathbf{\ell }}^{\mathit{wr}}}$	$\begin{array}{l}\frac{\mathbf{a}}{\mathbf{h}}\mathit{for}\\ {\mathbf{N}}_{\mathbf{X}}\mathbf{=}\mathbf{2}{\mathbf{N}}_{\mathbf{X}}^{\mathit{wr}}\end{array}$
5	−0.064	6.28	1.66
10	−0.090	7.48	1.66
15	−0.110	8.28	1.66
20	−0.127	8.90	1.66

Data: E = 70000 MPa, ν=0; L_X = L_Y = 200 mm, h = 0.05 mm.

We describe now the discretisation techniques and the solving algorithm for the reference model (30) (31) (32). Because this system is a second order partial differential system, any classical C⁰ finite element is acceptable. In the applications, quadratic quadrilateral finite elements (Q8) will be used. The resulting nonlinear discrete equations will be solved by the Asymptotic Numerical Method [70]. Some numerical tests will be performed with the pure membrane model (34); its discretisation being straightforward, the details will be omitted.

The procedure described in the previous section has been applied to two numerical tests: the wrinkling of a rectangular membrane submitted to a compressive-tensile loading studied previously in section “Clamped rectangular membrane submitted to tension and compression” and the wrinkling of long rectangular membrane submitted to a uniaxial traction that was studied by several authors in the literature. The main question is to evaluate the validity range of the reduced model (30-32) and (34).

Clamped rectangular membrane submitted to tension and compression

In this section, we study again the example of a clamped rectangular membrane under bi-axial tension-compression load (see Figure 3). The side lengths L_X and L_Y are respectively 400 mm and 200 mm, and the thickness h is 0.05 mm. The applied tension is N_Y = 10 N/mm and the compression force N_X increases in the path following procedure. In the macroscopic model, a wavenumber Q has to be chosen and we take the one predicted by the analytic formula (44). The Figure 6 presents the wrinkling pattern just after the bifurcation. Clearly the envelope is nearly sinusoidal in the two directions OX and OY, as predicted by the analytical solutions. It appears also that the envelope model is able to describe correctly the wrinkling shape predicted by the full shell model. A more quantitative picture is provided by Figure 7, where the results of the pure membrane model (34) are also reported. First the finite element study corroborates the analytical bifurcation study. Second the pure membrane model (34) underestimates very much the wrinkling load. Last the macroscopic model (30) (31) (32) is able to describe quite perfectly the initial post-bifurcation response. Let us underline that the macroscopic model requires less degrees of freedom than the full shell model.

As it is well known [43, 46] with the Ginzburg-Landau equation, the post-bifurcation profile evolves rapidly. Just after the bifurcation (w/h = 0.16), the envelope is nearly sinusoidal as predicted analytically, cf Figure 8. Next it changes rapidly for a hyperbolic tangent shape (w/h = 0.2, Figure 9), what is also predicted by Ginzburg-Landau theory. The numerical results show that the profile changes gradually and becomes more and more localized, with two peaks at the end of the boundary layers (w/h = 2, Figure 10).

The Figure 11 evaluates the ability of the macroscopic model to predict large sizes of wrinkles. It appears that correct amplitudes can be obtained for rather large wrinkles (w/h = 1). Nevertheless one has to be careful in the analysis of the responses for large deflections because these problems can have many stable and unstable solutions and it is not obvious that a path following method with a full shell model provides always the most relevant solution. In the same Figure 11, we reported the result predicted by the approximate analytical solution (61). It gives correctly the beginning of the bifurcated branch up to about w/h ≈ 0.3.

A rectangular membrane submitted to uniaxial traction

A thin rectangular membrane whose dimensions are h = 0.05 mm, L _X  = 1400 mm, L _Y  = 200 mm, is submitted to a uniaxial load (see Figure 12). The long sides are stress-free. Along the short sides, an increasing displacement is applied in the X-direction, the OY-displacement being locked. Linear bifurcation was studied in [56, 71] with the same boundary conditions and a nonlinear bifurcation as here in [33], but with a prescribed stress at the boundary. In this case, the wrinkling occurs for rather high axial stresses since it is caused by small transverse compressive stresses due to the boundary effects (for a finer analysis, see [71]). Comparing with the previous tests, it permits to evaluate the ability of the macroscopic model in a case of non uniform pre-buckling stresses.

In Figure 13, one sees that the post-bifurcation patterns obtained by the new reduced model (30) (31) (32) are quite similar as those provided by the full shell model. This establishes the relevance of this new reduced model to represent the wrinkling modes in a case with a non uniform pre-buckling stress field.

In Figure 14, a response curve is plotted for these two models: the macroscopic model (30) (31) (32) and the full shell model. As in the previous case, the new reduced model gives about the same bifurcation point as the reference model as well as the post-bifurcation response. The number of degrees of freedom is much smaller with the envelope models because they do not need to describe explicitly the full details of the wrinkles. Then a wrinkling pattern along the width is plotted in Figure 15 for the full and the reduced model. Globally they are quite similar but with slight differences for the amplitude and for large sizes of wrinkles (up to w/h = 4). One can wonder if these results could still be improved by keeping a complex envelope as in (36-39) or by keeping higher order harmonics.

To distinguish from the coefficients of the Fourier expansion (6) Ui(X), the indices of the asymptotic expansions (83) have been denoted by λ_(i),w_(i).