Mechanics of tension-induced film wrinkling and restabilization: a review

Abstract

Wrinkling of thin films under tension is omnipresent in nature and modern industry, a phenomenon which has aroused considerable attention during the past two decades because of its intricate nonlinear behaviours and intriguing morphology changes. Here, we review recent advancements in the mechanics of tension-induced film wrinkling and restabilization, by identifying three major stages of its progress: small-strain (less than $5\mathrm{\%}$) wrinkling of stiff sheets, finite-strain (up to $30\mathrm{\%}$) wrinkling and restabilization (isola-centre bifurcation) of soft films, and the effects of curved configurations and material properties on pattern formation. Growing demand for fundamental understanding, quantitative prediction and precise tracking of secondary bifurcation transitions in morphological evolution of thin films helps to advance finite-strain plate/shell theories and sophisticated modelling methods. This progress not only promotes our insightful understanding of complex instability behaviour but also reveals novel phenomena and sheds light on developing wrinkle-tunable membrane structures and functional surfaces.

1. Introduction

The stability of thin-walled structures continues to be scientifically fascinating and technically important. Structural applications, and developments in the life sciences and in the field of soft materials, motivate intensive investigations in this area [1]. Abundant intriguing wrinkling phenomena exist in nature induced by anisotropic growths, specific configurations and environmental confinements (figure 1a–e) and in our daily life (figure 1f–i). Confined thin films, membranes and shells are prone to lose stability towards catastrophic failure [13], which affects functions and mechanical properties of materials and structures in widespread applications [14]. One representative example of unconventional instability behaviours is the transverse wrinkling caused by longitudinal tension (figure 1f).

Figure 1.

(a) Shape buckling of a green algae (Acetabularia schenckii) [2]. (b) Dendrobium helix wavy morphology of orchid petals (image from www.orchidspecies.com/denhelix.htm). (c) Edge wrinkles in a floating lotus leaf upon growth [3]. (d) A free swimming sea slug (Hexabranchus Sanguineus) (image from www.youtube.com/watch?v=V6H01cUSpfQ). (e) Morphology of a bacterial biofilm (E. coli AR3110) [4]. (f) Transverse wrinkles in a polyethylene sheet under uniaxial tension [5]. (g) Transversal waviness on adhesive strips resulting from instabilities during twisting [6]. (h) Dynamic buckling of an elastic annulus [7]. (i) Dielectric elastomer film during pull-in wrinkling [8]. (j) Epidermal sensor [9]. (k) Solar cell [10]. (l) Three-dimensional integrated electric circuit [11]. (m) Micro light-emitting diode [12]. (Online version in colour.)

From another perspective, with the development of soft matter mechanics [15,16], the theory of nonlinear elastic finite deformation has been revived to quantitatively predict large deformations and instability mutation characteristics of soft materials and structures for functional applications such as shape morphing and configuration design [17–19]. Highly stretched membranes have also been widely used in soft robots, wearable technologies and biomedical engineering (figure 1j–m) [9–12,20–22].

Various wrinkling responses of film instability have raised considerable attention over the past 20 years [5,23–61]. Early works [5,24–31] on tension-induced film instability, starting from the beginning of the twenty-first century, unveiled that a stretched sheet with clamped–clamped boundary can induce compression somewhere within boundary layers, which makes the sheet wrinkling therein at a critical tensile stress. Most of the early studies, either dealing with traditional hard materials such as metal sheets that usually exhibit fracture or damage upon larger deformation, or being limited to small strain assumption that cannot accurately predict finite in-plane deformation, focused on the critical buckling conditions, while the post-buckling analysis was not considered for a while.

In 2009, Zheng [45] uncovered the restabilization phenomenon in a highly stretched sheet, showing that wrinkles appear first, then decrease, and eventually disappear with the continuous increase of applied stretching. To consider large deformations of soft films, some plate/shell theories accounting for finite in-plane strain [62–65] and advanced numerical methods [32,66–70] for capturing multiple bifurcations in nonlinear post-buckling paths were developed. General mechanics models to characterize large deformations of thin films can be placed into three categories: extended Föppl-von Kármán plate model (EFvK) [32–34], finite-strain plate model derived from three-dimensional nonlinear elasticity [71–74], and consistent finite-strain plate model [64,65,70,75]. Besides, an analytical method based on Koiter stability theory to predict the isola-centre bifurcation points [43], and numerical analyses via the finite-element method to investigate wrinkling evolution in thin sheets [37], were conducted.

Anisotropic membranes with fibrous textures are omnipresent both in organisms and manufacturing [76,77], e.g. biofilms, skins, biological tissues, woven fabrics and composite materials. Many soft tissues are naturally made of a deformable matrix and fibres with some privileged directions. Sipos & Fehér [41] revealed that material orthotropy affects the disappearance of transverse wrinkles in stretched thin sheets. Zhu et al. [46] experimentally showed that an orthotropic film remains flat when stretched in the high-stiffness direction, while wrinkles occur when it is stretched in the low-stiffness direction. Taylor et al. [78] investigated small-strain wrinkling deformation of incompressible fibre-reinforced plates under different loading types. Liu et al. [79] found that the degree of orthotropy and the shear modulus significantly affect wrinkling orientation and stability boundary for orthotropic, elastic films under uniaxial stretching. Yang et al. [80] derived a finite-strain EFvK-type plate model for anisotropic hyperelastic films under tension, showing that beyond a critical value, the shear modulus ratio of fibre/matrix prevents the appearance of wrinkles.

All the aforementioned studies are considering ideal planar geometry, without exploring the hidden rich and complex behaviours behind the wrinkling-restabilization response with curved surfaces. Curvature and mechanics are intimately connected in thin objects, and the interplay between geometry and material property has shown a variety of intriguing functions [81,82]. The effects of curvature on wrinkling behaviour of tensile curved shells, and of nonlinear interaction between material damage and curvature, were recently pursued [83]. Understanding the underlying mechanism of coupling effects of material and geometry on pattern formation and evolution is crucial for the effective use of wrinkling as a tool for realizing multifunctional surfaces.

Here, we review recent advancements in the mechanics of stretch-induced film wrinkling and restabilization since the beginning of the twenty-first century, by identifying three major stages of its progress: theoretical analyses on small-strain wrinkling of stiff sheets in §2–3, finite-strain wrinkling and restabilization of soft films in §4, and effects of curvature and material damage on pattern formation and evolution in §5. Concluding remarks and perspectives are provided in §6.

2. Tension-field theory for very thin sheets

Tension-field theory was first introduced by Wagner [84], which assumes the membrane with infinitely low bending stiffness to determine the stress field in flexible shear panels used in aircraft construction. This framework was then followed by Reissner [85] to calculate the torsion of an annular membrane. Then the tension-field theory was continuously extended for various applications [86–99]. Stein & Hedgepeth [88] derived a nonlinear model to predict the average displacements of wrinkled membranes. Mansfield [89] applied tension-field theory to obtain the distribution of tension rays based on a simple variational principle. Danielson & Natarajan [91] developed a generalized tension-field theory for membranes undergoing arbitrarily large deformations and used the theory to model operations on the human skin performed by plastic surgeons. Pipkin [93] demonstrated that tension-field theory can be incorporated into the ordinary theory of elastic membranes through replacing the strain energy density with a suitable relaxed energy density. Steigmann [94] established a general tension-field theory to analyse wrinkling in isotropic elastic membranes at finite deformations.

While tension-field theory that focuses on the linearized in-plane elastic response and neglects the flexural stiffness appears to be quite simple for assessing the stress distribution and location of wrinkling regions in very thin sheets [5,71,100], it cannot accurately resolve the physical attributes of wrinkling morphology due to the absence of bending energy [101]. More sophisticated plate/shell theories that account for both stretching and bending contributions to elastic strain energy are required for precise prediction of wrinkling morphology evolutions.

3. Small-strain wrinkling of stiff sheets

Friedl et al. [25] first uncovered that a stretched rectangular sheet induces transverse compression somewhere which thus wrinkles the sheet therein. Since the analytical solution of a buckling problem with complex stress fields may not be found, a model of a simply supported plate buckling under uniform membrane loads is, therefore, pursued to obtain analytical insights. Assuming that lateral compression and global tension are proportional, i.e. ${\sigma }_y=C{\sigma }_x$, the analytical expression of critical longitudinal stress as a function of the aspect ratio ($L/W$) and proportionality factor $C\le 0$ reads

$${\sigma }_x^{\text{cr}}=-\frac{{\pi }^{2}E{h}^2}{12(1-{\nu }^2)}\left[\frac{{(n_1/L)}^4+2{(n_2/W)}^2{(n_1/L)}^2+{(n_2/W)}^4}{C{(n_2/W)}^2+{(n_1/L)}^2}\right],$$ 3.1

where $h$ is the film thickness, $E$ denotes Young’s modulus, $\nu$ is Poisson’s ratio and $n_1,n_2$ are the half-buckled wave numbers along the longitudinal and transverse directions, respectively. Furthermore, to simply determine the critical condition, Friedl et al. [25] provided a diagram of the buckling coefficient $k_c$ to calculate the critical tensile stress based on the formula ${\sigma }_x^{\text{cr}}=k_{c}E{(h/W)}^2$.

Later, Cerda & Mahadevan [5,24] derived a scaling law for determining the transverse wrinkling wavelength $\ell$ and amplitude $A$ by minimizing bending and stretching energies, and pointed out that the essence of wrinkling lies in geometry. Considering that the film is subjected to a longitudinal stretching strain $\gamma$, and the condition of transverse inextensibility ${(A/\ell )}^2\sim \nu \gamma$, minimizing the energy $U\sim (E{h}^3/{\ell }^2+Eh\gamma {\ell }^2/L^2)\nu \gamma LW$ with respect to $\ell$ leads to the scaling law for the wrinkling wavelength and amplitude,

$$\ell \sim \frac{{(hL)}^{1/2}}{{\gamma }^{1/4}}\mathrm{and}A\sim {(\nu hL)}^{1/2}{\gamma }^{1/4}.$$ 3.2

Jacques & Potier-Ferry [26] further carried out an analytical calculation from an energy standpoint to explore wavelength selection and mode localization in a long tensile plate. Considering a thin plate model, the quadratic part of the potential energy can be expressed as

$$2{\mathcal{P}}_2={\int }_{\Omega }(D{(\mathrm{\Delta }w)}^2+N_x{\left(\frac{\mathrm{\partial }w}{\mathrm{\partial }x}\right)}^2+N_y{\left(\frac{\mathrm{\partial }w}{\mathrm{\partial }y}\right)}^2+2N_{xy}\frac{\mathrm{\partial }w}{\mathrm{\partial }x}\frac{\mathrm{\partial }w}{\mathrm{\partial }y})\hspace{0.17em}\text{d}x\hspace{0.17em}\text{d}y,$$ 3.3

where $w(x,y)$ is the out-of-plane deflection, $D=E{h}^3/[12(1-{\nu }^2)]$ represents the bending stiffness, $N_x$, $N_y$ are normal membrane forces, and $N_{xy}$ is the shear membrane force. Then the bifurcation can be predicted from the sign change of the quadratic part of the potential energy ${\mathcal{P}}_2$, i.e. ${\mathcal{P}}_2=0$ and $\delta {\mathcal{P}}_2=0$. The considered problem has no analytical solution, but the critical tensile load and instability mode can be determined by the Ritz method. Then the critical tensile stress ${\sigma }_x^{\text{cr}}$ and the associated transverse wavelength $\ell$ read

$${\sigma }_x^{\text{cr}}=\frac{N_x^{\text{cr}}}{h}=\frac{E{\pi }^2}{3(1-{\nu }^2)C^2}{\left(\frac{h}{l_c}\right)}^2\mathrm{and}\ell =\frac{2\pi }{k}=\sqrt{2|C|{\ell }_x{\ell }_c},$$ 3.4

in which ${\ell }_x$ denotes the longitudinal buckling wavelength, ${\ell }_c$ is the characteristic length defined as ${\ell }_c=\pi I_2/\sqrt{I_1{I}_3}$ where $I_1$, $I_2$ and $I_3$ are given in equation (4.40), and $k$ is the transverse buckling wavenumber.

Puntel et al. [102] and Kim et al. [103] revealed that the applied critical strain depends on aspect ratios and verified the scaling law obtained by Cerda & Mahadevan [5], providing the critical value of applied stretch,

$${\epsilon }_{\text{cr}}=\frac{{\pi }^2{h}^2}{48(1-{\nu }^2)L^2}[1+\frac{L^2}{W^2}({(n_k+2)}^2(1+\frac{L^2{n}_k^2}{W^2})+n_k^2)],$$ 3.5

in which $n_k\in \mathbb{N}$ is related to the wavenumber $k$, satisfying $n_k=k-2$.

4. Finite-strain wrinkling and restabilization of soft films

With the burgeoning use of extremely deformable materials and structures, the manipulation of large deformations and reversible instability responses could play a key role in functional applications. Zheng [45] first explored a restabilization behaviour in a highly stretched sheet, showing that wrinkles occur first, then decrease, and finally disappear with the continuous increase of stretching (see figure 2a). Such ‘birth’ and ‘death’ of a closed loop of bifurcating solutions from a singular point indicates a standard isola-centre bifurcation, which requires comprehensive understanding through a combination of theoretical, experimental and numerical investigations. However, soft films under overstretching generally exhibit large in-plane deformations, and the existing small-strain plate/shell models may become inadequate and fail to predict the isola-centre bifurcation response, which pushes forward novel finite-strain plate/shell theories and advanced computational methods to predict such highly nonlinear behaviour.

Figure 2.

Wrinkling and restabilization of highly stretched PDMS films: (a) the increasing tensile strains are $\epsilon =0,0.05,0.1,0.15,0.2,0.3$, respectively. (b) Bifurcation diagrams: maximum deflection $|w|$ versus applied strain $\epsilon$, $h=0.01$, $\beta =2$, $L=20$, $\text{dot-dash}=\mathrm{NH}\hspace{0.17em}(\mathrm{nHk})$, $\mathrm{dot}=\mathrm{MR}$, $\mathrm{dash}=\mathrm{SVK}$, $\mathrm{solid}=\mathrm{FvK}$ [34]. (c) Comparison of bifurcation diagrams for incompressible nHk material among three plate models ($L/W=3$, $W/h=1000$) [70]. (Online version in colour.)

(a) . Mechanics models of thin films

Differing from conventional buckling analysis usually based on the Föppl-von Kármán framework, models characterizing the large deformations of soft films should cover the post-buckling regime upon excess stretching. Here, we discuss an extended Föppl-von Kármán plate model proposed by Healey et al. [33,34], a finite-strain thin sheet model presented by Taylor et al. [71], and a consistent finite-strain plate model developed by Fu et al. [70] in the prediction of wrinkling behaviour of highly stretched films.

(i) . Extended Föppl-von Kármán plate model

Healey et al. [33] proposed an extended Föppl-von Kármán (EFvK) plate model with Saint-Venant Kirchhoff (SVK) constitutive law that accounts for moderate in-plane strains (nonlinear geometry but linear material law), to predict isola-centre bifurcation points.

The deformation of the sheet is given by

$$\mathbf{\text{f}}(\mathbf{\text{x}}):=\mathbf{\text{x}}+\mathbf{\text{u}}(\mathbf{\text{x}})+w(\mathbf{\text{x}}){\mathbf{\text{e}}}_3,$$ 4.1

where $\mathbf{x}\mathbf{:}=x_{\alpha }{\mathbf{e}}_{\alpha }$ represents the coordinates on undeformed midplane, $\{{\mathbf{e}}_1,{\mathbf{e}}_2,{\mathbf{e}}_3\}$ is a fixed orthonormal basis for the Euclidean vector space, $\mathbf{u}$ denotes the in-plane displacements. Then, the deformation gradient reads

$$\mathbf{\text{F:}}=\mathrm{\nabla }\mathbf{\text{f}}={\mathbf{\text{e}}}_{\alpha }\otimes {\mathbf{\text{e}}}_{\beta }+{\displaystyle \frac{\mathrm{\partial }{u}_{\alpha }}{\mathrm{\partial }{x}_{\beta }}{\mathbf{\text{e}}}_{\alpha }\otimes {\mathbf{\text{e}}}_{\beta }+\frac{\mathrm{\partial }w}{\mathrm{\partial }{x}_{\alpha }}{\mathbf{\text{e}}}_3\otimes {\mathbf{\text{e}}}_{\alpha },}$$ 4.2

in which $\mathrm{\nabla }(\cdot )$ represents the surface gradient and $\otimes$ stands for the tensor product. Without special elucidation, repeated Greek indices satisfy summation convention and $\alpha ,\beta ,\dots$ take values in $\{1,2\}$.

The Green–Lagrange membrane strain tensor can be expressed as

$$\mathbf{\text{E:}}=\frac{1}{2}({\mathbf{\text{F}}}^T\mathbf{\text{F}}-{\mathbf{\text{I}}}_s)=\frac{1}{2}(\mathrm{\nabla }\mathbf{\text{u}}+\mathrm{\nabla }{\mathbf{\text{u}}}^T+\mathrm{\nabla }{\mathbf{\text{u}}}^T\mathrm{\nabla }\mathbf{\text{u}}+\mathrm{\nabla }w\otimes \mathrm{\nabla }w),$$ 4.3

where ${\mathbf{I}}_s$ is ($2\times 2$) identity tensor. Note that when the in-plane nonlinear terms are neglected, it recovers the FvK model. The linear bending strain tensor is defined as

$$\mathbf{\text{K:}}=-{\mathrm{\nabla }}^{2}w,$$ 4.4

in which ${\mathrm{\nabla }}^2:=\mathrm{\nabla }\cdot \mathrm{\nabla }(\cdot )$. Variation of the energy yields the following Euler–Lagrange equilibrium equations,

$$\begin{array}{rl}& \mathrm{\nabla }\cdot [(\mathbf{\text{I}}+\mathrm{\nabla }\mathbf{\text{u}})\mathbf{\text{N}}]=0\\ \mathrm{and}& h^2{\mathrm{\Delta }}^{2}w-\mathrm{\nabla }\cdot (\mathbf{\text{N}}\mathrm{\nabla }w)=0,\end{array}\}$$ 4.5

where $\mathrm{\Delta }(\cdot )$ is the Laplace operator and $\mathbf{N}$ denotes the (in-plane) second Piola–Kirchhoff stress tensor.

(ii) . A finite-strain model for thin elastic sheets

Taylor et al. [71] presented a finite-strain thin sheet model derived from three-dimensional nonlinear elasticity which extends Koiter’s theory [104,105] to deformations that involve significant stretching [72–74].

The position of the plate in the reference configuration is parameterized as

$$\mathbf{\text{X}}=\mathbf{\text{x}}+\varsigma \mathbf{\text{k}},$$ 4.6

where $\mathbf{x}\in \Omega$, $\varsigma \in [-h/2,h/2],\mathbf{k}$ is unit normal, and the origin of position $\mathbf{X}$ lies on the midplane $\Omega$. The deformation gradient is expressed as [73]

$$\widetilde{\mathbf{\text{F}}}=\mathrm{\nabla }\widetilde{\mathbf{\text{y}}}+{\widetilde{\mathbf{\text{y}}}}^{\prime }\otimes \mathbf{\text{k}},$$ 4.7

in which ${(\cdot )}^{\prime }$ is the partial derivative with respect to $\varsigma$. The model includes the coefficient vectors in the expansion

$$\widetilde{\mathbf{\text{y}}}(\mathbf{\text{x}},\varsigma )=\mathbf{\text{r}}(\mathbf{\text{x}})+\varsigma \mathbf{\text{d}}(\mathbf{\text{x}})+{\displaystyle \frac{1}{2}{\varsigma }^2\mathbf{\text{g}}(\mathbf{\text{x}})+{\displaystyle \frac{1}{6}{\varsigma }^3\mathbf{\text{h}}(\mathbf{\text{x}})+\cdots ,}}$$ 4.8

where $\mathbf{r}(\mathbf{x})$ is the position of a material point on the deformed midplane, and the functions $\mathbf{d}(\mathbf{x})={\tilde{\mathbf{y}}}^{\prime }$, $\mathbf{g}(\mathbf{x})={\mathbf{d}}^{\prime }$ and $\mathbf{h}(\mathbf{x})={\mathbf{g}}^{\prime }$ are the directors. The projection $\mathbf{l}=\mathbf{I}-\mathbf{k}\otimes \mathbf{k}$ ($\mathbf{I}$ is ($3\times 3$) identity tensor) is used to expand $\tilde{\mathbf{F}}=\tilde{\mathbf{F}}\mathbf{I}$ in the form

$$\widetilde{\mathbf{\text{F}}}=\mathbf{\text{F}}+\mathbf{\text{d}}\otimes \mathbf{\text{k}}$$ 4.9

and

$$\mathbf{\text{F}}=\mathrm{\nabla }\mathbf{\text{r}}+\mathbf{\text{d}}\otimes \mathbf{\text{k}}\mathrm{and}{\mathbf{\text{F}}}^{\prime }=\mathrm{\nabla }\mathbf{\text{d}}+\mathbf{\text{g}}\otimes \mathbf{\text{k}}.$$ 4.10

Then, one introduces the Green–Lagrange strain tensor as [71]

$$\mathbf{\text{E}}=\frac{1}{2}({\mathbf{\text{F}}}^T\mathbf{\text{F}}-\mathbf{\text{I}}).$$ 4.11

Considering bending and stretching effects at leading order, the potential energy reads

$$W=h^3\{\frac{1}{2}\overline{\mathbf{\text{E}}}\cdot \mathcal{C}(\mathbf{\text{0}})[\overline{\mathbf{\text{E}}}]+\frac{1}{24}{\mathbf{\text{R}}}^T{\mathbf{\text{F}}}^{\prime }\cdot \mathcal{C}(\mathbf{\text{0}})[{\mathbf{\text{R}}}^T{\mathbf{\text{F}}}^{\prime }]\}+\mathcal{O}(h^3),$$ 4.12

in which $\overline{\mathbf{E}}$ satisfies the relation $\mathbf{E}=h\overline{\mathbf{E}}+\mathcal{O}(h)$, $\mathbf{R}$ is the rotation factor in the polar decomposition of the deformation gradient and $\mathcal{C}(\mathbf{0})$ is positive definite in the sense that $\mathbf{A}\cdot \mathcal{C}(\mathbf{0})[\mathbf{A}]>0$ for all non-zero symmetric $\mathbf{A}$, where $\mathcal{C}[\mathbf{A}]$ is the second-order tensor with components ${\mathcal{C}}_{ijkl}{A}_{kl}$. Owing to the characteristics of wrinkling of stretched sheets, the strain energy function (4.12) reduces to

$$W=\frac{1}{2}h\{\frac{2\lambda \mu }{\lambda +2\mu }{(\mathrm{tr}\epsilon )}^2+2\mu |\epsilon {|}^2\}+{\displaystyle \frac{1}{24}{h}^3\{\frac{2\lambda \mu }{\lambda +2\mu }{(\mathrm{tr}\kappa )}^2+2\mu |\kappa {|}^2\},}$$ 4.13

where $\lambda$ and $\mu$ are Lamé coefficients. The in-plane strain tensor $\epsilon$ satisfies

$$\epsilon =E_{\alpha \beta }{\mathbf{\text{e}}}_{\alpha }\otimes {\mathbf{\text{e}}}_{\beta }.$$ 4.14

The bending strain is expressed as

$$\kappa =-{(\mathrm{\nabla }\mathbf{\text{r}})}^T\mathbf{\text{b}}(\mathrm{\nabla }\mathbf{\text{r}}),$$ 4.15

where $\mathbf{b}$ is the curvature tensor on the deformed surface. In Cartesian coordinates on $\Omega$, one obtains

$$\kappa =-b_{\alpha \beta }{\mathbf{\text{e}}}_{\alpha }\otimes {\mathbf{\text{e}}}_{\beta },b_{\alpha \beta }=n_i{r}_{i,\alpha \beta },$$ 4.16

in which $\mathbf{n}$ is the unit normal field on the deformed surface, and Latin indices $i,j,\dots$ run from 1 to 3 unless special elucidation.

The Euler equations can now be derived as

$$\mathrm{\nabla }\cdot \mathbf{\text{T}}=\mathbf{\text{0}},$$ 4.17

where $\mathbf{T}$ is the tensor with nontrivial components

$$T_{i\alpha }=N_{i\alpha }-M_{i\alpha \beta ,\beta },$$ 4.18

in which

$$N_{i\alpha }=\frac{\mathrm{\partial }W}{\mathrm{\partial }{r}_{i,\alpha }}\mathrm{and}{M}_{i\alpha \beta ,\beta }=\frac{\mathrm{\partial }W}{\mathrm{\partial }{r}_{i,\alpha \beta }}.$$ 4.19

Compared to the EFvK plate model proposed by Healey et al. [33,34] that accounts for linearized curvature tensor, the finite-strain thin sheet model derived by Taylor et al. [71] considers more exact curvature tensor instead of its linearization in bending energy. For wrinkling with moderate amplitude ($w/h\sim 3$) in highly stretched sheets, it seems that the difference in curvature approximation does not have an apparent influence on the wrinkling profile [33,71].

(iii) . A consistent finite-strain plate model

The aforementioned plate models are generally established based on some a priori kinematic and/or scaling hypotheses, and thus a question is to what extent these assumptions do not affect the reliability of the results. If no three-dimensional theoretical or experimental results are available, it would be more reliable (and might be necessary) to develop a plate theory which is asymptotically consistent with three-dimensional field equations. Fu et al. [70] developed a finite-strain plate model which is asymptotically consistent with three-dimensional field equations and top/bottom traction conditions in a pointwise manner to compute the wrinkling behaviour of stretched films.

The deformation gradient tensor is expanded in a Taylor series along the film thickness,

$$\mathbf{\text{F}}={\mathbf{\text{F}}}^{(0)}+\varsigma {\mathbf{\text{F}}}^{(1)}+\frac{1}{2}{\varsigma }^2{\mathbf{\text{F}}}^{(2)}+\frac{1}{3!}{\varsigma }^3{\mathbf{\text{F}}}^{(3)}+\frac{1}{4!}{\varsigma }^4{\mathbf{\text{F}}}^{(4)}+\cdots ,$$ 4.20

where ${\mathbf{F}}^{(i)}$ is defined as

$${\mathbf{\text{F}}}^{(i)}=\mathrm{\nabla }{\mathbf{\text{y}}}^{(i)}+{\mathbf{\text{y}}}^{(i+1)}\otimes \mathbf{\text{k}},i=0,1,2,3,4,$$ 4.21

in which $\mathbf{y}$ is the position vector in the deformed configuration and yields ${\mathbf{y}}^{(i)}={\mathrm{\partial }}^i\mathbf{y}/\mathrm{\partial }{\varsigma }^i{|}_{\varsigma =-h/2}$ for $i=0,1,\dots ,5$. The nominal stress tensor $\mathbf{P}$ can also be expanded as

$$\mathbf{\text{P}}={\mathbf{\text{P}}}^{(0)}+\varsigma {\mathbf{\text{P}}}^{(1)}+\frac{1}{2}{\varsigma }^2{\mathbf{\text{P}}}^{(2)}+\frac{1}{3!}{\varsigma }^3{\mathbf{\text{P}}}^{(3)}+\frac{1}{4!}{\varsigma }^4{\mathbf{\text{P}}}^{(4)}+\cdots ,$$ 4.22

where

$${\mathbf{\text{P}}}^{(0)}=\frac{\mathrm{\partial }\Phi }{\mathrm{\partial }\mathbf{\text{F}}}({\mathbf{\text{F}}}^{(0)}),{\mathbf{\text{P}}}^{(1)}=\frac{{\mathrm{\partial }}^2\Phi }{\mathrm{\partial }\mathbf{\text{F}}\mathrm{\partial }\mathbf{\text{F}}}({\mathbf{\text{F}}}^{(0)})[{\mathbf{\text{F}}}^{(1)}],$$ 4.23

in which $\Phi (\mathbf{F})$ is the strain energy function, and the explicit expressions for ${\mathbf{P}}^{(j)}$ ($j=2,3,4$) are not needed.

Through the field equation $\mathrm{\nabla }\cdot \mathbf{P}=\mathbf{0}$ and the bottom and the top traction-free conditions (${\mathbf{P}}^T\mathbf{k}{|}_{\varsigma =-h/2}=\mathbf{0}$ and ${\mathbf{P}}^T\mathbf{k}{|}_{\varsigma =h/2}=\mathbf{0}$), the stress expansion coefficients satisfy the following equations,

$$\begin{array}{rl}& \mathrm{\nabla }\cdot {\mathbf{\text{P}}}_t^{(0)}+\mathrm{\nabla }\cdot {\mathbf{\text{P}}}_t^{(1)}=\mathbf{\text{0}}\\ \mathrm{and}& \mathrm{\nabla }\cdot ({\mathbf{\text{P}}}_{\mathbf{\text{k}}}^{(0)}-{({\mathbf{\text{P}}}^{(0)})}^T\mathbf{\text{k}})+h\mathrm{\nabla }\cdot ({\mathbf{\text{P}}}_{\mathbf{\text{k}}}^{(1)}-{({\mathbf{\text{P}}}^{(1)})}^T\mathbf{\text{k}})\\ & \hspace{0.17em}+\frac{h^2}{3}\mathrm{\nabla }\cdot \mathrm{\nabla }\cdot {\mathbf{\text{P}}}^{(1)}=0,\end{array}\}$$ 4.24

where ${\mathbf{P}}_t$ is the in-plane stress tensor.

(iv) . Numerical resolution methods

One challenge in film wrinkling at large strain is the resolution of highly nonlinear differential equations, since a large number of solution branches are possible and can be connected via multiple bifurcations. Path-following continuation techniques are the most popular class of methods to solve nonlinear static problems [3,32,79,81–83,106], which can predict intricate nonlinear responses in the presence of both subcritical and supercritical bifurcations as well as hysteresis loops, yet the convergence cannot always be ensured. For example, when solving a difficult buckling problem of a shearing membrane, Wong & Pellegrino [31] pointed out that ‘all attempts to use the arc-length solution method in Abaqus (Riks) were unsuccessful’. Besides, for instabilities that are extremely localized, e.g. smooth-wrinkle-ridge-sagging transition upon rolling up a sleeve (figure 3) [35,107], there may exist a local transfer of strain energy from one part of the model to the neighbouring parts, and global resolution strategies such as arc-length method might encounter difficulties in convergence.

Figure 3.

A sequence of morphological pattern transitions (smooth-wrinkle-ridge-sagging) upon rolling up a sleeve [107]. (Online version in colour.)

An alternative way to extend the range of numerical predictions turns out to be the dynamic relaxation method. Within this procedure, one introduces velocity-dependent damping and possibly inertial terms. Realistic definitions of mass and damping are not needed and thus these quantities are chosen to get an optimal convergence. Taylor et al. [42] compared the dynamic relaxation method with the finite-element-based general static analysis and Riks method, and mentioned that finite-element-based static methods strongly depend on the initial imperfection chosen from a priori appropriate buckling mode, while a random imperfection is sufficient for the dynamic relaxation method to generate low-energy solutions without requiring knowledge of the equilibrium solution beforehand. The dynamic relaxation method has been employed to solve various geometric and material nonlinear problems including wrinkled membranes [71,100], inflated cushions [108] and film-substrate systems [109,110]. This method supplements path-following techniques, while it cannot straightforwardly predict subcritical bifurcations, metastable solutions and hysteresis.

(b) . Isola-centre bifurcation

Through employing 4-node rectangular conformal finite elements [111] to solve the equations in §4i, the extended Föppl-von Kármán plate models [33,34] can predict the appearance and disappearance of wrinkles as well as the entire orbit of isola-centre bifurcation.

By considering the drastic contrast to the ever-increasing wrinkling amplitude versus applied stretching strain predicted by conventional Föppl-von Kármán (FvK) plate model, Li & Healey [34] demonstrated that the wrinkling amplitudes based on SVK, neo-Hookean (nHk) and Mooney–Rivlin (MR) models increase, then gradually decrease, and eventually become zero with the increasing stretching, i.e. isola-centre bifurcation, as shown in figure 2b. They pointed out that the nonlinear in-plane terms in the geometric equations play a crucial role in the disappearance of wrinkles, since both SVK and FvK models follow the same Hooke’s material law. Furthermore, Fu et al. [70] compared the computational results based on the consistent plate theory (CPT) in §4iii with EFvK model and shell models (S4R, S4R5 and S8R) in Abaqus for nHk and SVK materials, showing that the EFvK model may provide unreliable results in the post-buckling regime in some situations, as illustrated in figure 2c. Fu et al. [70] explained that the conventional thin plate models are almost based on some ad hoc kinematic and/or scaling assumptions, and as a result they are not consistent with the three-dimensional field equations. For example, the deformation gradient of these FvK-type plate models is not a $3\times 3$ matrix, and some plane stress assumptions (e.g. $S_{i3}=0$) are needed to get an approximation of the thickness deformation.

(c) . Effects of constitutive laws

Li & Healey [34] extended the work of Healey et al. [33] to nHk and MR constitutive laws, and argued that nHk and MR models are more realistic than FvK and SVK ones.

The potential strain energy functions of nHk and MR are respectively given by

$$\mathrm{nHk}:\hspace{0.17em}\varphi (\mathbf{\text{C}})=\frac{Eh}{6}[\mathrm{tr}\mathbf{\text{C}}+{(det\mathbf{\text{C}})}^{-1}-3]$$ 4.25

and

$$\mathrm{MR}:\hspace{0.17em}\varphi (\mathbf{\text{C}})=\frac{Eh}{6.6}\{[\mathrm{tr}\mathbf{\text{C}}+{(det\mathbf{\text{C}})}^{-1}-3]+0.1[\mathrm{tr}\mathbf{\text{C}}{(det\mathbf{\text{C}})}^{-1}+det\mathbf{\text{C}}-3]\},$$ 4.26

where $\mathbf{C}=2\mathbf{E}+\mathbf{I}$ is right Cauchy–Green deformation tensor. One can see from figure 2b that the SVK material shows a considerable discrepancy compared to nHk and MR models when the stretching strain $\epsilon >0.05$ due to its linear elasticity constraint, while all three constitutive laws (SVK, nHk and MR) can capture the characteristic of wrinkling and restabilization. This fact implies that the bifurcation scenario is inherently general and primarily determined by geometric nonlinearity rather than material nonlinearity.

Furthermore, Fu et al. [32] developed a systematic modelling strategy combining a path-following continuation technique by the asymptotic numerical method (ANM) and a spatial discretization by a spectral collocation method for wrinkling and restabilization of soft films made of various compressible and incompressible hyperelastic materials. In a departure from conventional hyperelastic material models, they explored the effect of strain-stiffening behaviour on pattern formation. Hence, they used the incompressible version of the Gent strain-energy function given by

$${\varphi }_{\mathrm{G}}(I_1,J,J_m)=-\frac{\mu }{2}{J}_m\ln (1-\frac{I_1-3}{J_m})-p(J-1),$$ 4.27

where $I_1=\mathrm{tr}(\mathbf{C})$, $J$ represents the elastic volume ratio, and $p$ denotes a Lagrange multiplier. It is found that the stiffening parameter $J_m$ has no apparent influence on the first bifurcation and a rather limited effect on the wrinkling amplitude. When $0.1<J_m<1.5$, the displacement ‘locks up’ at a certain value of stretching strain ($\epsilon <0.3$) before the disappearance of wrinkles. When $J_m\to \mathrm{\infty }$, the model recovers nHk constitution.

Apart from isotropic materials, Sipos & Fehér [41] experimentally revealed that orthotropy emerges during prestressing of a polyurethane film. Assuming the principal directions of the material orthotropy are aligned with the $x$ (stretching direction) and $y$ (transverse direction) coordinate axes, the stretching and bending energies can be respectively expressed as

$${\displaystyle {\Psi }_m(\mathbf{\text{E}})=\frac{1}{2}{\int }_{-h/2}^{h/2}\{\mathbf{\text{E}}\cdot \mathbf{\text{C}}\cdot \mathbf{\text{E}}\}\text{dz}=\frac{1}{2}\frac{Y_{0}h}{1-r{\nu }_{xy}^2}\mathbf{\text{E}}\cdot \widetilde{\mathbf{\text{C}}}\cdot \mathbf{\text{E}}}$$ 4.28

and

$${\displaystyle {\Psi }_b(\mathbf{\text{K}})=\frac{1}{2}{\int }_{-h/2}^{h/2}\{\mathbf{\text{K}}\cdot \mathbf{\text{C}}\cdot \mathbf{\text{K}}\}\text{dz}=\frac{1}{2}\frac{Y_{0}h}{1-r{\nu }_{xy}^2}\mathbf{\text{K}}\cdot \widetilde{\mathbf{\text{C}}}\cdot \mathbf{\text{K}},}$$ 4.29

where $r=Y_{90}/Y_0$ denotes the degree of orthotropy, $Y_0,Y_{90}$ are respectively the moduli of elasticity in directions $x$ and $y$, $\tilde{\mathbf{C}}$ depends on the dimensionless material parameters: ${\tilde{C}}_{1111}=1$, ${\tilde{C}}_{1122}={\tilde{C}}_{2211}=r{\nu }_{xy}$, ${\tilde{C}}_{2222}=r$, ${\tilde{C}}_{1212}=q(r{\nu }_{xy}^2-1)$, $q$ is a positive scalar satisfying $\mu =q{Y}_0$, and $\mu$ is the shear modulus. Then, the second Piola–Kirchhoff stress tensor can be calculated as

$$\mathbf{\text{N}}=\frac{\text{d}{\Psi }_m}{\text{d}\mathbf{\text{E}}}=12\left[\begin{array}{cc}{E}_{11}+r{\nu }_{xy}{E}_{22}& 2q(1-r{\nu }_{xy}^2)E_{12}\\ 2q(1-r{\nu }_{xy}^2)E_{12}& r{E}_{22}+r{\nu }_{xy}{E}_{11}\end{array}\right].$$ 4.30

They experimentally and numerically demonstrated the wrinkling and restabilization in stretched orthotropic films, and verified the stability boundary proposed by Healey et al. [34].

Besides, for orthotropic polypropylene films, Zhu et al. [46] experimentally demonstrated that wrinkles occur when the films are stretched in the low-stiffness direction, while no wrinkles are observed when they are stretched in the high-stiffness direction. They explained that the energy release in the transverse direction must exceed the energy barriers for bending, shearing, expanding and stretching in the loading direction, and the compressive stress in the transverse direction is the only way to relieve the energy.

Liu et al. [79] derived an EFvK plate model for orthotropic films, showing that a decrease of orthotropy and shear modulus significantly prevents or even suppresses wrinkles. Oriented oblique wrinkles, depending on the angle between orthotropy and stretching direction, were first revealed (figure 4a). Moreover, they revealed that the compressive stress ($S_{\text{com}}$) has a similar distribution as the wrinkling profile, and that the direction of maximum compression is parallel to the wrinkling orientation.

Figure 4.

(a) Oriented oblique wrinkles and contours of compressive stresses for different angles between material orthotropy and stretching direction [79]. (b) Uniaxial stretching of a fibrin gel sheet reinforced with a single family of fibres oriented at $0^{\circ }$, ${45}^{\circ }$ and ${90}^{\circ }$ to the horizontal axis [78]. (c) Uniaxial stretching of an orthotropic gel sheet reinforced with two families of fibres oriented at $0^{\circ }/{90}^{\circ }$ and ${45}^{\circ }/{135}^{\circ }$ [78]. (Online version in colour.)

Later, Taylor et al. [78] discussed anisotropic wrinkling deformation of incompressible fibre-reinforced plates at small strain, showing that wrinkles orient themselves towards the direction of strongest fibre reinforcement (figure 4b,c).

Lately, Yang et al. [80] have systematically developed a finite-strain model by introducing anisotropic, hyperelastic constitutive laws with multiple families of fibres into the EFvK theory. The strain energy function with one family of fibres embedded into isotropic matrix is expressed as

$$\Psi =\frac{\mu }{2}(I_1-3)+\frac{{\mu }_1}{2}{(I_4-1)}^2-p(J-1),$$ 4.31

where ${\mu }_1$ measures the degree of anisotropy and $I_4={\mathit{a}}_0\cdot \mathbf{C}\cdot {\mathit{a}}_0$, in which ${\mathit{a}}_0$ is the unit vector along the fibres in the reference configuration. Then, the second Piola–Kirchhoff (P-K) stress tensor is given by

$$\mathbf{\text{S}}=2\frac{\mathrm{\partial }\Psi }{\mathrm{\partial }\mathbf{\text{C}}}=\mu \mathbf{\text{I}}+2{\mu }_1(I_4-1){\displaystyle \frac{\mathrm{\partial }{I}_4}{\mathrm{\partial }\mathbf{\text{C}}}-p{\mathbf{\text{C}}}^{-1}.}$$ 4.32

For thin films, the direction of fibres is usually restricted in the plane and thus ${\mathit{a}}_0={(\cos {\phi }_0,\sin {\phi }_0,0)}^T$, in which ${\phi }_0\in [0,{90}^{\circ }]$ represents the angle between fibres and stretching direction ($x$-axis) in the initial reference configuration. Then

$$\frac{\mathrm{\partial }{I}_4}{\mathrm{\partial }\mathbf{\text{C}}}={\mathit{a}}_0\otimes {\mathit{a}}_0=\left[\begin{array}{ccc}{\cos }^2{\phi }_0& \sin {\phi }_0\cos {\phi }_0& 0\\ \sin {\phi }_0\cos {\phi }_0& {\sin }^2{\phi }_0& 0\\ 0& 0& 0\end{array}\right]=\left[\begin{array}{cc}{\mathit{a}}_{0s}\otimes {\mathit{a}}_{0s}& \mathbf{\text{0}}\\ \mathbf{\text{0}}& 0\end{array}\right],$$ 4.33

where ${\mathit{a}}_{0s}={(\cos {\phi }_0,\sin {\phi }_0)}^T$. The incompressibility condition $det(\mathbf{C})=1$ yields ${\text{C}}_{33}=1/A$, in which $A=det({\mathbf{C}}_s)$. Considering the plane stress condition $({\text{S}}_{i3}=0)$, one obtains

$$p=\mu {\text{C}}_{33}.$$ 4.34

Combining equations (4.32) and (4.33) yields the in-plane second P-K stress tensor as

$${\mathbf{\text{S}}}_s=\mu {\mathbf{\text{I}}}_s+2{\mu }_1(I_4-1){\mathit{a}}_{0s}\otimes {\mathit{a}}_{0s}-p{\mathbf{\text{C}}}_s^{-1}.$$ 4.35

Although only one family of fibres is accounted for here, equation (4.33) suggests that an additional family of fibres will add an independent term into the strain energy function. Consequently, an extra ${\mu }_{\alpha }(I_4^{(\alpha )}-1){\mathit{a}}_{0s}^{(\alpha )}\otimes {\mathit{a}}_{0s}^{(\alpha )}$ will contribute to the stress tensor ${\mathbf{S}}_s$ in equation (4.35), where $\alpha =1,2,\dots$. Therefore, this model can be flexibly extended to other types of hyperelastic energy functions with multiple families of fibres.

Apart from the aforementioned hyperelastic materials, viscoelastic responses of stretched films were examined [38,112]. Nayyar et al. [38] adopted two different viscoelastic models, i.e. a hyper-viscoelastic (HVE) model and a parallel network (PN) model, in finite-element simulations to study the effects of viscoelasticity on wrinkling, residual wrinkles and rate dependence. The HVE model describes constitutively linear viscoelastic behaviour that can be represented by a mechanical analogue with an elastic branch and any number of viscoelastic branches in parallel, while the nonlinear PN model consists of two branches representing two parallel networks of the polymer: network ‘A’ is hyperelastic and network ‘B’ is nonlinear viscoelastic.

The instantaneous Cauchy stress of incompressible neo-Hookean film under uniaxial tension reads

$${\sigma }_0={\mu }_0({\lambda }^2-\frac{1}{\lambda }),$$ 4.36

in which ${\mu }_0$ is the instantaneous shear modulus and $\lambda =1+\epsilon$. For the HVE model, it is assumed that the time-dependent shear relaxation modulus is in the form of a Prony series

$$\mu (t)={\mu }_0[1-\sum _{i=1}^N{g}_i(1-e^{-t/{\tau }_i})],$$ 4.37

where parameters ${\mu }_0,g_i$ and ${\tau }_i$ ($i=1,2,\dots$) are determined by experiments. For the nonlinear PN model, the total Cauchy stress is expressed as

$$\sigma ={\mu }_A({\lambda }_A^2-\frac{1}{{\lambda }_A})+{\mu }_B({({\lambda }_B^e)}^2-\frac{1}{{\lambda }_B^e}),$$ 4.38

where ${\lambda }_A=\lambda$ and ${\lambda }_B^e=\lambda /{\lambda }_B^{cr}$, where ${\lambda }_B^{cr}$ denotes the creep strain. Unlike hyperelastic materials, stretch-induced wrinkles in the polyethylene sheet cannot be fully flattened at large strain due to the nonlinear viscoelastic behaviour, and the PN model is shown to be capable of describing the viscoelastic affected wrinkling of polyethylene sheets (figure 5a,b).

Figure 5.

Wrinkle amplitude in a viscoelastic thin sheet subject to uniaxial stretching with three different strain rates, by post-buckling simulations using (a) HVE and (b) PN models, in comparison with the hyperelastic limit ($\alpha =1000,\beta =2.5$) [38]. (c) A heated sheet of thin glass that is undergoing the redraw process and its analogous experiment. Wrinkles are observed parallel to the edges and in the centre of the sheet [112]. (Online version in colour.)

Srinivasan et al. [112] investigated the shape and stability of a thin viscous sheet that is inhomogeneously stretched in an imposed nonuniform thermal field. They showed that the sheet can become unstable in two regions that are upstream and downstream of the heating zone where the minimum in-plane stress is negative. They pointed out that the wrinkling instability can be entirely suppressed when the surface tension is sufficiently large relative to the magnitude of the in-plane stress (figure 5c).

(d) . Energy state

Wang et al. [43] provided a semi-analytical stability analysis on the wrinkling and restabilization of a uniaxially stretched thin film, by examining the sign change of the second variation of potential energy ${\mathcal{P}}_2$, which can be expressed as

$$\frac{2{\mathcal{P}}_2}{W}=N_x{I}_3-\frac{I_2^2}{4D{I}_1},$$ 4.39

where

$$I_1={\int }_0^{m}F{(x)}^2\hspace{0.17em}\text{d}x,I_2={\int }_0^m{N}_{y}F{(x)}^2\hspace{0.17em}\text{d}x,\hspace{0.17em}{I}_3={\int }_0^m{\left[\frac{\mathrm{\partial }F(x)}{\mathrm{\partial }x}\right]}^2\text{d}x,$$ 4.40

in which $m=L/2$ due to geometric symmetry and $F(x)$ represents the buckling mode in the longitudinal direction. Based on the Koiter stability theory [113,114], bifurcations can be predicted by looking at the sign change of the quadratic terms of the potential energy in equation (4.39), where membrane forces can be obtained numerically so that the second variation of the potential energy ${\mathcal{P}}_2$ is achieved explicitly. With the rise of applied stretching, the second variation of potential energy ${\mathcal{P}}_2$ exhibits two sign changes, corresponding to two changes of stability [43].

5. Wrinkling and smoothing of curved films

Curvature and mechanics are intimately connected in thin objects, and the interaction between geometry and physical property gives rise to intriguing questions. Do tensional wrinkles in thin films depend on geometric curvature? How does curvature affect instability pattern formation and evolution? Does curvature advance or delay stretch-induced wrinkling behaviour? To answer these questions, advanced shell theories and sophisticated models are required for extremely deformable thin objects. However, conventional thin shell theories, e.g. the Donnell–Mushtari–Vlassov (DMV) model [115] and Sanders–Koiter model [105,116], normally treat cases with prescribed surface in cylindrical or polar coordinate systems, and can describe moderate rotation and deflection, but only small strains with linear Hooke’s Law. Wang et al. [81] derived generalized finite-strain shell models for widely used hyperelastic materials, which build on differential geometry [117] and thus can be extended to arbitrarily curved surfaces, to quantitatively explore curvature effects on the wrinkling of soft films.

(a) . Finite-strain shell model with constant curvature

The metric tensor $\mathbf{g}$ and curvature tensor $\mathbf{b}$ for cylindrical geometry with constant curvature (radius $R$, see figure 6a) are given in [81]

$$\mathbf{\text{g}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\mathrm{and}\mathbf{\text{b}}=\left[\begin{array}{cc}{\displaystyle \frac{1}{R}}& 0\\ 0& 0\end{array}\right].$$ 5.1

The Green–Lagrange membrane strain tensor of a cylindrical shell can be written as

$$\begin{array}{rl}{E}_{\alpha \beta }& \hspace{0.17em}=\frac{1}{2}(u_{\alpha ,\beta }+u_{\beta ,\alpha })+w{b}_{\alpha \beta })+\frac{1}{2}(u_{\gamma ,\alpha }{u}_{\gamma ,\beta }+u_{\gamma ,\beta }w{b}_{\gamma \alpha }+u_{\gamma ,\alpha }w{b}_{\gamma \beta }\\ & \hspace{0.17em}+w{b}_{\gamma \alpha }w{b}_{\gamma \beta })+\frac{1}{2}{w}_{,\alpha }{w}_{,\beta }.\end{array}$$ 5.2

The linearized bending strain tensor can be expressed as

$$K_{\alpha \beta }=-w_{,\alpha \beta }+\frac{1}{2}(u_{\gamma ,\alpha }{b}_{\beta }^{\gamma }+u_{\gamma ,\beta }{b}_{\alpha }^{\gamma })+w{b}_{\gamma \alpha }{b}_{\beta }^{\gamma }.$$ 5.3

Figure 6.

(a) Schematic of an open cylindrical shell [81]. (b) Geometry of $S$-shaped surface. (c) Geometry of $C$-shaped surface [82]. Wrinkles are smoothed with increasing curvature $\kappa =h/R$ at the same stretching strain $\epsilon =0.1$ [81]: (d) $\kappa =0$, wrinkling, (e) $\kappa =0.001$, a coupling behaviour of wrinkling and bending, (f) $\kappa =0.0017$, global bending with smooth surface. In three cases, $L=10\hspace{0.17em}\mathrm{cm}$, $W=5\hspace{0.17em}\mathrm{cm}$ and $h=50\hspace{0.17em}\mu \mathrm{m}$. (g) Comparison of bifurcation diagrams among three shell models: DMV and finite-strain models (small curvature and finite curvature). Finite-strain models well predict isola-centre bifurcation (entire orbit), while the conventional DMV model fails in divergence. (h) Evolutions of transverse compressive stresses for incompressible nHk shell with various curvatures (${\kappa }_{cr}\sim 0.0013$, purple dots). With increasing curvatures, compressive stresses decline in general. (i) A computed three-dimensional phase diagram of stability boundary. Theoretical prediction of critical surface (boundary between bending deformation and local wrinkling) remarkably agrees with experiments (dots) [81]. (Online version in colour.)

Based on the well-known Euler–Lagrange equation, one obtains the governing equilibrium equations,

$$\begin{array}{rl}& \hspace{0.17em}-\mathrm{\nabla }\cdot [({\mathbf{\text{I}}}_s+\mathrm{\nabla }\mathbf{\text{u}}+w\mathbf{\text{b}})\mathbf{\text{N}}({\mathbf{\text{E}}}^0)+\mathbf{\text{b}}\cdot \mathbf{\text{M}}(\mathbf{\text{K}})]=\mathbf{\text{0}},\\ & \hspace{0.17em}-\mathrm{\nabla }\cdot \mathrm{\nabla }\cdot \mathbf{\text{M}}(\mathbf{\text{K}})-\mathrm{\nabla }\cdot [\mathbf{\text{N}}({\mathbf{\text{E}}}^0)\cdot \mathrm{\nabla }w]+\mathbf{\text{M}}(\mathbf{\text{K}}):({\mathbf{\text{b}}}^T\cdot \mathbf{\text{b}})\\ \mathrm{and}& \hspace{0.17em}+\mathbf{\text{N}}({\mathbf{\text{E}}}^0):[\mathbf{\text{b}}+{\displaystyle \frac{1}{2}(\mathrm{\nabla }{\mathbf{\text{u}}}^T\cdot \mathbf{\text{b}}+{\mathbf{\text{b}}}^T\cdot \mathrm{\nabla }\mathbf{\text{u}})+w{\mathbf{\text{b}}}^T\cdot \mathbf{\text{b}}}]=0,\end{array}\}$$ 5.4

where $\mathbf{M}$ is the bending moment, ${\mathbf{E}}^0$ denotes the in-plane strain tensor, $\mathrm{\nabla }\mathbf{u}={\mathrm{\nabla }}_{\beta }{u}^{\alpha }{\mathbf{e}}_{\alpha }\otimes {\mathbf{e}}_{\beta }$ and $\mathrm{\nabla }\cdot \mathbf{u}={\mathrm{\nabla }}_{\beta }{u}^{\alpha }{\mathbf{e}}_{\alpha }\cdot {\mathbf{e}}_{\beta }$. Note that in the case of planar geometry ($\mathbf{b}=\mathbf{0}$), by eliminating the components of $\mathbf{u}$, the shell model (5.4) recovers the Föppl-von Kármán plate model. There is a numerical algorithm through coupling Chebyshev spectral collocation method for spatial discretization and the asymptotic numerical method (ANM) for nonlinear resolution can be adopted to solve the nonlinear partial differential equations (5.4) [81].

(b) . Finite-strain shell model with varying curvatures

The metric tensor and curvature tensor for $S$-shaped surface (curvatures with sign change, radii $R_1$ and $R_2$, see figure 6b) are given in [82]

$$\mathbf{\text{g}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\mathbf{\text{b}}=\left[\begin{array}{cc}{\displaystyle \frac{1}{R_1}}& 0\\ 0& 0\end{array}\right](s\le W)\mathrm{and}\mathbf{\text{b}}=\left[\begin{array}{cc}-{\displaystyle \frac{1}{R_2}}& 0\\ 0& 0\end{array}\right]\hspace{0.17em}(s>W).$$ 5.5

Similarly, the metric tensor and curvature tensor for $C$-shaped surface (figure 6c) can be expressed as

$$\mathbf{\text{g}}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],\mathbf{\text{b}}=\left[\begin{array}{cc}{\displaystyle \frac{1}{R_1}}& 0\\ 0& 0\end{array}\right]\hspace{0.17em}(s\le W),\hspace{0.17em}\mathbf{\text{b}}=\left[\begin{array}{cc}\frac{1}{R_2}& 0\\ 0& 0\end{array}\right](s>W).$$ 5.6

The Green–Lagrange membrane strain tensor, bending strain tensor and equilibrium equations remain the same as in equations (5.2), (5.3) and (5.4). Note that the curvature tensors in equations (5.5) and (5.6) for $S$-shaped and $C$-shaped surfaces are not continuous at the interface ($s=W$). To solve this problem, domain decomposition methods (DDM) [118,119] can be applied to deal with the surface ($(0,L)\times (0,2W)$) with discontinuous curvatures. The domain $\Omega$ is divided into two subdomains ${\Omega }_1$ ($(0,L)\times (0,W)$) and ${\Omega }_2$ ($(0,L)\times (W,2W)$). At the interface ($s=W$), the following continuity conditions are satisfied,

$$\begin{array}{rl}& \mathbf{\text{u}}{|}_{s=W}^{{\Omega }_1}=\mathbf{\text{u}}{|}_{s=W}^{{\Omega }_2},w{|}_{s=W}^{{\Omega }_1}=w{|}_{s=W}^{{\Omega }_2},\hspace{0.17em}{\frac{\mathrm{\partial }w}{\mathrm{\partial }s}|}_{s=W}^{{\Omega }_1}={\frac{\mathrm{\partial }w}{\mathrm{\partial }s}|}_{s=W}^{{\Omega }_2}\\ \mathrm{and}& \mathbf{\text{T}}\cdot {\mathbf{\text{e}}}_1{|}_{s=W}^{{\Omega }_1}=\mathbf{\text{T}}\cdot {\mathbf{\text{e}}}_1{|}_{s=W}^{{\Omega }_2},{\mathbf{\text{e}}}_1\cdot \mathbf{\text{M}}(\mathbf{\text{K}})\cdot {\mathbf{\text{e}}}_1{|}_{s=W}^{{\Omega }_1}\\ & \hspace{0.17em}={\mathbf{\text{e}}}_1\cdot \mathbf{\text{M}}(\mathbf{\text{K}})\cdot {\mathbf{\text{e}}}_1{|}_{s=W}^{{\Omega }_2},V{|}_{s=W}^{{\Omega }_1}=V{|}_{s=W}^{{\Omega }_2},\end{array}\}$$ 5.7

where the effective membrane stresses $\mathbf{T}$ and effective transverse shear force $V$ are given by

$$\begin{array}{rl}& \mathbf{\text{T}}=({\mathbf{\text{I}}}_s+\mathrm{\nabla }\mathbf{\text{u}}+w\mathbf{\text{b}})\mathbf{\text{N}}({\mathbf{\text{E}}}^0)+\mathbf{\text{b}}\cdot \mathbf{\text{M}}(\mathbf{\text{K}})\\ \mathrm{and}& V=[\mathbf{\text{N}}({\mathbf{\text{E}}}^0)\cdot \mathrm{\nabla }w+\frac{\mathrm{\partial }}{\mathrm{\partial }z}(\mathbf{\text{M}}(\mathbf{\text{K}}){\mathbf{\text{e}}}_2)+\mathrm{\nabla }\cdot \mathbf{\text{M}}(\mathbf{\text{K}})]\cdot {\mathbf{\text{e}}}_1.\end{array}\}$$ 5.8

(c) . A pseudo-elastic shell model

Wang et al. [83] further developed a pseudo-elastic shell model that accounts for the stress-softening and residual strain as a result of material damage, namely Mullins effect [120–124], in the loading–unloading cycle. The corresponding strain energy density function can be written as

$$\Psi =h{\Psi }_m({\mathbf{\text{C}}}_s,\eta )+{\Psi }_b+h\Phi (\eta ),$$ 5.9

in which $\eta$ is a state variable with $\eta \le 1$ and $\Phi$ denotes a dissipation function.

Considering the incompressible Mooney–Rivlin (MR) constitution [125], the corresponding membrane energy density function can be expressed as [126]

$$\begin{array}{rl}{\Psi }_m({\mathbf{\text{C}}}_s,\eta )& \hspace{0.17em}=p_1[((1+p_3)\eta -p_3)(C_{11}-1)+\eta (C_{22}-1)+\frac{1}{\text{det}{\mathbf{\text{C}}}_s}-1]\\ & \hspace{0.17em}+p_2\eta [\frac{\text{tr}{\mathbf{\text{C}}}_s}{\text{det}{\mathbf{\text{C}}}_s}+\text{det}{\mathbf{\text{C}}}_s-3],\end{array}$$ 5.10

where $p_1$ and $p_2$ are material constants, $p_3>0$ denotes a scalar parameter tuning the anisotropic damage ratio, and $p_3=0$ refers to isotropic damage. The state variable $\eta$ is given by

$$\eta :=1-p_4\tanh [p_5(W_{max}-W_i)],$$ 5.11

in which $p_4$ and $p_5$ are material parameters and one has

$$\begin{array}{rl}& W_i:=-\frac{\text{d}\Phi (\eta )}{\text{d}\eta }=\frac{\mathrm{\partial }{\Psi }_m({\mathbf{\text{C}}}_s,\eta )}{\mathrm{\partial }\eta }\\ \mathrm{and}& W_{\max }:=-\frac{\text{d}\Phi (\eta )}{\text{d}\eta }{|}_{\epsilon ={\epsilon }_{\max }}=\frac{\mathrm{\partial }{\Psi }_m({\mathbf{\text{C}}}_s,\eta )}{\mathrm{\partial }\eta }{|}_{\epsilon ={\epsilon }_{\max }},\end{array}\}$$ 5.12

where ${\epsilon }_{\text{max}}$ is the maximum stretching strain in the loading stage. When $\eta =1$, the membrane strain energy density function reduces to the MR material.

(d) . Curvature effect

Wang et al. [81] experimentally uncovered a nonlinear behaviour coupling global bending and local wrinkling in an open cylindrical shell upon axial stretching, and revealed that the wrinkling amplitude declines with the rise of curvature (figure 6d–f). They showed that the conventional DMV shell model without nonlinear terms of in-plane displacement gradients fails to predict the disappearance of wrinkles (figure 6g). Curvature induced restabilization of wrinkles can be understood by the transverse compressive stress that decreases with the increase of curvature (figure 6h), i.e. larger curvature leads to smaller compressive stress. A three-dimensional phase diagram of stability boundary (dimensionless curvature $h/R$, width/thickness ratio $\alpha =W/h$, length/width ratio $\beta =L/W$) was provided to guide quantitative wrinkle-tunable design (figure 6i). For films with variable curvatures, e.g. $S$-shaped and $C$-shaped geometries, wrinkles tend to distribute in areas with smaller curvature and can be reduced or inhibited by increasing the curvature of adjacent areas [82].

(e) . Competition between Mullins and curvature effects

Fehér et al. [126] reported an inelastic Mullins behaviour (stress softening and permanent residual strain) of stretched thin films in the loading–unloading process, demonstrating that planar films with certain aspect ratios keep flat upon loading, while they wrinkle upon unloading (figure 7a,b).

Figure 7.

(a) Inelastic behaviour of stretched thin films: a film remains flat upon first loading (left panel) but wrinkles appear upon cyclic loading (right panel). The residual strain is about $\epsilon =0.08$. (b) Maximum wrinkling amplitude $|w_{\max }|$ versus applied strain $\epsilon$, for sheets with $\beta =1.4$. There is no wrinkling during the first loading [126]. (c) Schematic of stability boundary for a purely hyperelastic shell. (d) Deflection of the neighbouring peaks (solid curves) and the valleys (stars) versus stretching strain upon loading (red curve and star) and unloading (blue curve and star). Inset indicates the peak and valley of wrinkles on cross section ($z=L/2$). (e) Evolutions of deformations in soft shells in the axial loading–unloading process. The shell exhibits global bending deformation with no surface wrinkle, while wrinkles appear upon unloading due to Mullins effect. Geometric parameters: $\alpha =800$, $\beta =2$ and $\kappa =0.0012$ [83].

Later, Wang et al. [83] revealed a nonlinear competition between Mullins and curvature effects in the wrinkling of stretched soft shells, i.e. the Mullins effect advances wrinkle formation, while curvature resists wrinkling occurrence. Such nonlinear competition can be qualitatively understood by a phase diagram of stability boundary in terms of curvature $\kappa$ and aspect ratio $\beta$, as illustrated in figure 7c. The grey zone indicates the location where wrinkles appear, while the shells keep smooth outside this region. Assuming that the loading produces a permanent change of length such that the increasing aspect ratio moves toward (${\beta }_1$ to ${\beta }_2$) and then locates inside the instability boundary, e.g. $P_2$ to $P_2^{\prime }$, no wrinkle appears upon loading while wrinkling occurs upon unloading. Naturally, there exist another two cases where wrinkles emerge upon both loading and unloading for smaller curvature (e.g. $P_1$ to $P_1^{\prime }$), and the shells remain smooth in the loading–unloading cycle for larger curvature (e.g. $P_3$ to $P_3^{\prime }$). Based on the finite-strain pseudo-elastic shell model with Mullins effect presented in §5c, the bifurcation ($P_2$ to $P_2^{\prime }$) was quantitatively predicted in figure 7d, as well as the evolution of deformations in soft shells during the loading–unloading process (figure 7e).

Understanding the coupling effects between material and curvature on morphology evolution is crucial for wrinkle-tunable design of functional surfaces.

6. Conclusion and perspective

Wrinkling of thin films has attracted considerable attention during the past two decades due to the associated rich instability morphologies and intriguing nonlinear behaviours. Early works focused on the critical buckling analyses on small-strain deformations of stiff films. With the development of soft materials, finite-strain plate/shell theories and sophisticated models were proposed to predict the wrinkling and restabilization of soft films at large deformation, e.g. extended Föppl-von Kármán plate model and consistent finite-strain plate model. Physical interpretation on the isola-centre bifurcation of wrinkling-restabilization was also pursued based on Koiter’s stability theory. Besides, the effects of constitutive laws on the wrinkling response were examined. Notably, oriented oblique wrinkles occur in stretched anisotropic films. Moreover, nonlinear interplays between geometric and material properties on pattern formation were thoroughly explored, e.g. curvature and Mullins effects on the cyclic stretching of soft films, and some insightful mechanisms were uncovered, e.g. Mullins effect advances wrinkle formation, while curvature prevents wrinkling occurrence.

In the presence of microstructures or defects, thin films at large deformation may result in localized failure such as cavitation and fracture, and these factors deserve further consideration in practical applications. In addition, macroscopic instability of thin films can be affected by material-level deformations at micro scale, involving microscopic topology, multi-scale interaction and hierarchical coupling, which is an important and complex topic.

The rapid development of metamaterials in recent years has greatly extended the scope of membrane structures. Is it possible to obtain ultrathin films with special mechanical properties through the ingenious mechanics-guided design of topological microstructures?

Besides, instability of functional films at large deformation under multi-physics stimuli requires further investigations. Is it feasible to programmatically control film wrinkling through electric, magnetic and optical fields, quantitatively guided by advanced mathematical models?

Efforts towards these directions would provide fundamental insights into fabricating morphology-related functional films, and promote the rational design and optimization of wrinkle-tunable membrane structures in broad applications [127].

Data accessibility

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Authors' contributions

T.W.: data curation, formal analysis, investigation, methodology, software, validation, visualization, writing—original draft, writing—review and editing; Y.Y.: data curation, formal analysis, investigation, methodology, software, validation, visualization, writing—review and editing; F.X.: conceptualization, formal analysis, funding acquisition, investigation, methodology, project administration, resources, supervision, validation, writing—original draft, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

This work is supported by the National Natural Science Foundation of China (grants no. 12122204, 11872150 and 11890673), Shanghai Shuguang Program (grant no. 21SG05), Shanghai Rising-Star Program (grant no. 19QA1400500), Shanghai Pilot Program for Basic Research-Fudan University (grant no. 21TQ1400100-21TQ010), and young scientist project of MOE innovation platform.