WGMwgm QEqe EPep PMSpms BECbec DEde

[1]

[1]This research project funded by the XS-META ITN: Marie Skłodowska-Curie
European Actions (Grant Agreement No. 956401).

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orcid=0000-0002-2409-7332]

Writing, methodology, numerical simulations

1]organization=School of Mathematical and Statistical Sciences, University of
Galway, addressline=University Road, city=Galway, postcode=H91 TK33,
country=Ireland

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Conceptualisation, writing

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2]organization=Key Laboratory of Soft Machines and Smart Devices of Zhejiang
Province and Department of Engineering Mechanics, Zhejiang University,
postcode=Hangzhou 310027, country=People’s Republic of China

Conceptualisation, writing

[style=english,orcid=0000-0002-1438-5015] [1] Conceptualisation, methodology,
writing [cor1]Corresponding author

# Designing necks and wrinkles in inflated auxetic membranes

Sairam Pamulaparthi Venkata<EMAIL_ADDRESS>[
Valentina Balbi<EMAIL_ADDRESS>Michel Destrade
<EMAIL_ADDRESS>[  Giuseppe Zurlo
<EMAIL_ADDRESS>
###### Abstract

Thin elastic membranes may undergo various types of instabilities, depending
on material properties, mechanical loading and boundary conditions. In this
work, we focus on axially symmetric and functionally graded membranes, and we
study their behavior upon inflation of an initially flat disk. Our findings
reveal that when the membrane is auxetic, its material properties can be
functionally graded to produce wrinkles and necks at desired locations upon
inflation.

To overcome the complexities arising from strong geometric and constitutive
nonlinearities, we use a semi-analytical approach to identify the right
functional grading required to produce nontrivial instabilities. Hence we use
the method of tension field theory to describe the onset of wrinkling in an
averaged way. We obtain a series of universal results providing powerful
insights into the formation of wrinkles and necks in inflated membranes. For
example, we prove analytically that necks and wrinkles may never overlap in
inflated, axially symmetric membranes.

Guided by these universal results, we implement the relaxed strain energy
method into a Finite Element solver (COMSOL). By tuning spatial
inhomogeneities of the material moduli, we are able to generate specific
instability patterns at desired locations. In particular, we obtain necks
alone, wrinkles alone, or alternating necking and wrinkling patterns.
Moreover, we are able to delay or anticipate the snap-through behavior of the
inflated membrane.

These results will help with the deployment of functionally-graded auxetic
membranes for applications where inflated or laterally pulled membranes may be
morphed and corrugated on demand.

###### keywords:

Auxetic materials Thin membranes Functionally graded materials Hyperelasticity
Relaxed strain energy Tension Field Theory Wrinkling Necking limit-point
instability Elastic Instabilities

##  1 Introduction

Artificially designed materials with negative Poisson’s ratio thicken when
stretched, in contrast to classical elastic materials, which become thinner in
the directions lateral to an applied loading direction. These so-called
auxetic materials have been studied in several fields, including
ferromagnetics (Popereka and Balagurov, 1970), crystal elasticity (Milstein
and Huang, 1979), foam structures (Lakes, 1987), microporous materials (Evans
and Caddock, 1989), and composites (Milton, 1992).

Theoretically, for isotropic materials which satisfy the pointwise energy
stability criterion, the Poisson ratio can take values between -1 and 0.5 for
3D solids (Love, 1944; Timoshenko, 1983). For 2D solids such as membranes, the
allowable Poisson’s ratio value lies between -1 and 1 (Wojciechowski and
Brańka, 1989). In contrast, Poisson’s ratio for anisotropic structures has no
bounds (Ting and Chen, 2005), which is why some biological tissues exhibit
auxeticity (Frolich et al., 1994; Bowick et al., 2001; Lakes, 2001).

Figure 1: Examples of applications and behaviors for auxetic materials. (a)
Hip implants can be designed to include regions with a negative Poisson ratio
to minimize retraction from the bone under biomechanical loading (Kolken et
al., 2018). (b) Auxetic tubular lattice stents exhibit increased ductility
compared to conventional diamond tubular lattices (Jiang et al., 2022). (c)
Auxetic thin membranes under tension wrinkle in the neighborhood of clamps, in
contrast to conventional membranes, which wrinkle in their central region
(Bonfanti and Bhaskar, 2019). (d) Auxetic patches glued onto balloon membranes
can undergo larger deformations than conventional patches, a property which
can help with the healing of puncture wounds to the bladder for example
(Chansoria et al., 2022).

Auxetic materials can be fabricated using additive manufacturing techniques,
such as powder bed fusion (King et al., 2015; Sun et al., 2017) or subtractive
manufacturing techniques, such as laser cutting for thin structures (Mueller
et al., 2013; Bhullar et al., 2017). With the rapid advancement in additive
manufacturing techniques, even complex auxetic structures can be produced at
large scales in a short time. With a careful design of voids or holes at the
micro-scale, auxetic properties can be obtained at the continuum level (Lakes,
1993; Bertoldi et al., 2017).

These structures have a wide range of applications, from soft robotics, with
compliant actuators (Lazarus and Reis, 2015) or compliant grippers (Kaur and
Kim, 2019), to biomedical applications, with auxetic stents (Dolla et al.,
2006), dynamic organ patches (Chansoria et al., 2022), or skin grafts (Gupta
et al., 2022). Figure 1 displays examples of recent applications.

Due to their thin-walled, lightweight, and impressive tensile properties,
nonlinear elastic membranes play a prominent role in many fields such as
automobile, aerospace, civil and biomedical engineering (Beatty, 1987; Evans,
2009; Fu et al., 2016). However, thin membranes lose their mechanical
stability under in-plane compressive stresses due to negligible bending
rigidity, leading to interesting behaviors (Timoshenko, 1983; Roddeman et al.,
1987; Cerda et al., 2002). Inflatable membranes under large elastic
deformations experience various kinds of bifurcation phenomena such as limit-
point (snap-through), wrinkling, and necking instabilities. Some of the most
common and significant applications of inflatable auxetic materials can be
seen in stent deployment for angioplasty (Amin et al., 2015), in morphing
structures (Sun et al., 2014), and for smart biomaterials (Bhullar et al.,
2015).

Over the past several decades, there has been extensive research on finite
inflation in incompressible circular membranes. Early experimental works by
Flint and Naunton (1937), Treloar (1944), and Rivlin and Saunders (1951)
present a detailed analysis of the deformation shape and strain distributions
over the surface of the inflated incompressible isotropic balloons. Adkins and
Rivlin (1952) studied theoretically pressurized circular and spherical thin
shells using incompressible neo-Hookean and Mooney-Rivlin strain energy
functions, to obtain the relationship between inflating pressure, extension
ratio, and radius of curvature near the pole. The book by Green and Adkins
(1960) covers the mathematical theory of finite strain deformations in
nonlinear membranes extensively for both isotropic and anisotropic materials.
More recent experimental works include that by Machado et al. (2012), who
proposed a method to determine curvatures and membrane stresses using an
axisymmetric bulge test for isotropic circular membranes and a three-
dimensional digital image correlation technique.

Finite element analysis (FEA) is very useful in understanding the behavior of
complex geometries and loading conditions. Many researchers have developed
numerical methods based on FEA to study finite inflation in nonlinear
membranes using different material models and for different geometries. For
example, see the works on axisymmetric membranes by Oden and Sato (1967),
Wriggers and Taylor (1990), Gruttmann and Taylor (1992), Jiang and Haddow
(1995), Rumpel et al. (2005), Eriksson and Nordmark (2012), and Selvadurai
(2022), on rectangular membranes using finite difference iterative scheme by
Yang and Lu (1973), and on square membranes using FEA by Adler and Mikulas
(2001), Lee and Youn (2006), Barsotti and Ligarò (2014), and Chen et al.
(2014).

Analytical solutions associated with the finite inflation of nonlinear
membranes are scarce in the literature due to strong material and geometrical
nonlinearities. However, analytical solutions play an important role in
providing simplified and direct solutions to predict the mechanical behavior
of inflated membranes. With an assumption of linear elastic constitutive
behavior and spherical deformation shape for a pressurized incompressible
isotropic circular membrane, Fichter (1997) provided closed-form analytical
solutions for small deformations, later extended by Coelho et al. (2014) for
finite strains. Relaxing the constraint of linear elastic material behavior
but still assuming the spherical deformation shape of the membrane, Yuan et
al. (2021), Foster (1967), and Yang et al. (2021) derived analytical solutions
for pre-stretched circular membranes under inflation using different
incompressible hyperelastic material models. Dropping the hypothesis of
spherical deformation shape and considering compressibility, pressure-
deflection formulae for inflated isotropic circular membranes with
compressible Mooney-Rivlin material model are provided by Pelliciari et al.
(2022) without pre-stretch, by Sirotti et al. (2023) with pre-stretch, and by
Pelliciari and Tarantino (2022) for anisotropic pressurized graphene membranes
without prestretch.

Many of the above-mentioned works dealt with limit-point instability in
inflatable circular membranes, but little attention has been drawn in the past
to the study of wrinkling and necking instabilities for functionally-graded
inflatable auxetic circular membranes under mechanical loads.

Several theories exist in the literature for studying wrinkles in nonlinear
elastic membranes. For example, the theory of incremental deformations
(Haughton and Ogden, 1978a, b), the Föppl–von Kármán theory of plates (Dym et
al., 1973; Puntel et al., 2011), the reduced-order finite element membrane
theory (Damil and Potier-Ferry, 2010), and the numerical bifurcation-
continuation analysis (Healey et al., 2013). Although these advanced and
refined models provide comprehensive details on the wavelength and amplitude
of wrinkles, they are computationally expensive to implement.

In this study, we focus on determining the average deformation in the wrinkled
region along with finding the location and orientation of wrinkles, but not
their amplitude and wavelength. Therefore, we employ tension field theory,
originally proposed by Wagner (1931) and Reissner (1938). Tension field theory
has the advantage of being computationally more viable and mathematically
elegant. According to tension field theory, membranes are assumed to have zero
out-of-plane bending stiffness and cannot sustain in-plane compressive
stresses. In the 1980s, in order to account for compressive stresses, Pipkin
(1986b) extended the theory by introducing the concept of “relaxed strain
energy function”, see also Pipkin (1986a), Steigmann and Pipkin (1989b),
Steigmann (1990), and Pipkin (1994). Relaxed models have been used to study
wrinkles in anisotropic membranes (Pipkin, 1995), electroactive elastomeric
membranes (De Tommasi et al., 2011; De Tommasi et al., 2012; Greaney et al.,
2019; Khurana et al., 2022a, b), pressurized magnetoelastic circular membranes
(Saxena et al., 2019), and inflatable isotropic membranes under uniform
pressure load (Dhavale et al., 2014; Barsotti, 2015; Pamulaparthi Venkata and
Saxena, 2019).

Alongside wrinkling, necking is another interesting bifurcation phenomenon
observed in nonlinear membranes, although less explored in hyperelastic
isotropic circular membranes. Chaudhuri and DasGupta (2014) found negative
Gaussian curvature and circumferential wrinkling at the fixed rim of the
hyperelastic isotropic circular membrane under inflation. Necking in
pressurized elasto-plastic spherical membranes has been investigated by
Needleman (1976) and in pressurized elasto-plastic circular membranes by
Chater and Neale (1983). However, a study on necking and multi-layered
bubbling phenomenon in pressurized functionally-graded hyperelastic isotropic
auxetic circular membranes is still missing in the literature.

In this work, as a proof-of-a-concept, we study the effect of varying material
properties such as the Young modulus and the Poisson ratio on limit-point
instabilities (snap-through), necking, and wrinkling in pressurized isotropic
auxetic membranes (circular and square geometries). We use the Blatz-Ko strain
energy function to model the membrane’s hyperelastic mechanical behavior. We
study the effect of pre-stretches on wrinkling instabilities in auxetic
circular membranes. For the onset of limit-point instabilities, we investigate
the effect of pre-stretches and material parameters for circular geometries.
Finally, we show how to obtain wrinkling patterns in specific areas of the
membrane by tuning the spatial distribution of Young’s modulus and Poisson’s
ratio, both for square and circular membranes.

The paper is organized as follows. In Section 2, we introduce the problems of
interest and the constitutive equations, and we derive the kinematics of the
deformation. We also briefly summarize the main features of the relaxed strain
energy functional based on tension field theory to derive the membrane
stresses. Finally, we write down the equilibrium conditions along with the
applied boundary conditions. In Section 3, with the help of membrane theory,
we derive the equations linking curvatures and principal stresses, which are
necessary for wrinkling and necking in the circular membrane. We establish
several universal insights, valid for all hyperelastic isotropic membranes.
These include the results that regions of necking and wrinkling cannot
overlap, and that necking and wrinkling cannot occur in the center of a pre-
stretched circular membrane.

In Section 4, we compare the results of our finite element simulations in
COMSOL (COMSOL Multiphysics® Version 6.0, 2021) with the universal predictions
of Section 3. Conclusions and limitations of the current work, along with
possible directions for future works are detailed in Section 5. These include
some preliminary results on square membranes which can be inflated to exhibit
a desired wrinkling pattern. These results could be used in applications
involving haptics.

##  2 Membrane deformations, energy and stress

In this work, we describe the behavior of inflated elastic membranes that are
rotationally symmetric about an axis. In the so-called “membrane
approximation”, the membrane thickness is small in comparison to its diameter
and bending effects are neglected. In this approximation, the 3D deformation
is deduced from the 2D deformation of the membrane mid-surface.

We consider a circular membrane with radius $R_{\text{in}}$ and we use a
cylindrical coordinate system to represent the kinematics of the deformation.
We identify the position of a point on the mid-surface of the membrane in its
undeformed configuration with $\pazocal{P}_{1}\left(R,\Phi,0\right)$. As the
membrane is radially stretched axisymmetrically, the point $\pazocal{P}_{1}$
moves to position $\pazocal{P}_{2}\left(\rho_{0},\Phi,0\right)$ in the
membrane with radius $R_{\text{fin}}$. Upon axisymmetric inflation of the pre-
stretched membrane (with a fixed circumference) under a uniform pressure $P$
from the side $Z=0^{-}$, the membrane bulges out of the plane towards
$Z=0^{+}$, and the point $\pazocal{P}_{2}$ is displaced to position
$\pazocal{P}_{3}\left(\rho(R),\Phi,\eta(R)\right)$. Here, $\rho(R)$ and
$\eta(R)$ represent the radial and transverse deflections of the point
$\pazocal{P}_{1}$ in the final configuration, respectively, as shown in Figure
2.

Figure 2: Deformation profiles of the axisymmetric auxetic circular membrane.
(a): Circular membrane with initial radius of $R_{\text{in}}$ and thickness
$H$. (b): Deformed profile of the circular membrane under radial pre-stretch
with the radius $R_{\text{fin}}$. Note that the thickness of the membrane is
increased with the in-plane stretching due to auxeticity. (c): Deformed
profile of the inflated circular membrane subjected to uniform pressure $P$
and fixed on its circumferential boundary. Point $\pazocal{P}_{1}$ in the
undeformed configuration is displaced to the positions $\pazocal{P}_{2}$ and
$\pazocal{P}_{3}$ with pre-stretch and inflation, respectively. A point in the
final configuration has coordinates $(\rho,\Phi,\eta)$, where $\rho$ is radial
position and $\eta$ is the vertical deflection.

The in-plane stretches of the 2D membrane are calculated as

$\lambda_{1}=\sqrt{\left(\derivative{\rho}{R}\right)^{2}+\left(\derivative{\eta}{R}\right)^{2}},\qquad\lambda_{2}=\frac{\rho}{R},$
(1)

where the index $1$ refers to meridians and $2$ to parallels in the current
configuration.

In the membrane approximation, it is assumed the deformation is normal
preserving. Therefore the gradient of deformation admits the diagonal
representation $\mathbf{F}=\text{diag}(\lambda_{1},\lambda_{2},\lambda_{3})$
in a local basis, where the eigenvalue $\lambda_{3}$ is relative to the
membrane normal direction. By introducing the volumetric variation coefficient
$J=\det\mathbf{F}$, we may therefore write

$\qquad\lambda_{3}=\frac{J}{\lambda_{1}\lambda_{2}}.$ (2)

With F written this way, we may now compute the 2D membrane energy from any 3D
energy. Specifically here, to model the compressible behavior of the membrane,
we use the three-dimensional Blatz-Ko strain energy density (Blatz and Ko,
1962; Brockman, 1986),

$W^{\text{3D}}=\alpha\frac{\mu}{2}\left[I_{1}-3+\frac{1-2\nu}{\nu}\left(I_{3}^{-\frac{\nu}{1-2\nu}}-1\right)\right]+(1-\alpha)\frac{\mu}{2}\left[\frac{I_{2}}{I_{3}}-3+\frac{1-2\nu}{\nu}\left(I_{3}^{\frac{\nu}{1-2\nu}}-1\right)\right],$
(3)

where $I_{1}=\text{tr}(\mathbf{FF}^{T})$, $I_{3}=J^{2}$,
$I_{2}=I_{3}\text{tr}(\mathbf{FF}^{T})^{-1}$, and $0<\alpha<1$, $\mu>0$,
$-1<\nu\leq 1/2$ are material constants. The infinitesimal shear modulus $\mu$
and infinitesimal Poisson ratio $\nu$ are related through
$\mu=\nicefrac{{E}}{{2(1+\nu)}}$, where $E$ is the infinitesimal Young
modulus.

With the help of the plane-stress state condition (Haughton and McKay, 1995,
1996), $P_{33}^{\text{3D}}=0$, where $\mathbf{P}^{\text{3D}}=\partial
W^{\text{3D}}/{\partial\mathbf{F}}$ is the first Piola-Kirchhoff stress, we
find the out-of-plane principal stretch ratio $\lambda_{3}$ as

$\lambda_{3}=\left(\lambda_{1}\lambda_{2}\right)^{-\frac{\nu}{1-\nu}}=\Lambda^{\frac{1}{2}},\qquad\text{
where
}\qquad\Lambda=\left(\lambda_{1}\lambda_{2}\right)^{-\frac{2\nu}{1-\nu}}.$ (4)

Now, by substituting Equation 4 in Equation 3, we obtain $W$, i.e. the
membrane strain energy function, in terms of the two in-plane principal
stretch ratios $\lambda_{1}$ and $\lambda_{2}$, as follows

$W=E\frac{\alpha\left(\lambda_{1}^{2}+\lambda_{2}^{2}\right)+\left(1-\alpha\right)\left(\lambda_{1}^{-2}+\lambda_{2}^{-2}\right)-1}{4(1+\nu)}+E\frac{\left(\left(\alpha-1\right)\Lambda^{-1}-\alpha\Lambda\right)\left(\nu-1\right)-1}{4\nu(1+\nu)}.$
(5)

Using Equation 5, we then compute the non-zero components of the first Piola-
Kirchhoff stress $\mathbf{P}=\partial W/\partial\mathbf{F}$ in the membrane as

$P_{i}=\partialderivative{W}{\lambda_{i}}=E\frac{\Lambda^{-1}\left(\lambda_{i}^{2}-\Lambda\right)\left(1+\alpha\left(\lambda_{i}^{2}\Lambda-1\right)\right)}{2\lambda_{i}^{3}\left(1+\nu\right)},\qquad
i=1,2,$ (6)

and the components of the principal Cauchy stress
$\mathbf{T}=J^{-1}\mathbf{P}\mathbf{F}^{T}$ associated with the membrane
energy, as

$T_{i}=E\frac{\Lambda^{-\frac{3}{2}}\left(\lambda_{i}^{2}-\Lambda\right)\left(1+\alpha\left(\lambda_{i}^{2}\Lambda-1\right)\right)}{2\lambda_{i}^{2}\lambda_{1}\lambda_{2}\left(1+\nu\right)},\qquad
i=1,2.$ (7)

Note that in axially symmetric membranes, the Cauchy stress has only in-plane
components $(T_{1},T_{2})$ representing the principal stresses along meridians
and parallels, respectively. With stretches given by Equation 1, these are
functions of $R$ only.

It is worth noting that the Cauchy stress components (7) satisfy the Baker-
Ericksen inequality $(T_{2}-T_{1})(\lambda_{2}-\lambda_{1})>0$ for all choices
of the material parameters. This condition is fundamental to ensure that an
inflated, elastically homogeneous spherical membrane in its reference
configuration remains spherical for all values of applied pressure, see De
Tommasi et al. (2012) and De Tommasi et al. (2013). In this paper we
demonstrate that an initially flat membrane with elastic inhomogeneities can
attain non-trivial geometries (other than the spherical configuration) under a
uniform pressure load. Furthermore, wrinkles and necks may be achieved at
desired locations.

###  2.1 Tension Field Theory: Relaxed strain energy functional

Ideally, membranes with negligible bending stiffness may only achieve non-
negative stress states as they cannot sustain in-plane compression. Real
membranes, however, possess a small (albeit non-negligible) bending stiffness
which effectively slightly delays the onset of wrinkling when compressive
stresses arise.

Lack of resistance to compression may be seen as a unilateral constitutive
constraint. This is embedded in tension field theory by constructing a
“relaxed strain energy density” $W^{\star}(\lambda_{1},\lambda_{2})$ from the
parent energy $W(\lambda_{1},\lambda_{2})$, which sets an in-plane stress
component to zero whenever it would be negative in the parent energy. Clearly,
in taut regions $W^{\star}\equiv W(\lambda_{1},\lambda_{2})$, whereas in
completely slack regions $W^{\star}=0$.

Following A.C. Pipkin (Pipkin, 1986b), we may formalize the above by taking

$W^{\star}=\begin{cases}\begin{array}[]{lllll}W\left(\lambda_{1},\lambda_{2}\right)&\text{
if
}&\lambda_{1}\geq\lambda_{1}^{\star}\left(\lambda_{2},\nu\right),&\lambda_{2}\geq\lambda_{2}^{\star}\left(\lambda_{1},\nu\right),&\textbf{Taut
region with biaxial tension},\\\
W\left(\lambda_{1},\lambda_{2}^{\star}\left(\lambda_{1},\nu\right)\right)&\text{
if }&\lambda_{1}\geq
1,&\lambda_{2}\leq\lambda_{2}^{\star}\left(\lambda_{1},\nu\right),&\textbf{Wrinkled
region with uniaxial tension},\\\
W\left(\lambda_{1}^{\star}\left(\lambda_{2},\nu\right),\lambda_{2}\right)&\text{
if }&\lambda_{2}\geq
1,&\lambda_{1}\leq\lambda_{1}^{\star}\left(\lambda_{2},\nu\right),&\textbf{Wrinkled
region with uniaxial tension},\\\ 0&\text{ if }&\lambda_{1}\leq
1,&\lambda_{2}\leq 1,&\textbf{Slack region with no
tension}.\end{array}\end{cases}$ (8)

The function $\lambda_{i}^{\star}(\lambda_{j},\nu)$ is called “natural width
in tension” and is the main player of the relaxed energy construction. Now
consider a strip of membrane with energy Equation 5, oriented along the
principal directions. If the membrane is pulled by so that $\lambda_{1}\geq 1$
along the direction $1$, while being free in the direction $2$, the membrane
contracts laterally with stretch ratio
$\lambda_{2}=\lambda_{2}^{\star}(\lambda_{1},\nu)$. This value is found by
solving $T_{2}(\lambda_{1},\lambda_{2})$ = 0 for $\lambda_{2}$. Therefore if
$\lambda_{2}\leq\lambda_{2}^{\star}(\lambda_{1},\nu)$, then $T_{2}\leq 0$, and
the membrane would be compressed in the non-relaxed energy. In tension field
theory, this problem is avoided by assuming that if the membrane is compressed
further than $\lambda_{2}^{\star}$, its energy does not change once
$\lambda_{2}^{*}$ is attained: namely,
$W^{\star}(\lambda_{1},\lambda_{2})=W(\lambda_{1},\lambda_{2}^{\star}(\lambda_{1},\nu))$.

Physically, if $\lambda_{1}$ is kept fixed, there would be no energetic
expenditure in shortening the membrane in the direction $2$, below the natural
width $\lambda_{2}^{\star}$. The same considerations apply to the
perpendicular direction, whereas in the fully slack region, the energy is
directly set to zero. For auxetic materials ($\nu<0$), “lateral contraction”
is changed to “lateral expansion”, while all the remaining considerations are
unchanged.

Remarkably, for membranes with energy defined by Equation 5, the natural width
depends on the Poisson ratio $\nu$ only, as follows,

$\begin{split}\lambda_{i}^{\star}\left(\lambda_{j},\nu\right)=\left(\lambda_{j}\right)^{-\nu},\qquad
i,j=1,2,\qquad i\neq j.\end{split}$ (9)

Note that for $\nu=\nicefrac{{1}}{{2}}$, we recover the expression for the
natural width obtained in Steigmann and Pipkin (1989a) and Steigmann and
Pipkin (1989b) for incompressible and isotropic membranes.

The dependence of the natural width in tension on the Poisson coefficient
$\nu$ has not been explored much so far. However, this feature can be
powerfully exploited to achieve non-trivial wrinkling patterns on demand.
Indeed, as already illustrated by Venkata et al. (2023) for stretched
membranes, by carefully tuning the spatial distribution of $E$ and $\nu$, one
can achieve unusual wrinkling patterns. This has a great potential in
technological applications (Naebe and Shirvanimoghaddam, 2016; Ren et al.,
2018).

Auxetic materials have interesting properties: when $\lambda_{1}>1$, then
$\lambda_{2}^{\star}>1$, indicating that the membrane expands in all
directions. This is a very strong difference between auxetic and classical
membranes. Classical membranes always contract laterally when pulled in the
perpendicular direction. In the next section, we discuss the implications of
these features for both classical and auxetic functionally graded membranes.

###  2.2 Equilibrium of a pressurized membrane

Equilibrium equations of axially symmetric pressurized membranes are written
along the meridian and normal directions (Green and Adkins, 1960; Gurtin and
Ian Murdoch, 1975; Libai and Simmonds, 1998). If the reference membrane is
flat, all involved fields depend on the reference radius $R$ only, and the
equilibrium equations take the form

$T_{1}^{\prime}+\frac{\rho^{\prime}}{\rho}\left(T_{1}-T_{2}\right)=0,\qquad\kappa_{1}T_{1}+\kappa_{2}T_{2}=-P,$
(10)

where $\left(\bullet\right)^{\prime}=d/dR$ and where the membrane curvatures
$\kappa_{1}$ (curvature along a meridian line) and $\kappa_{2}$ (curvature
along a parallel line) in axial symmetry may be calculated as

$\begin{split}\kappa_{1}=\frac{\rho^{\prime}\eta^{\prime\prime}-\rho^{\prime\prime}\eta^{\prime}}{\lambda_{1}^{3}}=\frac{(\lambda_{2}R)^{{}^{\prime}}\lambda_{1}^{{}^{\prime}}-(\lambda_{2}R)^{{}^{\prime\prime}}\lambda_{1}}{\lambda_{1}^{2}\sqrt{\lambda_{1}^{2}-\big{(}(\lambda_{2}R)^{{}^{\prime}}\big{)}^{2}}},\qquad\kappa_{2}=\frac{\eta^{\prime}}{\rho\lambda_{1}}=-\frac{\mathbf{n}\cdot\mathbf{e}_{r}}{\rho}=-\frac{\sqrt{\lambda_{1}^{2}-\big{(}(\lambda_{2}R)^{{}^{\prime}}\big{)}^{2}}}{\lambda_{1}\lambda_{2}R}.\end{split}$
(11)

Note that for $\kappa_{2}$ we have also used the alternative expression based
on the outward normal $\mathbf{n}$ to the current surface, and on the radial
unit vector $\mathbf{e}_{r}$ pointing outwards radially and perpendicularly to
the membrane axis $z$. This expression exemplifies that the curvature of a
parallel does not coincide, in general, with the curvature of the membrane
along the parallel unless of course $\mathbf{n}\equiv\mathbf{e}_{r}$.

Because the membrane is flat in its reference configuration, the fields
$\rho,\eta$ must satisfy the following boundary conditions,

$\rho(R_{\text{fin}})=0,\qquad\eta(R_{\text{fin}})=0.$ (12)

Also, because point loads are not applied to the membrane, the suitable
boundary conditions to be imposed on the functions $\rho,\eta$ to avoid
stretch and curvature singularities at the origin are

$\lambda_{1}(0)=\lambda_{2}(0),\qquad\kappa_{1}(0)=\kappa_{2}(0).$ (13)

Finally, to account for the presence of wrinkling, it is sufficient to write
the equilibrium equations in terms of the relaxed counterparts
$(T_{1}^{\star},T_{2}^{\star})$ as obtained from the relaxed energy through
$\mathbf{T}^{\star}=J^{-1}(\partial
W^{\star}/\partial\mathbf{F})\mathbf{F}^{T}$. This will automatically ensure
that no compressive states can be achieved on the inflated membrane.

##  3 General insights into necks and wrinkles

Due to constitutive and geometric non-linearities, it would appear that little
can be said in general on the placement of wrinkles and necks in an inflated
membrane undergoing large deformations. Surprisingly, a careful analysis of
the equilibrium equations of an inflated membrane, together with the
implementation of tension field theory, provides a deep and fully general
characterization of what type of instability patterns may or may not be
expected in an inflated membrane.

Even more remarkably, these characterizations are universal, in the sense that
they are independent on the choice of the (isotropic and elastic) constitutive
behavior of the material. One such remarkable universal result is that necks
and wrinkles can never overlap: this insight will be used in the sequel to
produce alternating patterns of regions with wrinkling and necking.

In this section we collect these universal characterizations, calling them
“insights”, that apply to all isotropic, axisymmetric and inflated membranes.
Their relevance to the present study is that one can use them as general
guidelines to design the spatial distribution of elastic moduli, in order to
obtain desired patterns of wrinkles and necks in the inflated membrane. The
insights also provide interesting characterizations of the membrane shape when
wrinkles or necks occur, therefore helping to understand what type of shapes
may, or may not, be obtained in an inflated membrane undergoing such
instabilities.

As we deal with thin membranes that offer no resistance to compression, we use
tension field theory and base our analysis on the equilibrium equations
Equation 10, expressed in terms of the relaxed Cauchy principal stress
components $T_{1}^{\star},T_{2}^{\star}$.

  * •

Insight 1: The membrane curvature along the parallels is always negative:
$\kappa_{2}<0$.

First note that the equilibrium of a cap above a parallel can be written, in
global form, by balancing pressure $P>0$ and membrane tension, to give

$T_{1}^{\star}\kappa_{2}+\frac{P}{2}=0.$ (14)

Because, by definition, $T_{1}^{\star}\geq 0$, equilibrium imposes
$T_{1}^{\star}>0$ and $\kappa_{2}<0$ everywhere, which concludes the proof.

In passing, note that $\kappa_{2}=\eta^{\prime}/(\rho\lambda_{1})$ so that
$\eta(R)$ is a decreasing function from $R=0$ to $R=R_{\text{fin}}$. But
$\eta(R_{\text{fin}})=0$, which means that $\eta(0)>\eta(R)>0$, in agreement
with the fact that a membrane inflated from below ($z<0$) displaces upwards.

  * •

Insight 2: Wrinkles can never be aligned with parallels.

If such wrinkles were to occur it would result in $T_{1}^{\star}=0$, but from
Insight 1 we know this is not possible. Therefore, wrinkles can only be
aligned with meridians.

  * •

Insight 3: The following inequality is always satisfied in the membrane:
$\kappa_{1}\geq-2|\kappa_{2}|$.

To prove this, we use Equation 10 and Equation 14 to get111Note sign
difference with respect to Hill (1950) and Machado et al. (2012) due to
different notations

$T_{2}^{\star}=-\frac{P}{\kappa_{2}}\left(1-\frac{\kappa_{1}}{2\kappa_{2}}\right).$
(15)

Because $T_{2}^{\star}\geq 0$ and $\kappa_{2}<0$, and by setting
$\kappa_{2}=-|\kappa_{2}|$, we obtain

$\kappa_{1}\geq-2|\kappa_{2}|.$ (16)

It can also be interpreted as ${\kappa_{1}}/{\kappa_{2}}\leq 2$, always.

  * •

Insight 4: When wrinkles occur, the membrane curvatures are linked through:
$\kappa_{1}=-2|\kappa_{2}|$.

This result follows from the previous point with $T_{2}^{\star}=0$.

  * •

Insight 5: Neck regions (where $\kappa_{G}=\kappa_{1}\kappa_{2}<0$) and
wrinkled regions can never overlap.

If there is wrinkling, then $\kappa_{1}=-2|\kappa_{2}|<0$; therefore the
Gaussian curvature becomes positive, $\kappa_{G}=\kappa_{1}\kappa_{2}>0$, and
thus, no necking can occur in the region of wrinkling.

  * •

Insight 6: For necking to happen, the principal stresses must satisfy:
$T^{\star}_{2}\geq 2T_{1}^{\star}$.

By using Equations 14 to 15, we obtain the principal curvatures in terms of
stresses

$\kappa_{1}=-\frac{P}{T_{1}^{\star}}\left(1-\frac{T_{2}^{\star}}{2T_{1}^{\star}}\right),\qquad\kappa_{2}=-\frac{P}{2T_{1}^{\star}},\qquad\kappa_{G}=\kappa_{1}\kappa_{2}=\frac{P^{2}}{2{T_{1}^{\star}}^{2}}\left(1-\frac{T_{2}^{\star}}{2T_{1}^{\star}}\right).$
(17)

For necking we need $\kappa_{G}<0$, and the result follows222Using this
condition, we can design a membrane material with alternating regions of low
and high Young’s modulus to create alternating regions of necking and
wrinkling under a uniform pressure load..

This insight may be also used to prove that wrinkles can never overlap necks.
Indeed, for necks $2T_{1}^{\star}\leq T^{\star}_{2}$, for wrinkles
$T^{\star}_{2}=0$, and combining both we get $2T_{1}^{\star}\leq
T^{\star}_{2}=0$, which cannot occur because $T_{1}^{\star}$ must be strictly
positive.

  * •

Insight 7: Neither wrinkling nor necking can take place at the apex of the
membrane.

According to Equation 13, both principal stretches are equal at the center of
the membrane, i.e. $\lambda_{1}=\lambda_{2}$ at $R=0$. Therefore, we have
$T_{1}^{\star}=T_{2}^{\star}$ and $\kappa_{1}=\kappa_{2}$ at $R=0$. Then, from
Insights 4 and 6, we conclude that neither wrinkling nor necking can occur at
the apex of the membrane.

Based on Insight 2, we therefore conclude that slack regions and regions of
wrinkles aligned with parallels can never occur in the inflated membranes.
These observations help us to simplify the relaxed energy function in Equation
8 as follows,

$W^{\star}=\begin{cases}\begin{array}[]{lllll}W\left(\lambda_{1},\lambda_{2}\right)&\text{
if
}&\lambda_{1}\geq\lambda_{1}^{*}\left(\lambda_{2},\nu\right),&\lambda_{2}\geq\lambda_{2}^{*}\left(\lambda_{1},\nu\right),&\\\
W\left(\lambda_{1},\lambda_{2}^{*}\left(\lambda_{1},\nu\right)\right)&\text{
if }&\lambda_{1}\geq
1,&\lambda_{2}\leq\lambda_{2}^{*}\left(\lambda_{1},\nu\right).\end{array}\end{cases}$
(18)

We emphasize here that not all insights are directly helpful in providing
guidance on how to design the functionally graded membrane to get desired
instabilities. From Insights 1, 2, 3, 4, 5, 7 we deduce important information
on the shape of an inflated membrane, with remarkable bounds on the membrane
curvatures in regions with and without instabilities. These insights dictate
what shapes may be expected (or not expected) in an inflated membrane,
regardless of its constitutive response.

From Insight 6, together with the consideration that $T_{2}^{\star}=0$ leads
to the appearance of wrinkles, we conclude that it is possible to create necks
and wrinkles by tuning the ratio between $T_{2}^{\star}$ and $T_{1}^{\star}$.
This method is used in the following sections to create instability patterns
at desired locations.

##  4 Designing instabilities

In principle, it is logical to assume that the spatial distribution of the
elastic moduli $E(R),\nu(R)$ may be used to achieve wrinkles and necks.
However, in practice, the inverse problem is extremely difficult due to
geometric and constitutive nonlinearities. To circumvent the analytical
complexity of this inverse problem, we use numerical simulations to devise
regions of spatial inhomogeneity with sharp variations of the elastic moduli
to get desired instabilities.

Across a boundary of radius $R$ between elastically inhomogeneous regions, the
following jump conditions have to be satisfied,

$[T_{1}^{\star}]=0,\qquad[\lambda_{2}]=0$ (19)

where $[f]:=\lim_{\epsilon\rightarrow 0}(f(R+\epsilon)-f(R-\epsilon))$. We see
that across such boundary, we may sharply act on the jump of the tension
$[T_{2}^{\star}]$ in the direction of parallels, while at the same time, the
value of $T_{1}^{\star}$ along meridians shall remain continuous across the
phase boundary. By tuning the ratio ${E_{\text{stiff}}}/{E_{\text{soft}}}$
between the Young’s modulus of two neighboring regions (stiff and soft), we
can act on the ratio $T_{2}^{\star}/T_{1}^{\star}$ which, as the insights
above reveal, regulates the onset of wrinkles (when
$T_{2}^{\star}/T_{1}^{\star}$ is decreased toward 0) or the onset of necks
(when $T_{2}^{\star}/T_{1}^{\star}$ is increased above 2).

The Poisson coefficient $\nu$ is also a key player. Indeed, by taking this
coefficient to be negative, the membrane will expand laterally and create
compressive stresses ($T_{2}^{\star}=0$).

The complexity of the problem requires a trial and error approach to determine
appropriate elastic inhomogeneity patterns to achieve the sought
instabilities. By using the commercial software COMSOL, we implement the
Blatz-Ko relaxed strain energy function, Equation 8, for functionally-graded,
inflated membranes.

In the simulations below, we inflate pre-stretched membranes by keeping the
external rim fixed. Then we solve Equation 10 to obtain the Pressure-Volume
(P-V) curves and the deformation profiles of the membranes. The numerical
implementation is explained in Appendix A. Our simulations are consistent with
the universal insights developed above: notably, we see that necking cannot
occur in wrinkling regions and that only wrinkles aligned with meridians can
occur. By playing on the ratio ${E_{\text{stiff}}}/{E_{\text{soft}}}$ and on
the location of elastically inhomogeneous regions, we show that a desired
number of wrinkling and necking regions can be obtained at prescribed
locations.

For the sake of brevity, the effects of pre-stretch and of the Blatz-Ko
material parameters are not discussed in detail. Here, we focus on the most
relevant design parameters: $\nu$ and ${E_{\text{stiff}}}/{E_{\text{soft}}}$.

###  4.1 Necks and no wrinkles

To ensure that the inflated membrane develops necks without wrinkles, we have
to create regions where the ratio $T_{2}^{\star}/T_{1}^{\star}$ exceeds 2,
while maintaining $T_{2}^{\star}>0$ everywhere in the membrane. To achieve
this goal, in Section 4.1, we use a one-step variation of the Young modulus.
The reference membrane has a central region with low Young’s Modulus
($E_{\text{soft}}$) surrounded by a region of high Young’s Modulus
($E_{\text{stiff}}$) which extends up to the boundary, as shown in Figure 3.
Moreover, to avoid that $T_{2}^{\star}$ approaches zero, we set $0<\nu<-0.6$
which ensures a low-to-moderate auxeticity.

Figure 3: Schematic representation of the one-step variation of the Young
modulus distribution $E$ in the reference configuration. The circular membrane
has a softer region in the center surrounded by a stiffer region that extends
to the disc circumference, see Equation B.1a. Figure 4: (a) Inflation process:
Pressure-Volume curves of a pre-stretched circular membranes with four
different cases of one-step variation of the Young modulus. From Case A to
Case D, the radius or size of $E_{\text{soft}}$ region is assumed to be
increasing (see Equation B.1b), (b) Deformation profiles of the membrane at
the end of inflation process are shown here for each case. Inflation process
of each membrane is continued until the apex of the membrane reaches a height
of five times the value of radius $R_{\text{in}}$. For each profile, the
region in blue highlights the necking area ($\kappa_{\pazocal{G}}<0$). Blatz-
Ko parameter $\alpha=0.4$, pre-stretch $\lambda_{p}=2$, Poisson’s ratio
$\nu=-0.4$, and referential radius $R_{\text{in}}=1\hskip 2.84526pt\text{cm}$.

Our simulations show that if $E_{\text{stiff}}/E_{\text{soft}}$ is small (¡
1.5), the membrane displays a smooth necking during inflation. If
$E_{\text{stiff}}/E_{\text{soft}}$ is moderate
($1.5<E_{\text{stiff}}/E_{\text{soft}}<3$), the necking region is more
prominent.

On the left panel of Figure 4, we plot the P-V profiles of pre-stretched
circular membranes with different sizes of the $E_{\text{soft}}$ region. From
Case A to Case D, the radius or size of $E_{\text{soft}}$ region increases
(see Equation B.1b). We observe that, as the size of the soft region
increases, the value of the limit-point pressure $P_{\text{lim}}$ decreases.
This clearly occurs because the bigger the soft region, the smaller the
effective Young modulus.

The deformation profile at the end of inflation process is shown for each case
on the right panel of Figure 4. It can be clearly noticed that, by adjusting
the size of the low Young modulus region, we can shift the boundary of the
necking region. Therefore necks may be obtained at any desired location. It
has to be noted that the simulations confirm that necking cannot occur at the
tip of the membrane, in agreement with Insight 7.

#### Special case: Double bubbling

Figure 5: Pressure-Volume curves of pre-stretched circular membranes for two
different cases of one-step variation of the Young modulus (see, Equation
B.1c) are shown: (a) One-step variation–I, (b) One-step variation–II. The
deformation profiles of the membrane are shown at four different points along
the inflation path. The black region in the deformation profiles (at the end
of inflation process) denotes wrinkling regions. Parameters:
$R_{\text{in}}=1\hskip 2.84526pt\text{cm},\lambda_{p}=2$, $\alpha=0.4$ and
$\nu=-0.9$.

Membranes with a high stiffness ratio (${E_{\text{stiff}}}/{E_{\text{soft}}}$
¿3) undergo a transition into a double-bubbled shape, due to a sudden
inflation of the softer region after the limit-point pressure, as shown in
Figure 5. In this figure, we plot the P-V curves for two different cases of
one-step variation of the Young modulus.

It is important to note that the behavior of the P-V curve is strongly
dependent on the size of the softer core. In Figure 5(a), the volume decreases
immediately after the limit point, whereas in Figure 5(b) the volume keeps
increasing while the pressure decreases after the limit point. Note that case
(a) is also studied by Selvadurai (2022) for pressurized incompressible
elliptical membranes with the Ogden strain energy model.

We also show the deformed shapes at four different stages of inflation. Note
that the formation of the second bubble corresponds to the necking point on
the P-V curve.

Highly auxetic membranes (with $\nu<-0.8$) develop wrinkles in the region
between the two bubbles. Indeed, high auxeticity results into high compressive
stresses. Note that in line with Insight 5, wrinkles (black color regions) are
never observed in the necking region, but only slightly above the neck.

###  4.2 Necks and wrinkles

In this section, we extend the findings from Section 4.1 to the general case
where the Young modulus has an $n$-step variation across the membrane, while
$\nu=-0.9$ is uniform throughout the membrane. Again, we use alternating stiff
and soft regions to create multiple necks. Depending on the size of these
regions we may or may not achieve lateral compression, and therefore
wrinkling, in the membrane. Interestingly, the complexity of the problem leads
to wrinkling regions appearing and disappearing during the inflation process.

Figure 6: First column: two-step (Equations B.2a to B.2b) and three-step
(Equations B.3a to B.3b) variations in the Young modulus $E$. Right column:
the corresponding wrinkling profiles appearing during the pre-stretch process.
Here, the pre-stretch is $\lambda_{p}=2$, with material parameters:
$\alpha=-0.4$ and $\nu=-0.9$. Figure 7: (a) Inflation process: Pressure-Volume
curves of a pre-stretched circular membrane with two-step (solid curve) and
three-step (dash-dotted curve) variations of the Young modulus, (b)
Deformation profiles of the membrane at three different locations are shown
for each variation case. For each profile, the left half highlights the
necking area in blue ($\kappa_{\pazocal{G}}<0$) and the right half highlights
the wrinkling area in black ($T_{2}=0$). Parameters: $\lambda_{p}=2$,
$\alpha=0.4$ and $\nu=-0.9$.

The simulations show that membranes with $n$-step variations of the Young
modulus develop $n-1$ wrinkled regions during the pre-stretching process,
consistent with the findings of Venkata et al. (2023). We recall that during
pre-stretching, the membrane remains flat. These simulations are reported in
Figure 6, showing the onset of wrinkling patterns in laterally pre-stretched
membranes with two- and three-step variations of the elastic moduli.

Following the pre-stretch, we simulate the inflation. The deformation profiles
for the inflation process are shown in Figure 7(b). Wrinkling and necking
regions are represented by black (right half of the figure) and blue (left
half) colors, respectively. Our simulations show that stiff regions inflate
relatively less than soft regions. Moreover, this difference leads to the
formation of a neck at the boundary between the two regions. Interestingly,
the wrinkles generated during the pre-stretch phase tend to be preserved
during the inflation process. Also, the wrinkles never overlap with the
necking regions, which is consistent with the findings of Insight 5.

Furthermore, depending on the stiffness ratio between soft and stiff regions,
the spacing between the necks and the wrinkles can be modulated. In
particular, the higher the ratio ${E_{\text{stiff}}}/{E_{\text{soft}}}$, the
closer the two patterns are, and vice-versa.

The spatial heterogeneity of the elastic moduli affects also the P-V curve, as
shown in Figure 7(a). Interestingly, for membranes with two-step variation of
the Young modulus, the pressure increases until the limit-point instability
and later drops, whereas it remains nearly constant in the three-step
variation case. In all simulations, no further increase of pressure past the
limit point is observed, irrespective of the material constant $0<\alpha<1$.

Finally, we highlight two important features. First, configurations containing
multiple regions of necks and wrinkles can be obtained by increasing the
number of stiff-soft regions in the original membrane. Second, the pre-stretch
$\lambda_{p}$ may be used as a parameter to unwrinkle, during inflation, the
regions of the membrane that are close to the boundary.

###  4.3 Wrinkles and no necks

Figure 8: Pressure-Volume curve of pre-stretched circular membranes with
constant material properties. The deformation profiles of the membrane are
shown at four different points along the inflation path. The black region in
the deformation profiles denotes wrinkling regions. Parameters:
$R_{\text{in}}=1\hskip 2.84526pt\text{cm},\lambda_{p}=2$, $\alpha=0.4,E=70$
kPa, and $\nu=-0.9$.

Our results from the previous sections suggest that a high contrast between
stiff and soft regions (i.e. high ${E_{\text{stiff}}}/{E_{\text{soft}}}$)
leads to necking. Therefore, to avoid necks, we now consider a homogeneous (or
nearly homogeneous) distribution of the Young modulus across the membrane.
Although different types of distributions (constant, linear, step, and
Gaussian) of the Poisson ratio are explored for this case, they all yield
similar deformation behaviors.

For the sake of brevity, we consider the case where the material properties
$E,\nu$ are spatially uniform throughout the membrane. We also compare our
results with existing solutions in the literature.

In Figure 8, we see that for an auxetic membrane with homogeneous Young’s
modulus, the pressure increases with the volume up to the limit point, where
the membrane loses stability and wrinkles appear near the base of the
membrane. Beyond the limit point, the pressure decreases with increasing
volume. The results in Figure 8 also show that the membrane attains a
spherical shape, and indeed no necks are developed. These findings are
consistent with previous experiments on pressurized incompressible circular
membranes (Treloar, 1944; Machado et al., 2012; Zhou et al., 2018) and with
several analytical and numerical studies (Yang and Feng, 1970; Patil and
DasGupta, 2013; Selvadurai, 2022).

For auxetic membranes, we predict the formation of wrinkles aligned with
meridians. We note that Chaudhuri and DasGupta (2014) predict wrinkles aligned
with parallels in inflated incompressible membranes with homogeneous material
properties. However, according to our Insight 2, such wrinkles are not
possible in any axisymmetrically inflated isotropic circular membranes.

Additionally, we find that for a Blatz-Ko material with homogeneous material
properties, tensile stresses exist everywhere in the membrane, except near the
fixed circumferential edge. Hence, wrinkles are obtained only near the fixed
boundary.

Finally, we mention that wrinkles, if present, may also be suppressed by
tuning the Poisson ratio of the membrane and the pre-stretch applied to the
membrane before inflating. We also observed separately that the limit-point
pressure $P_{\texttt{lim}}$ increases linearly with the Blatz-Ko material
constant $\alpha$ and nonlinearly with the pre-stretch, respectively.

##  5 Conclusion and perspectives

Figure 9: (a) Square membranes with two different Young’s modulus profiles
(top) and wrinkling patterns (bottom); (b) Associated Pressure-Volume curves.
FG-SQ-1: The circular region in the center and the rest of the membrane have
$E=0.3$ MPa and $E=0.03$ MPa, respectively. FG-SQ-2: The triangular and
rectangular strips have $E=0.63$ MPa and $E=4.03$ MPa, respectively, while the
rest of the membrane has $E=0.03$ MPa. The Poisson ratio is $\nu=-0.1$, pre-
stretch $\lambda_{p}=1$, and the Blatz-Ko coefficient is $\alpha=0.4$.
Wrinkling patterns at two points $\text{Q}_{1}$ and $\text{Q}_{2}$ on the
inflation curves corresponding to FG-SQ-1 and FG-SQ-2, respectively, are shown
at the bottom of (a).

With this work we investigated the possibility of inducing various forms of
instabilities in inflated hyperelastic membranes. Our results are valid for
axially symmetric configurations. In particular, we investigated the effects
of spatial material inhomogeneities on membranes with auxetic properties
($\nu<0$).

We showed that wrinkling and necking instabilities can be induced in the
membrane during inflation, by tuning the material inhomogeneities. We avoided
solving the complex inverse problem of identifying the elastic heterogeneity
patterns that deliver the desired instabilities at desired locations. Instead,
we developed the following novel methods:

  1. 1.

We implemented tension-field theory based on the relaxed strain energy
function into COMSOL to study the stability of an inflated compressible,
hyperelastic membrane of the Blatz-Ko type for an arbitrary geometry;

  2. 2.

We obtained a series of universal results for the formation of wrinkles and
necks in thin inflated membranes;

  3. 3.

Building upon the universal results and the numerical simulations, we
identified spatial inhomogeneity distributions across the undeformed membrane
that result in wrinkles alone, necks alone, and simultaneous (but not
overlapping) wrinkles and necks in the inflated membrane.

Our study is limited by various factors. For example, typical auxetic
membranes are anisotropic in nature and their material properties are
deformation-dependent: these features might greatly affect the results
presented in this work. This aspect is not addressed in the current work but
could be studied with the methods we have developed here.

Another avenue of interest is to investigate the inflation of non-axisymmetric
membranes. For example, in Figure 9, we show 2D distributions of material
properties in square membranes. The membrane is fixed on its edges and
inflated under a uniform pressure load. We show that material properties can
be tuned in the membranes to produce desired wrinkling patterns, as shown in
the Figure 9. This concept could potentially be used in Braille reading and
haptics.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Acknowledgments

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie Grant
Agreement No. 956401. GZ gratefully acknowledges the support of GNFM (Gruppo
Nazionale di Fisica Matematica) of the INdAM F. Severi.

## Supplementary data

No supplementary material is attached to this article.

##  Appendix A Appendix: Numerical model in COMSOL

We adjust the pre-stretch $\lambda_{p}$ by changing the radius of the
undeformed membrane $R_{\text{in}}$, while holding the radius of the pre-
stretched membrane $R_{\text{fin}}$ fixed. Following Gonçalves et al. (2009);
Patil and DasGupta (2013), we represent the radial position of point
$\pazocal{P}_{2}$ in the stretched membrane as

$\begin{split}\rho_{0}&=\lambda_{p}R,\qquad\lambda_{p}=\frac{R_{\text{fin}}}{R_{\text{in}}}.\end{split}$
(A.1)

We assume that the pre-stretched circular membrane is fixed on its boundaries,
i.e. $\mathbf{u}(R_{\text{fin}})=0$, where $\mathbf{u}$ is the displacement
field in the deformed membrane and $R_{\text{fin}}$ is the radius of the pre-
stretched circular membrane.

The initial thickness of the circular membrane is considered to be negligible
in comparison to its radius (we take $H=R_{\text{in}}/200$). We take the Young
modulus to vary spatially as a function of the referential radial coordinate
$R$.

From an experimental viewpoint, the volume-controlled inflation process is
performed by increasing the mass of the gas in the membrane (Ericksen, 1998).
However, in our simulations, we do not model the behavior of the gas. Instead,
we implement the inflation by using a volume-controlled (or displacement-
controlled) procedure as proposed by Yang and Feng (1970), Pujara and Lardner
(1978), and Patil and DasGupta (2013). We prescribe a vertical displacement at
the center of the membrane and calculate the corresponding pressure through a
global optimization process using an inbuilt function in COMSOL.

The volume enclosed by the surface of the deformed membrane is computed as:

$V=\frac{1}{3}\int_{\Omega}\left(\pazocal{Q}\cdot\mathbf{n}\right)\text{d}a,$
(A.2)

where $\Omega$ represents the surface domain of the deformed membrane. The
position vector of any point on the deformed surface of the membrane is
represented by $\pazocal{Q}$ with $\mathbf{n}$ being the outward unit normal
at that membrane point.

Here we briefly summarize the main features of that relaxed strain energy
implementation in COMSOL software for axisymmetric circular membrane (see
Venkata et al. (2023)):

  1. 1.

By taking advantage of axisymmetry around the Z-axis, we consider a line
segment (which lies on mid-plane of the membrane) in $(R,Z)$ plane with
membrane elements for our analysis in a cylindrical coordinate system.

  2. 2.

The relaxed energy functional (Equation 8) depends only on the principal
stretches and material properties. In COMSOL, by using variable definitions
under the component section in the model builder, we define mathematical
expressions for the material properties. Principal stretches and invariants of
the right Cauchy-Green tensor are then obtained by using internal variables in
the software. Once the expressions for natural widths are defined, the relaxed
strain energy functional is written in the definition window under the
component section, which will be assigned as a user-defined hyperelastic model
to the simulation geometry.

  3. 3.

Once the boundary conditions and loading are assigned to the geometry with a
desired mesh, different numerical solvers can be employed for the analysis.

  4. 4.

In the post-processing stage, we obtain the location and orientation of
wrinkles using logical operators based on the natural width conditions
mentioned in Equation 8.

##  Appendix B Appendix: Circular membranes

###  B.1 One-step variation in material properties

The variation in the Young modulus for the membrane is according to the
following function:

$\begin{split}E(R)&=E_{1}+E_{2}\Bigg{(}\frac{\exp{A\frac{(R-{Y}_{1})}{D}}-1}{\exp{A\frac{(R-{Y}_{1})}{D}}+1}\Bigg{)}-E_{2}\Bigg{(}\frac{\exp{A\frac{(R-{Y}_{2})}{D}}-1}{\exp{A\frac{(R-{Y}_{2})}{D}}+1}\Bigg{)}.\end{split}$
(B.1a)

####  B.1.1 Only necks

$\begin{split}E_{1}&=10\hskip 2.84526pt\text{kPa},\hskip
5.69054ptE_{2}=5\hskip 2.84526pt\text{kPa},\hskip 5.69054pt{A}=2,\hskip
5.69054pt{Y}_{2}=1.1\hskip 2.84526pt\text{cm},\hskip 5.69054pt{D}=0.003\hskip
2.84526pt\text{cm},\\\ {Y}_{1}&=0.07875\hskip
2.84526pt\text{cm},\qquad(\text{for Case A}),\qquad{Y}_{1}=0.105\hskip
2.84526pt\text{cm},\qquad(\text{for Case B}),\\\ {Y}_{1}&=0.16125\hskip
2.84526pt\text{cm},\qquad(\text{for Case C}),\qquad{Y}_{1}=0.45\hskip
2.84526pt\text{cm},\qquad(\text{for Case D}),\end{split}$ (B.1b)

####  B.1.2 Double bubbling

$\begin{split}E_{1}&=10\hskip 2.84526pt\text{kPa},\hskip
5.69054ptE_{2}=30\hskip 2.84526pt\text{kPa},\hskip 5.69054pt{A}=2,\hskip
5.69054pt{Y}_{2}=1.1\hskip 2.84526pt\text{cm},\hskip 5.69054pt{D}=0.003\hskip
2.84526pt\text{cm},\\\ {Y}_{1}&=0.115\hskip 2.84526pt\text{cm},\hskip
5.69054pt(\text{for one-step variation--I}),\qquad{Y}_{1}=0.145\hskip
2.84526pt\text{cm},\hskip 5.69054pt(\text{for one-step variation--
II}).\end{split}$ (B.1c)

###  B.2 Multi-step variations in material properties: Interplay of necking
and wrinkling behaviors

The two-step variation in the Young modulus for the membrane is expressed as

$\begin{split}E(R)&=E_{1}+E_{2}\Bigg{(}\frac{\exp{{A}\frac{(R-{Y}_{3})}{{D}_{1}}}-1}{\exp{{A}\frac{(R-{Y}_{3})}{{D}_{1}}}+1}\Bigg{)}-E_{2}\Bigg{(}\frac{\exp{{A}\frac{(R-{Y}_{4})}{{D}_{1}}}-1}{\exp{{A}\frac{(R-{Y}_{4})}{{D}_{1}}}+1}\Bigg{)}\\\
&+E_{2}\Bigg{(}\frac{\exp{{A}\frac{(R-{Y}_{5})}{{D}_{1}}}-1}{\exp{{A}\frac{(R-{Y}_{5})}{{D}_{1}}}+1}\Bigg{)}-E_{2}\Bigg{(}\frac{{N}_{1}\exp{{A}\frac{(R-{Y}_{6})}{{D}_{1}}}-1}{\exp{{A}\frac{(R-{Y}_{6})}{{D}_{1}}}+1}\Bigg{)},\end{split}$
(B.2a) where $\begin{split}E_{1}&=10\hskip 2.84526pt\text{kPa},\hskip
5.69054ptE_{2}=30\hskip 2.84526pt\text{kPa},\hskip 5.69054pt{A}=2,\hskip
5.69054pt{D}_{1}=0.0085\hskip 2.84526pt\text{cm},\hskip
5.69054pt{N}_{1}=1.2,\\\ {Y}_{3}&=0.05\hskip 2.84526pt\text{cm},\hskip
2.84526pt{Y}_{4}=0.1375\hskip 2.84526pt\text{cm},\hskip
2.84526pt{Y}_{5}=0.225\hskip 2.84526pt\text{cm},\hskip
2.84526pt{Y}_{6}=1.075\hskip 2.84526pt\text{cm}.\end{split}$ (B.2b)

Similarly, the three-step variation in the Young modulus for the membrane is
expressed as

$\begin{split}E(R)&=E_{1}+E_{2}\Bigg{(}\frac{\exp{{A}\frac{(R-{Y}_{7})}{{D}_{2}}}-1}{\exp{{A}\frac{(R-{Y}_{7})}{{D}_{2}}}+1}\Bigg{)}-E_{2}\Bigg{(}\frac{\exp{{A}\frac{(R-{Y}_{8})}{{D}_{2}}}-1}{\exp{{A}\frac{(R-{Y}_{8})}{{D}_{2}}}+1}\Bigg{)}\\\
&+E_{2}\Bigg{(}\frac{\exp{{A}\frac{(R-{Y}_{9})}{{D}_{2}}}-1}{\exp{{A}\frac{(R-{Y}_{9})}{{D}_{2}}}+1}\Bigg{)}-E_{2}\Bigg{(}\frac{\exp{{A}\frac{(R-{Y}_{10})}{{D}_{2}}}-1}{\exp{{A}\frac{(R-{Y}_{10})}{{D}_{2}}}+1}\Bigg{)}\\\
&+E_{2}\Bigg{(}\frac{\exp{{A}\frac{(R-{Y}_{11})}{{D}_{2}}}-1}{\exp{{A}\frac{(R-{Y}_{11})}{{D}_{2}}}+1}\Bigg{)}-E_{2}\Bigg{(}\frac{{N}_{1}\exp{{A}\frac{(R-{Y}_{12})}{{D}_{2}}}-1}{\exp{{A}\frac{(R-{Y}_{12})}{{D}_{2}}}+1}\Bigg{)},\end{split}$
(B.3a) where $\begin{split}E_{1}&=10\hskip 2.84526pt\text{kPa},\hskip
5.69054ptE_{2}=30\hskip 2.84526pt\text{kPa},\hskip 5.69054pt{A}=2,\hskip
5.69054pt{D}_{2}=0.006\hskip 2.84526pt\text{cm},\hskip
5.69054pt{N}_{1}=1.2,\hskip 5.69054pt{Y}_{7}=0.085\hskip
2.84526pt\text{cm},\\\ {Y}_{8}&=0.1625\hskip 2.84526pt\text{cm},\hskip
2.84526pt{Y}_{9}=0.325\hskip 2.84526pt\text{cm},\hskip
2.84526pt{Y}_{10}=0.5125\hskip 2.84526pt\text{cm},\hskip
2.84526pt{Y}_{11}=0.7375\hskip 2.84526pt\text{cm},\hskip
2.84526pt{Y}_{12}=1.0375\hskip 2.84526pt\text{cm}.\end{split}$ (B.3b)

##  Appendix C Appendix: Square membranes

###  C.1 Distribution of the Young modulus in the auxetic square membranes

The mathematical expression for the spatially varying Young’s modulus of a
quarter of the square membrane (FG-SQ-1) reads

$\begin{split}E(X,Y)&=\pazocal{F}\left(\frac{L}{6}-X,\delta\right)\pazocal{F}\left(\sqrt{\big{\lvert}\left(\frac{L}{6}\right)^{2}-X^{2}}\big{\rvert}-Y,\delta\right)E_{1}\\\
&+\pazocal{F}\left(\frac{L}{2}-X,\delta\right)\pazocal{F}\left(\frac{L}{2}-Y,\delta\right)E_{2},\\\
\text{where}\hskip
2.84526pt\pazocal{F}\left(M,N\right)&=0.5+0.9375\left(\frac{M}{N}\right)-0.625\left(\frac{M}{N}\right)^{3}+0.1875\left(\frac{M}{N}\right)^{5},\\\
\text{and}\hskip 2.84526ptE_{1}&=0.27\hskip 2.84526pt\text{MPa},\hskip
5.69054ptE_{2}=0.03\hskip 2.84526pt\text{MPa},\hskip
5.69054pt\delta=10^{-20},\hskip 5.69054ptL=1\hskip 2.84526pt\text{m},\hskip
5.69054pt0\leq X,Y\leq\frac{L}{2}.\end{split}$ (C.1a) Here, the left bottom
vertex of the quarter square membrane is located at ($X=0,Y=0$). The side
length of the quarter square membrane in the undeformed configuration is
$L/2$. The smoothed Heaviside function with a continuous second derivative is
represented by $\pazocal{F}$, it is an inbuilt function in the COMSOL
software. The function, $\lvert(\bullet)\rvert$, returns the absolute value of
any variable, $(\bullet)$.
Similarly, for the full square membrane (FG-SQ-2) with side length $L$ in the
undeformed configuration, the Young modulus can be mathematically expressed as
$\begin{split}E(X,Y)&=\pazocal{F}\left(X+\frac{2L_{1}}{3},\delta\right)\pazocal{F}\left(\varepsilon-X,\delta\right)\pazocal{F}\left(3X+L_{1}-Y,\delta\right)\pazocal{F}\left(Y+\varepsilon,\delta\right)E_{1}\\\
&+\pazocal{F}\left(\frac{2L_{1}}{3}-X,\delta\right)\pazocal{F}\left(X-\varepsilon,\delta\right)\pazocal{F}\left(-3X+L_{1}-Y,\delta\right)\pazocal{F}\left(Y+\varepsilon,\delta\right)E_{1}\\\
&+\pazocal{F}\left(\frac{L}{2}-X,\delta\right)\pazocal{F}\left(\frac{L}{2}-Y,\delta\right)E_{2}\\\
&+\pazocal{F}\left(X+\frac{L_{1}}{3},\delta\right)\pazocal{F}\left(-\frac{L_{1}}{6}-X,\delta\right)\pazocal{F}\left(-\frac{2L_{3}}{3}-Y,\delta\right)\pazocal{F}\left(Y+L_{2},\delta\right)E_{3}\\\
&+\pazocal{F}\left(\frac{L_{1}}{3}-X,\delta\right)\pazocal{F}\left(X-\frac{L_{1}}{6},\delta\right)\pazocal{F}\left(-\frac{2L_{3}}{3}-Y,\delta\right)\pazocal{F}\left(Y+L_{2},\delta\right)E_{3},\\\
E_{1}=0.6&\hskip 2.84526pt\text{MPa},\hskip 2.84526ptE_{2}=0.03\hskip
2.84526pt\text{MPa},\hskip 2.84526ptE_{3}=4\hskip
2.84526pt\text{MPa},L=1\hskip 2.84526pt\text{m},\hskip
2.84526ptL_{1}=\frac{2L}{7},\hskip 2.84526ptL_{2}=\frac{L_{1}}{2},\\\
L_{3}&=\frac{L_{1}}{7},\hskip 2.84526pt\delta=\varepsilon=10^{-20},\hskip
2.84526pt-\frac{L}{2}\leq X,Y\leq\frac{L}{2}.\end{split}$ (C.1b)

## References

  * Adkins and Rivlin (1952) Adkins, J.E., Rivlin, R.S., 1952\.  Large elastic deformations of isotropic materials IX. The deformation of thin shells.  Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 244, 505–531.  DOI:10.1098/rsta.1952.0013.
  * Adler and Mikulas (2001) Adler, A., Mikulas, M., 2001\.  Application of a wrinkled membrane finite element approach to advanced membrane structures.  p. 4646.  DOI:10.2514/6.2001-4646.
  * Amin et al. (2015) Amin, F., Ali, M.N., Ansari, U., Mir, M., Minhas, M.A., Shahid, W., 2015\.  Auxetic coronary stent endoprosthesis: Fabrication and structural analysis.  Journal of Applied Biomaterials & Functional Materials 13, 127–135.  DOI:10.5301/jabfm.5000213.
  * Barsotti (2015) Barsotti, R., 2015.  Approximated Solutions for Axisymmetric Wrinkled Inflated Membranes.  Journal of Applied Mechanics 82, 111007\.  DOI:10.1115/1.4031243.
  * Barsotti and Ligarò (2014) Barsotti, R., Ligarò, S.S., 2014\.  Static response of elastic inflated wrinkled membranes.  Computational Mechanics 53, 1001–1013.  DOI:10.1007/s00466-013-0945-5.
  * Beatty (1987) Beatty, M.F., 1987.  Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues—With Examples.  Applied Mechanics Reviews 40, 1699–1734.  DOI:10.1115/1.3149545.
  * Bertoldi et al. (2017) Bertoldi, K., Vitelli, V., Christensen, J., Van Hecke, M., 2017\.  Flexible mechanical metamaterials.  Nature Reviews Materials 2, 1–11.  DOI:10.1038/natrevmats.2017.66.
  * Bhullar et al. (2015) Bhullar, S.K., Lala, N.L., Ramkrishna, S., 2015.  Smart biomaterials-a review.  Reviews on Advanced Materials Science 40, 303–314.
  * Bhullar et al. (2017) Bhullar, S.K., Rana, D., Lekesiz, H., Bedeloglu, A.C., Ko, J., Cho, Y., Aytac, Z., Uyar, T., Jun, M., Ramalingam, M., 2017.  Design and fabrication of auxetic PCL nanofiber membranes for biomedical applications.  Materials Science and Engineering: C 81, 334–340.  DOI:10.1016/j.msec.2017.08.022.
  * Blatz and Ko (1962) Blatz, P.J., Ko, W.L., 1962\.  Application of finite elastic theory to the deformation of rubbery materials.  Transactions of the Society of Rheology 6, 223–252.  DOI:10.1122/1.548937.
  * Bonfanti and Bhaskar (2019) Bonfanti, A., Bhaskar, A., 2019\.  Elastic stabilization of wrinkles in thin films by auxetic microstructure.  Extreme Mechanics Letters 33, 100556\.
  * Bowick et al. (2001) Bowick, M., Cacciuto, A., Thorleifsson, G., Travesset, A., 2001\.  Universal negative Poisson ratio of self-avoiding fixed-connectivity membranes.  Physical Review Letters 87, 148103\.  DOI:10.1103/PhysRevLett.87.148103.
  * Brockman (1986) Brockman, R.A., 1986.  On the use of the Blatz-Ko constitutive model in nonlinear finite element analysis.  Computers & Structures 24, 607–611.  DOI:10.1016/0045-7949(86)90199-9.
  * Cerda et al. (2002) Cerda, E., Ravi-Chandar, K., Mahadevan, L., 2002.  Wrinkling of an elastic sheet under tension.  Nature 419, 579–580.  DOI:10.1038/419579b.
  * Chansoria et al. (2022) Chansoria, P., Blackwell, J., Etter, E.L., Bonacquisti, E.E., Jasiewicz, N., Neal, T., Kamal, S.A., Hoque, J., Varghese, S., Egan, T., Nguyen, J., 2022.  Rationally designed anisotropic and auxetic hydrogel patches for adaptation to dynamic organs.  Advanced Functional Materials 32, 2207590.  DOI:10.1002/adfm.202207590.
  * Chater and Neale (1983) Chater, E., Neale, K., 1983\.  Finite plastic deformation of a circular membrane under hydrostatic pressure—II: Strain-rate effects.  International Journal of Mechanical Sciences 25, 235–244.  DOI:10.1016/0020-7403(83)90027-9.
  * Chaudhuri and DasGupta (2014) Chaudhuri, A., DasGupta, A., 2014\.  On the static and dynamic analysis of inflated hyperelastic circular membranes.  Journal of the Mechanics and Physics of Solids 64, 302–315.  DOI:10.1016/j.jmps.2013.11.013.
  * Chen et al. (2014) Chen, L., Nguyen-Thanh, N., Nguyen-Xuan, H., Rabczuk, T., Bordas, S.P.A., Limbert, G., 2014\.  Explicit finite deformation analysis of isogeometric membranes.  Computer Methods in Applied Mechanics and Engineering 277, 104–130.  DOI:10.1016/j.cma.2014.04.015.
  * Coelho et al. (2014) Coelho, M., Roehl, D., Bletzinger, K.U., 2014.  Numerical and analytical solutions with finite strains for circular inflated membranes considering pressure–volume coupling.  International Journal of Mechanical Sciences 82, 122–130.  DOI:10.1016/j.ijmecsci.2014.03.012.
  * COMSOL Multiphysics® Version 6.0 (2021) COMSOL Multiphysics® Version 6.0, 2021\.  COMSOL AB, Stockholm, Sweden.  URL: https://www.comsol.com/comsol-multiphysics.
  * Damil and Potier-Ferry (2010) Damil, N., Potier-Ferry, M., 2010\.  Influence of local wrinkling on membrane behaviour: a new approach by the technique of slowly variable fourier coefficients.  Journal of the Mechanics and Physics of Solids 58, 1139–1153.  DOI:10.1016/j.jmps.2010.04.002.
  * De Tommasi et al. (2011) De Tommasi, D., Puglisi, G., Zurlo, G., 2011.  Compression-induced failure of electroactive polymeric thin films.  Applied Physics Letters 98, 123507\.  DOI:10.1063/1.3568885.
  * De Tommasi et al. (2012) De Tommasi, D., Puglisi, G., Zurlo, G., 2012.  A note on strong ellipticity in two-dimensional isotropic elasticity.  Journal of Elasticity 109, 67–74.  DOI:10.1007/s10659-011-9370-1.
  * De Tommasi et al. (2012) De Tommasi, D., Puglisi, G., Zurlo, G., 2012.  Taut states of dielectric elastomer membranes.  International Journal of Non-Linear Mechanics 47, 355–361.  DOI:10.1016/j.ijnonlinmec.2011.08.002.
  * De Tommasi et al. (2013) De Tommasi, D., Puglisi, G., Zurlo, G., 2013.  Inhomogeneous spherical configurations of inflated membranes.  Continuum Mech. Thermodyn. 25, 197–206.  DOI:10.1007/s00161-012-0240-2.
  * Dhavale et al. (2014) Dhavale, N.N., Tamadapu, G., DasGupta, A., 2014.  Finite Inflation Analysis of Two Circumferentially Bonded Hyperelastic Circular Flat Membranes.  Journal of Applied Mechanics 81, 091012\.  DOI:10.1115/1.4027972.
  * Dolla et al. (2006) Dolla, W.J.S., Fricke, B.A., Becker, B.R., 2006.  Structural and Drug Diffusion Models of Conventional and Auxetic Drug-Eluting Stents.  Journal of Medical Devices 1, 47–55.  DOI:10.1115/1.2355691.
  * Dym et al. (1973) Dym, C.L., Shames, I.H., et al., 1973\.  Solid mechanics.  Springer.  DOI:10.1007/978-1-4614-6034-3.
  * Ericksen (1998) Ericksen, J.L., 1998.  Introduction to the Thermodynamics of Solids. volume 275\.  Springer.
  * Eriksson and Nordmark (2012) Eriksson, A., Nordmark, A., 2012\.  Instability of hyper-elastic balloon-shaped space membranes under pressure loads.  Computer Methods in Applied Mechanics and Engineering 237, 118–129.  DOI:10.1016/j.cma.2012.05.012.
  * Evans and Caddock (1989) Evans, K., Caddock, B., 1989\.  Microporous materials with negative poisson’s ratios. II. Mechanisms and interpretation.  Journal of Physics D: Applied Physics 22, 1883.  DOI:10.1088/0022-3727/22/12/013.
  * Evans (2009) Evans, S.L., 2009.  On the implementation of a wrinkling, hyperelastic membrane model for skin and other materials.  Computer Methods in Biomechanics and Biomedical Engineering 12, 319–332.  DOI:10.1080/10255840802546762.
  * Fichter (1997) Fichter, W., 1997.  Some solutions for the large deflections of uniformly loaded circular membranes.  Technical Report.
  * Flint and Naunton (1937) Flint, C., Naunton, W., 1937\.  Physical testing of latex films.  Rubber Chemistry and Technology 10, 584–614.  DOI:10.5254/1.3539012.
  * Foster (1967) Foster, H.O., 1967.  Inflation of a Plane Circular Membrane.  Journal of Engineering for Industry 89, 403–407.  DOI:10.1115/1.3610067.
  * Frolich et al. (1994) Frolich, L., LaBarbera, M., Stevens, W., 1994.  Poisson’s ratio of a crossed fibre sheath: the skin of aquatic salamanders.  Journal of Zoology 232, 231–252.  DOI:10.1111/j.1469-7998.1994.tb01571.x.
  * Fu et al. (2016) Fu, B., Sperber, E., Eke, F., 2016.  Solar sail technology—A state of the art review.  Progress in Aerospace Sciences 86, 1–19.  DOI:10.1016/j.paerosci.2016.07.001.
  * Gonçalves et al. (2009) Gonçalves, P.B., Soares, R.M., Pamplona, D., 2009.  Nonlinear vibrations of a radially stretched circular hyperelastic membrane.  Journal of Sound and Vibration 327, 231–248.  DOI:10.1016/j.jsv.2009.06.023.
  * Greaney et al. (2019) Greaney, P., Meere, M., Zurlo, G., 2019.  The out-of-plane behaviour of dielectric membranes: Description of wrinkling and pull-in instabilities.  Journal of the Mechanics and Physics of Solids 122, 84–97.  DOI:10.1016/j.jmps.2018.09.006.
  * Green and Adkins (1960) Green, A.E., Adkins, J.E., 1960\.  Large elastic deformations and non-linear continuum mechanics.  Clarendon Press, Oxford.
  * Gruttmann and Taylor (1992) Gruttmann, F., Taylor, R., 1992\.  Theory and finite element formulation of rubberlike membrane shells using principal stretches.  International Journal for Numerical Methods in Engineering 35, 1111–1126.  DOI:10.1002/NME.1620350511.
  * Gupta et al. (2022) Gupta, S., Gupta, V., Chanda, A., 2022.  Biomechanical modeling of novel high expansion auxetic skin grafts.  International Journal for Numerical Methods in Biomedical Engineering 38, e3586.  DOI:10.1002/cnm.3586.
  * Gurtin and Ian Murdoch (1975) Gurtin, M.E., Ian Murdoch, A., 1975\.  A continuum theory of elastic material surfaces.  Archive for rational mechanics and analysis 57, 291–323.  DOI:10.1007/BF00261375.
  * Haughton and McKay (1995) Haughton, D., McKay, B., 1995\.  Wrinkling of annular discs subjected to radial displacements.  International Journal of Engineering Science 33, 335–350.  DOI:10.1016/0020-7225(94)00068-U.
  * Haughton and McKay (1996) Haughton, D., McKay, B., 1996\.  Wrinkling of inflated elastic cylindrical membranes under flexure.  International Journal of Engineering Science 34, 1531–1550.  DOI:10.1016/0020-7225(96)00059-6.
  * Haughton and Ogden (1978a) Haughton, D., Ogden, R., 1978a.  On the incremental equations in non-linear elasticity—I. Membrane theory.  Journal of the Mechanics and Physics of Solids 26, 93–110.  DOI:10.1016/0022-5096(78)90016-9.
  * Haughton and Ogden (1978b) Haughton, D., Ogden, R., 1978b.  On the incremental equations in non-linear elasticity—II. Bifurcation of pressurized spherical shells.  Journal of the Mechanics and Physics of Solids 26, 111–138.  DOI:10.1016/0022-5096(78)90017-0.
  * Healey et al. (2013) Healey, T.J., Li, Q., Cheng, R.B., 2013.  Wrinkling behavior of highly stretched rectangular elastic films via parametric global bifurcation.  Journal of Nonlinear Science 23, 777–805.  DOI:10.1007/s00332-013-9168-3.
  * Hill (1950) Hill, R., 1950.  C. a theory of the plastic bulging of a metal diaphragm by lateral pressure.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 41, 1133–1142.  DOI:10.1080/14786445008561154.
  * Jiang et al. (2022) Jiang, H., Ziegler, H., Zhang, Z., Zhang, H., Le Barbenchon, L., Atre, S., Chen, Y., 2022.  3D printed tubular lattice metamaterials for mechanically robust stents.  Composites Part B: Engineering 236, 109809.
  * Jiang and Haddow (1995) Jiang, L., Haddow, J., 1995\.  A finite element formulation for finite static axisymmetric deformation of hyperelastic membranes.  Computers & Structures 57, 401–405.  DOI:10.1016/0045-7949(94)00629-H.
  * Kaur and Kim (2019) Kaur, M., Kim, W.S., 2019.  Toward a smart compliant robotic gripper equipped with 3d-designed cellular fingers.  Advanced Intelligent Systems 1, 1900019\.  DOI:10.1002/aisy.201900019.
  * Khurana et al. (2022a) Khurana, A., Joglekar, M., Zurlo, G., 2022a.  Electromechanical stability of wrinkled dielectric elastomers.  International Journal of Solids and Structures 246-247, 111613.  DOI:10.1016/j.ijsolstr.2022.111613.
  * Khurana et al. (2022b) Khurana, A., Kumar, D., Sharma, A.K., Zurlo, G., Joglekar, M., 2022b.  Taut domains in transversely isotropic electro-magneto-active thin membranes.  International Journal of Non-Linear Mechanics 147, 104228.  DOI:10.1016/j.ijnonlinmec.2022.104228.
  * King et al. (2015) King, W.E., Anderson, A.T., Ferencz, R.M., Hodge, N.E., Kamath, C., Khairallah, S.A., Rubenchik, A.M., 2015.  Laser powder bed fusion additive manufacturing of metals; physics, computational, and materials challenges.  Applied Physics Reviews 2, 041304\.  DOI:10.1063/1.4937809.
  * Kolken et al. (2018) Kolken, H.M., Janbaz, S., Leeflang, S.M., Lietaert, K., Weinans, H.H., Zadpoor, A.A., 2018\.  Rationally designed meta-implants: A combination of auxetic and conventional meta-biomaterials.  Materials Horizons 5, 28–35.  DOI:10.1039/C7MH00699C.
  * Lakes (1987) Lakes, R., 1987.  Foam structures with a negative Poisson’s ratio.  Science 235, 1038–1040.  DOI:10.1126/science.235.4792.1038.
  * Lakes (1993) Lakes, R., 1993.  Advances in negative Poisson’s ratio materials.  Advanced Materials 5, 293–296.  DOI:10.1002/adma.19930050416.
  * Lakes (2001) Lakes, R., 2001.  A broader view of membranes.  Nature 414, 503–504.  DOI:10.1038/35107190.
  * Lazarus and Reis (2015) Lazarus, A., Reis, P.M., 2015\.  Soft actuation of structured cylinders through auxetic behavior.  Advanced Engineering Materials 17, 815–820.  DOI:10.1002/adem.201400433.
  * Lee and Youn (2006) Lee, E.S., Youn, S.K., 2006\.  Finite element analysis of wrinkling membrane structures with large deformations.  Finite Elements in Analysis and Design 42, 780–791.  DOI:10.1016/j.finel.2006.01.004.
  * Libai and Simmonds (1998) Libai, A., Simmonds, J.G., 1998\.  The Nonlinear Theory of Elastic Shells.  2 ed., Cambridge University Press.  DOI:10.1017/CBO9780511574511.
  * Love (1944) Love, A.E.H., 1944.  A Treatise on the Mathematical Theory of Elasticity.  Dover Publications I, New York .
  * Machado et al. (2012) Machado, G., Favier, D., Chagnon, G., 2012.  Membrane curvatures and stress-strain full fields of axisymmetric bulge tests from 3D-DIC measurements. Theory and validation on virtual and experimental results.  Experimental Mechanics 52, 865–880.  DOI:10.1007/s11340-011-9571-3.
  * Milstein and Huang (1979) Milstein, F., Huang, K., 1979\.  Existence of a negative Poisson ratio in FCC crystals.  Physical Review B 19, 2030\.  DOI:10.1103/PhysRevB.19.2030.
  * Milton (1992) Milton, G.W., 1992.  Composite materials with Poisson’s ratios close to -1.  Journal of the Mechanics and Physics of Solids 40, 1105–1137.  DOI:10.1016/0022-5096(92)90063-8.
  * Mueller et al. (2013) Mueller, S., Kruck, B., Baudisch, P., 2013.  Laserorigami: Laser-cutting 3d objects, in: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, Association for Computing Machinery, New York, NY, USA. p. 2585–2592.  DOI:10.1145/2470654.2481358.
  * Naebe and Shirvanimoghaddam (2016) Naebe, M., Shirvanimoghaddam, K., 2016\.  Functionally graded materials: A review of fabrication and properties.  Applied Materials Today 5, 223–245.  DOI:10.1016/j.apmt.2016.10.001.
  * Needleman (1976) Needleman, A., 1976.  Necking of pressurized spherical membranes.  Journal of the Mechanics and Physics of Solids 24, 339–359.  DOI:10.1016/0022-5096(76)90008-9.
  * Oden and Sato (1967) Oden, J., Sato, T., 1967.  Finite strains and displacements of elastic membranes by the finite element method.  International Journal of Solids and Structures 3, 471–488.  DOI:10.1016/0020-7683(67)90002-9.
  * Pamulaparthi Venkata and Saxena (2019) Pamulaparthi Venkata, S., Saxena, P., 2019\.  Instabilities in the free inflation of a nonlinear hyperelastic toroidal membrane.  Journal of Mechanics of Materials and Structures 14, 473–496.  DOI:10.2140/jomms.2019.14.473.
  * Patil and DasGupta (2013) Patil, A., DasGupta, A., 2013\.  Finite inflation of an initially stretched hyperelastic circular membrane.  European Journal of Mechanics-A/Solids 41, 28–36.  DOI:10.1016/j.euromechsol.2013.02.007.
  * Pelliciari et al. (2022) Pelliciari, M., Sirotti, S., Aloisio, A., Tarantino, A.M., 2022\.  Analytical, numerical and experimental study of the finite inflation of circular membranes.  International Journal of Mechanical Sciences 226, 107383.  DOI:10.1016/j.ijmecsci.2022.107383.
  * Pelliciari and Tarantino (2022) Pelliciari, M., Tarantino, A.M., 2022\.  A continuum model for circular graphene membranes under uniform lateral pressure.  Journal of Elasticity 151, 273–303.  DOI:10.1007/s10659-022-09937-w.
  * Pipkin (1986a) Pipkin, A., 1986a.  Continuously distributed wrinkles in fabrics.  Archive for Rational Mechanics and Analysis 95, 93–115.  DOI:10.1007/BF00281083.
  * Pipkin (1986b) Pipkin, A.C., 1986b.  The relaxed energy density for isotropic elastic membranes.  IMA Journal of Applied Mathematics 36, 85–99.  DOI:10.1093/imamat/36.1.85.
  * Pipkin (1994) Pipkin, A.C., 1994.  Relaxed energy densities for large deformations of membranes.  IMA Journal of Applied Mathematics 52, 297–308.  DOI:10.1093/imamat/52.3.297.
  * Pipkin (1995) Pipkin, A.C., 1995.  Relaxed energy densities for anisotropic membranes, in: Parker, D.F., England, A.H. (Eds.), IUTAM Symposium on Anisotropy, Inhomogeneity and Nonlinearity in Solid Mechanics, Springer Netherlands, Dordrecht. pp. 333–338.  DOI:10.1007/978-94-015-8494-4_45.
  * Popereka and Balagurov (1970) Popereka, M.Y., Balagurov, V., 1970\.  Ferromagnetic films having a negative Poisson ratio.  Soviet Physics, Solid state 11, 2938–2943.
  * Pujara and Lardner (1978) Pujara, P., Lardner, T., 1978\.  Deformations of elastic membranes—Effect of different constitutive relations.  Zeitschrift für angewandte Mathematik und Physik ZAMP 29, 315–327.  DOI:10.1007/BF01601526.
  * Puntel et al. (2011) Puntel, E., Deseri, L., Fried, E., 2011.  Wrinkling of a stretched thin sheet.  Journal of Elasticity 105, 137–170.  DOI:10.1007/s10659-011-9340-7.
  * Reissner (1938) Reissner, E., 1938.  On tension field theory.  Proceedings of the Fifth International Congress for Applied Mechanics, Harvard University & MIT , 88–92.
  * Ren et al. (2018) Ren, X., Das, R., Tran, P., Ngo, T.D., Xie, Y.M., 2018\.  Auxetic metamaterials and structures: A review.  Smart Materials and Structures 27, 023001.  DOI:10.1088/1361-665X/aaa61c.
  * Rivlin and Saunders (1951) Rivlin, R.S., Saunders, D., 1951\.  Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber.  Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 243, 251–288.  DOI:10.1098/rsta.1951.0004.
  * Roddeman et al. (1987) Roddeman, D., Drukker, J., Oomens, C., Janssen, J., 1987\.  The wrinkling of thin membranes: Part I—Theory. DOI:10.1115/1.3173133.
  * Rumpel et al. (2005) Rumpel, T., Schweizerhof, K., Haßler, M., 2005.  Efficient finite element modelling and simulation of gas and fluid supported membrane and shell structures.  Textile Composites and Inflatable Structures , 153–172. DOI:10.1007/1-4020-3317-6_10.
  * Saxena et al. (2019) Saxena, P., Reddy, N.H., Pradhan, S.P., 2019.  Magnetoelastic deformation of a circular membrane: Wrinkling and limit point instabilities.  International Journal of Non-Linear Mechanics 116, 250–261.  DOI:10.1016/j.ijnonlinmec.2019.07.006.
  * Selvadurai (2022) Selvadurai, A., 2022.  Mechanics of pressurized planar hyperelastic membranes.  Philosophical Transactions of the Royal Society A 380, 20210319.  DOI:10.1098/rsta.2021.0319.
  * Sirotti et al. (2023) Sirotti, S., Pelliciari, M., Aloisio, A., Tarantino, A.M., 2023\.  Analytical pressure–deflection curves for the inflation of pre-stretched circular membranes.  European Journal of Mechanics-A/Solids 97, 104831.  DOI:10.1016/j.euromechsol.2022.104831.
  * Steigmann (1990) Steigmann, D., 1990.  Tension-field theory.  Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 429, 141–173.  DOI:10.1098/rspa.1990.0055.
  * Steigmann and Pipkin (1989a) Steigmann, D., Pipkin, A., 1989a.  Finite deformations of wrinkled membranes.  The Quarterly Journal of Mechanics and Applied Mathematics 42, 427–440.  DOI:10.1093/qjmam/42.3.427.
  * Steigmann and Pipkin (1989b) Steigmann, D.J., Pipkin, A.C., 1989b.  Wrinkling of Pressurized Membranes.  Journal of Applied Mechanics 56, 624–628.  DOI:10.1115/1.3176137.
  * Sun et al. (2014) Sun, J., Gao, H., Scarpa, F., Lira, C., Liu, Y., Leng, J., 2014.  Active inflatable auxetic honeycomb structural concept for morphing wingtips.  Smart Materials and Structures 23, 125023.  DOI:10.1088/0964-1726/23/12/125023.
  * Sun et al. (2017) Sun, S., Brandt, M., Easton, M., 2017.  Powder bed fusion processes: An overview.  Laser Additive Manufacturing , 55–77DOI:10.1016/B978-0-08-100433-3.00002-6.
  * Timoshenko (1983) Timoshenko, S., 1983.  History of Strength of Materials: With a Brief Account of the History of Theory of Elasticity and Theory of Structures.  Dover Publications, New York.
  * Ting and Chen (2005) Ting, T.C.T., Chen, T., 2005\.  Poisson’s ratio for anisotropic elastic materials can have no bounds.  The Quarterly Journal of Mechanics and Applied Mathematics 58, 73–82.  DOI:10.1093/qjmamj/hbh021.
  * Treloar (1944) Treloar, L., 1944.  Strains in an inflated rubber sheet, and the mechanism of bursting.  Rubber Chemistry and Technology 17, 957–967.  DOI:10.5254/1.3546716.
  * Venkata et al. (2023) Venkata, S.P., Balbi, V., Destrade, M., Accoto, D., Zurlo, G., 2023.  Programmable wrinkling for functionally-graded auxetic circular membranes.  Extreme Mechanics Letters 63, 102045\.  DOI:10.1016/j.eml.2023.102045.
  * Wagner (1931) Wagner, H., 1931.  Flat sheet metal girders with very thin metal web. Part I: General theories and assumptions.  Technical Report.
  * Wojciechowski and Brańka (1989) Wojciechowski, K., Brańka, A., 1989\.  Negative Poisson ratio in a two-dimensional “isotropic”solid.  Physical Review A 40, 7222\.  DOI:10.1103/physreva.40.7222.
  * Wriggers and Taylor (1990) Wriggers, P., Taylor, R., 1990\.  A fully non-linear axisymmetrical membrane element for rubber-like materials.  Engineering Computations 7, 303–310.  DOI:10.1108/eb023817.
  * Yang and Feng (1970) Yang, W., Feng, W., 1970.  On axisymmetrical deformations of nonlinear membranes.  Journal of Applied Mechanics 37, 1002–1011.  DOI:10.1115/1.3408651.
  * Yang and Lu (1973) Yang, W.H., Lu, C.H., 1973.  General Deformations of Neo-Hookean Membranes.  Journal of Applied Mechanics 40, 7–12.  DOI:10.1115/1.3422977.
  * Yang et al. (2021) Yang, X., Yu, L., Long, R., 2021.  Contact mechanics of inflated circular membrane under large deformation: Analytical solutions.  International Journal of Solids and Structures 233, 111222.  DOI:10.1016/j.ijsolstr.2021.111222.
  * Yuan et al. (2021) Yuan, J., Liu, X., Xia, H., Huang, Y., 2021.  Analytical solutions for inflation of pre-stretched elastomeric circular membranes under uniform pressure.  Theoretical and Applied Mechanics Letters 11, 100243.  DOI:10.1016/j.taml.2021.100243.
  * Zhou et al. (2018) Zhou, L., Wang, S., Li, L., Fu, Y., 2018.  An evaluation of the Gent and Gent-Gent material models using inflation of a plane membrane.  International Journal of Mechanical Sciences 146, 39–48.  DOI:10.1016/j.ijmecsci.2018.07.035.