# From a distance: Shuttleworth revisited

Stefanie Heyden<EMAIL_ADDRESS>ETH Zürich, Institute for
Building Materials, 8093 Zürich, Switzerland.  Nicolas Bain
<EMAIL_ADDRESS>University of Lyon, Université Claude Bernard Lyon 1,
CNRS, Institut Lumière Matière, F-69622, Villeurbanne, France.

###### Abstract

The Shuttleworth equation: a linear stress-strain relation ubiquitously used
in modeling the behavior of soft surfaces. Its validity in the realm of
materials subject to large deformation is a topic of current debate. Here, we
zoom out to derive the constitutive behavior of the surface from the general
framework of finite kinematics. We distinguish cases of finite and
infinitesimal surface relaxation prior to an infinitesimal applied
deformation. The Shuttleworth equation identifies as the Cauchy stress measure
in the fully linearized setting. We show that in both finite and linearized
cases, measured elastic constants depend on the utilized stress measure. In
addition, we discuss the physical implications of our results and analyze the
magnitude of surface relaxation in the light of two different test cases.

††preprint: APS/123-QED

## Introduction

In most solids, small forces lead to small deformations. A soft solid, on the
contrary, can experience large deformations under minute forces: a phenomenon
as minor as depositing a millimetric droplet leads to divergent deformations
of its surface Style  _et al._ (2017); Andreotti and Snoeijer (2020); Style
_et al._ (2013a); Hui and Jagota (2014) (Fig. 1). Such materials, which can
take the form of gels, pastes, or elastomers, are ubiquitous in our lives:
They amount to most of our body tissues, and can be used as lubricants, glues,
and water-repellent coatings. In past years, it has been shown that surface
stresses are essential in wetting, adhesion, and fracture, and can be
harvested in composites Style  _et al._ (2017); Andreotti and Snoeijer (2020);
Style  _et al._ (2013a); Hui and Jagota (2014); Smith-Mannschott  _et al._
(2021); Schulman  _et al._ (2018); Hui  _et al._ (2020); Style  _et al._
(2013b); Jensen  _et al._ (2015); Grzelka  _et al._ (2017); Style  _et al._
(2015). Yet, the physical origins of surface stresses are poorly understood.
This statement is especially true in gels, in which cohabitation of a
crosslinked polymer network and a liquid solvent complexifies the link between
molecular structure and mechanical properties.

To tackle this fundamental question, the dominant approach is to investigate
surface elastic properties Bain  _et al._ (2021); Heyden  _et al._ (2021,
2022); Carbonaro  _et al._ (2020); Xu  _et al._ (2018). For instance, surface
topography measurements of a stretched patterned silicone gel revealed an
elastic surface, with surface stresses that increase with surface deformations
Bain  _et al._ (2021). This result hints towards a role of the crosslinked
polymeric network in the surface constitutive behavior of silicone gels. On
the contrary, deformation measures of a spinning hydrogel bead evidenced a
constant surface stress, independent of surface deformations, akin to the
surface tension of the solvent Carbonaro  _et al._ (2020).

In its simplest form, the most common description for the surface mechanics of
soft solids relates the surface stresses $\bm{\Upsilon}_{s}$ to the surface
strains $\bm{\epsilon}_{s}$ and free energy $W_{s}$,

$\bm{\Upsilon}_{s}=W_{s}\bm{I}+\frac{\partial
W_{s}}{\partial\bm{\epsilon}_{s}},$ (1)

and is commonly called the Shuttleworth equation Shuttleworth (1950). This
description is restricted to the linear regime, in which small deformations
prevail. It is, however, largely applied to surfaces that undergo large
deformations. In this setting, its validity and the physical interpretations
that follow have been rightfully questioned Gutman (2022); Bain  _et al._
(2022); Pandey  _et al._ (2020); Masurel  _et al._ (2019). In parallel,
extensive works in continuum mechanics laid out the framework of surface
elasticity (see, e.g., Gurtin and Murdoch (1975); Gurtin  _et al._ (1998);
Huang and Wang (2007); Krichen  _et al._ (2019); Mozaffari  _et al._ (2020)).
Here, we aim to link both paths.

Figure 1: Experimental surface profiles of a initially flat soft solids in
contact with (a) a drop of glycerol, (b) a silica bead and (c) a glass
coverslip. The data were obtained from confocal imaging of fluorescent beads
deposited on a silicone gel of shear modulus $\mu\sim 2\,\mbox{kPa}$ (same
protocol as Jagota  _et al._ (2012); Style  _et al._ (2013a); Xu  _et al._
(2017, 2018)). Each blue circle represents a detected fluorescent bead, and
the light blue areas schematically represent the contacting objects.

The major drawback of the Shuttleworth equation is that it neglects the bigger
picture. Hence, any features stemming from a differentiation between initial
and final configurations are lost. In the following, we take advantage of the
framework of finite kinematics to rigorously derive the surface stress-strain
relation from a strain energy density. Prior to an applied deformation, soft
solids are usually detached from a container and their surfaces undergo an
initial relaxation, which induces residual bulk stresses. Because soft solids
are easy to deform, this surface relaxation can be large Jagota  _et al._
(2012); Mora  _et al._ (2013). We distinguish cases of infinitesimal and
finite surface relaxation. We highlight cases in which different stress
measures do not coincide upon linearization. The given framework should incite
experimentalists to choose the most suitable stress-strain relation and
carefully interpret measured surface elastic parameters. We exemplify this
interpretation considering two different test cases.

## Finite kinematics

### General case

We start with a brief outline of the basic finite kinematics setting. Let us
consider a solid in two states. A reference state, prior to being deformed,
and a deformed state (Fig. 2). In the reference state, it is well established
that the surface exhibits mechanical stresses, which in the simplest case have
the form of a uniform and isotropic surface tension Style  _et al._ (2017);
Andreotti and Snoeijer (2020). Let us furthermore assume that the surface
stresses can be thermodynamically defined by a surface strain energy density
$W^{\rm R}$ in the reference configuration.

Following the Piola transform specialized to two dimensions, we can define a
free energy density expressed in the current configuration $W^{\rm C}$ as

$\int_{S}W^{\rm C}\,dS=\int_{S_{\rm R}}W^{\rm C}J_{s}\,dS_{\rm
R}=\int_{S_{R}}\,W^{\rm R}\,dS_{R},$ (2)

with $S_{R}$ and $S$ the areas in the reference- and current configurations,
respectively. Furthermore,
$J_{s}=\text{det}(\bm{F}_{s})=\text{det}(\bm{I}+\bm{\nabla}_{s}\bm{u})$
measures the local change in area, where $\bm{F}_{s}$ is the deformation
gradient and $\bm{u}$ the displacement field (details are given in
Supplementary Section 1). Based on material frame indifference, any strain
energy density can be written as a function of the Green-Lagrange strain
tensor $\bm{E}_{s}=\frac{1}{2}(\bm{F}_{s}^{\rm T}\bm{F}_{s}-\bm{I})$. Here, as
in the rest of the manuscript, all vector quantities, such as displacement
fields, and all tensor quantities, such as stress- and strain fields, are
projected onto the surface (see Supplementary Section 1 for details).

Figure 2: Sketch of reference configuration (left) and deformed configuration
(right).

Depending on the experimental conditions, different stress measures may be
used for comparison to theory. The first Piola-Kirchhoff stress measure
$\bm{P}_{s}$, also known as the engineering stress, represents forces in the
current configuration per unit referential area. Therefore, it is likely the
quantity of interest in traction-controlled experiments Mozaffari  _et al._
(2020). In contrast, the second Piola-Kirchhoff stress measure $\bm{S}_{s}$
constitutes forces mapped back to the reference configuration per unit
referential area. Finally, the Cauchy stress measure $\bm{\sigma}_{s}$, also
denoted as true stress, refers to forces in the current configuration per unit
current area. It is then the stress measure used in a Eulerian setting, or
phenomena where the reference state is ill defined, as in liquid flows. By
definition, the second Piola-Kirchhoff stress derives directly from the free
energy density in the reference configuration,

$\bm{S}_{s}=\frac{\partial W^{\rm R}}{\partial\bm{E}_{s}},$ (3)

and the two other stress measures account for the change of reference frame by
multiplication with the deformation gradient

$\bm{P}_{s}=\bm{F}_{s}\frac{\partial W^{\rm R}}{\partial\bm{E}_{s}}\text{, and
}\bm{\sigma}_{s}=J_{s}^{-1}\bm{F}_{s}\frac{\partial W^{\rm
R}}{\partial\bm{E}_{s}}\bm{F}_{s}^{\rm T}.$ (4)

If one considers the free energy in the current configuration Eq.(2), as is
implicitly assumed in the Shuttleworth equation, the Cauchy stress takes a
form similar to Eq.(1)

$\bm{\sigma}_{s}=W^{\rm C}\mathbf{I}+\bm{F}_{s}\frac{\partial W^{\rm
C}}{\partial\bm{E}_{s}}\bm{F}_{s}^{\rm T},$ (5)

and the other stresses measures also have a similar expression (see
Supplementary Section 2 for details).

At this point, it is instructive to define a constitutive equation for the
surface free energy. If we assume the surface energy to be strain-independent
$W^{\rm C}=\gamma$, as in liquids, the Cauchy stresses Eq. (5) correspond to a
liquid-like isotropic surface tension $\bm{\sigma}_{s}=\gamma\mathbf{I}$.

For a strain-dependent surface energy, without loss of generality we use the
St. Venant-Kirchhoff model, which is the simplest extension of linear
elasticity that accounts for geometric nonlinearities. To account for the
surface tension in the reference state, we enrich this model with a constant
surface energy in the current configuration,

$W^{R}(\bm{E}_{s})=\gamma\,J_{s}+\mu_{s}\,\text{tr}(\bm{E}_{s}\bm{E}_{s})+\frac{1}{2}\lambda_{s}(\text{tr}(\bm{E}_{s}))^{2}.$
(6)

Here, $\mu_{s}$ and $\lambda_{s}$ are the surface shear modulus and surface
Lamé parameter, respectively. Physically speaking, $\gamma$ is equivalent to
the surface tension of a liquid, while $(\mu_{\rm s},\lambda_{\rm s})$ are the
surface elastic constants attributing an energetic cost to elastic deformation
from a reference state. In this framework, a simple derivation leads to the
Cauchy stress as a function of the elastic parameters

$\bm{\sigma}_{s}=\gamma\bm{I}+\bm{F}_{s}\left(2\mu_{s}\bm{E}_{s}+\lambda_{s}\text{tr}(\bm{E}_{s})\bm{I}\right)\bm{F}_{s}^{T},$
(7)

and similarly to the first and second Piola-Kirchhoff stresses

$\displaystyle\bm{P}_{s}$ $\displaystyle=\gamma
J_{s}\bm{F}_{s}\left(\bm{F}_{s}^{\rm
T}\bm{F}_{s}\right)^{-1}+2\mu_{s}\bm{F}_{s}\bm{E}_{s}+\lambda_{s}\text{tr}(\bm{E}_{s})\bm{F}_{s},$
(8) $\displaystyle\bm{S}_{s}$ $\displaystyle=\gamma J_{s}\left(\bm{F}_{s}^{\rm
T}\bm{F}_{s}\right)^{-1}+2\mu_{s}\bm{E}_{s}+\lambda_{s}\text{tr}(\bm{E}_{s})\bm{I}.$
(9)

### The case of soft solids

Soft solids are often cured within a mold, from which they are detached before
being used. Upon detachment, soft solids undergo a surface relaxation due to
the change of interface, and enhanced wherever the local curvature is nonzero
Jagota  _et al._ (2012); Mora  _et al._ (2013); Hui  _et al._ (2020). We
therefore distinguish three mechanical states (Fig. 3). The soft solid with
the shape of the mold before it relaxes, $\Omega^{*}$, after relaxed from the
mold but prior to any external load or displacement, $\Omega_{0}$, and after
being deformed by external loads $\Omega$.

In the first, fictitious state $\Omega^{*}$, the bulk of the soft solid is
stress-free. For this reason, we will use this state as the reference
configuration (Fig. 2). The surface, however, is not stress-free, as is
accounted for in our definition of the surface energy Eq. (6). While the
second state, $\Omega_{0}$, is the one experimentalists physically work with,
we will consider it as an intermediate state because it contains potentially
finite residual bulk stresses due to prior surface relaxation. We note,
however, that these two states coincide when the surface does not undergo
prior relaxation, which happens when the soft solid is tested in the state it
was prepared.

In the framework of finite kinematics, the deformation gradient $\bm{F}_{s}$
maps material points from the stress-free configuration $\Omega^{*}$ to the
deformed configuration $\Omega$. We also have
$\bm{F}^{*}_{s}=\bm{I}+\bm{\nabla}_{s}\bm{u}^{*}$ mapping from the stress-free
configuration $\Omega^{*}$ to the relaxed configuration $\Omega_{0}$, and
$\bm{F}^{0}_{s}=\bm{I}+\bm{\nabla}_{s}\bm{u}^{0}$ mapping from the relaxed
configuration $\Omega_{0}$ to the deformed configuration $\Omega$. By
composition of mappings, the total deformation follows as
$\bm{F}_{s}=\bm{F}^{0}_{s}\bm{F}^{*}_{s}$.

Figure 3: Sketch of fictitious stress-free configuration $\Omega^{*}$, e.g., a
cured sample within a mold), relaxed configuration $\Omega_{0}$, e.g., a
sample after removal from the mold, and deformed configuration $\Omega$. The
stress-free and the current configurations have a surface strain energy
density $W^{\rm R}$ and $W^{\rm C}$, respectively.

## Linearized Kinematics

In practice, experiments most easily impose deformations $\bm{E}^{0}_{s}$,
from the relaxed state $\Omega_{0}$ to the deformed state $\Omega$. Here, we
will focus on the case where the imposed deformations are small, for which
$\bm{E}^{0}_{s}\sim\bm{\epsilon}^{0}_{s}$. We then elucidate how the measured
stresses vary with imposed surface strain
$\bm{\epsilon}^{0}_{s}=[\bm{\nabla}_{s}\bm{u}^{0}+(\bm{\nabla}_{s}\bm{u}^{0})^{T}]/2$,
both when the relaxation deformation $\bm{E}^{*}_{s}$ is infinitesimal and
finite.

### Infinitesimal surface relaxation

Let us assume that the deformation due to the relaxation of the surface is
small $\bm{E}^{*}_{s}\sim\bm{\epsilon}^{*}_{s}$. In this scenario, the
resultant total strain is a simple additive decomposition
$\bm{\epsilon}_{s}=\bm{\epsilon}_{s}^{*}+\bm{\epsilon}_{s}^{0}$.

After linearization, stress measures are usually assumed to coincide. This
assumption, however, is not true when initial stresses are present Krichen
_et al._ (2019), as in the case of solid surfaces. We thus need to carefully
distinguish different stress measures in the realm of linearized kinematics.
At first order in strains
$\bm{\epsilon}_{s}=(\bm{\nabla}_{s}\bm{u}+\bm{\nabla}_{s}\bm{u}^{T})/2$, the
linearized Cauchy stress $\tilde{\bm{\sigma}}$ simplifies to the Hookean form

$\tilde{\bm{\sigma}}_{s}=\gamma\bm{I}+2\mu_{s}\bm{\epsilon}_{s}+\lambda_{s}\text{tr}(\bm{\epsilon}_{s})\bm{I}.$
(10)

Based on the additive decomposition of strains, Cauchy stresses result in a
contribution from the surface relaxation $\tilde{\bm{\sigma}}_{s}^{*}$ and a
contribution from the imposed deformations $\tilde{\bm{\sigma}}_{s}^{0}$

$\tilde{\bm{\sigma}}_{s}=\gamma\bm{I}+\tilde{\bm{\sigma}}_{s}^{*}+\tilde{\bm{\sigma}}_{s}^{0},$
(11)

where
$\tilde{\bm{\sigma}}_{s}^{\alpha}=2\mu_{s}\bm{\epsilon}_{s}^{\alpha}+\lambda_{s}\text{tr}(\bm{\epsilon}_{s}^{\alpha})\bm{I}$.
Unsurprisingly, the surface relaxation here only comes as an additive stress.
It will therefore not influence the estimation of the surface elastic moduli
$\mu_{s}$ and $\lambda_{s}$.

Similarly, the second Piola-Kirchhoff stress takes the form

$\tilde{\bm{S}}_{s}=\gamma\bm{I}+2\mu_{s}^{S}\bm{\epsilon}_{s}+\lambda_{s}^{S}\text{tr}(\bm{\epsilon}_{s})\bm{I},$
(12)

with effective elastic constants $\mu_{s}^{S}=\mu_{s}-\gamma$ and
$\lambda_{s}^{S}=\lambda_{s}+\gamma$. The first Piola-Kirchhoff stress is non-
symmetric

$\tilde{\bm{P}}_{s}=\gamma\bm{I}+2\mu_{s}^{P}\bm{\epsilon}_{s}+\lambda_{s}^{P}\text{tr}(\bm{\epsilon}_{s})\bm{I}+\gamma^{C}\bm{\omega}_{s},$
(13)

with $\mu_{s}^{P}=\mu_{s}-\gamma/2$, $\lambda_{s}^{P}=\lambda_{s}^{S}$, and
$\bm{\omega}_{s}=(\bm{\nabla}_{s}\bm{u}-\bm{\nabla}_{s}\bm{u}^{T})/2$ the
infinitesimal rotation tensor (see Supplementary Section 3 for details of the
linearization). Both the first and second Piola-Kirchhoff stress measures can
also be decomposed akin to the Cauchy stress Eq. (11).

From Eqs. (10) to (13), we note that the three linearized stress measures are
only equal in two scenarios. First, when the solid is unstretched
$\bm{\epsilon}_{s}=0$. All stress measures are then trivially equal to the
prestress $\gamma\mathbf{I}$, which is the surface tension of the solid at
rest. Second, when the prestress is much smaller than the surface elastic
constants $\gamma\ll(\mu_{\rm s},\lambda_{\rm s})$. In this case, which
primarily pertains to hard solids, all stress measures are equal to the
Hookean form Eq.(10). Besides these cases, however, linearized stress measures
need to be carefully distinguished from each other. This applies not only to
soft solids, for which surface tension and surface elasticity can be of the
same order of magnitude Bain  _et al._ (2021); Heyden  _et al._ (2022), but
also for complex fluid-fluid interfaces Fuller and Vermant (2012).

### Finite surface relaxation

If the surface relaxation induces large deformations, we expand to linear
order in $\bm{\nabla}\bm{u}^{0}$ the imposed deformations and keep the
relaxation deformations $\bm{F}^{*}$ finite. At first order, we write the
dependence of Cauchy stress with imposed deformations

$\displaystyle\bar{\bm{\sigma}}_{s}=$
$\displaystyle\,\gamma\bm{I}+\bar{\bm{\sigma}}^{*}_{s}+\bm{\nabla}\bm{u}^{0}\bar{\bm{\sigma}}^{*}_{s}+\bar{\bm{\sigma}}^{*}_{s}(\bm{\nabla}\bm{u}^{0})^{T}$
$\displaystyle+2\mu_{s}[\bm{F}_{s}^{*}(\bm{F}_{s}^{*})^{T}]\bm{\epsilon}^{0}_{s}[\bm{F}_{s}^{*}(\bm{F}_{s}^{*})^{T}]$
$\displaystyle+\lambda_{s}\text{tr}[(\bm{F}_{s}^{*})^{T}\bm{\epsilon}^{0}_{s}\bm{F}_{s}^{*}]\bm{F}_{s}^{*}(\bm{F}_{s}^{*})^{T},$
(14)

where
$\bar{\bm{\sigma}}^{*}_{s}=\bm{F}^{*}_{s}\left(2\mu_{s}\bm{E}^{*}_{s}+\lambda_{s}\text{tr}(\bm{E}^{*}_{s})\bm{I}\right)(\bm{F}^{*}_{s})^{T}$
is the stress contribution due to surface relaxation (see Supplementary
Section 3 for details, and Supplementary Section 4 for a full derivation in
the absence of surface shear). Here, not only does the relaxation impose an
additional stress term $\bar{\bm{\sigma}}^{*}_{s}$, it also mixes non-
trivially into the terms that include the surface elastic parameters
$(\lambda_{s},\mu_{s})$. In practice, the exact contribution from the finite
surface relaxation depend on the sample geometry and has to be estimated
accordingly. For the sake of completeness, we estimate the strain dependence
of the Second Piola-Kirchhoff stress as

$\displaystyle\bar{\bm{S}}_{s}$ $\displaystyle=\bar{\bm{S}}_{s}^{*}+\gamma
J_{s}^{*}(\bm{F}_{s}^{*})^{-1}\left[\text{tr}(\bm{\epsilon}^{0})-2\bm{\epsilon}^{0}_{s}\right](\bm{F}_{s}^{*})^{-T}$
$\displaystyle+2\mu_{s}(\bm{F}_{s}^{*})^{T}\bm{\epsilon}^{0}_{s}\bm{F}_{s}^{*}+\lambda_{s}\text{tr}((\bm{F}_{s}^{*})^{T}\bm{\epsilon}^{0}_{s}\bm{F}_{s}^{*})\bm{I}.$
(15)

Finally, the First Piola-Kirchhoff stress tensor follows as

$\bar{\bm{P}}_{s}=\bm{F}_{s}^{*}\bar{\bm{S}}_{s}+\bm{\nabla}_{s}\bm{u}^{0}\bar{\bm{S}}_{s}^{*},$
(16)

where $\bar{\bm{S}}_{s}^{*}=\gamma J_{s}^{*}\left[(\bm{F}_{s}^{*})^{\rm
T}\bm{F}_{s}^{*}\right]^{-1}+2\mu_{s}\bm{E}_{s}^{*}+\lambda_{s}\text{tr}(\bm{E}_{s}^{*})\bm{I}$
is the Second Piola-Kirchhoff stress contributions to surface relaxation.
Although the expressions (14) to (16) are cumbersome, we note that they do not
coincide even in the case of no imposed deformations
$\bm{\epsilon}_{s}^{0}=\bm{0}$.

Overall, whether the surface relaxation induces small or finite deformations,
the different stress measures differ from each other. While we can extract
effective surface elastic coefficients in the case of infinitesimal surface
relaxation (Eqs. (10) to (13)), accounting for finite relaxation (Eqs. (14) to
(16)) prevents having a simple constitutive equation between stresses and
imposed deformation $\bm{\nabla}_{s}\epsilon^{0}_{s}$.

### Estimation of the surface relaxation

To understand when surface relaxation induces infinitesimal or finite surface
strains, we examine how an incompressible soft solid of shear modulus $\mu$
and surface tension $\gamma$ responds in two canonical examples. As we are
looking for an estimate of the surface strain we do not account for surface
elasticity effects, which would be of higher order.

(a)(b)(c)(d)$\bm{F}^{*}$$\bm{F}^{*}$$R$$r$$L$$l$$a_{0}$$w$$a_{1}$$w$ Figure 4:
(a) Surface profile of periodic grooves of wavelength $w$. Left: initial
profile, with amplitude $a_{0}$. Right: after relaxation, with amplitude
$a_{1}$, estimated for a material of shear modulus $\mu=2\,\rm kPa$, surface
tension $\gamma=20\,\rm mN/m$, wavelength $w=50\,\mu\rm m$ and initial
amplitude $a_{0}=2\,\mu\rm m$. (b) Cylindrical rod before (left) and after
relaxation (right). (c) Contour plot for the surface relaxation axial strain
$\epsilon_{s\parallel}$ as a function of shear modulus and cylinder radius,
computed from Eq.(19) with surface tension $\gamma=20\,\rm mN/m$. The black
dashed line represents stands for $R=L_{\rm ec}$. (d) Deviation factors $P$
and $L$, quantifying the effect of surface relaxation on estimated surface
modulus (Eqs.(17) and (20)).

First, we assume that it is cured into a mold with periodic rectangular
grooves of wavelength $w$ and initial amplitude $a_{0}$. For simplicity, we
assume that the solid is much thicker than the pattern wavelength, akin to the
experimental system in Bain  _et al._ (2021). After demolding, the surface
topography relaxes to a nearly rectangular wave with final amplitude
$a_{1}\sim a_{0}/(1+|q|L_{\rm ec})$, where $q=2\pi/w$ is the pattern
wavevector and $L_{\rm ec}=\gamma/2\mu$ is the elastocapillary length Jagota
_et al._ (2012); Hui  _et al._ (2020); Bain  _et al._ (2021) (Fig. 4a). During
this process, one period of the surface goes from its initial length
$l_{0}=w+2a_{0}$ to a final length $l_{1}\sim w+2a_{1}$. We define the surface
relaxation strain $\epsilon_{s}^{*}=(l_{1}-l_{0})/l_{0}$ from the difference
in surface length before and after demolding.

In this simplified example, we distinguish two regimes. When the pattern is
very flat, $a\ll w$, the surface strain $\epsilon_{s}^{*}\sim(a_{1}-a_{0})/w$
is always infinitesimal and we can use the fully linearized framework (Eqs.
(10) to (13)). When the pattern is not flat, i.e., the amplitude is either
comparable to or larger than the wavelength, the surface strain
$\epsilon_{s}^{*}\sim(a_{1}-a_{0})/a_{0}\sim qL_{\rm ec}/(1+qL_{\rm ec})$ is
infinitesimal when the wavelength is much larger than the elastocapillary
length, $qL_{\rm ec}\ll 1$, and finite when the wavelength is smaller than or
comparable to $L_{\rm ec}$. In this last case, we have to account for finite
relaxations (Eqs. (14) to (16)). To do so, we assume the surface relaxation
strains are tangential to the surface profile $\left(\epsilon_{\rm
s\parallel}^{*},\epsilon_{\rm s\perp}^{*}\right)=\left(\epsilon_{\rm
s}^{*},0\right)$, and that we impose an external deformation that manifests as
a longitudinal strain $\epsilon_{\rm s}^{0}$. The longitudinal Cauchy stress,

$\bar{\sigma}_{\rm s\parallel}=\gamma+\bar{\sigma}_{\rm
s\parallel}^{*}+(2\mu_{\rm s}+\lambda_{\rm s})P(\epsilon_{\rm
s}^{*})\epsilon_{\rm s}^{0},$ (17)

is then linear in imposed strain $\epsilon_{\rm s}^{0}$ and polynomial in
relaxation strains
$P(\epsilon)=(1+\epsilon)^{2}\left(1+4\epsilon+2\epsilon^{2}\right)$. When the
surface relaxation strain is infinitesimal $\epsilon_{\rm s}^{*}\ll 1$, we
recover the surface modulus $(2\mu_{\rm s}+\lambda_{\rm s})$ that can be
calculated from the fully linear Cauchy stress Eq.(10). Otherwise, the
estimated surface modulus deviates from the true modulus by a factor
$P(\epsilon_{\rm s}^{*})$ that increases rapidly with increasing relaxation
surface strain (Fig. 4d).

Second, we assume the soft solid is cured into a slender cylindrical mold of
length $L$ and radius $R$, with $L\gg R$. Once removed from the mold, the
length and radius change to $l$ and $r$ respectively (Fig. 4b). At first
order, this relaxation follows the uniform deformation field

$\bm{F^{*}}=\begin{pmatrix}\lambda^{*}&0&0\\\ 0&\lambda^{*}&0\\\
0&0&\lambda^{*-2}\end{pmatrix},$ (18)

where $\lambda^{*}=r/R$ is the radial stretch and $\lambda^{*-2}=l/L$ the
axial stretch.

In this framework, the stretch induced by the surface relaxation that
minimizes the total elastic energy,

$\lambda^{*}=\left(L_{\rm ec}/R+\sqrt{1+\left(L_{\rm
ec}/R\right)^{2}}\right)^{1/3},$ (19)

depends only on the ratio of elastocapillary length and cylinder radius
$L_{\rm ec}/R$ (see Supplementary Section 5 for details). Here, we denote two
types of surface strains: the axial strain $\epsilon_{\rm
s\parallel}^{*}=\lambda^{*-2}-1$, and the circumferential strain
$\epsilon_{\rm s\perp}^{*}=\lambda^{*}-1$. From Eq. (19), surface strains are
infinitesimal when the elastocapillary length is much smaller than the
cylinder radius, $L_{\rm ec}\ll R$, where bulk elasticity dominates.

Otherwise, the relaxation-induced deformations are finite (Fig. 4c). If we
assume they have no shear component, and impose a longitudinal deformation
that results in small surface strains $(\epsilon_{\rm
s\parallel}^{0},\epsilon_{\rm s\perp}^{0})=(\epsilon_{\rm
s}^{0},-\epsilon_{\rm s}^{0}/2)$, the longitudinal Cauchy stress

$\bar{\sigma}_{\rm s\parallel}=\gamma+\bar{\sigma}_{\rm
s\parallel}^{*}+\left(2\mu_{\rm s}P(\epsilon_{\rm
s\parallel}^{*})+\frac{\lambda_{\rm s}}{2}L(\epsilon_{\rm
s\parallel}^{*})\right)\epsilon_{\rm s}^{0}$ (20)

is linear in imposed strain $\epsilon_{\rm s}^{0}$ and non-linear in
relaxation strains $\epsilon_{\rm s\parallel}^{*}$ through $P(\epsilon_{\rm
s\parallel}^{*})$ and the polynomial function
$L(\epsilon)=(1+\epsilon)(1+8\epsilon+12\epsilon^{2}+4\epsilon^{3})$ (derived
from Supplementary Section 4). We recover the correct surface modulus
$(2\mu_{\rm s}+\lambda_{\rm s}/2)$ calculated from Eq.(10) in the case of
infinitesimal surface relaxation, and increasing deviation factors when
surface relaxation is finite (Fig. 4d).

## Conclusion

In this work, we have shown that the first step in determining surface elastic
constants of a soft solid is to assess how the sample reached its undeformed
state. For the case of finite surface relaxations, the constitutive behavior
cannot be reduced to the Shuttlework equation. Instead, measured surface
elastic constants will depend on the deformations induced by the prior
relaxation. In the fully linearized setting, surface relaxation will not
influence the estimation of surface elastic moduli. The measured elastic
constants, however, will depend on the used stress measure, even in the linear
regime.

## Acknowledgements

S.H. gratefully acknowledges funding via the SNF Ambizione grant PZ00P2186041.
The authors thank Eric R. Dufresne for useful discussions and suggestions.
Author contributions: N.B. and S.H. contributed equally to the development of
the theory, and to the redaction of the manuscript. N.B. performed the
experimental measurements and data analysis of surface deformations.

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