# Internal sites of actuation and activation in thin elastic films and
membranes of finite thickness

Tyler Lutz<EMAIL_ADDRESS>Institut für Physik, Otto-von-Guericke-
Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
Department of English Language and Literature, University of Chicago, Chicago,
IL 60637, USA  Andreas M. Menzel<EMAIL_ADDRESS>Institut für Physik, Otto-
von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg,
Germany  Abdallah Daddi-Moussa-Ider<EMAIL_ADDRESS>School of Mathematics and Statistics, The Open University, Walton Hall, Milton
Keynes MK7 6AA, UK

(August 28, 2024)

###### Abstract

Functionalized thin elastic films and membranes frequently feature internal
sites of net forces or stresses. These are, for instance, active sites of
actuation, or rigid inclusions in a strained membrane that induce
counterstress upon externally imposed deformations. We theoretically analyze
the geometry of isotropic, flat, thin, linearly elastic films or membranes of
finite thickness, laterally extended to infinity. At the mathematical core of
such characterizations are the fundamental solutions for localized force and
stress singularities associated with corresponding Green’s functions. We
derive such solutions in three dimensions and place them into the context of
previous two-dimensional calculations. To this end, we consider both no-slip
and stress-free conditions at the top and/or bottom surfaces. We provide an
understanding for why the fully free-standing thin elastic membrane leads to
diverging solutions in most geometries and compare these situations to the
truly two-dimensional case. A no-slip support of at least one of the surfaces
stabilizes the solution, which illustrates that the divergences in the fully
free-standing case are connected to global deformations. Within the mentioned
framework, our results are important for associated theoretical
characterizations of thin elastic films, whether supported or free-standing,
and of membranes subject to internal or external forces or stresses.

##  I Introduction

Under various circumstances, we meet thin elastic films and membranes being
exposed to the action of internal forces and stresses. One obvious situation
arises when such membranes are mechanically reinforced, for instance, by the
inclusion of relatively firmer fibers to increase the overall stability under
loading [1]. When the overall system is stretched, mechanical counterstresses
are generally exerted by these fibers on the elastic environment.

The same is true for any rigid inclusion upon functionalization of thin
elastic films or membranes. A specific example is given by elastic membranes
containing rigid magnetic or magnetizable inclusions [2]. The action of
loudspeakers can be mimicked in such cases by directly exerting magnetic
forces on the membrane itself, without additional external mechanical
components of stimulation.

Situations of supported elastic films on a substrate are encountered
frequently during measurements of atomic force microscopy (AFM). In the field
of soft matter, scales of the probes are given by lengths up to micrometers
[3, 4]. The tip used for this purpose directly exerts forces onto the film.
Depending on the situation, one can consider such forcing as internal, with
the AFM tip pressed into the membrane.

In a biological context, living cells generate internal and external stresses.
Depending on the time scales, layers of growing and/or migrating biological
cells [5, 6, 7, 8] can be considered as thin elastic or viscoelastic films
that generate internal stresses. Similarly, biofilms typically consist of
living bacteria inside a thin elastic extracellular matrix [9, 10, 11, 12],
supported by a substrate.

On the theoretical side, there have been some recent observations concerning
two-dimensional elastic systems as representations of free-standing thin
elastic membrane systems. Net forces acting in an in-plane direction within
infinitely extended two-dimensional systems lead to a formal logarithmic
divergence of the induced displacement field as a function of the distance
from the force center [13]. The two-dimensional elastic membrane, even if
infinitely extended, does not offer sufficient resistance against a net force.
Instead, the whole two-dimensional system will get displaced. This is in
contrast to three-dimensional materials, which do not experience such
diverging displacements in bulk in response to persistent point-like force
centers [14].

The divergence in two dimensions naturally cancels if the overall force on the
elastic membrane vanishes [15]. Particularly, this is the case when so-called
symmetric force dipoles [16] act on the material. Similarly, it vanishes in
the presence of, for example, a no-slip boundary [17, 18]. This means that, no
matter how distant a boundary is, it always has a decisive influence on the
displacement field resulting from a net force in the two-dimensional membrane
system.

Here, we employ a two-dimensional Fourier transformation technique to
investigate thin elastic films. Yet and notably, we here do not confine
ourselves to strictly two-dimensional systems. Instead, we consider elastic
films and membranes of finite thickness, that is, of finite extension in the
third dimension. The two-dimensional Fourier transformation technique has been
widely utilized in elucidating the behavior of elastic sheets and deformable
membranes. Its fundamental concept revolves around harnessing the
translationally invariant symmetry along horizontal planes. The approach
involves transformations of the partial differential equations governing
membrane displacements or fluid flows — particularly in fluid dynamics — into
remaining ordinary differential equations with respect to the out-of-plane
coordinate. Previously, this technique has been effectively used to describe
in low-Reynolds-number hydrodynamics the behavior near walls and interfaces
[19, 20, 21, 22, 23]. The same applies to the behavior of elastic membranes
undergoing shear and/or bending deformation modes [24, 25, 26, 27, 28, 29, 30,
31, 32].

When in the present work we consider flat elastic membranes or thin films that
are infinitely extended in the lateral direction, yet of finite thickness in
the normal, third direction, the situation is intermediate between genuinely
two-dimensional and three-dimensional, space-filling, bulk systems. An
important question from the conceptual perspective is the role of force-
induced divergences in this intermediate situation. How does it compare to the
results for two- and three-dimensional systems?

To address this question, the remainder of the paper is structured as follows.
In Sec. II, we present the two-dimensional Fourier transformation technique
applied to the governing equations of elasticity, along with solutions for
various boundary conditions. Specifically, these are no-slip and/or stress-
free conditions at the top and/or bottom surfaces. We explore both monopole
and dipole types of singularity and address the solutions in terms of
resulting Green’s functions. Our results are discussed in Sec. III, where we
also delineate the conditions under which the solutions converge and diverge.
The paper concludes in Sec. IV. Complex expressions pertaining to the solution
are presented in tables within Appendix A, while additional background
concerning the calculations is included in Appendix B.

##  II Mathematical formulation

The displacement field $\bm{u}(\bm{r})$ of the material points in the elastic
medium is governed by the Navier-Cauchy equation

$\mu\bm{\Delta}\bm{u}(\bm{r})+\left(\lambda+\mu\right)\bm{\nabla}\left(\bm{\nabla}\cdot\bm{u}(\bm{r})\right)+\bm{f}(\bm{r})=\mathbf{0}\,,$
(1)

with $\mu$ and $\lambda$ denoting the shear modulus and Lamé parameter,
respectively, and $\bm{f}(\bm{r})$ the force density field acting on the
elastic material. Both coefficients are related to each other via

$\lambda=\frac{2\mu\nu}{1-2\nu}\,,$ (2)

where $\nu\in(-1,1/2]$ is the Poisson ratio. The components of the stress
tensor read

$\sigma_{ij}=2\mu\varepsilon_{ij}+\lambda\varepsilon_{kk}\delta_{ij}\,.$ (3)

In this expression, $\delta_{ij}$ represents the Kronecker delta and equal to
the components of the unit matrix.
$\varepsilon_{ij}=(\nabla_{i}u_{j}+\nabla_{j}u_{i})/2$ includes the components
of the linearized strain tensor.

To derive solutions for the displacement field, we utilize a two-dimensional
Fourier transformation in both the $x$\- and $y$-direction. This
transformation is defined as follows. The forward two-dimensional Fourier
transformation of a function $g(\bm{\rho},z)$ is denoted as
$\widetilde{g}(\bm{k},z)$ and is expressed as

$\widetilde{g}(\bm{k},z)=\mathscr{F}\left\\{g(\bm{\rho},z)\right\\}=\int_{\mathbb{R}^{2}}g(\bm{\rho},z)\,e^{-i\bm{k}\cdot\bm{\rho}}\,\mathrm{d}^{2}\bm{\rho}\,,$
(4)

where $\bm{k}$ represents the wavevector in the two-dimensional plane of the
Fourier transformation. Here, $\bm{\rho}=(x,y)$ denotes the projection of the
position vector into the $xy$-plane. We do not perform a Fourier
transformation with respect to the $z$-component. Similarly, we define the
inverse two-dimensional Fourier transformation as

$g(\bm{\rho},z)=\frac{1}{\left(2\pi\right)^{2}}\int_{\mathbb{R}^{2}}\widetilde{g}(\bm{k},z)\,e^{i\bm{k}\cdot\bm{\rho}}\,\mathrm{d}^{2}\bm{k}\,.$
(5)

Additionally, we introduce the wavenumber $k=|\bm{k}|$, representing the
magnitude of the two-dimensional wavevector, and we define the unit vector
$\hat{\bm{k}}=\bm{k}/k$.

To address the elastic equations, we adopt a cylindrical coordinate system and
represent the unit wavevector as
$\hat{\bm{k}}=\cos\phi\,\hat{\bm{x}}+\sin\phi\,\hat{\bm{y}}$. Additionally, we
define the two-dimensional unit vector $\hat{\bm{t}}$ in the $xy$-plane
perpendicular to $\hat{\bm{k}}$ such that
$\hat{\bm{t}}=\sin\phi\,\hat{\bm{x}}-\cos\phi\,\hat{\bm{y}}$. Consequently,
the Fourier-transformed displacement field and force density field can be
projected onto the basis consisting of the unit vectors $\hat{\bm{k}}$ and
$\hat{\bm{t}}$. The longitudinal and transverse components of the Fourier-
transformed displacement field are defined as
$\widetilde{u}_{l}=\widetilde{\bm{u}}\cdot\hat{\bm{k}}$ and
$\widetilde{u}_{t}=\widetilde{\bm{u}}\cdot\hat{\bm{t}}$, respectively [33]. We
refer to the $z$-component $\widetilde{u}_{z}$ of the Fourier-transformed
displacement field as the normal component. Similarly, the longitudinal,
transverse, and normal components of the Fourier-transformed force density are
denoted as $\widetilde{f}_{l}$, $\widetilde{f}_{t}$, and $\widetilde{f}_{z}$,
respectively.

Next, in the new orthogonal basis, we derive the two-dimensional Fourier-
transformed elastic equations. We find that the transverse component of the
displacement field is completely independent of the longitudinal and normal
components,

$\left(\partial_{zz}-k^{2}\right)\widetilde{u}_{t}+\frac{\widetilde{f}_{t}}{\mu}\,=0\,.$
(6)

The longitudinal and normal components of the displacement field are governed
by a system of second-order differential equations,

$\displaystyle\left(\partial_{zz}-\frac{2\left(1-\nu\right)}{1-2\nu}\,k^{2}\right)\widetilde{u}_{l}+\frac{ik}{1-2\nu}\,\partial_{z}u_{z}+\frac{\widetilde{f}_{l}}{\mu}$
$\displaystyle=0\,,$ (7a)
$\displaystyle\left(\frac{2\left(1-\nu\right)}{1-2\nu}\,\partial_{zz}-k^{2}\right)\widetilde{u}_{z}+\frac{ik}{1-2\nu}\,\partial_{z}\widetilde{u}_{l}+\frac{\widetilde{f}_{z}}{\mu}$
$\displaystyle=0\,.$ (7b)

Although our focus lies in seeking solutions to the elastic equations
involving force singularities concentrated at one point, we remark that Eqs.
(6) and (7) apply to any arbitrary force density $\bm{f}(\bm{r})$, given that
its Fourier transformation is known.

###  II.1 Solution in a bulk elastic medium

We begin by revisiting the scenario involving a force singularity within an
infinitely extended, three-dimensional, bulk elastic medium. Thus, we examine
the effect of a force density represented by a Dirac delta function located at
the origin of our Cartesian coordinate system,
$\bm{f}(\bm{r})=\bm{F}\,\delta^{3}\left(\bm{r}\right)$. The corresponding two-
dimensionally Fourier-transformed quantity, according to Eq. (4), follows as
$\widetilde{\bm{f}}(z)=\bm{F}\,\delta(z)$. Below, $\bm{F}$ is likewise
expressed in the orthogonal basis $(\hat{\bm{k}},\hat{\bm{t}},\hat{\bm{z}})$.

Equation (6) represents a second-order differential equation. The
characteristic polynomial of the homogeneous equation features two distinct
roots, $\pm k$. As a consequence, the solution for the transverse component of
the displacement field takes the form $\widetilde{u}_{t}=Ae^{-k|z|}$, where
the constant $A$ is obtained from the jump in the first $z$-derivative due to
the Dirac delta function. Specifically,

$\left.\partial_{z}\widetilde{u}_{t}^{\infty}\right|_{z=0^{+}}-\left.\partial_{z}\widetilde{u}_{t}^{\infty}\right|_{z=0^{-}}=-\frac{F_{t}}{\mu}\,.$
(8)

This leads to

$\widetilde{u}_{t}^{\infty}=\frac{F_{t}}{2k}\,e^{-k|z|}\,.$ (9)

Figure 1:  (a) Illustration of the system setup. A force $\bm{F}$ acts onto
the thin elastic film or membrane of finite thickness $H$ at the position
$(0,0,h)$. The bottom surface of the film or membrane is located at $z=-H/2$,
the top surface at $z=H/2$. For this situation, three different sets of
boundary conditions are addressed. These are (b) no-slip boundary conditions
at both the top and bottom surface (2NOS), (c) the bottom boundary exhibiting
a no-slip condition while the top boundary remains stress-free (NOS-STF), and
(d) stress-free boundary conditions at both the top and bottom surfaces
(2STF). No-slip boundary conditions imply vanishing displacements
$\bm{u}=\bm{0}$, while stress-free conditions imply
$\bm{\sigma}\cdot\hat{\bm{z}}=\bm{0}$.

The characteristic polynomials associated with the homogeneous parts of Eqs.
(7) similarly yield two distinct roots $\pm k$, each with a multiplicity of 2.
Consequently, the longitudinal and normal components of the displacement field
take on the forms
$\widetilde{u}_{l}^{\infty\,\pm}=\left(\alpha_{1}^{\pm}+\alpha_{2}^{\pm}z\right)e^{\mp
kz}$ and
$\widetilde{u}_{z}^{\infty\,\pm}=\left(\beta_{1}^{\pm}+\beta_{2}^{\pm}z\right)e^{\mp
kz}$, where the superscripts ‘plus’ and ‘minus’ correspond to the regions
above and below the singularity. Continuity of the displacements across $z=0$
implies $\alpha_{1}^{+}=\alpha_{1}^{-}$ and $\beta_{1}^{+}=\beta_{1}^{-}$.
Further relations follow by introducing these functional forms into Eqs. (7)
away from the singularity position. Moreover, the Dirac delta function at
$z=0$ implies the jump conditions

$\displaystyle\left.\partial_{z}\widetilde{u}_{l}^{\infty}\right|_{z=0^{+}}-\left.\partial_{z}\widetilde{u}_{l}^{\infty}\right|_{z=0^{-}}$
$\displaystyle=-\frac{F_{l}}{\mu}\,,$ (10a)
$\displaystyle\left.\partial_{z}\widetilde{u}_{z}^{\infty}\right|_{z=0^{+}}-\left.\partial_{z}\widetilde{u}_{z}^{\infty}\right|_{z=0^{-}}$
$\displaystyle=-\frac{1-2\nu}{2\left(1-\nu\right)}\frac{F_{z}}{\mu}\,.$ (10b)

Upon solving the resulting system of linear equations, we acquire the final
expressions

$\displaystyle\widetilde{u}_{l}^{\infty}$
$\displaystyle=\frac{e^{-k|z|}}{8\mu\left(1-\nu\right)k}\left(\left(3-4\nu-k|z|\right)F_{l}-ikzF_{z}\right)\,,$
(11a) $\displaystyle\widetilde{u}_{z}^{\infty}$
$\displaystyle=\frac{e^{-k|z|}}{8\mu\left(1-\nu\right)k}\left(\left(3-4\nu+k|z|\right)F_{z}-ikzF_{l}\right)\,.$
(11b)

###  II.2 Image solution technique

Our goal is to quantify the static deformational response of an elastic
membrane of finite thickness $H$ to an applied static force acting at position
$h\in(-H/2,H/2)$, see Fig. 1(a). Initially, in the undeformed state, the
membrane features two flat, parallel, infinitely extended (top and bottom)
surfaces at $z=\pm H/2$. We examine three scenarios concerning the possible
confinement of these surfaces when a force singularity is applied between
these surfaces. These are (i) no-slip conditions both at the top and bottom
boundaries, see Fig. 1(b), (ii) one no-slip condition (top boundary) and one
stress-free condition (bottom boundary), see Fig. 1(c), and (iii) stress-free
conditions both at the top and bottom boundaries, see Fig. 1(d).

Utilizing the bulk solution derived in Fourier space in Sec. II.1, we express
the result as a combination of the bulk term and an additional contribution
necessary to meet the prescribed boundary conditions. This technique is
commonly known as the image solution technique. Specifically,

$\widetilde{\bm{u}}=\widetilde{\bm{u}}^{\infty}+\widetilde{\bm{u}}^{*}\,,$
(12)

with $\widetilde{\bm{u}}^{*}$ representing image solution.

We remark that the position of the force center in the bulk elastic medium
considered in Sec. II.1 was located at the origin. Obtaining the corresponding
expressions for the force acting at $z=h$ is straightforward in bulk through
simple translation. To this end, $z$ is replaced by $z-h$ in the expressions
for $\widetilde{u}_{t}$, $\widetilde{u}_{l}$, and $\widetilde{u}_{z}$ in Eqs.
(9) and (11).

The image solution $\widetilde{\bm{u}}^{*}$ satisfies the homogeneous parts of
Eqs. (6) and (7). It can be expressed in the form

$\widetilde{u}_{t}^{*}=c_{1}^{-}e^{-kz}+c_{1}^{+}e^{kz}\,$ (13)

for the transverse component and

$\displaystyle\widetilde{u}_{l}^{*}$
$\displaystyle=\left(c_{2}^{-}+c_{3}^{-}z\right)e^{-kz}+\left(c_{2}^{+}+c_{3}^{+}z\right)e^{kz}\,,$
(14a) $\displaystyle\widetilde{u}_{z}^{*}$
$\displaystyle=\left(c_{4}^{-}+c_{5}^{-}z\right)e^{-kz}+\left(c_{4}^{+}+c_{5}^{+}z\right)e^{kz}\,$
(14b)

for the longitudinal and normal components, where

$\displaystyle c_{4}^{\pm}$
$\displaystyle=\frac{i}{k}\left(\left(3-4\nu\right)c_{3}^{\pm}\mp
kc_{2}^{\pm}\right)\,,$ (15a) $\displaystyle c_{5}^{\pm}$ $\displaystyle=\mp
ic_{3}^{\pm}\,.$ (15b)

The wavenumber-dependent coefficients $c_{1}^{\pm}$, $c_{2}^{\pm}$, and
$c_{3}^{\pm}$ will subsequently be determined based on the boundary conditions
specified at $z=\pm H/2$, see Fig. 1.

At each no-slip surface, we enforce the vanishing components of the
displacement field, namely $\bm{u}=\bm{0}$, ensuring
$\widetilde{u}_{t}=\widetilde{u}_{l}=\widetilde{u}_{z}=0$ at these boundaries.
For a stress-free surface, we prescribe $\bm{\sigma}\cdot\hat{\bm{z}}=\bm{0}$,
implying the boundary conditions

$\displaystyle\partial_{z}\widetilde{u}_{t}$ $\displaystyle=0\,,$ (16a)
$\displaystyle\partial_{z}\widetilde{u}_{l}+ik\widetilde{u}_{z}$
$\displaystyle=0\,,$ (16b)
$\displaystyle\left(1-\nu\right)\partial_{z}\widetilde{u}_{z}+i\nu
k\widetilde{u}_{l}$ $\displaystyle=0\,.$ (16c)

Particularly, we note that, if $\nu=1/2$, Eq. (16c) becomes the
incompressibility condition for the elastic medium.

Subsequently, we express lengths in dimensionless units by normalizing them
with respect to the thickness of the membrane $H$. Effectively we thus set a
unit membrane thickness. In this way, the resulting expressions simplify.

Imposing the relevant boundary conditions on the top and bottom surfaces, we
obtain a system of six linear equations for each combination of boundary
conditions. Their solutions yield the expressions for the remaining unknown
coefficients $c_{j}^{\pm}$, $j\in\\{1,2,3\\}$. Due to their complexity and
lengthiness, and since it is straightforward to formulate the system of
equations using computer algebra systems, we do not include the details of
evaluation here.

First, as previously indicated, the transverse component of the displacement
field, see Eq. (13), remains unaffected by normal and longitudinal forces, see
Eq. (8). This results in a set of equations for $c_{1}^{\pm}$ when imposing
the relevant boundary conditions. For both types of boundary conditions, the
expressions for the coefficients $c_{1}^{\pm}$ follow in the form

$c_{1}^{\pm}=\frac{F_{t}}{2kd}\left(w_{0}e^{\mp kh}+w_{1}e^{k\left(1\pm
h\right)}\right).$ (17)

Here, the specific expressions for $w_{0}$, $w_{1}$, and $d$ are determined by
the specified set of boundary conditions. As depicted in Fig. 1, we introduce
the abbreviations 2NOS (two no-slip walls, NOS-STF (bottom no-slip and top
stress-free), and 2STF (two stress-free surfaces). We find that
$d=\mu\left(e^{2k}-1\right)$ for 2NOS and 2STF, and
$d=\mu\left(e^{2k}+1\right)$ for NOS-STF. In addition, $(w_{0},w_{1})=(1,-1)$,
$(-1,\pm 1)$, and $(1,1)$ for 2NOS, NOS-STF, and 2STF, respectively.

The coefficients $c_{2}^{\pm}$ and $c_{3}^{\pm}$ entail more complex
expressions, as they specify both the longitudinal and normal components of
the displacement field, see Eqs. (14) and (15). This results from their
combination in the coupled differential equations, see Eqs. (7). For ease of
reference, we introduce the abbreviations

$\sigma=3-4\nu\,\in\,[1,7)\,,$ (18)

together with $\delta_{\pm}=1\pm h$ and $\eta_{\pm}=1\pm 2h$. Imposing the
different sets of boundary conditions, the expressions for the coefficients
$c_{2}^{\pm}$ and $c_{3}^{\pm}$ determining the expressions for the
longitudinal and normal components of the image displacement field, see Eqs.
(14) and (15), for all sets of boundary conditions can be formulated as

$\displaystyle c_{2}^{\pm}$
$\displaystyle=\frac{1}{D}\sum_{n=0}^{3}\left(a_{n}^{\pm}F_{l}+ib_{n}^{\pm}F_{z}\right)e^{k\left(n\mp(-1)^{n}h\right)}\,,$
(19a) $\displaystyle c_{3}^{\pm}$
$\displaystyle=\frac{2k}{D}\sum_{n=0}^{3}\left(g_{n}^{\pm}F_{l}+iq_{n}^{\pm}F_{z}\right)e^{k\left(n\mp(-1)^{n}h\right)}\,.$
(19b)

Here, the specific expressions for the coefficients $a_{n}^{\pm}$,
$b_{n}^{\pm}$, $g_{n}^{\pm}$, and $q_{n}^{\pm}$ together with the denominator
$D$ depend on the specific boundary conditions. They are provided in Tab. A1
of Appendix A.

We now define the displacement Green’s function as

$\bm{u}(\bm{r})=\bm{\mathcal{G}}(\bm{r})\cdot\bm{f}\,.$ (20)

It is composed of the Green’s function associated with the bulk displacement
field, $\bm{\mathcal{G}}^{\infty}(\bm{r})$, and the Green’s function
associated with the image displacement field, $\bm{\mathcal{G}}^{*}(\bm{r})$,

${\bm{\mathcal{G}}(\bm{r})=\bm{\mathcal{G}}^{\infty}(\bm{r})+\bm{\mathcal{G}}^{*}(\bm{r}).}$
(21)

Based on Eqs. (9) and (11), we deduce that in Fourier space the components of
the Green’s function associated with the bulk displacement field for a force
acting at a distance $h$ above the center plane are

$\displaystyle\widetilde{\mathcal{G}}_{tt}^{\infty}(k,z)$
$\displaystyle=\frac{1}{2k}\,e^{-k|z-h|}\,,$ (22a)
$\displaystyle\widetilde{\mathcal{G}}_{ll}^{\infty}(k,z)$
$\displaystyle=\frac{\sigma-k|z-h|}{8\mu\left(1-\nu\right)k}\,e^{-k|z-h|}\,,$
(22b) $\displaystyle\widetilde{\mathcal{G}}_{lz}^{\infty}(k,z)$
$\displaystyle=-\frac{ik\left(z-h\right)}{8\mu\left(1-\nu\right)k}\,e^{-k|z-h|}\,,$
(22c) $\displaystyle\widetilde{\mathcal{G}}_{zz}^{\infty}(k,z)$
$\displaystyle=\frac{\sigma+k|z-h|}{8\mu\left(1-\nu\right)k}\,e^{-k|z-h|}\,.$
(22d)

These expressions can simply be read off from Eqs. (9) and (11), recalling the
vertical shift due to the position of the force center at $z=h$ instead of
$z=0$ and regarding Eq. (18). In addition, it follows that
$\widetilde{\mathcal{G}}_{zl}^{\infty}(k,z)=\widetilde{\mathcal{G}}_{lz}^{\infty}(k,z)$.

Next, in a similar way, we infer the the corresponding expressions for the
Fourier-transformed image solution of the Green’s function from the previous
results for the image displacement field. Utilizing Eqs. (13) and (17), we may
express the $tt$-component of the image Green’s function for all sets of
boundary conditions as

$\widetilde{\mathcal{G}}_{tt}^{*}(k,z)=\frac{1}{2kd}\left(w_{0}\left(\psi_{0}^{-}+\psi_{0}^{+}\right)+w_{1}\left(\psi_{1}^{-}+\psi_{1}^{+}\right)\right),$
(23)

where

$\psi_{n}^{\pm}=e^{k\left(n\mp\left(-1\right)^{n}h\pm z\right)}\,.$ (24)

Moreover, it follows from Eqs. (14), (15), and (19) that

$\widetilde{\mathcal{G}}_{ij}^{*}(k,z)=\frac{1}{D}\sum_{n=0}^{3}\left({A_{n}^{-}}_{ij}\psi_{n}^{-}+{A_{n}^{+}}_{ij}\psi_{n}^{+}\right)$
(25)

for $i,j\in\\{l,z\\}$ for all sets of boundary conditions. We here defined the
abbreviations

$\displaystyle{A_{n}^{\pm}}_{ll}$
$\displaystyle=a_{n}^{\pm}+2kzg_{n}^{\pm}\,,$ (26a)
$\displaystyle{A_{n}^{\pm}}_{lz}$
$\displaystyle=i\left(b_{n}^{\pm}+2kzq_{n}^{\pm}\right),$ (26b)
$\displaystyle{A_{n}^{\pm}}_{zl}$ $\displaystyle=\mp
i\left(a_{n}^{\pm}+2\left(kz\mp\sigma\right)g_{n}^{\pm}\right),$ (26c)
$\displaystyle{A_{n}^{\pm}}_{zz}$
$\displaystyle=\pm\left(b_{n}^{\pm}+2\left(kz\mp\sigma\right)q_{n}^{\pm}\right).$
(26d)

The expressions for $w_{0}$, $w_{1}$, $d$, $a_{n}^{\pm}$, $b_{n}^{\pm}$,
$g_{n}^{\pm}$, $q_{n}^{\pm}$, and $D$ again depend on the specific set of
boundary conditions and are listed below Eq. (17) and in Tab. A1 of Appendix
A.

It is beneficial to have available the expressions of the Green’s functions in
the standard Cartesian coordinate system as well. To this end, we invert the
transformation outlined below Eq. (5). This implies

$\displaystyle\widetilde{\mathcal{G}}_{xx}$
$\displaystyle=\widetilde{\mathcal{G}}_{ll}\cos^{2}\phi+\widetilde{\mathcal{G}}_{tt}\sin^{2}\phi\,,$
(27a) $\displaystyle\widetilde{\mathcal{G}}_{yy}$
$\displaystyle=\widetilde{\mathcal{G}}_{ll}\sin^{2}\phi+\widetilde{\mathcal{G}}_{tt}\cos^{2}\phi\,,$
(27b) $\displaystyle\widetilde{\mathcal{G}}_{xy}$
$\displaystyle=\left(\widetilde{\mathcal{G}}_{ll}-\widetilde{\mathcal{G}}_{tt}\right)\sin\phi\cos\phi\,.$
(27c)

Besides, $\widetilde{\mathcal{G}}_{xz}=\widetilde{\mathcal{G}}_{lz}\cos\phi$,
$\widetilde{\mathcal{G}}_{yz}=\widetilde{\mathcal{G}}_{lz}\sin\phi$,
$\widetilde{\mathcal{G}}_{zx}=\widetilde{\mathcal{G}}_{zl}\cos\phi$, and
$\widetilde{\mathcal{G}}_{zy}=\widetilde{\mathcal{G}}_{zl}\sin\phi$. Moreover,
$\widetilde{\mathcal{G}}_{yx}=\widetilde{\mathcal{G}}_{xy}$, which is dictated
by the geometric symmetry in the $xy$-plane.

We next proceed to the inverse Fourier transformation back to real space [34].
Given that the expressions in Fourier space are formulated utilizing polar
coordinates $(k,\phi)$, we anticipate that the expressions in real space,
after inverse Fourier transformation, can likewise be expressed in polar
coordinates $(\rho,\theta)$. Here, $\rho:=\sqrt{x^{2}+y^{2}}$ represents the
radial distance from the origin, while $\theta:=\arctan(y/x)$ denotes the
angle formed between the positive $x$-direction and the radial direction of
the considered position in space. A comprehensive collection of associated
formulas can be found in Ref. 35. In Appendix B, we include an overview of the
pertinent formulas utilized in the present computation. For later reference
and for the ease of computing the inverse Fourier transformation, we present
the expressions of the Green’s function itself in Cartesian coordinates.
Conversely, its dependence on the position for convenience is maintained in
polar coordinates $\rho$ and $\theta$.

First, we list the components of the Green’s function associated with the
$z$-component of the applied force. A force only featuring a $z$-component is
oriented perpendicular to the surfaces, implying an axisymmetric geometry. The
inverse Fourier transformation yields

$\displaystyle\mathcal{G}_{\rho z}$
$\displaystyle=\frac{i}{2\pi}\int_{0}^{\infty}\widetilde{\mathcal{G}}_{lz}(k,z)J_{1}(\rho
k)\,k\,\mathrm{d}k\,,$ (28) $\displaystyle\mathcal{G}_{zz}$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{\infty}\widetilde{\mathcal{G}}_{zz}(k,z)J_{0}(\rho
k)\,k\,\mathrm{d}k\,.$ (29)

In our notation, $J_{n}$ refers to the Bessel function of the first kind of
order $n$ [36]. It follows that $\mathcal{G}_{xz}=\mathcal{G}_{\rho
z}\cos\theta$ and $\mathcal{G}_{yz}=\mathcal{G}_{\rho z}\sin\theta$ in
Cartesian coordinates.

The remaining components of the Green’s function are calculated in analogy to
that. We introduce the notation

$\widetilde{\mathcal{G}}^{\pm}_{n}(k,\rho,z):=\frac{k}{4\pi}\left(\widetilde{\mathcal{G}}_{tt}(k,z)\pm\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{n}(\rho
k)\,.$ (30)

Then, we obtain

$\displaystyle\mathcal{G}_{xx}$
$\displaystyle=\int_{0}^{\infty}\left(\widetilde{\mathcal{G}}^{+}_{0}(k,\rho,z)+\widetilde{\mathcal{G}}^{-}_{2}(k,\rho,z)\cos\left(2\theta\right)\right)\mathrm{d}k\,,$
(31a) $\displaystyle\mathcal{G}_{yy}$
$\displaystyle=\int_{0}^{\infty}\left(\widetilde{\mathcal{G}}^{+}_{0}(k,\rho,z)-\widetilde{\mathcal{G}}^{-}_{2}(k,\rho,z)\cos\left(2\theta\right)\right)\mathrm{d}k\,.$
(31b)

Moreover,

$\mathcal{G}_{xy}=\int_{0}^{\infty}\mathcal{G}^{-}_{2}(k,\rho,z)\sin\left(2\theta\right)\,\mathrm{d}k\,.$
(32)

Again, $\mathcal{G}_{yx}=\mathcal{G}_{xy}$, which agrees with the geometric
symmetry within the $xy$-plane. Furthermore,
$\mathcal{G}_{zx}=\mathcal{G}_{z\rho}\cos\theta$ and
$\mathcal{G}_{zy}=\mathcal{G}_{z\rho}\sin\theta$, where

$\mathcal{G}_{z\rho}=\frac{i}{2\pi}\int_{0}^{\infty}\widetilde{\mathcal{G}}_{zl}(k,z)J_{1}(\rho
k)\,k\,\mathrm{d}k\,.$ (33)

In a bulk elastic medium, which is infinitely extended in all space
directions, the Green’s function reads[37]

$\widetilde{\mathcal{G}}_{ij}^{\infty}=\frac{1}{16\pi\mu\left(1-\nu\right)}\left(\frac{\sigma}{r}\,\delta_{ij}+\frac{r_{i}r_{j}}{r^{3}}\right),$
(34)

where $r:=|\bm{r}|=\sqrt{\rho^{2}+\left(z-h\right)^{2}}$ denotes the distance
from the position of the applied force singularity. The result in Eq. (34) is
obtained by inverse Fourier transformation of Eqs. (22). For $\nu=1/2$,
implying $\sigma=1$ in Eq. (18), representing an incompressible elastic medium
that does not allow any divergence of the displacement field at any position
in space, we recover the classical solution recognized in fluid mechanics as
the Oseen tensor [38, 39]. This tensor characterizes the behavior of an
incompressible fluid under the influence of a force singularity under low-
Reynolds-number conditions. There, $\mu$ assumes the role of the (dynamic)
shear viscosity and $\bm{u}(\bm{r})$ takes the role of the flow field of the
fluid.

###  II.3 Convergence of the Green’s functions for force monopoles

We here consider the displacement fields arising in a laterally infinitely
extended elastic film or membrane of finite thickness. The expressions for the
Green’s functions quantifying these displacements in real space in response to
the localized application of a force were derived in Sec. II.2, see Eqs.
(28)–(33). They are provided by integral expressions that result from the
inverse two-dimensional Fourier transformation back to real space, see Eq.(5).
We discuss the divergence of these expressions for the different types of
boundary conditions.

Considering that the Green’s functions exhibit a screened exponential decay as
the wavenumber $k$ approaches infinity, which results from the Bessel
functions in Eqs. (28)–(33), problems of convergence of the integrals do not
arise when $k$ approaches infinity. Our primary concern remains the
convergence of the integrals determining the Green’s functions in Eqs.
(28)–(33) as $k$ tends towards zero. Near $k=0$, the Bessel functions
$J_{n}(\rho k)$ can be approximated as $\left(\rho
k/2\right)^{n}/n!+\mathcal{O}\left(k^{n+2}\right)$. Thus,
$\widetilde{\mathcal{G}}$ must exhibit $\mathcal{O}\left(k^{-1}\right)$ or
higher-order behavior in $k$ near $k=0$ to warrant convergence of the
integrals. The cause lies with the factor $k$ that the Green’s functions
$\widetilde{\mathcal{G}}$ are multiplied with in polar coordinates when
performing the two-dimensional inverse Fourier transformation in Eq. (5).
Overall, this requires an exponent of $k$ of at least $-1$ in
$\widetilde{\mathcal{G}}$ to hinder divergence of the integrals at $k=0$.

For boundary conditions involving at least one no-slip surface, that is, for
the categories 2NOS and NOS-STF, we observe that the diagonal components
$\widetilde{\mathcal{G}}_{tt}$, $\widetilde{\mathcal{G}}_{ll}$, and
$\widetilde{\mathcal{G}}_{zz}$ all exhibit behavior of the order
$\mathcal{O}\left(k^{-1}\right)$, see Eqs. (34)–(26) together with the
expressions of the parameters listed in Tab. A1 of Appendix A. Moreover, the
off-diagonal components $\widetilde{\mathcal{G}}_{lz}$ and
$\widetilde{\mathcal{G}}_{zl}$ show behavior on the order of
$\mathcal{O}\left(1\right)$, as equally inferred from this set of equations.
Overall, this indicates convergence of the Green’s function for all cases
involving at least one no-slip boundary.

The situation becomes markedly different in the case of two stress-free
boundaries (2STF). In this situation, from Eqs. (23)–(26) together with the
coefficients listed in Tab. A1 in Appendix A, we observe that
$\widetilde{\mathcal{G}}_{tt}$ and $\widetilde{\mathcal{G}}_{ll}$ exhibit
behavior on the order of $\mathcal{O}\left(k^{-2}\right)$. Even more so,
$\widetilde{\mathcal{G}}_{lz}$ and $\widetilde{\mathcal{G}}_{zl}$ show
behavior of the order of $\mathcal{O}\left(k^{-3}\right)$, and
$\widetilde{\mathcal{G}}_{zz}$ is of the order of
$\mathcal{O}\left(k^{-4}\right)$. These relations indicate a non-convergent
Green’s function in real space when the integrals in Eqs. (28)–(33) are
carried out.

###  II.4 Convergence of the Green’s functions for force dipoles

Having analyzed the behavior in response to singularities consisting of force
monopoles, our focus shifts to singularities of force dipoles. Again, we
investigate how the behavior varies for the different boundary conditions
enforced at the surfaces.

First, we introduce the $\hat{\bm{a}}$-directed Green’s function as

$\bm{G}(\bm{r};\hat{\bm{a}})=\bm{\mathcal{G}}(\bm{r},{\bm{r}_{0}})\cdot\hat{\bm{a}}\,,$
(35)

where $\bm{a}$ is a unit vector and $\bm{r}_{0}$ is the position of applying
the force singularity. Next, we calculate the gradient $\bm{\nabla}_{0}$ of
the $\hat{\bm{a}}$-directed Green’s function with respect to the position
$\bm{r}_{0}$. In this way, we define the Green’s function for the force dipole
composed of $\hat{\bm{a}}$-directed forces as

$\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{a}},\hat{\bm{b}})=8\pi\mu\left(\hat{\bm{b}}\cdot\bm{\nabla}_{0}\right)\bm{G}(\bm{r};\hat{\bm{a}})\,.$
(36)

This force dipole extends along the direction given by the unit vector
$\bm{b}$.

In this expression, the differentiation $\bm{\nabla}_{0}(\cdot)$ with respect
to the position $\bm{r}_{0}$ of the force singularity stems from casting the
discrete setup of two nearby force centers into a differential form. During
this process, as a limit, the distance between the two centers of the point
forces tends to zero. This is how we turn from the two infinitesimally
separated force centers to the force dipole. In contrast to
$\bm{\nabla}_{0}(\cdot)$, the regular $\bm{\nabla}(\cdot)$ is the
differentiation with respect to the regular space coordinate. Due to the
translational symmetry along the $xy$ plane, it follows from Eq. (36) in
Cartesian coordinates that $\partial/\partial x_{0}=-\partial/\partial x$ and
$\partial/\partial y_{0}=-\partial/\partial y$. However, in the presence of a
boundary, $\partial/\partial h$ does not equate to $-\partial/\partial z$. In
fact, $\bm{\mathcal{G}}(\bm{r},{\bm{r}_{0}})$ in Eq. (35) depends on $x-x_{0}$
and $y-y_{0}$, but not on $z-z_{0}$, where $\bm{r}_{0}=(x_{0},y_{0},z_{0})$.

Next, we define the stresslet as the symmetric part of the force dipole,
expressed as

$\bm{G}_{\mathrm{S}}(\bm{r};\hat{\bm{a}},\hat{\bm{b}})=\frac{1}{2}\left(\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{b}},\hat{\bm{a}})+\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{a}},\hat{\bm{b}})\right)\,.$
(37)

Likewise, we define the rotlet as the antisymmetric part,

$\bm{G}_{\mathrm{R}}(\bm{r};\hat{\bm{c}})=\frac{1}{2}\left(\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{b}},\hat{\bm{a}})-\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{a}},\hat{\bm{b}})\right)\,,$
(38)

where $\hat{\bm{c}}=\hat{\bm{a}}\times\hat{\bm{b}}$.

It is more convenient to derive the expressions of the force dipole in Fourier
space according to Eq. (4). Then, we obtain

$\displaystyle\widetilde{\bm{G}}_{\mathrm{D}}(\bm{k},z;\hat{\bm{a}},\hat{\bm{x}})$
$\displaystyle=-ik\cos\phi\,\,\widetilde{\bm{G}}(\bm{k},z;\hat{\bm{a}})\,,$
(39a)
$\displaystyle\widetilde{\bm{G}}_{\mathrm{D}}(\bm{k},z;\hat{\bm{a}},\hat{\bm{y}})$
$\displaystyle=-ik\sin\phi\,\,\widetilde{\bm{G}}(\bm{k},z;\hat{\bm{a}})\,,$
(39b)
$\displaystyle\widetilde{\bm{G}}_{\mathrm{D}}(\bm{k},z;\hat{\bm{a}},\hat{\bm{z}})$
$\displaystyle=\partial_{h}\widetilde{\bm{G}}(\bm{k},z;\hat{\bm{a}})\,,$ (39c)

recalling that the Fourier transformation is performed only in two dimensions
so that the arguments are given by the two-dimensional wavevector $\bm{k}$
together with the real-space $z$ coordinate. The minus signs in Eqs. (39a) and
(39b) result from the relations $\partial/\partial x_{0}=-\partial/\partial x$
and $\partial/\partial y_{0}=-\partial/\partial y$ noted above.

When turning to the inverse two-dimensional Fourier transformation according
to Eq. (5), the solution is expressed in integral form

$\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{a}},\hat{\bm{b}})=\int_{0}^{\infty}k\,\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})\,\mathrm{d}k\,.$
(40)

Integration with respect to the variable $\phi$ has already been performed in
the function $\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})$. This
calculation can be supported by conventional computer algebra systems. In
Cartesian coordinates, the expression for
$\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})$ needs to be evaluated with
respect to all
$\hat{\bm{a}},\hat{\bm{b}}\in\\{\hat{\bm{x}},\hat{\bm{y}},\hat{\bm{z}}\\}$.
Each such combination of $\hat{\bm{a}}$ and $\hat{\bm{b}}$ implies three
Cartesian components of
$\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{a}},\hat{\bm{b}})$ in Eq. (40). We
present the corresponding components of
$\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{a}},\hat{\bm{b}})$ in Tab. A2 of Appendix
A.

Next, we investigate the convergence of the integral in Eq. (40) in analogy to
the procedure outlined in Sec. II.3. To this end, we examine whether
$\bm{\Gamma}$ is of order $k^{-1}$ or higher as $k$ approaches $0$. From the
relations in Eqs. (35) and (39), we infer convergence for
$\bm{G}_{\mathrm{D}}$ whenever the integrals for $\bm{\mathcal{G}}$ converge.
We found convergence for the pure force singularity in all cases that involve
at least one no-slip boundary (2NOS and NOS-STF). Thus, we may assume converge
of the solution under these boundary conditions also for
$\bm{G}_{\mathrm{D}}$.

Therefore, the remaining task is to consider the situation of two stress-free
boundaries (2STF). It turns out that in this case the solution for the force
dipole generally can still show divergence. For certain geometries, it
converges while the solution for the pure force singularity diverges.

| $\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})$ |  Expression for 2STF up to $\mathcal{O}\left(k^{-3}\right)$
---|---|---
1 | $\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{x}})$ | $12\left(1-\nu\right)hk^{-2}\,\hat{\bm{z}}$
2 |  $\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{x}})$ |  $12\left(1-\nu\right)k^{-2}\left(-z\,\hat{\bm{x}}+x\,\hat{\bm{z}}\right)$
3 | $\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{y}})$ | $12\left(1-\nu\right)hk^{-2}\,\hat{\bm{z}}$
4 | $\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{y}})$ | $12\left(1-\nu\right)k^{-2}\left(-z\,\hat{\bm{y}}+y\,\hat{\bm{z}}\right)$
5 | $\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{z}})$ | $12\left(1-\nu\right)k^{-2}\left(z\,\hat{\bm{x}}-x\,\hat{\bm{z}}\right)$
6 | $\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{z}})$ | $12\left(1-\nu\right)k^{-2}\left(z\,\hat{\bm{y}}-y\,\hat{\bm{z}}\right)$
7 | $\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{z}})$ | $-24\,\nu hk^{-2}\,\hat{\bm{z}}$
Table 1: Leading-order expressions in $k$ of the function
$\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})$ around $k=0$ with
$\hat{\bm{a}},\hat{\bm{b}}\in\\{\hat{\bm{x}},\hat{\bm{y}},\hat{\bm{z}}\\}$ up
to $\mathcal{O}\left(k^{-3}\right)$. The function $\bm{\Gamma}$ determines the
Green’s function $\bm{G}_{D}$ for the force dipole, see Eq. (40). Here, we
refer to the scenario of a free-standing elastic film or membrane of finite
thickness featuring two parallel surfaces of stress-free boundary conditions
(2STF). The combinations $\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{y}})$ and
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{x}})$ are omitted because both are
of $\mathcal{O}\left(1\right)$ to leading order in $k$ at $k=0$ and thus imply
convergent integrals in Eq. (40).

The leading-order behavior of $\bm{\Gamma}$ with respect to $k$ near $k=0$ can
be determined from the expressions in Tab. A2 in Appendix A systematically
using computer algebra. Upon examination, we observe that only the
configurations $\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{y}})$ and
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{x}})$ satisfy the criteria
necessary for the integral in Eq. (40) to generally converge. Thus, only the
Green’s functions $\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{x}},\hat{\bm{y}})$ and
$\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{y}},\hat{\bm{x}})$ for the force dipole
are well defined in general. We present in Tab. 1 the leading-order terms in
the series expansion of the remaining combinations of
$\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})$ around $k=0$ with
$\hat{\bm{a}},\hat{\bm{b}}\in\\{\hat{\bm{x}},\hat{\bm{y}},\hat{\bm{z}}\\}$.
From there, it is obvious that no other combination of $\hat{\bm{a}}$ and
$\hat{\bm{b}}$ provides the required leading-order behavior of $k^{-1}$ or
higher so that convergence of the integral in Eq. (40) does not generally
result.

Concerning the Green’s function for the stresslet $\bm{G}_{\mathrm{D}}$, see
Eq. (37), it is evident that convergence is achieved, nevertheless, if the
stresslet is positioned precisely at the center plane of the elastic film or
membrane. This implies $h=0$. Then, we find from Tab. 1 that the leading
orders in $k^{-2}$ cancel for
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{x}})$,
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{y}})$, and
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{z}})$. Moreover, upon
symmetrization in Eq. (37), the leading orders in $k^{-2}$ cancel for
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{z}})+\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{x}})$
and
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{z}})+\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{y}})$.
Therefore, the leading order is shifted to at least $k^{-1}$ around $k=0$,
thus satisfying the required criterion of convergence for the integral in Eq.
(40). Besides, this implies that for the Green’s function of the rotlet
$\bm{G}_{\mathrm{R}}$, see Eq. (38), only the configuration of
$\hat{\bm{c}}=\hat{\bm{z}}$ implies general convergence from the integrals in
Eq. (40).

##  III Discussion

We touched here the subtle question of possible divergences in the
displacements of elastic films and membranes when exposed to mechanical force
density fields. Conversely to many previous works, we here focused on systems
of finite, non-vanishing thickness. In deriving associated Green’s functions,
we concentrated on the response to imposed forces, see Secs. II.1–II.3,
stresslets (symmetric contributions to force dipoles), see Sec. II.4 and
rotlets implying torques (antisymmetric contributions to force dipoles), see
Sec. II.4 as well.

During these investigations, we encountered possible divergences in the
resulting displacement fields. Generally, we found that the presence of at
least one no-slip boundary stabilizes the system so that no divergence is
observed, see Sec. II.3. However, divergences in the displacement field emerge
for free-standing elastic membranes that are infinitely extended to the
lateral sides and of finite thickness in the normal direction. That is,
stress-free boundary conditions apply on both the top and bottom surface. The
divergences are connected to small wavenumbers and thus large length scales,
see Sec. II.3. In fully three-dimensional, bulk systems such divergences are
not observed [14].

We first compare to the situation of two-dimensional elastic systems, that is,
infinitely thin, flat elastic membranes. In that case, logarithmic divergences
are found for displacement fields in response to net forces [40], but not in
response to force dipoles [15], all applied along in-plane directions.

In our considerations, the two-dimensional situation is recovered when
applying forces and force dipoles that are oriented in the $xy$-plane and
positioned in the center plane at $z=0$ ($h=0$), see Fig. 1. In that case, in
strictly two dimensions, net forces lead to diverging displacement fields, but
force dipoles do not [15]. Indeed, we recover this convergence for force
dipoles, if $\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{x}},\hat{\bm{x}})$ and
$\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{y}},\hat{\bm{y}})$ are evaluated for
positioning the force dipoles in the center plane. This is observed from Eq.
(40) together with entries 1 and 3 in Tab. 1. For $h=0$, the leading order in
$k$ around $k=0$ for $\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{x}})$ and
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{y}})$ shifts to at least $k^{-1}$,
thus ensuring convergence of the integral in Eq. (40), see also Sec. II.4.
Along the same lines, from the orders of $\bm{\Gamma}$ mentioned in the
caption of Tab. 1, we infer that the expressions of
$\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{y}},\hat{\bm{x}})$ and
$\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{y}},\hat{\bm{x}})$ are always well
defined. Therefore, the symmetric positioning of force dipoles at $h=0$
recovers the situation of strictly two-dimensional considerations [15], where
all resulting displacement fields remain finite. Conversely, off-center
positioning of force dipoles, implying $h\neq 0$, can lead to diverging
displacement fields, signaled by the nonconverging integral in Eq. (40) when
$k$ approaches $0$ as highlighted by the expressions in Tab. 1.

Additionally, we note that displacements according to stresslets corresponding
to the Green’s functions
$\bm{G}_{\mathrm{S}}\left(\bm{r};\hat{\bm{x}},\hat{\bm{z}}\right)$ and
$\bm{G}_{\mathrm{S}}\left(\bm{r};\hat{\bm{y}},\hat{\bm{z}}\right)$ are always
well defined, also for off-center positioning at $h\neq 0$. The cause lies
with the symmetrization in Eq. (37). In combination, as is obvious from
entries 2 and 5 in Tab. 1, the leading order of the symmetrized $\bm{\Gamma}$
shifts to at least order $k^{-1}$, thus ensuring convergence of the integral
in Eq. (40) around $k=0$. The analogous behavior is observed when considering
entries 4 and 6 in Tab. 1. Therefore, the displacements associated with
$\bm{G}_{\mathrm{S}}\left(\bm{r};\hat{\bm{x}},\hat{\bm{z}}\right)$ and
$\bm{G}_{\mathrm{S}}\left(\bm{r};\hat{\bm{y}},\hat{\bm{z}}\right)$ remain
finite.

The remaining question is, how the diverging displacement fields can arise
even in the case of symmetric force dipoles. In fact, considering elastic
films or membranes in three spatial dimensions, a bending contribution can
generally be involved, as soon as the force dipole is not placed to the center
plane at $h=0$. We recall absence of anchoring at infinity. Consequently, we
expect bending modes associated with infinite magnitudes of normal
displacements at infinite distance from the center of the force dipole.
Moreover, antisymmetric parts of force dipoles that involve the normal
direction lead to overall rotations of the membrane, and therefore again to
diverging displacement fields at infinity. Once more, the cause lies with the
absence of anchoring of the lateral ends at infinity. We refer to the remark
at the end of Sec. II.4. Only for in-plane rotations, we found converging
integrals in Eq. (40), implying finite displacements. Thus, again, for the
scenario of in-plane rotations, we recover the two-dimensional situation of
well-defined displacement fields [15].

Finally, we remark that in the context of linear elasticity, small strains do
not necessarily equate to small displacements. Rather, they signify a small
symmetrized gradient in displacement, that is, strain. In other words, we may
reside well in the regime of linear elasticity (small strains) but still
encounter divergences in the displacement field (for instance, at infinity in
an infinitely extended system). Nonlinear elastic response may suppress such
divergences. To our knowledge, there is limited analytical exploration of
nonlinear elasticity theory concerning singularities within the framework of
free-standing elastic films and membranes. Investigating the challenging
nonlinear problem in future research could provide further insight.

##  IV Conclusions

In summary, we here consider the mechanical displacements and deformations of
linearly elastic, initially flat films and membranes exposed to forces and
force dipoles. The systems are infinitely extended to the sides, yet of
finite, non-vanishing thickness in their normal direction. We address
supported films and membranes subject to no-slip boundary conditions on at
least one of the two surfaces (top and/or bottom). In these situations, the
solutions for resulting displacements in response to both net applied forces
and/or stresslets are well defined. For free-standing elastic films and
membranes, that is, for stress-free boundary conditions applying to both the
top and bottom surface, we observe diverging solutions for most geometries.
These divergences can be related to global deformations and large-scale
displacements of the whole (infinitely extended) system. When positioning the
force dipoles at the center plane of the elastic sheet, our expressions are in
line with corresponding considerations for genuinely two-dimensional systems
[15, 18].

Our results are significant for any theoretical approach that addresses the
deformation of thin, isotropic elastic films and membranes of infinite lateral
extension within the linear regime. Particularly, we find that, unless for
very symmetric situations of stresslets included symmetrically at the center
plane of the membrane, infinitely extended free-standing membranes cannot
serve as immediate theoretical model systems without further precaution. In
contrast to genuinely two-dimensional treatments, even off-center stresslets
imply divergence of the solutions. Therefore, some stabilization of the
system, presumably by introduction of lateral boundaries and/or counterforces,
is necessary in this case.

Overall, the analysis in this study relies on linear elasticity, primarily due
to the employed formalism of Green’s functions and the principle of
superposition. Yet, linear elasticity does not always accurately capture the
observed response of various materials, particularly in the fields of soft
matter and biology. Defining parameters like Young’s moduli for biological
tissues poses significant challenges, especially when their stiffness is very
small in the absence of loading or pre-stretch. We opt for a linear framework
as an initial approach to understand the behavior within the regime of small
deformations and to facilitate analytical advancements. Addressing
nonlinearities represents an important avenue for future investigations,
potentially through perturbative methods. Still, we note that the observed
divergences in displacement do not necessarily imply that the regime of linear
elasticity is violated. Since strains follow as symmetrized gradients in
displacements, strains can be small in magnitude even if displacements diverge
in infinitely extended systems.

Finally, it will be interesting to analyze experimental data in view of our
results. Particularly, this concerns the analysis and quantification of
observed displacement fields. Naturally, real-space experimental systems are
always of finite extent, in contrast to our mathematical picture. It is
crucial to acknowledge that the role of thermal fluctuations may become
significant with increasing extension. Potentially, they obscure certain
aspects of the described effects, especially in large systems. Introducing
lateral boundaries into our investigations, such as clamping of elastic films
and membranes, comparison with experiments is facilitated. Mathematically
speaking, such additional boundary conditions play a pivotal role. Our study
offers valuable insight in this regard, aiming to draw attention to this
aspect. We infer that in experiments the role of lateral clamping is dominant
for free-standing systems of stress-free top and bottom surfaces, as they are
the ones that maintain displacements finite. Eventually, if the system is
stabilized by a no-slip boundary condition at the bottom surface, while
stress-free conditions apply at the top surface, we recover the situation of
atomic force microscopy of thin films [3, 4]. It is our plan to provide
further theoretical tools in this framework.

###### Acknowledgements.

We acknowledge support of this work by the DFG through the research grant no.
ME 3571/5-1. A.M.M. thanks the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) for support through the Heisenberg grant no. ME 3571/4-1.

2NOS
---
$D$ | $4\mu k\left(1+\sigma\right)\left(\sigma^{2}e^{4k}-2\left(\sigma^{2}+2k^{2}\right)e^{2k}+\sigma^{2}\right)$
$a_{0}^{\pm}$ | $\pm 2\sigma^{2}\left(kh\mp\sigma\right)$
$a_{1}^{\pm}$ | $\sigma\left(k^{2}\eta_{\pm}+2\sigma k\delta_{\pm}+2\sigma^{2}\right)$
$a_{2}^{\pm}$ |  $-2k^{3}\eta_{\pm}+4\sigma k^{2}\delta_{\pm}-2\sigma^{2}\left(2\pm h\right)k+2\sigma^{3}$
$a_{3}^{\pm}$ | $-\sigma\left(k^{2}\eta_{\mp}-2\sigma k\delta_{\mp}+2\sigma^{2}\right)$
$b_{0}^{\pm}$ | $-2\sigma^{2}kh$
$b_{1}^{\pm}$ | $\sigma k\left(2\sigma h\pm k\eta_{\pm}\right)$
$b_{2}^{\pm}$ | $2k\left(\pm k^{2}\eta_{\pm}-2\sigma kh+\sigma^{2}h\right)$
$b_{3}^{\pm}$ | $k\sigma\left(\mp k\eta_{\mp}-2\sigma h\right)$
$g_{0}^{\pm}$ | $\mp\sigma^{2}$
$g_{1}^{\pm}$ | $\pm\sigma\left(k\eta_{\pm}+\sigma\right)$
$g_{2}^{\pm}$ | $\pm\left(2k^{3}\eta_{\pm}-\sigma\left(2k-\sigma\right)\right)$
$g_{3}^{\pm}$ | $\pm\sigma\left(k\eta_{\mp}-\sigma\right)$
$q_{0}^{\pm}$ | $\sigma^{2}$
$q_{1}^{\pm}$ | $\sigma\left(k\eta_{\pm}-\sigma\right)$
$q_{2}^{\pm}$ | $-2k^{2}\eta_{\pm}-\sigma\left(2k+\sigma\right)$
$q_{3}^{\pm}$ | $\sigma\left(k\eta_{\mp}+\sigma\right)$
2STF
$D$ | $4\mu k\left(1+\sigma\right)\left(e^{4k}-2\left(1+2k^{2}\right)e^{2k}+1\right)$
$a_{0}^{\pm}$ | $2\left(\pm kh-\sigma\right)$
$a_{1}^{\pm}$ | $-\left(k^{2}\eta_{\pm}+2\sigma k\delta_{\pm}+1+\sigma^{2}\right)$
$a_{2}^{\pm}$ |  $2\left(-k^{3}\eta_{\pm}+2\sigma k^{2}\delta_{\pm}-k\left(\sigma^{2}+\delta_{\pm}\right)+\sigma\right)$
$a_{3}^{\pm}$ | $k^{2}\eta_{\mp}-2\sigma k\delta_{\mp}+1+\sigma^{2}$
$b_{0}^{\pm}$ | $-2kh$
$b_{1}^{\pm}$ | $\mp k^{2}\eta_{\pm}-2\sigma kh\pm\sigma^{2}\mp 1$
$b_{2}^{\pm}$ | $2k\left(\pm k^{2}\eta_{\pm}-2\sigma kh\mp\sigma^{2}\pm\delta_{\pm}\right)$
$b_{3}^{\pm}$ | $\pm k^{2}\eta_{\mp}+2\sigma kh\mp\sigma^{2}\pm 1$
$g_{0}^{\pm}$ | $\mp 1$
$g_{1}^{\pm}$ | $\mp\left(k\eta_{\pm}+\sigma\right)$
$g_{2}^{\pm}$ | $\pm\left(2k^{2}\eta_{\pm}-2\sigma k+1\right)$
$g_{3}^{\pm}$ | $\mp\left(k\eta_{\mp}-\sigma\right)$
$q_{0}^{\pm}$ | $1$
$q_{1}^{\pm}$ | $\sigma-k\eta_{\pm}$
$q_{2}^{\pm}$ | $-\left(2k^{2}\eta_{\pm}+2\sigma k+1\right)$
$q_{3}^{\pm}$ | $-\left(k\eta_{\mp}+\sigma\right)$

NOS-STF
---
$D$ | $4\mu k(1+\sigma)\left(\sigma\left(1+e^{4k}\right)+\left(1+\sigma^{2}+4k^{2}\right)e^{2k}\right)$
$a_{0}^{\pm}$ | $-2\sigma\left(\sigma\pm kh\right)$
$a_{1}^{+}$ | $k^{2}\eta_{+}+2\sigma k\delta_{+}+2\sigma^{2}$
$a_{1}^{-}$ | $-\sigma\left(k^{2}\eta_{-}+2k\sigma\delta_{-}+1+\sigma^{2}\right)$
$a_{2}^{+}$ | $2k^{3}\eta_{+}-4\sigma k^{2}\delta_{+}+\left(3\sigma^{2}+\eta_{+}\right)k-\sigma^{3}-\sigma$
$a_{2}^{-}$ |  $2k^{3}\eta_{-}-4\sigma k^{2}\delta_{-}+\left(1+(3-2h)\sigma^{2}\right)k-\sigma^{3}-\sigma$
$a_{3}^{+}$ | $\sigma\left(k^{2}\eta_{-}-2\sigma k\delta_{-}+1+\sigma^{2}\right)$
$a_{3}^{-}$ | $-k^{2}\eta_{+}+2\sigma k\delta_{+}-2\sigma^{2}$
$b_{0}^{\pm}$ | $-2\sigma kh$
$b_{1}^{+}$ | $k\left(k\eta_{+}+2\sigma h\right)$
$b_{1}^{-}$ | $\sigma\left(k^{2}\eta_{-}-2\sigma kh+1-\sigma^{2}\right)$
$b_{2}^{+}$ | $-2k^{3}\eta_{+}+4\sigma k^{2}h+\left(\sigma^{2}-\eta_{+}\right)k+\sigma^{3}-\sigma$
$b_{2}^{-}$ | $2\eta_{-}k^{3}+4\sigma k^{2}h+\left(1-\eta_{+}\sigma^{2}\right)k+\sigma^{3}-\sigma$
$b_{3}^{+}$ | $\sigma\left(k^{2}\eta_{-}+2\sigma kh+1-\sigma^{2}\right)$
$b_{3}^{-}$ | $k\left(k\eta_{+}-2\sigma h\right)$
$g_{0}^{\pm}$ | $\mp\sigma$
$g_{1}^{+}$ | $k\eta_{+}+\sigma$
$g_{1}^{-}$ | $\sigma\left(k\eta_{-}+\sigma\right)$
$g_{2}^{+}$ | $-2k^{2}\eta_{+}+\sigma\left(2k-\sigma\right)$
$g_{2}^{-}$ | $2k^{2}\eta_{-}-2\sigma k+1$
$g_{3}^{+}$ | $\sigma\left(\sigma-k\eta_{-}\right)$
$g_{3}^{-}$ | $\sigma-k\eta_{+}$
$q_{0}^{\pm}$ | $\sigma$
$q_{1}^{+}$ | $k\eta_{+}-\sigma$
$q_{1}^{-}$ | $\sigma\left(\sigma-k\eta_{-}\right)$
$q_{2}^{+}$ | $2k^{2}\eta_{+}+2\sigma k+\sigma^{2}$
$q_{2}^{-}$ | $2k^{2}\eta_{-}+2\sigma k+1$
$q_{3}^{+}$ | $-\sigma\left(\sigma+k\eta_{-}\right)$
$q_{3}^{-}$ | $k\eta_{+}+\sigma$
Definition of coefficients
$\sigma$ | $3-4\nu$
$\delta_{\pm}$ | $1\pm h$
$\eta_{\pm}$ | $1\pm 2h$

Table A1: Expressions for $a_{n}^{\pm}$, $b_{n}^{\pm}$, $g_{n}^{\pm}$, and $q_{n}^{\pm}$, where $n\in\\{0,1,2,3\\}$, defining the coefficients $c_{2}^{\pm}$ and $c_{3}^{\pm}$ introduced in Eqs. (19). These coefficients specify the longitudinal and normal components of the two-dimensionally Fourier-transformed image solutions for the displacement field, $\widetilde{u}_{l}^{*}$ and $\widetilde{u}_{z}^{*}$, respectively, see Eqs. (14) and (15). The expressions vary depending on the boundary conditions imposed at the top and bottom surface. We address geometries of two no-slip boundaries (2NOS), two stress-free boundaries (2STF), and the combination of one bottom no-slip and one top stress-free boundary (NOS-STF). For convenience, the definitions of $\sigma$, $\delta_{\pm}$, and $\eta_{\pm}$, see Eq. (18) and thereafter, are repeated on the bottom right.  $\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})$ | Exact expression in terms of the monopole solution
---|---
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{x}})\cdot\hat{\bm{x}}$ |  $k\left(\left(3\widetilde{\mathcal{G}}_{ll}(k,z)+\widetilde{\mathcal{G}}_{tt}(k,z)\right)J_{1}(\rho k)\cos\theta+\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{3}(\rho k)\cos\left(3\theta\right)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{x}})\cdot\hat{\bm{y}}$ | $k\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)\left(J_{3}(\rho k)\sin(3\theta)-J_{1}(\rho k)\sin\theta\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{x}})\cdot\hat{\bm{z}}$ | $2ik\,\widetilde{\mathcal{G}}_{zl}(k,z)\left(J_{2}(\rho k)\cos(2\theta)-J_{0}(\rho k)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{x}})\cdot\hat{\bm{x}}$ | $k\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)\left(J_{3}(\rho k)\sin(3\theta)-J_{1}(\rho k)\sin\theta\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{x}})\cdot\hat{\bm{y}}$ | $k\left(\left(3\widetilde{\mathcal{G}}_{tt}(k,z)+\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{1}(\rho k)\cos\theta-\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{3}(\rho k)\cos\left(3\theta\right)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{x}})\cdot\hat{\bm{z}}$ | $2ik\,\widetilde{\mathcal{G}}_{zl}(k,z)J_{2}(\rho k)\sin(2\theta)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{x}})\cdot\hat{\bm{x}}$ | $2ik\,\widetilde{\mathcal{G}}_{lz}(k,z)\left(J_{2}(\rho k)\cos(2\theta)-J_{0}(\rho k)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{x}})\cdot\hat{\bm{y}}$ | $2ik\,\widetilde{\mathcal{G}}_{lz}(k,z)J_{2}(\rho k)\sin(2\theta)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{x}})\cdot\hat{\bm{z}}$ | $4k\,\widetilde{\mathcal{G}}_{zz}(k,z)J_{1}(\rho k)\cos\theta$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{y}})\cdot\hat{\bm{x}}$ | $k\left(\left(3\widetilde{\mathcal{G}}_{tt}(k,z)+\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{1}(\rho k)\sin\theta+\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{3}(\rho k)\sin\left(3\theta\right)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{y}})\cdot\hat{\bm{y}}$ | $-k\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)\left(J_{1}(\rho k)\cos\theta+J_{3}(\rho k)\cos(3\theta)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{y}})\cdot\hat{\bm{z}}$ | $2ik\,\widetilde{\mathcal{G}}_{zl}(k,z)J_{2}(\rho k)\sin(2\theta)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{y}})\cdot\hat{\bm{x}}$ | $-k\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)\left(J_{1}(\rho k)\cos\theta+J_{3}(\rho k)\cos(3\theta)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{y}})\cdot\hat{\bm{y}}$ | $k\left(\left(3\widetilde{\mathcal{G}}_{ll}(k,z)+\widetilde{\mathcal{G}}_{tt}(k,z)\right)J_{1}(\rho k)\sin\theta-\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{3}(\rho k)\sin(3\theta)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{y}})\cdot\hat{\bm{z}}$ | $2ik\,\widetilde{\mathcal{G}}_{zl}(k,z)\left(J_{0}(\rho k)+J_{2}(\rho k)\cos(2\theta)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{y}})\cdot\hat{\bm{x}}$ | $2ik\,\widetilde{\mathcal{G}}_{lz}(k,z)J_{2}(\rho k)\sin(2\theta)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{y}})\cdot\hat{\bm{y}}$ | $-2ik\,\widetilde{\mathcal{G}}_{lz}(k,z)\left(J_{0}(\rho k)+J_{2}(\rho k)\cos(2\theta)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{y}})\cdot\hat{\bm{z}}$ | $4k\,\widetilde{\mathcal{G}}_{zz}(k,z)J_{1}(\rho k)\sin\theta$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{z}})\cdot\hat{\bm{x}}$ | $2\left(\partial_{h}\left(\widetilde{\mathcal{G}}_{tt}(k,z)+\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{0}(\rho k)+\partial_{h}\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{2}(\rho k)\cos(2\theta)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{z}})\cdot\hat{\bm{y}}$ | $2\partial_{h}\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{2}(\rho k)\sin(2\theta)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{x}},\hat{\bm{z}})\cdot\hat{\bm{z}}$ | $4i\,\partial_{h}\widetilde{\mathcal{G}}_{zl}(k,z)J_{1}(\rho k)\cos\theta$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{z}})\cdot\hat{\bm{x}}$ | $2\partial_{h}\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{2}(\rho k)\sin(2\theta)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{z}})\cdot\hat{\bm{y}}$ | $2\left(\partial_{h}\left(\widetilde{\mathcal{G}}_{tt}(k,z)+\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{0}(\rho k)-\partial_{h}\left(\widetilde{\mathcal{G}}_{tt}(k,z)-\widetilde{\mathcal{G}}_{ll}(k,z)\right)J_{2}(\rho k)\cos(2\theta)\right)$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{y}},\hat{\bm{z}})\cdot\hat{\bm{z}}$ | $4i\,\partial_{h}\widetilde{\mathcal{G}}_{zl}(k,z)J_{1}(\rho k)\sin\theta$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{z}})\cdot\hat{\bm{x}}$ | $4i\,\partial_{h}\widetilde{\mathcal{G}}_{lz}(k,z)J_{1}(\rho k)\cos\theta$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{z}})\cdot\hat{\bm{y}}$ | $4i\,\partial_{h}\widetilde{\mathcal{G}}_{lz}(k,z)J_{1}(\rho k)\sin\theta$
$\bm{\Gamma}(\bm{r},k;\hat{\bm{z}},\hat{\bm{z}})\cdot\hat{\bm{z}}$ | $4\,\partial_{h}\widetilde{\mathcal{G}}_{zz}(k,z)J_{0}(\rho k)$
Table A2: Expressions for the various combinations of the function
$\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})$, where
$\hat{\bm{a}},\hat{\bm{b}}\in\\{\hat{\bm{x}},\hat{\bm{y}},\hat{\bm{z}}\\}$,
are listed componentwise by projecting them for each combination on the
Cartesian unit vectors. The function $\bm{\Gamma}$ upon integration defines
the Green’s function of the force dipole in real space, see Eq. (40). We here
express $\bm{\Gamma}$ in the system of polar coordinates $(\rho,\theta)$. In
these expressions, $\widetilde{\mathcal{G}}_{ij}$ represents the two-
dimensionally Fourier-transformed Green’s function quantifying the
displacements induced in the elastic system in response to a force acting at
position $(0,0,h)$ on the elastic film or membrane of finite thickness, see
Eqs. (21)–(26).

##  Appendix A Appendix: Tables of expressions

In this Appendix, we present tables containing the remaining expressions that
specify the Green’s functions introduced in the main text. While not essential
for comprehending the contents, they detail the Green’s functions for
reference. In Tab. A1, we include the expressions for $D$, $a_{n}^{\pm}$,
$b_{n}^{\pm}$, $g_{n}^{\pm}$, and $q_{n}^{\pm}$, $n\in\\{0,1,2,3\\}$, as
defined in Eqs. (19). They determine the coefficients $c_{2}^{\pm}$ and
$c_{3}^{\pm}$ introduced in Eqs. (14) and (15), thus governing the image
solution for the longitudinal and normal components of the displacement field.
Consequently, they also enter the corresponding part of the Green’s function
in Eqs. (25) and (26). The expressions depend on the specific set of boundary
conditions and are tabulated accordingly. Moreover, Tab. A2 includes
expressions of the Cartesian components of the function
$\bm{\Gamma}(\bm{r},k;\hat{\bm{a}},\hat{\bm{b}})$,
$\hat{\bm{a}},\hat{\bm{b}}\in\\{\hat{\bm{x}},\hat{\bm{y}},\hat{\bm{z}}\\}$.
Via Eq. (40), this function determines the Green’s function for the force
dipole $\bm{G}_{\mathrm{D}}(\bm{r};\hat{\bm{a}},\hat{\bm{b}})$.

##  Appendix B Two-dimensional inverse Fourier transformation in polar
coordinates

In this Appendix, we summarize expressions for calculating two-dimensional
inverse Fourier transformations in polar coordinates. For a comprehensive
cover and associated proofs, classical textbooks on Fourier analysis [34, 41,
42] offer detailed insight. Additionally, a concise summary of the methods is
found in Ref. 35.

The two-dimensional inverse Fourier transformation of a given function
expressed in polar coordinates as $\widetilde{f}(k,\phi)$ can be represented
through Fourier series as

$f(\rho,\theta)=\mathscr{F}^{-1}\left\\{\widetilde{f}(k,\phi)\right\\}=\sum_{n=-\infty}^{\infty}f_{n}(\rho)\,e^{in\theta}\,,$
(41)

where

$f_{n}(\rho)=\frac{i^{n}}{2\pi}\int_{0}^{\infty}\widetilde{f}_{n}(k)J_{n}(\rho
k)\,k\,\mathrm{d}k\,,$ (42)

with

$\widetilde{f}_{n}(k)=\frac{1}{2\pi}\int_{0}^{2\pi}\widetilde{f}(k,\phi)\,e^{-in\phi}\,\mathrm{d}\phi\,.$
(43)

Here, again, $J_{n}$ refers to the Bessel function of the first kind of order
$n$ [36].

When dealing with a radially symmetric function $\widetilde{f}(k)$ independent
of the azimuthal angle $\phi$, only the term $n=0$ persists in the series
described by Eq. (41). As a result, the inverse Fourier transformation
simplifies to

$f(\rho)=\mathscr{F}^{-1}\left\\{\widetilde{f}(k)\right\\}=\frac{1}{2\pi}\int_{0}^{\infty}\widetilde{f}(k)J_{0}(\rho
k)\,k\,\mathrm{d}k\,.$ (44)

This specific formulation applies to the component $\mathcal{G}_{zz}$ of the
Green’s function, as indicated by Eq. (29). For the other components, we
observe that only terms of $n\in\\{0,1,2\\}$ remain for the force monopole and
terms of $n\in\\{0,1,2,3\\}$ for the force dipole. The entries of Tab. A2 were
derived using Eqs. (41)– (43).

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