# Elastic disk with isoperimetric Cosserat coating

Matteo Gaibotti  DICAM, University of Trento, via Mesiano 77, I-38050 Trento,
Italy  Davide Bigoni Corresponding author: Davide Bigoni fax: +39 0461 282599;
phone: +39 0461 282507; web-site: http://www.ing.unitn.it/$\sim$bigoni/;
e-mail<EMAIL_ADDRESS>DICAM, University of Trento, via Mesiano 77,
I-38050 Trento, Italy  Sofia G. Mogilevskaya  Departement of Civil,
Environemental and Geo- Engineering, University of Minnesota, 500 Pillsbury
Drive S.E. Minneapolis, MN 55455-0116, USA.

###### Abstract

A circular elastic disk is coated with an elastic beam, absorbing shear and
normal forces without deformation and linearly reacting to a bending moment
with a change in curvature. The inexstensibility of the elastic beam
introduces an isoperimetric constraint, so that the length of the initial
circumference of the disk is constrained to remain fixed during the loading of
the disk/coating system. The mechanical model for this system is formulated,
solved for general loading, and particularized to the case of two equal and
opposite traction distributions, each applied on a small boundary segment
(thus modelling indentation of a coated fiber). The stress fields, obtained
via complex potentials, are shown to evidence a nice correspondence with
photoelastic experiments, ad hoc designed and performed. The presented results
are useful for the design of coated fibers at the micro and nano scales.

Dedicated to
Professors Natasha and Sasha Movchan on the occasion of their 60th birthday

##  1 Introduction

Several technologies involve coating of the surface of a bulk material with a
thin layer made up of another material. This technique was developed in order
to enhance electrical conductivity [1, 2, 3], achieve electrical insulation
[4, 5], protect from heating [6, 7], enhance biocompatibility [8, 9, 10],
increase strength [11] or wear, fatigue, or corrosion resistance [12].

A coating layer diffuses the load on an attached solid in a nonlocal way,
introduces a characteristic length, and deeply affects the mechanics of the
coated object. As a consequence, a strong research effort has been devoted to
the modelling of coatings and to the analysis of associated problems. In the
framework of the nonlinear theory of elasticity, the coating may be modelled
as a surface possessing a membrane stiffness [13], possibly enhanced with a
flexural and torsional stiffness [14, 15]. Applications of these theories have
been presented to bifurcation [16, 17] and wave propagation [18] in coated
elastic blocks.

Within the realm of linear elasticity, the coating has been idealized as an
elastic shell (becoming a curved beam in a two-dimensional formulation) [19,
20]. In composites, the presence of a film separating two elastic media has
been reduced to a thin interface, suitably describing the contact conditions.
This interface model has been used to estimate the effective mechanical
properties of composites with inclusions [21, 19, 22] and for applications in
thermal conductivity [23].

The advent of nanotechnologies has strongly fueled the development of
interface models. These have been used to estimate thermo-mechanical
properties of nano-composites, of mono-layered-graphene based materials [24,
25], to analyze the interaction between nano- inhomogeneities [26] and the
mechanics of reinforcements, such as coated fibers [27, 28] or nanoplatelets
embedded in a core matrix [29, 30]. The presence of coatings or thin layers
strongly influences failure mechanisms, such as crack nucleation and
propagation inside the coating/substrate [31, 32, 33], or delamination induced
by mechanical/thermal mismatch and residual stresses existing between coating
and substrate [34, 35, 36, 37, 38, 39, 40, 41, 42], or by curvature changes or
buckling [43, 44, 45].

Figure 1:  Qualitative comparison between the in-plane deviatoric stress
($\left|\sigma_{I}-\sigma_{II}\right|/\mu$, analytically evaluated with the
disk/coating model introduced in the present article) and the photoelastic
fringes resulting during a diametral compression test of a coated disk. The
sample before testing is reported in the upper part of Fig. 5. The applied
external load distribution $p$ is modelled with a Fourier series expansion,
enhanced with the Lanczos smoothing method (Section 4.2.1).

In the present article a circular disk made up of a (linear, isotropic)
elastic material is analyzed, covered with a ‘beam-like’ model of coating, of
the type introduced by Benveniste and Miloh [20]. The beam model is assumed to
satisfy inextensibility, so that an isoperimetric constraint is introduced of
the Cosserat type, as the beam transmits bending111 More precisely, an Euler-
Bernoulli beam is an example of constrained Cosserat material, whereas a
Timoshenko beam corresponds to the unconstrained Cosserat case, called also
micropolar unconstrained theory.. The cylindrical geometry considered here may
be important in view of the development of nanowire technology (where a coated
cylinder is obtained, when a nanowire is grown inside a single-wall carbon
nanotube [46]). We show that, when the exterior of the coated disk is loaded
by a generic (but self-equilibrated) force distribution, the problem can be
analytically solved via complex potentials, thus obtaining the displacement,
strain, and stress fields within the disk, together with the axial and shear
forces and bending moment in the coating.

The analytical result is complemented with the explicit treatment of the case
in which the coating/disk system is subject to a load distribution modelling
two equal and opposite concentrated forces. This load corresponds to that
applied during a nanoindentation test (often performed on nanofibers to
measure their mechanical properties [47]) and may easily be reproduced
experimentally. On this vein, we have designed two coated disks, manufactured
(with a CNC engraving machine) from a single block of polymethyl methacrylate,
so that the bonding between the coating and disk is perfect and residual
stresses are absent. The samples have been tested in a circular polariscope
and the results strongly support the coated disk model, so that the
photoelastic fringes are very well captured by the elastic solution, as
anticipated in Fig. 1. The figure shows the strong effect of the coating,
where the stress distribution is typical of that forming inside a curved beam
subject to bending moment.

The article is organized as follows. After the coating/disk system is modelled
in Section 2, the equations governing its behaviour are presented (Section 3).
The special case of loading consisting of two equal and opposite radial force
distributions applied to a small area is solved in Section 4.1 and the
solution is compared with photoelastic experimental results in Section 4.3.

##  2 Modelling of the complex coating/disk

A (linear and isotropic) elastic circular disk is examined, enclosed in a
perfectly-bonded annular elastic beam, which obeys the Euler-Bernoulli model,
so that it is axially inextensible, unshearable and reacts linearly to a
curvature variation through the development of a bending moment. The
mechanical model of the coating/disk system follows from the combination of
the separate equations holding for its two components, which are introduced
below.

###  2.1 The coating, a circular annular beam

A circular Euler-Bernoulli (unshearable) elastic beam of bending stiffness
$EJ$ and radius $R$ is considered. The center of the circle is located at the
origin of the Cartesian coordinate system with axes $x_{1}$ and $x_{2}$ and
the polar system $r$ and $\theta$ (the latter assumed positive when
counterclockwise), equipped with the two radial and circumferential unit
vectors ${\bf e}_{r}$ and ${\bf e}_{\theta}$ (Fig. 2). The beam is subjected
to radial and tangential forces, respectively $p$ and $q$, that will be
applied by both the external environment and the elastic disk. The internal
forces along the beam, comprised of normal and shear components, $N$ and $T$,
as well as bending moment $M$, satisfy equilibrium

$\frac{dN}{ds}+\frac{T}{R}=-q,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ -\frac{dT}{ds}+\frac{N}{R}=-p,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \frac{dM}{ds}=-T,$ (1)

where $ds=Rd\theta$ is the elementary arclength defined anticlockwise
accordingly with $\theta$.

Figure 2: The coating of the disk is modelled with an inextensible and
unshearable elastic beam (with bending stiffness $EJ$). The beam introduces an
isoperimetric closed contour $L$, so that the perimeter of the coating cannot
change its total volume, when a load of components $p$ (radial) and $q$
(tangential) is applied. This load generates internal normal and shear forces,
$N$ and $T$, and bending moment $M$.

The kinematics of the beam is defined by a radial $u_{r}$ and tangential
$u_{\theta}$ displacement and a rotation $\Phi$, the latter positive when
clockwise. Pure kinematic considerations, sketched in Fig. 3, show that the
axial deformation $\epsilon$, the rotation $\Phi$, and the curvature $\chi$
(the latter additional and opposite to the natural curvature $1/R$ of the
circle) are given by [48]

$\epsilon=\frac{u_{r}}{R}+\frac{du_{\theta}}{ds},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Phi=-\frac{u_{\theta}}{R}+\frac{du_{r}}{ds},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \chi=-\frac{d\Phi}{ds}.$ (2)

The beam represents an example of constrained Cosserat solid, capable of
reacting to a change in curvature with a bending moment according to the
linear constitutive equation

$\chi=-\frac{M}{EJ}.$ (3)

It is assumed now that this beam be axially inextensible, so that
$\epsilon=0$, namely,

$\frac{du_{\theta}}{d\theta}=-u_{r},$ (4)

a relation which introduces a local isoperimetric constraint for the area
enclosed inside the annular region, which perimeter cannot change. Here,
differently from the mathematical ‘Queen Dido problem’, the curve enclosing
the area is an elastic beam, equipped with a finite bending stiffness and
subjected to external loads.

Equations (1)–(3), determining the mechanics of an annular beam under quasi-
static loading, can now be particularized for the inextensibility constraint
$\epsilon=0$ to hold, so that they become functions of the radial displacement
$u_{r}$ only, as follows.

  * •

Kinematics governing the displacements and rotation, $u_{r}$, $u_{\theta}$,
and $\Phi$:

$u_{\theta}=-\int u_{r}(\theta)\,d\theta,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
R\Phi=-u_{\theta}+\frac{du_{r}}{d\theta},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\chi=-\frac{1}{R}\frac{d\Phi}{d\theta}.$ (5)

  * •

Constitutive equations for the internal forces $M$, $T$, and $N$:

$\begin{array}[]{ll}\displaystyle\frac{M}{EJ}=\frac{d^{2}u_{r}}{ds^{2}}+\frac{u_{r}}{R^{2}},\\\\[14.22636pt]
\displaystyle\frac{T}{EJ}=-\frac{d^{3}u_{r}}{ds^{3}}-\frac{1}{R^{2}}\frac{du_{r}}{ds},\\\\[14.22636pt]
\displaystyle\frac{N}{EJ}=-R\left(\frac{d^{4}u_{r}}{ds^{4}}+\frac{1}{R^{2}}\frac{d^{2}u_{r}}{ds^{2}}+\frac{p}{EJ}\right).\end{array}$
(6)

  * •

Differential equation for the radial displacement $u_{r}$:

$\frac{d^{5}u_{r}}{d\theta^{5}}+2\frac{d^{3}u_{r}}{d\theta^{3}}+\frac{du_{r}}{d\theta}=\frac{R^{4}}{EJ}\left(q-\frac{dp}{d\theta}\right).$
(7)

Equations (4) and (7) define the so-called inextensional shell type interface
model introduced by Benveniste and Miloh [20], their equations (2.15)3 and
(2.15)4 with $N=3$, in the particular case when the radius of curvature is
constant. More in detail, equation (4.9) in [20] coincides with the above
equation (7).

Figure 3:  Kinematics of a circular beam modelling the coating of the elastic
disk. a) On an infinitesimal beam element the sum of both radial $u_{r}$ and
tangential $u_{\theta}$ displacement generate a total length variation
$\epsilon$ described by Eq. 2. b) The rotation $\phi$ of a cross section of
the beam is related to $d\theta/d\phi$, the tangential displacement
$u_{\theta}$, and the increment $du_{r}$.

###  2.2 The coated disk

A two-dimensional (thus subject to conditions of either plane stress or plane
strain) elastic disk of radius $R$, made up of an isotropic material (defined
by a shear modulus $\mu$ and Poisson’s ratio $\nu$), is coated with the above
introduced beam (Euler-Bernoulli, axially inextensible and unshearable, Fig.
4).

Figure 4: Elastic disk coated with an inextensible and unshearable elastic
beam (with bending stiffness $EJ$).

The coating is perfectly connected to the disk and hence conditions of
continuity of displacement at the contact between the two impose the following
conditions

$u^{\left(b\right)}_{r}=u^{\left(d\right)}_{r}\big{|}_{r=R},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
u^{\left(b\right)}_{\theta}=u^{\left(d\right)}_{\theta}\big{|}_{r=R}\,,$ (8)

where $(b)$ and $(d)$ denote the ‘beam’ and the ‘disk’, respectively. These
superscripts will be in the following avoided, as the distinction between the
two will be not needed, as they have been assumed to coincide.

The axial inextensibility of the coating enforces the isoperimetric constraint
(5)1 on the points of the boundary $L$ of the disk

$u_{\theta}=-\frac{1}{R}\int_{L}u_{r}\,ds,$ (9)

defining a nonlocal relation between displacement components.

The load on the coating, of components $p$ and $q$ in equations (1)1,2, (6)3,
and (7), is partially applied externally to the disk, but also transmitted by
the disk in terms of stress components $\sigma_{rr}$ and $\sigma_{r\theta}$
(multiplied by the thickness of the disk $b$, which becomes unity in a plane
strain problem). Therefore, the loads $p$ and $q$ present in equations (1)1,2,
(6)3, and (7), have now to be interpreted as

$q\longrightarrow q-b\sigma_{r\theta},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ p\longrightarrow p+b\sigma_{rr},$
(10)

so that in this way $p$ and $q$ are identified as external loads applied to
the coated disk. In a complex notation, the total load applied on the coating
can be represented as

$P=p+b\sigma_{rr}+i\,\left(q-b\sigma_{r\theta}\right).$ (11)

In the elastic disk, the strain components, expressed in polar coordinates $r$
and $\theta$, are

$\epsilon_{rr}=u_{r,r},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\epsilon_{\theta\theta}=\frac{u_{r}+u_{\theta,\theta}}{r},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon_{r\theta}=\frac{u_{r,\theta}-u_{\theta}}{2r}+\frac{u_{\theta,r}}{2},$
(12)

so that, using Hooke’s laws, the stress components become

$\sigma_{rr}=\mu\frac{(1+k)\epsilon_{rr}+(3-k)\epsilon_{\theta\theta}}{k-1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma_{\theta\theta}=\mu\frac{(1+k)\epsilon_{\theta\theta}+(3-k)\epsilon_{rr}}{k-1},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma_{r\theta}=2\mu\epsilon_{r\theta},$ (13)

and vice-versa

$\epsilon_{rr}=\frac{(1+k)\sigma_{rr}+(k-3)\sigma_{\theta\theta}}{8\mu},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon_{\theta\theta}=\frac{(1+k)\sigma_{\theta\theta}+(k-3)\sigma_{rr}}{8\mu},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon_{r\theta}=\frac{\sigma_{r\theta}}{2\mu},$ (14)

where

$\kappa=\left\\{\begin{array}[]{ll}\displaystyle 3-4\nu,&\mbox{ for plane
strain,}\\\\[14.22636pt] \displaystyle\frac{3-\nu}{1+\nu},&\mbox{ for plane
stress.}\end{array}\right.$ (15)

###  2.3 Complex potential representation for the disk

The elastic disk of radius $R$, enclosed by the annular beam defining the
coating, has a smooth, non intersecting boundary, so that the tangential and
normal directions can be used and respectively denoted by the unit vectors
${\bf e}_{r}$ and ${\bf e}_{\theta}$. Every point of the disk can be
identified by a complex number $z=x_{1}+ix_{2}$, so that points belonging to
the boundary $L$ will be denoted as $\tau=R\,e^{i\,\theta}$. The following
notation is introduced

$\displaystyle g\left(\tau\right)=\frac{R}{\tau},$
$\displaystyle\overline{g\left(\tau\right)}=\frac{R}{\overline{\tau}}=g^{-1}\left(\tau\right),$
$\displaystyle
g^{\prime}\left(\tau\right)=-\frac{1}{R}\,g^{2}\left(\tau\right),$ (16)

where the prime, $(\,)^{\prime}$, denotes differentiation with respect to the
variable $\tau$, while a superscript bar indicates complex conjugate.

In a complex variable formulation, complex combinations are introduced for
displacements, boundary tractions, and external load. These combinations allow
the governing equations for the problem (1), (4) and (6) to be rewritten in a
way that the elastic fields are determined in terms of complex Fourier series
involving unknown complex coefficients. The knowledge of the latter
coefficients permits the evaluation of displacements and stresses everywhere
in the disk via Kolosov-Muskhelishvili complex potentials
$\varphi\left(z\right)$ and $\psi\left(z\right)$ defined as [49]

$\displaystyle\left\\{\begin{array}[]{lll}2\mu{u\left(z\right)}=\kappa\varphi\left(z\right)-z\overline{\varphi^{\prime}\left(z\right)}-\overline{\psi\left(z\right)},\\\
\sigma_{11}+\sigma_{22}=4\,\mathrm{Re}\\!\left(\varphi^{\prime}\left(z\right)\right),\\\
\sigma_{22}-\sigma_{11}+2i\sigma_{12}=2\left[\overline{z}\varphi^{\prime\prime}\left(z\right)+\psi^{\prime}\left(z\right)\right],\end{array}\right.$
(17)

where $\mathrm{Re}\\!\left(\,\,\right)$, and in the following
$\mathrm{Im}\\!\left(\,\,\right)$, represent the real and the imaginary part
of the enclosed quantity.

The components of the strain tensor can be expressed via the inverse Hooke’s
law as

$\displaystyle\left\\{\begin{array}[]{lll}\displaystyle\varepsilon_{11}+\varepsilon_{22}=2\frac{1-2\nu}{\mu}\mathrm{Re}\\!\left(\varphi^{\prime}\left(z\right)\right),\\\
\\\
\displaystyle\varepsilon_{22}-\varepsilon_{11}+2i\varepsilon_{12}=\frac{1}{\mu}\left[\overline{z}\varphi^{\prime\prime}\left(z\right)+\psi^{\prime}\left(z\right)\right].\end{array}\right.$
(18)

When dealing with a circular disk, the general expression for the complex
potentials $\varphi\left(z\right)$ and $\psi\left(z\right)$ has been stated by
Mogilevskaya et al. [26] as

$\displaystyle\varphi\left(z\right)=\frac{2\mu}{\kappa-1}\,\mathrm{Re}\\!\left(A_{1}\right)\,g^{-1}\left(z\right)+\frac{2\mu}{\kappa}\sum_{n=1}^{\infty}{A_{n+1}\,g^{-\left(n+1\right)}\left(z\right)},$
(19)
$\displaystyle\psi\left(z\right)=-\frac{2\mu}{\kappa-1}\,\mathrm{Re}\\!\left(A_{1}\right)\,\frac{\overline{z_{c}}}{R}-\frac{2\mu}{\kappa}\left[\frac{\overline{z_{c}}}{R}+g\left(z\right)\right]\sum_{n=1}^{\infty}{\left(n+1\right)A_{n+1}\,g^{-n}\left(z\right)}$
$\displaystyle\phantom{\psi\left(z\right)=}-2\mu\sum_{n=2}^{\infty}{\overline{A_{1-n}}\,g^{-\left(n-1\right)}\left(z\right)},$

where $z_{c}$ denotes the centre of the disk and the functions
$g\left(z\right),g^{\prime}\left(z\right)$ ,$g^{\prime\prime}\left(z\right)$,
and the conjugate $\overline{g\left(z\right)}$ are defined as

$\displaystyle
g\left(z\right)=\frac{R}{z}=\frac{R}{\left(x_{1}+ix_{2}\right)},$ (20)
$\displaystyle g^{\prime}\left(z\right)=-\frac{1}{R}\,g^{2}\left(z\right),$
$\displaystyle
g^{\prime\prime}\left(z\right)=\frac{2}{R^{2}}\,g^{3}\left(z\right),$
$\displaystyle\overline{g\left(z\right)}=\frac{R^{2}}{r^{2}}\,g^{-1}\left(z\right),$
$\displaystyle r=\sqrt{x_{1}^{2}+x_{2}^{2}}\,.$

It will be shown that some of the coefficients $A_{j}$ ($j=1,...,+\infty$ and
$j=-1,...,-\infty$) have to be evaluated by imposing additional conditions. In
particular, the rigid body motion will be eliminated by fixing points
belonging to $L$. When the latter condition is imposed, the expression for the
displacement field in equation (17) becomes

$u\left(z\right)=\frac{1}{2\mu}\left[\kappa\varphi\left(z\right)-z\overline{\varphi^{\prime}\left(z\right)}-\overline{\psi\left(z\right)}\right]+A_{0}+i\,z\,\mathrm{Im}\\!\left(A_{1}\right).$
(21)

##  3 Solution for the coated disk

In a Cartesian coordinate system, the displacement at any point of the
boundary of the disk, which coincides with the coating, $\tau\in{L}$ can be
expressed through the complex Fourier series expansion

$u\left(\tau\right)=u_{1}\left(\tau\right)+i\,u_{2}\left(\tau\right)=\sum_{n=1}^{\infty}{A_{-n}\,g^{n}\left(\tau\right)}+\sum_{n=0}^{\infty}{A_{n}\,g^{-n}\left(\tau\right)},$
(22)

where $u_{1}\left(\tau\right)$ and $u_{2}\left(\tau\right)$ are displacement
components respectively parallel to the $x_{1}$ and $x_{2}$ axes and the
complex coefficients $A_{\pm{n}}$ are for the moment unknown.

Recalling the relation between Cartesian and polar coordinates,

$u_{r}\left(\tau\right)+i\,u_{\theta}\left(\tau\right)=\left[u_{1}\left(\tau\right)+i\,u_{2}\left(\tau\right)\right]g\left(\tau\right),$
(23)

it is possible to obtain the displacement at every point $\tau$ in the polar
coordinate system $\left(r;\theta\right)$ as

$\displaystyle
u_{r}\left(\tau\right)=\frac{1}{2}\left[u\left(\tau\right)\,g\left(\tau\right)+\overline{u\left(\tau\right)}\,g^{-1}\left(\tau\right)\right],$
(24) $\displaystyle
u_{\theta}\left(\tau\right)=\frac{1}{2i}\left[u\left(\tau\right)\,g\left(\tau\right)-\overline{u\left(\tau\right)}\,g^{-1}\left(\tau\right)\right],$

so that the final representation for displacements follows from the equation
(22)

$\displaystyle
u_{r}\left(\tau\right)=\frac{1}{2}\left[\sum_{n=1}^{\infty}{A_{-n}\,g^{n+1}\left(\tau\right)}+\sum_{n=0}^{\infty}{A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)}\right.+\left.\sum_{n=1}^{\infty}{\overline{A_{-n}}\,g^{-\left(n+1\right)}\left(\tau\right)}+\sum_{n=0}^{\infty}{\overline{A_{n}}\,g^{n-1}\left(\tau\right)}\right],$
(25) $\displaystyle
u_{\theta}\left(\tau\right)=\frac{1}{2i}\left[\sum_{n=1}^{\infty}{A_{-n}\,g^{n+1}\left(\tau\right)}+\sum_{n=0}^{\infty}{A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)}\right.-\left.\sum_{n=1}^{\infty}{\overline{A_{-n}}\,g^{-\left(n+1\right)}\left(\tau\right)}-\sum_{n=0}^{\infty}{\overline{A_{n}}\,g^{n-1}\left(\tau\right)}\right].$

In a similar vein, the tractions at any point of the coating $\tau$ can be
expressed through the complex Fourier series

$\sigma\left(\tau\right)=\sigma_{rr}\left(\tau\right)+i\,\sigma_{r\theta}\left(\tau\right)=\sum_{n=1}^{\infty}{B_{-n}\,g^{n}\left(\tau\right)}+\sum_{n=0}^{\infty}{B_{n}\,g^{-n}\left(\tau\right)},$
(26)

where $\sigma_{rr}$ and $\sigma_{r\theta}$ are the components of the traction
at the point $\tau\in{L}$ respectively directed parallel to the normal and the
tangential direction.

The coefficients $A_{\pm{n}}$ and $B_{\pm{n}}$ are interrelated through the
following relations (see Zemlyanova and Mogilevskaya [50])

$\displaystyle B_{-1}=0,$ (27)
$\displaystyle\frac{\kappa-1}{2\mu}\,B_{0}=\frac{2}{R}\,\mathrm{Re}\\!\left(A_{1}\right),$
$\displaystyle\frac{1}{2\mu}\,B_{-n}=\frac{n-1}{R}\,A_{1-n},\,$ for
$\displaystyle n\geq{2},$
$\displaystyle\frac{\kappa}{2\mu}\,B_{n}=\frac{n+1}{R}\,A_{n+1},\,$ for
$\displaystyle n\geq{1}.$

The representation of the load acting on the external surface of the coating
is introduced in its local coordinates system, using the following complex
Fourier series as (Fig. 4):

$p+iq=\sum_{n=1}^{\infty}{D_{-n}}\,g^{n}\left(\tau\right)+\sum_{n=0}^{\infty}{D_{n}\,g^{-n}\left(\tau\right)},$
(28)

where the components of the load $p$ and $q$ are assumed to be single-valued
on $L$ and variable with the angle $\theta$, while coefficients $D_{\pm{n}}$
remain determined as the series expansion of $p$ and $q$, so that they are
treated as known complex coefficients. The complex representation of the total
load affecting the coating, $P$, equation (11), can be computed by recalling
equations (26) and (28) to obtain

$\displaystyle
P\left(\tau\right)=\sum_{n=1}^{\infty}{\left(D_{-n}+b\,\overline{B_{n}}\right)\,g^{n}\left(\tau\right)}+\sum_{n=0}^{\infty}{\left(D_{n}+b\,\overline{B_{-n}}\right)\,g^{-n}\left(\tau\right)}.$
(29)

###  3.1 Modelling of inextensible coating

The axial strain in the coating, equation (2)1, can be translated into the
complex notation as shown in [50], their equation (99)2, namely,

$\displaystyle
u_{\theta,\theta}\left(\tau\right)+u_{r}\left(\tau\right)=R\,\mathrm{Re}\\!\left(\frac{\partial{u}\left(\tau\right)}{\partial{\tau}}\right),$
(30)

so that the inextensibility condition for the annular beam, in other words the
isoperimetric constraint, equation (4), becomes

$\mathrm{Re}\\!\left(\frac{\partial{u\left(\tau\right)}}{\partial{\tau}}\right)=0.$
(31)

Differentiating equation (22) with respect to $\tau$ and using the resulting
$g^{\prime}$ into equation (16)3 yields

$\displaystyle\frac{\partial{u\left(\tau\right)}}{\partial{\tau}}=\frac{1}{R}\sum_{n=1}^{\infty}{\left(-n\,A_{-n}\,g^{n+1}\left(\tau\right)+n\,A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)\right)},$
(32)

so that, a substitution into equation (31) leads to

$\sum_{n=1}^{\infty}\left(n\,A_{-n}\,g^{n+1}\left(\tau\right)-n\,A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)+n\,\overline{A_{-n}}\,g^{-\left(n+1\right)}\left(\tau\right)-n\,\overline{A_{n}}\,g^{n-1}\left(\tau\right)\right)=0.$
(33)

By collecting terms with the same power of $g^{\pm{n}}\left(z\right)$ in the
left hand side of equation (33) and by equating them to zero leads to the
following conditions.

  1. (i.)

For $n=0$:

$\mathrm{Re}\left(A_{1}\right)=0,$ (34)

  2. (ii.)

For $n=-1$:

$\displaystyle A_{2}=0,$ (35)

  3. (iii.)

For $n\neq{0}$ and $n\neq{-1}$:

$\displaystyle A_{n+1}=\frac{n-1}{n+1}\,\overline{A_{1-n}}.$ (36)

Recalling the relation between coefficients $A_{n}$ and $B_{n}$ expressed by
equations (27), $B_{0}=0$ follows from equation (34). Moreover, setting $n=1$
in equation (27)4 and recalling equation (35), $B_{1}=0$ follows.

###  3.2 Equilibrium and kinematic condition

The 5-th order differential equation (7) describing the radial displacement
couples with equilibrium and kinematics of the beam representing the coating,
under the constraint of axial inextensibility (4). The perfect bonding between
coating and disk, equation (8) implies that the radial displacement entering
equation (7) coincides with the radial displacement at the boundary of the
disk. In particular, it is possible to rewrite the differential equation (7)
through an identification of the components of the external load $p$ and $q$
provided by equation (10). At the right hand side of equation (7), the
quantity $\left(q-d{p}/d\theta\right)$ assumes the form

$q-\frac{dp}{d\theta}=q-p_{,\theta}-b\left[\sigma_{r\theta}\left(\tau\right)+\sigma_{rr,\theta}\left(\tau\right)\right],$
(37)

where the terms on the right hand side can be particularized by exploiting the
similarity with equation (98) in [50] as

$\displaystyle\frac{q-p_{,\theta}}{R}=\mathrm{Im}\\!\left(\frac{\partial}{\partial{\tau}}\left[{\left(p+i\,q\right)g^{-1}\left(\tau\right)}\right]\right),$
(38)
$\displaystyle\sigma_{r\theta}\left(\tau\right)+\sigma_{rr,\theta}\left(\tau\right)=2\sigma_{r\theta}\left(\tau\right)-R\,\mathrm{Im}\\!\left(\frac{\partial}{\partial{\tau}}{\left[\sigma\left(\tau\right)g^{-1}\left(\tau\right)\right]}\right).$

Equation (20)2 for the derivatives and expressions (26) and (28) lead to

$\displaystyle q-p_{,\theta}=$
$\displaystyle-\frac{1}{2i}\left\\{\sum_{n=1}^{\infty}{\left[\left(n-1\right)D_{-n}+\left(n+1\right)\overline{D_{n}}\,\right]g^{n}\left(\tau\right)}\right.$
(39)
$\displaystyle\left.-\sum_{n=1}^{\infty}{\left[\left(n+1\right)D_{n}+\left(n-1\right)\overline{D_{-n}}\,\right]g^{-n}\left(\tau\right)}\right\\}+\mathrm{Im}\\!\left(D_{0}\right),$
$\displaystyle\sigma_{r\theta}\left(\tau\right)+\sigma_{rr,\theta}\left(\tau\right)=$
$\displaystyle\frac{1}{2i}\left\\{\sum_{n=1}^{\infty}{\left[\left(n+1\right)B_{-n}+\left(n-1\right)\overline{B_{n}}\,\right]g^{n}\left(\tau\right)}\right.$
$\displaystyle\left.-\sum_{n=1}^{\infty}{\left[\left(n-1\right)B_{n}+\left(n+1\right)\overline{B_{-n}}\,\right]g^{-n}\left(\tau\right)}\right\\}+\mathrm{Im}\\!\left(B_{0}\right).$

Through a substitution of equations (39), the expression (37) can now be cast
in the form

$\displaystyle q-\frac{dp}{d\theta}$
$\displaystyle=-\frac{1}{2i}\left\\{\sum_{n=1}^{\infty}{\left[\left(n+1\right)\left(\overline{D_{n}}+b\,B_{-n}\right)+\left(n-1\right)\left(D_{-n}+b\,\overline{B_{n}}\right)\right]g^{n}\left(\tau\right)}\right.$
(40)
$\displaystyle\left.-\sum_{n=1}^{\infty}{\left[\left(n+1\right)\left(D_{n}+b\,\overline{B_{-n}}\right)+\left(n-1\right)\left(\overline{D_{-n}}+b\,B_{n}\right)\right]g^{-n}\left(\tau\right)}\right\\}+\mathrm{Im}\\!\left(D_{0}-b\,B_{0}\right).$

The left hand side of equation (7) can be written in terms of coefficients
$A_{\pm{n}}$ by computing the complex derivatives of the displacement in
equation (24)2. In particular, the same procedure reported in [50], their
Appendix C, yields now

$\displaystyle\frac{d^{5}u_{r}}{d\theta^{5}}+2\frac{d^{3}u_{r}}{d\theta^{3}}+\frac{du_{r}}{d\theta}=$
$\displaystyle-R^{5}\,\mathrm{Im}\\!\left(\frac{\partial^{5}{u\left(\tau\right)}}{\partial{\tau^{5}}}g^{-4}\left(\tau\right)\right)-5R^{4}\,\mathrm{Im}\\!\left(\frac{\partial^{4}{u\left(\tau\right)}}{\partial{\tau^{4}}}g^{-3}\left(\tau\right)\right)$
(41)
$\displaystyle-3R^{3}\,\mathrm{Im}\\!\left(\frac{\partial^{3}{u\left(\tau\right)}}{\partial{\tau^{3}}}g^{-2}\left(\tau\right)\right),$

where, in agreement with [50] [their equation (100)], the derivatives of the
complex representation of the displacement field reads as

$\displaystyle\frac{\partial^{3}u\left(\tau\right)}{\partial{\tau^{3}}}=$
$\displaystyle\frac{1}{R^{3}}\left[-\sum_{n=1}^{\infty}{n\left(n+1\right)\left(n+2\right)A_{-n}\,g^{n+3}\left(\tau\right)}\right.$
(42)
$\displaystyle\left.+\sum_{n=3}^{\infty}{n\left(n-1\right)\left(n-2\right)A_{n}\,g^{-\left(n-3\right)}\left(\tau\right)}\right],$
$\displaystyle\frac{\partial^{4}u\left(\tau\right)}{\partial{\tau^{4}}}=$
$\displaystyle\frac{1}{R^{4}}\left[+\sum_{n=1}^{\infty}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)A_{-n}\,g^{n+4}\left(\tau\right)}\right.$
$\displaystyle\left.+\sum_{n=4}^{\infty}{n\left(n-1\right)\left(n-2\right)\left(n-3\right)A_{n}\,g^{-\left(n-4\right)}\left(\tau\right)}\right],$
$\displaystyle\frac{\partial^{5}u\left(\tau\right)}{\partial{\tau^{5}}}=$
$\displaystyle\frac{1}{R^{5}}\left[-\sum_{n=1}^{\infty}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)\left(n+4\right)A_{-n}\,g^{n+5}\left(\tau\right)}\right.$
$\displaystyle\left.+\sum_{n=5}^{\infty}{n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right)A_{n}\,g^{-\left(n-5\right)}\left(\tau\right)}\right].$

A complete form of equation (41) can be computed after substitution of
expressions (42) as

$\displaystyle\frac{d^{5}u_{r}}{d\theta^{5}}+2\frac{d^{3}u_{r}}{d\theta^{3}}+\frac{du_{r}}{d\theta}=$
$\displaystyle\frac{1}{2i}\left\\{\sum_{n=1}^{\infty}{n^{2}\left(n+1\right)\left(n+2\right)^{2}\left[A_{-n}\,g^{n+1}\left(\tau\right)-\overline{A_{-n}}\,g^{-\left(n+1\right)}\left(\tau\right)\right]}\right.$
(43)
$\displaystyle\left.-\sum_{n=3}^{\infty}{n^{2}\left(n-1\right)\left(n-2\right)^{2}\left[A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)-\overline{A_{n}}\,g^{n-1}\left(\tau\right)\right]}\right\\}.$

Using expressions (40) and (43) and collecting terms with the same summation
index, the differential equation (7) becomes

$\displaystyle-\sum_{n=1}^{\infty}{\frac{R^{4}}{EJ}\left\\{\left[\left(n+1\right)\left(\overline{D_{n}}+b\,B_{-n}\right)+\left(n-1\right)\left(D_{-n}+b\,\overline{B_{n}}\right)\right]g^{n}\left(\tau\right)\right.}$
(44)
$\displaystyle\left.-\left[\left(n+1\right)\left(D_{n}+b\,\overline{B_{-n}}\right)+\left(n-1\right)\left(\overline{D_{-n}}+b\,B_{n}\right)\right]g^{-n}\left(\tau\right)-2i\,\mathrm{Im}\\!\left(D_{0}-b\,B_{0}\right)\right\\}$
$\displaystyle-n^{2}\left(n+1\right)\left(n+2\right)^{2}\left[A_{-n}\,g^{n+1}\left(\tau\right)-\overline{A_{-n}}\,g^{-\left(n+1\right)}\left(\tau\right)\right]$
$\displaystyle+\sum_{n=3}^{\infty}{n^{2}\left(n-1\right)\left(n-2\right)^{2}\left[A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)-\overline{A_{n}}\,g^{n-1}\left(\tau\right)\right]}=0,$

from which, collecting terms with the same power $g^{\pm{n}}\left(\tau\right)$
in equation (44) and equating them to zero, the following expression is
obtained

$\begin{array}[]{ll}\left(n+1\right)\left(\overline{D_{n}}+b\,B_{-n}\right)+\left(n-1\right)\left(D_{-n}+b\,\overline{B_{n}}\right)\\\\[14.22636pt]
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\displaystyle+\frac{n\left(n+1\right)^{2}\left(n-1\right)^{2}EJ}{R^{4}}\left[A_{1-n}+\overline{A_{n+1}}\,\right]=0,\end{array}$
(45)

of which the following two particular cases can be highlighted

$\mathrm{Im}\\!\left(D_{0}-bB_{0}\right)=0,\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{ for }\leavevmode\nobreak\ n=0\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{ and }\leavevmode\nobreak\ \leavevmode\nobreak\
\overline{B_{-1}}=-\frac{1}{b}D_{1},\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{ for }\leavevmode\nobreak\ n=-1.$ (46)

Recalling equations (27)3-4 and (36), it becomes possible to express the
coefficients $B_{\pm{n}}$ involved in equation (45) as functions of the
coefficients $A_{\pm{n}}$

$\displaystyle\overline{B_{n}}=\frac{2\mu}{\kappa{R}}\left(n+1\right)\overline{A_{n+1}}=\frac{2\mu}{\kappa{R}}\left(n-1\right)A_{1-n},\,$
$\displaystyle\text{for}\,n\geq{1}$ (47) $\displaystyle
B_{-n}=\frac{2\mu}{R}\left(n-1\right)A_{1-n},\,$
$\displaystyle\text{for}\,n\geq{2}.$

A substitution of equations (47) into equation (45) yields

$\displaystyle\left(n+1\right)\overline{D_{n}}+\frac{2\mu{b}}{R}\left(n^{2}-1\right)A_{1-n}+\left(n-1\right)D_{-n}+\frac{2\mu{b}}{\kappa{R}}\left(n-1\right)^{2}A_{1-n}$
(48) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+\frac{2n^{2}\left(n+1\right)\left(n-1\right)^{2}EJ}{R^{4}}A_{1-n}=0,$

so that collecting terms involving $A_{1-n}$ leads to

$\displaystyle
A_{1-n}=-\frac{\kappa{R^{4}}\left[\left(n+1\right)\overline{D_{n}}+\left(n-1\right)D_{-n}\right]}{2\left(n-1\right)\left[\kappa{EJ}n^{2}\left(n^{2}-1\right)+R^{3}\mu{b}\left(n+\kappa{n}+\kappa-1\right)\right]},$
$\displaystyle\text{for}\,n\geq{2}.$ (49)

###  3.3 Fixing rigid body motion

Equation (21) shows that restrictions on coefficients $A_{0}$ and $A_{1}$ can
be inferred from the imposition of a rigid-body roto-translation. Following
the procedure adopted in [26], the latter will be eliminated by assuming the
displacement of the disk to be zero at one point of it and requiring vanishing
of the vertical displacement component at another point of the disk on the
same horizontal line. It will now be imposed $u\left(z\right)=0$ at points
$z=\tau_{0}=R+i\,0$ and $z=z_{c}=0$, see Fig. (4) a. Recalling equations
(34)-(36) and setting $z_{c}=0$ in relations (19), the complex potentials for
the disk are determined as

$\displaystyle\varphi\left(z\right)=\frac{2\mu}{\kappa}\sum_{n=1}^{\infty}{A_{n+1}\,g^{-\left(n+1\right)}\left(z\right)},$
(50)
$\displaystyle\psi\left(z\right)=-2\mu\sum_{n=2}^{\infty}{\left(\frac{n+\kappa-1}{\kappa}\right)\overline{A_{1-n}}\,g^{-\left(n-1\right)}\left(z\right)}.$

At points $z=z_{c}$ and $z=\tau_{0}$ the following results can be derived from
equations (21).

  * •

At $z=z_{c}$:

$\displaystyle{u}\left(z_{c}\right)=\frac{1}{2\mu}\left[\kappa\varphi\left(z_{c}\right)-\overline{\psi\left(z_{c}\right)}\right]+A_{0},$
(51)

so that equations (20)1-2, (34) and (35) lead to the particularization of
equation (50) at $z=z_{c}$

$\displaystyle\varphi\left(z_{c}\right)=\psi\left(z_{c}\right)=0,$ (52)

in addition to which equations (52), substituted into equation (51) with
$u\left(z_{c}\right)=0$, provide $A_{0}=0$.

  * •

At $z=\tau_{0}$:

$\displaystyle{u}\left(\tau_{0}\right)=\frac{1}{2\mu}\left[\kappa\varphi\left(\tau_{0}\right)-\tau_{0}\,\overline{\varphi^{\prime}\left(\tau_{0}\right)}-\overline{\psi\left(\tau_{0}\right)}\right]+i\,\tau_{0}\,\mathrm{Im}\\!\left(A_{1}\right),$
(53)

from which the following specific expressions for the complex potentials (50)
can be derived

$\displaystyle\varphi\left(\tau_{0}\right)=\frac{2\mu}{\kappa}\sum_{n=1}^{\infty}{A_{n+1}},$
(54)
$\displaystyle\varphi^{\prime}\left(\tau_{0}\right)=\frac{2\mu}{R\kappa}\sum_{n=1}^{\infty}{\left(n+1\right)}A_{n+1},$

and

$\displaystyle\psi\left(\tau_{0}\right)=-2\mu\sum_{n=2}^{\infty}{\left(\frac{n+\kappa-1}{\kappa}\right)\overline{A_{1-n}}},$
(55)

where the latter expressions have been obtained by setting $\tau_{0}=R$ in
equations (16), i.e
$g\left(\tau_{0}\right)^{-n}=\left(\tau_{0}/R\right)^{n}=1\,\,\forall{n\geq{0}}$.
An explicit expression for $u\left(\tau_{0}\right)$ can be obtained through a
substitution of equations (54) and (55) into equation (53)

$\displaystyle{u}\left(\tau_{0}\right)$
$\displaystyle=\sum_{n=1}^{\infty}{A_{n+1}}-\frac{1}{\kappa}\sum_{n=1}^{\infty}{\left(n+1\right)\overline{A_{n+1}}}+\sum_{n=2}^{\infty}{\left(\frac{n+\kappa-1}{\kappa}\right)A_{1-n}}$
(56) $\displaystyle+iR\,\,\mathrm{Im}\\!\left(A_{1}\right),$

which, recalling equations (35) and (36), provides

$u\left(\tau_{0}\right)=\sum_{n=2}^{\infty}{\left[\frac{n-1}{n+1}\,\overline{A_{1-n}}+A_{1-n}\right]+iR\,\mathrm{Im}\\!\left(A_{1}\right)}.$
(57)

Imposition of the condition
$u_{2}\left(\tau_{0}\right)=\mathrm{Im}\\!\left(u\left(\tau_{0}\right)\right)$
in equation (57) yields

$u_{2}\left(\tau_{0}\right)=\mathrm{Im}\\!\left(\sum_{n=2}^{\infty}{\left[\frac{n-1}{n+1}\,\overline{A_{1-n}}+A_{1-n}\right]}\right)+R\,\mathrm{Im}\\!\left(A_{1}\right).$
(58)

The requirement that the vertical displacement component
$u_{2}\left(\tau_{0}\right)$ be zero at $z=\tau_{0}$ is finally established
from the vanishing of the right hand side of equation (58),

$\mathrm{Im}\\!\left(A_{1}\right)=-\frac{2}{R}\,\mathrm{Im}\\!\left(\sum_{n=2}^{\infty}{\frac{1}{n+1}\,A_{1-n}}\right).$
(59)

The condition $A_{0}=0$ at $z=z_{c}$ and equation (59) allow to derive all
additional terms in equation (21), describing a rigid body movement.

The above obtained solution is summarized in terms of the evaluated
coefficients reported in Tables 1 and 2. These coefficients completely define
the displacement and stress fields for the coated disk.

Coefficient | Value | Rule
---|---|---
$A_{1-n}$ |  $-\frac{\kappa{R^{4}}\left[\left(n+1\right)\overline{D_{n}}+\left(n-1\right)D_{-n}\right]}{2\left(n-1\right)\left[\kappa{EJ}n^{2}\left(n^{2}-1\right)+R^{3}\mu{b}\left(n+\kappa{n}+\kappa-1\right)\right]}$ |  $n\geq{2}$
$A_{0}=A_{2}$ |  $0$ | -
$A_{1}$ |  $-\frac{2\,i}{R}\,\mathrm{Im}\\!\left(\sum_{n=2}^{\infty}{\frac{1}{n+1}\,A_{1-n}}\right)$ | -
$A_{n+1}$ |  $\frac{n-1}{n+1}\,\overline{A_{1-n}}$ |  $n$>${1}$

Table 1: Coefficients $A_{\pm{n}}$ of the complex power series defining the
displacement field within the disk.

Coefficients $A_{\pm{n}}$ listed in Table 1 are given in terms of the known
complex coefficients $D_{\pm{n}}$, which define the loads applied to the
external surface of the coated disk.

Coefficient | Value | Rule
---|---|---
$B_{-n}$ |  $\frac{2\mu}{R}\left(n-1\right)A_{1-n}$ |  $k\geq{2}$
$B_{-1}=B_{0}=B_{1}$ |  $0$ | -
$B_{n}$ |  $\frac{2\mu}{\kappa{R}}\left(n-1\right)\overline{A_{1-n}}$ |  $k\geq{1}$

Table 2: Coefficients $B_{\pm{n}}$ of the complex power series defining the
stress distribution inside the disk.

Table 2 shows that all coefficients $B_{\pm{n}}$ can be written in terms of
coefficients $A_{1-n}$ and hence from equation (49) they are found to depend
on the known complex coefficients $D_{\pm{n}}$. Once $A_{\pm{n}}$ and
$B_{\pm{n}}$ are known, it becomes possible to compute the expressions for
elastic displacement and stress fields.

###  3.4 Elastic fields for the disk

Displacement, stress, and strain at any point inside the disk and on its
boundary are known in terms of complex potentials $\varphi\left(z\right)$ and
$\psi\left(z\right)$, equations (17), through coefficients $A_{\pm{n}}$ listed
in Table 1, as

$\displaystyle\varphi\left(z\right)=\frac{2\mu}{\kappa}\sum_{n=1}^{\infty}{\frac{n-1}{n+1}\,\overline{A_{1-n}}}\,g^{-\left(n+1\right)}\left(z\right),$
(60)
$\displaystyle\varphi^{\prime}\left(z\right)=\frac{2\mu}{R\kappa}\sum_{n=1}^{\infty}{\left(n-1\right)\overline{A_{1-n}}\,g^{-n}\left(z\right)},$
$\displaystyle\varphi^{\prime\prime}\left(z\right)=\frac{2\mu}{R^{2}\kappa}\sum_{n=1}^{\infty}{n\left(n-1\right)\overline{A_{1-n}}\,g^{-n+1}\left(z\right)},$
$\displaystyle\psi\left(z\right)=-2\mu\sum_{n=2}^{\infty}{\left(\frac{n+\kappa-1}{\kappa}\right)\overline{A_{1-n}}\,g^{-n+1}\left(z\right)},$
(61)
$\displaystyle\psi^{\prime}\left(z\right)=-\frac{2\mu}{R}\sum_{n=2}^{\infty}{\left(n-1\right)\left(\frac{n+\kappa-1}{\kappa}\right)\,\overline{A_{1-n}}\,g^{-n+2}\left(z\right)}.$

Except for a rigid roto-translation, the displacement $u\left(z\right)$ at
every point of the disk is determined by relation (17)1, through equation (21)
and the coefficients listed in Tables 1 and 2 entering the following equation

$\displaystyle
u\left(z\right)=\sum_{n=1}^{\infty}{A_{n+1}g^{-n-1}\left(z\right)}+\sum_{n=2}^{\infty}{A_{1-n}g^{n-1}\left(z\right)}-2i\,g^{-1}\left(z\right)\,\mathrm{Im}\\!\left(\sum_{n=2}^{\infty}{\frac{1}{n+1}\,A_{1-n}}\right),$
(62)

which, recalling equations (35) and (36) and collecting terms, can be
rewritten as

$\displaystyle u\left(z\right)=$
$\displaystyle\sum_{n=2}^{\infty}{\left[\frac{n-1}{n+1}\overline{A_{1-n}}\,g^{-\left(n+1\right)}\left(z\right)+A_{1-n}\,g^{n-1}\left(z\right)\right.}$
(63)
$\displaystyle{\left.-2i\,g^{-1}\left(z\right)\mathrm{Im}\\!\left(\frac{1}{n+1}\,A_{1-n}\right)\right]}.$

The stress is determined from a combination between equations (17)2-3, with a
chain substitution of equations (35) and (36) into the potentials, equations
(50)1, as

$\displaystyle\sigma_{11}+\sigma_{22}=\frac{8\mu}{R\kappa}\mathrm{Re}\\!\left(\sum_{n=2}^{\infty}{\left(n-1\right)\overline{A_{1-n}}\,g^{-n}\left(z\right)}\right),$
(64)

while equation (17)3 leads to

$\displaystyle\sigma_{22}-\sigma_{11}+2i\sigma_{12}=\frac{4\mu}{R}\sum_{n=2}^{\infty}{\left(n-1\right)\left[\frac{r^{2}}{R^{2}\kappa}n-\frac{1}{\kappa}\left(n-1\right)-1\right]\overline{A_{1-n}}\,g^{-n+2}\left(z\right)}.$
(65)

##  4 Experiments vs. theoretical predictions

The aim of this section is to introduce the experimental set-up and results
and compare the latter with the general solution for the coated disk, to be
particularized now to a specific applied loading, tailored to model the
experiments.

###  4.1 Photoelastic experiments

The previously derived analytic solution is compared with photoelastic
experiments performed at the Instabilities Lab of the University of Trento on
two coated disks, subject to two radial compressive force distributions
applied at two diametrically opposed small portions of the coating. The two
coated disks, shown in Fig. 5 on the left, have been machined with radii
$R_{1}$ and $R_{2}$ equal to 60 and 35 mm and thickness 5 mm in the internal
portion (with a CNC engraving machine) from a single thick plate of polymethyl
methacrylate (Perspex Clear, Lucite International, Young modulus $E=$ 3200 MPa
and Poisson’s ratio $\nu=0.36$). The coatings have in-plane $\times$ out-of-
plane thicknesses equal to 14 mm $\times$ 19 mm and 5 mm $\times$ 21 mm,
respectively.

The fact that the coated disks have been obtained by carving a single block of
material implies that the bonding between disk and coating is perfect and that
the sample is obtained without introducing any residual stress, so that
detachments are excluded until failure. Note that failure cannot be
investigated with the proposed experimental set-up, because it involves out-
of-plane buckling and may break the polariscope.

Figure 5: The two families of samples used for diametrical compression tests:
the coated disk (left), the coating alone, representing a circular beam
(center) and disk without coating (right). The coating has a rectangular
cross-section equal to $14\times 19$ mm (upper part) and $5\times 21$ mm
(lower part), resulting in a strongly different bending stiffness for the
coating, when the two families of samples are compared.

For comparison, both the interior disk and the external coating of the
coating/disk complex have also been machined into separate pieces, shown in
Fig. 5 on the center and on the right. In this way, the inner disk, the
annular beam representing the coating, and the coated disk have all been
individually tested.

Vertical load has been quasi-statically increased through compression against
two horizontal steel plates, equipped with a loading cell (TH-KN2D load cell
RC 20 kN, from Gefran) to measure (signal was acquired with a NI CRio
interfaced via software Labview, ver. 2018 from National Instruments) the
applied load, under displacement control (imposed through an electromechanical
testing machine, ELE Tritest 50, by ELE International Ltd). The samples under
loading have been analyzed with a linear and circular polariscope (with
quarterwave retarders for 560nm, dark field arrangement; equipped with a white
and sodium vapor lightbox at $\lambda$ = 589.3nm, purchased from Tiedemann &
Betz), designed and manufactured at the Instabilities Lab of the University of
Trento.

Photos at white and monochromatic light were taken with a Nikon D200 digital
camera, equipped with a AF-S micro Nikkor (105 mm, 1:2.8G ED) or with a AF-S
micro Nikkor (70180 mm, 1:4.55.6 D) lens. Monitored with a thermocouple
connected to a Xplorer GLX Pasco©, temperature near the samples during
experiments was found to lie around 22.5 ∘C, without sensible oscillations.

####  4.1.1 Analysis of the experimental results

Photos taken during the tests at increasing values of loading are reported in
Fig. 6 at monochromatic light, and in Fig. 7 at white light (for the coated
disk samples reported in Fig. 5).

Starting the discussion with Fig. 6 (referred to the samples visible in Fig.
5, upper part), it should be noted that the loads for the three samples are
different, higher for the coated disk (on the left, 10 kN), low for the ring
(central part, 0.6 kN), and intermediate for the uncoated disk (on the right
4.6 kN). This choice has been guided by the fact that at the value used for
the coated disk the ring would break and the uncoated disk would suffer an
out-of-plane buckling, while at the value used for the ring the coated disk is
loaded so little that the fringes are barely visible.

Figure 6: Monochromatic photoelastic fringes under diametrical compression of:
a coated disk (left), its external ring (center), and the inner disk without
coating (right). Samples are shown in Fig. 5 (upper part).

Experiments on the uncoated disk are well-known [51], so that only one
additional is reported in Appendix A, used to calibrate the other photoelastic
analyses.

Further experimental results on the ring, together with comparisons with the
calculated stress state, are deferred to Appendix B. These photoelastic
analyses show that the ring behaves as a circular Euler-Bernoulli beam, where
flexural deformation prevails. Therefore, the coating can be modelled with
excellent approximation as inextensible, because the ratio between flexural
and axial stiffnesses becomes negligible. To substantiate this statement with
an evaluation, an annular beam of radius $R$ is assumed to be both flexurally
and axially deformable and subject to two diametrical forces $F$. The
shortening of the diameter of the rod can be written as

$\Delta=\frac{(\pi^{2}-8)\,R^{3}}{4\pi\,EJ}\left[1+\underbrace{\frac{\pi^{2}}{\pi^{2}-8}\left(\frac{\rho}{R}\right)^{2}}_{axial\leavevmode\nobreak\
deformability}\right],$ (66)

where $\rho$ is the radius of inertia of the cross section of the beam.
Considering the data pertaining to our experiments, Fig. 5, effects related to
the axial deformability of the coating can be evaluated as 0.024 and 0.009, to
be compared with 1, representing the flexural deformability.

Figure 7: Photoelastic fringes generated during (vertical) diametrical
compression in the coated disks with bending stiffness of the coating EJ1=1.4
$\cdot$10-2 kNm2 (upper part) and EJ2=7 $\cdot$10-4 kNm2 (lower part). Four
increasing values of compression are reported: 3.0 kN (1 kN), 7.0 kN (2.5 kN),
8.5 kN (7.0 kN), and 10.0 kN (8.5 kN) in the upper part (lower part), for the
samples shown in Fig. 5; photos have been taken at white circularly polarized
light.

The fringe pattern reported in Fig. 6 evidences the deep effect connected to
the presence of the coating, which introduces a nonlocal distribution of the
external load, thus strongly affecting the stress state in the disk.

Considering now Fig. 7 the photographs refer to four different values of
loading, namely, 3.0 kN (1 kN), 7.0 kN (2.5 kN), 8.5 kN (7.0 kN), and 10.0 kN
(8.5 kN) in the upper part (in the lower part).

Although the stress states for the two experiments reported in the figure are
qualitatively similar, it is clear that the effect of coating is strongly
increased in the case of thick coating (upper part of the figure). The fringes
visualized in the above reported photoelastic experiments provide a measure of
the difference between the in plane principal stresses (for more details see
[51])

$\displaystyle\left|\sigma_{I}-\sigma_{II}\right|={\sqrt{\left(\sigma_{11}-\sigma_{22}\right)^{2}+4\sigma_{12}^{2}}},$
(67)

thus permitting a comparison with the analytical solution derived in the
previous Sections, that will be applied to a loading distribution modelling
the load during the experiments and considered in the following Sections. The
comparison, anticipated in Fig. 1, shows that the analytical solution
excellently models the experimental results.

###  4.2 An analytical solution for the coated disk modelling the experiments

In order to apply the general solution for the coated disk obtained in Section
3, the traction distribution acting on the external portion of the coating has
to be modelled and implemented. These are the subjects of the present section.

####  4.2.1 Model for the external load applied on the coated disk

The external load will be modelled as a constant normal traction $p$ ($q=0$)
applied along two diametrically opposed small arcs of equal amplitude,
$s=\alpha\,R$, where $\alpha$ is an angle centred at the vertical diameter of
the disk. Using the complex Fourier series expansion (28), multiplying both
sides of it by $e^{-mi\theta},\,\,m=\pm{1},\pm{2},...\pm{n}$ and integrating
over the whole circle leads to

$\int_{\frac{\pi}{2}-\alpha}^{\frac{\pi}{2}+\alpha}p\,e^{-mi\theta}d\theta+\int_{\frac{3\pi}{2}-\alpha}^{\frac{3\pi}{2}+\alpha}p\,e^{-mi\theta}d\theta=\int_{0}^{2\pi}{\left[\sum_{n=1}^{\infty}{D_{-n}e^{-\left(n+m\right)i\theta}}+\sum_{n=0}^{\infty}{D_{n}e^{-\left(n-m\right)i\theta}}\right]d\theta}.$
(68)

The right hand side of equation (68) is different from zero and equal to
$2\pi$ if and only if $n=m$. Hence, at every fixed $m$, one non-null
coefficient is determined. Collecting terms with the same power of
$e^{\pm{ni\theta}}$ and inverting equation (68) yield the values of all
coefficients $D_{\pm{n}}$.

Figure 8: Convergence of the series approximation for the applied load,
equation (28), using $N=25$ and $N=100$ complex coefficients $D_{\pm{n}}$
(left). Severe oscillations are evident, as related to the presence of a
discontinuity in the applied loads, which should vanishes at
$\theta=\pi/2\mp{\alpha}$ and $\theta=3\pi/2\pm{\alpha}$. Both global and
local oscillation can be filtered out by means of a smoothing technique
(right) resulting in a better approximation for the applied external load
using $N=100$ complex coefficients $D^{*}_{\pm{n}}$ having the form of
equation (75)

.

The convergence of the series approximation for an applied load of unit
resultant is analyzed in Fig. 8 (central part). A satisfactory representation
for the applied concentrated load can be obtained with an arc length
$\alpha=5\degree$ and a truncation of the complex Fourier series at $N=100$.
The convergence of the series is analyzed in Fig. 8 (on the left), where
results are reported for $N=25$ and $N=100$. Here it may be concluded that:
(i.) the maximum load does not exceed the unit value up to $N=100$ (for $N=25$
the maximum value is $\approx 0.65$) and (ii.) severe sign variations of the
represented function are visible, a phenomenon known as ‘Gibbs oscillations’.

Although the approximation obtained with $N=100$ is judged as satisfactory and
will be used in the following, the local smoothing technique proposed by
Lanczos (1966) has been also implemented to further increase the precision in
the representation. The Lanczos technique can be explained with reference to a
given a Fourier series representation of a function $f\left(\theta\right)$,
which displays a jump discontinuity, with partial sum

$f_{N}\left(\theta\right)=a_{0}+\sum_{n=1}^{N}{a_{n}\cos{\left(n\theta\right)}}+\sum_{n=1}^{N}{b_{n}\sin{\left(n\theta\right)}}.$
(69)

Lanczos (1966) introduced the smoothed value for the partial sum (69) as

$f^{*}_{N}\left(\theta\right)=a^{*}_{0}+\sum_{n=1}^{N}{\frac{\sin{\left(n\pi/N\right)}}{n\pi/N}\left[a^{*}_{n}\cos{\left(n\theta\right)}+b^{*}_{n}\sin{\left(n\theta\right)}\right]}.$
(70)

Using now the complex notation,

$\displaystyle\cos{n\theta}=\frac{1}{2}\left(e^{i\,n\theta}+e^{-i\,n\theta}\right),$
$\displaystyle\sin{n\theta}=\frac{1}{2i}\left(e^{i\,n\theta}-e^{-i\,n\theta}\right),$
(71)

the partial sum expressed by equation (70) can be rewritten as

$f^{*}_{N}\left(\theta\right)=a^{*}_{0}+\sum_{n=1}^{N}{\frac{\sin{\left(n\pi/N\right)}}{n\pi/N}\left[\frac{a^{*}_{n}}{2}\left(e^{i\,n\theta}+e^{-i\,n\theta}\right)+\frac{b^{*}_{n}}{2i}\left(e^{i\,n\theta}-e^{-i\,n\theta}\right)\right]},$
(72)

so that by collecting terms yields

$f^{*}_{N}\left(\theta\right)=a^{*}_{0}+\sum_{n=1}^{N}{\frac{\sin{\left(n\pi/N\right)}}{n\pi/N}\left[\left(\frac{a^{*}_{n}-i\,b_{n}^{*}}{2}\right)e^{in\theta}+\left(\frac{a^{*}_{n}+i\,b_{n}^{*}}{2}\right)e^{-in\theta}\right]},$
(73)

and adopting the notation used in equation (16) finally leads to

$f^{*}_{N}\left(\theta\right)=a^{*}_{0}+\sum_{n=1}^{N}{\frac{\sin{\left(n\pi/N\right)}}{n\pi/N}\left[\left(\frac{a^{*}_{n}-i\,b_{n}^{*}}{2}\right)g^{-n}\left(\tau\right)+\left(\frac{a^{*}_{n}+i\,b_{n}^{*}}{2}\right)g^{n}\left(\tau\right)\right]}.$
(74)

Equations (74) and (68) allow the determination of the new coefficients
$D_{n}^{*}$ for the smoothed function

$\left\\{\begin{array}[]{l}D_{0}^{*}=a_{0}^{*},\\\\[10.0pt] \displaystyle
D_{-n}^{*}=\left(\frac{a_{n}+ib_{n}}{2}\right)\frac{\sin{\left(n\pi/N\right)}}{n\pi/N},\\\\[10.0pt]
\displaystyle
D_{n}^{*}=\left(\frac{a_{n}-ib_{n}}{2}\right)\frac{\sin{\left(n\pi/N\right)}}{n\pi/N},\end{array}\right.$
(75)

providing a sufficiently accurate representation of the applied load, for
appropriate choice of $N$, as shown in Fig. 8 on the right.

####  4.2.2 Solution for the coated disk subject to opposite force
distributions

All ingredients are now ready to obtain the solution of a coated disk
subjected to two equal and opposite distributions of surface tractions of the
type shown in Fig. 8. This solution is derived below and will be used for
comparison with the experimental results.

A substitution of equations (36) and (49) into equation (50), provides the
complex potentials for the coated disk when subjected to two opposite radial
traction distributions, written in terms of the coefficients $D_{\pm{n}}$

$\displaystyle\varphi\left(z\right)=-\mu{R^{4}}\sum_{n=2}^{\infty}{\left\\{\frac{\left(n-1\right)\overline{D_{-n}}+\left(n+1\right)D_{n}}{\left(n+1\right)\left[\kappa{EJ}n^{2}\left(n^{2}-1\right)+R^{3}\mu{b}\left(n+\kappa{n}+\kappa-1\right)\right]}\right\\}}\,g^{-\left(n+1\right)}\left(z\right),$
(76)
$\displaystyle\psi\left(z\right)=\mu{R^{4}}\sum_{n=2}^{\infty}{\left\\{\frac{\left(n-1+\kappa\right)\left[\left(n-1\right)\overline{D_{-n}}+\left(n+1\right)D_{n}\right]}{\left(n-1\right)\left[\kappa{EJ}n^{2}\left(n^{2}-1\right)+R^{3}\mu{b}\left(n+\kappa{n}+\kappa-1\right)\right]}\right\\}}\,g^{-\left(n-1\right)}\left(z\right),$

so that the displacement follows as

$\displaystyle{u}\left(z\right)=$
$\displaystyle-\frac{R^{4}\kappa}{2}\sum_{n=2}^{\infty}\left\\{\frac{\left[\left(n-1\right)\overline{D_{-n}}+\left(n+1\right)D_{n}\right]}{\left(n+1\right)\left[\kappa{EJ}n^{2}\left(n^{2}-1\right)+R^{3}\mu{b}\left(n+\kappa{n}+\kappa-1\right)\right]}\,g^{-\left(n+1\right)}\left(z\right)\right.$
(77)
$\displaystyle\left.-\frac{\left[\left(n-1\right){D_{-n}+\left(n+1\right)\overline{D_{n}}}\right]\left[r^{2}\left(n-1\right)-R^{2}\left(n-1+\kappa\right)\right]R^{-2n}}{\kappa\left(n-1\right)\left[\kappa{EJ}n^{2}\left(n^{2}-1\right)+R^{3}\mu{b}\left(n+\kappa{n}+\kappa-1\right)\right]r^{-2\left(n-1\right)}}\,g^{n-1}\left(z\right)\right\\},$

and the stress components, equation (17)2, as

$\displaystyle\frac{\sigma_{11}+\sigma_{22}}{4\mu{R^{3}}}=-\sum_{n=2}^{\infty}{\mathrm{Re}\\!\left(\frac{\left[\left(n-1\right)\overline{D_{-n}}+\left(n+1\right)D_{n}\right]}{\kappa{EJ}n^{2}\left(n^{2}-1\right)+R^{3}\mu{b}\left(n+\kappa{n}+\kappa-1\right)}\,g^{-n}\left(z\right)\right)},$
(78)

while, from equation (17)3, as

$\displaystyle\frac{\sigma_{22}-\sigma_{11}+2i\sigma_{12}}{2\mu{R}}=$
$\displaystyle\sum_{n=2}^{\infty}{\frac{\left[\left(n-1\right)\overline{D_{-n}}+\left(n+1\right)D_{n}\right]\left[R^{2}\left(n+\kappa-1\right)-r^{2}n\right]}{\kappa{EJ}n^{2}\left(n^{2}-1\right)+R^{3}\mu{b}\left(n+\kappa{n}+\kappa-1\right)}\,g^{-\left(n-2\right)}\left(z\right)}.$
(79)

The in-plane deviatoric stress can be calculated using equation (67) from the
knowledge of $\sigma_{11}$, $\sigma_{22}$, and $\sigma_{12}$.

The results are reported in Fig. 1, where they are compared with the
photoelastic experiments performed on the coated disks, shown in Fig. 5. The
internal forces in the coating, equations (6), are completely determined by
the displacement component $u_{r}$ in terms of the coefficients $A_{1-n}$ in
equation (49). Recalling equations (35) and (36), the derivatives of complex
displacement involved in equation (6) are found as

$\displaystyle\frac{du_{r}}{ds}=-\frac{1}{R}\sum_{n=1}^{\infty}{\mathrm{Im}\\!\left(-\left(n+1\right)A_{-n}\,g^{n+1}\left(\tau\right)+\left(n-1\right)A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)\right)},$
(80)
$\displaystyle\frac{d^{2}{u_{r}}}{ds^{2}}=-\frac{1}{R^{2}}\sum_{n=1}^{\infty}{\mathrm{Re}\\!\left(\left(n+1\right)^{2}A_{-n}\,g^{n+1}\left(\tau\right)+\left(n-1\right)^{2}A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)\right)},$
$\displaystyle\frac{d^{3}u_{r}}{ds^{3}}=\frac{1}{R^{3}}\sum_{n=1}^{\infty}{\mathrm{Im}\\!\left(-\left(n+1\right)^{3}A_{-n}\,g^{n+1}\left(\tau\right)+\left(n-1\right)^{3}A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)\right)},$
$\displaystyle\frac{d^{4}u_{r}}{ds^{4}}=\frac{1}{R^{4}}\sum_{n=1}^{\infty}{\mathrm{Re}\\!\left(\left(n+1\right)^{4}A_{-n}\,g^{n+1}\left(\tau\right)+\left(n-1\right)^{4}A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)\right)}.$

A substitution of equations (80) into expressions (6) leads to the internal
forces along the coating as

$\displaystyle
M=-\frac{EJ}{R^{2}}\sum_{n=1}^{\infty}{n\,\mathrm{Re}\\!\left(\left(n+2\right)A_{-n}\,g^{n+1}\left(\tau\right)+\left(n-2\right)A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)\right)},$
(81) $\displaystyle
T=-\frac{EJ}{R^{3}}\sum_{n=1}^{\infty}{n\,\mathrm{Im}\\!\left(-\left(n+1\right)\left(n+2\right)A_{-n}\,g^{n+1}\left(\tau\right)+\left(n-1\right)\left(n-2\right)A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)\right)},$
$\displaystyle
N=-\frac{EJ}{R^{3}}\sum_{n=1}^{\infty}{n\,\mathrm{Re}\\!\left(\left(n+2\right)\left(n+1\right)^{2}A_{-n}\,g^{n+1}\left(\tau\right)+\left(n-2\right)\left(n-1\right)^{2}A_{n}\,g^{-\left(n-1\right)}\left(\tau\right)\right)}$
$\displaystyle\,-R\,\mathrm{Re}\\!\left(\sum_{n=1}^{\infty}{\left[D_{-n}+\frac{2\mu{b}}{\kappa{R}}\left(n-1\right)A_{1-n}\right]g^{n}\left(\tau\right)+\sum_{n=0}^{\infty}{\left[D_{n}+\frac{2\mu{b}}{R}\left(n-1\right)\overline{A_{1-n}}\right]g^{-n}\left(\tau\right)}}\right).$

The two coatings of the disks used in the experiments are characterized by two
strongly different bending stiffnesses, namely, EJ1=1.4 $\cdot$10-2 kNm2 and
EJ2=7 $\cdot$10-4 kNm2. Analytical results referred to these two rings,
coating a disk, are reported in Fig. 9 (only a quarter of the coating is
shown) and discriminated with the indices 1 and 2, referring to EJ1 and EJ2,
respectively. Normal, $N$, and shear, $T$, forces and bending moments, $M$ are
reported in panels (a), (b), and (c), as calculated from equations (81). The
displacement of the coating $u$, equation (77), is reported in Panel (d).

Fig. 9 shows that the internal forces and the displacement are quantitatively,
but not qualitatively, affected by the stiffness of the coating. The
displacement of the coating evidences that a stiff coating leads to a higher
non-locality effect.

From the knowledge of the internal forces, $N$, $T$, and $M$, the state of
stress at every point of the cross section of the coating can be obtained by
considering the model of an annular beam [48]. From the stress distribution,
the in-plane deviatoric stress $\sigma_{I}-\sigma_{II}$ in the coating can be
obtained, as reported inside the coatings visible in Figs. 1 and 10.

Additional deviatoric stress distributions, determined inside the annular
beams so far considered, but taken as isolated, are reported in Appendix B,
where they are compared to the results from photoelastic experiments.

Figure 9:  Force distributions and displacements of the coating for the two
coated disks used in the experiments and loaded by two equal and opposite
force distributions (reported in red, with resultant $F=7$kN). The coatings
are characterized by bending stiffness EJ1=1.4 $\cdot$10-2 kNm2 (index 1) and
EJ2=7 $\cdot$10-4 kNm2 (index 2), representative of the samples reported on
the left of Fig. 5. (a) Normalized normal forces $N_{(1)}/F$, $N_{(2)}/F$; (b)
shear forces $T_{(1)}/F$, $T_{(2)}/F$; (c) normalized bending moments
$M_{(1)}/(F\cdot{R_{1}})$, $M_{(2)}/(F\cdot{R_{2}})$, equations (81); (d)
Normalized displacements $u_{(1)}/R_{1}$ and $u_{(2)}/R_{2}$, equation (77)
along the external surface of the coated disk.

####  4.2.3 Special case: the disk with a coating imposing only an
isoperimetric constraint, but not transmitting bending

At vanishing $EJ$ for the coating, equation (49) reduces to the solution of a
disk coated with a device (which may be imagined as an inextensible string)
maintaining the isoperimetric constraint.

The determination of the elastic fields inside a disk coated with the above-
mentioned axially inextensible constraint can be pursued by using equations
(17) for the coated disk, in the limit for $EJ\to{0}$. In particular, the
Kolosov-Muskhelisvili potentials $\varphi$ and $\psi$, equations (60)1 and
(61)1, assume the same expressions (76) as for the coated case, but now the
new coefficients $A_{1-n}^{disk}$ for the disk without coating will be
generated by requiring that the bending stiffness $EJ$ in equation (49)
vanishes

$\displaystyle
A_{1-n}^{disk}=\lim_{EJ\to{0}}{A_{1-n}}=-\frac{\kappa{R}}{2\mu{b}}\frac{\left(n+1\right)\overline{D_{n}}+\left(n-1\right)D_{-n}}{\left(n-1\right)\left(n+\kappa{n}+\kappa-1\right)},$
$\displaystyle\text{for}\,n\geq{2},$ (82)

and hence potentials, equations (76), assume the form

$\displaystyle\varphi^{disk}\left(z\right)=-R\sum_{n=2}^{\infty}{\left\\{\frac{\left(n-1\right)\overline{D_{-n}}+\left(n+1\right)D_{n}}{b\left(n+1\right)\left(n+\kappa{n}+\kappa-1\right)}\right\\}}\,g^{-\left(n+1\right)}\left(z\right),$
(83)
$\displaystyle\psi^{disk}\left(z\right)=R\sum_{n=2}^{\infty}{\left\\{\frac{\left(n-1+\kappa\right)\left[\left(n-1\right)\overline{D_{-n}}+\left(n+1\right)D_{n}\right]}{b\left(n-1\right)\left[\left(n+\kappa{n}+\kappa-1\right)\right]}\right\\}}\,g^{-\left(n-1\right)}\left(z\right).$

All the elastic fields which solve the problem of the disk with the
isoperimetric coating without bending stiffness can now be computed through
relations (17), with the substitutions
$\varphi\rightarrow\varphi^{disk},\psi\rightarrow\psi^{disk}$.

####  4.2.4 Stress distributions for different models of coating

Contour plots for the in-plane principal deviatoric stress
$\left|\sigma_{I}-\sigma_{II}\right|$ and for the von Mises stress are
reported inside the disk in Fig. 10, (a) and (b), respectively, both made
dimensionless through division by the elastic shear modulus $\mu$. Four models
of disks are considered (only a quarter of the domain is reported), all loaded
with the two radial compressive stress distributions shown in the figure.
Results refer to: two coated disks, with (i.) EJ1=1.4 $\cdot$10-2 kNm2 and
(ii.) EJ2=7 $\cdot$10-4 kNm2 (upper parts); (iii.) the disk coated with the
isoperimetric constrain without bending stiffness; (iv.) the ‘nude’ disk,
namely, unconstrained.

Figure 10:  Various models of disks loaded under diametrical compression: with
two coatings stiff under bending, EJ1, and EJ2; with a coating imposing an
isoperimetric constraint, but without bending stiffness; without any coating.
Dimensionless stress contours for (a) in-plane deviatoric stress and (b)
normalized von Mises stress are reported for disk and coating (sketched as a
purple line when it corresponds to a mere isoperimetrical constraint). The
coating introduces a non local stress diffusion, an effect which tends to
vanish when the bending stiffness of the coating is decreased, is strongly
reduced for a disk with isoperimetric constraint, and vanishes when boundary
constraints on the disk are absent.

Note that the deviatoric and von Mises stress distributions are similar, as it
may be expected as both are providing measures of distortional stress. Numbers
reported on the contours facilitate comparisons and same colors refer to the
same stress level.

The figure highlights the role of the bending stiffness of the coating,
showing that the stress strongly increases at the mid point of load
distribution, when the bending stiffness vanishes, as in the case of the
isoperimetric constrained disk. This effect is even more pronounced in the
case of the unconstrained disk. A non-local diffusion of the stresses are
clearly visible in the coated cases.

###  4.3 Comparison between photoelastic experiments and mechanical modelling

####  4.3.1 Determination of stresses from fringe colours

The photoelastic analysis of the coated disk is reported below, referred to
the same vale of the load resultant $F=7$kN, applied at the top of the
vertical diameter of the samples shown in Fig. 5. Results are reported in Fig.
11, where the photoelastic image is reported on the right half of the sample,
while the analytical solution is depicted on the left half. The in-plane
deviatoric stress as determined from the analysis of the transmitted colours
is reported in Table 3.

Figure 11: Determination of the in-plane deviatoric stress from the
photoelastic fringes. The dashed lines on the right half of the sample refer
to Table 3, while the location of the same stress level is reported on the
left half of the sample, as evaluated with the analytic solution. The two
samples are those reported in Fig. 5.

|  |  Coated disk sample upper part of Fig. 5 |  Coated disk sample lower part of Fig. 5
---|---|---|---
Transmitted color | $\bf N\\!\cdot\\!\lambda$ | Line | $\bf\left|\sigma_{I}-\sigma_{II}\right|/\mu$ | Line | $\bf\left|\sigma_{I}-\sigma_{II}\right|/\mu$
$[-]$ | $[\text{nm}]$ | $[-]$ | $[-]$ | $[-]$ | $[-]$
Gray | 1$\cdot$218 | A | 0.0083 | L | 0.0083
Brilliant yellow | 1$\cdot$390 | B | 0.0148 | M | 0.0148
Orange | 1$\cdot$505 | C | 0.0192 | - | -
Red | 1$\cdot$536 | D | 0.0203 | N | 0.0203
Indigo-violet | 1$\cdot$575 | E | 0.0219 | O | 0.0219
Sky blue | 1$\cdot$664 | F | 0.0252 | P | 0.0252
Brilliant Yellow | 2$\cdot$390 | - | - | Q | 0.0296
Orange | 2$\cdot$505 | - | - | R | 0.0384
Red | 2$\cdot$536 | G | 0.0108 | - | -
Indigo-violet | 2$\cdot$575 | H | 0.0116 | S | 0.0438
Sky blue | 2$\cdot$664 | I | 0.0133 | T | 0.0505
Orange | 3$\cdot$505 | - | - | U | 0.0576
Indigo-violet | 3$\cdot$575 | - | - | V | 0.0657

Table 3:  Determination of the in-plane deviatoric stress
$\left|\sigma_{I}-\sigma_{II}\right|/\mu$ from the photoelastic experiments
performed in the sample reported in the upper part of Fig. 5.

The stress field inside the disk shows an excellent agreement between theory
and experiments, while the in-plane deviatoric stress in the external coating
evidences some discrepancies. The latter are due to the fact that the coating
has been modelled as an interface, so that the evaluation of the stresses is
conducted starting from the knowledge of the internal force resultants,
equations (81), using the classical de Saint-Venant theory, which is an
approximation. Moreover, the stress level in the external part of the coating
near the zone where the load is applied makes the reading of the transmitted
colors difficult. For this reason, a quantitative evaluation of the stresses
inside the external coating is not reported for the sample in the lower part
of Fig. 5.

Although detachment is not possible in our experiments, points at the
interface where detachment between disk and coating may occur are those
subject to tensile tractions. The distribution of the latter along the
disk/coating interface are depicted in Fig. 12, where the two samples shown in
Fig. 5 are subject to the resultant force $F=7$kN. Note that red line refers
to tensile tractions, while blue refers to compressive.

Figure 12:  The traction applied to the disk is reported along the interface
between coating and disk. When tensile (red line), the traction determines the
zone where detachment of coating might occur (it does not in our experiments).
The samples are those shown in Fig. 5.

##  5 Conclusions

Bulk materials having their external surface coated with a film of mechanical
properties different from the base material are commonly used in a number of
technologies. Within this context, a paradigmatic mechanical problem has been
formulated in the present article, solved, and validated against experiments.
This is a linear elastic (isotropic) circular disk, coated with a thin and
stiff film, modelled as a linear elastic circular beam unshearable and axially
inextensible, but linearly deformable under bending. The coating therefore
imposes an isoperimetric constraint to the disk and a nonlocal load diffusion
on its boundary. The latter effect is related to the introduction of an
internal length scale and implies that concentrated forces applied to the
external of the coating/disk complex do not introduce a singularity. It has
been shown that the considered problem can be analytically solved via complex
potentials formalism, so that the full displacement, strain, and stress fields
have been derived for a generic applied external load. Finally, it has been
shown how to practically realize the coated disk with a photoelastic material,
thus producing two models which have been used to validate the analytical
solution, in the case when two equal and opposed load distributions are
applied on a small circular segment. The presented results open new
possibilities in the design of coated solids of cylindrical geometry, which
may find applications in micro and nano technologies, for instance in the
characterization of nanowires via nanoindentation.

## Acknowledgements

D.B. remembers with great pleasure and emotion the longstanding, intense,
fruitful, and beautiful cooperation with Natasha and Sasha on so many topics
of science. He looks forward to continue enjoying scientific collaboration
with them and sincere friendship for many more years to come.

D.B. and M.G. gratefully acknowledges the funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Sklodowska-
Curie grant agreement No 955944-REFRACTURE2. S.G.M. gratefully acknowledges
support from National Science Foundation, award NSF CMMI - 2112894.

##  Appendix A Calibration of the photoelastic model

Polimethyl Metacrilate, as other materials, exhibits the phenomenon known as
temporary birefringence [51]. The photoelastic parameters for our experiments
have been determined by testing under diametral compression a disk with radius
$R=$ 35 mm and thickness $b=$ 5 mm. The usual identification of the
photoelastic parameters is performed by using the elastic solution for a disk
subject to opposite concentrated forces, for which the in-plane deviatoric
stress $\left|\sigma_{I}-\sigma_{II}\right|$ at the centre of the disk is
known [52]. In this way, the photoelastic constant $f_{\sigma}$ of the
material can be evaluated, once the fringe number $N$ at the centre of the
disk has been determined. From a photoelastic experiment performed on the
uncoated disk (reported in the lower part of Fig. 5), the fringe number $N$
appearing at the centre of the disk is found to be $N=2$ from the left part of
Fig. 13, referred to an applied force $F=7.1$ kN. Therefore, the photoelastic
constant of the material is found to be approximately $f_{\sigma}=129.1$ N/mm.

Figure 13:  Left: A polymethyl methacrylate disk (subject to two opposite
forces $F=7.1$kN) exhibits temporary birefringence displaying a fringe number
$N=2$ near the centre, where the corresponding indigo-violet fringes are
visible. Right: the state of in-plane deviatoric stress (dashed line) compared
with the theoretical results pertaining to the reported load distribution.
Observation has been carried with a circular polariscope at a white light;
quantitative results are reported in Table 4.

Line | Transmitted colour | N$\lambda$ |  $\bf\left|\sigma_{I}-\sigma_{II}\right|/\mu$
---|---|---|---
$[-]$ |  $[-]$ |  $[\text{nm}]$ |  $\left[-\right]$
A | Brilliant yellow | 390 | 0.0148
B | Red | 536 | 0.0203
C | Indigo-violet | 575 | 0.0219
D | Sky blue | 664 | 0.0252
E | Brilliant yellow | 2$\cdot$390 | 0.0296
F | Orange | 2$\cdot$505 | 0.0384
G | Indigo-violet | 2$\cdot$575 | 0.0438
H | Sky blue | 2$\cdot$664 | 0.0505
I | Orange | 3$\cdot$505 | 0.0576
L | Indigo-violet | 3$\cdot$575 | 0.0657
M | purple | 4$\cdot$565 | 0.0859

Table 4:  Determination of the in-plane deviatoric stress
$\left|\sigma_{I}-\sigma_{II}\right|/\mu$ from the photoelastic experiment
performed in the uncoated disk sample reported in the lower part of Fig. 5.

Having evaluated $f_{\sigma}$, the in-plane deviatoric stress inside the disk
can be evaluated for each integer fringe number $N$ (corresponding to indigo-
violet transition colour, visible at a white light). In order to determine the
order of fractional fringes, the in-plane deviatoric stress has to be
evaluated for a particular transmitted colour, once its wave length $\lambda$
is known. In Table 4 the dimensionless in-plane deviatoric stress for the
coloured fringes are reported, as obtained during the photoelastic experiment
shown in Fig. 13 (dashed line in the right part), contrasted with the analytic
results (obtained for the sketched traction distribution and drawn as
continuous lines) obtained from [49].

##  Appendix B The ring, an annular beam

The problem of a loaded circular beam is a classical exercise in structural
mechanics and can completely be solved by means of the relations (6), once the
expressions for the radial displacement $u_{r}$ in equation (25) is known in
terms of the $A_{\pm{n}}$ coefficients. The stress distribution inside the
circular beam can be determined once the internal forces have been determined
from equations (81) in the limit $\mu\to{0}$ for the disk.

Figure 14: Photoelastic fringes generated during (vertical) diametrical
compression of an annular beam, with bending stiffness EJ1=1.4 $\cdot$10-2
kNm2 (upper part) and EJ2=7 $\cdot$10-4 kNm2 (lower part). Four increasing
values of compression are reported (from left to right): 0.30 kN (0.06 kN),
0.40 kN (0.15 kN), 0.50 kN (0.30 kN), and 0.60 kN (0.40 kN) in the upper part
(in the lower part) for the samples shown in Fig. 5; photos have been taken at
white circularly polarized light.

Photoelastic experiments on the annular beams shown in Fig. 5 are reported in
Fig. 14. The photos refer to four different compressive values of load,
namely, 0.30 kN (0.06 kN), 0.40 kN (0.15 kN), 0.50 kN (0.30 kN), and 0.60 kN
(0.40 kN) in the upper part (in the lower part). Maps of the normalized in-
plane deviatoric stress $\left|\sigma_{I}-\sigma_{II}\right|/EJ$ are compared
in Fig. 15 with the photoelastic experiments, loaded at 0.60 kN and 0.40 kN.

Figure 15: Distribution of the normalized in-plane deviatoric stress
$(\sigma_{I}-\sigma_{II})/EJ_{j}$ (j=1,2) compared with the photoelastic
fringes during a diametral compression test of two photoelastic circular beams
with bending stiffness EJ1=1.4 $\cdot$10-2 kNm2 (left) and EJ2=7 $\cdot$10-4
kNm2 (right). The applied external load distribution $p$ is modelled with a
Fourier series expansion truncated at $N=100$ and enhanced with the Lanczos
smoothing method. The load distribution results in a compression of 0.60 kN
(left) and 0.40 kN (right).

The results demonstrate the high accuracy of the beam theory to produce the
stress distribution in a circular beam. This is a known result and can be
found in [51].

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