# A Critical Study of Cottenden _et al._ ’s _An Analytical Model of the Motion
of a Conformable Sheet Over a General Convex Surface in the Presence of
Frictional Coupling_

Kavinda Jayawardana<EMAIL_ADDRESS>
###### Abstract

In our analysis, we show that what Cottenden _et al._ [1, 2] and Cottenden [3]
accomplish is the derivation of the ordinary capstan equation, and a solution
to a dynamic membrane with both a zero-Poisson’s ratio and a zero-mass density
on a rigid right-circular cone. The authors states that the capstan equation
holds true for an elastic obstacle, and thus, it can be used to calculate the
coefficient of friction between human skin and fabrics. However, using data
that we gathered from human trials, we show that this claim cannot be
substantiated as it is unwise to use the capstan equation (i.e. belt-friction
models in general) to calculate the friction between in-vivo skin and fabrics.
This is due to the fact that such models assume a rigid foundation, while
human soft-tissue is deformable, and thus, a portion of the applied force to
the fabric is expended on deforming the soft-tissue, which in turn leads to
the illusion of a higher coefficient of friction when using belt-friction
models.

###### keywords:

Capstan Equation , Contact Mechanics , Coefficient of Friction , Mathematical
Elasticity

##  1 Introduction

Consider a situation where two elastic bodies that are in contact with each
other, where the contact area exhibits friction. A common area where modelling
of such problems can be found in the field of tire manufacturing [4, 5]. Now,
consider the scenario where one of the elastic bodies is very thin and almost
planar in a curvilinear sense, in comparison to the other body. Then the thin
body can be approximated by a shell or a membrane, and such models can be used
to model skin abrasion caused by fabrics as a result of friction. Need for
valid modelling techniques are immensely important in fields such as field of
incontinence associated dermatitis [2], sports related skin trauma [6] and
cosmetics [7]. It is documented that abrasion damage to human skin in cases
such as the jogger’s nipple [8] and dermatitis from clothing and attire [9]
are caused by repetitive movement of fabrics on skin, and in cases such as
pressure ulcers [10] and juvenile plantar dermatitis [11], friction may worsen
the problem. In an attempt to model such problem mathematically, Cottenden _et
al._ [2] put forward a model in their publication _An analytical model of the
motion of a conformable sheet over a general convex surface in the presence of
frictional coupling_. We show that, regardless of the authors’ claims, what
they derive is the ordinary capstan equation and a solution for a membrane
with a zero-Poisson’s ratio and a zero-mass density on a rigid right-circular
cone. Cottenden _et al._ [1, 2] and Cottenden [3] also claims that capstan
equation can be used to calculate the friction between in-vivo skin and
fabrics. With data gathered from human trials, we show that it is unwise to
use _belt-friction models_ (e.g. capstan equation) to calculate the
coefficient of friction between in-vivo skin and fabrics, as such models
assume a rigid foundation, while human soft-tissue is elastic; thus, a portion
of the applied force to the fabric is expended on deforming the soft-tissue,
which in turn leads to the illusion of a higher coefficient of friction when
using belt-friction models.

##  2 Capstan Equation

The capstan equation, otherwise known as Euler’s equation of tension
transmission, is the relationship governing the maximum applied-tension
$T_{\text{max}}$ with respect to the minimum applied-tension $T_{0}$ of an
elastic string wound around a rough cylinder. Thus, the governing equation can
be express by the following equation,

$\displaystyle T_{\text{max}}=T_{0}\exp(\mu_{F}\theta)~{},$ (1)

where $\theta$ is the contact angle and $\mu_{F}$ is the coefficient of
friction. By _string_ we mean a one-dimensional elastic body, and _rough_ we
mean that the contact area exhibits friction, where the coefficient of
friction is the physical ratio of the magnitude of the shear force and the
normal force between two contacting bodies. The capstan equation is the most
perfect example of a _belt-friction model_ , which describes behaviour of a
belt-like object moving over a rigid-obstacle subjected to friction [12]. In
engineering, the capstan equation describes a body under a load equilibrium
involving friction between a rope and a wheel like circular object, and thus,
it is widely used to analyse the tension transmission behaviour of cable-like
bodies in contact with circular profiled surfaces, such as in rope rescue
systems, marine cable applications, computer storage devices (electro-optical
tracking systems), clutch or brake systems in vehicles, belt-pulley machine
systems and fibre-reinforced composites [13].

For a rigorous study of the capstan equation (a membrane or a string over a
rigid-obstacle) generalised to non-circular geometries, we refer the reader to
chapter 2 of Jayawardana [14]. There, as an example, the author present a
solution to the capstan equation generalised for a rigid elliptical-prism,
i.e. given a prism with a horizontal diameter of $2a$ and the vertical
diameter of $2b$, parametrised by the map
$\boldsymbol{\sigma}(x^{1},\theta)=\boldsymbol{(}x^{1},~{}a\sin(\theta),~{}b\cos(\theta)\boldsymbol{)}$,
where $\theta$ is the acute angle that the vector
$\boldsymbol{(}0,0,1\boldsymbol{)}_{\text{C}}$ makes with the vector
$\boldsymbol{(}0,\varphi(\theta),0\boldsymbol{)}$, and where
$\varphi(\theta)=(b^{2}\sin^{2}(\theta)+a^{2}\cos^{2}(\theta))^{\frac{1}{2}}$,
the capstan equation takes the following form,

$\displaystyle
T_{\text{elliptical}}(\theta)=T_{0}\exp\left(\mu_{F}\arctan\left(\frac{b}{a}\tan(\theta)\right)\right)~{}.$
(2)

Note that equation (2) assumes that minimum applied-tension, $T_{0}$, is
applied at $\theta=0$, and the vector brackets
$\boldsymbol{(}\cdot\boldsymbol{)}$ implies that the vectors are in the
Euclidean space and $\boldsymbol{(}\cdot\boldsymbol{)}_{\text{C}}$ implies
that the vectors are in the curvilinear space.

Figure 1: Tension ratio against $\delta b$.

Equation (2) implies that the maximum applied-tension, $T_{\text{max}}$, is
dependent on the mean curvature of the rigid prism. To investigate this matter
further, assume that $T_{0}$ is applied at $\theta=-\frac{1}{4}\pi$ and
$T_{\text{max}}$ applied at $\theta=\frac{1}{4}\pi$, and thus, equation (2)
implies that

$\displaystyle\delta\tau=\exp\left(2\mu_{F}\arctan\left(\frac{b}{a}\right)\right)~{},$
(3)

where $\delta\tau=T_{\text{max}}/T_{0}$, which is defined as the _tension
ratio_. As the reader can see that for a fixed contact interval and a fixed
coefficient friction, equation (3) implies a non-constant tension ratio for
varying $\delta b$, where $\delta b=b/a$. As the mean curvature of the prism
is $H(\theta)=\frac{1}{2}ab(\varphi(\theta))^{-3}$, one can see that the
tension ratio is related to the mean curvature by the following relation,

$\displaystyle\delta\tau=\exp\left(2\mu_{F}\arctan\left[\max_{\theta\in[-\frac{1}{4}\pi,\frac{1}{4}\pi]}(2aH(\theta),1)+\min_{\theta\in[-\frac{1}{4}\pi,\frac{1}{4}\pi]}(2aH(\theta),1)-1\right]\right)~{}.$

Figure 1 presents a visualisation of the tension ratio against $\delta b$
(which is analogous to the mean curvature), which is calculated with
$\mu_{F}=\frac{1}{2}$ and $\delta b\in[\frac{1}{2},\frac{3}{2}]$, and it shows
that, for a fixed contact interval, as $\delta b$ increases (i.e. as the mean
curvature of the contact region increases), the tension ratio also increases.
This is an intuitive result as the curvature of the contact region increases,
the normal reaction force on the membrane (or the string) also increases,
which in turn leads to a higher frictional force, and thus, a higher tension
ratio. Now, this is a fascinating result as this effect cannot be observed
with the ordinary capstan equation (1).

For a rigorous study of belt friction models, we refer the reader to section
6.2 of Jayawardana [14], and for a rigorous study of the capstan equation
generalised to elastic obstacles, we refer the reader to Konyukhov’s [15], and
Konyukhov’s and Izi’s [16].

##  3 Cottenden _et al._ ’s Work

Cottenden _et al._ [2] (the principal author is D. J. Cottenden) attempt to
derive a mathematical model to analyse a frictionally coupled membrane
(defined as a nonwoven sheet) on an elastic foundation (defined as a
substrate) based on the research findings of D. J. Cottenden’s PhD thesis
[3]111https://discovery.ucl.ac.uk/id/eprint/1301772/. It is assumed that
friction (abrasion in the context of the publication) is the cause of some
pressure ulcers in largely immobile patients, and abrasion due to friction
contributes to the deterioration of skin health in incontinence pad users,
especially in the presence of liquid. The current literature shows very little
research in the area of frictional damage on skin due to fabrics, and thus,
the authors’ goal is to present a mathematical model to investigate this
phenomenon in a purely geometrical setting. Thus, the authors propose a model
for a general class of frictional interfaces, which includes those that obey
Amontons’ three laws of friction.

Cottenden _et al._ ’s method [2] for calculating the kinetic frictional force
induced on human skin due to nonwoven fabrics is as follows. The human body
part in question is modelled as a homogeneous isotropic ‘convex surface
[_sic_]’ [2] (i.e. the substrate) and the nonwoven fabric is modelled as an
isotropic membrane (i.e. the nonwoven sheet). The goal is to find the stresses
acting on the nonwoven sheet, including determining the friction acting on the
substrate. The contact region between the fabric and the skin is defined as
‘An _Instantaneous Isotropic Interface_ , [which] is an interface composed of
a pair of surfaces which have no intrinsically preferred directions and no
directional memory effects, so that the frictional force acts in the opposite
direction to the _current_ relative velocity vector … or to the sum of
_current_ applied forces acting to initiate motion …’ (see section 2.2 of
Cottenden _et al._ [2]): this simply implies that the contact region is
isotropic and nothing more. Also, consider the contact body in question: it is
modelled as a surface, i.e. a two-dimensional manifold. However, in reality,
it must be modelled as a three-dimensional object as a two-dimensional object
cannot describe the elastic properties of a fully three-dimensional object
such as a human body part, as a two-dimensional surface has measure zero in
three-dimensional space (for an introduction on measure theory, please consult
chapter 1 of Kolmogorov _et al._ [17]); Unless suitable assumptions are made
as one finds in shell theory (for a comprehensive mathematical study of the
shell theory, please consult sections B of Ciarlet [18]); however, this is not
what the authors are considering. Now, consider the authors’ statement
regarding the modelling assumptions carefully, particularly the term ‘convex
surface’. The authors definition of convexity is
$\boldsymbol{\eta}\cdot\boldsymbol{\nabla}_{\\!\text{E}}\hat{\textbf{N}}\cdot\boldsymbol{\eta}\geq
0$ (see section 3.1 of Cottenden _et al._ [2]), where $\hat{\textbf{N}}$ and
$\boldsymbol{\eta}$ are unit normal and tangential vectors surface
respectively. However, the authors’ definition is erroneous. Convexity has a
very precise mathematical definition, i.e. we say the functional
$f:X\to\mathbb{R}$ is convex, if $f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$,
$\forall~{}t\in[0,1]$ and $\forall~{}x,y\in X$ (for more on the definition of
convexity, please see chapter 1 of Badiale and Serra [19]). Also, the very
idea of a convex surface is nonsensical as definition of convexity is only
applicable to functionals. A simple example of a convex functional is
$\exp(\cdot):\mathbb{R}\to\mathbb{R}_{>0}$. One is left to assume that what
the authors mean by convexity is surfaces (i.e. manifolds) with a positive
mean-curvature. For more on elementary differential geometry, please consult
do Carmo [20] or Lipschutz [21].

In their analysis, the authors defines a membrane with the following property,
‘always drapes, following the substrate surface without tearing or puckering’
(see section 2.1 of [2]). The authors’ definition is erroneous, as one cannot
guarantee that the given property will hold for a flat-membrane (in a
Euclidean sense) over an arbitrary curved surfaces. To illustrate the flaw,
consider a flat elastic-membrane (i.e. a film) over a rigid sphere. The only
way one can keep the membrane perfectly in contact with the sphere in a two-
dimensional region with nonzero measure is by deforming the membrane by
applying appropriate boundary stresses and or external loadings. Otherwise,
the membrane only makes contact with the sphere at a single point or a line.
Also, the authors do not specify whether the membrane is elastic or not. One
is left to assume that the membrane is elastic as the proposed frame work does
not acknowledge plastic deformations. Note that the authors never referred to
their nonwoven sheet as a membrane, but a membrane (or a film) is the closest
mathematical definition for modelling such objects. Please consult chapters 4
and 5 of Ciarlet [18] or chapter 7 of Libai and Simmonds [22] for a
comprehensive analysis on the theory of membranes.

To find the stresses acting on the membrane, consider Cauchy’s momentum
equation in the Euclidean space, which the authors define as follows,

$\displaystyle\boldsymbol{\nabla}_{\\!\text{E}}\cdot\textbf{T}+\textbf{f}=\rho\ddot{\boldsymbol{\chi}}~{},$
(4)

where T is Cauchy’s stress tensor,
$\textbf{f}=\textbf{f}(\textbf{T},\boldsymbol{\nabla}_{\\!\text{E}}\textbf{T})$
is the force density field and $\rho$ is the material mass density of the
membrane, $\boldsymbol{\nabla}_{\\!\text{E}}$ is the Euclidean differential
operator and $\boldsymbol{\chi}$ is given as a ‘time-dependent deformation
function mapping the positions of points in their undeformed _reference_
configuration to their deformed positions and the superposed double dot
denotes a double material description time derivative’ (see section 2.1 of
Cottenden _et al._ [2]). It is unclear what $\boldsymbol{\chi}$ represent from
the authors’ definition, whether it is the displacement field of the membrane
or some time-dependent mapping from one manifold to another. If the latter is
true, then equation (4) has a very different meaning, i.e. it means that the
space is dependent of time, and such problems are encountered in the field of
cosmology. If the reader consults section 5.4 of Cottenden’s thesis [3], then
it becomes evident that $\boldsymbol{\chi}$ is, indeed, a time dependent map.
However, if one consults Cottenden _et al._ [1, 2] and Cottenden [3], then one
concludes that the authors do not put forward the framework to handle the 3+1
decomposition in general relativity, with any mathematical rigour. If the
reader is interested in the 3+1 formalism in general relativity, then please
consult the publications [23, 24, 25] or Dr J. A. V. Kroon (QMUL) on his LTCC
lecture notes on _Problems of General Relativity_ 222
http://www.maths.qmul.ac.uk/$\sim$jav/LTCC.htm, where the reader can find an
extraordinary solution for two merging black-holes (i.e. the Brill-Lindquist
solution).

Assuming that the foundation is static and rigid, and the mass density of the
membrane is negligible, i.e. $\rho\approx 0$, the authors state that Cauchy’s
momentum equation (4) can be expressed as

$\displaystyle\textbf{P}_{s}\cdot(\boldsymbol{\nabla}_{\\!\text{E}}\cdot\textbf{T})+\textbf{P}_{s}\cdot\textbf{f}$
$\displaystyle=\boldsymbol{0}~{},$ (5)
$\displaystyle-(\boldsymbol{\nabla}_{\\!\text{E}}\hat{\textbf{N}}):\textbf{T}+\hat{\textbf{N}}\cdot\textbf{f}$
$\displaystyle=0~{},$ (6)

where $\textbf{P}_{s}$ projection matrix to the substrate (the explicit form
is not defined by the authors), $\hat{\textbf{N}}$ is the unit normal to the
surface, and $\cdot$ and $:$ are defined as a contraction and a double
contraction in the Euclidean space respectively. Although it is not explicitly
defined, one must assume that the authors use the fact that membranes cannot
support any normal stresses, i.e.
$\hat{\textbf{N}}\cdot\textbf{T}=\boldsymbol{0}$, to obtain equation (6). The
authors give equations (5) and (6) as the state of the ‘general case’ of the
problem. However, their assertion cannot hold as the system is
underdetermined. Consider the vector f, which consists of three unknowns.
Also, consider the tensor $\textbf{T},$ which is a symmetric tensor with six
unknowns. Thus, using the condition
$\hat{\textbf{N}}\cdot\textbf{T}=\boldsymbol{0}$ the number of unknowns can be
reduced by three: leaving six remaining unknowns. Now, direct one’s attention
to equations (5) and (6), which provide three additional equations. Thus, one
comes to the conclusion that one has an underdetermined system, with three
equations and six unknowns. Furthermore, there is no description of any
boundary conditions for the ‘general case’, which are essential in obtaining a
unique solution.

The derivation of Cottenden _et al._ ’s governing equations [3] can be found
on section 2.2 to 2.4 of the publication. Upon examination, the reader may
find that the methods put forward by the authors’ are inconsistent with
mathematical elasticity, contact mechanics and differential geometry. Thus, we
refer reader to Kikuchi and Oden [26] to see how to model friction with
mathematical competence and to show how incredibly difficult modelling such
problems are. We further refer the reader to Ciarlet [18] to see how to model
mathematical elasticity in a differential geometry setting with mathematical
rigour.

To find explicit solutions, the authors direct their attention to only ‘surfa-
ces that are isomorphic to the plane; that is, those which have the same first
fundamental form as the plane; the identity matrix in the case of plane
Cartesian coordinates. [_sic_]’ [2]. Found in section 4.1 of Cottenden _et
al._ [2], this is the basis for their entire publication (also the basis for
Cottenden’s [3] thesis). However, the authors’ statement is nonsensical. An
_isomorphism_ (preserves form) is at least a homomorphism, i.e. there exists
at least a continuous bijective mapping, whose inverse is also continuous,
between the two manifolds in question [20, 21]. Thus, a surface that is
isomorphic to the plane simply implies that there exists a continuous
bijective map, with a continuous inverse, between the surface in question and
the Euclidean plane, and it does not automatically guarantee that the surface
have the same metric as the Euclidean plane under the given map. The latter
part of the authors’ statement is clearly describing surfaces that are
_isometric_ (preserves distance) to the Euclidean plane, i.e. surfaces with a
zero-Gaussian curvature. However, the statement is still erroneous as being a
surface that is isometric to the Euclidean plane does not guarantee that the
surface have the same metric as the Euclidean plane. Being isometric to the
Euclidean plane simply implies that, if $f:U\subset\mathbb{R}^{2}\to
W\subset\textbf{E}^{3}$ is a 2D manifold that is isometric to the Euclidean
plane, then there exists a map $g:V\subset\mathbb{R}^{2}\to
U\subset\mathbb{R}^{2}$ such that the first fundamental form of the isometry
$f\circ g:V\subset\mathbb{R}^{2}\to W\subset\textbf{E}^{3}$ is the $2\times 2$
identity matrix [20, 21]. One is left to assume that the surfaces that are in
question by the authors belongs a subgroup of surfaces with zero-Gaussian
curvature that has the same metric as the Euclidean plane with respect to the
authors’ coordinate system, i.e. cylinders and right-circular cones: as one
later see that these are the only possible manifolds that can generate any
valid solutions. Note that Cottenden [3] accredits Pressley [27] for his
differential geometry results. However, Pressley’s publication [27] is a
widely accepted and verified mathematical publication in the field of
differential geometry, and it does not contain such provably false statements
as given by Cottenden [3].

Now, consider the equation

$\displaystyle\textbf{P}_{s}\cdot\textbf{f}+\mu_{d}(\hat{\textbf{N}}\cdot\textbf{f})\dot{\boldsymbol{\chi}}=0~{},$
(7)

which the authors define as Amontons’ law friction, where $\mu_{d}$ is the
coefficient of dynamic friction and $\dot{\boldsymbol{\chi}}$ is the relative
velocity vector between the membrane and the foundation. The inclusion of the
two equations implied by condition (7) still does not guarantee that the
system is fully determined, as the system requires one more equation to be
fully determined.

Now, assume that one is dealing with a rectangular membrane whose orthogonal
axis defined by the coordinates $\boldsymbol{(}x,y\boldsymbol{)}$, where $y$
defines the longer dimension, that is placed over a surface defined by the
regular map $\boldsymbol{\sigma}$. Also, assume that Poisson’s ratio of the
membrane is zero to prevent any lateral contractions due to positive tensile
strain. To reduce the complexity, the authors modify the problem by letting
$\dot{\boldsymbol{\chi}}$ be parallel to $\boldsymbol{\sigma}_{\\!,y}$. Also,
by applying a boundary stress of $T_{0}$ at some point $\phi_{1}$ whilst
applying a even greater stress at $\phi_{2}$ so that $T_{yy}(y)$ is an
increasing function in $y$, where $\phi_{1}$ and $\phi_{2}$ are angles of
contact such that $\phi_{1}<\phi_{2}$. Due to zero-Poisson’s ratio and the
boundary conditions, one finds $T_{xx}=0$, $T_{xy}=0$, where $T_{ij}$ are
stress tensor components. Thus, the governing equations finally reduce to a
fully determined system, i.e. is only now the number of unknowns equals to the
number of governing equations. Therefore, one must understand that having
zero-Gaussian curvature and zero-Poisson’s ratio is a necessity for this
model, and it is not some useful tool for deriving explicit equations as
stated by the authors. Upon integrating equation (7), under the specified
boundary conditions, one finds solutions of the following form (see equation
4.4 of Cottenden _et al._ [2]),

$\displaystyle
T_{yy}(y)=T_{0}\exp\left(\mu_{d}\int^{y}|C_{yy}(\eta)|~{}d\eta\right)~{},$ (8)

where
$C_{\alpha\beta}=F_{I\alpha\gamma}^{-1}F_{II\gamma\delta}F_{I\beta\delta}^{-1}$
is defined as the curvature tensor, ${\boldsymbol{F}}_{\\!I}$ is the first
fundamental form tensor and $F_{IIyy}$ is the only nonzero component of the
second fundamental form tensor. However, equation (8) is erroneous. This is
because, whatever is inside the $\exp(\cdot)$ term must be non-dimensional,
but this is not the case with equation (8). To illustrate this flaw, let $L$
be an intrinsic Euclidean length scale and $\ell$ be an intrinsic length scale
of the curvilinear coordinate $y$. Now, with the definition of $C_{yy}$ (see
equation 3.3 of Cottenden _et al._ [2]), one finds that the length scale
inside the $\exp(\cdot)$ term in equation (8) to be ${(\ell/L)}^{3}$. Given
that $y=\theta$ (i.e. the contact angle, which is dimensionless), one finds
that the length scale inside the $\exp(\cdot)$ term to be $L^{-3}$, which is
not mathematically viable. This flaw of the authors’ work is a result of not
correctly following tensor contraction rules, and not discerning between
covariant and contravariant tensors (see sections 3.2 and 4.1 of Cottenden _et
al._ [2]). For elementary tensor calculus, please consult Kay [28].

If one assumes a 2D manifold that is _isometric_ to the Euclidean plane, has a
positive mean-curvature (with respect to the unit-outward normal) and whose
first fundamental form tensor is diagonal (after a change of coordinates or
otherwise), and should one follow the derivation with mathematical rigour,
then one finds a solution of the following form,

$\displaystyle
T_{y}^{y}(y)=T_{0}\exp\left(-\mu_{d}\int^{y}\sqrt{F_{Iyy}(\eta)}F_{IIy}^{~{}~{}y}(\eta)~{}d\eta\right)~{},$
(9)

where a rigorous derivation can be fond in chapter 2 of Jayawardana [14]. As
the reader can see that unlike equation (8), no dimension violations can be
possible with equation (9).

To find the explicit solution for the general _prism_ case, the authors
present the following map (see section 4.2 of Cottenden _et al._ [2])

$\displaystyle\boldsymbol{\sigma}(x,y)$
$\displaystyle=\boldsymbol{(}R(\phi)\cos(\phi),~{}R(\phi)\sin(\phi),~{}x\cos(\zeta)+y\sin(\zeta)\boldsymbol{)}~{},$
(10)

where

$\displaystyle d\phi$
$\displaystyle=\frac{\cos(\zeta)dy-\sin(\zeta)dx}{\sqrt{R(\phi)^{2}+(R^{\prime}(\phi))^{2}}}~{}.$

From the authors’ definition, $\zeta$ appears to be the acute angle that the
vector $\hat{\boldsymbol{\sigma}}_{\\!,y}$ makes with the vector
$\hat{\boldsymbol{\sigma}}_{\\!,\phi}$, and $R$ and $\phi$ appear to be the
radius of curvature and the angle of the centre of rotation respectively. One
can clearly see that map (10) is only valid for cylinders (i.e. prisms with a
constant radius) as it must have the same metric as the Euclidean plane. To be
precise,

$\displaystyle\boldsymbol{F}_{I}=$
$\displaystyle\begin{pmatrix}{\boldsymbol{\sigma}}_{\\!,x}\cdot{\boldsymbol{\sigma}}_{\\!,x}&{\boldsymbol{\sigma}}_{\\!,x}\cdot{\boldsymbol{\sigma}}_{\\!,y}\\\
{\boldsymbol{\sigma}}_{\\!,x}\cdot{\boldsymbol{\sigma}}_{\\!,y}&{\boldsymbol{\sigma}}_{\\!,y}\cdot{\boldsymbol{\sigma}}_{\\!,y}\end{pmatrix}$
$\displaystyle=$
$\displaystyle[R(\phi)^{2}+(R^{\prime}(\phi))^{2}]\begin{pmatrix}(\phi_{,x})^{2}+\cos^{2}(\zeta)&\phi_{,x}\phi_{,y}+\sin(\zeta)\cos(\zeta)\\\
\phi_{,x}\phi_{,y}+\sin(\zeta)\cos(\zeta)&(\phi_{,y})^{2}+\sin^{2}(\zeta)\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}~{},$

which, in turn, implies the following,

$\displaystyle\phi_{,x}$
$\displaystyle=-\frac{\sin(\zeta)}{\sqrt{R(\phi)^{2}+(R^{\prime}(\phi))^{2}}}~{},$
$\displaystyle\phi_{,y}$
$\displaystyle=\frac{\cos(\zeta)}{\sqrt{R(\phi)^{2}+(R^{\prime}(\phi))^{2}}}~{},$

and thus, the following,

$\displaystyle(\phi_{,x})^{2}+(\phi_{,y})^{2}$
$\displaystyle=\frac{1}{R(\phi)^{2}+(R^{\prime}(\phi))^{2}}~{}.$ (11)

As $\phi$ is a real function (i.e. not complex) and noting that equation (11)
must hold for all $x$ and $y$, one finds that $R^{\prime}=0$, and thus,
equation (10) reduces to the following,

$\displaystyle\boldsymbol{\sigma}(x,y)$
$\displaystyle=\boldsymbol{(}c\cos(\phi),~{}c\sin(\phi),~{}x\cos(\zeta)+y\sin(\zeta)\boldsymbol{)}~{},$
$\displaystyle\phi(x,y)$
$\displaystyle=\frac{1}{c}[\cos(\zeta)y-\sin(\zeta)x]~{},$

which represents a parametrisation of a cylinder, where $c$ is a positive
constant (and $R=c$). Now, given that a solution exists in the interval
$[\phi_{1},\phi_{2}]$, the authors state that the solution can be expressed as
follows (see equation 4.7 of Cottenden _et al._ [2]),

$\displaystyle
T_{yy}(\phi_{2})=T_{0}\exp\left(\mu_{d}\cos(\zeta)\left[\phi-\arctan\left(\frac{R(\phi)_{,\phi}}{R(\phi)}\right)\right]\big{|}^{\phi_{2}}_{\phi_{1}}\right)~{}.$
(12)

Figure 2: Cross sections of elliptical-prisms (and elliptical-cones for the
$z=1$ case).

Regardless of the authors’ claims, solution (12) is only valid for cylinders,
i.e. the capstan equation (1). To see why equations (10) and (12) are not
valid for general prisms, one only needs to consider an example with
noncircular cross section. If the reader wishes to, then consider a
positively-oriented elliptical-prism (for the $\zeta=0$ case) that is defined
by the map

$\displaystyle\boldsymbol{\sigma}(\phi,z)=\boldsymbol{(}a\cos(\phi),~{}b\sin(\phi),~{}z\boldsymbol{)}~{},$
(13)

where $z\in\mathbb{R}$ and $a,b>0$, and let
$\phi\in[\frac{1}{4}\pi,\frac{3}{4}\pi]$ be the contact interval (see figure
2, and see equation (2) for the capstan equation for an elliptical-prism).
Now, the reader can see that both map (10) and solution (12) are no longer
valid for the elliptical-prism mapping (13).

To find the explicit solution for the _cone_ case, the authors present the
following map (see equation 4.8 of Cottenden _et al._ [2]),

$\displaystyle\boldsymbol{\sigma}(x,y)$
$\displaystyle=\frac{r}{\sqrt{1+R(\phi(\theta))^{2}}}\boldsymbol{(}R(\phi(\theta))\cos(\phi(\theta)),~{}R(\phi(\theta))\sin(\phi(\theta)),~{}1\boldsymbol{)}~{},$
(14)

where

$\displaystyle d\theta$
$\displaystyle=\frac{\sqrt{R^{2}+(R^{\prime})^{2}+R^{4}}}{1+R^{2}}d\phi=\frac{R\sqrt{1+R^{2}}}{\sqrt{(1+R^{2})^{2}-(R_{,\theta})^{2}}}d\phi~{},$
$\displaystyle r$ $\displaystyle=\sqrt{x^{2}+y^{2}}~{}~{}\text{and}$
$\displaystyle\theta$ $\displaystyle=\arctan(\frac{y}{x})-\zeta~{}.$

From the authors’ definition, $\zeta$ appears to be the acute angle that the
vector $\hat{\boldsymbol{\sigma}}_{\\!,y}$ makes with the vector
$\hat{\boldsymbol{\sigma}}_{\\!,\phi}$, $R$ is given as a ‘cylindrical polar
function’ and $\phi$ appears to be the angle of the centre of rotation. One
can clearly see that map (14) is only valid for right-circular cones (i.e.
cones with a circular cross section) as it must have the same metric as the
Euclidean plane. To be precise

$\displaystyle\boldsymbol{F}_{I}=$
$\displaystyle\begin{pmatrix}{\boldsymbol{\sigma}}_{\\!,x}\cdot{\boldsymbol{\sigma}}_{\\!,x}&{\boldsymbol{\sigma}}_{\\!,x}\cdot{\boldsymbol{\sigma}}_{\\!,y}\\\
{\boldsymbol{\sigma}}_{\\!,x}\cdot{\boldsymbol{\sigma}}_{\\!,y}&{\boldsymbol{\sigma}}_{\\!,y}\cdot{\boldsymbol{\sigma}}_{\\!,y}\end{pmatrix}$
$\displaystyle=$
$\displaystyle\begin{pmatrix}1&\frac{(1-R)}{1+R^{2}}(\phi^{\prime}R^{\prime})^{2}\\\
\frac{(1-R)}{1+R^{2}}(\phi^{\prime}R^{\prime})^{2}&\frac{(\phi^{\prime}R^{\prime})^{2}(1-2R)}{(1+R^{2})^{2}}+\frac{(\phi^{\prime})^{2}[R^{2}+(R^{\prime})^{2}]}{1+R^{2}}\end{pmatrix}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}~{},$

which, in turn, implies that $R^{\prime}=0$ (as $\phi^{\prime}\neq 0$). Thus,
equation (14) reduces to the following form,

$\displaystyle\boldsymbol{\sigma}(x,y)$
$\displaystyle=r\boldsymbol{(}\sin(\alpha)\cos(\phi(\theta),~{}\sin(\alpha)\sin(\phi(\theta)),~{}\cos(\alpha)\boldsymbol{)}~{},$
$\displaystyle\phi(\theta)$
$\displaystyle=\operatorname{cosec}(\alpha)\theta~{},$

which represents a parametrisation of a right-circular cone, where $2\alpha$
is the (constant-) angle of aperture. Now, given that a solution exists in the
interval $[\theta_{1},\theta_{2}]$, the authors states that the solution can
be expressed as follows (see equation 4.20 of Cottenden _et al._ [2]),

$\displaystyle
T_{yy}(\phi_{2})=T_{0}\exp\left(\frac{\mu_{d}}{R(\phi(\theta))}\sin\left(\theta+\zeta\right)\big{|}^{\theta_{2}}_{\theta_{1}}\right)~{}.$
(15)

Regardless of the authors’ claims, solution (15) is only valid for right-
circular cones, i.e. valid for $R=\tan(\alpha)$ where $2\alpha$ is the
(constant-) angle of aperture. To see why equations (14) and (15) is invalid
for a general cone, one only needs to consider an example with noncircular
cross section. If the reader wishes to, then consider a positively-oriented
elliptical-cone (for the $\zeta=0$ case) that is defined by the following map,

$\displaystyle\boldsymbol{\sigma}(\phi,z)=\boldsymbol{(}az\cos(\phi),~{}bz\sin(\phi),~{}z\boldsymbol{)}~{},$
(16)

where $z\in\mathbb{R}_{>0}$ and $a,b>0$, and let
$\phi\in[\frac{1}{4}\pi,\frac{3}{4}\pi]$ be the contact interval (see figure 2
and consider the $z=1$ case). Now, the reader can that both map (14) and
solution (15) is no longer valid for the elliptical-cone mapping (16).

The authors conclude by stating that their experimental results agreed almost
perfectly with equation (12) for the cylinder case. One expects that the
solution to agree with experiment data for the cylinder case as the solution
is only valid for the cylinder case. The authors further state the
‘Experimental data gathered on [right-circular] cones constructed from plaster
of Paris and Neoprene … with half-angles ranging up to $12^{\circ}$ and
contact angles in the range $[70^{\circ},120^{\circ}]$ show good agreement
with the simple cylindrical model at their error level (around $\pm 10\%$ for
most samples)’ [2]. Again, one expects this be the case as solution (15) is
only valid for right-circular cones. Also, it is given by the authors that in
the limit $R\rightarrow 0$, the cone case is asymptotically equals to the
cylinder case. This result is just a trivial mathematical result that follows
directly from the Maclaurin series, i.e. $\sin(\theta)\approx\theta$, when
$\theta\approx 0$.

In Cottenden _et al._ ’s publication [2], the authors fail to demonstrate a
sufficient knowledge in the subject of differential geometry, mathematical
elasticity and contact mechanics to tackle this problem with any mathematical
rigour, and this is evident in D. J. Cottenden’s thesis [3] as the publication
Cottenden _et al._ [2] is a summary of all the mathematical results from
Cottenden’s thesis [3]. For example, in section 2.15 of the thesis, the
compatibility conditions for the left Cauchy-Green deformation tensor is given
as a general condition (see page 8 of Cottenden [3]). However, there exists no
general compatibility condition for the left Cauchy-Green deformation tensor,
and the given compatibility conditions exist for the two-dimensional case only
[29]. Another example is that the entire section 5.4 of the thesis (see pages
132 to 137 of Cottenden [3]) is based on the assumption that one can invert a
$3\times 2$ matrix (see equation 5.15 of Cottenden [3]), i.e. given a
sufficiently differentiable 2D manifold
$\boldsymbol{\lambda}:\mathbb{R}^{2}\to\textbf{E}^{3}$ (e.g. a map of a
cylinder), the author asserts that the Jacobian matrix,
$\boldsymbol{(}\partial_{\beta}\lambda^{j}\boldsymbol{)}_{3\times 2}$ (where
$\beta\in\\{1,2\\}$ and $j\in\\{1,2,3\\}$), is invertible: note that the
author is considering regular inverse and not one-sided inverse. Should the
reader consult section 5.4.1 of Cottenden [3], it is evident that the author
failed to understand the difference between the inverse of a bijective mapping
and a preimage (which need not be bijective), as both definitions expressed
with the same mathematical notation, $\boldsymbol{\lambda}^{-1}$, in
Pressley’s publication [27]: recall that Cottenden [3] accredits Pressley [27]
for his differential geometry results. This misunderstanding of Pressley’s
work [27] leads to a substantial part of Cottenden’s work [3] being incorrect,
as Section 5.4 of Cottenden’s thesis [3] is based on Cottenden’s assumption
[3] that a $3\times 2$ matrix is invertible (where the author is considering
regular inverse).

In a different publication, a precursor to the one we discussed, Cottenden _et
al._ [1] present experimental framework to calculate the coefficient of
friction, based on the master’s thesis of Skevos-Evangelos Sp. Karavokyros333
_Validating a mathematical model with a simple laboratory model_. MSc. UCL.
2017.. The authors state that ‘The model generalizes the common assumption of
a cylindrical arm to any convex prism’ [1]. Coefficients of friction are
determined from experiments conducted on Neoprene-coated Plaster of Paris
prisms of circular and elliptical cross-sections (defined as arm phantoms) and
a nonwoven fabric. The authors state experimental results agreed within $\pm
8\%$, i.e. $16\%$. They also state that the coefficients of friction varied
very little with the applied weight, geometry and contact angle. Thus, the
authors conclude by asserting that accurate values of the coefficient of
friction can be obtained by applying the cylindrical model (i.e. the capstan
equation) to the experimental data on human arms. They further assert that the
coefficient of friction is independent of the substrate geometry, applied
weights and contact angles, and claims that both their mathematical model and
experimental results are in complete agreement.

Unfortunately, none of Cottenden _et al._ ’s mathematical results [1] can be
mathematically justified, mostly for the reasons that we described before. For
example, the reader may try to derive an arc-length of an ellipse with the use
of the definition of an arc length from section 2.4 of the publication:
although, the formula $\text{d}[\text{arc
length}]=\sqrt{R(\theta)^{2}+\frac{\text{d}R(\theta)}{d\theta}}\text{d}\theta$
holds true when calculating an arc-length of a curve (which can be derived
with simple differential geometry techniques), the term $\text{d}[\text{arc
length}]^{2}=(R\text{d}\theta)^{2}+\text{d}R^{2}$ (see directly above equation
12 of Cottenden _et al._ [1]) does not imply the former equation nor does it
have any mathematical context. Another example is the equation
$1/\tan(0.5\pi)=0$, which is from the latter part of section 4.1 of the
publication (see directly below equation 17 of Cottenden _et al._ [1]).

Figure 3: Coefficient of friction against applied mass in grams: (a)
Cylindrical body; (b) Elliptical prism with horizontal major axis; (c)
Elliptical prism with vertical major axis; (d) Elliptical prism with major
axis making $+135^{\circ}$ to the horizontal [1].

As for the experimental results, consider figure 3, which shows the
coefficients of friction in relation to different geometries, applied weights,
and contact angles (see figure 11 of Cottenden _et al._ [1]). One can see that
there are clear discrepancies between each calculated coefficients of friction
as the coefficients of friction vary between different geometries, weights,
and contact angles. The authors only acknowledge the dependence of coefficient
of friction relative to the applied weight (see section 4 of Cottenden _et
al._ [2]), but hastily dismiss this effect by asserting that ‘… the increase
[coefficient of friction relative to the applied weight] is small compared to
the scatter in the data’ [1].

Mass g | Tension $10^{-3}$N
---|---
1st | 2nd | 3rd | 4th | 5th | Mean
$10$ | $16.0g$ | $15.0g$ | $15.0g$ | $15.0g$ | $16.0g$ | $15.6g$
$30$ | $51.0g$ | $54.0g$ | $51.0g$ | $50.0g$ | $51.0g$ | $51.4g$
$50$ | $88.0g$ | $87.0g$ | $89.0g$ | $87.0g$ | $90.0g$ | $88.2g$
$70$ | $125g$ | $124g$ | $128g$ | $122g$ | $124g$ | $125g$
Table 1: Tensometer readings: Cylinder with $\frac{127}{360}\pi$ contact
angle, where $g$ is the acceleration due to gravity.

If one consults Karavokyros’ masters thesis for the experimental data, then
one can find the raw data for the cylinder, $\frac{127}{360}\pi$ contact angle
case (see table 2a of Karavokyros’ masters thesis), which is displayed in
table 1. Now, using this data, if one plot the tension ratio against the
applied mass, then one gets figure 4. Note that the capstan equation implies
that the tension ratio is constant for all applied masses, i.e.
$\delta\tau=\exp(\mu_{d}\theta_{0})$ is constant given that $\mu_{d}$ and
$\theta_{0}$ are constants. However, this is not what the experimental results
are implying, as the reader can clearly see from figure 4 that as the applied
mass increases, tension ratio also increases, and this is documented
phenomenon in the literature [13], which cannot be simply dismissed. Thus,
this implies that, for the given experiments, it is unsuitable to use the
standard capstan equation to find the coefficient of friction with a
significant degree of accuracy. Now, this is direct evidence that shows the
flaws in Cottenden _et al._ ’s [2] data analysis methods and their
interpretation of the experimental results.

Figure 4: Tension ratio against applied mass.

Unfortunately no raw data is available for the other experiments in the theses
of Karavokyros or Cottenden [3] to conduct further rigorous analysis, as we
did with the $\frac{127}{360}\pi$-cylinder case.

As a result of the flawed mathematics and data analysis techniques, Cottenden
_et al._ [1, 2] and Cottenden [3] assert that the tension observed on the
membrane is independent of the geometry and the elastic properties of the
foundation, and thus, the stress profile at the contact region and the
coefficient of friction can be calculated with the use of the ordinary capstan
equation. They further assert that the experimental methodology for
calculating the coefficient of friction between fabrics and in-vivo (i.e.
within the living) human skin is identical to the capstan model. However, we
found no experimental evidence in the body of the authors’ publications to
verify their assertion, i.e. no evidence is given for the assertion that
foundation’s geometric and elastic properties are irrelevant when calculating
the coefficient friction between an elastic foundation and an overlying
membrane. Thus, our next subject of investigation is the authors’ experimental
methodology.

##  4 Experimental Data: June 2015 Human Trial

We now analyse the data obtained from human trials based on Cottenden _et al._
[1, 2] and Cottenden’s [3] experimental methodology. We recruit $10$ subjects,
$5$ males and $5$ females, all between the ages of $18$ to $60$, and the
approval was granted by the UCL Research Ethics Committee: UCL Ethics Project
ID number 5876/001. Collected data of the subjects can be found in table 2, if
the reader wishes to reproduce any results, where BMI is the body mass index
(calculated with NHS BMI calculator 444https://www.nhs.uk/live-well/healthy-
weight/bmi-calculator/), Radius is the radius of the upper arm and $\delta l$
is a measure of how _flaccid_ the subject’s tissue is (see equation (17)). For
more comprehensive set of raw data, please consult Dr N. Ovenden (UCL) at
<EMAIL_ADDRESS>

Subject | Gender | Age (Years) | BMI | Radius (cm) | $\delta l$
---|---|---|---|---|---
F19 | Female | $19$ | $21.0$ | $3.98$ | $0.994$
F34 | Female | $34$ | $22.0$ | $4.22$ | $0.991$
F40 | Female | $40$ | $23.4$ | $3.82$ | $1.01$
F53 | Female | $53$ | $27.3$ | $4.54$ | $1.02$
F60 | Female | $60$ | $22.5$ | $4.46$ | $1.02$
M18 | Male | $18$ | $17.5$ | $3.50$ | $0.98$
M23 | Male | $23$ | $24.7$ | $4.77$ | $1.04$
M25 | Male | $25$ | $22.6$ | $4.22$ | $1.01$
M26 | Male | $26$ | $22.8$ | $4.50$ | $0.988$
M54 | Male | $54$ | $26.2$ | $5.09$ | $1.00$
Table 2: Experimentee data 2015.

For our experimental configuration, we place subjects upper arm horizontally,
bicep facing upwards, on custom-designed armrest. Then, we place a fabric (a
nonwoven fabric: $95\%$ polypropylene and $5\%$ cotton) over their bicep and
attach the upper end to the tensometer (Dia-Stron MTT170 provided by Dr S.
Falloon, UCL), and the free hanging end is reserved for hanging set of
weights, such that the contact angle between the bicep and the fabric (if
measured from the humerus) is approximately $\frac{1}{2}\pi$. The dimensions
of the fabric are $4$cm$\times 50$cm, and from our measurements, the fabric
has an approximate thickness of $0.5$mm and an approximate mass of $0.6$g.
Also, we mark the skin with a $3$cm$\times 5$cm grid with $1$cm$\times 1$cm
grid spacing, $0.5$cm away from either side of the fabric and starting from
the highest part (of the horizontal axis) of the arm, then placing semi-
hemispherical markers with a radius of $2$mm. See figure 5 for a visual
representation.

Figure 5: Experimental configuration on subject F53 (.stl file).

For each run, we pull the fabric with a constant speed of
$\frac{1}{6}\text{cms}^{-1}$ with the use the tensometer and we use a
static-3D scanner (3dMD Photogrammetric System555http://3dmd.com provided by
Mr C. Ruff, UCLH) to record the before and after deformation of the upper arm.
Note that we use the metric

$\displaystyle\delta
l=\frac{\sum_{10\text{g},\dots,140\text{g}}||\text{deformed}(\textbf{A4}-\textbf{E6})||}{\sum_{10\text{g},\dots,140\text{g}}||\text{undeformed}(\textbf{A4}-\textbf{E6})||}~{},$
(17)

to measure the _flaccidity_ (analogous to $1/(\text{Young's modulus})$) of the
subject’s soft tissue. Also, Radius (from table 2) is calculated by measuring
the girth around the bicep and then dividing the measurement by $2\pi$

Subject | $\delta\tau$
---|---
$40$g |  $60$g |  $80$g |  $100$g |  $120$g |  $140$g
F19 | $2.19$ | $1.70$ | $1.52$ | $1.92$ | $1.88$ | $1.86$
F34 | $2.08$ | $\cdots$ | $1.93$ | $1.89$ | $1.83$ | $1.83$
F40 | $1.99$ | $1.96$ | $1.92$ | $1.89$ | $1.88$ | $1.79$
F53 | $2.13$ | $2.31$ | $2.23$ | $2.15$ | $1.84$ | $1.70$
F60 | $2.29$ | $2.20$ | $2.06$ | $1.99$ | $1.96$ | $1.95$
M18 | $1.99$ | $1.90$ | $1.88$ | $1.84$ | $1.68$ | $1.76$
M23 | $2.46$ | $2.28$ | $2.24$ | $2.19$ | $2.33$ | $1.99$
M25 | $2.14$ | $1.98$ | $1.81$ | $1.80$ | $1.98$ | $\cdots$
M26 | $2.41$ | $2.31$ | $2.26$ | $2.18$ | $2.31$ | $1.99$
M54 | $\cdots$ | $2.12$ | $2.03$ | $1.77$ | $1.91$ | $1.96$
Table 3: Experimental data from the 2015 human trial.

Table 3 shows the tension ratio, $\delta\tau=T_{\text{max}}/T_{0}$, for each
subject with respect to each applied mass, where $T_{\text{max}}$ are the
tensometer readings, $T_{0}=\text{Mass}\times g$ are the weight of the applied
mass and $g=9.81$ is the acceleration due to gravity. Note that the tensometer
data of F34 for $60$g, M25 for $140$g and M54 for $40$g are corrupted, and
thus, omitted.

Cottenden _et al._ [1, 2] and Cottenden [3] assert that the
$T_{\text{max}}=T_{0}\exp(\mu_{F}\theta_{0})$ equation holds true, regardless
of the geometric and elastic properties of the substrate (i.e. human soft-
tissue). If this is indeed true then $\delta\tau=\exp(\mu_{F}\theta_{0})$ is
only a function of the coefficient of friction, regardless of the geometric
and elastic properties of human soft-tissue, for a fixed contact angle
$\theta_{0}$. Also, one of the major assumption of the authors is that
coefficient of friction between skin and fabrics positively correlated with
the age of an individual, as they observe higher occurrence of skin damage in
elderly subjects. Thus, now we plot the tension ratio against various
properties to test the authors’ claims.

(a) Tension ratio against the age (in years).

(b) Tension ratio against the BMI.

(c) Tension ratio against the $\delta l$.

(d) Tension ratio the radius of the upper arm (in cm).

Figure 6: Tension ratio for varying age, BMI, $\delta l$ and radius of the
upper arm.

Figure 6(a) shows the tension ratio against subjects’ age, where the linear
regression line is $\delta\tau=0.00117\times\text{Age}+1.96$ and the age is in
years ($\text{R}^{2}=0.0512$, where $R^{2}$ is the coefficient of
determination). As the reader can see there is no obvious relationship between
the age of the subject and the tension ratio. In fact, the highest tension
ratios are observed for M23 and M26 (two of the youngest subjects).

Figure 6(b) shows the tension ratio against the body mass index (BMI), where
the linear regression line is $\delta\tau=0.0169\times\text{BMI}+1.61$
($\text{R}^{2}=0.0961$). As the reader can see that there is a vague positive
correlation between the BMI and the tension ratio. The reason we observe this
correlation is because that those who have a higher BMI tend to have a greater
fat content, i.e. higher volume of flaccid tissue. Thus, during the
experiments, a higher tension needs to be applied to the fabric as the as a
portion of the applied-tension is expended on deforming the flaccid tissue of
the subject.

Figure 6(c) shows the tension ratio against $\delta l$, where the linear
regression line is $\delta\tau=4.39\times\delta l-2.41$
($\text{R}^{2}=0.282$). As the reader can see that there is a positive
correlation between $\delta l$ and the tension ratio. This implies that for
more flaccid soft-tissue, larger portion of the applied-tension is expended on
deforming it.

Figure 6(d) shows the tension ratio against the radius of the upper arm, where
the linear regression line is $\delta\tau=0.205\times\text{Radius}+1.12$ and
the radius is in centimetres ($\text{R}^{2}=0.412$). As the reader can see
that see there is a strong positive correlation between the radius and the
tension ratio. The reason why we observe this correlation is exactly same as
the cases before: the larger the radius is, the larger the volume of soft-
tissue that needs to be deformed, and thus, a larger applied-tension.

From figures 6(b), 6(d) and 6(c), we observe higher tension ratios for
subjects with higher BMI, more flaccid soft-tissue, and larger biceps. Should
one use the capstan equation (or any belt-friction model in general) to
calculate the coefficient of friction, then one would observe a higher
coefficient of friction for subjects with higher BMI, more flaccid soft-
tissue, and larger biceps. However, this does neither imply nor does not imply
that those with a higher BMI, more flaccid soft-tissue, and larger biceps have
a greater risk of skin abrasion, i.e _correlation does not imply causation_
[30]; It merely implies that belt-friction models are not suitable for
modelling such problems, which directly contradicts Cottenden _et al._ [1, 2]
and Cottenden’s [3] assertion regarding the efficacy of the capstan model.

For a mathematically rigorous way of modelling this problem, we refer the
reader to section 6.6 and 6.7 of Jayawardana [14]. There, both numerical-
modelling (shell-membrane frictionally coupled to an elastic-foundation) and
human trial data imply that given a constant coefficient of friction, a higher
volume of soft-tissue (high radius) and more compliant soft-tissue (lower
Young’s modulus) would result in higher deformation of the skin, and a higher
volume of soft-tissue would result in more shear-stress generated on the skin.

##  5 Conclusions

In conclusion, we showed that no mathematical claim of Cottenden _et al._ [1,
2] and Cottenden [3] can be mathematically justified, and some fundamental and
trivial results in mathematical elasticity, differential geometry and contact
mechanics are misrepresented. Only the ordinary capstan equation, and a
solution to a dynamic membrane with both a zero-Poisson’s ratio and a zero-
mass density on a right-circular cone is given. Finally, limited experimental
data is given to show the trivial asymptotic nature of $\sin(\theta)$ near
$\theta=0$.

Also, Cottenden _et al._ [1, 2] and Cottenden [3] claim that the capstan
equation (1) holds true, regardless of the geometric and elastic properties of
the obstacle, and thus, it can be used to calculate the coefficient of
friction between human skin and fabrics. However, the data gathered from human
trials implies that it is unwise to use the capstan equation (e.g. belt-
friction models in general) to calculate the friction between in-vivo skin and
fabrics. This is because that such models assume a rigid obstacle while human
soft-tissue is elastic, and thus, a portion of the applied force to the fabric
is expended on deforming the soft-tissue, which in turn leads to the illusion
of a higher coefficient of friction when such models are used to calculate the
coefficient of friction.

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