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
aa r X i v : . [ phy s i c s . m e d - ph ] J a n A Critical Study of Cottenden et al. ’s An AnalyticalModel of the Motion of a Conformable Sheet Over aGeneral Convex Surface in the Presence ofFrictional Coupling
Kavinda Jayawardana ∗ Abstract
In our analysis, we show that what Cottenden et al. [1, 2] and Cotten-den [3] accomplish is the derivation of the ordinary capstan equation, anda solution to a dynamic membrane with both a zero-Poisson’s ratio and azero-mass density on a rigid right-circular cone. The authors states thatthe capstan equation holds true for an elastic obstacle, and thus, it can beused to calculate the coefficient of friction between human skin and fabrics.However, using data that we gathered from human trials, we show that thisclaim 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-vivoskin and fabrics. This is due to the fact that such models assume a rigidfoundation, while human soft-tissue is deformable, and thus, a portion of theapplied force to the fabric is expended on deforming the soft-tissue, whichin turn leads to the illusion of a higher coefficient of friction when usingbelt-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 witheach other, where the contact area exhibits friction. A common area where ∗ Corresponding author
Email address: [email protected] (Kavinda Jayawardana)
January 20, 2021 odelling 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 thinand 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, andsuch models can be used to model skin abrasion caused by fabrics as a resultof friction. Need for valid modelling techniques are immensely important infields such as field of incontinence associated dermatitis [2], sports relatedskin trauma [6] and cosmetics [7]. It is documented that abrasion damageto human skin in cases such as the jogger’s nipple [8] and dermatitis fromclothing 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 problemmathematically, Cottenden et al. [2] put forward a model in their publication
An analytical model of the motion of a conformable sheet over a generalconvex surface in the presence of frictional coupling . We show that, regardlessof the authors’ claims, what they derive is the ordinary capstan equation anda solution for a membrane with a zero-Poisson’s ratio and a zero-mass densityon a rigid right-circular cone. Cottenden et al. [1, 2] and Cottenden [3] alsoclaims that capstan equation can be used to calculate the friction betweenin-vivo skin and fabrics. With data gathered from human trials, we show thatit is unwise to use belt-friction models (e.g. capstan equation) to calculatethe coefficient of friction between in-vivo skin and fabrics, as such modelsassume a rigid foundation, while human soft-tissue is elastic; thus, a portionof 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 whenusing belt-friction models.
2. Capstan Equation
The capstan equation, otherwise known as Euler’s equation of tensiontransmission, is the relationship governing the maximum applied-tension T max with respect to the minimum applied-tension T of an elastic stringwound around a rough cylinder. Thus, the governing equation can be ex-press by the following equation, T max = T exp( µ F θ ) , (1)where θ is the contact angle and µ F is the coefficient of friction. By string we mean a one-dimensional elastic body, and rough we mean that the con-2act area exhibits friction, where the coefficient of friction is the physicalratio of the magnitude of the shear force and the normal force between twocontacting bodies. The capstan equation is the most perfect example of a belt-friction model , which describes behaviour of a belt-like object movingover a rigid-obstacle subjected to friction [12]. In engineering, the capstanequation describes a body under a load equilibrium involving friction betweena rope and a wheel like circular object, and thus, it is widely used to anal-yse the tension transmission behaviour of cable-like bodies in contact withcircular profiled surfaces, such as in rope rescue systems, marine cable appli-cations, computer storage devices (electro-optical tracking systems), clutchor brake systems in vehicles, belt-pulley machine systems and fibre-reinforcedcomposites [13].For a rigorous study of the capstan equation (a membrane or a string overa rigid-obstacle) generalised to non-circular geometries, we refer the readerto chapter 2 of Jayawardana [14]. There, as an example, the author presenta solution to the capstan equation generalised for a rigid elliptical-prism, i.e.given a prism with a horizontal diameter of 2 a and the vertical diameter of2 b , parametrised by the map σ ( x , θ ) = ( x , a sin( θ ) , b cos( θ ) ) , where θ isthe acute angle that the vector ( , , ) C makes with the vector ( , ϕ ( θ ) , ) ,and where ϕ ( θ ) = ( b sin ( θ ) + a cos ( θ )) , the capstan equation takes thefollowing form, T elliptical ( θ ) = T exp (cid:18) µ F arctan (cid:18) ba tan( θ ) (cid:19)(cid:19) . (2)Note that equation (2) assumes that minimum applied-tension, T , is ap-plied at θ = 0, and the vector brackets ( · ) implies that the vectors are in theEuclidean space and ( · ) C implies that the vectors are in the curvilinear space.Equation (2) implies that the maximum applied-tension, T max , is depen-dent on the mean curvature of the rigid prism. To investigate this matterfurther, assume that T is applied at θ = − π and T max applied at θ = π ,and thus, equation (2) implies that δτ = exp (cid:18) µ F arctan (cid:18) ba (cid:19)(cid:19) , (3)where δτ = T max /T , which is defined as the tension ratio . As the reader cansee that for a fixed contact interval and a fixed coefficient friction, equation3 .5 1 1.5 δ b δ τ Figure 1: Tension ratio against δb .(3) implies a non-constant tension ratio for varying δb , where δb = b/a . Asthe mean curvature of the prism is H ( θ ) = ab ( ϕ ( θ )) − , one can see that thetension ratio is related to the mean curvature by the following relation, δτ = exp µ F arctan " max θ ∈ [ − π, π ] (2 aH ( θ ) ,
1) + min θ ∈ [ − π, π ] (2 aH ( θ ) , − . Figure 1 presents a visualisation of the tension ratio against δb (whichis analogous to the mean curvature), which is calculated with µ F = and δb ∈ [ , ], and it shows that, for a fixed contact interval, as δb increases(i.e. as the mean curvature of the contact region increases), the tension ratioalso increases. This is an intuitive result as the curvature of the contactregion increases, the normal reaction force on the membrane (or the string)also increases, which in turn leads to a higher frictional force, and thus, ahigher tension ratio. Now, this is a fascinating result as this effect cannot beobserved with the ordinary capstan equation (1).For a rigorous study of belt friction models, we refer the reader to section6.2 of Jayawardana [14], and for a rigorous study of the capstan equationgeneralised to elastic obstacles, we refer the reader to Konyukhov’s [15], and4onyukhov’s and Izi’s [16].
3. Cottenden et al. ’s Work
Cottenden et al. [2] (the principal author is D. J. Cottenden) attemptto derive a mathematical model to analyse a frictionally coupled membrane(defined as a nonwoven sheet) on an elastic foundation (defined as a sub-strate) based on the research findings of D. J. Cottenden’s PhD thesis [3] .It is assumed that friction (abrasion in the context of the publication) is thecause of some pressure ulcers in largely immobile patients, and abrasion dueto friction contributes to the deterioration of skin health in incontinence padusers, especially in the presence of liquid. The current literature shows verylittle research in the area of frictional damage on skin due to fabrics, andthus, the authors’ goal is to present a mathematical model to investigate thisphenomenon in a purely geometrical setting. Thus, the authors propose amodel for a general class of frictional interfaces, which includes those thatobey Amontons’ three laws of friction.Cottenden et al. ’s method [2] for calculating the kinetic frictional forceinduced on human skin due to nonwoven fabrics is as follows. The humanbody part in question is modelled as a homogeneous isotropic ‘convex sur-face [ sic ]’ [2] (i.e. the substrate) and the nonwoven fabric is modelled as anisotropic membrane (i.e. the nonwoven sheet). The goal is to find the stressesacting on the nonwoven sheet, including determining the friction acting onthe substrate. The contact region between the fabric and the skin is definedas ‘An Instantaneous Isotropic Interface , [which] is an interface composedof a pair of surfaces which have no intrinsically preferred directions and nodirectional memory effects, so that the frictional force acts in the oppositedirection to the current relative velocity vector ... or to the sum of current applied forces acting to initiate motion ...’ (see section 2.2 of Cottenden etal. [2]): this simply implies that the contact region is isotropic and nothingmore. 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 asa three-dimensional object as a two-dimensional object cannot describe theelastic properties of a fully three-dimensional object such as a human body https://discovery.ucl.ac.uk/id/eprint/1301772/ et al. [17]); Unless suitable assumptions are made as one findsin shell theory (for a comprehensive mathematical study of the shell theory,please consult sections B of Ciarlet [18]); however, this is not what the au-thors are considering. Now, consider the authors’ statement regarding themodelling assumptions carefully, particularly the term ‘convex surface’. Theauthors definition of convexity is η · ∇ E ˆ N · η ≥ et al. [2]), where ˆ N and η are unit normal and tangential vectors surfacerespectively. However, the authors’ definition is erroneous. Convexity has avery precise mathematical definition, i.e. we say the functional f : X → R isconvex, if f ( tx + (1 − t ) y ) ≤ tf ( x ) + (1 − t ) f ( y ), ∀ t ∈ [0 ,
1] and ∀ x, y ∈ X (for more on the definition of convexity, please see chapter 1 of Badiale andSerra [19]). Also, the very idea of a convex surface is nonsensical as defini-tion of convexity is only applicable to functionals. A simple example of aconvex functional is exp( · ) : R → R > . One is left to assume that what theauthors mean by convexity is surfaces (i.e. manifolds) with a positive mean-curvature. For more on elementary differential geometry, please consult doCarmo [20] or Lipschutz [21].In their analysis, the authors defines a membrane with the following prop-erty, ‘always drapes, following the substrate surface without tearing or puck-ering’ (see section 2.1 of [2]). The authors’ definition is erroneous, as onecannot guarantee that the given property will hold for a flat-membrane (ina 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 onlyway one can keep the membrane perfectly in contact with the sphere ina two-dimensional region with nonzero measure is by deforming the mem-brane by applying appropriate boundary stresses and or external loadings.Otherwise, the membrane only makes contact with the sphere at a singlepoint or a line. Also, the authors do not specify whether the membrane iselastic or not. One is left to assume that the membrane is elastic as the pro-posed frame work does not acknowledge plastic deformations. Note that theauthors never referred to their nonwoven sheet as a membrane, but a mem-brane (or a film) is the closest mathematical definition for modelling suchobjects. Please consult chapters 4 and 5 of Ciarlet [18] or chapter 7 of Libaiand Simmonds [22] for a comprehensive analysis on the theory of membranes.6o find the stresses acting on the membrane, consider Cauchy’s momen-tum equation in the Euclidean space, which the authors define as follows, ∇ E · T + f = ρ ¨ χ , (4)where T is Cauchy’s stress tensor, f = f ( T , ∇ E T ) is the force density fieldand ρ is the material mass density of the membrane, ∇ E is the Euclidean dif-ferential operator and χ is given as a ‘time-dependent deformation functionmapping the positions of points in their undeformed reference configurationto their deformed positions and the superposed double dot denotes a dou-ble material description time derivative’ (see section 2.1 of Cottenden et al. [2]). It is unclear what χ represent from the authors’ definition, whether itis the displacement field of the membrane or some time-dependent mappingfrom one manifold to another. If the latter is true, then equation (4) hasa 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 readerconsults section 5.4 of Cottenden’s thesis [3], then it becomes evident that χ 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 notput forward the framework to handle the 3+1 decomposition in general rel-ativity, with any mathematical rigour. If the reader is interested in the 3+1formalism in general relativity, then please consult the publications [23–25]or Dr J. A. V. Kroon (QMUL) on his LTCC lecture notes on Problems ofGeneral Relativity , where the reader can find an extraordinary solution fortwo merging black-holes (i.e. the Brill-Lindquist solution).Assuming that the foundation is static and rigid, and the mass densityof the membrane is negligible, i.e. ρ ≈
0, the authors state that Cauchy’smomentum equation (4) can be expressed as P s · ( ∇ E · T ) + P s · f = , (5) − ( ∇ E ˆ N ) : T + ˆ N · f = 0 , (6)where P s projection matrix to the substrate (the explicit form is not definedby the authors), ˆ N is the unit normal to the surface, and · and : are definedas a contraction and a double contraction in the Euclidean space respectively. ∼ jav/LTCC.htm N · T = ,to obtain equation (6). The authors give equations (5) and (6) as the stateof the ‘general case’ of the problem. However, their assertion cannot hold asthe system is underdetermined. Consider the vector f , which consists of threeunknowns. Also, consider the tensor T , which is a symmetric tensor with sixunknowns. Thus, using the condition ˆ N · T = the number of unknownscan be reduced by three: leaving six remaining unknowns. Now, direct one’sattention 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 descriptionof any boundary conditions for the ‘general case’, which are essential in ob-taining a unique solution.The derivation of Cottenden et al. ’s governing equations [3] can be foundon section 2.2 to 2.4 of the publication. Upon examination, the reader mayfind that the methods put forward by the authors’ are inconsistent withmathematical elasticity, contact mechanics and differential geometry. Thus,we refer reader to Kikuchi and Oden [26] to see how to model friction withmathematical competence and to show how incredibly difficult modellingsuch problems are. We further refer the reader to Ciarlet [18] to see how tomodel mathematical elasticity in a differential geometry setting with math-ematical 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 thesame first fundamental form as the plane; the identity matrix inthe case of plane Cartesian coordinates. [ sic ]’ [2]. Found in section 4.1of Cottenden et al. [2], this is the basis for their entire publication (also thebasis for Cottenden’s [3] thesis). However, the authors’ statement is non-sensical. An isomorphism (preserves form) is at least a homomorphism, i.e.there exists at least a continuous bijective mapping, whose inverse is alsocontinuous, between the two manifolds in question [20, 21]. Thus, a surfacethat is isomorphic to the plane simply implies that there exists a continuousbijective map, with a continuous inverse, between the surface in questionand the Euclidean plane, and it does not automatically guarantee that thesurface have the same metric as the Euclidean plane under the given map.The latter part of the authors’ statement is clearly describing surfaces that8re isometric (preserves distance) to the Euclidean plane, i.e. surfaces with azero-Gaussian curvature. However, the statement is still erroneous as beinga surface that is isometric to the Euclidean plane does not guarantee thatthe surface have the same metric as the Euclidean plane. Being isometric tothe Euclidean plane simply implies that, if f : U ⊂ R → W ⊂ E is a 2Dmanifold that is isometric to the Euclidean plane, then there exists a map g : V ⊂ R → U ⊂ R such that the first fundamental form of the isometry f ◦ g : V ⊂ R → W ⊂ E is the 2 × P s · f + µ d ( ˆ N · f ) ˙ χ = 0 , (7)which the authors define as Amontons’ law friction, where µ d is the coeffi-cient of dynamic friction and ˙ χ is the relative velocity vector between themembrane and the foundation. The inclusion of the two equations impliedby 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 whoseorthogonal axis defined by the coordinates ( x, y ) , where y defines the longerdimension, that is placed over a surface defined by the regular map σ . Also,assume that Poisson’s ratio of the membrane is zero to prevent any lateralcontractions due to positive tensile strain. To reduce the complexity, theauthors modify the problem by letting ˙ χ be parallel to σ ,y . Also, by applyinga boundary stress of T at some point φ whilst applying a even greaterstress at φ so that T yy ( y ) is an increasing function in y , where φ and φ are angles of contact such that φ < φ . Due to zero-Poisson’s ratioand the boundary conditions, one finds T xx = 0, T xy = 0, where T ij are9tress tensor components. Thus, the governing equations finally reduce toa fully determined system, i.e. is only now the number of unknowns equalsto the number of governing equations. Therefore, one must understand thathaving zero-Gaussian curvature and zero-Poisson’s ratio is a necessity forthis model, and it is not some useful tool for deriving explicit equations asstated by the authors. Upon integrating equation (7), under the specifiedboundary conditions, one finds solutions of the following form (see equation4.4 of Cottenden et al. [2]), T yy ( y ) = T exp (cid:18) µ d Z y | C yy ( η ) | dη (cid:19) , (8)where C αβ = F − Iαγ F IIγδ F − Iβδ is defined as the curvature tensor, F I is thefirst fundamental form tensor and F IIyy is the only nonzero component ofthe second fundamental form tensor. However, equation (8) is erroneous.This is because, whatever is inside the exp( · ) term must be non-dimensional,but this is not the case with equation (8). To illustrate this flaw, let L bean intrinsic Euclidean length scale and ℓ be an intrinsic length scale of thecurvilinear coordinate y . Now, with the definition of C yy (see equation 3.3 ofCottenden et al. [2]), one finds that the length scale inside the exp( · ) termin equation (8) to be ( ℓ/L ) . Given that y = θ (i.e. the contact angle, whichis dimensionless), one finds that the length scale inside the exp( · ) term to be L − , which is not mathematically viable. This flaw of the authors’ work isa result of not correctly following tensor contraction rules, and not discern-ing between covariant and contravariant tensors (see sections 3.2 and 4.1 ofCottenden 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) andwhose first fundamental form tensor is diagonal (after a change of coordinatesor otherwise), and should one follow the derivation with mathematical rigour,then one finds a solution of the following form, T yy ( y ) = T exp (cid:18) − µ d Z y q F Iyy ( η ) F yIIy ( η ) dη (cid:19) , (9)where a rigorous derivation can be fond in chapter 2 of Jayawardana [14]. Asthe reader can see that unlike equation (8), no dimension violations can bepossible with equation (9). 10o find the explicit solution for the general prism case, the authors presentthe following map (see section 4.2 of Cottenden et al. [2]) σ ( x, y ) = ( R ( φ ) cos( φ ) , R ( φ ) sin( φ ) , x cos( ζ ) + y sin( ζ ) ) , (10)where dφ = cos( ζ ) dy − sin( ζ ) dx p R ( φ ) + ( R ′ ( φ )) . From the authors’ definition, ζ appears to be the acute angle that the vectorˆ σ ,y makes with the vector ˆ σ ,φ , and R and φ appear to be the radius ofcurvature and the angle of the centre of rotation respectively. One can clearlysee that map (10) is only valid for cylinders (i.e. prisms with a constantradius) as it must have the same metric as the Euclidean plane. To beprecise, F I = (cid:18) σ ,x · σ ,x σ ,x · σ ,y σ ,x · σ ,y σ ,y · σ ,y (cid:19) =[ R ( φ ) + ( R ′ ( φ )) ] (cid:18) ( φ ,x ) + cos ( ζ ) φ ,x φ ,y + sin( ζ ) cos( ζ ) φ ,x φ ,y + sin( ζ ) cos( ζ ) ( φ ,y ) + sin ( ζ ) (cid:19) = (cid:18) (cid:19) , which, in turn, implies the following, φ ,x = − sin( ζ ) p R ( φ ) + ( R ′ ( φ )) ,φ ,y = cos( ζ ) p R ( φ ) + ( R ′ ( φ )) , and thus, the following,( φ ,x ) + ( φ ,y ) = 1 R ( φ ) + ( R ′ ( φ )) . (11)As φ is a real function (i.e. not complex) and noting that equation (11) musthold for all x and y , one finds that R ′ = 0, and thus, equation (10) reducesto the following, σ ( x, y ) = ( c cos( φ ) , c sin( φ ) , x cos( ζ ) + y sin( ζ ) ) ,φ ( x, y ) = 1 c [cos( ζ ) y − sin( ζ ) x ] , c is a positive con-stant (and R = c ). Now, given that a solution exists in the interval [ φ , φ ],the authors state that the solution can be expressed as follows (see equation4.7 of Cottenden et al. [2]), T yy ( φ ) = T exp (cid:18) µ d cos( ζ ) (cid:20) φ − arctan (cid:18) R ( φ ) ,φ R ( φ ) (cid:19)(cid:21) (cid:12)(cid:12) φ φ (cid:19) . (12) -1.5 -1 -0.5 0 0.5 1 1.5 x-axis -2.5-2-1.5-1-0.500.511.522.5 y - a x i s a = 2, b = 1.5 Contact region: φ ∈ [0.25 π , 0.75 π ] Contact free region -1.5 -1 -0.5 0 0.5 1 1.5 x-axis -2.5-2-1.5-1-0.500.511.522.5 y - a x i s a = 2, b = 2.5 Figure 2: Cross sections of elliptical-prisms (and elliptical-cones for the z = 1case).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 validfor general prisms, one only needs to consider an example with noncircularcross section. If the reader wishes to, then consider a positively-orientedelliptical-prism (for the ζ = 0 case) that is defined by the map σ ( φ, z ) = ( a cos( φ ) , b sin( φ ) , z ) , (13)where z ∈ R and a, b >
0, and let φ ∈ [ π, π ] be the contact interval (seefigure 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 longervalid for the elliptical-prism mapping (13).12o find the explicit solution for the cone case, the authors present thefollowing map (see equation 4.8 of Cottenden et al. [2]), σ ( x, y ) = r p R ( φ ( θ )) ( R ( φ ( θ )) cos( φ ( θ )) , R ( φ ( θ )) sin( φ ( θ )) , ) , (14)where dθ = p R + ( R ′ ) + R R dφ = R √ R p (1 + R ) − ( R ,θ ) dφ ,r = p x + y and θ = arctan( yx ) − ζ . From the authors’ definition, ζ appears to be the acute angle that the vectorˆ σ ,y makes with the vector ˆ σ ,φ , R is given as a ‘cylindrical polar function’and φ appears to be the angle of the centre of rotation. One can clearly seethat map (14) is only valid for right-circular cones (i.e. cones with a circularcross section) as it must have the same metric as the Euclidean plane. To beprecise F I = (cid:18) σ ,x · σ ,x σ ,x · σ ,y σ ,x · σ ,y σ ,y · σ ,y (cid:19) = (1 − R )1+ R ( φ ′ R ′ ) − R )1+ R ( φ ′ R ′ ) φ ′ R ′ ) (1 − R )(1+ R ) + ( φ ′ ) [ R +( R ′ ) ]1+ R ! = (cid:18) (cid:19) , which, in turn, implies that R ′ = 0 (as φ ′ = 0). Thus, equation (14) reducesto the following form, σ ( x, y ) = r ( sin( α ) cos( φ ( θ ) , sin( α ) sin( φ ( θ )) , cos( α ) ) ,φ ( θ ) = cosec( α ) θ , which represents a parametrisation of a right-circular cone, where 2 α is the(constant-) angle of aperture. Now, given that a solution exists in the interval[ θ , θ ], the authors states that the solution can be expressed as follows (seeequation 4.20 of Cottenden et al. [2]), T yy ( φ ) = T exp (cid:18) µ d R ( φ ( θ )) sin ( θ + ζ ) (cid:12)(cid:12) θ θ (cid:19) . (15)13egardless of the authors’ claims, solution (15) is only valid for right-circular cones, i.e. valid for R = tan( α ) where 2 α is the (constant-) angle ofaperture. 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 thereader wishes to, then consider a positively-oriented elliptical-cone (for the ζ = 0 case) that is defined by the following map, σ ( φ, z ) = ( az cos( φ ) , bz sin( φ ) , z ) , (16)where z ∈ R > and a, b >
0, and let φ ∈ [ π, π ] be the contact interval (seefigure 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 al-most perfectly with equation (12) for the cylinder case. One expects that thesolution to agree with experiment data for the cylinder case as the solution isonly valid for the cylinder case. The authors further state the ‘Experimentaldata gathered on [right-circular] cones constructed from plaster of Paris andNeoprene ... with half-angles ranging up to 12 ◦ and contact angles in therange [70 ◦ , ◦ ] show good agreement with the simple cylindrical model attheir error level (around ±
10% for most samples)’ [2]. Again, one expectsthis be the case as solution (15) is only valid for right-circular cones. Also, itis given by the authors that in the limit R →
0, the cone case is asymptot-ically equals to the cylinder case. This result is just a trivial mathematicalresult that follows directly from the Maclaurin series, i.e. sin( θ ) ≈ θ , when θ ≈ et al. ’s publication [2], the authors fail to demonstrate asufficient knowledge in the subject of differential geometry, mathematicalelasticity and contact mechanics to tackle this problem with any mathemat-ical rigour, and this is evident in D. J. Cottenden’s thesis [3] as the publica-tion Cottenden et al. [2] is a summary of all the mathematical results fromCottenden’s thesis [3]. For example, in section 2.15 of the thesis, the com-patibility conditions for the left Cauchy-Green deformation tensor is given asa general condition (see page 8 of Cottenden [3]). However, there exists nogeneral compatibility condition for the left Cauchy-Green deformation ten-sor, and the given compatibility conditions exist for the two-dimensional caseonly [29]. Another example is that the entire section 5.4 of the thesis (see14ages 132 to 137 of Cottenden [3]) is based on the assumption that one caninvert a 3 × λ : R → E (e.g. a map of a cylinder),the author asserts that the Jacobian matrix, ( ∂ β λ j ) × (where β ∈ { , } and j ∈ { , , } ), is invertible: note that the author is considering regularinverse and not one-sided inverse. Should the reader consult section 5.4.1 ofCottenden [3], it is evident that the author failed to understand the differencebetween the inverse of a bijective mapping and a preimage (which need notbe bijective), as both definitions expressed with the same mathematical no-tation, λ − , in Pressley’s publication [27]: recall that Cottenden [3] accreditsPressley [27] for his differential geometry results. This misunderstanding ofPressley’s work [27] leads to a substantial part of Cottenden’s work [3] beingincorrect, as Section 5.4 of Cottenden’s thesis [3] is based on Cottenden’s as-sumption [3] that a 3 × et al. [1] present experimental framework to calculate the coefficient offriction, based on the master’s thesis of Skevos-Evangelos Sp. Karavokyros .The authors state that ‘The model generalizes the common assumption ofa cylindrical arm to any convex prism’ [1]. Coefficients of friction are de-termined from experiments conducted on Neoprene-coated Plaster of Parisprisms of circular and elliptical cross-sections (defined as arm phantoms) anda nonwoven fabric. The authors state experimental results agreed within ± et al. ’s mathematical results [1] canbe mathematically justified, mostly for the reasons that we described be- Validating a mathematical model with a simple laboratory model . MSc. UCL. 2017. q R ( θ ) + d R ( θ ) dθ d θ holds truewhen calculating an arc-length of a curve (which can be derived with simpledifferential geometry techniques), the term d[arc length] = ( R d θ ) + d R (see directly above equation 12 of Cottenden et al. [1]) does not imply theformer equation nor does it have any mathematical context. Another exam-ple is the equation 1 / tan(0 . π ) = 0, which is from the latter part of section4.1 of the publication (see directly below equation 17 of Cottenden et al. [1]).Figure 3: Coefficient of friction against applied mass in grams: (a) Cylindricalbody; (b) Elliptical prism with horizontal major axis; (c) Elliptical prismwith vertical major axis; (d) Elliptical prism with major axis making +135 ◦ to the horizontal [1].As for the experimental results, consider figure 3, which shows the co-efficients of friction in relation to different geometries, applied weights, and16ontact angles (see figure 11 of Cottenden et al. [1]). One can see that thereare clear discrepancies between each calculated coefficients of friction as thecoefficients of friction vary between different geometries, weights, and con-tact angles. The authors only acknowledge the dependence of coefficient offriction relative to the applied weight (see section 4 of Cottenden et al. [2]),but hastily dismiss this effect by asserting that ‘... the increase [coefficientof friction relative to the applied weight] is small compared to the scatter inthe data’ [1].Mass g Tension 10 − N1st 2nd 3rd 4th 5th Mean10 16 . g . g . g . g . g . g
30 51 . g . g . g . g . g . g
50 88 . g . g . g . g . g . g
70 125 g g g g g g Table 1: Tensometer readings: Cylinder with π contact angle, where g isthe acceleration due to gravity.If one consults Karavokyros’ masters thesis for the experimental data,then one can find the raw data for the cylinder, π contact angle case (seetable 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 onegets figure 4. Note that the capstan equation implies that the tension ratio isconstant for all applied masses, i.e. δτ = exp( µ d θ ) is constant given that µ d and θ are constants. However, this is not what the experimental results areimplying, as the reader can clearly see from figure 4 that as the applied massincreases, tension ratio also increases, and this is documented phenomenonin the literature [13], which cannot be simply dismissed. Thus, this impliesthat, for the given experiments, it is unsuitable to use the standard capstanequation to find the coefficient of friction with a significant degree of accu-racy. 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.Unfortunately no raw data is available for the other experiments in thetheses of Karavokyros or Cottenden [3] to conduct further rigorous analysis,17 Mass (grams) δ τ Mean
Figure 4: Tension ratio against applied mass.as we did with the π -cylinder case.As a result of the flawed mathematics and data analysis techniques, Cot-tenden et al. [1, 2] and Cottenden [3] assert that the tension observed onthe membrane is independent of the geometry and the elastic properties ofthe foundation, and thus, the stress profile at the contact region and thecoefficient of friction can be calculated with the use of the ordinary capstanequation. They further assert that the experimental methodology for calcu-lating the coefficient of friction between fabrics and in-vivo (i.e. within theliving) human skin is identical to the capstan model. However, we found noexperimental evidence in the body of the authors’ publications to verify theirassertion, i.e. no evidence is given for the assertion that foundation’s geo-metric and elastic properties are irrelevant when calculating the coefficientfriction 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 1018ubjects, 5 males and 5 females, all between the ages of 18 to 60, and theapproval was granted by the UCL Research Ethics Committee: UCL EthicsProject ID number 5876/001. Collected data of the subjects can be found intable 2, if the reader wishes to reproduce any results, where BMI is the bodymass index (calculated with NHS BMI calculator ), Radius is the radius ofthe upper arm and δl is a measure of how flaccid the subject’s tissue is (seeequation (17)). For more comprehensive set of raw data, please consult DrN. Ovenden (UCL) at [email protected] Gender Age (Years) BMI Radius (cm) δl F19 Female 19 21 . .
98 0 . . .
22 0 . . .
82 1 . . .
54 1 . . .
46 1 . . .
50 0 . . .
77 1 . . .
22 1 . . .
50 0 . . .
09 1 . π . The dimensions of thefabric are 4cm × . . × × . Figure 5: Experimental configuration on subject F53 (.stl file).For each run, we pull the fabric with a constant speed of cms − with theuse the tensometer and we use a static-3D scanner (3dMD PhotogrammetricSystem provided by Mr C. Ruff, UCLH) to record the before and afterdeformation of the upper arm. Note that we use the metric δl = P ,..., || deformed( A4 − E6 ) || P ,..., || undeformed( A4 − E6 ) || , (17)to measure the flaccidity (analogous to 1 / (Young’s modulus)) of the sub-ject’s soft tissue. Also, Radius (from table 2) is calculated by measuring thegirth around the bicep and then dividing the measurement by 2 π Table 3 shows the tension ratio, δτ = T max /T , for each subject withrespect to each applied mass, where T max are the tensometer readings, T =Mass × g are the weight of the applied mass and g = 9 .
81 is the accelerationdue to gravity. Note that the tensometer data of F34 for 60g, M25 for 140g http://3dmd.com δτ
40g 60g 80g 100g 120g 140gF19 2 .
19 1 .
70 1 .
52 1 .
92 1 .
88 1 . . · · · .
93 1 .
89 1 .
83 1 . .
99 1 .
96 1 .
92 1 .
89 1 .
88 1 . .
13 2 .
31 2 .
23 2 .
15 1 .
84 1 . .
29 2 .
20 2 .
06 1 .
99 1 .
96 1 . .
99 1 .
90 1 .
88 1 .
84 1 .
68 1 . .
46 2 .
28 2 .
24 2 .
19 2 .
33 1 . .
14 1 .
98 1 .
81 1 .
80 1 . · · · M26 2 .
41 2 .
31 2 .
26 2 .
18 2 .
31 1 . · · · .
12 2 .
03 1 .
77 1 .
91 1 . et al. [1, 2] and Cottenden [3] assert that the T max = T exp( µ F θ )equation holds true, regardless of the geometric and elastic properties of thesubstrate (i.e. human soft-tissue). If this is indeed true then δτ = exp( µ F θ )is only a function of the coefficient of friction, regardless of the geometric andelastic properties of human soft-tissue, for a fixed contact angle θ . Also, oneof the major assumption of the authors is that coefficient of friction betweenskin and fabrics positively correlated with the age of an individual, as theyobserve higher occurrence of skin damage in elderly subjects. Thus, now weplot the tension ratio against various properties to test the authors’ claims.Figure 6a shows the tension ratio against subjects’ age, where the lin-ear regression line is δτ = 0 . × Age + 1 .
96 and the age is in years(R = 0 . R is the coefficient of determination). As the readercan see there is no obvious relationship between the age of the subject andthe tension ratio. In fact, the highest tension ratios are observed for M23and M26 (two of the youngest subjects).Figure 6b shows the tension ratio against the body mass index (BMI),where the linear regression line is δτ = 0 . × BMI + 1 .
61 (R = 0 . Age (years) δ τ MeanRegression (R =0.0512) (a) Tension ratio against the age (inyears).
16 18 20 22 24 26 28
BMI δ τ MeanRegression (R =0.0961) (b) Tension ratio against the BMI. δ l δ τ MeanRegression (R =0.282) (c) Tension ratio against the δl . Radius (cm) δ τ MeanRegression (R =0.412) (d) Tension ratio the radius of the upperarm (in cm). Figure 6: Tension ratio for varying age, BMI, δl and radius of the upper arm.As the reader can see that there is a vague positive correlation between theBMI and the tension ratio. The reason we observe this correlation is be-cause 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 highertension 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 6c shows the tension ratio against δl , where the linear regressionline is δτ = 4 . × δl − .
41 (R = 0 . δl and the tension ratio. This implies that22or more flaccid soft-tissue, larger portion of the applied-tension is expendedon deforming it.Figure 6d shows the tension ratio against the radius of the upper arm,where the linear regression line is δτ = 0 . × Radius + 1 .
12 and the radiusis in centimetres (R = 0 . correlation does not imply causation [30]; It merely impliesthat belt-friction models are not suitable for modelling such problems, whichdirectly contradicts Cottenden et al. [1, 2] and Cottenden’s [3] assertion re-garding the efficacy of the capstan model.For a mathematically rigorous way of modelling this problem, we referthe reader to section 6.6 and 6.7 of Jayawardana [14]. There, both numerical-modelling (shell-membrane frictionally coupled to an elastic-foundation) andhuman trial data imply that given a constant coefficient of friction, a highervolume of soft-tissue (high radius) and more compliant soft-tissue (lowerYoung’s modulus) would result in higher deformation of the skin, and ahigher volume of soft-tissue would result in more shear-stress generated onthe skin.
5. Conclusions