AN ANALYTICAL MODEL OF THE MOTION OF A
CONFORMABLE SHEET OVER A GENERAL
CONVEX SURFACE IN THE PRESENCE OF
FRICTIONAL COUPLING
by D. J. COTTENDEN† and A. M. COTTENDEN
[Received 20 November 2008. Revised 17 April 2009. Accepted 22 April 2009]
Summary
Friction is important across a wide range of applications. In particular, in health care, friction is
thought to be 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. Some of these frictional forces are due to stress in materials wrapped
around curved anatomical surfaces, which are often complicated shapes. The little work to date
that has considered friction arising by this mechanism has assumed very simplified geometries
(prisms, or even cylinders), which have enabled coefficients of friction to be extracted from
laboratory tests on arms, but which are certainly not applicable to, for example, the diaper
region. This work describes the development of a much more general mathematical model
for friction between a draped, stressed sheet and the substrate, relating geometry, material
mechanical properties and stress for essentially any convex surface. A general, wide, class of
frictional interfaces is described (which includes those which obey Amontons’ law), and the
model is presented in differential form for a generic member of this class. Finally, an analytical
solution is developed for convex, instantaneously rigid substrates isomorphic to the plane
draped with a low-density sheet exhibiting no Poisson contraction, a fair approximation to some
anatomical situations. The solution is explicitly calculated for a general prism and a general
cone, producing expressions consistent with previous published models and with limited new
experimental data.
1. Introduction
Frictionally coupled surfaces are relevant in several health care applications and more widely. For
example, friction produces some of the shear stress thought to cause pressure ulcers (1); and abrasion due to friction between incontinence pads and the skin contributes to poor skin health in wearers
(2), exacerbated by the presence of liquid (3, 4).
Considering this, it is perhaps surprising that there has been little theoretical work on the dynamics of non-trivial surfaces (such as a non-flat body part and a conformable fabric sheet) interacting
under general stresses via friction. In particular, the situation in which the primary stress is a tension
applied to the boundaries of the sheet (Fig. 1) appears to have been overlooked in spite of the large
frictional stresses that can ensue (5).
†hd.cottenden@ucl.ac.uki
c The author 2009. Published by Oxford University Press;
Q. Jl Mech. Appl. Math, Vol. 62. No. 3 all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org
Advance Access publication 22 June 2009. doi:10.1093/qjmam/hbp012
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(Department of Medical Physics and Bioengineering, University College London,
Gower Street, London WC1E 6BT)
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D. J. COTTENDEN AND A. M. COTTENDEN
The present authors are aware of three papers that have employed quantitative models to friction
arising at a skin–fabric interface due to tensile stress acting on a sheet wrapped around a curved
surface (6, 4, 5). Gwosdow et al. (6) and Cottenden et al. (4) sought to measure the coefficients of
dynamic (µd ) and static (µs ) friction (respectively) using the apparatus illustrated in Fig. 2(a) and
applying the well-known formula
1
F
(1.1)
µ = log
2
mg
(see, for example, (7, p. 301) for a derivation) to their results, where 2 is the angle of contact,
F is the pulling force and mg is the resisting force. However, since this result assumes that the
surface around which fabric is wrapped is a rigid cylinder (Fig. 2(b)), the remarkable agreement
that Cottenden et al. (4) found between their µs measurements and those calculated by a classic
‘straight pull’ method in their experiments on volar forearms was remarkable. This was partially
explained by a subsequent paper (5) in which it was shown that (subject to an appropriate choice
of coordinate centre about which to measure 2) this result continues to hold for all strictly convex
prisms.
However, the true significance of the generalised model of Cottenden et al. (5) subsists in its
predictive power: the authors calculated the frictional shear stress as a function of angle and coefficients of friction. Though this is clearly of limited quantitative value (a situation so ‘pure’ as that
which they described is rarely encountered outside of a laboratory), the strength of an analytical
solution is to elucidate what is and is not important, and in this case to show that the shear varied
approximately exponentially with angle and coefficient of friction, and so that such forces could be
far larger than might be anticipated.
Both these models assumed that Amontons’ law holds; that is, that the maximum static and the
dynamic friction forces are proportional to the local normal force and that the constants of proportionality (the coefficients of friction) are invariant with velocity (see, for example, (8, chapter 6)).
There appears to be no published evidence of any skin–fabric system deviating from Amontons’
law: Comaish and Bottoms (3) only presented load results for polythene sheet against the skin;
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Fig. 1 When a conformable sheet is placed in tension over a convex surface, a normal reaction force is generated. For ‘rough’ surfaces, this causes friction, which in turn modifies the stress within the sheet
FRICTION ON GENERAL CONVEX SURFACES
347
Zhang and Mak (9) reported load and velocity data for all their varied test materials but did not say
which was which; and Cottenden et al. (4) found no disagreement with Amontons’ law in terms of
load, though their experiment was very insensitive to any variation of µ, which may have occurred
at the low end of their 0·36–2·23 kPa pressure range. The common prejudice against Amontons’
law in this context is apparently without foundation.
This work seeks to develop both the interpretive and predictive aspects of previous models by deriving a similar model for more general and complicated surfaces, such as present on many anatomical sites. The model assumes Amontons’ law, though it could readily be adapted to other friction
laws. Having derived the general relationship between surface geometry and sheet stress, means
of description and parametrisation are considered, and finally, analytical solutions for some limited
cases are obtained and compared with those used and derived in earlier work (6, 4, 5).
2. A general geometrical friction model
The most general model to date (5) applies to prisms. It cannot readily be extended to deal with
even a simple or minor deviation from a constant cross section because it makes no admission of
the existence of the third dimension. Additionally, a third dimension (and thus a two-dimensional
surface) necessitates the substitution of stress for force and introduces the question of the orientation
of the principal directions. A much more general approach is required.
2.1 Continuum mechanics
The most appropriate tool to deal with the general surface and stresses presented above is continuum
mechanics. Cauchy was the first to derive the continuum analogue of Newton’s second law (see (10),
for example):
∇ · T + f = ρ χ̈,
(2.1)
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Fig. 2 An experimental method for obtaining coefficients of friction used by Gwosdow et al. (6) and Cottenden
et al. (4). (a) The experimental method involved pulling a weighted strip of a test fabric around the volar
forearm of a test subject and measuring the force required to initiate or sustain slip. (b) Definitions of the
coordinate and variables
348
D. J. COTTENDEN AND A. M. COTTENDEN
•
•
•
•
has no through thickness and can be represented as a two-dimensional object;
always drapes, following the substrate surface without tearing or puckering;
is of sufficiently low density that its weight makes a negligible contribution to the forces acting;
does not resist bending in the sense that a beam does.
Representing the nonwoven sheet as a two-dimensional object requires some interpretation of
the three-dimensional quantities in Cauchy’s law of motion. Describing the nonwoven in this way
is equivalent to assigning some finite thickness L, allowing no change through the thickness and
normalising quantities’ dimensions by L. A further consequence of the lower dimension and the
neglect of the nonwoven’s weight is that the only body force (f) to act is due to the normal reaction
from and frictional interaction with the substrate.
The substrate requires no such interpretation. For convenience, its weight will be considered
balanced by a prestress, and both are henceforth neglected: the substrate’s f is now also purely due
to friction and has support only at the surface. For simplicity, the substrate is assumed to be convex
everywhere; the contact region is thus simple and connected.
2.2
Instantaneous isotropic interfaces
Before assuming Amontons’ law, it is well to consider frictional interfaces more generally.
D EFINITION An instantaneous isotropic interface (III) 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 v (for v 6= 0) or
to the sum of current applied forces acting to initiate motion (for v = 0).
The last aspect of this definition addresses an interesting ambiguity in the common statement
of Amontons’ law: the direction of frictional forces does not appear to have been specified by
Amontons and no mention is made in a variety of standard texts (11, 12, 8, 13). Though the point
seems superficially unimportant, the independence of the relative velocity and force vectors makes
it clear that it is vital. The importance is further elevated shortly.
It is thus possible to write down a general friction law for IIIs, which depends only on two
materially determined functions, the friction scalar functions, one for statics and one for dynamics:
(
−ψs F̂, v = 0,
PS · f =
(2.2)
−ψd v̂, v 6= 0,
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where T is Cauchy stress, f is body force per unit volume, ρ is material density, χ is a timedependent 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.
It is worth briefly describing the properties of the Cauchy stress. Cauchy stress is defined as force
area density, where both the magnitudes and directions of force and area are measured in the current
material configuration. It differs from the Piola–Kirchhoff (P-K) stress tensors (see (10), for example) in the latter respect: the first P-K stress refers the vectorial area to the reference configuration;
the second P-K tensor is obtained from the first by transforming the forces by the inverse of the
deformation suffered by the material body in question. The Cauchy stress is the most appropriate
choice when all interesting measures and conditions are applied in the deformed configuration.
Specialising to the problem at hand, consider first the draped sheet. Assume that the sheet:
FRICTION ON GENERAL CONVEX SURFACES
349
where PS is a projection matrix that removes components outside of a surface S and the circumflex
indicates normalisation. Substituting these definitions into Cauchy’s law of motion gives
∇ · T1 + PN̂ · f1
∇ · T1 + PN̂ · f1 − ψs
= ρ1 χ̈1 ,
v = 0,
(2.3)
|∇ · T1 + PN̂ · f1 |
PS · (χ̇1 − χ̇2 )
= ρ1 χ̈1 ,
v 6= 0,
(2.4)
∇ · T1 + PN̂ · f1 − ψd
|PS · (χ̇1 − χ̇2 )|
|∇ · T + PN̂ · f1 | = ψs + ρ1 |χ̈1 |.
(2.5)
Clearly, since the stress is fundamentally determined by the deformation vector field χ and its
derivatives (10), this scalar equation cannot be solved. Further, ψs is generally not a single-valued
function of local pressure; it scales with applied force up to a limit, beyond which slip occurs. These
observations imply that for an III there are not unique stable static stress or deformation fields,
even if the materials show no memory effects and are linear elastic: information on the history
of the interface is required. It is still, of course, possible to determine whether a particular configuration is stable, most simply by allowing ψs to denote its maximum value and considering
|∇ · T + PN̂ · f1 | 6 ψs + ρ1 |χ̈1 |;
(2.6)
if this is true then local slip will not occur. The only case in which the static system may be solved
is in the case of high symmetry in which the situation is really a one-dimensional problem. The
dynamic equation is, at this stage, in principle solvable.
2.3
Contact forces at a dynamic interface
In general, both the nonwoven sheet and the substrate that constitute an interface will deform in
response to the tensile forces imposed upon the sheet. In the presence of acceleration, the simple
‘equal and opposite’ relationship between normal components of ∇ · T and normal reaction breaks
down: a more general relationship must be considered.
Consider the abbreviated forms of (2.4) for the nonwoven sheet (n) and the substrate (s):
∇ · Ts + PN̂ · fs + PS · fs = ρs χ̈s ,
x ∈ S,
∇ · Tn + PN̂ · fn + PS · fn = ρn χ̈n .
(2.7)
(2.8)
Also note that fs = −fn by Newton’s third law. Further, for the surfaces to remain in contact both
χ̇n · N̂ = χ̇s · N̂ and χ̈n · N̂ = χ̈s · N̂: Appendix A justifies and discusses these equations further.
Consider the dot product of (2.7) and (2.8) with N̂:
(∇ · Ts ) · N̂ + fs · N̂ = ρs χ̈s · N̂,
(2.9)
(∇ · Tn ) · N̂ + fn · N̂ = ρn χ̈n · N̂.
(2.10)
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where the subscript 1 indicates the body in question, 2 the contacting body and PN̂ is a projection
matrix onto the normal to the surface S. The right-hand side of (2.3) need not be zero since a lack
of relative motion does not preclude motion of material particles. From these equations, it is clear
that if there is any stress, the static case is underdetermined: both terms on the left-hand side are
proportional to ∇ · T1 + PN̂ · f1 , which implies the right-hand side also is, so reducing (2.3) to the
scalar equation
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D. J. COTTENDEN AND A. M. COTTENDEN
By summing these equations,
(∇ · Ts + ∇ · Tn ) · N̂ = (ρs + ρn )χ̈ · N̂,
(2.11)
the normal force per unit area exerted by the substrate upon the sheet. Further substituting this result
into the abbreviated equations of motion, (2.7) and (2.8),
ρ s ∇ · T n − ρ n ∇ · Ts
∇ · Ts + PN̂ ·
(2.13)
+ PS · fs = ρs χ̈s , x ∈ S,
ρ s + ρn
ρ s ∇ · T n − ρ n ∇ · Ts
(2.14)
∇ · Tn − PN̂ ·
+ PS · fn = ρn χ̈n .
ρ s + ρn
A significant simplification of (2.13) and (2.14) is effected if the nonwoven (sheet) has negligible
inertia (ρn ≈ 0):
∇ · Ts + PN̂ · {∇ · Tn } + PS · fs = ρs χ̈s ,
∇ · Tn − PN̂ · {∇ · Tn } + PS · fn = 0.
x ∈ S,
(2.15)
(2.16)
As PN̂ + PS = I, the rank 2 identity tensor, (2.16) reduces to
PS · {∇ · Tn } + PS · fn = 0.
(2.17)
In this limit it is therefore the case that the nonwoven sheet behaves precisely as it would if there
were no normal acceleration at all (setting χ̈n · N̂ = 0 in (2.10)). This is perhaps as would be
expected: in the absence of inertia, all systems respond instantaneously and all their states are steady.
These equations are complete but for the specification of the friction scalars. They generally depend on normal force density (for example, Amontons’ law asserts proportionality between friction
and normal force density), and, as shown above, the normal force density depends on the normal
components of ∇ · Ts and ∇ · Tn . The substrate has through thickness and provides normal forces
by deforming; the nonwoven sheet provides them by virtue of the curvature of the surface.
2.4
Amontons’ law and normal forces
As the motivation for the above definition of IIIs, Amontons’ law (with the stated assumptions upon
the direction of the described force) is a member of this class with ψ∗ = µ∗ |f† · N̂|, where the *
denotes s or d as appropriate and f† is the force per unit volume on the appropriate surface. It is thus
necessary to determine f† · N̂.
This relationship can be derived by first obtaining the normal component of force density due to
surface stress, considering a generalisation of the argument employed in deriving friction formulae
for cylinders and prisms (7, 5); that is, considering the rotation of the surface as described by the
unit normal vector (Fig. 3). Following this lead and considering Fig. 4, it follows that
δN = η · ∇N̂ + O(|η|2 ).
(2.18)
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where the subscript has been dropped from χ̈ · N̂ because it is equal on both sides of the interface.
Equation (2.11) determines χ̈ · N̂. Substituting this result back into (2.9) and (2.10) gives
ρ s ∇ · T n − ρ n ∇ · Ts
· N̂,
(2.12)
−fn · N̂ = fs · N̂ =
ρs + ρ n
FRICTION ON GENERAL CONVEX SURFACES
351
Fig. 4 The ‘angle’ δ N̂ is equal to the change in unit normal between x and x + η. Its direction is generally not
the same as that of η
From this, it is possible to calculate the total inward directed force dR := −(∇ · T) · N̂ due to stress
in the draped sheet around the periphery ∂A of a region A (Fig. 5):
dR(x) = lim
I
|η |→0 ∂ S
dF(x + η) · [(η · ∇)N̂(x)].
(2.19)
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Fig. 3 The tension forces in a draped sheet over a curved surface are always parallel to that surface. It follows
that the two forces T (θ ) and T (θ + dθ) will not be parallel to each other and so exert a net force on an inwards
normal, exciting a normal reaction force dR
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D. J. COTTENDEN AND A. M. COTTENDEN
As discussed in section 2.1, on a two-dimensional surface, the Cauchy stress T within the material
of the surface and an element of force dF are related by dF = LT· d x̌, where d x̌ is an axial distance
vector (by analogy with an axial area vector) and L is the nonwoven sheet thickness. Substituting
this into (2.19),
I
[T(x) · d η̌] · [(η · ∇)N̂(x)]
dR(x) = L lim
|η |→0 ∂ S
= L lim
Z
|η |→0 S
∇η · [(η · ∇)N̂(x) · T(x)]d S,
(2.20)
where the latter step involves applying the divergence theorem in two dimensions. Note that the
divergence contracts between ∇ and the second index of T, which is the only free index inside the
square brackets. The divergence can be expanded using the product rule and in the limit given only
the term in which η is differentiated survives:
Z
dR(x)
= Tr[∇N̂(x) · T(x)].
dR(x) = L
Tr[∇N̂(x) · T(x)]d S ⇒
dV
S
Defining the curvature tensor C := ∇N̂(x),
(∇ · Tn ) · N̂ = −Tr [C(x) · T(x)] .
(2.21)
It is simple in general to use (2.21) and the surface acceleration to write down f† · N̂ using (2.10),
but since only the situation with negligible nonwoven density is considered henceforth, it suffices
to note that Amontons’ law becomes
ψ∗ = µ∗ |Tr [C(x) · T(x)] |
(2.22)
in the low-density limit.
This function closes the system of equations for Amontons’ law dynamic friction of a low-density
conformable sheet over an arbitrary convex surface.
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Fig. 5 The region considered when calculating the pressure at a point. dF is the element of force that acts
along a small length of the boundary ∂A
FRICTION ON GENERAL CONVEX SURFACES
353
An alternative derivation of normal stress force is offered in Appendix B. It is more mathematically rigorous, but is not presented as the primary method as it provides no physical insight.
3. Framework for solutions
3.1
Simplifying assumptions
The immediate interest of the present authors is in modelling low-density nonwoven fabrics (such as
those used as incontinence pad coverstocks) moving over skin, so the assumptions made hereafter
are apt to this situation; their validity must be judged independently for any new areas of work to
which they are applied. Allowing this caveat, proceed.
A SSUMPTION 1 The inertia of the sheet may be neglected, ρn = 0.
Nonwovens of the type described have area densities of the order of 15–25 g m−2 (14) and are a
few hundred micrometres thick. Experimental experience implies that they have negligible inertia
in circumstances such as those described here. Additionally, the classic cylinder model (7) and the
model of Cottenden et al. (5) ignore the inertia of the sheet, yet Cottenden et al. (4) found excellent
agreement between theory and experiment, implying that inertia can safely be neglected.
Note that confining attention to the steady state does not eliminate the need for this assumption:
spatial description steady state does not preclude the acceleration of material particles, and so if
inertia is not negligible even the steady state may be influenced. However, once inertia has been
neglected, there is no distinction between transient and steady states.
A SSUMPTION 2 The substrate behaves rigidly.
Although Assumption 1 breaks most aspects of the dynamic connection between the substrate and
the nonwoven sheet (2.17), it does not remove the effect of substrate velocity within the surface.
However, since the nonwoven has been assumed to have no inertia, it responds instantly so that
PS · (χ̇1 − χ̇2 ) is parallel to PS · (∇ · Tn ). Whether the relative velocity is due to the nonwoven
alone or also to the substrate is irrelevant so long as the substrate velocity is the same everywhere.
These two assumptions are adequate for the development of some analytical solutions.
3.2
A planar description of surfaces
So far no description of the surface in question has been advanced, and one is clearly needed in order
to proceed. The simplest description is that commonly used in differential geometry (15) where a
surface is described by a surface patch, a function σ : U → S, U ⊆ R2 , S ⊂ R3 , which maps the
plane onto some other surface embedded in three dimensions. Adopting this description enables a
number of other helpful definitions to be made.
Denote coordinates u α in U with Greek suffices; subscripts should not be interpreted as derivatives unless preceded by the customary comma. Following the convention stated by Pressley (15),
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The model developed so far describes the fundamental interactions, and upon the assumption of a
constitutive relationship for the materials it is amenable to solution for a general surface, but no
general analytical solution has been found to date for any such relationship; the complexity of the
resulting equations suggests that no such analytical solution exists. However, various assumptions
and simplifications can be made that make the development of some solutions possible.
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D. J. COTTENDEN AND A. M. COTTENDEN
tensors within the surface will be written in terms of the parameter derivatives of the patch σ so that
t = tα σ,α
⇒
t · s = tα F I αβ sβ
T = Tαβ σ,α ⊗ σ,β
⇒
T · S = Tαβ F I βγ Sγ δ σ,α ⊗ σ,δ ,
where F I αβ = σ,α · σ,β , the first fundamental form of the surface patch. A key advantage of
adopting this notation is that Weingarten’s theorem (15) can readily be adopted and adapted for the
calculation of C:
⇒
∇N̂ = −(F I I F I −1 )αβ ∇u α ⊗ σ,β ,
where F I I αβ = σ,αβ · N̂, the second fundamental form. (Note that in his excellent book Pressley
(15) unusually chooses to multiply column vectors by matrices on the right in his statement of
Weingarten’s theorem—the above statement has reordered the matrices so as to adopt the more
usual convention on matrix multiplication.) Further,
Cγ δ σ,γ ⊗ σ,δ = −F I I αη F I −1ηβ ∇u α ⊗ σ,β
(3.1)
Cγ δ F I γ F I δζ = −F I I αη F I −1ηβ (σ, · ∇u α )(σ,ζ · σ,β ) = −F I I ζ
(3.2)
Cγ δ = −F I−1
−1
γ F I I ζ F I δζ ,
(3.3)
since σ, · ∇u α = δα and σ,ζ · σ,β = F I ζβ .
It is now possible to define the meaning of the word ‘convex’ more tightly. In this context, it
should be interpreted as η · ∇N̂ · η > 0 for all η ∈ {aα σ,α , aα ∈ R}; that is, the change in the unit
normal in any direction should have a positive component in that direction.
Thus provisioned with simplifying assumptions and a well-developed language in which to express the relevant quantities, some simple solutions may be considered.
4. Solutions on simple surfaces
As stated before, the model defined in section 2 is applicable to all convex surfaces. However, the
only surfaces on which analytical solutions are obtainable are those with a high degree of symmetry. After a brief discussion of the mode by which approximate solutions have been obtained
(section 4.1), the solutions for a prism (section 4.2) and a general cone (section 4.3) are derived.
The first offers the opportunity for comparison with the solution of Cottenden et al. (5), and slightly
generalises it, while the second enables the better approximation of limbs.
Since only the nonwoven sheet is dynamic, the subscript n is dropped from quantities corresponding to it without ambiguity.
4.1
Geodesic flow around surfaces isomorphic to the plane
Most materials exhibit Poisson contraction: a positive tensile strain produces a lateral contraction.
This has the effect of linking the components of strain together and thus rendering material behaviour much more complicated and interesting. However, this effect is not included in either the
classic cylinder model or the newer model of Cottenden et al. (5), and yet they give excellent
agreement with experiment. This suggests that to a first approximation, materials may be modelled
without Poisson contraction. The authors are not aware of experimental studies that either support
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N̂,α = −(F I I F I −1 )αβ σ,β
355
FRICTION ON GENERAL CONVEX SURFACES
ˆ⊥ = 0
T · χ̇
⇒
ˆ ⊥ = Tαβ σ,α ⊗ σ,β · σ,x = 0
T · χ̇
⇒
Tαx σ,α = 0
⇒
Tαx = 0,
since F I = I . By the symmetry of Tαβ under exchange of suffices (which follows immediately
from its usual symmetry), this condition requires that its only non-zero component is Tyy . This
great simplification reduces the governing equation (2.17),
µd |Tr(T · C)|σ̇ˆ = PS · (∇ · T) = PS · [ (∇u γ ∂γ ) · (Tαβ σ,α ⊗ σ,β ) ]
= PS · [Tαβ,γ ∇u γ · σ,β σ,α + Tαβ ∇u γ · σ,βγ σ,α + Tαβ σ,β · ∇u γ σ,αγ ]
= PS · [Tαβ,β σ,α + Tαβ ∇u γ · σ,βγ σ,α + Tαβ σ,αβ ],
(4.1)
Fig. 6 A simple situation in which a rectangular strip of conformable fabric is draped over the surface S and
only subjected to tensile forces. In the absence of Poisson contraction, there can be no strain perpendicular to
the force-free boundaries (‘lateral strain’). The flow vector χ̇ is parallel to the long side of the rectangle and
has a perpendicular χ̇⊥
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or contradict this assumption for nonwoven in this situation; casual observation suggests that for the
skin/nonwoven system at hand it is at least fair.
Additionally, attention will be limited for the duration of this work to surfaces 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.
In the absence of Poisson contraction, situations similar to the experimental methods of Gwosdow
et al. (6) and Cottenden et al. (4) (Fig. 6) simplify considerably. There are no forces along the ‘sides’
of the samples, and in the absence of Poisson contraction longitudinal forces cannot generate them.
There cannot therefore be any lateral forces; T · χ̇⊥ = 0. All forces are therefore tensile.
An important feature of unidirectional stretches is that geodesics parallel to the principal stretch
axis are mapped onto themselves. Further, if the surface S is isomorphic to the plane then geodesics
on S are also geodesics of the plane. This implies that geodesics of the deformed sheet are identical
to those of the plane, which are straight lines and thus readily parametrised.
ˆ is known. Further, since the flow lines follow straight lines in the
Under these assumptions, χ̇
ˆ and σ,x is parallel to χ̇
ˆ ⊥ , so the lateral force
plane, a patch can be chosen so that σ,y is parallel to χ̇
condition implies
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D. J. COTTENDEN AND A. M. COTTENDEN
since ∇u γ · σ,β = δγβ for all patches. Further, when F I = I it follows that ∇u γ = σ,γ . As
σ,γ · σ,βγ = ( 12 σ,γ · σ,γ ),β = ( 12 Iγ γ ),β = 0, the second term in (4.1) vanishes:
µd |Tr(T · C)|σ̇ˆ = (σ,γ ⊗ σ,γ ) · [Tαβ,β σ,α + Tαβ σ,αβ ]
= σ,γ [Tγβ,β + Tαβ σ,αβ · σ,γ ].
(4.2)
Using
σ,α · σ,βγ = (σ,α · σ,β ),γ − σ,αγ · σ,β = −σ,αγ · σ,β ,
and Tαβ = Tβα , it follows that
Tαβ (σ,αβ · σ,γ ) = −Tαβ (σ,α · σ,βγ ) = − 12 Tαβ (σ,α · σ,βγ + σ,β · σ,αγ ) = 0.
(4.3)
It therefore follows from (4.2) that
µd |Tr(T · C)|σ̇ˆ = σ,γ Tγβ,β .
This equation simplifies after recalling that the only non-zero component of stress is Tyy :
µd |Tyy C yy |σ,y = Tyy,y σ,y .
By assumption Tyy > 0, so this equation can be further simplified and solved in integral form:
Z
(4.4)
Tyy,y − µd |C yy |Tyy = 0 ⇒ Tyy = T0 exp µd |C yy |dy .
This form is valid for any convex surface isomorphic to the plane (with an appropriate choice of
patch) for a low-density fabric with a Poisson ratio equal to zero. Solutions for specific examples of
such surfaces can now be considered.
4.2
Prisms
New models must be consistent with older, established ones. It is therefore important to derive the
solution for tangential flow around a prism, of which flow around a cylinder is clearly a special
case. The assumptions made herein are essentially those made for the ‘classic’ solution (7) and by
Cottenden et al. (5), so agreement should be obtained. It is simple with this model to generalise
the previous models slightly and consider flow at an angle ζ to the prism’s plane of cross section
(Fig. 7), so this generalisation is made.
In essence, once a patch σ(x, y) has been defined all subsequent quantities up to and including
Tyy itself follow by formal manipulation. Some waypoints are noted for ease of reading. Define
σ(x, y) = (R(φ) cos φ, R(φ) sin φ, x cos ζ + y sin ζ ),
cos ζ dy − sin ζ d x
dφ = p
,
R(φ)2 + R 0 (φ)2
with respect to a standard Cartesian basis in R3 . In principle, φ could be calculated for a given R(φ),
but in practice this is not required. By differentiation and the cross product, it is easy to determine
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σ,αβ · σ,γ = (σ,α · σ,γ ),β − σ,α · σ,βγ = −σ,α · σ,βγ ,
FRICTION ON GENERAL CONVEX SURFACES
357
σ,x , σ,y and N̂:
σ,x = φ,x (R 0 cos φ − R sin φ, R 0 sin φ + R cos φ, 0) + (0, 0, cos ζ ),
σ,y = φ,y (R 0 cos φ − R sin φ, R 0 sin φ + R cos φ, 0) + (0, 0, sin ζ ),
N̂ = √
1
R 02
(−{R 0 sin φ + R cos φ}, R 0 cos φ − R sin φ, 0).
+
From these, it is simple to confirm the isomorphism between the plane and a general prism by
checking that F I = I . Further differentiation of the patch and contraction with the unit normal
produce the second fundamental form,
!
− sin ζ cos ζ
sin2 ζ
−R(R 00 − R) + 2R 02
FI I =
3
− sin ζ cos ζ
cos2 ζ
(R 2 + R 02 ) 2
!
− sin ζ cos ζ
sin2 ζ
(d/dφ)(φ − tan−1 (R 0 /R))
√
,
(4.5)
=
− sin ζ cos ζ
cos2 ζ
R 2 + R 02
R2
where the matrices are with respect to the {x, y} coordinates of U . Further simplification can be
effected by changing the differentiation variable to y:
!
"
#
sin2 ζ
− sin ζ cos ζ
d φ − tan−1 (R 0 /R)
.
(4.6)
FI I =
dy
cos ζ
− sin ζ cos ζ
cos2 ζ
Recalling that F I = I , substituting (4.6) into (4.4) produces
Z
d
[φ − tan−1 (R 0 /R)] dy
Tyy = T0 exp µd cos ζ
dy
φ
= T0 exp µd cos ζ [φ − tan−1 (R 0 /R)]φ21 ,
(4.7)
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Fig. 7 The angle ζ is defined as the angle between the flow vector and the prism’s plane of cross section as
measured on the surface
358
D. J. COTTENDEN AND A. M. COTTENDEN
where φ1 and φ2 are the limits of contact. This is the result derived by Cottenden et al. (5) for ζ = 0,
and reduces to the classic cylindrical solution for ζ = 0, R 0 = 0.
4.3
Cones
r
σ=p
1 + R(φ(θ))2
(R(φ(θ)) cos(φ(θ )), R(φ(θ )) sin(φ(θ)), 1),
(4.8)
where
p{r, θ } are plane polar coordinates for U derived from the Cartesian {x, y} coordinates by
r = x 2 + y 2 and θ = tan−1 (y/x) − ζ , where ζ is the angle between the direction of slip and the
tangential direction when θ = φ = 0 (Fig. 9). The connection between φ and θ is more subtle, but
consideration of Fig. 10 shows that
√
√
R 1 + R2
R 2 + R 02 + R 4
dφ = q
dφ.
(4.9)
dθ =
1 + R2
(1 + R 2 )2 − R 2
,θ
Differentiate the patch:
R R 0r θ,α φ,θ
1
√
σ,α = r,α −
(R cos φ, R sin φ, 1)
2
1+ R
1 + R2
θ,α φ,θ r
+√
(R 0 cos φ − R sin φ, R 0 sin φ + R cos φ, 0).
1 + R2
(4.10)
Fig. 8 The plane maps onto a general cone as shown. The plane polar coordinates {r, θ } relate to the plane
Cartesian coordinates in the usual way
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The simplest generalisation of a prism is a cone, so it is a logical surface to consider for an analytical
solution. Additionally, most limbs can be fairly well modelled as general cones, so providing additional incentive: Cottenden et al. (5) went some way towards explaining the unreasonable accuracy
of the cylindrical model applied to volar forearms (4); the results of a conical model may further
clarify this surprising result.
Again, the starting point for a solution is to state the patch for a general cone. This is specified by
a cylindrical polar function R(φ), so by considering Fig. 8, the patch is
FRICTION ON GENERAL CONVEX SURFACES
359
Fig. 10 (a)
p Elemental increases in φ and θ are connected by the two interrelated triangles. (b) the relationship
between 1 + R 2 dθ (the arc length at constant radius) and the other quantities
It is more convenient in the ensuing derivation to change to the non-constant orthonormal basis
{(cos φ, sin φ, 0), (− sin φ, cos φ, 0), (0, 0, 1)},
(4.11)
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Fig. 9 The angle ζ is defined as the angle between the flow vector and the cone’s cross-section plane measured
along the surface at θ = φ = 0
360
D. J. COTTENDEN AND A. M. COTTENDEN
and in this basis the components of σ,α read
r θ,α φ,θ R 0
R R 0r θ,α φ,θ
1
σ,α = √
,
Rθ
φ
r,
r
−
Rr,α +
.
,α
,θ
,α
1 + R2
1 + R2
1 + R2
(4.12)
It is useful to note that by definition σ,α · N̂ = 0 and so σ,α · N̂,β + σ,αβ · N̂ = 0. Therefore,
σ,αβ · N̂ = −σ,α · N̂,β , which (given the relative complexities of σ,α and N̂) somewhat simplifies
the derivation of F I I for general cones. Recalling that the new basis is not constant,
N̂,β = f N̂ + √
φ,θ θ,β
+
R2
R 02
+
R4
2R 0 , R − R 00 , −2R R 0 ,
(4.14)
where the coefficient f of the N̂ term (obtained by differentiation of the scale factor) need not be
calculated as it is orthogonal to σ,α . Proceeding formally with the calculation of σ,α · N̂,β , the terms
depending upon r,α cancel, leading to
F I I αβ = −σ,α · N̂,β = √
2θ θ
r φ,θ
,α ,β
√
(R R 00 − 2R 02 − R 2 ).
2
2
1 + R R + R 02 + R 4
(4.15)
Recalling that θ,y = x/(x 2 + y 2 ) = cos(θ + ζ )/r , the only relevant component of F I I αβ can be
obtained:
(
)
p
R R 00 − 2R 02 − R 2
2
F I I yy dy = (θ y dy) cos(θ + ζ )φ,θ 1 + R
.
(4.16)
R 2 + R 02 + R 4
Observing that
(
)
p
R R 00 − 2R 02 − R 2
d
1
R0
−1
2
√
= 1+ R
+√
tan
,
2
02
4
dφ
R +R +R
R 1 + R2
1 + R2
(4.17)
(4.16) can be further simplified to
F I I yy dy = dθ cos(θ + ζ )
d
dθ
tan−1
√
R0
R 1 + R2
−√
φ,θ
1 + R2
.
(4.18)
This expression has several shortcomings. Most obviously, it is still not directly integrable; even
the perfect differential is accompanied by a cosine. Additionally, although it splits off a term that
vanishes in the case of circular cross section, the second term is not independent of R 0 . However,
(4.18) is the most compact and enlightening form obtained.
In the absence of a generally integrable form for F I I , it is worth
√ looking at the simpler situation
of a circular cone, where R is a constant. In this situation φ = ( 1 + R 2 /R)θ , and (4.18) reduces
to
F I I yy dy = −dθ R −1 cos(θ + ζ ).
(4.19)
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Again, it is easy to show from here that F I = I , as expected. The normal vector can again be
obtained by taking the cross product of the two σ,α and normalising:
h
i
1
R, −R 0 , −R 2 .
(4.13)
N̂ = √
R 2 + R 02 + R 4
361
FRICTION ON GENERAL CONVEX SURFACES
Since the first fundamental form is the identity, C yy = −F I I yy , so substituting this expression
into (4.4),
Z
µ
cos(θ + ζ )
d
θ
Tyy = T0 exp µd
(4.20)
dθ = T0 exp
[sin(θ + ζ )]θ21 ,
R
R
φ1
(If ζ 6= 0 then the exponent in (4.21) attracts a factor of cos ζ , and another term of order φ 2 R sin ζ
arises.)
Experimental data gathered on cones constructed from plaster of Paris and Neoprene (after the
fashion of the prisms reported by Cottenden et al. (5)) with half-angles ranging up to 12◦ and contact
angles in the range [70◦ , 120◦ ] show good agreement with the simple cylindrical model at their error
level (around ±10% for most samples). Substituting R = tan(12◦ ) ≈ 0 · 20 into (4.21) gives
h
iφ2
(4.22)
Tyy = T0 exp (µd φ) exp −0 · 02µd φ 1 + 13 φ 2
φ1
[70◦ ,
120◦ ]
≈ [1 · 2, 2 · 1] radians the exponent in the second
to quadratic order in R. In the range
exponential varies in the range [−0 · 104µd , −0 · 036µd ], so the exponential function itself varies in
[∼ 0 · 90µd , ∼ 0 · 96µd ] ≈ [0 · 95, 0 · 98] (in these experiments µ ≈ 0 · 5). This degree of variation
is small in comparison with the experimental error, so the agreement of the conical experiments
with the prismical theory is consistent with the conical theory. However, as these experiments were
focused on establishing the validity of the model of Cottenden et al. (5) for anatomically representative conical half-angles, they cannot provide any more substantial verification of this new model.
Further work to test the solution more thoroughly is planned.
5. Summary
A novel and very general description of the frictional interaction between a stressed, compliant sheet
and a substrate has been developed. The model does not intrinsically assume any particular friction
law, but friction at a defined class of interfaces (instantaneous isotropic interfaces, IIIs) has been
studied in more detail. The model shows that for IIIs there is no unique stable static stress field, but
that for dynamic situations a solution can be found. The further specialised case of Amontons’ law
has been further considered, and a complete equation has been written down for this case.
The work to this point has made very few assumptions (see section 2.1). It would be perfectly
valid to include the forces described as a contribution along with other forces. Within any of these
scenarios, the assumption of a constitutive relationship for the sheet would make the problem
amenable to numerical solution.
The latter portion of this paper has considered some analytical solutions that can be obtained by
making assumptions about the materials and selecting simple surfaces. The material assumptions
have served well in the past, and the surfaces have been selected as both tractable and reasonably
representative of some anatomical surfaces. The solution for a prism is consistent with established
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where θ1 and θ2 are the limits of contact.
To compare this solution with the solution for a ζ = 0 pull around a cylinder, it is useful to
change variables to φ and then expand √
the exponent in terms of φ and R. Thus, in (4.20), use
sin θ = θ − θ 3 /6 + O(θ 5 ) and θ = Rφ/ 1 + R 2 to give
h
n
oiφ2
Tyy = T0 exp µd φ 1 − 12 R 2 1 + 13 φ 2 + O R 4
.
(4.21)
362
D. J. COTTENDEN AND A. M. COTTENDEN
Acknowledgements
The authors acknowledge with thanks SCA Hygiene Products AB, and the Engineering and Physical
Sciences Research Council who funded the work. The experimental results of Skevos Karavokiros
pertaining to circular cones are also gratefully acknowledged.
References
1. B. P. J. A. Keller, J. Wille, B. van Ramshorst and C. van den Werken, Pressure ulcers in intensive
care patients: a review of risks and prevention, Intensive Care Med. 28 (2002) 1379–1388.
2. R. W. Berg, Etiology and pathophysiology of diaper dermatitis, Adv. Dermatol. 3 (1988) 75–98.
3. S. Comaish and E. Bottoms, The skin and friction: deviations from Amontons’ laws, and the
effects of hydration and lubrication, Br. J. Dermatol. 84 (1971) 37–43.
4. A. M. Cottenden, W. K. R. Wong, D. J. Cottenden and A. Farbrot, Development and validation
of a new method for measuring friction between skin and nonwoven materials, J. Eng. Med.
222 (2008) 791–803.
5. A. M. Cottenden, D. J. Cottenden, S. Karavokiros and W. K. R. Wong, Development and experimental validation of a mathematical model for friction between fabrics and a volar forearm
phantom, ibid. 222 (2008) 1097–1106.
6. A. R. Gwosdow, J. C. Stevens, L. G. Berglund and J. A. J. Stolwijk, Skin friction and fabric
sensation in neutral and warm environments, Text. Res. J. 56 (1986) 574–580.
7. I. H. Shames, Engineering Mechanics: Statics and Dynamics (Prentice Hall, Upper Saddle
River, NJ 1996).
8. B. Bhushan, Principles and Applications of Tribology (Wiley, New York 1999).
9. M. Zhang and A. F. T. Mak, In vivo friction properties of human skin, Prosthet. Orthot. Int. 23
(1999) 135–141.
10. C. Truesdell and W. Noll, The Non-linear Field Theories of Mechanics (Springer, Berlin 1965).
11. F. Bowden and D. Tabor, The Friction and Lubrication of Solids (Oxford University Press,
Oxford 1986).
12. T. F. J. Quinn, Physical Analysis for Tribology (Cambridge University Press, Cambridge 1991).
13. G. W. Stachowiak and A. W. Batchelor, Engineering Tribology, 2nd edn. (ButterworthHeinemann, Woburn, MA 2001).
14. J. Lünenschloss and W. Albrecht (eds), Non-woven Bonded Fabrics (Ellis Horwood, Chichester
1985).
15. A. Pressley, Elementary Differential Geometry (Springer, London 2001).
Downloaded from http://qjmam.oxfordjournals.org/ at University College London on November 21, 2011
solutions (7, 5). That for a general cone has only been obtained in integral form, though in the special
case of a circular cone, an expansion around the prism solution has shown that the deviation over a
12◦ half-angle cone with a contact angle in the range [70◦ , 120◦ ] is small, with a prefactor varying
by about ±1 · 5%. No firm experimental verification of the conical solution has yet been obtained,
though experiments on shallow angle plaster of Paris/Neoprene cones show that the deviation from
prismical behaviour is certainly small, as predicted.
Although no closed form solution was found for the general cone problem, the minor variation
of the circular cone solution from the cylinder solution suggests a further reason why the results of
Cottenden et al. (4) showed such good agreement between coefficients of friction calculated from
‘curved’ experiments (of the type shown in Fig. 6) and standard ‘straight pull’ experiments.
363
FRICTION ON GENERAL CONVEX SURFACES
APPENDIX A
Normal velocity and acceleration at interfaces
In section 2.3, the relationship between stress and contact forces at an accelerating contact was established,
subject to the assumption that the two surfaces remained in contact throughout. It was stated there that this
required that the normal components of both acceleration and velocity for the two surfaces were the same,
χ̇n · N̂ = χ̇s · N̂,
χ̈n · N̂ = χ̈s · N̂.
(A.1)
A.1
Elucidation of the apparent contradiction
At first sight, the equations (A.1) appear to force rather stringent constraints on the evolution of the unit normal,
but do not provide or describe a mechanism for enforcing them: a naı̈ve differentiation of the velocity condition
would appear to produce
˙ = χ̈ · N̂ + χ̇ · N̂
˙ ”,
“χ̈n · N̂ + χ̇n · N̂
s
s
(A.2)
χ̇n (Xn , t) · N̂n (Xn , t) = χ̇s (Xs , t) · N̂s (Xs , t)
(A.3)
˙ = χ̇ · N̂
˙ ”. This is in fact fallacious, though the reason is subtle. Recall that the
apparently requiring “χ̇n · N̂
s
superposed dot represents a material picture derivative, and consider the changes in
over a time increment dt. In this increment, the changes on either side are
χ̇n (Xn , t + dt) · N̂n (Xn , t + dt) − χ̇n (Xn , t) · N̂n (Xn , t)
= χ̇s (Xs , t + dt) · N̂s (Xs , t + dt) − χ̇s (Xs , t) · N̂s (Xs , t),
(A.4)
but in this time, the spatial location of the particles with reference position Xn and Xs have also changed:
χn (Xn , t) → χn (Xn , t + dt) = χn (Xn , t) + χ̇n (Xn , t)dt + O(dt 2 )
(A.5)
χs (Xs , t) → χs (Xs , t + dt) = χs (Xs , t) + χ̇s (Xs , t)dt + O(dt 2 ).
(A.6)
There is certainly no requirement that the velocity components orthogonal to the normal are equal, so the
spatial locations described by either side of (A.4) are generally not the same. Equation (A.2) is not incorrect,
but it does not mean what it might be supposed to: it relates to the behaviour of particles on their own respective
flowlines, which were coincident at time t, not to the behaviour of material at a fixed location at time t.
Now that it is clear why there is no contradiction in section 2.3, the reasons why (A.1) hold can be considered.
A.2
Demonstration of the mutual necessity of (A.1)
Consider material particles N and S at positions x N and x S in the nonwoven and substrate, respectively, at
time t − dt. Require that (x N − x S ) · N̂ = 0, that is that the two particles are both at the interface, and consider
the requirements on the local velocity fields such that the particles coincide at time t,
x N → x N + χ̇n (x N , t − dt) dt,
x S → x S + χ̇s (x S , t − dt) dt,
x N + χ̇n (x N , t − dt) dt = x S + χ̇s (x S , t − dt) dt = x.
(A.7)
(A.8)
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No explanation of the apparent contradiction was offered in section 2.3: it is given here.
364
D. J. COTTENDEN AND A. M. COTTENDEN
Now take the dot product of (A.8) with N̂:
{x N + χ̇n (x N , t − dt)dt} · N̂ = {x S + χ̇s (x S , t − dt)dt} · N̂.
As (x N − x S ) · N̂ = 0, it follows that
χ̇n (x N , t − dt) · N̂ = χ̇s (x S , t − dt) · N̂.
(A.9)
χ̇n (x N , t − dt) · N̂ = χ̇n (x − χ̇n (x − χ̇n (· · · )dt, t − dt)dt, t − dt) · N̂
= {χ̇n (x, t) − dt χ̇n (x, t) − dt∂t χ̇n (x, t) + O(dt 2 )} · N̂
= {χ̇n (x, t) − χ̈n (x, t) + O(dt 2 )} · N̂
(A.10)
Substituting this back into (A.9),
{χ̇n (x, t) − χ̈n (x, t) + O(dt 2 )} · N̂ = {χ̇s (x, t) − χ̈s (x, t) + O(dt 2 )} · N̂
(A.11)
Equation (A.11) makes it clear that for particles on the mutual boundary to remain on the mutual boundary, the
condition of matched normal velocity does not contradict but rather implies matched normal acceleration.
APPENDIX B
Formal derivation of normal force per unit area
A more rigorous but less physical method for obtaining the normal component of stress than in section 2.4 is
simply to calculate (∇ · Tn ) · N̂. Representing T in terms of the derivatives of the surface patch σ,
∇ · Tn = (∇u γ ∂γ ) · (Tn αβ σ,α ⊗ σ,β )
= (∇u γ · σ,β ) Tn αβ,γ σ,α + Tn αβ σ,αγ + (∇u γ · σ,βγ )Tn αβ σ,α
= Tn αβ,β σ,α + Tn αβ σ,αβ + (∇u γ · σ,βγ )Tn αβ σ,α
(B.1)
since ∇u γ · σ,β = δβγ . Consider the last term by defining cβγ δ = σ,βγ · σ,δ
∇u γ · σ,βγ = (∇u γ · σ,δ )cβγ δ = δγ δ cβγ δ = cβγ γ = (σ,βγ · σ, )F I γ .
(B.2)
Substituting this into (B.1),
∇ · Tn = Tn αβ,β σ,α + Tn αβ σ,αβ + (σ,βγ · σ, )F I γ Tn αβ σ,α .
(B.3)
The normal component of the stress force is thus
(∇ · Tn ) · N̂ = Tn αβ σ,αβ · N̂ = Tn αβ F I I αβ ,
where F I I αβ is the second fundamental form of the surface patch σ.
To demonstrate the equivalence of this expression to (2.21), use (3.3) to write it in terms of Cγ δ :
Tn αβ F I I αβ = −Tn αβ F I αγ Cγ δ F I δβ = −Tn αβ (σ,α · σ,γ )(σ,β · σ,δ )Cγ δ
= −Tr({Tn αβ σ,α ⊗ σ,β } · {Cγ δ σ,γ ⊗ σ,δ }) = −Tr(Tn · C).
The two expressions are thus equal.
(B.4)
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In order to avoid the pitfall exposed in section A.1, the flux vectors must be expressed in terms of the location
x. Consider for the moment the left-hand side of (A.9). Recalling that x N = x − χ̇n (x N , t − dt) dt,