The capstan equation or, as otherwise known, Euler's equation of tension transmission, is the relationship governing the maximum applied tension T_max with respect to the minimum applied tension T₀ of an elastic string wound around a rough cylinder. The governing equation is given by

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 is an engineering term implying that the contact area exhibits friction. Note that the coefficient of friction is the physical ratio of the shear force and the normal force between two contacting bodies. In engineering, the capstan equation describes a body under a load, in equilibrium, involving friction between rope and a wheel-like circular object, and thus it is widely used to analyse the tension transmission behavior of cable-like bodies in contact with circular profiled surfaces (Jung et al., 2008; Baser and Konukseven, 2010) as well as in the field of robotics (Behzadipour and Khajepour, 2006).

As an example of the application of a capstan-type equation to experimental results, consider the results of Cottenden et al. (2008a) where the coefficient of friction is determined from experiments as that shown in Figure 1 of a non-woven fabric strip in contact with various arm phantoms made from Plaster of Paris and covered in Neoprene. Note, in particular, Figure 11 of Cottenden et al. (2008a) which shows the coefficients of friction obtained with different geometries, applied weights, and contact angles. While the capstan equation proves quite successful in obtaining coefficients of friction, the authors observe a steady increase in the mean coefficient of friction as the applied weight increases. Such dependence contradicts the assumptions of Equation (1). There is also variation in the coefficients of friction measured across different geometries and contact angles which seems to get wider as the applied mass is reduced. The authors acknowledge the apparent dependence of the coefficient of friction on the applied weight in Cottenden et al. (2008a) but highlight that the mean variation is small compared to the scatter of the data. The authors suggest that the departure from the capstan equation is likely to be caused by an interaction between the Neoprene on the underlying body and the moving nonwoven fabric at large tension.

The raw data of the case of a fabric strip in contact with an arm phantom of cylindrical cross section with a 127360π contact angle can be found in Karavokiros' masters thesis (see table 2a of Karavokiros, 2007); for convenience, this raw data is reproduced here in Table 1. The table shows five repeated measurements involving dragging the fabric strip over the arm phantom with different applied masses (m in Figure 1). Figure 3 is a plot using these data of the measured tension ratio δτ = T_max/T₀, vs. the applied mass, where the mean ratio obtained for each applied mass is also shown. Note that Equation (1) implies that the tension ratio is constant for all applied masses, i.e., δτ = exp(μ_dθ₀), where μ_d and θ₀ are constants. However, Figure 3 suggests as the applied mass increases, the tension ratio increases also. Of course, such an effect could be attributed to a number of factors, including experimental errors, but one possibility is that the standard capstan equation is simply not valid in these cases. To test this, we now develop two, more sophisticated, three-dimensional numerical models of a thin compliant sheet over an underlying body and vary the geometry and material properties of both the thin sheet and underlying body, as well as the applied tensions.

In this section, we derive a pure-traction belt-friction model to describe the behavior of dynamic membranes supported by static rigid foundations.

Let ω ⊂ ℝ² be a simply connected open bounded domain with the sufficiently smooth boundary ∂ω and let σ ∈ C²(ω; E³) be an injective immersion. Now assume that an isotropic elastic membrane is in contact with a rigid surface with positive mean curvature initially such that, in the stress-free configuration of the membrane, the contact area can be parameterized by the immersion σ(x¹, x²) in curvilinear coordinates. Now we assert that the membrane is dynamic but the contact area remains static and constant as ω. Invoking a similar terminology to that of Cottenden and Cottenden (2009), the governing Cauchy momentum equations for the membrane stress τ^αβ (α, β = 1, 2) in tangential and normal directions can be derived thus

with the boundary conditions

where u ∈ C²(ω; ℝ²) is the displacement field and ϱ is the mass density of the membrane, gr∈C0(ω;ℝ3) is an external loading field, frβ(u) are the shear force densities, fr3(u) is the normal reaction density, n ∈ C⁰(ω; ℝ²) is the unit outward normal to the boundary and τ0∈C0(ω;ℝ2) is the traction field applied to the boundaries of the membrane. The covariant second fundamental form tensor of σ with respect to the curvilinear coordinates is defined as

where N=||∂1σ×∂2σ||-1(∂1σ×∂2σ) is the unit normal to the surface σ, × is the Euclidean cross product and || · || is the Euclidean norm.

Notice that this is a pure-traction problem. This implies that the boundary traction field τ₀ cannot be arbitrary chosen. To proceed, we assume that the velocity, the acceleration and the force density fields are known and fixed prior to the problem. Typically here, in line with the tensometer experiments, we assume that the membrane moves at a given constant speed. The time-dependent translational part of the displacement is thus easily subtracted out, leaving the residual displacement u^β. We now invoke the compatibility condition for pure-traction problems (see Section 1.3.4 of Necas et al., 2011 or Section 1.8 of Ciarlet, 2000) to find

where ϵ(·) is the strain tensor of a true-membrane. The compatibility condition implies that the internal forces are balanced by all the applied external forces.

Suppose now that the contact area is rough and the friction law governing this region is given by the model in chapter 10 of Kikuchi and Oden (1988) for Coulomb's law of static friction for the slip case. For a flat contact surface with normal (compressive) and tangential stress fields given by σ_n and σ_T respectively, Coulomb's law of static friction can be expressed by the following relationship between these surface stresses and the tangential displacement field u_T:

Here, ν_F is the coefficient of friction in respect to Coulomb's law of friction. Kikuchi and Oden point out that it is not possible to mathematically analyse a variational problem for Coulomb's law using conventional mathematical methods. As a result, the question of existence of solutions to such friction problems remains an open one. To circumvent this difficulty, Kikuchi and Oden propose the following regularized version of the static friction law involving a small parameter ε:

In the limit ε → 0, Coulomb's original law of static friction is recovered. For the slip case, we assume the first condition holds for large tangential displacements which, on converting this to a membrane in contact with a rigid surface in curvilinear coordinates, yields

We now rearrange the compatibility condition Equation (5) and use Coulomb's law of friction Equation (8) to find a relationship between the coefficient of friction and the external loadings, given by

The residual displacement u^β and frj(u) give us five unknowns and Equations (2), (3), and (8) provide us with five equations. Thus, the system is fully determined with boundary conditions Equation (4). Furthermore, Equation (9) provides us with a stronger system as if the coefficient of friction is unknown then a known traction can close the system and vice versa.

While it is a closed system, it is not straightforward to prove that a solution exists for this model. The problem of proving the existence of solutions arises from the function frβ(u), as it is a function of both u and ∇u. If frβ(u) was purely a function of u, then the existence of solutions may be proved by variational methods for semi-linear elliptic equations (see Badiale and Serra, 2010) but with the ∇u dependence, our model is not even a variational problem. Nevertheless, a numerical finite-difference solution is pursued in the next section.

To conduct numerical experiments, assume that we are dealing with a surface of revolution case where both the contact surface and the unstressed membrane are parameterized by the same immersion. Let this immersion be σ(x1,x2)=(x1,φ(x1) sin(x2),φ(x1) cos(x2))E, where x¹ ∈ (0, l) and x2∈(-12π,0). To permit changes in lateral curvature we assert that φ(x1)=r0-16c(l-1x1-12)4, where, initially, to keep the contact area as a surface of positive mean curvature, c is a positive parameter with c < r₀. Note that l, r₀ are some positive real constants that are specified later. With some calculations, we find the first fundamental form tensor to be F[I]=diag((ψ1)2,(ψ2)2), where ψ1=(1+(φ′(x1))2)12 and ψ2=φ(x1). With a few more calculations one can find that

where Γ​αβγ are the Christoffel symbols of the second kind and F_[II] is the second fundamental form tensor. Now given that u = (u¹(x¹, x²), u²(x¹, x²)) is the displacement field, one can derive the following:

Now assume that our membrane is subjected to the acceleration of gravity i.e., subject to (0, 0, −g)_E in Cartesian coordinates. With coordinate transforms, from Euclidean to curvilinear, one may re-express acceleration due to gravity in the curvilinear coordinates as gJ, where

and x² is the angle that the vector (ψ₁, ψ₂, 0) makes with the vector (0, 0, 1)_E.

Now given that ϱ is the mass density, (0, 0, 0) is the acceleration field and (0,(ψ2)-1V,0) is the velocity field of the membrane, then we can express the governing equations of the membrane as

and

where Λ = 2λμ(λ + 2μ)⁻¹ and λ and μ are the first and second Lamé parameters respectively. Assuming that our contact area is rough, the final governing equation required is the friction law Equation (8) where the coefficient of friction ν_F is considered to be an unknown.

Now divide the boundary into sub-boundaries, so that

and we assert that the boundary conditions are

where τ_max > τ₀ are positive real constants.

Finally, we take Equation (9) and modify it ever so slightly to make it easier to numerically model, thus obtaining the following relation,

Now we are ready to conduct some numerical experiments. Our goal in this section is to investigate how variables such as the Gaussian curvature, Young's modulus, Poisson's ratio, speed and the mass density of the membrane and the traction may affect the value of the coefficient of friction obtained. Note that for our experiments, we keep the values τ₀ = 1, l = 1, r₀ = 1 and g = 9.81 fixed.

To conduct numerical experiments, we employ a second-order accurate finite-difference method in conjunction with Newton's method for nonlinear systems. On discretising the domain, as we are dealing with curvilinear coordinates, we find that Δx2≤ψ0Δx1, for all ψ0∈{ψ1/ψ2∣x1∈[0,l]} where Δx^β is a small increment in the x^β direction. For our purposes we use Δx2=1N-1 and ψ0=ψ1/ψ2|x1=12l, where N = 250. We also choose to terminate our iterating process once |1 − ν_Fm+1/ν_Fm| falls below a certain specified tolerance, where ν_Fm is the m^th iterative solution for the coefficient of friction. Note that to numerically model Equation (10), we use the prismoidal formula (Meserve and Pingry, 1952). Unfortunately, as this is a pure-traction problem iterative schemes can be highly unstable and so, to ensure convergence, the condition u2|∂ωT0≤0 is strictly enforced.

We initially ran the numerical code using the following values: τmax=32, c = 0, E = 10³, ν=14, V = 0.01, ϱ = 0.01 and with a grid of 160 × 250 points. The grid size N was varied to confirm accuracy of the converged solution. The coefficient of friction calculated in this case is ν_F = 0.195 to three significant figures. We now proceed to investigate how the variation in certain parameters may change this value and thus see whether this differs from the value predicted by the classical capstan model Equation (1).

We begin by varying the tension applied to the membrane. Figure 4 is calculated with the values of τ_max ∈ {1.25, 1.30, 1.35, … 2.00}, c = 0, E = 10³, ν=14, V = 0.01 and ϱ = 0.01. It shows that as the tension ratio increases, the coefficient of friction also increases. This is intuitive because, as the maximum applied tension increases, the coefficient of friction must increase to maintain a constant speed. Equation (1) predicts a similar trend and so agreement with the classical capstan model seems quite good although the figure shows that the numerical code produces consistently lower values. The principal reason for the discrepancy is the non-zero mass density of the fabric which is investigated below.

Varying Poisson's ratio ν, the Young's modulus E and the speed of the membrane V do not lead to any significant changes in the coefficient of friction. But when we examine varying the mass density of the fabric we find significant alterations to the coefficient of friction determined by the model. Figure 5 is calculated for three different applied tensions: τ_max ∈ {1.50, 1.75, 2.00}, with values c = 0, E = 10³, ν=14, V = 0.01 and ϱ ∈ {0, 0.003, 0.006, …, 0.03}. This shows as the mass density (with respect to the volume) of the membrane increases, the coefficient of friction decreases markedly in all cases. As ρ → 0 the coefficient of friction obtained from Equation (1), e.g., (2/π) ln(3/2) = 0.258… , is attained, but even a small mass density, ρ = O(10⁻³) for instance, can lead to the accuracy being significantly reduced to only one significant figure. This feature is ignored in typical capstan equation calculations.

Finally, let us examine what effect varying the Gaussian curvature K has on the coefficient of friction. Note that, for our given geometry, K is a function of x¹, and thus it varies across the contact surface. As our curvature parameter c changes the Gaussian curvature changes across the contact surface. For c = 0, K = 0 everywhere. For c > 0 the contact surface is a barrel-shape with non-negative (i.e., positive or zero) Gaussian curvature whereas for c < 0 the contact surface is a saddle-shape with non-positive (i.e., negative or zero) Gaussian curvature across the contact area. This implies that there exists a positive correlation between c and K, i.e., as c increases so does K and so we investigate how c relates to ν_F. Figure 6 is calculated for non-negative Gaussian curvatures with the values of τ_max ∈ {1.50, 1.75, 2.00}, and c∈{0,140,240,340,440,540,640,740,840,940}, E = 10³, ν=14, V = 0.01, and ϱ = 0.01. The three plots in the figure for different tension ratios demonstrate that as c increases (i.e., Gaussian curvature increases) the coefficient of friction decreases and there exists an optimum value of c, where the coefficient of friction is at a minimum, given that all other variables are constant. For our simulations, this value is observed around c=18. The initial decrease in ν_F as c increases is intuitive. To illustrate this, consider a membrane pulled over a surface with high curvature. The higher curvature would leads to higher normal reaction force which, in turn, results in a higher friction force and finally a higher maximum applied tension. Also inclusion of a nonzero lateral curvature (note that we are considering a case with two positive principal curvatures) means that a higher maximum applied tension is required to support the strains in the lateral direction. Thus, for even a relatively small coefficient of friction, a higher maximum applied tension can be observed. Hence, if one kept every variable fixed, except for the Gaussian curvature and the coefficient of friction, then one would expect to see a low coefficient of friction for a high Gaussian curvature. However, the existence of a minimum ν_F for positive c is a surprising outcome suggesting a possible optimum Gaussian curvature that can lead to a minimum coefficient of friction. Conversely, it is possible numerically to extend our range of c values to include {-140,-240,-340,-440} that represent saddle-type contact regions. These simulations suggest that such saddle-type contact regions lead to significantly higher values of ν_F than for the zero-curvature case (Figure 7). Extending that range yet further to c = −0.02 (not shown) suggests that no maximum coefficient of friction is attained but that the coefficient of friction continues to increase as c becomes more negative. Such intriguing variations in ν_F captured by our model here cannot be simulated by the classical capstan model Equation (1).

Recall Kikuchi and Oden's model (chapter 10 of Kikuchi and Oden, 1988) for Coulomb's law of friction for slip that we extended to curvilinear coordinates Equation (8). For the static case, by eliminating the regularization parameter ε from their original Equations, (6) and (7), the friction law can be expressed in the following form

Unlike the rigid substrate case, this equation is extremely difficult to impose on a free contact boundary between shell-membrane and substrate in its present form. Therefore, a more computationally tractable displacement-based approximation is sought. Assume that u^β ≥ 0 and contract the above equation with u^β. Noting that in our framework g₃₃ = 1, we thus find

Now assume that this body is in contact with an elastic foundation, thus it permits normal displacements at the boundary, so we assert that only normal derivatives are of any consequence. Hence, we may approximate the above relation as

To simplify matters further, we assert that the condition is independent of any elastic properties of the overlying body. This may be achieved by assuming λ = 0, and thus we find

By approximating the above condition even further, we arrive at the following hypothesis:

Hypothesis 1. A shell supported by an elastic foundation with a rough contact area, that has thickness small relative to the radius of mean curvature, satisfies the following displacement-based friction condition

where ν_F is the coefficient of friction between shell and the foundation, and u is the displacement field of the shell. If  2νFu3+(uαuα)12<0, then we say that the the shell is bonded to the foundation, and if  2νFu3+(uαuα)12=0, then we say that the shell is at limiting equilibrium.

The justifications for introducing the hypothesis are as follows. Kikuchi and Oden (1988) assert that the variational form of a linear elastic body subjected to Coulomb's law of static friction, is non-convex and non-differentiable. Thus, the existence of a (unique or otherwise) solution is an open question that cannot be proven with conventional means. But we can show that the variational form, i.e., the energy functional of a linear elastic body, subjected to the displacement-based friction condition from this hypothesis is convex, coercive and differentiable, and thus proving the existence of solutions is perfectly possible. Also, unlike the model of Kikuchi and Oden (1988), our displacement-based condition is independent of the regularization parameter ε and it is well defined for all finite values of u. Furthermore, we show that our problem can be numerically modeled without an initial guess of the purely normal stress, which is something Kikuchi and Oden's model is incapable of. Note, however, that we can guarantee that the condition from our hypothesis will hold as we have already asserted that the lower-surface of the shell is not hyperbolic and its mean curvature is positive. Thus, for sensible boundary conditions, we can always expect the normal displacement to be non-positive.

Does our hypothesized displacement-based friction law make physical sense? Well, consider two elastic blocks in a zero-gravity scenario, block-A at the top and block-B at the bottom, where the base of block-B satisfies the zero-Dirichlet boundary condition. Now assume that the contact area of both blocks is rough and that one is applying forces to both the top and to one side of block-A to mimic respectively compression and shear in the contact region. The higher the compression, the higher the normal displacement is toward the bottom, i.e., u³ < 0, and, the higher the shear, the higher the tangential displacement is in the direction of the applied tangential force. Just as for the case of Coulomb's friction, where the bodies are in respective equilibrium given that the magnitude of the normal stress is above a certain factor of the magnitude of the tangential stress, i.e., |T33(u)|>νF−1|T3α(u)Tα3(u)|12, we assert that the bodies are in equilibrium given that the normal displacement is below a certain factor of the magnitude of the tangential displacement, i.e., u3<−12νF−1(uαuα)12, if u3<0. Note that this factor may or may not be 12νF-1, but this is the most mathematically logical factor we have derived.

Let Ω ⊂ ℝ³ be a connected open bounded domain that satisfies the segment condition with a uniform C¹(ℝ³; ℝ²) boundary ∂Ω such that ω, ∂Ω₀ ⊂ ∂Ω with ω̄ ∩ ∂ Ω0̄= Ø and meas(∂Ω0;ℝ2)>0, and let ω ⊂ ℝ² be a connected open bounded plane that satisfies the segment condition with a uniform C¹(ℝ²; ℝ) boundary ∂ω. Let X̄∈C2(Ω̄;E3) be a diffeomorphism and σ∈C3(ω̄;E3) be an injective immersion. Let f ∈ L²(Ω), f0∈L2(ω) and τ0∈L2(∂ω).

For this section, we assume that u ∈ C²(Ω; ℝ³), uα|ω∈C3(ω), u3|ω∈C4(ω) and 2νFu3+(uαuα)12≤0 everywhere in ω. For the elastic foundation we define Tij(u)=AijklEkl(u) to be the second Piola-Kirchhoff stress tensor, Eij(u)=12(gik∇̄juk+gjk∇̄iuk) to be the linearized Green-St Venant stress tensor, Aijkl=λ̄gijgkl+μ̄(gikgjl+gilgjk) to be the elasticity tensor, λ̄=(1-ν̄-2ν̄2)-1ν̄Ē as the first Lamé parameter, μ̄=12(1+ν̄)-1Ē as the second Lamé parameter, Ē as the Young's modulus and ν̄ as Poisson's ratio of the elastic foundation. Furthermore, f is some external force density field acting on the elastic foundation. The covariant first fundamental form tensor of σ with respect to the curvilinear coordinates is defined as

Also we regard the indices α, β, γ, δ ∈ {1, 2}. Furthermore, F[I]αγF[I]γβ=δαβ, ∀ α, β ∈ {1, 2}. The second fundamental form tensor is as defined above in Section 2.2.

The governing equations of the elastic foundation are given by

with the following boundary conditions:

where n̄ is the unit outward normal to the boundary ∂Ω in curvilinear coordinates.

Turning now to the overlying body, we assume that the body is so thin, and the bending moments are so small, that it can be approximated by a shell-membrane. For the shell-membrane we define ταβ(u)=Bαβγδϵγδ(u) to be the stress tensor and ηαβ(u)=Bαβγδργδ(u) to be the negative of the change in the moments density tensor.

is then half the change in the first fundamental form tensor,

is the change in the second fundamental form tensor,

is the elasticity tensor, λ = (1 − ν − 2ν²)⁻¹ νE is the first Lamé parameter, μ=12(1+ν)-1E is the second Lamé parameter, E is the Young's modulus and ν is Poisson's ratio of the frictionally coupled shell. Tr(Tj3(u))=Tj3(u)|ω is the normal stress of the foundation in the contact region, and f₀ is some external force density field acting on the overlying shell.

The governing equations for the shell-membrane are determined by considering a variational problem based on the following energy functional:

where u is subject to the displacement-based friction condition Equation (11) and the region in which the shell is at limiting equilibrium is unknown prior to solving the problem. Following the detailed analysis in chapter 4 of Jayawardana (2016), it is possible to obtain the following governing equations for the shell-membrane:

if [2νFu3+(uαuα)12]|ω<0 (the bonded case), then

if [2νFu3+(uαuα)12]|ω=0 (the limiting equilibrium case), then

where u¯|ω=(u1,u2,−12νF−1(uαuα)12)|ωand (∂3u¯1,∂3u¯2,∂3u¯3)|ω=(∂3u1,∂3u2,∂3u3)|ω. Finally, the boundary conditions of the overlying shell are

where n is the unit outward normal vector to the boundary ∂ω in curvilinear coordinates and τ₀ is the external traction field acting on the boundary of the overlying shell.

To conduct numerical experiments, assume that we are dealing with a shell-membrane of thickness h, supported by an elastic foundation, where the unstrained configuration of the foundation has an annular cross-section, characterized by the diffeomorphism X¯(x,θ,r)=(x,r sin(θ),r cos(θ))E,where (x1,x2,x3)=(x,θ,r),x∈(−L,L),θ∈(−π,π], and r∈(a0,a), and assume that the contact region lies within x ∈ (−ℓ, ℓ), θ∈(-12π,0), where 0 < ℓ < L. Let the sufficiently smooth field u = (u¹(x, θ, r), u²(x, θ, r), u³(x, θ, r)) be the displacement field of the foundation. With some calculations one can find the metric tensor to be g = diag(1, r², 1) and the covariant derivatives to be

With further calculations, one can express the governing equations of the foundation as

The boundary of the foundation can be decomposed into sub-boundaries as

Thus, we can express the boundary conditions of the foundation as

Let u|ω=(u1(x,θ,a),u2(x,θ,a),u3(x,θ,a)) be the displacement field of the shell-membrane. With some calculations, one can find the first fundamental form tensor to be F[I]=diag(1,a2) and the covariant derivatives to be

Considering the case h2ραγ(u)ργα(u)≪ϵαγ(u)ϵγα(u), with some further calculations one can express the governing equations of the shell-membrane as:

If [2νFu3+(u1u1+r2u2u2)12]|ω<0, then

If [2νFu3+(u1u1+r2u2u2)12]|ω=0, then

The boundary of the shell-membrane can be decomposed into sub-boundaries as

Thus, we can express the boundary conditions of the shell-membranes as

A second-order accurate finite-difference method is again employed in conjunction with Newton's method for nonlinear systems. A modestly fine grid is chosen and the iterative process is terminated once |1-||um+1||ℓ2/||um||ℓ2| falls below a certain value (10⁻¹⁰ in the calculations shown here). We attempt to model a stiff shell-membrane on a relatively flaccid foundation with a large coefficient of friction. To do so, we keep the values ν_F = 1, h = 0.001, a = 1, ℓ=14, L=12, E = 10³, ν=14, ν̄=0, τ₀ = 1 and τ_max = 2 fixed for all experiments. Note that some preliminary work in chapter 4 of Jayawardana (2016) found that if (i) the overlying body is relatively thin, (ii) it has a relatively high Young's modulus, (iii) the coefficient of friction is high, and (iv) the mean curvature is a constant, then the solution for the shell model with friction is in relatively good agreement with a numerical model using Kikuchi and Oden's original friction law in chapter 10 of Kikuchi and Oden (1988), thus justifying the choice of our parameters.