## Abstract

Friction testing of microfibre arrays is typically conducted under displacement boundary conditions. For example, after the application of a compressive preload, the translation stage is fixed in the vertical direction while a shear displacement is applied by translating the stage laterally. A nonlinear rod model is used to compute the normal and shear forces acting on a typical microfibre during such a test. The normal load acting on a typical fibre is found to switch from compression to tension as the shear displacement increases. The critical shear force to detach a fibre is found to depend linearly on the compressive preload. The fibre is also found to become more stable as it is sheared, hence it never buckles. On the other hand, instead of fixing the vertical displacement, when a fibre is subjected to constant normal load, it becomes unstable upon shearing. We show that the buckling load (the applied normal load to make a fibre unstable) of a microfibril is reduced by the application of a shear displacement.

## 1. Introduction

Many small animals, e.g. Gecko, use fine hairs on their feet to climb and to stick to surfaces. Their unique ability to climb and to stick to different surfaces has inspired researchers to fabricate surfaces consisting of arrays of microfibres. Many of these man-made surfaces use stiff materials such as carbon-nanotube (CNT) as fibres (e.g. Kinoshita *et al.* 2004; Dickrell *et al*. 2005; Majidi *et al*. 2006; Aksak *et al.* 2007; Ge *et al*. 2007; Schubert *et al* 2008; Qu *et al* 2008; Maeno & Nakayama 2009). Although most stiff fibre arrays show poor normal adhesion, the friction coefficients of these arrays, *μ*, are found to be much greater than flat surfaces of the same material. For example, Dickrell *et al*. (2005) slid vertically aligned CNT (VACNT), grown (average length = 50 μm) on quartz substrate, on a glass hemisphere (7.78 mm radius) and found *μ*≈0.9, whereas *μ*=0.1 for CNT dispersed flat on the same substrate. By sliding a gold tip with apex radii of (4.5–30 μm) on a 6 μm thick VACNT film, Kinoshita *et al*. (2004) found *μ*=1.0–2.2. Aksak *et al*. (2007) have found that *μ* ranges from 0.8 to 1 for VACNT fibres (3, 25 μm average lengths) on a compliant polyurethane (PU) backing. The same trend for sliding friction was observed by Majidi *et al*. (2006) in stiff polymer fibre systems. It should be noted that many friction experiments have been performed on natural systems using setae on the toe of a gecko and on synthetic fibre arrays. In this article, we focus on synthetic surfaces. Readers interested in the friction behaviour of natural systems can consult Autumn *et al*. (2006), Zhao *et al*. (2008), Gravish *et al*. (2009) and the references within.

Adhesion is particularly poor in stiff fibre systems at low preloads because the real contact area is very small owing to unevenness in fibre length and high fibre stiffness. This motivates the fabrication of soft elastomeric fibres that are terminated with either a thin film or spatula to enhance contact and adhesion (Gorb *et al*. 2006; Kim & Sitti 2006; del Campo *et al*. 2007; Greiner *et al*. 2007; Reddy *et al*. 2007; Schubert *et al*. 2007; Shen *et al*. 2008; Varenberg & Gorb 2008; Parsaiyana *et al*. 2009). Many research groups have demonstrated that these arrays have adhesion considerably greater than that of a flat surface of the same material (e.g. Kim & Sitti 2006; Noderer *et al*. 2007). Friction of soft fibre arrays have also been studied by many researchers (e.g. Majidi *et al*. 2006; Bhushan & Sayer 2007; Kim *et al.* 2007*b*; Varenberg & Gorb 2007; Yao *et al*. 2007; Shen *et al*. 2008, 2009; Gravish *et al*. 2009; Vajpayee *et al*. 2009). Varenberg & Gorb (2007) studied the friction of polyvinylsiloxane (PVS) patterned surfaces. Their sample was a circular PVS disc of diameter 2 mm and 1 mm in height. Mushroom-shaped pillars of approximately 100 μm in height bearing terminal contact plates of approximately 40 μm in diameter were distributed uniformly on the surface of the disc. The control specimen was a flat PVC disc of the same geometry. The sample was brought into contact with a flat glass slide that was fixed on a translation stage. They applied a constant normal load *N*_{o} ranging from 40 to 160 mN throughout the test and moved the sample laterally at a velocity of 100 μm s^{−1} for distances ranging from 25 μm to 7 mm. Finally, the translation stage was withdrawn to measure the force required to pull the sample off. Their flat control samples exhibited stick–slip behaviour (large peak shear force followed by a significant drop in force during sliding). By contrast, sliding of their structured samples was smooth and stable. Both structured and flat control samples exhibited a peak friction force; for the structured sample, sliding occurred after a slight drop in friction force from its peak value. If one identifies the peak force as static friction, then static friction of the structured sample was always lower than the flat control. Varenberg & Gorb (2007) determined the coefficient of static friction *μ* using a modified coulomb friction law, which accounts for adhesion; i.e.
1.1
where *F*_{s} is the peak shear force, *F*_{N0}(< 0) is the compressive preload on the sample and *F*_{pull-off}(> 0) is the tensile force required to pull-off the sample. Their data for both flat and structured samples appear to be consistent with equation (1.1), giving *μ*=1.34 and 0.37 for the flat and the structured surfaces, respectively. In §3, we will provide a rationale for the usage of equation (1.1). We will refer to Varenberg & Gorb (2007) as V.G. and Kim *et al.* (2007*a*) as K.A.S. below.

During shear, many of the terminal plates on V.G.’s pillars lost contact with the glass slide, reducing the real contact area. Once the terminal plate losses contact, bending of the pillars occurs as the tangential and normal load is concentrated at the edge. Since the number of terminal plates with bad contact increases with shear displacement (this is because of the increase in normal force acting on the plate, see analysis below), the real contact area decreases with increasing shear displacement. As a result, most of the shear load is borne by pillars that are still adhered to the glass slide. The drop in shear force at the static peak corresponds to the rapid drop in contact area as surviving pillars bear more and more load. During sliding, all pillars are bent and contact the glass slide either on their sides or on their edges. Thus, the pull-off forces measured at the end of their tests were practically zero.

The experiments of V.G. and K.A.S. are actually quite similar, so it is interesting to compare them. Instead of PVS, K.A.S. used PU elastomeric pillars (pillar height, 20 μm; pillar diameter, 4.5 μm) with spatulated tips (diameter, 9 μm). The flat glass plate in V.G.’s experiments was replaced by a glass hemispherical indenter of 6 mm radius. The thickness of backing layer of K.A.S.’s samples was less than 0.1 mm to reduce the effect of backing layer deformation. As shown by Kim *et al*. (2007*b*) and Long *et al.* (2008), backing layer deformation increases stress concentration at the contact edge between the indenter and the fibrillar surface, leading to reduction of adhesion (e.g. pull-off force). As in V.G., K.A.S.’s samples were mounted on a translation stage. In the beginning of an experiment, the stage was moved vertically upwards to make contact with the indenter until a prescribed normal compressive preload (typical preload = –4 mN) was reached; the sample was then translated horizontally with the position of the indenter fixed. They also used a flat PU sample of approximately the same thickness (0.1 mm) as a control. In their flat control experiments, K.A.S. did not observe significant differences between static and sliding friction. In addition, the static friction peak force of their microfibre array was at least three times higher than the peak force observed in their flat control test—in contrast to the observations of V.G. Microscopic images of the microfibres near the static friction peak show why this is the case. Many of the microfibres inside the contact region are well adhered to the indenter, and are highly stretched. Since stretched fibres are in tension, the normal force acting on the indenter switches from compression to tension during shear. The normal force versus time curve of K.A.S. shows that the initial compression drops rapidly to tension as the shear displacement increases. Further increase in shear displacement causes some of the fibres to detach (most probably starting from fibres near the contact edge, which are subjected to higher tension than fibres near the centre of the contact) and the normal force becomes compressive again. Near the static peak, the force is compressive (but still substantially smaller than the preload). The compressive normal force continues to increase after the static peak, consistent with their observation that all the fibres were detached and made side or edge contact with the indenter. Because of poor contact and low fibre density, the pull-off force at the end of their test was also practically zero, similar to the observations of V.G.

It must be noted that since the translation stage maintains a *fixed* vertical separation between the sample and the glass slide during shear; the normal force *F*_{N} acting on the sample will increase with lateral displacement in K.A.S.’s experiment. However, in V.G’s experiments, the normal force is kept constant throughout the test, thus the vertical displacement of the sample will decrease with increasing lateral displacement.

The following friction behaviour of soft fibre arrays is consistent with most experiments:

— Friction of soft elastic microfibre arrays is strongly dependent on adhesion; in general, good adhesion is needed to achieve high static friction. Sliding of the countersurface (e.g. a spherical indenter or a flat hard surface) occurs due to loss in adhesion.

— Sliding friction is usually lower than static friction owing to loss in adhesion which results in side and edge contact. Edge contact leads to a reduction in the true area of contact, which decreases friction and adhesion. While one may argue that side contact will increase the true area of contact, this effect is countered by the fact that soft arrays have low fibre density to avoid lateral collapse; also, the absence of plastic deformation means that the stored elastic energy owing to fibre bending can readily cancel the increase in adhesion and friction energy owing to the increase in contact area, if any.

Friction testing of hard unstructured surfaces is usually conducted under a *fixed normal load* *F*_{N} and the friction coefficient is defined by *μ*=*F*_{s}/|*F*_{N}|, where *F*_{s} is the maximum shear force acting on the surface. However, friction testing of microfibre arrays (soft or stiff fibres) is usually carried out under *fixed normal displacement conditions*. Specifically, after a normal preload *F*_{N0} is applied to bring the surfaces into contact, the normal displacement is fixed during shear. As pointed out in our recent work, fibres in the array can buckle during shear if the normal load is fixed during shear (Nadermann *et al.* 2010). In general, buckling will cause the fibres to detach and will lower the static friction (Shen *et al*. 2009).

While normal adhesion of a microfibre array is well analysed (see, for example, Persson 2003; Hui *et al*. 2004; Gao *et al*. 2005; Chen *et al*. 2008) much less attention has been given to model fibre arrays under shear. Deformation of fibres under shear is typically studied using linear beam theory (Schubert *et al*. 2007), where the deflections are assumed to be small and the effect of normal and shear loads is decoupled. However, in a shear test, fibre deflection can be very large; on the order of the fibre length. These bring into doubt the use of linear theory. In addition, buckling is a nonlinear phenomena that cannot be analyzed using linear theory. In previous works (Liu *et al*. 2009; Nadermann *et al.* 2010), we employed a large deformation rod theory to bypass these difficulties. This theory takes into account large deflection, as well as the extensibility of a microfibril. Rod extensibility is important as the fibres in a test where the normal displacement is fixed are stretched appreciably.

The goal of this work is to use nonlinear rod theory to study how a single fibre in an array behaves during a friction experiment. In §2, we summarize the governing equations of a nonlinearly elastic rod relevant to this work. Numerical results are given in §3. Discussion and summary are given in §4.

## 2. Bending and buckling of a sheared and compressed fibre

In the literature, bending and buckling are sometimes used interchangeably. However, there are very important distinctions between them. Bending is usually a stable deformation; mathematically, this means that a *unique* solution exists for a given set of boundary conditions. This is not the case for buckling. For a fixed set of boundary conditions, multiple solutions *always* exist. Some of these solutions are stable in the sense that their potential energies correspond to local minima in the energy landscape. There can be many local minima; the rod can be trapped in any one of these equilibrium states. A difficulty is to identify which one of the stable equilibrium states the rod will occupy. After an elastic rod buckles, further increase in shear or normal compression leads to post-buckling behaviour. Again, multiple solutions exist in this regime and the rod can assume very complicated configurations.

When an initially straight elastic rod is subjected to *uniaxial* compression, it will remain straight until the Euler buckling load
2.1
is reached. Here, *I* is the moment of inertia of the cross section of a rod about the bending axis; *c* is a numerical constant that depends on the boundary conditions, e.g. *c*=1 for a pinned–pinned rod, *c*=4 for a clamped–clamped rod, *c*=1/4 for a clamped-free rod and *c*=2.046 for a clamped–pinned rod. We shall use the terms ‘rod’ and ‘fibre’ interchangeably in this paper. Also, compressive force is assumed to be negative throughout this work.

The situation becomes much more complex when an initially straight rod is subjected to *both* compression and shear. As the rod is sheared, it bends; bending is stable for sufficiently small applied compression and shear. At some point, the rod buckles. The question is how to compute this critical buckling load. Computing the critical buckling load is non-trivial since one has to solve a nonlinear elasticity problem. One further has to decide which of the solutions in the buckled regime is stable or has the lowest potential energy (Kumar & Healey 2010; Nadermann *et al.* 2010).

We consider two cases that are relevant to the interpretation of experiments. In the first case, we determine the buckling load of a sheared fibre under boundary conditions encountered in typical friction experiments, i.e. under fixed normal load. Post-buckling behaviour of fibres is also briefly examined. In the second, the fibre is subjected to fixed normal displacement. Under this condition, a fibre becomes more stable as it is sheared, hence it never buckles. The later simulates a fibre in a displacement controlled test in which a compressive preload is applied in the beginning. For this case, we show that the normal force on a fibre under initial compression can switch to tension, consistent with the observations of K.A.S.

### (a) Geometry and governing equations

The undeformed rod is assumed to be straight and the arc-length coordinate of a material point on the centre line of the undeformed rod is specified by *S* (figure 1). The position vector of the material point *S* after deformation is denoted as ** R**(

*S*). A local coordinate system with orthonormal basis vectors is attached at each cross section, with aligned with the centre line tangent, while the other two vectors are aligned with the principal flexural axes of a rod’s cross section. This body-fixed frame describes the orientation of a cross section with respect to the inertial frame . The curvature vector

**is the rotation of the body-fixed frame with respect to the inertial frame per unit of undeformed arc-length**

*K**S*, 2.2

The internal moment ** Q** is related to the curvature by a constitutive law for bending. For this paper, the initial rod curvature is zero and we use the standard linear relation:
2.3
Since we have chosen to coincide with the principal torsional–flexure axes of a cross section, the torsion–flexure stiffness tensor

**is diagonal with respect to the body-fixed frame.**

*B*Vector quantities such as the force, ** F**, acting on the cross section of a rod can be expressed in terms of their components once we fix the basis. In the following, quantities such as are three tuples consisting of the three components of these vectors with respect to the body-fixed frame . In the body-fixed reference frame, the static forms of the linear momentum and the angular momentum balance equations for an extensible rod are the following:
2.4and
2.5
Here while denotes stretching of the centre line of a rod. We also have the following equations enforcing unshearability, i.e. line elements perpendicular to the centre line remain so even when a rod is deformed,
2.6

For an elaborate discussion of the equations (2.4)–(2.6) and issues associated with their numerical solution, we refer to Healey & Mehta (2005). The deformation of the fibrillar array while undergoing indentation and friction tests is assumed to be planar, i.e. independent of the out-of-plane coordinate . This assumption is consistent with the loading conditions in shearing and indentation experiments. As a result of this simplification, *F*_{1},*F*_{3},*R*_{1},*R*_{3} and *K*_{2} are the only non-zero physical quantities. Here, *F*_{1} is the shear force acting perpendicular to the centre line tangent while *F*_{3} is the axial force acting along the centre line. In this paper, *F*_{3} is also called normal force as it acts normal to the cross section. A tensile normal force is taken to be positive. As the deformation is restricted to a plane, the angular momentum balance reduces to a scalar equation. Furthermore, as there is no rotation of a cross section about the or axes, the only element of ** B** that comes into play is the bending stiffness

*EJ*about the principal flexural axis . Here,

*E*is the small strain Young’s modulus of the rod and

*J*is the second moment of area. The following four scalar equations survive from their three-dimensional vector counterparts in equations (2.4) through (2.6). 2.7 2.8 2.9and 2.10 Here,

*θ*denotes an angle that the tangent to the centre line makes with the axis and ∂

*θ*/∂

*S*is the curvature

*K*

_{2}. The following normalized variables are defined: 2.11 Here,

*L*is the length of the undeformed rod. Also, with this normalization, the Euler buckling load for a non-stretchable clamped–clamped rod is −4

*π*

^{2}. Similarly, the Euler buckling load for a non-stretchable pinned–pinned rod is −

*π*

^{2}.

A nonlinear constitutive model is used to relate *λ* (stretching of the centre line of a rod) to the axial force *F*_{3} by assuming that the material is neo-Hookean. Under uniaxial tension, the engineering stress *σ* and the stretch ratio *λ* satisfy
2.12
Accordingly, the axial force *F*_{3} is given by
2.13
where *A*_{o} is the undeformed cross-sectional area of a fibre. It can be shown that there exists only one positive real root for *λ* for a given value of axial force *F*_{3} in equation (2.13). The normalized governing equations for an extensible rod are accordingly
2.14
2.15
2.16
2.17and
2.18

To solve the nonlinear partial differential equations (2.14)–(2.18) for the unknowns *r*_{1}, *r*_{3}, *θ*,*f*_{1} and *f*_{3} we use AUTO, a bifurcation software (Doedel 2000). It may be noted that the aspect ratio of the fibre comes into picture only in equation (2.18) as part of the stretch modulus (e.g. for a circular cross section, *A*_{o}*L*^{2}/3*J*=(16/3)(*L*/*D*)^{2}, *D* being the diameter of a fibre). The numerical accuracy of any solver is accordingly affected depending on the aspect ratio. The present model also neglects effect of viscoelasticity, which can be important for some fibre systems. Twisting and shearing of a fibre can also get coupled to stretching and bending, respectively, owing to inherent chirality in the microstructure of a fibre (Kumar & Mukherjee 2010). However, this coupling effect has not been studied in this work.

### (b) Boundary conditions

A reasonable boundary condition is to assume that the bottom end of the rod is *fixed* with respect to the top end, which is in contact with the countersurface. This boundary condition implies that the bottom end can neither rotate nor move. This boundary condition for the bottom end will be assumed throughout this work.

Two types of boundary conditions are used to describe adhesion at the top end of a fibre. If this end is assumed to be *well adhered* to the countersurface, then a *clamped* boundary condition is used. If adhesion is lost at this end, then the rod is assumed to contact the countersurface on its edge, and a *pinned* boundary condition is used. In this work, the pinned boundary condition models the case where there is no adhesion between the top end of the rod and the countersurface. More details about the connection between the pinned boundary condition and buckling of fibres can be found in Hui *et al*. (2007).

A fibre deforms owing to forces and/or displacements applied at the top end. We consider two cases:

Case I: The normal force

*N*is prescribed, corresponding to a test where the normal force is fixed. We also prescribe the tangential displacement*U*at this end.Case II: Both tangential and normal displacements (

*U*,*V*) are prescribed; this corresponds to a displacement-controlled test.

## 3. Results

### (a) A clamped fibre subjected to a fixed normal load and sideway displacement

We first present the solution of a well-adhered fibre whose top end is subjected to a fixed normal load and is sheared sideways. The assumption that the fibre is well adhered to the countersurface implies that this end is clamped. Figure 2*a*,*b* plots the normalized shear force *t* and negative of normalized vertical displacement −*v*=−*V*/*L* versus the normalized shear displacement *u*=*U*/*L* for several different normalized compressive loads *n*. Note that a vertical displacement (upward in the paper) that creates tension is taken to be positive. Similarly, a tangential shearing of a fibre to the right is taken to be positive while shear force at the top end acting to the right is considered positive. All forces are normalized by the Euler buckling load *P*_{Euler} for a clamped–clamped rod, e.g. *t*=*T*/|*P*_{Euler}|and*n*=*N*/|*P*_{Euler}|. Specifically, *N* and *T* are the values of *F*_{3} and *F*_{1} evaluated at the top end of the rod. Figure 2*b* shows that the downward displacement of the rod (fibre) increases as the shear displacement increases, implying that the normal compliance increases with shear displacement, as shown in an earlier work by Liu *et al*. (2008). In figure 2*a*,*b*, the solid curves represent the non-buckled stable solutions, whereas the dotted lines represent unstable buckled states. The dotted lines in figure 2*a* show how applied shear displacement reduces the buckling load. Figure 2*a* shows that the shear force is positive for sufficiently small compression. According to our sign convention, a positive shear force is in the same direction as the applied shear displacement. As compression increases, e.g. *n*=−0.48, shear force changes its sign from positive to negative, that is, the shear force acts in the direction opposite to the applied shear displacement. This observation is counterintuitive. In fact, an argument is given by Nadermann *et al.* (2010) explaining why under clamped–clamped boundary conditions, shear force is opposite to the shear displacement whenever the magnitude of normal compressive load at the top end is greater than the Euler buckling load corresponding to the pinned–pinned case. We refer the readers to Nadermann *et al.* (2010) for its elaborate discussion.

The dependence of the critical buckling load (the superscript cc denotes clamped–clamped boundary conditions) on the shear displacement is shown in figure 3. Here, *n*^{cc}_{b} denotes the dimensionless buckled load, i.e. *N*^{cc}_{B} normalized by the Euler load for a clamped–clamped rod. It shows that shearing of a fibre reduces its buckling load. We mention two possible scenarios associated with normal loading of a fibre. If a fibre is well adhered to the countersurface (e.g. a rigid indenter), then buckling will occur under the clamped–clamped boundary condition. Once buckling occurs, a fibre will be subjected to large rotation, which may cause peeling of the fibre end (Kim *et al.* 2007*a*). If this occurs, the boundary condition at this end will change suddenly (from clamped to pinned). Since the applied normal load is fixed, this sudden change will give a post-buckled shape corresponding to that of a clamped–pinned rod. The other possibility is that loss of rotation constraint occurs earlier; before the rod buckles (e.g. if adhesion is weak). This will induce a buckling instability if the applied load is higher than the buckling load of a clamped–pinned rod. The significance of these results is that the normal stiffness of a fibre drops suddenly to zero as it buckles. In a fibre array indented by a fixed normal force, this sudden loss of normal stiffness will shed normal load to unbuckled fibres, causing some of them to buckle; as a result, the contact area of the indenter can increase suddenly during shear (e.g. if the countersurface is a spherical indenter). This increase in contact area will bring more fibres into contact and may cause stiffening of the shear force versus shear displacement curve in experiments (Shen *et al*. 2009). If however, all the fibres underneath the indenter collapse (which may occur for a flat indenter), then the normal contact compliance of the sample is determined by the compliance of the backing layer (Shen *et al*. 2008). For example, if the backing layer is thick when compared with the indenter radius, then the contact compliance is approximately determined by the standard JKR theory.

Figure 3 also plots the normalized critical buckled load *n*^{cp}_{b} (superscript cp denotes clamped–pinned boundary condition, divided by the magnitude of Euler buckling load for a clamped–clamped case) versus normalized shear displacement. As expected, for the same applied shear displacement, the buckling load for the clamped–pinned boundary condition is always lower than the buckling load for the clamped–clamped case. In particular, figure 3 shows that for the same normal load, the minimum shear displacement necessary to make a rod unstable is very low in the case of the clamped–pinned rod. Figure 4 shows the first unstable configuration of a clamped–clamped rod and a clamped–pinned rod as the rod is being sheared (subjected to the same normal load, *n*=−0.44).

### (b) A clamped fibre subjected to fixed normal and shear displacement

As mentioned in §1, K.A.S. observed the normal force on their sample change from compression to tension during their displacement-controlled friction test. We examine the deformation of a typical fibre (before detachment) in such an array using the following boundary conditions for the top end: (i) the fibre is clamped (good adhesion), (ii) a fixed normal displacement *V* is applied, and (iii) a sideways shear displacement *U* is imposed. Figure 5*a*,*b* plots the normalized normal force *n* and normalized shear force *t, respectively,* versus the normalized shear displacement *u* for different *initial* normalized preload *n*_{o}. The values of the initial preload for different curves in figure 5*a* can be read by looking at the intersection of these curves with the *y*-axis. Note that a fixed normal displacement *V* (*downward in the beginning*) corresponds to preloading the fibre. Specifically, the initial preload *N*_{o} is related to the applied normal displacement *V* by
3.1
In experiments, the preload is sufficiently small so that the rod does not buckle, i.e. |*N*_{o}|<|*P*_{Euler}|≡4*π*^{2}*EI*/*L*^{2}.

Figure 5*a* shows that the normal force, acting on a fibre at its top end, changes from compression (< 0) to tension (> 0) as the fibre is sheared. As expected, at a fixed preload, the tension *n* increases with the applied shear displacement. This is also consistent with the observation of K.A.S. Our numerical results show that a fibre, in fact, becomes more stable as it is sheared. At some critical shear displacement, which we denoted by *U*_{c}(*N*_{o})(*u*_{c}(*n*_{o}) normalized), *N* reaches a critical tensile value, *N*_{c}(*n*_{c}), causing detachment. When this happens, the fibre will snap from the countersurface and release tension. It is difficult to predict the equilibrium configuration of the rod at the end of this episode as the final equilibrium configuration depends on the dynamics of detachment and the details of contact and friction. Three configurations are possible: (i) the rod is pinned at the top but does not buckle, (ii) the rod is pinned and buckled, and (iii) a portion of the rod is in side contact with the countersurface. It may be mentioned that in order to have side contact, the rod must move down appreciably so that it can bend a lot and hence make a side contact with the counterface. Usually in a displacement-controlled test, the vertical movement of a fibre is restricted and hence the rod is more likely to make an edge contact. However, if the detachment occurs for fixed normal load test, the fibre could make side contact as the fibre can now move down vertically by an appreciable amount. In any case, after the fibre snaps, the rod will be under compression, consistent with the experiments of Kim *et al*. (2007*a*). Fibre snapping should cause a jump in the normal force as it suddenly changes from tension to compression.

Figure 5*b* shows a very interesting feature of our solution, for sufficiently large preloads (e.g. the three lowest curves)—the shear force first decreases (negative) for small shear displacement, reaches a minimum, then eventually becomes positive. Since the tangential displacement is always increasing in the simulation, our result shows that *initially* the shear force on the fibre is in the *opposite* direction to the applied shear displacement. Thus, the slope of the shear load versus the shear displacement is *negative* in this regime. This feature was also observed in the experiment of Shen *et al*. (2009).

As pointed out in §1, equation (1.1) was used by V.G. to interpret their friction data. Some insight into this equation, in a displacement-controlled test, can be obtained by understanding how the shear force *T* varies with *N*, especially in the regime where *N*≥0 (tension). Figure 6 plots the normalized shear force *t* versus the normalized normal force *n*. Each curve in figure 6 corresponds to a normalized preload *n*_{o}. A vertical line in this figure represents a constant *N* or *n*. If we identify this value of *N*(*n*) with *N*_{c} (*n*_{c}), then the critical shear force *T*_{c} (*t*_{c}) associated with fibre detachment is the intersection of this vertical line with these curves. Figure 6 shows that *T*_{c} increases with *N*_{c}. Also, the critical shear force *T*_{c} increases as more compressive preload is applied in the beginning.

Figure 7 shows how the normalized critical shear force *t*_{c} depend on *n*_{o} and *n*_{c}. Note that for a fixed critical normal force *n*_{c}, the critical shear force is a *linear function* of the initial compressive preload. The numerical results in figure 7, although for the displacement-controlled case, agrees well with equation (1.1). A mathematical derivation is carried out in the appendix, which further establishes validity of equation (1.1) and numerical results in figure 7.

## 4. Summary and discussion

We computed the deformed configuration and forces acting on a typical fibre in a microfibre array as it is sheared by a countersurface (e.g. a spherical indenter) using nonlinear rod theory that takes into account large deflection of fibres associated with their stretching as well as bending. We show that the normal force acting on a fibre switches from compression to tension during a typical displacement-controlled friction test. It is also found that a fibre remains stable as it is sheared during a displacement-controlled test, in fact a fibre gets more stable upon shearing. It must be emphasized that, in an actual fibre array, fibres are subjected to slightly different boundary conditions (even though the indenter is rigid and hence all fibres are subjected to the same sideway displacement). However, because of the finite curvature of the indenter, fibres are subjected to different normal forces during preload. In addition, the shape of each fibre is slightly different (e.g. fibre length) and adhesion to the countersurface varies from fibre to fibre. As a result, loads are not equally distributed among fibres, and not all fibres detach at the peak shear load. This feature is evident from the experiments of K.A.S. Nevertheless, the prediction of our model is in qualitative agreement with experimental observation.

We also consider the situation where a fixed normal load acts on a fibre. In this case, shearing reduces the buckling load. The dependence of the buckling load on the amount of shear displacement is determined numerically. Post-buckling analysis was carried out to study the deformed shape of these fibres. In the experiments of V.G., the sample is in contact with a flat glass substrate. If the sample and the glass substrate are perfectly aligned and the fibres are identical in length, then in theory, the friction force of their sample can be computed using our single fibre theory. The fact that shearing reduces the buckling load suggests that in a constant normal load experiments, the fibres in the array can buckle during shear. This would result in loss of adhesion leading to a much lower static friction coefficient, despite the fact that the fibres in V.G. and K.A.S. are quite similar. We hope our analysis can lead to better interpretation of friction testing of microfibre arrays.

## Acknowledgements

C.Y.H. is supported by the US Department of Energy, Office of Basic Energy Science, Division of Material Sciences and Engineering under Award (DE-FG02-07ER46463).

## Appendix A

Here, we carry out a mathematical derivation to determine how critical shear load, required to pull-off a single fibre in a displacement-controlled test, depends on initial preload (*n*_{o}) and the prescribed vertical displacement (*u*). From dimensional considerations, the normalized shear force and the normalized normal force must be a function of *n*_{o} and *u*, that is,
A1a
and
A1b
where *f* and *g* are functions determined by solving the rod equations. Assume that the normal force necessary to detach a fibre is a material constant for a given type of fibre; that is, detachment occurs when *n*=*n*_{c}. Using equation (A 1*b*) and the critical shear displacement *u*_{c}(*n*_{o}) at which detachment occurs, for a given preload *n*_{o}, we get
A2
According to equation (A 1*a*), the critical normalized shear force at detachment is
A3
Consider the situation where *n*_{o} is zero, there is no normal preload. Denote physical quantities associated with this case by *t*_{c0},*u*_{c0}, respectively. For small preloads, we can expand *t*_{c}=*f*(*n*_{o},*u*_{c}(*n*_{o})) about *n*_{o}=0, this gives
A4
Equation (A4) has the form
A5
where
A6
Physically, we expect *χ*<0 since we expect the critical shear force to increase with decreasing preload or as more compressive preload is applied (figure 7). The dimensional form of equation (A5) is
A7
Note that equation (A7) has the same mathematical form as equation (1.1) if we identify |*χ*| with *μ*, *T*_{c} with *F*_{s}, *T*_{c0}/|*χ*| with *F*_{pull-off}. Finally, (d*u*_{c}/d*n*_{o})|_{no=0} can be evaluated using equation (A2), that is, since *n*_{c} is constant, we have
A8
It is interesting to note that the ‘friction’ coefficient |*χ*| is controlled by the adhesion of the fibril to the countersurface. Equation (A7) illustrates the intimate relation between friction and adhesion.

- Received August 25, 2010.
- Accepted October 25, 2010.

- This journal is © 2010 The Royal Society