Effective macroscopic adhesive contact behavior induced by small surface roughness

Journal of the Mechanics and Physics of Solids 59 (2011) 2488–2510

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Effective macroscopic adhesive contact behavior induced by small surface roughness Haneesh Kesari, Adrian J. Lew  Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-4040, United States

a r t i c l e in f o

abstract

Article history: Received 8 April 2011 Received in revised form 18 July 2011 Accepted 30 July 2011 Available online 16 August 2011

In this paper we study a model contact problem involving adhesive elastic frictionless contact between rough surfaces. The problem’s most notable feature is that it captures the phenomenon of depth-dependent-hysteresis (DDH) (e.g., see Kesari et al., 2010), which refers to the observation of different contact forces during the loading and unloading stages of a contact experiment. We specifically study contact between a rigid axi-symmetric punch and an elastic half-space. The roughness is represented as arbitrary periodic undulations in the punch’s radial profile. These undulations induce multiple equilibrium contact regions between the bodies at each indentation-depth. Assuming that the system evolves so as to minimize its potential energy, we show that different equilibrium contact regions are selected during the loading and unloading stages at each indentation-depth, giving rise to DDH. When the period and amplitude of our model’s roughness is reduced, we show that the evolution of the contact force and radius with the indentation-depth can be approximated with simpler curves, the effective macroscopic behavior, which we compute. Remarkably, the effective behavior depends solely on the amplitude and period of the model’s roughness. The effective behavior is useful for estimating material properties from contact experiments displaying DDH. We show one such example here. Using the effective behavior for a particular roughness model (sinusoidal) we analyze the energy loss during a loading/ unloading cycle, finding that roughness can toughen the interface. We also estimate the energy barriers between the different equilibrium contact regions at a fixed indentation-depth, which can be used to assess the importance of ambient energy fluctuations on DDH. & 2011 Elsevier Ltd. All rights reserved.

Keywords: A. Adhesion and adhesives B. Contact mechanics B. Elastic material

1. Introduction The topography of solid surfaces can span many length scales, ranging from its physical dimensions to the size of its molecular constituents. On engineered surfaces it is possible to restrict topography variations to below a certain length scale that is small when compared to the physical dimensions of the body. Typically, these small-scale variations are of the order of micrometers. In microfabricated devices, these can be smaller than a nanometer. These small-scale variations are generally referred to as surface roughness. The effects of surface roughness in some scenarios, however, are far from small. Surface roughness has been known to govern a number of important phenomena, such as friction, adhesion, and wear during mechanical contact between surfaces (e.g., see Bowden and Tabor, 2001; Briggs and Briscoe, 1976). It is also known

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E-mail addresses: [email protected] (H. Kesari), [email protected] (A.J. Lew). 0022-5096/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2011.07.009

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to govern surface wetting, contact angle hysteresis (Jung and Bhushan, 2006), biocompatibility and electro-optical behavior (Assender et al., 2002). More recently, it was found that small-scale surface topography is also at the root of some very interesting surface phenomena observed in biological systems such as optical iridescence (Parker and Townley, 2007), lotus effect (Barthlott and Neinhuis, 1997), petal effect (Feng et al., 2008), and locomotion in geckos (Autumn et al., 2000; Gao and Yao, 2004). Recently, it was shown by Kesari et al. (2010) that surface roughness can give rise to depth-dependent-hysteresis (DDH). A typical contact experiment consists of measuring the force between two bodies as they are moved towards (loaded), and then away (unloaded) from, one another. Classical contact models, such as Hertz (1881) or JKR (Johnson et al., 1971), predict that the contact forces between bodies depend primarily on the separation between them irrespective of the history of the contact process. In some experiments, however, the contact force is found to additionally depend on whether the bodies are being moved towards or away from one another. These observed differences in the measured contact forces for a given separation between the bodies are what we refer to as DDH. Depth-dependent hysteresis was usually thought to appear due to effects such as the material’s plasticity or viscoelasticity, or to moisture on the surfaces. Kesari et al. (2010) showed that DDH could be caused by adhesion and roughness alone. They further showed that roughness could actually increase the energy loss during contact due to DDH. This energy loss scales with the energy required for separating a pair of bodies in contact, and therefore, an increase in the energy loss due to DDH is equivalent to an adhesive toughening of the contact interface. In this paper we present a model contact problem that captures the DDH due to adhesion and roughness. It showcases a mechanism by which adhesion and roughness cause DDH. Furthermore, it also captures the adhesive toughening effect of surface roughness. We pay special attention to the solutions of this model problem for small roughness sizes, for which we derive an approximate or effective macroscopic behavior in the limit of small roughness size. The model contact problem we consider is between an axi-symmetric rigid punch and an isotropic linear elastic halfspace (see Fig. 1). The rigid punch’s radial profile is assumed to be made of a large-scale and a small-scale topography. We take the small-scale topography to be an arbitrary smooth periodic function that acts as a model for the composite roughness of the contacting surfaces. The contact is assumed to be frictionless but adhesive. We also assume that the contact region is always simply connected. Adhesion is modeled as done in the JKR theory of contact, see Section 2.1. We first derive a parametric equation between the equilibrium contact force and indentation-depth in terms of the equilibrium contact radius. We then reduce the equilibrium contact force-indentation-depth equations to a simpler, asymptotic form by using that the period of the small-scale topography of the punch is small in comparison to the radius of the contact region, and by scaling the amplitude of the small-scale topography together with its period. We term this the small roughness size limit. This simpler form enables us to perform a lot of the subsequent analysis. The equilibrium indentation-depth and contact force equations show that for a prescribed indentation-depth, there may be many configurations of the half-space that are in equilibrium. Each of these locally stable configurations corresponds to a different equilibrium contact force and contact radius (the radius of the contact region). In fact, the number of different equilibrium configurations at each indentation-depth grows as the period of the small-scale topography is decreased. We proceed further and consider a hypothetical experiment in which the half-space is first loaded to a maximum indentation-depth, and then unloaded until it detaches from the punch. The existence of multiple equilibrium contact regions for each indentation-depth makes it necessary to clearly define what the evolution of its radius is as the loading/ unloading of the half-space proceeds. To this end, we formulate the evolution of the system by arguing that, at each indentation-depth, the contact radius always evolves in the direction in which the potential energy decreases. This definition naturally leads to different evolutions of the contact radius with the indentation-depth upon loading and unloading. Correspondingly, different evolutions for the contact force as a function of the indentation-depth appear as well, and this is what we call the ‘‘measured’’ contact force-indentation-depth curve. The appearances of different evolutions of the contact radii upon loading and unloading are the essential reason behind the onset of DDH. It is then possible to identify the effective evolution of the contact force and contact radius in the small roughness size limit. The effective evolution does not display oscillations with the period of the small-scale topography of the measured curve, and is uniformly close to the latter at essentially all indentation-depths. Therefore they can be used to model the

Undeformed half-space surface

Rigid axi-symmetric punch

r

h Deformed surface z

a

Elastic isotropic half-space

Fig. 1. Sketch of the contact problem studied in this paper. The symbols h and a denote the indentation-depth and the contact radius, respectively.

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macroscopic behavior of the problem. Remarkably, only the amplitude and period of the small-scale topography participate in this effective behavior. The equations for the effective contact force as a function of the indentation-depth have a simple algebraic form, and hence can be used to estimate material properties, such as Young’s modulus and adhesion energy, from contact experiments displaying DDH, see Section 4.3. Currently, classical contact models, such as Hertz, JKR, DMT (Derjaguin et al., 1975), Maugis (1992), are being routinely used to interpret experiments displaying DDH. However, these classical contact models do not capture DDH, and therefore, only the loading or the unloading phase of the experiment can be fit to these models at a time. This leads to different estimates for material properties depending on which phase of the experiment is chosen for the fitting. In contrast, data from both the loading and unloading phases of the experiment can be simultaneously fit to the effective contact force equations presented in this work, see Section 4.3. This is especially attractive since it leads to unique estimates for the material properties. Finally, the possibility of observing the postulated evolutions would depend, among other things, on the size of the energy barriers between different equilibrium configurations with different contact radii for any given indentation-depth. With this in mind, we also obtain a coarse estimate for the size of the energy barriers in the limit of small roughness size. This estimate can be used to check whether or not roughness is a likely factor to be considered while interpreting DDH in an experiment. There are several studies reported in the literature that present models for contact between rough surfaces. Some of these models do capture DDH but are lacking in one or more aspects, which we discuss below. Greenwood and Williamson (1966) were among the first to study the effect of roughness in contact mechanics. They modeled contact between rough surfaces as the contact between a smooth surface and an ensemble of non-interacting asperities. In their work, the contact between each asperity and the rigid body was assumed to be a Hertzian-type contact, which is non-adhesive and frictionless. Fuller and Tabor (1975) extended Greenwood and Williamson’s (1966) approach to study the effect of roughness during contact between adhesive surfaces. They did this by modeling the contact between the smooth punch and each asperity as a JKR-type contact, which is adhesive and frictionless. There have been several other models proposed by adopting ideas similar to Greenwood and Williamson’s (1966) models, e.g., see the works of Greenwood and Tripp (1967), Refs. [11]–[21] in Bhushan (1998), and Wei et al. (2010). These models are collectively referred in the literature as asperity-type contact models. Apart from the model of Wei et al. (2010), none of the other asperity-type contact models capture DDH. In these other models roughness is ignored when studying the evolution of the contact region. The model in Wei et al. (2010) is the same as that in Fuller and Tabor (1975), but in the former they additionally describe a DDH mechanism. This mechanism is similar to the one proposed by Zappone et al. (2007). However, Wei et al. (2010) additionally provide equations for the measured contact forces predicted by their model. Notably, none of the asperity-type contact models capture the effect of adhesive toughening due to roughness. This is because, due to the assumption of non-interaction between asperities, these models do not apply at small roughness sizes, and this is precisely the regime where the experiments in Kesari et al. (2010) indicate that roughness can cause adhesive toughening. The contact problem studied in this paper captures the adhesive toughening effect of roughness, primarily because we assumed that the contact region is always simply connected. This assumption is expected to be valid in the regime where roughness is small (Johnson, 1995). We evaluate the adhesive toughening by estimating the energy loss in a complete loading/unloading cycle. We do this through the use of the effective evolution of the contact force as a function of the indentation-depth. Recently, Li and Kim (2009) considered the adhesive contact problem between two half-spaces, in which one of them was elastic and the other one was rigid with two-dimensional sinusoidal undulations on its surface, assumed to represent its roughness. They showed that surface topography could give rise to more than one possible equilibrium configuration at a given separation between the bodies. They discussed that as the system transited between such states through surface instabilities it would dissipate energy leading to an effective toughening of the contact interface. The mechanism through which small-scale roughness gives rise to DDH herein is the same as that causing adhesive toughening in Li and Kim (2009). Similar surface instabilities appear in the axi-symmetric contact problem studied in Guduru (2007), which considered JKR-type adhesion between an elastic half-space and a rigid paraboloidal punch with superimposed sinusoidal undulations. A primary outcome of this work was that surface topography can increase the maximum adhesive force between contacting bodies. Additionally, Guduru also notes that the surface instabilities lead to an effective toughening of the interface. The mixed boundary value problem we solve in this work is a generalized version of the problem considered in this reference. It is from the solution of this problem that we then obtain the effective macroscopic response. In this work we show that DDH could essentially be the smeared out effect of a large number of surface instabilities, of the type found in Guduru (2007) and Li and Kim (2009), occurring at a much smaller scale. A further distinguishing aspect of our work is that we analyze and derive the effective contact behavior of the system as the roughness size is decreased. The structure of the paper is as follows, we describe the problem and its solution in Section 2. By using Sneddon theorems and the relations given by Barquins and Maugis (1982) we derive the equilibrium contact force and indentationdepth as a function of the contact radius. The approximate or asymptotic representation of these curves is obtained in Section 2.3. In Section 3 we discuss how DDH arises in this model problem. We first highlight the multiple equilibrium configurations for each indentation-depth in Section 3.1, and then define the evolution in Section 3.2. The effective evolutions or approximations to the measured contact force-indentation-depth curves are obtained in Sections 3.3 and 3.4.

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In Section 4, we particularize these ideas for a model problem involving a paraboloidal punch with sinusoidal roughness. For this problem we compute the energy loss during the loading/unloading cycle in Section 4.1, estimate the energy barrier between neighboring stable equilibrium configurations in Section 4.2, and conclude by showcasing how the results here are used to fit some experimental results from the literature in Section 4.3. Throughout the paper we have included the main ideas behind the proofs of some of the statements in sections labeled ‘‘Justification,’’ which could be skipped by the uninterested reader. 2. The model contact problem 2.1. Geometry and governing equations A sketch of the contacting bodies, the axi-symmetric rigid punch and the elastic half-space, is shown in Fig. 1. To describe the deformation of the half-space, we adopt a cylindrical system of coordinates ðr, y, zÞ in which the z-axis coincides with the axi-symmetry axis of the punch, and is positive towards the half-space. In its undeformed configuration, the elastic half-space O is the set of points for which z 4 0, while its surface Gs is defined as the set of points where z ¼0. The configuration of the punch is completely specified by its radial profile u~ z ðr; hÞ, where h, the indentation-depth, denotes its z-displacement. We assume that there is clear separation of the length scales describing the punch’s radial profile. More precisely, we take the punch’s radial profile to be of the form, u~ z ðr; hÞ ¼ h þf ðrÞ þ lRðr=lÞ,

ð2:1Þ

where the function f(r) describes the large-scale topography of the punch. For concreteness, this function will be assumed to be of the form f ðrÞ ¼ r a =Ra1 for a Z1 and R 40, which include the cone and the paraboloid. However, similar results to those shown here apply to more general profiles, as long as some conditions on f and its derivative are imposed. The smallscale topography of the surface is described by a parameter l and the function R, which is assumed to be a non-constant, smooth (for simplicity C 1 , but this condition can be substantially relaxed) periodic function with period 1 and such that Rð0Þ ¼ 0. We will also assume that R r 0, given that this makes the description of the pull-in instability simpler, but it is not an essential assumption. We study here the behavior of the problem discussed next as l-0. We describe the deformation of the elastic half-space with a displacement field u: O-R3 , which in a standard cylindrical basis has components ur, uz, and uy , but uy ¼ 0 because of the axi-symmetry. All displacements are measured with respect to the initially undeformed configuration. The half-space is assumed to be an isotropic, initially unstressed, linear elastic solid with Young’s modulus E and Poisson’s ratio n. The plane strain elastic modulus will be denoted En ¼ E=ð1n2 Þ. The surface of the punch and Gs are assumed to adhere to each other upon contact, as described below. In the following we are going to consider the following loading–unloading program of the half-space by the punch. Starting from an initial position just before touching Gs (h o 0), the punch is moved towards the half-space (loaded) until it reaches a maximum indentation-depth hmax . The unloading phase then begins and continues until the punch and the halfspace are no longer in contact. Owing to the adhesion, the two bodies may remain in contact when h o 0. The contact region Gc is the set of points in Gs which lie on the surface of the punch in the deformed configuration, more precisely, Gc ¼ fðr, y, z ¼ 0Þ9uz ðr, y, z ¼ 0Þ ¼ u~ z ðr; hÞg. In this work we assume that the contact region is always simply connected. Consequently, the contact region will be a circular region. We refer to its radius as the contact radius a. For a given indentation-depth, the solution of this contact problem consists in determining a contact radius a and a corresponding displacement field u for which the potential energy of the system is locally minimized for a class of small variations in a and u. This gives rise to two conditions which have to be simultaneously satisfied. The first is that given h and a the displacements of the half-space satisfy the mixed boundary value problem, divðrÞ ¼ 0 uz ¼ u~ z tz ¼ 0

in O,

ð2:2Þ

on Gc ,

ð2:3Þ

on Gs \Gc ,

ð2:4Þ

tr , ty ¼ 0

on Gs ,

ð2:5Þ

where r is the Cauchy stress tensor, tr ,ty ,tz are the components in the standard cylindrical basis of the traction vector t ¼ r  n, with n being the unit outward normal to O on Gs . It is also required that u  Oððr 2 þz2 Þ1=2 Þ and r  Oððr 2 þ z2 Þ1 Þ as r 2 þ z2 -1. The second condition is that a be a local minimizer of the potential energy, Z 1 Pða; hÞ ¼ tz uz dGpwa2 , ð2:6Þ 2 Gc where for each a, tz and uz are the solutions of the mixed boundary value problem defined in (2.2)–(2.5), and w is the Dupre´’s work of adhesion (Maugis, 2000, p. 30). The first term in (2.6) is the elastic energy stored in the half-space, while the second term is the adhesion energy of the contact interface. This methodology of incorporating an adhesion energy term in

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the potential energy of the system to model adhesive contact was first used by Johnson et al. (1971) using a thermodynamic approach. It was later generalized by Barquins and Maugis (1982) using a fracture mechanics analogy. After determining a and u, the contact force P is computed as, Z tz dG: ð2:7Þ P¼ Gc

Considering the co-ordinate system in our problem, a positive P signifies a compressive force between the bodies through which they push each other, and a negative P signifies an attractive force between the bodies through which they pull each other. Finally, for the solution to be admissible, the half-space surface should not touch the punch away from Gc , i.e., uz 4 u~ z

almost everywhere on Gs \Gc :

ð2:8Þ

2.2. Solution A procedure for solving the mixed boundary value problem in (2.2)–(2.5) was given by Sneddon (1965). Using that procedure we conclude that Z a ^ hÞ da, ^ Pða,hÞ ¼ En p wða; ð2:9aÞ 0

where the function wð; hÞ is an inverse Abel transform of u~ z ð; hÞ, and takes the form " # Z a^ 2 u~ 0z ðr; hÞ ^ hÞ ¼ wða; h þ a^ dr p ^ 2 r 2 Þ1=2 0 ða

ð2:9bÞ

for a^ 40, where u~ 0z ¼ @u~ z =@r. For a ¼0, PðaÞ ¼ 0 and hðaÞ 2 ð1,0Þ. Eqs. (2.9a) and (2.9b) are essentially a single equation in the three variables h, a, and P. An additional equation, relating h and a, is obtained from the necessary condition needed for a to be a stationary point of Pða; hÞ with respect to a, i.e., 0¼

@P p2 En ða; hÞ ¼ wða; hÞ2 2paw: @a 4

ð2:10Þ

The solution uz that gives rise to (2.9) and (2.10) is not guaranteed to satisfy the admissibility condition (2.8). Close to the boundary of the contact region, admissibility is guaranteed by setting (Maugis, 2000, p. 214)

wða; hÞ r0:

ð2:11Þ

However, it is evident that without changing the solution above, the shape of the punch for r 4 a can be changed so that the admissibility condition is violated. It is of course possible to check whether condition (2.8) is satisfied a posteriori, i.e., once the form of the punch profile is specified. For additional details and a simplified derivation of the procedure used for obtaining (2.9) and (2.10) see Kesari and Lew (2011). It follows from (2.9) and (2.10) that for u~ z ðr; hÞ as in (2.1) and a 40 we can write Z a 0 R ðr=lÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dr, hðaÞ ¼ hM ðaÞa ð2:12Þ a2 r 2 0   Z a 0 Z a R ðr=lÞ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi dr þ R ðr=lÞða2 r 2 Þ1=2 dr , ð2:13Þ PðaÞ ¼ PM ðaÞ þ 2En a2 a2 r 2 0 0 where rffiffiffiffiffiffiffiffiffiffiffiffiffi Z a 2pwa f 0 ðrÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dr, a 2 r 2 En a 0

hM ðaÞ ¼ 

 Z PM ðaÞ ¼ 2En hM ðaÞa þ

a

 f 0 ðrÞða2 r 2 Þ1=2 dr :

ð2:12aÞ

ð2:13aÞ

0

Again, for a¼0 we have Pð0Þ ¼ 0 and hð0Þ 2 ð1,0Þ. In hM and PM we have lumped all the terms in (2.12) and (2.13) that depend solely on the large-scale topography of the punch. Thus, PM–hM will be the equilibrium P–h curve when there is no small-scale topography on the surfaces. For the class of punches under consideration, f ðrÞ ¼ r a =Ra1 with a Z1, we have rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi  a a 2pwa p aGða=2Þ R , ð2:14aÞ hM ðaÞ ¼  þ n E 2 Gða=2 þ 1=2Þ R pffiffiffiffi   a a þ 1  p aGða=2Þ R2 , PM ðaÞ ¼ 2En hM ðaÞa R 4 Gða=2þ 3=2Þ where GðxÞ is Euler’s gamma function.

ð2:14bÞ

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2.3. An approximate representation for the equilibrium P–h curve The equilibrium P–h curve (2.12) and (2.13) often displays several interesting features, even for simple choices of R, as we showcase in Section 4. However, considering that R is a model for roughness, it is possible to compute a simpler approximate representation of the P–h curve by considering the limit when l 5a. This is done in Appendix B, where we show that (2.12) and (2.13) can be written as rffiffiffiffiffiffiffiffiffi pal hðaÞ ¼ hM ðaÞ rða=lÞ þaOðl=aÞ, ð2:15Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðaÞ ¼ PM ðaÞ 2pa3 lEn rða=lÞ þ a2 Oðl=aÞ,

ð2:16Þ

as l=a-0. Here r ¼ L D1=2 R, the semi-derivative of R, see Appendix B. The function r is continuous and periodic of period 1, satisfies that r ¼ 0 if R is a constant, and it has a zero mean value over its period. At least formally, the semi-derivative of R can be expressed using the Fourier transform F over R as h i r ¼ F 1 ðıxÞ1=2 F ½RðxÞ : ð2:17Þ We give specific examples of R, r in Sections 4 and Fig. 6. The identities in (2.15) and (2.16) provide approximate equations for the P–h curve when l=a is small enough. Therefore, in this limit the equilibrium P–h curve can be approximated as rffiffiffiffiffiffiffiffiffi pal rða=lÞ, ð2:18aÞ hðaÞ  hM ðaÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi PðaÞ  PM ðaÞ 2pa3 lEn rða=lÞ:

ð2:18bÞ

A length scale l for the roughness can be estimated through roughness characterization. It is usually, however, not possible to a priori know the order of magnitude of the contact radius for arbitrary punch profiles, since as we shall see later, it may depend on the loading program as well. For a wide class of punches a grows as the indentation-depth does, and hence the approximations in (2.18) will be useful for h large enough. For the initial stages of loading, the value of a is often correlated to the natural length scales of the problem, R and w=En . Consequently, these should be much larger than l for (2.18) to be a good approximation in this case. Examples comparing the P–h curves given by (2.12) and (2.13) and (2.18a) and (2.18b) are shown in Figs. 2 and 6. As expected, it can be seen that the approximation improves as l decreases. Since we are interested in the limit of small roughness size, we will use the simpler approximate P–h curve given by (2.18a) and (2.18b) to analyze the contact process in the remainder of the paper. We begin by studying the appearance of depth-dependent hysteresis (DDH) next.

3. Depth-dependent hysteresis due to roughness 3.1. The equilibrium P(h) curve is multiply valued In typical contact mechanics problems, such as those studied in Maugis and Barquins (1983), the equilibrium P–h curves always display a unique value for the equilibrium contact force for any indentation-depth (h 40). Consider, for example, the equilibrium P–h curve of the JKR contact problem (Johnson et al., 1971) shown in Fig. 3(b). Only when h o 0 two values of P can be found for each h. A similar observation can be made for r ¼ 0 and any a Z0 in the profiles considered here. The introduction of ra0, however, changes this observation substantially. In fact, for l small enough, the number of 1=2 values of P for each value of h grows at least as l with l. The appearance of multiple values, and the increase in its number as l decreases, is evident from the plots in Figs. 2 and 6. Justification: The essential reason behind it is illustrated next. For the class of punches under consideration, for each h 40 the equation hM ðah Þ ¼ h has only one solution ah, and h0M ðah Þ 4 0 in a neighborhood of ah. Let then r þ ¼  min r, and for simplicity, assume that r þ 4 0. Then, for l small enough set Da 40 so that rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi plðah DaÞ þ r ¼ h, ð3:1Þ hM ðah DaÞ þ 2 1=2

which clearly exists due to the monotonicity of hM. It then follows from here that Da ¼ Oðl Þ as l-0. Let then a0 be such that ah Da oa0 oa0 þ l o ah and rða0 =lÞ ¼ 0. Clearly the number of such points grows at least as Da=l, or l1=2 , as l-0. Furthermore, notice that for any such point pffiffiffiffiffiffiffiffiffiffiffi ð3:2Þ hða0 Þ ¼ hM ða0 Þ pla0 2rða0 =lÞ ¼ hM ða0 Þ ohM ðah Þ ¼ h,

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0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0

0

−0.02

−0.02

−0.04

−0.04

−0.06 −0.2

−0.1

0

0.1

0.2

0.3

−0.06 −0.2

−0.1

0

0.1

0.2

0.3

0.1

0.2

0.3

0.08 0.06 0.04 0.02 0 −0.02 −0.04 −0.06 −0.2

−0.1

0

Fig. 2. Comparison of the P–h curve (2.12) and (2.13), (thick grey curve) with its asymptotic form (2.18a) and (2.18b) (thin black curve). The model for roughness R is taken to be a pure sinusoid as given in (4.2), and the punch is a paraboloid as described in (4.1). In all P–h curves, A¼ 0.14, 2pw=REn ¼ 0:05, and the maximum contact radius, amax =R, is 0.6. The parameter lEn =w is 12.6 in (a), 6.3 in (b), and 1.3 in (c).

since hM is increasing there. Notice as well that there exists a1 2 ða0 ,a0 þ lÞ such that rða1 =lÞ ¼ r þ . For any such point a1 we have pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3:3Þ hða1 Þ ¼ hM ða1 Þ þ pla1 2r þ 4 hM ðah DaÞ þ plðah DaÞ2r þ ¼ h: Hence, hða0 Þ o hða0 þ lÞ o h ohða1 Þ,

ð3:4Þ a1þ

hða17 Þ ¼ h.

and since r and hM are continuous, we conclude that there exist 2 ða0 ,a1 Þ and 2 ða1 ,a0 þ lÞ such that Since 1=2 as mentioned earlier the number of such intervals grows at least as l , it follows that as l-0 the number of roots of hðaÞ ¼ h for each h grows at least in the same way. For each interval we get the values of Pða17 Þ, and hence the P(h) curve is 1=2 multivalued with the number of values of P for each h growing at least as l as l-0. A similar argument can be made with respect to the maximum value of r, and for the parts of the hM curve with h0M o0. Fig. 4, referenced later, can be useful in interpreting this explanation. a 1

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0.02 0.01

G E A

0

F C

−0.01 −0.02

D

B K J

−0.03

H

I

−0.1

−0.05

0

0.05

0.1

0.15

0.1

0.15

0.02 0.01 C E

0 −0.01

A

D B

−0.02 −0.03 −0.1

−0.05

0

0.05

Fig. 3. Effect of the model’s roughness on the equilibrium and measured P–h curves in a displacement controlled experiment, shown in (a). In contrast, (b) shows the result when no roughness is considered. In both cases the equilibrium P–h curves are shown as grey curves with alternating solid and dashed segments. The solid and dashed segments denote stable and unstable set of equilibrium states, respectively. Closed and open symbols mark stable and unstable states on the equilibrium P–h curve, respectively. The measured curves are shown as thin black curves with alternating solid and dashed segments. In this case the dashed segments represent mechanical instabilities. These curves were plotted by using (4.3) and (4.4) with the sinusoidal roughness R given in (4.2) and the paraboloidal punch in (4.1). We set l=R ¼ 0:03 and 2pw=REn ¼ 0:05, and let A ¼0.1 in (a) and A ¼ 0 in (b).

3.2. Model of the evolution resulting in depth-dependent hysteresis (DDH) The presence of multiple static equilibrium configurations for each value of h raises the question of which force will be measured in a hypothetical contact experiment. To this end, the quasi-static evolution for the system needs to be defined, or modeled. This is what we do next. Notice first that the equilibrium P–h curve is parameterized by the contact radius. So each point in the curves shown in Figs. 2 and 6 corresponds to a distinct contact radius and hence to a distinct configuration of the contacting bodies. To define the evolution, it is necessary to specify how the contact radius a evolves as h is changed. To illustrate the ideas, we first discuss the case r ¼ 0, a ¼ 2, which corresponds to the JKR problem. The measured P–h curve for this case has been discussed in earlier works (e.g., Johnson et al., 1971; Maugis, 2000, pp. 266–272). Following Maugis, we see from the equilibrium P–h curve, Fig. 3(b), that there is a unique value for P for all h except for some negative values of h, for which there are two possible values for P. Of these, the zero value corresponds to the bodies being out of contact (a¼0) and the negative value to the bodies being in contact (aa0). The experiment measures the zero values (E–A) during loading and the negative values (B–D) during unloading. For h 40, the experiment measures the same set of forces (B–C) during both loading and unloading. The instances of sudden drop (A - B) and increase (D - E) in the measured force are called the pull-in and pull-off instabilities, respectively. It is possible to interpret this evolution from an energetic point of view. The idea is that whenever at a given indentation-depth h the contact radius a 4 0 is not in equilibrium according to (2.10), then the contact radius will evolve in

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0.15

0.1

G E

F

0.05 C 0

A

D H

B J

−0.05

I

K

−0.1 0

0.1

0.2

0.3

0.4

0.5

0.04 0.03 0.02 0.01

G E

0

A

F C B

−0.01

D

K

−0.02

J −0.03

H I

−0.04 0

0.1

0.2

0.3

0.4

0.5

Fig. 4. Approximate envelopes for the indentation-depth (top) and contact force (bottom). Their measured evolutions during an indentation-depth controlled loading and unloading process are shown as well. The equilibrium indentation-depth and contact force as functions of the contact radius are shown with thick gray lines. Solid and dashed lines indicate stable and unstable equilibria, respectively. The approximate envelopes P 7 and h 7 are shown in thin black lines. The measured evolution of the contact radius a þ ðhÞ(resp., a ðhÞ) during the loading (resp., unloading) process is shown in solid pink (resp., blue) lines (top). The evolutions a 7 ðhÞ are strictly increasing functions for a 40, with possible jump discontinuities (shown with dashes) whenever dh=da ¼ 0. The corresponding measured evolution of the contact force is shown at the bottom as Pða þ ðhðaÞÞÞ (resp., Pða ðhðaÞÞ), with the dashed portions connecting the values at both sides of each discontinuity. These curves were obtained from the example in Fig. 3(a), so corresponding points are 7 7 labeled with the same letters. For simplicity, the effective evolutions aap ðhÞ and P 7 ðaap ðhÞÞ are shown in Fig. 5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

the direction in which the potential energy of the system is minimized, until it reaches a value that satisfies (2.10), and of course, the admissibility condition (2.11). At the initial loading stage h o 0 we have a¼0, and any pull-in instability would require a finite sized perturbation of the system. As soon as h 40, from (2.9b) we have that lima-0 þ wða; hÞ ¼ 2 h=p 4 0. Contact configurations for which wða; hÞ 4 0 are not admissible, i.e., they do not satisfy (2.11). For h4 0, the smallest admissible contact radius satisfies wða; hÞ ¼ 0, and for such contact radius we can see from (2.10) that @P=@a o 0. The contact radius then grows from here until it reaches the first and in this case unique stable equilibrium, point B. As shown in Appendix A, portions of the P–h curve for which dh=da 4 0 are locally stable, since @2 P=@a2 40 therein. Consequently, while the system lies along the D–B–C curve in Fig. 3(b), small perturbations of the contact radius at each fixed h lead to increases in the potential energy of the system. When reaching point D upon unloading, however, any further decrease in the value of h leaves a ¼0 as the only equilibrium configuration. As shown in Appendix A, at any

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equilibrium configuration ða; hðaÞÞ with aa0 1 @P ða; hðaÞ þ DhÞ  En pwða; hðaÞÞ o 0, Dh @a

ð3:5Þ

since wðaÞ o 0 therein by (2.10) and (2.11). This implies that for Dho 0, the contact radius a should decrease to follow the direction in which the potential energy of the system decreases. It does so until it reaches complete detachment at a¼ 0. With these ideas in mind, we proceed to define the evolution in the presence of roughness, as shown in Fig. 3(a). Beginning with h o0 small enough so that the two bodies are not in contact, the indentation-depth is increased. The minimum value of the contact radius here is zero, and it remains that way until it reaches h¼ 0 (point A). Any further increase in the value of h leads to a sudden jump in the size of the contact region, similar to the JKR case. As shown in the figure, there are several equilibrium configurations at h¼0. Those for which the value of a is a local minimizer of the potential energy are shown with closed circles, while local maximizers are showcased with open marks. These are the stable and unstable equilibria, respectively. Following the idea that, when out of equilibrium, the contact radius evolves so as to lower the value of the potential energy of the system at each fixed h, the new configuration of the contact regions will be that in point B. This is the first stable equilibrium configuration that the growing contact radius will find starting from a¼0. Notice that point B is also the configuration with the smallest stable equilibrium contact radius. As the loading proceeds, the radius of the contact region grows continuously along the thin black line. Since dh=da 40, these are all stable equilibria. It does so until it reaches an unstable equilibrium when the equilibrium P–h curve folds back with respect to h, i.e., dh=da ¼ 0 (point C). According to (3.5) then, for any Dh 40 the potential energy decreases as a grows, and hence the contact radius grows until it reaches the first stable equilibrium, point D. The loading stage continues with several of these discontinuities in the P–h curve appearing in an almost periodic fashion. The loading program finishes at point G, where h ¼ hmax . The unloading stage begins with a specially long interval of values of a in which discontinuity in the P–h curve appears. As a decreases the configuration of the two bodies at F is recovered. However, all configurations along the curve G–F–H are stable equilibria, so the system evolves along it as a decreases. This is similar to what is found for JKR at point B. At this point the loading and unloading stages begin to differ, giving rise to hysteresis. After h decreases below point H, a number of instabilities are found at which there are sudden decreases in the contact radius similar to that found at point D in the JKR description. For example, at point I, any further decrease in h by Dh o0 leads the contact radius to decrease to follow the direction in which the potential energy does, as (3.5) implies. The decrease in the size of the contact radius stops at the first stable equilibrium, in this case point J. The unloading process finishes with a final pull-out instability at K. This is similar to the one at D in the JKR problem in which the value of a decreases suddenly from a positive value to zero. Clearly, the loading and unloading curves are different, giving rise to hysteresis. The curves meet on the left where the bodies are out of contact, and on the right where h¼hmax. Taken together they form a loop whose area gives the energy lost in the experiment. Since the size of the unloading path and hence the loop depends on hmax, the energy loss depends on hmax as well. Thus, the model for the evolution of the contact radius captures DDH. As formulated, the evolution does not account for the possibility of skipping local stable equilibria due to thermal fluctuations. These will clearly be relevant in many circumstances, and we provide a brief discussion about it in Section 4.2. Finally, the same ideas could be applied to other types of experiments, in which for example the displacement is not exactly controlled, or a different type of loading/unloading program is sought.

3.2.1. Evolutions of the contact radius during loading and unloading The evolution of the contact radius during loading and unloading can be described with functions a þ ðhÞ and  a ðhÞ, respectively. These are the evolutions seen during this hypothetical experiment, and hence we call them measured evolutions. A simple characterization of these curves is that during the loading stage the radius of the contact region grows as little as possible, while during the unloading stage the radius of the contact region decreases as little as possible. More precisely, for h(a) in (2.12), the evolution of the radius of the contact region during the loading stage is ( min fa Z0: hðaÞ ¼ hg if h 40 a þ ðhÞ ¼ ð3:6Þ 0 if h r0: This function embodies the idea that the radius of the contact region grows as little as necessary during the loading stage. To define the evolution during the unloading stage we need information about the maximum depth of penetration, so we set amax ¼ a þ ðhmax Þ. Additionally, we set hmin ¼ min hðaÞ, a2½0,amax 

and clearly hmin r 0. The evolution of the contact region radius during unloading is then given by the function ( max fa 2 ½0,amax : hðaÞ ¼ hg if hZ hmin  a ðhÞ ¼ 0 if ho hmin :

ð3:7Þ

ð3:8Þ

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The function a ðhÞ materializes the idea that the value of a should decrease as little as possible during unloading. These two functions are sketched in Fig. 4. Justification: The equivalence between the evolution in Section 3.2 and the two functions introduced here is based on a few facts that are evident from Section 3.2. At all indentation-depths it holds that dh=da Z0, for stability. Since 7 dh=da o þ 1 as well for aa0, it follows that da =dh 40 for a 40. As discussed earlier, a þ ðhÞ 40 if h 4 0, and a ðhÞ 4 0 if h Zhmin . In particular, this means that a þ ðhÞ and a ðhÞ are strictly increasing functions of h whenever h 40 and hZ hmin , respectively, see Fig. 4.1 Notice then that this also implies that for any h1 40, a þ ðh2 Þ with h2 2 ½0,h1 Þ takes all values in ½0,a þ ðh1 ÞÞ. Consequently, if 0 o a o a þ ðh1 Þ there exists h2 2 ½0,a þ ðh1 ÞÞ such that a þ ðh2 Þ ¼ a. Then, since a þ is strictly increasing for h 40, we can conclude that a þ ðh2 Þ o a þ ðh1 Þ implies that hða þ ðh2 ÞÞ ¼ h2 o h1 ¼ hða þ ðh1 ÞÞ, and hence a þ ðh1 Þ satisfies (3.6). A similar argument can be constructed to show that (3.8) is a correct characterization of a ðhÞ.

3.3. Approximate equations of the measured P–h curve Given the evolution of the contact radius, the force as a function of the indentation-depth during loading is computed as Pða þ ðhÞÞ, and during unloading as Pða ðhÞÞ. Together, we call them the measured P–h curve. These are the thin black lines in Fig. 3. We now obtain approximate expressions for these two curves when l is very small. To this end, we define the approximate envelopes of the P(a) and h(a) curves in (2.12) and (2.13). These are pffiffiffiffiffiffiffiffiffiffiffiffiffiffi P þ ðaÞ ¼ PM ðaÞ þ 2pa3 lEn r þ , ð3:9Þ

h þ ðaÞ ¼ hM ðaÞ þ

rffiffiffiffiffiffiffiffiffi pal þ r , 2

ð3:10Þ

and P  ðaÞ ¼ PM ðaÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2pa3 lEn r ,

h ðaÞ ¼ hM ðaÞ

rffiffiffiffiffiffiffiffiffi pal  r , 2

ð3:11Þ

ð3:12Þ

for a 40, and P 7 ð0Þ ¼ 0 and h 7 ð0Þ 2 ð1,0, where

r þ ¼  minx rðxÞ and r ¼ maxx rðxÞ:

ð3:13Þ

Since ra0 and has zero mean value, both r þ and r are positive numbers. These envelopes are illustrated in Figs. 4 and 6. After the discussion on the evolution of the contact region, the examples in Figs. 5 and 6 illustrate that the P þ 2h þ envelope can be used to approximate Pða þ ðhÞÞ as l=a-0 during the loading stage, and that the P  2h envelope does the same for Pða ðhÞÞ for most of the unloading stage. A part is missing during the initial stages of unloading, segment GH in Fig. 3(a), which we discuss in Section 3.4. Taken all together, the approximate envelopes and the early unloading stage form the effective macroscopic response of the system. Effective evolutions of the contact radius can also be computed from the approximate envelopes, and shown to approximate the measured evolutions. We specify the sense in which the effective evolutions approximate the measured contact force and measured contact radius evolutions below. However, a sketch of the effective and measured evolutions is shown in Fig. 5. 1=2 1=2 Notice that P þ P  ¼ Oðl Þ and h þ h ¼ Oðl Þ, so the amplitude of the hysteresis loop goes to zero with l. The reason the approximate envelopes are good approximations of the loading and unloading curves is that the distance between them and the measured loading and unloading curves is OðlÞ for essentially all h. Thus, they approximate the measured curves faster than the amplitude of the hysteresis loop decays. A distinctive feature of the approximate envelopes is that the only information about the small-scale features involved are l, r þ and r . The power law dependence of the envelopes on the contact radius is only scaled by constants when r is changed. As we demonstrate in Section 4.3, this makes it possible to fit experimental results without specific knowledge about the underlying roughness. Instead, some information about r 7 and l is obtained as a product of the fitting. þ The effective evolutions of the contact radius aap and a ap during loading and unloading, respectively, are defined as in (3.6) and (3.8). For the loading stage ( min fa Z 0: h þ ðaÞ ¼ hg if h4 0 þ aap ðhÞ ¼ ð3:14Þ 0 if hr 0:

1

This fact can be proved directly from (3.6) and (3.8) without appealing to the values of dh/da.

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0.15

0.1

0.05

0

−0.05

−0.1 0

0.1

0.2

0.3

0.4

0.5

−0.05

0

0.05

0.1

0.15

0.02

0

−0.02

−0.04 −0.1

Fig. 5. Comparison between the effective and measured contact radius and contact force evolutions during an indentation-depth controlled experiment. 7 7 The figure on top (resp., bottom) illustrates the approximation result stated in (3.16) (resp., (3.17)). The effective evolutions (aap and P 7 ðaap ðhÞÞ) do not display the oscillations of the measured curves (a 7 and Pða 7 ðhÞÞ). ap þ For the unloading stage, we set aap h ðaÞ, and let max ¼ aap ðhmax Þ and hmin ¼ mina2½0,aap max  ( ap   max fa 2 ½0,aap max : h ðaÞ ¼ hg if hmin r hr hmax a ap ap ðhÞ ¼ 0 if h o hmin ,

ð3:15Þ

 ap where h max ¼ h ðamax Þ. The measured evolutions during loading and unloading, Pða þ ðhÞÞ and Pða ðhÞÞ, are approximated by the effective þ evolutions P þ ðaap ðhÞÞ and P  ða ap ðhÞÞ, respectively. In fact, we have that 7 aap ðhÞa 7 ðhÞ ¼ OðlÞ,

ð3:16Þ

7 P 7 ðaap ðhÞÞPða 7 ðhÞÞ ¼ OðlÞ,

ð3:17Þ

as l-0. These approximations are uniform for h 2 during   min ½h and h 0 ,h1  during unloading, with h0 4 hM 1 o hmax independent proximity between the measured and effective evolutions is showcased in Fig. 5. Justification: The possible values that h(a) and P(a) can take are approximately contained between the two approximate envelopes. We term them approximate envelopes because at an increasing number of points the envelopes approach the ½h0þ ,hmax 

loading, with h0þ 4 0 independent of l, and for h 2 of l. Here hmin is the unique minimizer of hM. The M

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0.08

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0

0

−0.02

−0.02

−0.04

−0.04

−0.06

−0.06

−0.2

−0.1

0

0.1

0.2

0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.08 0.06 0.04 0.02 0 −0.02 −0.04 −0.06 −0.2

−0.1

0

0.1

0.2

0.3

Fig. 6. Comparison of the equilibrium P–h curves (2.12) and (2.13) (shown as thick grey curves) with their asymptotic form (2.18a) and (2.18b) (shown pffiffiffi as thin black curves) for the paraboloid punch f(r) in (4.1) and R ¼ A1 cosð2pxÞþ A2 cosð2px=c þ fÞ, where A1 ¼ 0.14, A2 ¼ 0.025, c ¼0.25 and f ¼ p= 2. The n parameter l=R is progressively reduced in (a), (b) and (c) as 0.1, 0.05 and 0.01, respectively. In all curves 2pw=E R ¼ 0:05 and the maximum contact radius amax =R ¼ 0:54. The red curves show the approximate envelopes computed using (3.9)–(3.12). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

exact curves as l-0. These features are evident in Figs. 4 and 6, and are precisely stated for a 4 0 as P  ðaÞa2 9Oðl=aÞ9r PðaÞ r P þ ðaÞ þ a2 9Oðl=aÞ9, h ðaÞa9Oðl=aÞ9 rhðaÞ r h þ ðaÞ þ a9Oðl=aÞ9,

ð3:18Þ 7

which are an immediate consequence of (2.15) and (2.16). Additionally, if x , x 2 ð0,1 are such that r 7 ¼ 8 rðx Þ, then 7 for any n 2 N0 set bn7 ¼ ðn þ x Þl to get that þ



9P 7 ðbn7 ÞPðbn7 Þ9 rðbn7 Þ2 9Oðl=bn7 Þ9 9h 7 ðbn7 Þhðbn7 Þ9 rbn7 9Oðl=bn7 Þ9:

ð3:19Þ

The basic idea behind (3.11) and (3.12) is illustrated next for the loading stage. The first observation is that for h 2 ½h0þ ,hmax , we have that 0 o a0 r a þ ðhÞ ramax , for a0 independent of l. Additionally, hM ða þ ðhÞÞ is a strictly increasing function of h, and it satisfies mðbaÞ r hM ðbÞhM ðaÞ rMðbaÞ for m,M 4 0. Because for the range of h considered a þ ðhÞ is bounded from below by a0, in the following we omit the dependence of the residual terms on a, keeping only that with respect to l. The key observation here is (3.16), which is illustrated in Fig. 4. To see this, notice that the function rða=lÞ attains its maximum value r þ somewhere in ½a þ ðhÞl,a þ ðhÞÞ. This means that sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pa þ ðhÞl þ r þ OðlÞ, ð3:20Þ hða þ ðhÞÞ ZhM ða þ ðhÞÞM l þ 2

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where we have used (2.15). Notice that the first and third terms on the right hand side of (3.20) form h þ ða þ ðhÞÞ. It follows from here that h þ ða þ ðhÞÞhða þ ðhÞÞ ¼ OðlÞ, since from the definition of h loading stage, namely,

þ

ð3:21Þ þ

þ

we have h ðaÞ Z hðaÞ þ OðlÞ. Given that h ¼ hða ðhÞÞ ¼ h

þ

þ ðaap ðhÞÞ,

we conclude (3.16) for the

þ þ ðhÞ9r 9h þ ða þ ðhÞÞh þ ðaap ðhÞÞ9 ¼ OðlÞ: m9a þ ðhÞaap

ð3:22Þ

þ

Now, the smoothness of P ðaÞ implies that þ þ 9P þ ðaap ðhÞÞP þ ða þ ðhÞÞ9 rC9a þ ðhÞaap ðhÞ9 ¼ OðlÞ,

ð3:23Þ

for some C 4 0, and from their definitions it is possible to write P þ ða þ ðhÞÞPða þ ðhÞÞ ¼ 2a þ ðhÞEn ½h þ ða þ ðhÞÞhða þ ðhÞÞ þOðlÞ:

ð3:24Þ

Together with (3.21), we conclude that 9P þ ða þ ðhÞÞPða þ ðhÞÞ9 ¼ OðlÞ:

ð3:25Þ

The result (3.17) for the loading stage follows from (3.23) and (3.25). A similar construction can be made for the unloading stage. 3.4. Effective evolution during early unloading During the initial stages of the unloading phase, the measured P–h curve will be a segment of the equilibrium P–h curve lying between two consecutive folds. For example, in Fig. 3 the part of the curve from point G to point H is measured at the beginning of unloading. This is the part of the loading program in which the system changes the evolution from jumping across the top folds in Fig. 4 to do it across the bottom folds. Thus, the evolution during this stage is approximately described as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi paap max l hn ðzÞ ¼ hM ðaap z þ OðlÞ, ð3:26aÞ max Þ 2 Pn ðzÞ ¼ PM ðaap max Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 n 2pðaap max Þ lE z þ OðlÞ,

ð3:26bÞ

with r r z r r . The contact radius changes by less than a period l along this early stage of unloading. In this case it is possible to explicitly write P as a function of h as þ



ap n ap n Pn ðhn Þ ¼ PM ðaap max Þ2amax E ðhM ðamax Þh Þ þ OðlÞ, n h max r h r hmax .

where These give

These limits follow from those after (3.26) by recalling that h

ð3:27Þ þ

ðaap max Þ ¼ hmax

and

h max

¼ h ðaap max Þ. 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ap þ  h max ¼ hmax  pamax l=2ðr þ r Þ: In the next section we determine the specific form of (2.12) and (2.13), and (2.18a) and (2.18b) for the case where R is a sinusoid. We will then use those equations to derive some more predictions about DDH. 4. A paraboloidal punch with sinusoidal roughness In this section we particularize our results to a punch that is a paraboloid with superimposed sinusoidal undulations. Specifically, we take f(r) and RðxÞ to be f ðrÞ ¼ r 2 =2R,

ð4:1Þ

RðxÞ ¼ Að1cosð2pxÞÞ

ð4:2Þ

for some A 4 0. The punch profile corresponding to these particular f(r) and RðxÞ has been used in Guduru (2007), Guduru and Bull (2007) to study the effect of surface topography on adhesive strength. They considered length scales of the sinusoidal undulations of the same order as the overall size of the punch (10  2 m). Substituting the f(r) and RðxÞ given in (4.1) and (4.2) into (2.12) and (2.13), we get for a 4 0   2pa , ð4:3Þ hs ðaÞ ¼ hsM ðaÞ þap2 AH0

l

     2pa apAl 2pa s H1 Ps ðaÞ ¼ PM ðaÞ þ 2En a2 p2 AH0  , l 2 l

ð4:4Þ

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where hsM ðaÞ ¼ s ðaÞ ¼ PM

rffiffiffiffiffiffiffiffiffiffiffiffiffi a2 2pwa  , En R

ð4:3aÞ

ffi 4En 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a  8pwEn a3 : 3R

ð4:4aÞ

The symbols H0 and H1 denote Struve functions of zeroth and first order, respectively. We use the superscript s to denote that the equations are for the particular case of sinusoidal roughness. Eqs. (4.3) and (4.4) are equivalent to Eqs. (2) and (3) in Guduru and Bull (2007). The approximations to the equilibrium P–h curve are obtained by substituting f(r) and RðxÞ in (2.18a) and (2.18b), which renders rffiffiffiffiffiffiffiffiffi pal s hs ðaÞ  hsM ðaÞ r ða=lÞ, ð4:5Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ðaÞ 2pa3 lEn rs ða=lÞ ð4:6Þ P s ðaÞ  PM for a 40, with pffiffiffiffiffiffi



rs ðxÞ ¼  2pA sin 2px

p 4

:

ð4:7Þ

A comparison of the curves given by (4.3) and (4.4) and (4.5) and (4.6) is shown in Fig. 2. In agreement with the discussion in Section 2.3, the figure shows that the approximation of Ps–hs by (4.5) and (4.6) improves as l is decreased. The approximate envelopes of the Ps–hs curve are, from (3.9)–(3.12), rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi a2 2pwa  ð4:8Þ 7 pA la, hs 7 ðaÞ ¼ En R P s 7 ðaÞ ¼

ffi pffiffiffiffiffiffiffiffi 4En 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a  8pwEn a3 7 2pEn A la3 3R

ð4:9Þ

for a 4 0. The þ and  signs correspond to the loading and unloading envelopes, respectively. For the case of no roughness (A¼0) the envelope (4.8) and (4.9) collapses to the single curve PsM–hsM (4.3a) and (4.4a), which is the equilibrium P–h curve corresponding to the JKR contact problem. The approximate envelopes are completed by the early unloading stage approximation in Section 3.4. This follows from (4.3a) and (4.4a) into (3.27), namely, P  2En aap max h

3 2En ðaap max Þ , 3R

ð4:10Þ

ap sþ where aap is the max pffiffiffiffiffiffiffiffiffiffiffiffi ffi approximate maximum contact radius during the experiment, which satisfies h ðamax Þ ¼ hmax , and rh r h hmax 2pA laap . max max

4.1. Energy loss during a displacement controlled contact experiment Since the loading and unloading path differ, not all the work done during loading is recovered upon unloading. This leads to a net energy loss during the cycle, measured by the area enclosed by the measured P–h curves. In the following we estimate the net energy loss for the paraboloid punch with the sinusoidal model for roughness (Section 4). An example plot of the approximate equilibrium P–h curve (4.5)–(4.7) and the corresponding effective evolution (4.8)–(4.10) is shown in Fig. 7. An approximation to the net energy loss H due to DDH can then be evaluated by taking advantage of the effective evolution as Z H ¼ P dh, ð4:11Þ C

The integration path C is marked using numbers 0–4 in Fig. 7. The integration is detailed in Appendix C, and renders  5 4 1=3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiw7 R8 1=6 w R 2 2 3 n þ 29:6A l þ Aððaap ð4:12Þ H ¼ 7:1 max Þ a2 Þ 8p E wl þ OðlÞ, n 2 n E E where a2 4 0 is the contact radius just after initial contact. It satisfies hs þ ða2 Þ ¼ 0, which leads to  2=3 1=2 1=2 pARl : a2 ¼ 2pwR2 =En

ð4:13Þ

The expression for H illustrates a number of interesting features. First, notice that when l 5a, the root mean square measure of roughness is proportional to A. This is what we will henceforth refer to as the roughness size. So, (4.12) implies that H will be larger in experiments with rougher surfaces. This prediction runs counter to what is general expected from

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0.08

2503

3

0.06

0.04

0.02

0

0

1

3’ 2

−0.02

4 2’

−0.04 −0.1

0

0.1

0.2

Fig. 7. Energy loss in a displacement controlled experiment. The approximate equilibrium P–h curve, grey curve, was plotted using (4.5) and (4.6) for the sinusoidal R given in (4.2) and the paraboloidal punch in (4.1), with a/R ranging from 0 to 0.6. The thin black loop, marked using numbers 0–4, is the effective macroscopic evolution. The Section 2–3 was plotted using the P s þ –hs þ curve (4.8) and (4.9), with a=R ranging from a2 ¼ 0.28 to amax ¼ 0:6. Section 3–30 is a segment of the equilibrium P–h curve between two consecutive folds, approximated with (4.10). Section 30 –4 was plotted using the P s 2hs curve (4.8) and (4.9), with a=R ranging from amax to a4 ¼ 0.18. The point 20 on 30 –4 corresponds to a ¼ a2 . In all curves, we used A ¼ 0.14, 2pw=En R ¼ 0:05, and l=R ¼ 0:03.

roughness—which is that it decreases adhesion. However, recent experiments by Kesari et al. (2010) show that for small roughness sizes, H can indeed increase with it. The prediction of energy loss increasing with roughness is primarily a consequence of assuming a simply connected contact region. This is a good assumption for small enough roughness sizes, in which the contact region will be the union of a few simply connected patches. For large enough roughness sizes, however, the contact region will become multiply connected. In this case, the mechanics of the contact will likely be better described by asperity-type contact models. Both regimes were observed by Kesari et al. (2010). What type of contact region is obtained is also a function of the material properties, such as En and w. Eq. (4.12) further predicts that H is affine with the maximum contact area in a contact cycle. From (4.8) it follows that at large indentation-depths, a2 ph. Thus, the energy loss is predicted to scale linearly with hmax at large indentationdepths. This was also observed to be in good agreement with experiments in Kesari et al. (2010). A final observation from (4.12) is that as l-0, H approaches the energy loss for JKR, the first term in the right hand 1=2 side. The net energy loss due to roughness decays as l . This is accounted for by the second and third terms in the right hand side of (4.12). These terms are negligible with respect to the JKR term when lEn =w 5 1. Thus, DDH is relevant only when l is comparable to the w=En length scale. 4.2. Energy barrier between metastable states As discussed in Section 3, DDH appears because roughness induces multiple configurations of the two bodies to be locally stable at a given indentation-depth. For DDH to be observable, however, the minimum energy required for the system to transit between these locally stable configurations should be large in comparison with ambient energy fluctuations, such as those induced by thermal energy. Borrowing terminology from chemical kinetics we refer to this energy as the energy barrier, dP. In order to check whether or not roughness would be relevant in an experiment, we next derive a coarse estimate for dP from the sinusoidal roughness model (Section 4). The minimum potential energy P in our contact problem at a given indentation-depth and contact radius can be written as Z En p2 a ^ hÞ2 da ^ pwa2 : Pða; hÞ ¼ wða; ð4:14Þ 4 0 The function w is defined in (2.9b). The first term in (4.14) is the strain energy of the half-space. This strain energy corresponds to the solution of (2.2)–(2.5) for given h and a. The second term is the interface energy due to adhesion. For the proof of (4.14) see Kesari and Lew (2011, Section 5.2).

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For a prescribed h, the energy barrier dP between configurations with different contact radii can be extracted by studying P as a function of a. Example plots of P as a function of a are shown in Fig. 8. The equilibrium configurations correspond to the contact radii at which P is locally stationary. The stable and unstable configurations among these correspond to the points where P is locally minimized and maximized, respectively. The energy barrier dP to get from one stable configuration to another is the maximum magnitude of the energy difference between the starting configuration and any intermediate one. ^ follows after substituting u~ z from (4.1) For the particular case of the sinusoidal roughness and paraboloidal punch, wðaÞ and (4.2) into (2.9b), namely,    2 2pa^ 2 ^ hÞ ¼ ^ 0 wða; ha^ =Rp2 AaH : ð4:15Þ

p

l

x 10−3 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 −0.5 0.1

0.2

0.3

0.4

x 10−3

0.6

x 10−3

4.5

4.5

4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

−0.5

−0.5 0.1

0.5

0.2

0.3

0.4

0.5

0.6

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 8. Potential energy as a function of the contact radius at zero indentation-depth (h ¼ 0), for the sinusoidal roughness model with a paraboloidal punch in Section 4. Black circles are the exact potential energy of the system computed using (4.14), while plain solid black curves are approximations of the potential energy computed using the first four terms in (4.16). The grey curves are approximate bounds of the potential energy computed using (4.16) by replacing the fourth, oscillatory, term in it by its maximum and minimum value. These plots were obtained by setting A ¼ 0.14, 2pw=REn ¼ 0:05 and l=R to be 0.5 in (a), 0.25 in (b), and 0.1 in (c). The dashed black curve in (a), (b), or (c) is the potential energy of the system when there is no roughness, i.e., when A ¼0.

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In general the values of a where the stable equilibria, as well as dP, will depend on the indentation-depth. Since we are looking for a sample of the order of magnitude of dP, we next estimate it for h¼0. A different value of h results in more complex expressions, but are not difficult to compute. Substituting w from (4.15) into (4.14), replacing H0 by its asymptotic expansion (B.4), evaluating the integral, and then expanding around l ¼ 0 renders ! En a5 p2 En A2 a2 2En Aa3 2 þ Pða; 0Þ ¼  p wa þ l 4 3R 5R2 

En A sin R

 2pa

l



þ

p 4

a5=2 l

3=2

2

þ Oðl Þ:

ð4:16Þ

In Fig. 8 we compare the approximate value of P computed from (4.16) with its exact value computed numerically from (4.14). As expected, the approximation becomes better as l decreases. It can be seen from (4.16), and also from Fig. 8, that as l decreases the value of dP is essentially determined by the amplitude of the fourth, oscillatory term in (4.16). Thus, 3=2 5=2

dP 

En Al

a

R

,

ð4:17Þ

as l-0. This estimate for dP is a very coarse one, considering the various assumptions made in its derivation. However, we think that (4.17) will be useful to evaluate whether roughness could be important in a particular experiment. For example, in the glass-PDMS DDH experiments presented by Kesari et al. (2010), En  0:75 MPa, w  26 mJ=m2 and A  102 , l  1000 nm, and a  5 mm. For which we get dP  0:01 pJ. This value is much larger than the average thermal energy fluctuations which are of the order of 1021 J ¼ kB T, where kB is the Boltzmann constant and T is the temperature. Thus, thermal noise would be insufficient to stop the roughness-adhesion DDH mechanism from operating in this specific case. The energy barrier can thus be similarly compared with other types of energy fluctuations, such as those due to the vibrational noise in the actuators moving the bodies.

4.3. Experimental comparison of the P–h curves If a contact experiment displays hysteresis, and if there is reason to believe that the hysteresis is due to the roughnessadhesion mechanism discussed in Section 3, then in that case (4.8) and (4.9) will be more useful for estimating material properties than the JKR equations (4.3a) and (4.4a). As an example, we estimated the material properties of gelatin by fitting the data reported by Guduru and Bull (2007) to (4.8) and (4.9). The P–h data given in Guduru and Bull (2007) displays hysteresis. These curves are the result of measuring contact between a polycarbonate paraboloidal punch and a rectangular slab of gelatin. In that experiment, the strain rate was of the order of 10  3 s  1; and A and l were of the order of 0.04 and 50 mm, respectively (these were estimated by performing a discrete Fourier transform of the punch’s topography given in Fig. 9 of Guduru and Bull (2007)). We fitted the distinct branches of the P–a data measured during the loading and unloading phases to the upper and lower envelopes given by (4.9). The fits are shown in Fig. 9(a). From the fits, we estimated En and w to be 17.08 KPa and 80.4 mJ/m2, respectively. A comparison of these estimates with values obtained from other alternate testing methods is shown in Table 1. As can be seen from the comparison, our estimates obtained through fitting indentation data to (4.8) and (4.9) are quite close to those obtained through alternate methods. Additionally, using these estimated values we computed the P–h curve predicted by (4.8) and (4.9). We show this curve alongside the measured P–h curve in Fig. 9(b). The predicted and measured P–h curves are especially close to one another during the unloading phase. The match is not that good for the loading phase, and there are several plausible reasons for the differences between the two curves. A discussion about these can be found in Guduru and Bull (2007). To bring out the advantage of using (4.8) and (4.9) we compare our estimates with those reported by Guduru and Bull (2007), which were obtained by fitting the same data to the JKR theory. The first noticeable difference is that, since the JKR theory does not capture DDH, only sections of the experimental data can be fit to the JKR equations. Usually, quite different estimates for w are obtained depending on the section chosen. For example, fitting the loading branch of the P–a data to the JKR equations gives an estimate for w of 8 mJ/m2, whereas fitting them to the data from the unloading branch gives an estimate of 220 mJ/m2. Both values are substantially different from the values reported in the literature, as well as the values estimated here for the very same data. By fitting the JKR equations, Guduru and Bull (2007) report En ¼ 16:5 KPa. In contrast with w, this value is close to the value measured through other methods (see pffiffiffi Table 1) and the one obtained through the use of (4.8) and (4.9). From the fit we also get a value for A l as 1.1  10  4 m1/2 which is consistent with the values for A and l reported earlier for the polycarbonate punch by Guduru and Bull (2007). Thus, in addition to the surface energy and material stiffness information we also get information about the roughness of the contacting bodies. Eqs. (4.8) and (4.9) have also been used for estimating material properties from contact experiments between glassbeads and polydimethylsiloxane samples (Kesari et al., 2010). Good qualitative and quantitative matches of the P–h curves and mechanical properties were also obtained.

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1.2 1

Experimental unloading Experimental loading Theory, envelope equations

Load, P (N)

0.8 0.6 0.4 0.2 0 -0.2

0. 00 8 0. 01 0. 01 2 0. 01 4 0. 01 6 0. 01 8 0. 02 0. 02 2 0. 02 4 0. 02 6

-0.4

Radius of the contact area, a (m)

1 Experimental Theory, envelope equations

Load, P (N)

0.8 0.6 0.4 0.2 0 -0.2 -6

-4

-2 0 2 4 6 8 10 Indentation-depth, h (10-4m)

12

14

Fig. 9. Comparison with the experimental results in Guduru and Bull (2007) on indentation of gelatin. The P–a data was fit to (4.9), as shown in (a). From the fit, values for En and w were estimated, and used to calculate the P–h curve (4.8) and (4.9). This predicted P–h curve is compared with the experimentally measured one in (b). (Guduru, P.R., Bull, C., 2007. Detachment of a rigid solid from an elastic wavy surfa Experiments. Journal of the Mechanics and Physics of Solids 55, 473–488. Copyright (2007), with permission from Elsevier.)

Table 1 Comparison of theoretical and experimental estimates for plain strain Young’s modulus En and surface energy, w. Source

En (KPa)

w (mJ/m2)

Fitting data from Guduru and Bull (2007) to (4.9) Fitting data from Guduru and Bull (2007) to JKR (4.3a) and (4.4a) Samani and Plewes (2007) Bialopiotrowicz and Janczuk (2002) Lavielle et al. (1991)

17.08 16.5 (loading) 16.5 (unloading) 19.85 – –

80.4 8 (loading) 220 (unloading) – 42.25 63.1

Notice that the use of the sinusoidal roughness model is not important to obtain a good fitting of the P–a curve. The same fit would have been obtained with the general roughness model by setting r þ ¼ r . In this case, instead of pffiffiffi pffiffiffiffiffiffiffiffiffi recovering a value for A l, we would have computed r þ 2pl ¼ 1:1  104 m1=2 .

5. Closing remarks The purpose of (3.9)–(3.12) is to facilitate estimating material properties from DDH data. Real surfaces have roughness patterns that are clearly not axi-symmetric or periodic. However, the experimental comparisons here and in Kesari et al. (2010) suggest that the approximate envelopes (3.9)–(3.12) match the measured curves well, and give better estimates for material properties than the JKR equations in the presence of DDH. Fitting DDH data to (4.8) and (4.9) also has the added

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pffiffiffi advantage of yielding some information about surface roughness, i.e., the value of the parameter A l, as in Section 4.3 and Kesari et al. (2010). In this paper we considered roughness patterns at length scales much smaller than the dimensions of the contact radius. Some roughness models argue that there is no such a clear separation of length scales, such as those developed by Majumdar and Bhushan (1991) and Persson (2002). This is often true for many naturally formed solids, such as rocks and geological formations. On engineered surfaces, however, it is possible to restrict topography variations below a certain small length scale. This length scale is a characteristic of the fabrication process. So, the problem discussed here would be more appropriate for modeling contact between engineered surfaces. We believe that the roughness-adhesion mechanism causing DDH in our problem is fairly general. Despite the axisymmetry and periodicity, the result here holds for a very large class of roughness patterns. Additionally, we have seen this same roughness-adhesion mechanism in more sophisticated contact models that use a Lennard–Jones type interaction for modeling contact, and more complicated surface topographies for modeling roughness. These other models were studied using finite element and molecular statics simulations (Kesari, 2011).

Acknowledgments This work was partly supported by the Center for Probing the Nanoscale (CPN), an NSF NSEC, NSF Grant No. PHY0425897, and by an NSF-Career Award, NSF Grant No. CMMI-0747089. HK is supported by the Herbert Kunzel Stanford Graduate Fellowship. Appendix A. Stability of the equilibrium configurations The minimum potential energy for our contact problem at a given indentation-depth and contact radius can be written as in (4.14). At a fixed h, a contact radius a defines a local equilibrium configuration if it is a stationary point of P, namely, it satisfies (2.10). We show next that at a local equilibrium configuration ða; hðaÞÞ we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh @2 P ðaÞ: ða; hðaÞÞ ¼ 8paEn w da @a2

ðA:1Þ

Additionally, at any configuration ða; hÞ it holds that @2 P ða; hÞ ¼ En pwða; hÞ: @h@a

ðA:2Þ

Since locally stable equilibrium configurations are local minimizers of the potential energy, (A.1) implies that a local equilibrium is locally stable if and only if dh=da 4 0. The cross-derivative of the potential energy (A.2) implies (3.5), which is used to estimate the gradient of P with respect to a when the indentation-depth is changed slightly from an equilibrium configuration. More precisely, by expanding the value of @P=@a we have @P @P @2 P ða; h þ DhÞ ¼ ða; hÞ þ ða; hÞDhþ OðDh2 Þ: @a @a @h@a

ðA:3Þ

Replacing by (A.2) and evaluating it at a local equilibrium ða; hðaÞÞ we get @P ða; hðaÞ þ DhÞ ¼ En pwða; hðaÞÞDh þ OðDh2 Þ, @a from where (3.5) follows. To prove (A.1) and (A.2), we use that at any equilibrium configuration ða; hðaÞÞ we have rffiffiffiffiffiffiffiffiffiffi 8aw wða; hðaÞÞ ¼  , pEn which follows from (2.10) and (2.11). Therefore, rffiffiffiffiffiffiffiffiffiffiffi d 2w @w @w dh ðaÞ: ða; hðaÞÞ þ ða; hðaÞÞ ¼ wða; hðaÞÞ ¼  da da @h paEn @a

ðA:4Þ

ðA:5Þ

ðA:6Þ

Since from (2.9b) we have @w 2 ða; hÞ ¼ , @h p

ðA:7Þ

it follows that rffiffiffiffiffiffiffiffiffiffiffi @w 2w 2 dh ðaÞ: ða; hðaÞÞ ¼   @a paEn p da

ðA:8Þ

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The results are obtained by direct differentiation of @P=@a from (2.10). First, @2 P p2 En @w ða; hðaÞÞ ¼ wða; hðaÞÞ ða; hðaÞÞ2pw 2 2 @a @a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh ðaÞ ¼ 8paEn w da

ðA:9Þ

Similarly, @2 P p2 En @w ða; hÞ ¼ wða; hÞ ða; hÞ @h@a 2 @h ¼ En pwða; hÞ

ðA:10Þ

Notice that (A.4) can be simplified further by taking advantage of (A.5) to get pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @P ða; hðaÞ þ DhÞ ¼  8awEn p Dhþ OðDh2 Þ: @a

ðA:11Þ

Appendix B. Asymptotic form of the equilibrium P–h curve (2.12) and (2.13) We next derive the expansion (2.15) and (2.16) of the P–h curve given by (2.12) and (2.13). Recalling that R is periodic with period 1, we use the following representation for R in terms of Fourier series:

RðxÞ ¼

1 X a0 þ ½an cosðn2pxÞ þ bn sinðn2pxÞ, 2 n¼1

ðB:1Þ

where an ¼ 2

Z

1=2

RðxÞcosðn2pxÞ dx, n Z 0,

ðB:1aÞ

RðxÞsinðn2pxÞ dx, n Z 1:

ðB:1bÞ

1=2

bn ¼ 2

Z

1=2

1=2

Since R was assumed to be C 1 ½0,1, it is possible to substitute this representation into (2.12) and (2.13) and evaluate the integrals individually on each term to give    

nX ¼1 2pa 2pa n an H0 n þ bn J0 n , ðB:2Þ hðaÞ ¼ hM ðaÞap2

l

n¼1

" PðaÞ ¼ PM ðaÞ þ 2En a2 p2 þ

alp 2

nX ¼1 n¼1

nX ¼1

l

   

2pa 2pa n an H0 n þ bn J0 n

l

n¼1

    # 2pa 2pa an H1 n þ bn J1 n ,

l

l

l

ðB:3Þ

where hM and PM are given in (2.12a) and (2.13a). The symbols H0 and H1 denote the zeroth and first Struve functions, respectively, and J0 and J1 denote the zeroth and first Bessel functions of the first kind, respectively. We derive (2.15) and (2.16) by substituting the asymptotic expansion of the Struve and Bessel functions as l=a-0 in (B.2) and (B.3). The Struve functions attain the asymptotic forms (Abramowitz and Stegun, 1970, p. 497, Eqs. (12.1.30), (12.1.31)) !     2pa 2pa l l2 H0 n ¼ Y0 n þ 2 þO 2 2 , ðB:4Þ l l p na n a       2pa 2pa 2 l , ¼ Y1 n þ þO H1 n l l p na

ðB:5Þ

as l=na-0, for all n 4 1. The functions Y0 and Y1 are the zeroth and first Bessel functions of the second kind. The Bessel functions attain the asymptotic form (Abramowitz and Stegun, 1970, p. 364, Eqs. (9.2.1), (9.2.2))   rffiffiffiffiffiffiffiffiffiffiffin  o 2pa l a np p  þOðl=anÞ , sin n2p  ¼ ðB:6Þ Yn n 2 l l 2 4 np a  

  rffiffiffiffiffiffiffiffiffiffiffi 2pa l 2pa np p  þ Oðl=anÞ , cos n Jn n  ¼ 2 l l 2 4 np a

ðB:7Þ

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for n ¼ 0,1. Replacing H0 and H1 in (B.2) and (B.3) with their asymptotic forms (B.4) and (B.5), and then replacing the Bessel functions in the resulting by (B.6) and (B.7), we get rffiffiffiffiffiffiffiffiffi pal hðaÞ ¼ hM ðaÞ rða=lÞ þaOðl=aÞ, ðB:8Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB:9Þ PðaÞ ¼ PM ðaÞ 2pa3 lEn rða=lÞ þ a2 Oðl=aÞ as l=a-0, where

rðxÞ ¼

nX ¼ 1 pffiffiffiffiffiffiffiffiffin n¼1

  p po n2p an sin n2px þ bn cos n2px : 4 4

ðB:10Þ

Notice that in obtaining the asymptotic form for P(a), (B.9), the terms containing H1 and J1 in (B.3) are multiplied by l, so they only contribute higher-order terms in l=a. The asymptotic expansions for both h(a) and P(a) are a consequence of the asymptotic behavior of H0 and J0, and this is why r plays essentially the same role in both quantities. If the next terms in the expansion of H0 and H1 are kept, it is possible to show that the last term in (B.9) is Oððl=aÞ3=2 Þ rather Oðl=aÞ. The function rðxÞ is well-defined, since R was assumed to be C 1 ðRÞ. This also implies that r is continuous. Clearly the smoothness assumption on R can be further relaxed without harming the definition of r. It is not hard to verify that as defined in (B.10), r is the semi-derivative2 of ðRðxÞa0 =2Þ as defined by Weyl (see, e.g., Zygmund, 1959, p. 134), which we denote as

r ¼ L D1=2 R:

ðB:11Þ

Appendix C. Computation of the energy loss during a loading–unloading cycle The expression for H in (4.12) is obtained by direct computation. To this end, we split the integration as Z Z H¼ P dh þ P dh, C1

ðC:1Þ

C2

where the two closed paths C1 and C2 are shown in Fig. 7, marked as 02122220 2420 and 223230 220 22, respectively. We discuss first the integration over C1 . In it, there are no contribution from the segments 0–1, 1–2, and 4–0, since P ¼0 over 0–1 and there is no change in h over 1–2 and 4–0. The integral over 2–20 can be evaluated using (4.10) with aap max 0 replaced by a2, which is the contact radius at point 2. The limits of integration are h2 ¼ hs ða2 Þ and h2 ¼ hs þ ða2 Þ. The integral over 20 –4 can be evaluated using the Ps 2hs curve (4.8) and (4.9) with a varying from a2 to a4. The parameter a4 s is the contact radius at point 4 at which the bodies detach unstably. We obtain a4 by solving dh ða4 Þ=da ¼ 0 to get a4 ¼ ½ðpwR2 =8En Þ1=2 þ pARl

1=2

=42=3 :

Evaluating the integrals over 2–20 and 20 24 and expanding around l ¼ 0 leads to  5 4 1=3 Z pffiffiffiw7 R8 1=6 w R P dh ¼ 7:1 þ 29:6A l þ OðlÞ: En En 2 C1

ðC:2Þ

ðC:3Þ

We turn next to the integration over C2 . The integral over the segment 2-3 can be evaluated using the P s þ 2hs þ curve, ap ap (4.8) and (4.9), with a varying from a2 to amax. The value of amax is the approximate maximum contact radius, which is ap sþ attained at point 3, and satisfies h ðamax Þ ¼ hmax . The integral over the segment 3-30 can be evaluated using (4.10), with 0 0 30 s ap s s integration limits h3 ¼ hs þ ðaap curve, (4.8) max Þ and h ¼ h ðamax Þ. The integral over 3 22 can be evaluated using the P 2h ap 0 and (4.9), with a varying from amax to a2. The integral over 2 –2 was discussed earlier. This leads to Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 n ðC:4Þ P dh ¼ Aððaap max Þ a2 Þ 8p E wl: C2

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The 1/2-fractional derivative.

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