## Abstract

We report on the elastic contact between a spherical lens and a patterned substrate, composed of a hexagonal lattice of cylindrical pillars. The stress field and the size of the contact area are obtained by means of numerical methods: a superposition method of discrete pressure elements and an iterative bisection-like method. For small indentations, a transition from a Hertzian to a soft-flat-punch behaviour is observed when the surface fraction of the substrate that is covered by the pillars is increased. In particular, we present a master curve defined by two dimensionless parameters, which allows one to predict the stress at the centre of the contact region in terms of the surface fraction occupied by pillars. The transition between the limiting contact regimes, Hertzian and soft-flat-punch, is well described by a rational function. Additionally, a simple model to describe the Boussinesq–Cerruti-like contact between the lens and a single elastic pillar, which takes into account the pillar geometry and the elastic properties of the two bodies, is presented.

## 1. Introduction

The contact mechanics between an elastic sphere and a flat elastic solid, with frictionless, perfectly smooth and non-conforming surfaces, is beautifully provided by the Hertz theory [1]. From the knowledge of the geometry (sphere radius *R*), the elastic properties (Young’s modulus and Poisson’s ratio) and the total load *F* compressing the two elastic solids, one can forecast the radius of contact

However, real surfaces are not smooth and the contact mechanics of two real bodies can bitterly deviate from the Hertzian predictions since only partial contact can occur [1–3]. Attention has been paid to this problem from two different points of departure, the first based on a statistical approach of real surfaces and the second on the deterministic behaviour of periodic surfaces.

On the one hand, the statistical approach to the contact mechanics of real surfaces bifurcates into three branches. First, considering the contact between a plane and an isotropically rough surface composed of non-interacting ‘spherical asperities’ with a vertical position given by a distribution of heights (exponential and Gaussian) [4–6], from which estimations of the number of contact spots, the contact area and the total load were obtained in terms of the apparent contact area, the density of asperities, the radius of curvature of the asperities and a function of the heights distribution. In particular, it has been found that both the number of asperities in contact and the real contact area are proportional to the total load [4], and the ratio of the roughness to the indentation has been proposed as a parameter to account for the influence of the roughness on the contact mechanics [5,6]. Additionally, useful empirical elasto-plastic models can be found in the recent literature [7,8], which consider nominally rough surfaces and are based on the above-mentioned statistical approaches. Second, a similar approach to the previous one was developed, but considering a distribution for the shapes of the asperities, i.e. an isotropically rough surface composed of non-interacting ‘paraboloid-like asperities’ with a Gaussian distribution of the slopes and curvatures [9]. It has been shown that considering roughness, given by a bandwidth of wavelengths, the real contact area is strictly proportional to the load and, thus, the stress distribution at the average asperity contact region remains unchanged. Nevertheless, the two above-mentioned approaches break down for large loads, at which the separation between the contact surfaces becomes small and the interaction between adjoining asperities cannot be neglected. Finally, the surface roughness power spectrum, for instance, self-affine fractal surfaces and frozen capillary waves, was employed to estimate the ratio between the real and apparent contact areas [10]. This description is particularly accurate for high loads, and it is based on the knowledge of the stress distribution, which depends on the average pressure and the magnification (the ratio between size of the apparent contact area and the shortest wavelength roughness). It also predicts a linear dependence of the real contact area on the total load, for small loads and small real-to-apparent contact area ratios. Even though the aforementioned statistical methods provide average quantities and global insight of the rough contact phenomenon, the local stress and displacement fields at the contact spots remain inaccessible since the surface topography may not be persistent from one body to another.

On the other hand, in order to gain insight on the local response of the bodies in contact, periodic contact problems have been addressed [11,12]. For instance, the contact between a flat surface and a well-characterized one-dimensional wavy surface, of wavelength *λ* and amplitude *δ*, has been addressed by superposition of concentrated normal forces at the surface of an infinite half-space (Boussinesq–Cerruti approach). In this situation, a close-form solution for the stress and displacement fields is obtained and we should encourage the interested reader to review the aforementioned bibliography. However, we should mention that for a given line load *F*, the local contact occurs in bands of increasing width 2*a* that obey the scaling

Understanding the contact mechanics of rough surfaces, starting with well-controlled patterned surfaces, is fundamental in order to clarify the effect of pattern covering on friction [13–18], or to identify the influence of texturation on the adhesive properties of surfaces [19–22]. This query has received a lot of attention since patterned surfaces have been extensively used as biomimetic surfaces [23–28]. From this point of view, one of the important issues is the contact formation, which fixes the relationship between the measured adhesion and the preload before the detachment [29,30].

In the simplest case of a spherical lens on well-controlled patterned surface made on elastomeric PDMS with circular bumps and cylindrical pillars, a transition from suspended (the lens touches only the top of the bumps/pillars) to intimate contact (the lens gets in contact not only with the bumps/pillars but also with the underlying substrate in between the patterns) has been observed. The contact formation has been shown to depend strongly on the geometry and elastic properties of the textured surface [3,31–33]. In particular, Verneuil and co-workers [31] have shown that the one leading parameter is the surface fraction *Φ* occupied by the pillars, since the critical force, which marks the threshold between both contact regimes, scales as *F*_{c}∼*Φ*^{3}.

Therefore, there is no doubt about the central role that the surface fraction *Φ* plays in the distribution of stresses and in ascertaining the size of the contact region. The geometry of a sphere in contact with an array of cylindrical pillars is represented graphically in figure 1. When

In this paper, we discuss the elastic contact between a spherical lens and patterned surface with cylindrical pillars. The presented analysis, which corresponds to a first approach on the real contact mechanics of periodic rough surfaces, considers two fundamental hypotheses: (i) frictionless contact and (ii) no adhesion effects. The impact of elastic properties and of the geometry of the system is analysed, with special focus on the surface fraction occupied by the pillars. A detailed description of the transition from Hertzian contact (short pitch distance) to soft-flat-punch (SFP) regime (large pitch distance), for the aforementioned lens-patterned substrate system, is presented. We will discuss the contact behaviour for a wide range of elastic parameters (combination of Young’s modulus) and displacement of the bodies (total indentation), and analyse the effects of the pitch between pillars in terms of the stress concentration and the number of pillars (real contact area) within the contact region.

## 2. System description

Consider an elastic system formed by a semi-infinite substrate, the upper surface of which is patterned with a hexagonal lattice of cylindrical pillars, and a spherical lens. We imagine a situation in which the lens is pushed against the substrate, whereas the latter is fixed at the far-field, with respect to the lens initial position. A discrete mechanical contact region (for small loads) occurs, which produces and conveys stress and displacement fields at the lens and the substrate. In brief, once a load is applied, the system evolves from an initial undeformed state, at which the lowest point of the lens touches in a single point the centre of the top of a central pillar, towards a final deformed state, where multiple contact regions appear. We expect to observe deviations from this general trend, depending on the specific geometry of the system and the magnitude of the applied load.

Herein, we consider another important hypothesis: the effect of lateral displacement, due to the discrete stress field is negligible with respect to normal displacement. Our intuition leads us to imply that the lateral displacement provoked by a single pillar is cancelled by the lateral displacement generated by the surrounding pillars and, as a result, the superposition of the effects of all the pillars can be neglected. In addition, also due to the aforementioned idea, the patterned substrate can be further separated. To be more precise, the system, represented in figure 2, can be decomposed into several independent bodies: (i) a spherical lens, a cap that corresponds to the lower region of a sphere of radius *R*; (ii) a flat substrate, which we will take as an elastic half-space; and (iii) an array of cylinders, where *d* and *h* are, respectively, the diameter and height of each pillar, whereas *e* is the pitch, the distance between the centre of two pillars.

### (a) Array of pillars

When the lens is pushed against the patterned substrate, the number of pillars in contact *Q* can be estimated by using the Wigner–Seitz cell of an hexagonal lattice of pillars (figure 2). Computing the fraction of the surface occupied by the pillars:
*a* as the apparent contact zone, the number of pillars in contact becomes
*Q* because the apparent contact radius *a* is unknown *a priori*. Despite this fact, in the present study we will consider that *R*≫*d*/2 and *R*≫*a* which corresponds to a lot of practical situations previously studied in the literature.

Considering a polar system (*r*,*θ*), the position of the centre of the *i*th pillar is given by the coordinates (*r*_{i},*θ*_{i}), where the value of *i*=0 corresponds to the pillar whose centre is placed at the origin of the coordinate system. A detailed description of the pillar array and the precise method to enumerate the pillars and to count the number of pillars in contact is given in the electronic supplementary material.

### (b) Displacement and indentation

Before the lens and patterned substrate are squeezed and the deformation of both bodies takes place, the centre of the spherical lens is placed at *z*=*h*+*R*. In turn, the equation of the pillars-substrate surface is
*H* is a Heaviside step function, taking either the value of 0 if *ρ*_{i}≤*d*/2 or 1 if *ρ*_{i}>*d*/2. As described in the electronic supplementary material relative to the centre of the *i*th pillar, which is placed at (*r*_{i},*θ*_{i}), *ρ*_{i} is the distance to the point (*r*,*θ*), given by
*r*=0 where the lowest position of the lens touches the centre of the pillar upper surface.

Once the load is applied and the system has attained equilibrium, the final gap between the surfaces, the space separating the lens and the pillar-substrate body, is now given by
*w*(*r*,*θ*) is the total displacement field and *ζ*=*w*(0,*θ*) is the indentation, i.e. the displacement at the centre. For light loads and within the apparent contact region, only the top of the pillars can be in contact with the lens, whereas the substrate not covered by pillars will be separated from the lens by a gap. In addition, one may note that, within the contact region, the top of each pillar may be either in total or partial contact. Whatever the contact situation of each pillar, at the local contact regions, one has *Δ*(*r*,*θ*)=0 and *g*^{°}(*r*,*θ*)=*h*, and thus the total displacement field *w*(*r*,*θ*) is related to the total indentation *ζ* as follows:

## 3. Discrete contact

For each body *j* (lens *j*=*l*, array of pillars *j*=*p* and substrate *j*=*s*), considering its Young’s elastic modulus *E*_{j} and Poisson’s ratio *ν*_{j}, we define the parameter *E*_{p}=*E*_{s} and *ν*_{p}=*ν*_{s}. As mentioned above, when these bodies are brought into contact, i.e. the lens is approached towards the pillars, which in turn pushes the substrate, they suffer a deformation. Under these circumstances, recalling that the vertical displacement at the surface of a half-space *u*_{z} due to a concentrated point force *Σ* is given by [1]
*j*=*l*,*s*) reads
*σ*=*σ*(*ϱ*,*φ*) is the stress field over the apparent contact area *ϱ* and *φ*, the radial and angular coordinates relative to the point (*r*,*θ*). In other words, the displacement *w*_{j} at the point (*r*,*θ*) is the effect of the stress field applied over a region described by the set of relative positions (*ϱ*,*φ*). Owing to the discrete nature of the contact, the normal displacement of each surface can be decomposed as follows:
*W*_{i} is the reduced displacement at (*r*,*θ*) due to the contact with the *i*th pillar. Additionally, we make emphasis on the fact that the displacement field *W*_{i} depends explicitly on the position of the *i*th pillar, since the integration limits *ϱ*_{i1}, *ϱ*_{i2}, *φ*_{i1} and *φ*_{i2} must be determined for each combination of pillar location (*r*_{i} ,*θ*_{i}) and point (*r*,*θ*). For instance, *ϱ*_{i1} and *ϱ*_{i2} are the solutions for *ϱ* of quadratic equation
*ρ*_{i}=*ρ*_{i}(*r*,*θ*,*r*_{i},*θ*_{i}) given in equation (2.4), whereas *φ*_{i1} and *φ*_{i2} depend on the location of the point (*r*,*θ*). On the one hand, if (*r*,*θ*) lies inside the contact region of the *i*th pillar, we have *φ*_{i1}=0 and *φ*_{i2}=*π*, and, on the other hand, if (*r*,*θ*) is located outside the contact region of the *i*th pillar, we find

In turn, each pillar is compressed because of its confinement between the lens and the substrate. Herein, we consider that the compression of the *i*th pillar is approximated by Hooke’s Law, which gives a change of height equal to *δ*_{i}=4*hF*_{i}/(*πd*^{2}*E*_{p}). As well, *F*_{i} is the load exerted over the *i*th pillar in the contact region, which reads
*ρ*_{i} and *ϕ*_{i} represent, respectively, the radial and angular coordinates at the surface of the *i*th pillar, relative to the pillar centre (*r*_{i},*θ*_{i}) and restricted to the ranges *ρ*_{i}∈[0,*d*/2] and *ϕ*_{i}∈[0,2*π*]. In order to transform, the stress field over the entire domain *σ*(*r*,*θ*) into *σ*(*r*_{i},*θ*_{i},*ρ*_{i},*ϕ*_{i}), one should express *r* and *θ* for a given relative position (*ρ*_{i},*ϕ*_{i}) and a pillar (*r*_{i},*θ*_{i}) using the following expressions:

Additionally, the total load *F*_{T} reads

Since, the total displacement field is also composed of
*δ*_{i}, imposing a total indentation, given by equation (2.6) and evaluated at the top of the pillars where *g*^{°}(*r*,*θ*)=*h*, allows one to recover the stress field *σ* inside the contact region. Herein, *σ* is computed numerically using a superposition method of discrete pressure elements, for which a segmentation of each contact region (the top of each pillar) into square subregions of constant constraint is performed. Details are given in the electronic supplementary material. A similar approach has been introduced and validated in previous studies [38,39], for which a Hookean hypothesis and a uniform pressure has been applied to each pillar. Even though our methodology retains the Hookean behaviour for each pillar, it allows the appearance of non-axysimmetric pressure fields for the non-centred pillars (*i*>0), thanks to the discretization of each contact region. This procedure provides a more realistic approach for several cases, including small total loads *F*_{T} and small surface fractions *Φ*, but also the effect of the finite size of the contact area, for any load and any surface fraction. The corresponding results are presented in the following section.

## 4. Results

In figure 3*a*, the contact region and the stress field for a small surface fraction *Φ*=0.03 is depicted for *E*_{p}=*E*_{s}=1.6×10^{6} Pa, *ζ*=0.3 μm, two particular indentations and all other parameter values given in table 1. The contact radius is approximately equal to *a*≈*e*+*d*/2=26 μm. For the pillar at the centre of the contact region, it turns out that the stress field is equivalent to a soft-flat-punch (SFP) stress field. The stress field is axissymetric, with a large gradient, i.e. small *σ* around the central zone *r*≃0 and very large *σ* near the edge of the pillar *θ*=0. This field is eccentric with respect to the centre of the pillar, leading to slightly oblate iso-stress contour lines, which recover the centric and circular shape as the border of the pillar is reached. The increase of stress magnitude, from the centre to the border of the pillar, is very similar to the one observed for the central pillar. The stress *σ* at the central region of both pillars seems to follow a Hertzian-like behaviour, a square root profile calculated with the stress *σ*_{0} at *r*=0 and the contact radius *a* obtained from the simulations. As expected, relatively high magnitudes of the ‘discrete’ stress field are observed in this case, since the total load is concentrated over seven pillars, considering the geometry of the system. For comparison, the Hertzian stress profile is plotted in the figure.

For the same indentation and Young’s modulus, the contact region and the stress field for a large surface fraction *Φ*=0.82 is illustrated in figure 3*b*. The contact radius is approximately equal to *a*≈5*e*+*d*/2=23 μm, slightly smaller than the radius of the small *Φ*=0.03 case. The behaviour of *σ* at the central pillar deviates strongly from the SFP profile, with a nearly constant stress magnitude at the centre region 0≤*r*≤*d*/4, that diverges rapidly as the border is reached (a stress profile similar to the ones found by Gecit [37]). Nevertheless, the stress profile remains axisymmetric and its magnitude is relatively small over the entire lid of the pillar. For the other pillars, which are not at *r*=0, the local stress field is only symmetric with respect to the corresponding plane *θ*=*θ*_{i} of the *i*th pillar. Oblate iso-stress contour zones, nearly crescent-like shapes, are found, mainly for the pillars near the border of the contact region. Once more, considering *σ* at the central region of each pillar, a Hertzian stress field is observed, with the usual square root profile computed with a stress *σ*_{0} at *r*=0, very close to the true Hertz value, and a contact radius *a* that is significantly smaller than the true Hertz contact radius.

It is important to note that, while a constant indentation *ζ* is kept for the cases presented in figure 3, a reduction of *σ*_{0}, of one order of magnitude (from 33 to 8 kPa) when *Φ* is increased (from 0.03 to 0.82), is observed. As well, the same increase of *Φ* produces a reduction of the apparent contact region and the contact radius *a* (from 26 to 23 μm), but the real contact area given by the number of pillars in contact has increased, even though the total load *F*_{T} has risen (from 3 to 10 μN).

In figure 4*a*, the dependence of the stress at the centre of the contact region *σ*_{0} on the surface fraction *Φ*, for an indentation of *ζ*=0.3 μm and the remaining parameter values given in table 1, is depicted. Going from right to left, when the surface fraction is *Φ*=1, the Hertzian value of *σ*_{0} is found. As *Φ* decreases, the total load distribution over a smaller number of pillars induces a slightly larger apparent contact region, but it provokes a stress concentration, represented by an increase of *σ*_{0}, due to the reduction of the real contact area and the number of pillars in contact. If we continue to diminish *Φ*, a monotonic growth of *σ*_{0} is observed until it saturates to a plateau. This saturation, which corresponds to an SFP behaviour, is completely attained when the surface fraction *Φ* becomes small enough to leave just a single pillar within the contact region. In figure 4*a*, the Hertz-to-SFP transition is presented for different values of the ratio *E*_{p}/*E*_{l}, for the same indentation *ζ*. A reduction of the system stiffness *E*_{p}/*E*_{l} enhances a decrease of the *σ*_{0} stress levels observed for both limits, Hertz and SFP. Despite this vertical shift, the general trend remains unchanged whatever the value of the ratio *E*_{p}/*E*_{l}.

In figure 4*b*, the real contact area *πa*^{2}, for a Young’s moduli ratio *E*_{p}/*E*_{l}=1 and all other parameters given in table 1. The following description remains qualitatively valid for any value of the ratio *E*_{p}/*E*_{l}, presented in this work. Note that data corresponding to very small values of the surface fraction *Φ*<0.01 are included but not visible, because they are superimposed near the origin since the real and apparent contact areas are equal and correspond to the top surface of a pillar *ζ* is small, both the apparent *πa*^{2} and the real *F*_{T}, the indentation *ζ* and the apparent contact area *πa*^{2} are increased, *Φ*[*πa*^{2}], for all the values of *Φ*. For the highest values of the surface fraction *Φ*, the real contact area *Φ*, the number of pillars in contact is reduced, provoking a drop of *Φ*, which is smaller for the highest Young’s moduli ratios *E*_{p}/*E*_{l}, the real and apparent contact areas suddenly take the value of the SFP contact, i.e. the area of the top of a single pillar

## 5. Asymptotic cases

There are special situations for which the size of the contact region, the contact radius *a*, can be predicted analytically. They correspond to the two limiting cases of the surface fraction, which are depicted in figure 1:

### (a) Hertzian contact

Hertzian contact [1] occurs either when *a*≪*d*/2. Under these conditions, the problem simplifies to an axisymmetric geometry for which the pillar array becomes a layer of finite thickness [40]. Considering infinite friction between the two bodies, the well-known stress field for Hertzian contact reads
*σ*_{0H}, perceived at the centre of the contact region *r*=0, is computed from the total indentation *ζ* as follows:
*σ*_{0H} is equally sensitive to a change in stiffness of any component of the system (lens and compound substrate). In addition, as it is explicitly indicated in equation (5.2), a larger value of the stress *σ*_{0H} is expected when an important indentation *ζ* is imposed. It is also well known that the total force is *F*_{T}=(2*π*/3)*Rζσ*_{0H}.

### (b) Soft-flat-punch contact

When *E*_{p}≫*E*_{l}, we can consider that the pillars/substrate system is rigid *δ*_{0}=*w*_{s}(*r*,*θ*)=0 and, if the top of the pillar is completely in contact with the lens *a*=*d*/2, the problem is described by a uniform normal displacement generated with a rigid flat punch [1,35,36]. The solution for the aforementioned problem corresponds to the superposition of the SFP solution, within a circular region at which uniform displacement is imposed. This situation, for which the well-known stress field reads
*ζ* that is smaller than the pillar height *h*. Herein, the stress magnitude *r*=0, the centre of the pillar, is computed from the total indentation *ζ* as follows:

By contrast, the case for which *E*_{p}≃*E*_{l} does not presents an analytical model, and must be studied numerically [37]. Herein, we will name this case the ‘soft-flat-punch’ contact problem, for which we propose the following simple model. Considering that the total indentation *ζ* equals the addition *w*_{l}(0,*θ*)+*w*_{s}(0,*θ*)+*δ*_{0}, and the expressions *F*_{0}=*πd*^{2}*σ*_{0S}/2 and *w*_{l,s}(0,*θ*)=*πdγ*_{l,s}*σ*_{0S}/2, which come from the rigid-flat-punch solution (the latter by making *ζ*=*w*_{l,s}(0,*θ*) and *δ*_{0}=4*hF*_{0}/(*πd*^{2}*E*_{p}), we can write
*γ** constructed as follows:
*E*_{p}≫*E*_{l} and recalling that *E*_{s}=*E*_{p}, which corresponds to a rigid flat punch and an elastic half-space since the pillar/substrate can be taken as a rigid body, and also considering that *ν*_{l}∼*ν*_{p} and *h*∼*d*, leads us to recover

For the contact between the lens and a soft pillar, employing the effective elastic constant, we can define the stress at *r*=0 as
*ζ* and considering *E*_{s}=*E*_{p}, the comparison between the previous definition and equation (5.4) yields the following expression for *σ*_{0S} in terms of the ratio of Elastic moduli *E*_{l}/*E*_{p}:
*E*_{l}/*E*_{p}<10^{−3}, a saturation towards the Boussines–Cerruti solution is observed *E*_{l}/*E*_{p} increases, the value of *σ*_{0S} falls down, reaching a new plateau of level *σ*_{0S}≃0, when *E*_{l}/*E*_{p}>10^{3}. Therefore, in the latter regime, for which the lens is extremely soft compared with the pillar/substrate stiffness, the stress *σ*_{0S} is negligible.

## 6. Discussion

Starting with a relatively slight load, provoking the contact of the lens with a single pillar, the SFP contact behaviour is observed until a second pillar (six pillars at different angular position but the same radial coordinate) is touched by the lens, which is due to an increase of the load and, consequently, the total indentation. Under this circumstances, from the contact condition *Δ*(*r*,*θ*)=0, we find the relationship between the indentation *ζ* and the radial position *r*_{S}, at which a second pillar must be placed in order to be barely touched by the lens
*Π*=*π*+4*h*/[*d*(*γ*_{s}+*γ*_{l})*E*_{s}]. For a given *ζ*, we can estimate the radial position *r*_{S} at which a second pillar may be reached and, consequently, the corresponding pitch distance *e*_{S}=*r*_{S} and the SFP threshold surface fraction
*ζ* and surface fraction *Φ*, is such that *Φ*<*Φ*_{S}, only one pillar must be in contact with the lens.

Considering that *R*≫*r*_{S}≫*d*/2, for relatively large indentations *ζ*≫(3/2)^{3}*R*(*d*/*Π*)^{2}, we can approximate the radial position with the expression *Φ*_{S}∼*d*^{2}/(*Rζ*).

Now, from the opposite situation, we define *Φ*_{H} as another threshold surface fraction, above which a Hertzian contact can be used to approximate the magnitude of the stress field at the centre of each pillar. For high values of *Φ*, this behaviour has been discerned qualitatively from the numerical results, as it was shown in figure 3*b*. Nevertheless, for a given indentation and a set of elastic parameters, a quantitative estimation of *Φ*_{S} is only achieved numerically.

Based on this insight, we introduce the reduced variables *Φ**, the reduced surface fraction, defined as
*σ*_{0H} and *σ*_{0S} are the limiting stresses at *r*=0 for the Hertz and the SFP contact conditions, i.e. when *Φ*<*Φ*_{H} and *Φ*>*Φ*_{S}, respectively. Herein, both stress values *σ*_{0H} and *σ*_{0S} are obtained numerically for *Φ*∼1 and *Φ*∼0, respectively, whereas *Φ*_{H} and *Φ*_{S} result from fitting the data of the curves shown in figure 4*a* and their equivalents for the different values of indentation *ζ* and Young’s modulus ratio *E*_{p}/*E*_{l} given in table 1.

As shown in figure 6*a*, all the data, including the curves in figure 4*a* and the results for different values of *ζ*, collapse to a master curve when the reduced variables given in equation (6.3) are used. The following rational function,
*Φ**≥1, and an excellent quantitative agreement for *Φ**∈[0,1−*Φ*_{S}/*Φ*_{H}]. Therefore, the stress concentration and the growth of *σ*_{0}, perceived during the transition from a Hertzian contact, along a discrete contact region, to a single punch-like contact, behaves as the inverse of the cube of a quantity that measures the separation from the limiting regimes: the difference of the surface fraction from the SFP threshold divided by the Hertz threshold value. Outside this transition zone, we recover the limiting cases: on one hand, *Φ**=0 implies that *σ*_{0}(*Φ*)=*σ*_{0S} is expected, whereas on the other hand, *Φ**=(1−*Φ*_{S})/*Φ*_{H} implies that *σ*_{0}(*Φ*)≈*σ*_{0H} should represent a good illustration of the discrete contact phenomenon.

The dependencies of *Φ*_{S} and *Φ*_{H} on the indentation *ζ* and Young’s modulus ratio *E*_{p}/*E*_{l} are depicted in figure 7*a* and *b*, respectively. For a fixed value of the ratio *E*_{p}/*E*_{l}, both threshold parameters *Φ*_{S} and *Φ*_{H} follow a similar monotonical decreasing tendency as the reduced indentation *ζ*/*h* increases. The effect of the ratio *E*_{p}/*E*_{l} seems to be restricted to a shift towards larger values of both thresholds, strongly for *Φ*_{H} and almost indistinguishably for *Φ*_{S}. It is also important to note that the values that *Φ*_{S} takes are an order of magnitude smaller than the values presented by *Φ*_{H}.

For very small indentations *ζ*/*h*<0.025, the determination of the values of *Φ*_{S} and *Φ*_{H} becomes delicate, since the Hertzian and SFP stresses at *r*=0 present a very close magnitude *σ*_{0S}∼*σ*_{0H}. Matching equations (5.2) and (5.7) gives the threshold indentation *ζ*_{T}/*h*=[*γ**/(*γ*_{l}+*γ*_{s})]^{2}[*d*^{2}/*Rh*], which for the parameters given in table 1 varies between 0.004≤*ζ*_{T}/*h*≤0.014 and acts as an inferior boundary for the swept range of indentations. Nevertheless, an estimation of both parameters may be provided for *ζ*/*h*<0.025. From the curve of *σ** as a function of *Φ** for the last performed fit, i.e. for *ζ*/*h*=0.025, *Φ*_{H} corresponds to the value of *Φ* for which *Φ*_{S} is to the value of *Φ* that makes *Φ*_{H} and *Φ*_{S} can be approximated for *ζ*/*h*<0.025. This results are shown in figure 8 and will be discussed in the conclusion section.

We introduce the reduced variable *a*=*d*/2 for the SFP situation. In figure 6*b*, all the data, including the curves in figure 4*b* and the results for different values of *ζ*, show a similar behaviour when the reduced variable, given in equation (6.5), is used in combination with the surface fraction *Φ*. The following power law
*Φ*∈[10^{−2},10^{0}]. Note that only the contact area data for which *b*, with logarithmic scales, since for an SFP contact situation

## 7. Conclusion

In the present work, the elastic contact between a spherical lens and a patterned substrate (hexagonal lattice of pillars) has been studied in depth, using a superposition method of discrete pressure elements. For relatively small indentations, a transition from a Hertzian elastic contact behaviour towards a SFP contact, considering a uniform displacement of the pillars, was observed when the surface fraction of pillars on the substrate is increased. With the use of reduced variables for the stress at the centre of the central pillar and the surface fraction occupied by the pillars, all the results collapse into a master curve, for which a rational function describes the passage between the aforementioned limiting contact regimes. Additionally, a simple model to describe the behaviour of a soft-flat-punch contact between the lens and a single elastic pillar has been presented as a function of the pillar geometry and the elastic properties of the lens and the pillar-substrate. Special attention has been made on the effect of the ratio between Young’s moduli of both bodies.

Finally, from the reduced surface traction, two threshold values are obtained: the Hertzian *Φ*_{H} and the SFP *Φ*_{S} limits. To summarize these results, a phase diagram is depicted in figure 8 in which axes correspond to the surface fraction *Φ* and the reduced indentation *ζ*/*h*. The SFP and Hertzian contact regions, which had been discerned from the numerical results, are shown. In brief, for a given indentation *ζ*, if the surface fraction *Φ* is greater than *Φ*_{H}, the Hertzian contact offers a good approximation of the discrete contact phenomenon, whereas, if *Φ* is smaller than *Φ*_{S}, the system must have only one pillar in contact with the lens. In addition, also illustrated in figure 8, an analytical expression for the SFP threshold has been deducted from purely geometric considerations, which shows a good agreement with the numerical results, mainly in the large indentations regime. In this region, a simple theoretical approximation is also represented, for which the effect of the elastic parameters seems to be negligible. According to the theory developed by Greenwood and colleagues [5,6] based on a statistical approach, the dimensionless parameter *ζ*, indicates the influence of the roughness on the contact mechanics. For the hexagonal array of cylinders, as the sphere is considered to be smooth, *α*, which has been given the empirical value of *α*=0.05 [6], below which the Hertz theory can be applied with confidence, a vertical line in the phase diagram shown in figure 8 at the value of *ζ*/*h*=(2*α*)^{−1} is found. Even though this assumption should be valid for large indentations *ζ*/*h*≥10, also corresponding to large contact regions, this procedure wipes out completely the impact of the surface fraction *Φ*, at the indentation regime presented here. Fortunately, a second dimensionless parameter *μ* is also proposed by Greenwood and colleagues [5,6], required to forecast precisely the contact behaviour. For rough nominally flat surfaces, *μ* depends only on the geometry, i.e. the roughness, the asperity density and the RMS curvature of the asperities, and only generates a secondary effect on the stress and displacement fields when the contact zone spans infinitely compared with the nominal size of the asperities. For the case presented herein, we expect that *μ* will take the form *Φ* appears directly.

Thus, this analysis, which is a result of the qualitative comparison with Greenwood’s theory, indicates that *Φ* and *ζ*/*h* are the correct dimensionless parameters required to describe the passage from Hertzian to SFP contact on periodically rough surfaces.

The methodology that we have developed allowed us to gain some insight on the discrete contact phenomenon. Nevertheless, it only represents some first steps and several important issues remain to be addressed. For instance, the case of large indentations, which are comparable in size with the height of the pillars *ζ*∼*h*, must be analysed separately. Under these circumstances, an intimate contact between the lens and the patterned surface, which has been studied with experiments [3,31,33,41], may be triggered, and the role of adhesion should be taken into account. Our efforts are currently being focused on this problem, and the corresponding analysis will be presented in a future publication.

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## Authors' contributions

R.L.A. carried out the theoretical work, performed the numerical simulations, participated in the analysis of results and drafted the manuscript; E.R., L.L., F.R. and C.P. conceived of the study, participated in the analysis of results; F.R. and C.P. coordinated the study and helped draft the manuscript. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This project was carried out thanks to funding by the ANR (ANR-11-BS04-0030 project).

## Disclaimer

The views and opinions expressed in this article are those of the authors.

## Acknowledgements

We wish to thank T. Salez for fruitful and very interesting discussions.

- Received March 31, 2016.
- Accepted August 19, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.