Fractal surface-based three-dimensional modeling to study the role of morphology and physiology in human skin friction

Human skin plays an important role in our perception of contact made throughout the day. In this work, we study the interplay of various morphological and physiological factors that dictate its contact mechanics. A hybrid computational-empirical approach is developed to model skin friction and to understand the role of roughness in contact mechanics of human skin variations in structural properties. A fractal rough surface is considered to model the skin surface. A layered three-dimensional finite element model is generated with stratum corneum, viable epidermis, and dermis which is further used to determine its mechanical response under normal loading. An empirical relationship is then used to predict the coefficient of friction. The effects of varying the Young's modulus, roughness parameters, thickness of stratum corneum and domain size are studied. Simulations are performed for multiple realizations to quantify statistical variations. Our results show that the proposed approach can replicate several experimental findings from the literature such as the decrease in skin friction with humidity and increasing roughness. The study provides qualitative and quantitative insight into the role of roughness in the contact mechanics of human skin while accounting for the effects of micro-level interfacial phenomena.


Introduction
Human skin acts as the first line of interaction for any human contact with a counter-surface.As such, it plays an important role in the underlying contact mechanics of such interactions.Skin is a complex organ made of different layers and its response in any kind of interaction is a function of many individual factors like age, sex and ethnicity as well as environmental conditions like temperature and moisture.Thus, the mechanics of skin and its tribological properties are an important area of research [1][2][3].
The relative humidity to which human skin is exposed affects its contact properties [4][5][6].Age is another important factor that affects the biomechanics of the skin.This is due to the changes in physiological contents, the roughness of skin, moisture-holding capacity, etc [7][8][9][10][11][12][13]. For example, human skin undergoes a transformation from geometric isotropy to anisotropy as a person ages [14][15][16][17].Also, the physiology and structure of male skin are different from female skin.For instance, female skin is less thick than male skin as a result of the composition of collagen fibers [18].Other factors affecting the morphological and physiological properties of human skin include day-to-day activities, use of cosmetics or treatments, etc.
In general, laboratory experiments on the response of skin include the effects of many or all the aforementioned factors, so understanding the role of individual factors is difficult.Here, computational modeling provides an efficient alternative to experiments to understand the role of individual factors such as age, sex, etc.Also, advances in constitutive modeling of skin, finite element solvers, and the more widespread availability of computational resources have made it easier to realize detailed models of skin.Another advantage of computational modeling is the ability to study the role of individual skin surface parameters on its friction response.
An important ingredient of a computational model of skin is its constitutive model.Since skin undergoes large deformations, the use of nonlinear constitutive models to describe the behavior of skin becomes important.This has been recognized by many researchers.Limbert [14] provides a detailed review of microstructural and phenomenological approaches, including nonlinear constitutive theories for human skin and its inelasticity, growth and thermoelasticity.Li [19] in a review article points to existing work on constitutive modeling of human skin and the lack of nonlinear continuum models related to damage.Flynn and McCormack [20] propose a nonlinear three-layer finite element model for skin wrinkling and compare their results with experiments.Hendriks et al [21] combine finite element studies with suction experiments to determine the nonlinear material model parameters for skin.Such an approach based on suction experiments is also used by Diridollou et al [22], who find mechanical parameters such as Youngʼs Modulus, initial stress, and inelasticity of skin.
Like any other surface, the human skin surface is not perfectly flat.It is composed of primary, secondary, and tertiary Langer lines that lead to surface roughness.They are crucial in determining the geometric isotropic and anisotropic features of human skin [11].The roughness of human skin and that of the contacting surface both play a significant role in the contact mechanics of human skin.This is one of the driving factors in our perception of objects and applications related to haptics.Arvidsson et al [23] account for both material properties and changes in wavelength of roughness in their experimental work on the perception of surfaces.They report that the coefficient of friction is a function of contact area between the skin and the surface.Chimata and Shwartz [24] find in their experimental studies that surface roughness is the primary factor controlling coefficient of friction.Sergachev et al [25] propose an empirical approach to determine skin elasticity through surface transition from asperity-level contact to full contact.
Recognizing the importance of roughness, experimental studies and micromechanical models of human skin that take into consideration the role of roughness and the individual layers of skin have been receiving more attention in the literature.There are studies to quantify the contact parameters on fingers during touch and grip [26][27][28].Levya-Mendivil et al [4,29,30] describe an image-based two-dimensional nonlinear computational model of human skin and use it to understand the parameters that influence skin friction.They find that skin roughness and its layered structure indeed influence its contact behavior.Diosa et al [31] show the importance of roughness in how aging impacts the mechanical and tribological behavior of human skin using an image-based computational model.Given the randomness of roughness over skin and the complexities involved in obtaining actual skin surfaces in large numbers, representative models of rough surfaces come as a convenient tool.To this end, Perssonʼs theory [32,33] provides directions to evaluate the friction response of human skin based on a fractal model of roughness.
Given the complexity of human skin contact, our goal is to understand the role of individual factors such as skin roughness, thickness, mechanical properties of individual layers, location on the body, interfacial conditions, etc.As mentioned earlier, these parameters vary from one individual to another, making it difficult to accurately predict the contact outputs of human skin.
In this work, we develop a hybrid fractal-based computational-empirical approach to model the contact response of rough human skin in contact with a flat rigid surface.The study focuses on the structural and geometrical impacts of the roughness of human skin under various conditions.
Skin is modeled as a fractal surface, which is used to generate a three-layered finite element model that includes the stratum corneum (SC), the viable epidermis (VE), and the dermis as shown in figures 1 and 2. Each layer is treated as an individual hyperelastic material.The nonlinear finite element model is used to predict the response of human skin under a normal displacement.Then, an empirical relationship based on the results reported in the literature is used to predict the coefficient of friction from the normal force.This hybrid approach has two major advantages.First, the nonlinear FE model is able to fully capture the geometrical and material complexities associated with a rough, layered, hyperelastic domain.Second, the use of an empirical relationship makes it possible to capture interfacial effects in a simple way and makes it unnecessary to simulate the tangential response of the interface.In addition, we simulate multiple realizations of each surface to fully represent the random nature of a rough surface like skin.With the use of a fractal model to define skin roughness, the role of magnification also becomes important.The proposed approach helps us to determine the effect of skin roughness at magnified scales.
Using this hybrid approach, we conduct studies to determine the roles of several factors in human skin friction: humidity, the thickness of the stratum corneum, roughness, and sample size (i.e.scaling).Our results show that the proposed approach is able to capture a wide range of experimental observations.
In section 2, we describe the development of the computational model of skin, including the geometry, finite element discretization, constitutive model, and loading conditions employed to predict the skin coefficient of friction.Results from simulations and discussion are presented in section 3, followed by concluding remarks in section 4.

Morphology of human skin
Human skin is a complex organ with its different layers composed of various types of cells [34,35].The top layers are the stratum corneum (SC), viable epidermis (VE) and dermis, as shown in figure 2. In general, the SC is 5-30 μm thick [36].The SC in males is found to be 25% thicker than in females [34,35].The SC is followed by the VE which is 30-80 μm thick, and the dermis, which is 1-4 mm thick [36], below which the blood vessels lie [34,37].We model a structure that is 1.2 mm in height, of which the first 0.026 mm is the SC and the rest is VE and dermis as in [4].

Fractal surface generation
Many rough surfaces exhibit geometric self-similarity and self-affinity, i.e. similar appearances of surfaces are observed under various degrees of magnification.One way to describe a fractal rough surface is using its fractal dimension, D, which describes the space occupancy of an object.For a three-dimensional fractal surface the value of D is 2 <D < 3. Alternately, one can use the Hurst exponent, H, which quantifies the relative tendency of a time series to either regress strongly to the mean or to cluster in a direction.The Hurst exponent can be calculated as H = 3-D, 0 < H < 1 [38].
The statistical properties of a fractal surface are not affected by scale transformation [32].Consider a surface that is nominally in the x − y plane.For a magnification factor Ω, the change in the x, y, and z coordinates can be represented in terms of the Hurst exponent as: To illustrate fractal scale transformation, figure 3 shows a quantitative example of fractal surface magnification.A surface profile of size 5 mm with RMS roughness 25 μm and fractal dimension of 2.11.This is accompanied by a close-up view at 5x magnification or 1 mm size with the same properties.The smaller and sharper asperities become visible as the magnification is increased, i.e. as the length scale is reduced.
In this work, we generate the height coordinates (z) of rough surfaces using a MATLAB [39]-based script as described by Kanafi et al [40].This script requires the surfaceʼs RMS roughness, Hurst exponent, the in-plane dimensions, and the number of coordinates.The generated surface is then converted into a CAD model using Ansys [41].
Persson [38] applied this approach to various natural and engineering surfaces and found that the fractal dimension of such surfaces, over a wide range of length scales, is D f = 2.15 ± 0.15.For human skin, Persson finds the value of Hurst exponent to be equal to 0.89, D f = 2.11.For our studies of volar forearm we assume representative values of RMS roughness, σ = 25 μm and fractal dimension, D f = 2.11 [5,20,42].For human finger skin, Ayyildiz et al [33] shows D f = 2.14 and H = 0.86, where H = 3-D f , is the Hurst exponent for a three-dimensional surface.
To understand the contribution of asperities of different length scales towards friction, we perform simulations on surfaces of two sizes, 5 mm × 5 mm and 1 mm × 1 mm (as shown in figure 2) with the same statistical properties.
Based on Greenwood and Tripp's sum surface assumption [43], contact between two rough surfaces is modeled as contact between an equivalent rough surface and a rigid plate.In this study, we only consider the geometric properties of the skin for our simulations.The study is limited to understanding the deformation of the skin surface with the assumption that the counter surface is an asperity of infinite radius.

Constitutive modeling
Human skin has a complex layered structure which is defined by a network of collagen fibers.As such, it is challenging to identify a constitutive model that can perfectly replicate its behavior.To this end, researchers have used different objective-guided approaches.Limbert [11] in his extensive review of constitutive modeling of human skin describes various aspects of isotropic/anisotropic hyperelastic and viscoelastic modeling.In particular, hyperelastic models of human skin have been used by some researchers to capture skin nonlinearity.Levya-Mendivil et al [4,29,30] use a Neo-Hookean hyperelastic model in their studies.Anniadh et al [44] describe an anisotropic hyperelastic model for human skin.
In this study our interest is in the effects of structural and geometrical properties of human skin at slow loading rates.We therefore choose the Neo-Hookean hyperelastic strain energy potential [45] to model the mechanical response of skin, in line with other recent studies.For example, Levya-Mendivil et al [4,29,30] use this model to understand contact interactions of microscopic skin using a two-dimensional model.Limbert and Kuhl [46] use this material model to understand the effect of compression induced wrinkling using a three-dimensional model.Recently Diosa et al [31] use this approach for three-dimensional modeling of human skin contact with a spherical indenter.
The Neo-Hookean hyperelastic strain energy potential is given by where I 1 and J are the first and third invariants of the right Cauchy-Green deformation tensor C = F  F, with F being the deformation gradient.The constants c 10 and κ 0 are material parameters that can be defined in terms of the Youngʼs modulus, E, and the Poisson's ratio, ν, as [4,45]: In the context of contact between surfaces, the modulus E is given by [4]: ( ) with E k and ν k representing the Young's modulus and Poisson ratio of surface k.In our case, since we are interested in contact between the stratum corneum layer of skin (surface 1) and a rigid surface (surface 2), this reduces to: Levya-Mendivil et al [29] provide an extensive review of reported values of Young's modulus and Poisson's ratio for human skin.They propose two values of Young's modulus for the stratum coerneum based on the level of relative humidity, as shown table 1.We adopt these values in our studies.In addition to this, to understand the effect of the value of Poisson's ratio for SC, we perform additional studies of wet skin with ν = 0.49.For all the finite element simulations, we consider the VE and dermis to have identical mechanical properties, E dermis = E VE = 0.6 MPa and ν dermis = ν VE = 0.3 [4,29,30].

Finite element model
We simulate normal contact between a sample of rough skin and a flat analytical rigid surface using the finite element software ABAQUS [48].In order to establish the mesh required for the skin component, we perform a mesh convergence study for three different in-plane mesh sizes 50 × 50, 100 × 100 and 150 × 150.In the thickness direction, a biased mesh is used with a higher mesh density at the top.The geometries are meshed using Hypermesh [49].
We perform a quasistatic simulation to model contact between the surfaces.This is to avoid any inertial effects in the simulations.In such simulations, the velocity of the rigid plate compressing the skin should not induce dynamic effects.In general, it is desirable to keep the kinetic energy at or below 1% of the internal energy.A study to establish the quasi-static nature of the simulations is performed as in [50,51].
Figure 4 shows a typical realization of skin with the associated boundary conditions and the finite element mesh.The bottom of the skin is fixed to replicate the physical process of skin attachment to the body, signifying that the effect of contact is restricted to the top layers of the skin.Symmetry conditions are used on the lateral edges to replicate an infinitely long problem.

Empirical model for friction
The coefficient of friction is a macroscopic measure of various complex interfacial phenomena and depends on many variables.This involves the complex physiological properties of skin, environmental conditions, the nature of interacting surfaces, the phenomenon of adhesion, etc.To avoid the complexity associated with modeling and simulating the tangential interaction at the interface, we use an empirical approach to determine the coefficient of friction from the normal force predicted by the finite element model.Thus, we use macroscopic quantities from experiments to calculate CoF using the normal force obtained from compression of microscopic asperities.This not only greatly simplifies the CoF calculation, but has the added advantage of encapsulating in it information about the various interfacial complexities.
The force of friction in human skin interactions is generally described [52,53] as a combination of interfacial forces like adhesion (denoted F μ,int ) and deformation forces (denoted F μ,def ): These contributions are due to interfacial conditions and deformations depend on many complex factors like sebum content, hydration, sweat, effects of creams and conditions, hair density [54,55], etc.These conditions differ from one individual to another.In this study we seek to evaluate the combined effect of these conditions on friction.Toward this end, we consider an alternative approach, one that has been adopted by many researchers, consisting of a power-law type model that describes the overall response of human skin combining the interfacial and deformation contributions: where C and m are constants determined by the tribological properties of the system and P is the normal force.Then using Coulomb's law, the coefficient of friction can be written as where n = m − 1.This power law approach to modeling the combined effects of adhesion and deformation has been used by many researchers.For example, van der Heide et al [54] suggest the power-law model for skin friction with specific coefficients for adhesiondominated (n = −1/3) and deformation-dominated (n = 1/3) problems.Experimental results from [28,36,52,53,56] confirm that the power-law function is able to describe skin friction adequately.While C is found to be 1 in most experiments, n varies.Koudline et al [57] reported a value of n = −0.28 for forearm skin.Sivamani et al [58] reported a value of n = −0.32 for experiments with skin on the back of the finger.Comaish and Bottoms [59] conducted experiments on the dorsum of the hand and found the value of n to be less than 1.For the hand, El- shimi [60] reported the value of n to be between −0.33 to 0. All these reported values were in line with Wolfram's [61] value of −0.33, for general skin.This huge variation is highly dependent on the individual subjects of each experiment.Factors like age, sex, ethnicity, type of cosmetic use, location on the body, etc., drive the resulting value.The counter surface and the adhesion at lower loads also play an role in this.In general, dry skin is reported to have a higher value of n than wet skin [28,36].
In this study, we use representative values for C and n, depending on the condition of skin and the location of interest.Here, the role of deformation (which depends on the roughness of skin and its constitutive model) is taken into account by evaluating the normal load from finite element simulations.The values of C and n used to evaluate the CoF are representative of the interfacial conditions of the skin surface, since these are experimentally obtained.Derler and Gerdhart [36] conducted an extensive review of all such experiments and reported a set of average values for n depending on location and state of skin.They plotted these values for forearm skin and their analysis showed that wet forearm skin has a n value of −0.53, and standard error of 0.37.Dry skin on the other hand has n value of −0.1, and standard error of 0.21.Therefore, we use a value of n, −0.53 and −0.1 for wet and dry state of skin respectively, while C is considered to be 1.Derler et al, [56] also showed that for pushing experiments of glass against wet rough finger skin, the average value of n is −0.35.We use the same for our comparisons of finger and forearm skin.

Description of studies
Figure 5 shows a description of the studies we perform.Each box shows the factor being considered and the corresponding model parameters being varied.
Additionally a study is performed to understand the role of anatomical location by comparisons between responses of forearm and finger skin.This takes into account the combined effect of variations in roughness properties and exponents used for calculating CoF.

Results and discussion
This section describes the results in detail.First, the results of the studies done to establish the computational model with mesh convergence and loading rate studies are described.Then, results from the various parametric variations of the model are presented.These variations are replicas of real-world scenarios in terms of values.Wherever necessary, multiple realizations of the skin surface with same properties are performed.The response curves shown in this section represent the average of these realizations and the error bars represent the variance.

Establishing the model
We perform a mesh convergence study to determine the number of skin surface coordinates required for the simulations.The sample of skin is loaded normally for three in-plane mesh sizes: 50 × 50, 100 × 100, and 150 × 150.The resulting force-displacement response in figure 6 shows there is no significant difference from the three mesh sizes.Based on this we consider 100 × 100 as the mesh size required for subsequent simulations.
To establish the loading rate to ensure the quasistatic nature of the simulations, a velocity study is performed as described in [51].A loading rate of 10 mm/s leads to a smooth response and is in agreement with lower loading rates.Also, the kinetic energy is within 1% of the internal energy, which means that inertial effects are negligible at these rates.Based on these results, we consider 10 mm/s as the loading rate for subsequent simulations.

Role of roughness
No surface is truly flat and this is especially true of skin.
To study the role of skin roughness, we perform finite element simulations of wet skin using the model described in section 2. We use two values of roughness, σ = 25 μm and σ = 50 μm.Skin roughness is one of many factors that change with ageing.Roughness increases with age [11,42,62] and there is a visible increase in wrinkles.Understanding how roughness affects friction is therefore important.Experimental studies by Mabuchi et al [27] show that the coefficient of friction decreases with age.As a person ages, it is common to have problems  holding objects, which is generally due to age-related weakness and other factors [63].The results in figure 7(b) show that the decrease in the CoF from increasing roughness could be an important contributor.There is a significant difference in the predicted coefficient of friction between the results from the two roughness values and this difference remains considerable even as the displacement increases.This phenomenon is discussed further in section 3.7 in the context of finger skin and the ability to grip objects.

Role of hydration
Simulations for understanding the effects of moisture content on human skin are carried out by varying the elastic modulus, as described in section 2.3.The moisture content is a function of relative humidity.Dry skin surfaces have a lower relative humidity compared to wet surfaces, and therefore a higher Youngʼs modulus.Figure 8(a) shows the normalized load-displacement comparison for dry and wet skin.The responses are similar, showing an increase in the load with displacement.The average normalized forces are quite comparable at different levels of displacement despite the large difference in the Young's modulus values of dry (370 MPa) and wet skin (0.6 MPa). Figure 8(b) shows the corresponding variation in the coefficient of friction with displacement.Wet skin has a noticeably higher coefficient of friction than dry skin, but the difference between the two decreases with the normal load.The variance also decreases with increasing load.This finding is consistent with the experimental observations of finger skin by Mabuchi et al [27] and other researchers like O'Meara and Smith [64], Derler et al [36], Gerhardt et al [65] for contact between steel and human skin.The large difference in the coefficient of friction between wet and dry skin (despite the comparable normalized force values) is due to the power law exponent n in equation (8).For wet skin, this value is −0.53, compared to a value of −0.1 for dry skin, a change that reflects the increased role of adhesive forces in wet skin.
To further explore this, we compare the normalized area of contact for dry and wet skin as a function  of the normalized displacement in figure 9(a), where the real area of contact from the finite element simulation is normalized by the apparent area of contact.The contact patches for dry and wet skin are shown in figure 9(b) for one realization.There is a dramatic difference in the magnitude and nature of contact area for the two cases.For dry skin, the high stiffness leads to multiple discrete local contact zones.However, the lower stiffness of wet skin leads to larger deformation of the asperities and therefore larger patches of contact.This in turn means adhesive forces play a larger role in wet skin contact than with dry skin, which is reflected in the different values used for the exponent n.
This is somewhat different from results reported by Levya-Mendivil et al [4], whose computational model predicts that in some cases dry skin has a higher coefficient of friction than wet skin.This disagreement could be due to the fundamental difference in modeling methodologies.In their two-dimensional finite element model, a local coefficient of friction is used to describe the interaction between rough skin and a rigid indenter.This local value represents level of local adhesion as an interplay of humidity and surface energy.The global coefficient of friction is then determined from the normal and shear forces predicted by the FE simulation.In this approach the dependence of adhesion on contact pressure is ignored.In contrast, in our approach, only the normal force is determined from the three-dimensional finite element simulation and an empirical relationship (equation ( 8)) is used to calculate the coefficient of friction.Additionally, the use of a three-dimensional model allows us to obtain a more realistic picture of contact between skin and the flat indenter.Another consequence of our approach is that at small displacements, small differences in the normal force caused by a factor (such as humidity) are amplified by equation (8), while large differences are attenuated.
Two features about figure 8(b) are worth noting.First, the difference in the coefficient of friction between wet and dry skin is higher at lower displacements.This has been observed experimentally by [36,65].Initially, at lower normal displacements, the resistance to the compression comes from the contacting asperities that form the top layer of the stratum corneum.Consequently, the difference in the modulus of wet and dry stratum corneum leads to the large difference in the coefficient of friction.At higher normal displacements, the inner layers of skin are also engaged in resisting the loading, leading to a lower difference in the coefficient of friction.
Second, the variation in the coefficient of friction is higher at lower displacements.This is due to the stochastic nature of our approach.As the surfaces are generated randomly for each realization, the number of asperities coming in contact with the flat plate varies significantly at lower displacements, resulting in a higher variance in the coefficient of friction.But with an increase in the normal displacement, the number of contacting asperities become similar because all realizations have the same statistical parameters.Therefore, the variance in the coefficient of friction decreases.This behavior was also observed experimentally by Derler and Gerhardt [36].This study confirms that the higher variation at lower displacement is due to the difference in contact area, which is a function of roughness and Young's modulus.
It should be noted that to account for the effect of humidity, we only consider a change the Youngʼs modulus of the SC in this study.Though, this is consistent with the previously conducted mechanical modeling of human skin in [4,[29][30][31], in practice, a change in the humidity can also affect the chemical bonds at molecular level [66] and physiological properties of skin like lipid barriers [67].Therefore, while our results point to the importance of humidity as a factor affecting skin friction, the question of humidity needs to be investigated further.

Role of the thickness of stratum corneum
The thickness of stratum corneum is another quantity that varies from one person to another and also between areas of the body [18].In order to understand the effect of this quantity on the contact mechanics properties of human skin, we compare two models under wet conditions with different stratum corneum thickness values: one with a nominal thickness of t = 0.026 mm and another with a thickness 0.8t.
Figure 10(a) shows a comparison of the normal loaddisplacement response between the two models.As expected, the thicker model has a slightly stiffer response.Figure 10(b) compares the coefficient of friction for the two models, which shows a decreasing trend.The difference in the CoF is higher for smaller normal displacements, but as the displacement increases the two models give similar values of coefficient of friction.A 20% reduction in the SC thickness leads to a reduction of roughly 16% in the coefficient of friction for a normalized displacement of 2, from 3.7 to 3.1.There is even less reduction at higher compression levels.
Differences in the thickness of stratum corneum are common between male and female subjects [18].The available literature does not define a clear distinction between the friction response of male and female skin.Vilhena et al [68] compared the experimental friction response of male and female skin against polyamide, polyester, silk, cotton, and wool.They found that for the former four surfaces, the average coefficient of friction for male skin was higher than female skin, though the error bar for female skin was higher.For wool, the average coefficient of friction was nearly the same.Savescu et al [69] found that the CoF for male skin was higher than female skin for sandpaper, silk and polyester.However, female skin had a higher coefficient of friction compared to male skin for cotton and rayon.These aforementioned results involve friction between skin and fabric rather than skin and steel, so a direct comparison with our results is not possible.But the trends from the results in this study suggest the thickness of the stratum corneum may be one of the reasons for female skin to have a slightly lower coefficient of friction than male skin.

Role of Poisson's ratio
As mentioned earlier, studies have reported various values of Poisson's ratio for stratum corneum.To understand the effect of this parameter, we perform two simulations under wet conditions of skin: one for a surface with ν sc = 0.49 and other with ν sc = 0.3.The value of ν for the other layers is set to 0.3 for both studies.
Figure 11 shows a comparison of the normalized load displacement response and the coefficient of friction for the two cases.There is very little difference in the normal load or the CoF between the two cases.This is in agreement with the observations of Levya-Mendivil et al [29], who also reported no significant difference in results due to a change in the Poisson's ratio of SC.

Role of scale
The advantage of using a fractal model to define rough surfaces is the ability to understand the effect of surface roughness at various levels.By decreasing the size of the domain while holding the fractal parameters constant, we virtually magnify the resolution of the surface.In this study, we simulate the response of a 5 mm × 5 mm surface and a 1 mm × 1 mm surface in dry conditions, as shown in figure 2.
Figure 12(a) shows the normalized load-displacement responses from the two domain sizes considered, with the 1 mm × 1 mm surface displaying a considerably stiffer response than the 5 mm× 5 mm surface.Also, the variance increases with increasing displacement.Persson [32] using his theory shows that the coefficient of friction increases with magnification.Our results in figure 12(b) are in agreement with this assertion.For a smaller domain size, at a given normal displacement, smaller asperities come into contact with the rigid surface, leading to a lower normal force.Consequently, the coefficient of friction is higher for the smaller domain size, leading to the agreement with Perssonʼs theory [32].

Role of location: roughness parameters and exponents of friction
In general, skin from different body sites has different friction responses.This is due to the variation in mechanical properties and morphological features.
These variations arise from exposure to sunlight, functional properties, etc [36].
In this study we compare the friction response of forearm skin and finger skin.Our focus is limited to the change in morphological features keeping all other parameters constant.The RMS roughness for forearm skin and finger skin are taken to be 25 μm and 22 μm, respectively, while the fractal dimensions for forearm skin and finger skin are set to 2.11 and 2.14, respectively [5,38].In addition to this, the value of n used in the calculation of friction from the normal force (equation ( 8)) is varied based on previously reported experimental results.The values are −0.53 and −0.32 for forearm skin and finger skin, respectively [36,56].
Figure 13 shows the normalized load-displacement and area-displacement responses for the simulations of finger skin and forearm skin realizations.The normalized load for a given displacement is higher for forearm skin than finger skin.This difference increases from a value of 0.002 at a normalized displacement of 2 to to 0.01 at a normalized displacement of 4. The  average normalized contact area is roughly the same for all the values of displacement.The combination of higher RMS roughness and lower fractal dimension is responsible for the higher normal load response for forearm skin for similar contact area compared to finger skin.Our studies for understanding the role of RMS roughness in section 3.2 show that a higher RMS roughness leads to a higher normal load for a given displacement.Also, recently Yuan et al [70] found that given all other properties are constant, a surface with a higher fractal dimension has a lower normal force for the same contact area.Thus, the combination of these two roughness parameters leads to a higher normal load for forearm skin.
Figure 13(c) shows the coefficient of friction response for forearm skin and finger skin, where the responses are different but mainly at lower loads (in contrast to the normal load response).This can be attributed to the lower value of n used in equation (8) for forearm skin.This is similar to the results from the hydration study in section 3.3, where a lower exponent leads to a higher difference in the CoF at smaller loads.At higher loads, the inner layers contribute more to the overall friction response, leading to essentially equal CoF values.In other words, regional variations in skin topography affect the friction response only at small loads.
In figure 13(c), the decrease in the coefficient of friction with increasing load has a practical implication.In the context of finger skin, this means that increasing the pinching force in an attempt to improve the ability to hold an object may be counter-productive.For instance, there is a two-fold difference in the predicted average coefficient of friction for finger skin between normalized displacements of 2 (CoF=2.63)and 4 (CoF=1.3).This is similar to the results reported in [28,36,53].

Discussion of the role of fractal surface parameters
There are three studies in this work that address the role of the fractal surface parameters used to model skin: 1. Section 3.2: Here, we study the role of skin roughness by varying the RMS value (25 μm and 50 μm).
The fractal dimension is held fixed at 2.11.The static coefficient of friction decreases with increasing roughness, as seen in figure 7.
2. Section 3.6: Here, we study the role of magnification.As the magnification level is increased, more smaller asperities contribute to the tribological response, leading to an increase in the coefficient of friction.This is shown in figure 12.
3. Section 3.7: This study addresses the role of location, and both RMS roughness and fractal dimension are varied.In addition, the exponent used in the empirical relationship (equation ( 8)) is also varied.The differences in RMS roughness and fractal dimension values are not large: 12% for roughness and 1.4% for the fractal dimension.The difference in the empirical exponent is more significant (40%).
We see in figure 13 that the normal load and contact area responses, which are computed from the finite element simulation, are similar.This is consistent with the small differences in the fractal surface parameters.However, the coefficient of friction shows large differences (particularly at small displacements) due to the considerable difference in the empirical exponent.
To stay consistent with experimental findings, we do not consider large variations in the fractal dimension in these studies.As mentioned in section 2.2, the fractal dimension for naturally occurring surfaces is found to be 2.15 ± 0.15 [38].
It is important to note here that the fractal surfaces used in this study ignore the presence of tension lines on skin surface or the presence of ridges and valleys on the skin from fingers.These are geometric features that will contribute to the normal force (and therefore the coefficient of friction).Taking these into consideration would result in some deviations from the results presented here.Including these features requires experiments to obtain roughness coordinates of skin or image based modeling [30,31].

Limitations of the current study
The current study does have some limitations.Our model assumes that skin is an isotropic and hyperelastic material rather than an anisotropic and viscoelastic material.Taking these complexities into account is part of our future work.This also includes plasticizing and capillary effects that would increase the fidelity of the computational models.This would require rigorous experiments under various conditions.The use of the empirical relationship (equation ( 8)) to determine the CoF has its advantages and disadvantages.On the one hand, it simplifies the calculation of the CoF using the normal force obtained from the finite element model.This negates the need for a simulation of the tangential interaction at the interface, which is complex and requires either detailed microscale parameters or assumptions.On the other hand, this approach is not useful when the tangential interaction is itself the goal of the simulation.For instance, how does the area of contact change during sliding?As mentioned earlier, to model tangential interaction would require a detailed understanding of interfacial mechanics as well as significant experimentation to support model development.
Given the huge number of factors on which the properties of skin depend, it is hard to obtain specific parameters for a required study.This study therefore is an attempt to gain a qualitative understanding of certain effects based on representative values.Specific experiments to obtain certain parameters like the Young's modulus, friction exponents, etc., can help create a more robust model.
While this study is primarily computational and no new experimental results are reported, the predictions from our computational-empirical approach are consistent with several experimental studies.In all the studies reported in this work, the coefficient of friction decreases as the normal load increases.This is in agreement with the results reported by André et al [28], Derler et al [36], Adams et al [53].The proposed approach shows that friction decreases with skin roughness.This is consistent with experimental studies that show that skin roughness increases with age as in Limbert et al [11], Korn et al [42], Trojahn et al [62] and that the coefficient of friction decreases with age as shown by Mabuchi et al [27].The increase in friction due to relative humidity is consistent with the findings of Mabuchi et al [27], O'Meara and Smith [64], Derler et al [36].

Conclusion
This work describes a study based on fractal modeling of human skin in conjunction with detailed threedimensional finite element modeling.We investigate the role of skin roughness in contact interactions and its effect in combination with various physiological features in its overall friction response.The approach is based on an empirical friction model to simplify the complex biophysical aspects of skin and to examine the role of complex morphological and physiological features.
The proposed model predicts that a lower RMS roughness leads to a higher coefficient of friction.The manner in which the area of contact develops for dry and wet skin with the roughness of skin is also shown, leading to a higher coefficient of friction for wet skin.The role of thickness and Poisson's ratio of stratum corneum is small towards influencing the coefficient of friction.With increase in magnification, more smaller asperities come in contact, which leads to higher friction.At lower loads, the difference in morphological features of skin from different body sites leads to a difference in the coefficient of friction.However, at higher loads this difference is minimal as inherently the underlying features of skin all over the body remain the same.
The approach proposed in this work offers a practical way to include the effects of roughness in modeling human skin friction.This can be used to obtain a more accurate picture of the response of human skin and can improve product design involving human skin contact.

Figure 1 .
Figure 1.A schematic of how fractal model of roughness is used to represent human frontal forearm skin in this study.The generated fractal surface is further used for simulation, where a normal compression is performed with a rigid analytical flat surface.

Figure 2 .
Figure 2. A 5 mm × 5 mm and 1 mm × 1 mm, three-dimensional model of rough human skin.The grey region denotes the stratum corneum, the white region denotes the viable epidermis and the black region denotes the underlying dermis.

Figure 3 .
Figure 3.A fractal surface of RMS roughness 25 μm and fractal dimension of 2.11 is shown at different levels of magnification.A higher magnification or smaller length scale reveals smaller asperities on the surface.

Figure 4 .
Figure 4. Finite element model of human skin: (a) Three-dimensional model of rough skin.(b) A flat plate compressing the rough surface with symmetry boundary conditions on edges and a constrained bottom.(c) Sample finite element mesh used in the study along with a close up view.

Figure 7 (
a) shows the normalized load-displacement response for the two values.The difference in the average response increases from 0.005 for a normalized displacement of 2 to 0.04 for a normal displacement of 4, showing that the contact behavior is highly sensitive to skin roughness.Figure 7(b) shows the variation of the coefficient of friction with displacement for both values of RMS roughness.The simulations with σ = 50 μm display a noticeably lower coefficient of friction.There is more variability in the values for σ = 25 μm.Interestingly, the variability in the CoF values for σ = 50 μm is consistently lower and almost disappears at higher displacements.

Figure 5 .
Figure 5. Summary of studies performed using the hybrid computational-empirical approach to determine skin coefficient of friction.The model parameters corresponding to individual factors are shown.

Figure 6 .
Figure 6.Mesh convergence results for compression: Normal force as a function of applied normal displacement for various mesh sizes.A mesh size of 100 × 100 elements on the top surface provides a good balance between efficiency and accuracy.A biased mesh is used in the thickness direction, with the density of elements increasing towards the top surface (SC).

Figure 7 .
Figure 7. Role of surface roughness: (a) Normalized load displacement response for skin surfaces with RMS roughness 25 μm and 50 μ m.(b) Their static friction response.

Figure 8 .
Figure 8. Role of hydration: (a) Normalized force displacement response for dry and wet skin.(b) Coefficient of friction versus normalized displacement response for dry and wet skin.

Figure 9 .
Figure 9. Role of hydration: (a) Normalized area of contact for dry and wet skin as a function of normalized displacement.(b) Contact patches over the surface at a displacement of 3σ.Light color indicates area of contact.

Figure 10 .
Figure 10.Effect of stratum corneum thickness: (a) Normalized force displacement response for models with stratum corneum thickness t and 0.8t (b) Coefficient of friction vs normalized displacement response for the models with change of stratum corneum thickness.There is no significant difference in the results from the two models.

Figure 12 (
Figure 12(b) shows the variation in the CoF for the two domain sizes.The average difference between the values of coefficient of friction from the two domain sizes varies from 0.3 at lower loads to 0.38 at higher loads.The variance decreases for both domain sizes.Persson[32] using his theory shows that the coefficient of friction increases with magnification.Our results in figure12(b) are in agreement with this assertion.For a smaller domain size, at a given normal displacement, smaller asperities come into contact with the rigid surface, leading to a lower normal force.Consequently, the coefficient of friction is higher for the smaller domain size, leading to the agreement with Perssonʼs theory[32].

Figure 11 .
Figure 11.Effect of stratum corneum Poisson's ratio: (a) Normalized force displacement response for models with SC Poisson's ratio ν = 0.3 and ν = 0.49 (b) Coefficient of friction vs normalized displacement response for the models with change of SC Poisson's ratio.There is no significant difference in the results from the two models.

Figure 12 .
Figure 12.Effect of magnification: (a) Normalized force displacement response for domain sizes of 5 mm × 5 mm and 1 mm × 1 mm.(b) Coefficient of friction vs normalized displacement response for the two domain sizes.

Figure 13 .
Figure 13.Role of location: (a) Normalized force displacement response for finger and forearm skin.(b) Normalized area of contact for finger and forearm skin as a function of normalized displacement.(c) Coefficient of friction versus normalized displacement response for finger and forearm skin.