Influence of surface topography on three-dimensional fractal model of sliding friction

Pan, Wujiu; Li, Xiaopeng; Wang, Linlin; Mu, Jiaxin; Yang, Zemin

doi:10.1063/1.4999036

The purpose of this paper is to establish a three-dimensional model of sliding friction and to study the influence of surface topography fractal parameters on the model. Firstly, the analysis of the contact between two asperities is completed, for according to the classical molecular-mechanical friction theory, the sliding friction among rough surfaces should be the sum of mechanical force and molecular adhesion. Then based on the fractal theory, the three-dimensional fractal model of sliding friction is deduced. Finally, the influence of the maximum contact area of asperity a_l, the fractal roughness G and the fractal dimension D on the sliding friction is analyzed by a simulation example, and the analysis results show that the sliding friction F has positive correlation with a_l and G, and there is an optimal fractal dimension D which minimizes F. The study of the paper can be used to explain the existing experimental results and the friction theory reasonably.

I. INTRODUCTION

A lot of engineering practices show that the main reason for the machine failure is not the damage of the machine structure, but the movable connection and the wear of the components under the friction force. In the process of designing the new machine, engineers must calculate the strength of important components but rarely consider the calculation and check of wear performance of movable connection. In fact, it seems that the traditional method of using solid strength to solve the friction and wear problem is invalid. Therefore, Janahmadov and Javadov¹ proposed a solution that the synergetics and fractals are applied to explain and study the comprehensive phenomenon and quantitative characteristics of material friction. Many mechanical structures are assembled by various components according to some certain requirements, not a continuous whole. The surface between two components which contact with each other is called “joint surfaces”. The surface topography of joint surfaces has significant influence on the friction,^2,3 wear,^4,5 vibration and noise,⁶ electro-mechanical behavior.⁷

Many researchers made many experiments and conducted theoretical study on the friction characteristics of the joint surfaces. Minet et al.² and Wei et al.⁶ studied the mechanical seal face about the influence of surface roughness and surface topography fractal parameters on the dynamic friction coefficient, and their study could be used to predict the friction coefficient of mechanical seal face under some certain conditions and to improve the contact performance of the end face. By experiments, Mofidi and Prakash⁴ studied the friction and wear performance of four kinds of rubber contacting with steel surface and analyzed the influence of steel surface with different roughness on the wear performance of different elastomer materials. Mulvihill et al.⁸ used the finite element model affected by the asperity overlaps and the material failure to analyze the elasto-plastic contact between two spherical asperities to predict the dynamic friction coefficient of relative sliding surfaces. Arakawa⁹ studied the dynamic sliding friction between polyurethane rubber surface and polymethylmethacrylate surface, and the result showed that the friction force depended on the actual contact area and sliding speed. Yastrebov et al.^10,11 deduced the pressure-dependent friction coefficient based on the adhesive theory of friction and used mesh discretization technique to accurately calculate the actual contact area; these two papers provide a unifying picture of rough elastic contact. AI-Bender and Moerlooze¹² studied the relationship between the friction force and the normal load in the presliding zone by theoretical analysis, and the results showed that the increase of the normal load decreased friction coefficients and increased the presliding distance. Due to the fractal characteristic of machine surface processed, the fractal theory was often applied to the contact modeling and tribology modeling.^13,14 Tian et al.¹⁵ improved the fractal theory, and built a prediction model of static friction coefficient, making relative simulation analysis. Chen et al.¹⁶ built a sliding friction model based on the fractal theory and provided good reference for other researchers.

However, the above theoretical study on the friction of joint surfaces based on the fractal theory is on the basis of the two-dimensional fractal (1<D_s<2). It is unreasonable in theory that the two-dimensional fractal model is applied to calculate the contact performance of the three-dimensional surface topography.¹⁷ Therefore, in this paper, the three-dimensional fractal characteristic of surface topography is considered, and a new sliding friction fractal model is built and the influence of fractal parameters on this model is analyzed. The study of this paper can be used to explain the existing experiment results effectively, and to provide reference for the reasonable design of surface topography.

II. CONTACT ANALYSIS

According to the molecular-mechanical friction theory,¹⁸ when the relative sliding occurs between two surfaces contacting with each other under the external load, the sliding friction force should be the sum of the mechanical force and the molecular adhesion. These two asperities contacting with each other will be analyzed on the contact stress below.

In Fig. 1, two asperities contacting with each other generate relative sliding. Two asperities initially contact with each other at two points, D₁ and D₂, however, due to the deformation of asperities, they contact with each other at point D₀. The angle α is between the micro contact surface and y direction. The contact area is so small that the length of arc A₁A₂ can be approximately equal to the length of line segment A₁A₂, so there is the following relationship, expressed as

cot α = \frac{ω_{y 1}}{ω_{x 1}}

(1)

FIG. 1.

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Sliding diagram of asperity.

Similarly, the angle between the contact surface and the direction of z is β, and the following relation is given by

cot β = \frac{ω_{z 1}}{ω_{x 1}}

(2)

Under the external load, two asperities contacting with each other generate the relative motion. Due to the relative sliding and elasto-plastic deformation, the distance in the direction of y and z of contact surface changes, and the variation value is given by

Δ h_{y} = A_{1} A_{2} \frac{ω_{y 1}}{ω_{x 1}} - (ω_{y 1} + ω_{y 2})

(3)

Δ h_{z} = A_{1} A_{2} \frac{ω_{z 1}}{ω_{x 1}} - (ω_{z 1} + ω_{z 2})

(4)

The derivation of both sides of Eqs. (3) and (4) can calculate the relative sliding speed of contact surface in the direction of y and z, and the sliding speed is given by

V_{y} = V_{x} \frac{ω_{y 1}}{ω_{x 1}} - \frac{\partial ω_{y}}{\partial t}

(5)

V_{z} = V_{x} \frac{ω_{z 1}}{ω_{x 1}} - \frac{\partial ω_{z}}{\partial t}

(6)

where $V_{y} = \frac{\partial h_{y}}{\partial t}$ ⁠, $V_{z} = \frac{\partial h_{z}}{\partial t}$ ⁠, $V_{x} = \frac{\partial A_{1} A_{2}}{\partial t}$ ⁠, $ω_{y} = ω_{y 1} + ω_{y 2}$ ⁠, $ω_{z} = ω_{z 1} + ω_{z 2}$ ⁠.

In Fig. 2, the contact surfaces between two asperities are analyzed on the stress. The projections value of F_mec1 and F_mec2 in the direction of x, y and z are the same while the directions of them are opposite, so are that of F_mol1 and F_mol2. Therefore, in the contact zone between two asperities, the relations of their slope and force are as follows:

\frac{ω_{y 1}}{ω_{x 1}} = - \frac{ω_{y 2}}{ω_{x 2}} = \frac{F_{mec y 1}}{F_{mec x 1}} = \frac{F_{mec y 2}}{F_{mec x 2}} = \frac{F_{mol x 1}}{F_{mol y 1}} = \frac{F_{mol x 2}}{F_{mol y 2}}

(7)

\frac{ω_{z 1}}{ω_{x 1}} = - \frac{ω_{z 2}}{ω_{x 2}} = \frac{F_{mec x 1}}{F_{mec z 1}} = \frac{F_{mec x 2}}{F_{mec z 2}} = \frac{F_{mol z 1}}{F_{mol x 1}} = \frac{F_{mol z 2}}{F_{mol x 2}}

(8)

FIG. 2.

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Stress analysis on the contact surfaces of two mutual contact asperities.

Substitute of Eqs. (7) and (8) into Eqs. (5) and (6) respectively can obtain

V_{y} = V_{x} \frac{F_{mec y 1}}{F_{mec x 1}} - \frac{\partial ω_{y}}{\partial t}

(9)

V_{y} = V_{x} \frac{F_{mol x 1}}{F_{mol y 1}} - \frac{\partial ω_{y}}{\partial t}

(10)

V_{z} = V_{x} \frac{F_{mec x 1}}{F_{mec z 1}} - \frac{\partial ω_{z}}{\partial t}

(11)

V_{z} = V_{x} \frac{F_{mol z 1}}{F_{mol x 1}} - \frac{\partial ω_{z}}{\partial t}

(12)

Through the integration of both sides of Eqs. (9) and (11) on the contact surface A_j and the summation of the whole contact surface A_i (i=1, 2,…,j,..,n) in sequence, the following relations are obtained, given by

\sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{y}}{\partial t} F_{mec x 1} d a_{j} = \sum_{i} \underset{A_{j}}{∭} V_{x} F_{mec y 1} d a_{j} - \sum_{i} \underset{A_{j}}{∭} V_{y} F_{mec x 1} d a_{j}

(13)

\sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{z}}{\partial t} F_{mec z 1} d a_{j} = \sum_{i} \underset{A_{j}}{∭} V_{x} F_{mec x 1} d a_{j} - \sum_{i} \underset{A_{j}}{∭} V_{z} F_{mec z 1} d a_{j}

(14)

Eq. (13) is added to Eq. (14) to obtain

\begin{matrix} \sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{y}}{\partial t} F_{mec x 1} d a_{j} + \sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{z}}{\partial t} F_{mec z 1} d a_{j} & = & V_{x} \sum_{i} \underset{A_{j}}{∭} F_{mec y 1} d a_{j} - V_{z} \sum_{i} \underset{A_{j}}{∭} F_{mec z 1} \\ \times d a_{j} + (V_{x} - V_{y}) \sum_{i} \underset{A_{j}}{∭} F_{mec x 1} d a_{j} \end{matrix}

(15)

Similarly, the Eqs. (10) and (12) can be transformed into the following equation, expressed as

\begin{matrix} \sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{y}}{\partial t} F_{mol y 1} d a_{j} + \sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{z}}{\partial t} F_{mol x 1} d a_{j} & = & V_{x} \sum_{i} \underset{A_{j}}{∭} F_{mol z 1} d a_{j} - V_{y} \sum_{i} \underset{A_{j}}{∭} F_{mol y 1} d a_{j} \\ + (V_{x} - V_{z}) \sum_{i} \underset{A_{j}}{∭} F_{mol x 1} d a_{j} \end{matrix}

(16)

In Eq. (15), three summation items on the right side respectively represent the mechanical force component in the direction of x, y and z on the contact surface among the whole asperities, and they are expressed as S_mecx, S_mecy, S_mecz respectively, and the equation of left side are expressed as U_mec, given by

U_{m e c} = (V_{x} - V_{y}) S_{m e c x} + V_{x} S_{m e c y} - V_{z} S_{m e c z}

(17)

In Eq. (16), three summation items on the right side respectively represent the molecular adhesion component in the direction of x, y, and z on the contact surface among the whole asperities, and they are expressed as T_molx, T_moly, T_molz respectively, and the equation of left side are expressed as W_mol, given by

W_{m o l} = (V_{x} - V_{z}) T_{m o l x} - V_{y} T_{m o l y} + V_{x} T_{m o l z}

(18)

Through the integration of both sides of Eqs. (5) and (6) on the contact surface A_j and the summation of the whole contact surface A_i(i=1,2,…,j,..,n) in sequence, two relations are obtained. Finally, these two relations are added together to obtain

\begin{matrix} (V_{y} + V_{z}) \sum_{i} \underset{A_{j}}{∭} d a_{j} & = & V_{x} (\sum_{i} \underset{A_{j}}{∭} \frac{ω_{y 1}}{ω_{x 1}} d a_{j} + \sum_{i} \underset{A_{j}}{∭} \frac{ω_{z 1}}{ω_{x 1}} d a_{j}) \\ - (\sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{y}}{\partial t} d a_{j} + \sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{z}}{\partial t} d a_{j}) \end{matrix}

(19)

On the left side of Eq. (19), the summation integral item represents the actual contact area A_r of joint surfaces. On the right side of Eq. (19), $\sum_{i} \underset{A_{j}}{∭} \frac{ω_{y 1}}{ω_{x 1}} d a_{j} + \sum_{i} \underset{A_{j}}{∭} \frac{ω_{z 1}}{ω_{x 1}} d a_{j}$ is the sum of area component of the whole asperities in the direction of y and z, and its value should be equal to part of A_r, A_y=bA_r and A_z=cA_r where both of the value of b and c are 0∼1, and assuming A_h=A_y+A_z. On the right side, $\sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{y}}{\partial t} d a_{j} + \sum_{i} \underset{A_{j}}{∭} \frac{\partial ω_{z}}{\partial t} d a_{j}$ represents the time variation rate of deformation in the direction of y and z, which is expressed by Q. Then the Eq. (19) can be expressed as

(V_{y} + V_{z}) A_{r} = V_{x} A_{h} - Q

(20)

Eqs. (17), (18) and (20) are combined together to obtain

V_{x} (Q S_{mec x} + Q S_{mec y} - U_{mec} A_{h}) + V_{y} (U_{mec} A_{r} - Q S_{mec x}) + V_{z} (U_{mec} A_{r} - Q S_{mec z}) = 0

(21)

V_{x} (Q T_{mol x} + Q T_{mol z} - W_{mol} A_{h}) + V_{y} (W_{mol} A_{r} - Q T_{mol y}) + V_{z} (W_{mol} A_{r} - Q T_{mol x}) = 0

(22)

When two asperities contacting with each other generate the relative sliding on joint surfaces, the relative speed component on joint surfaces in the direction of x, y and z is not zero. Therefore, the condition of making Eqs. (21) and (22) valid is that each item in their brackets should be zero, and the following relation can be obtained, expressed as

S_{mec x} + S_{mec y} - e S_{mec z} = 0

(23)

T_{mol x} - e T_{mol y} + T_{mol z} = 0

(24)

where 0 < e ≤ 1.

The sliding friction F of joint surfaces is expressed as

F = S_{mec x} + T_{mol x}

(25)

Substitute of Eqs. (23) and (24) into Eq. (25) can obtain

F = e F_{z} - F_{y} + (1 + e) T_{mol y} - (1 + e) T_{mol z}

(26)

where F_z = S_mecz + T_molz represents the normal load of joint surfaces, and F_y = S_mecy + T_moly represents the resultant force of joint surfaces in the direction of y.

If the adhesive shear strength of asperities is τ, the molecular adhesion is equal to the product of shear strength τ and plastic contact area A_p, and then the following relation can be obtained, expressed as

(1 + e) T_{mol y} - (1 + e) T_{mol z} = (1 + e) (b - c) τ A_{p}

(27)

Substitute of Eq. (27) into Eq. (26) can obtain the sliding friction of joint surfaces, given by

F = e F_{z} - F_{y} + η τ A_{p}

(28)

where η = (1 + e)(b − c).

III. THREE-DIMENSIONAL FRACTAL MODEL OF SLIDING FRICTION

According to the Hertz theory, two asperities contacting with each other are equivalent to a rigid plane and an equivalent spherical asperity, as shown in Fig. 3.

FIG. 3.

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Equivalent contact diagram of asperity.

According to the Hertz theory, the normal load of single asperity at the elastic deformation stage is given by

P_{e} = \frac{4}{3} E R^{0.5} ω^{1.5}

(29)

where E represents equivalent elastic modulus, $E = {(\frac{1 - ν_{1}^{2}}{E_{1}} + \frac{1 - ν_{2}^{2}}{E_{2}})}^{- 1}$ ⁠; E₁ and E₂ represent elastic modulus of two asperities respectively; ν₁ and ν₂ represent their Poisson’s ratio respectively; R represents the curvature radius of equivalent asperity; ω represents the deformation value.

In Fig. 3, when equivalent asperity without deformation contacts with rigid plane, the cross sectional area is given by

A = π {r^{'}}^{2} = π [R^{2} - {(R - ω)}^{2}] \approx 2 π R ω

(30)

However, in Fig. 3, the actual contact area is half of the cross sectional area,¹⁹ given by

a = π R ω

(31)

Yan and Komvopoulos²⁰ improved the single-variable W-M function, then the function improved can be used to simulate the three-dimensional surface topography. The modified double-variable W-M function is given by

z (x, y) = L {(\frac{G}{L})}^{(D - 2)} {(\frac{ln γ}{M})}^{1 / 2} \sum_{m = 1}^{M} \sum_{n = 0}^{n_{max}} γ^{(D - 3) n} \{cos φ_{m, n} - cos [\begin{gathered} \frac{2 π γ^{n} {(x^{2} + y^{2})}^{1 / 2}}{L} \times \\ cos ({tan}^{- 1} (\frac{y}{x}) - \frac{π m}{M}) + φ_{m, n} \end{gathered}]\}

(32)

where L is the sampling length of surface topography; D is the surface topography fractal dimension of joint surface (2 < D < 3), and G is fractal roughness, and both of them are obtained by measuring surface topography; γ(⁠ $γ > 1$ ⁠) is a parameter that determines the density of frequency; M is the number of surface peak ridge superposition; n denotes the frequency index, n_max = Int[log(L/L_S)/logγ]; L_s is the minimum truncation length; φ_m,n is random phase. Some parameters are given to obtain the three-dimensional surface topography, as shown in Figs. 4 and 5. The contact surface topography has fractal characteristics, which mainly depends on the fractal dimension D and the fractal roughness G. That the larger D means more high-frequency components of the three-dimensional surface topography so the surface is much finer in the macro view; that the larger G means amplitude of three-dimensional surface topography is larger so the surface is more rough.

FIG. 4.

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Surface topography diagram with different dimension (G=5.1372e-5mm, L=256, L_s=1μm, γ=1.5, M=5). (a) D=2.15, (b) D=2.55, (c) D=2.95.

FIG. 5.

$FIG. 5. Surface topography diagram with different fractal roughness (D=2.4241, L=256, Ls=1μm, γ=1.5, M=5. (a) G=5.42e-9, (b) G=5.42e-5, (c) G=5.42e-1.$

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Surface topography diagram with different fractal roughness (D=2.4241, L=256, L_s=1μm, γ=1.5, M=5. (a) G=5.42e-9, (b) G=5.42e-5, (c) G=5.42e-1.

Available by Eq. (32), the deformation value of asperity is equal to the height difference between the peak and valley, given by

ω = 2 G^{(D - 2)} {(ln γ)}^{1 / 2} {(2 r^{'})}^{(3 - D)}

(33)

According to Eqs. (30) and (31), Eq. (33) is transformed into the following equation, expressed as

ω = 2^{(11 - 3 D) / 2} G^{(D - 2)} {(ln γ)}^{1 / 2} π^{(D - 3) / 2} a^{(3 - D) / 2}

(34)

Substitute of Eq. (34) into Eq. (31) to obtain the radius of curvature, given by

R = \frac{2^{(3 D - 11) / 2} π^{(1 - D) / 2} G^{(2 - D)} a^{(D - 1) / 2}}{{(ln γ)}^{1 / 2}}

(35)

The critical elastic deformation value²¹ of asperity is given by

ω_{c} = {(\frac{33 π k_{μ} ϕ}{40})}^{2} R

(36)

where ϕ = σ_y/E represents material characteristic parameter; σ_y represents the yield strength of material; k_μ represents the corrected parameter of friction. When 0 ≤ μ ≤ 0.3, k_μ = 1 − 0.228μ; when 0.3 < μ < 1, k_μ = 0.932e^{−1.58(μ−0.3)}; μ is the dynamic friction coefficient.

According to Eqs. (34)–(36), the critical elastic deformation area of asperity can be deduced, expressed as

a_{c} = 2^{(3 D - 11) / (2 - D)} {(\frac{33 k_{μ} ϕ}{40})}^{2 / (2 - D)} π^{(4 - D) / (2 - D)} {(ln γ)}^{1 / (D - 2)} G^{2}

(37)

According to Eqs. (29), (34) and (35), the normal load of asperity at the stage of elastic deformation can be obtained, and the relation is expressed by the contact area a, given by

P_{e} (a) = \frac{1}{3} E π^{(D - 4) / 2} 2^{(15 - 3 D) / 2} {(ln γ)}^{1 / 2} G^{D - 2} a^{(4 - D) / 2}

(38)

The normal load of asperity at the stage of plastic deformation is given by

P_{p} (a) = λ σ_{y} a

(39)

where λ = H/σ_y is a defined parameter, H is the hardness of softer material.

The relationship between the area distribution function of the whole asperity n(a)²⁰ of joint surfaces and the maximum contact area of asperity a_l is given by

n (a) = \frac{D - 1}{2} a_{l}^{(D - 1) / 2} a^{- (D + 1) / 2}, 0 < a \leq a_{l}, 2 < D < 3

(40)

The continuous integral of n(a) in each deformation region of asperities is used to calculate actual contact area A_r on the whole joint surfaces, and A_r is given by

A_{r} = \int_{0}^{a_{c}} n (a) a d a + \int_{a_{c}}^{a_{l}} n (a) a d a = \frac{D - 1}{3 - D} a_{l}

(41)

Similarly, the plastic contact area of joint surfaces A_p is given by

A_{p} = \int_{0}^{a_{c}} n (a) a d a = \frac{D - 1}{3 - D} a_{l}^{(D - 1) / 2} a_{c}^{(3 - D) / 2}

(42)

According to the product of the load and area distribution function of asperity at elastic and plastic deformation stage and the integral in each corresponding deformation region, the normal load F_z of joint surfaces can be obtained.

(1) when 2 < D < 3 and D $\neq$ 2.5, Eqs. (38)–(40) are combined together to obtain

\begin{gathered} F_{z} = \int_{0}^{a_{c}} P_{p} n (a) d a + \int_{a_{c}}^{a_{l}} P_{e} (a) n (a) d a = \frac{D - 1}{3 - D} λ σ_{y} a_{l}^{0.5 D - 0.5} a_{c}^{1.5 - 0.5 D} \\ + \frac{(D - 1)}{3 (2.5 - D)} 2^{6.5 - 1.5 D} E π^{0.5 D - 2} {(ln γ)}^{0.5} G^{D - 2} a_{l}^{0.5 D - 0.5} (a_{l}^{2.5 - D} - a_{c}^{2.5 - D}) \end{gathered}

(43)

(2) when 2 < D < 3 and D = 2.5, Eqs. (38)–(40) are combined together to obtain

\begin{matrix} F_{z} = \int_{0}^{a_{c}} P_{p} n (a) d a + \int_{a_{c}}^{a_{l}} P_{e} (a) n (a) d a & = & 3 λ σ_{y} a_{l}^{0.75} a_{c}^{0.25} + 0.125 E π^{- 0.75} 2^{4.75} \\ \times {(ln γ)}^{0.5} G^{0.5} a_{l}^{0.75} (ln a_{l} - ln a_{c}) \end{matrix}

(44)

According to the above analysis, substitute of Eqs. (42)–(44) into Eq. (28) can obtain the sliding friction fractal model of joint surfaces.

IV. RESULTS AND DISCUSSION

As shown in Fig. 6, there is a cylinder assembling unit and its main material and length parameters are shown in Table I. Its material is 45 steel, the elastic modulus is 2.1×10¹¹Pa, and the equivalent elastic modulus can be obtained by E in Eq. (29). The adhesion shear strength²² of joint surfaces is $τ = σ_{y} / \sqrt{3}$ ⁠; the nominal contact area A_a is 19939mm²; the actual contact area²³ is A_r=0.01A_a. The value of other parameters are that the radial force F_y is 2kN, e is 0.5, η is 0.5. The roughness R_a of the joint surfaces is 6.2. The relationship between fractal roughness and roughness²⁴ is $G = 1 0^{(- 5.26 / R_{a}^{0.042})}$ ⁠; the relationship between two-dimensional fractal dimension and roughness is $D_{s} = 1.54 / R_{a}^{0.042}$ ⁠. Therefore the fractal roughness G is 1.34×10^-11m, the two-dimensional fractal dimension D_s is 1.426. The three-dimensional fractal dimension¹⁷ is D=D_s+1=2.426. These parameters will be applied to simulation analysis.

FIG. 6.

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Cylinder assembling unit.

TABLE I.

Material and length parameters.

Equivalent elastic		Density
modulus (Pa)		(kg/m³)	Poisson’s ratio	Yield strength(Pa)	Size dimension(mm)
Shaft					Diameter 63.5	Length 150
	1.154e11	7850	0.3	3.55e8
Strutting					Length 100	Height 84
piece					Width 72

Equivalent elastic		Density
modulus (Pa)		(kg/m³)	Poisson’s ratio	Yield strength(Pa)	Size dimension(mm)
Shaft					Diameter 63.5	Length 150
	1.154e11	7850	0.3	3.55e8
Strutting					Length 100	Height 84
piece					Width 72

As shown in Fig. 7, the sliding friction F increases with the increase of the maximum asperity contact area a_l. The reason is that the increase of a_l will lead to the increase of actual contact area of joint surfaces, and then the plastic contact area, part of actual contact area, increases; the molecular adhesion, part of sliding friction, is related to plastic contact area so that the increase of molecular adhesion eventually manifests itself as the increase of sliding friction of joint surfaces.

FIG. 7.

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The relationship between the maximum contact area of asperity and sliding friction (D=2.426, G=1.34×10^-11m).

As shown in Fig.8, the relationship between the surface fractal roughness G and the sliding friction F is that F increases with G increasing. As shown in Fig. 5, that the larger G means that amplitude of surface topography is larger so the surface is more rough, and then asperities are interlocked and mechanical deformation dominates in contact; furthermore, the increase of roughness decreases the contact area of micro contact point, then the elastic deformation area decreases, finally the wear of joint surfaces increases. Therefore, the surface fractal roughness should be specially designed and processed.

FIG. 8.

$FIG. 8. The relationship between the fractal roughness and sliding friction (D=2.426).$

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The relationship between the fractal roughness and sliding friction (D=2.426).

As shown in Fig.9, the relationship between the fractal dimension D, another important fractal parameter of surface topography, and the sliding friction F is that with D increasing, F firstly decreases, and then increases, that is to say, there is an optimal fractal dimension(D=2.54) which can minimize the sliding friction. As shown in Fig. 4, that the larger D means that the surface is more smooth in macroscopic view. Therefore, the Fig. 9 can be used to explain the surface friction theory reasonably from the perspective of the three-dimensional fractal surface topography:since the fractal dimension is less than the optimal fractal dimension, the surface roughness is larger, and the sliding friction is mainly expressed by mechanical force, and F decreases with D increasing; since the fractal dimension is larger than the optimal fractal dimension, the surface is very smooth, and the sliding friction is mainly expressed by molecular adhesion, and F increases with D increasing. The simulation results are consistent with the experimental results in the relevant literatures,^6,25,26 providing theoretical support for the experimental results.

FIG. 9.

$FIG. 9. The relationship between the fractal dimension and sliding friction (G=1.34×10-11m).$

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The relationship between the fractal dimension and sliding friction (G=1.34×10^-11m).

V. CONCLUSIONS

In this paper, the three-dimensional fractal model of sliding friction among dry-friction rough surfaces is deduced in detail. The present model not only shows the relative motion of asperities on surfaces, but also shows the three-dimensional fractal characteristic of rough surface topography, so the model is characterized by the scale independence. The effects of two important fractal parameters of surface topography, D and G, on the sliding friction model are analyzed by the numerical simulation. The main conclusions are as follows:

The three-dimensional fractal model of sliding friction among dry-friction rough surfaces is built based on the contact mechanics and the fractal theory.
The fractal dimension D has influence on the sliding friction, and in the case of this paper, there is an optimal value, D=2.54, which can minimize the sliding friction.
The sliding friction is positively correlated with the maximum contact area a_l of asperity and the fractal roughness G respectively.

ACKNOWLEDGMENTS

The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (NSFC) (Grant No.51275079 and No.51575091); Fundamental Research Funds for the Central Universities (N160306003). The authors also thank the anonymous reviewers for their valuable comments.

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Author(s)

Influence of surface topography on three-dimensional fractal model of sliding friction

I. INTRODUCTION

II. CONTACT ANALYSIS

III. THREE-DIMENSIONAL FRACTAL MODEL OF SLIDING FRICTION

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

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Influence of surface topography on three-dimensional fractal model of sliding friction Open Access

I. INTRODUCTION

II. CONTACT ANALYSIS

III. THREE-DIMENSIONAL FRACTAL MODEL OF SLIDING FRICTION

IV. RESULTS AND DISCUSSION

V. CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

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Influence of surface topography on three-dimensional fractal model of sliding friction