Theoretical Study of the Friction Coefficient in the M-B Model

Abstract

In order to study the influencing factors of friction coefficient in an M-B model, based on the basic model of fractal theory, the distribution function and probability distribution density of the micro-convex body truncation area are derived by using mathematical and statistical means, and a new model of critical truncation area and friction coefficient in fractal surface contact process are proposed. Considering the differences between the actual contact area and the truncated area during plastic deformation of the micro-convex body, a correction factor is introduced. Focusing on the mechanism of the elastic-plastic transition phase, and finally a friction coefficient model based on the fractal dimension, the normal force and correction factor is derived. Finally, the friction coefficient of fractal surface is simulated and verified by taking nickel as an example, and it is proved that the new model is correct in predicting the change trend of friction coefficient in the M-B model.

1. Introduction

There are a large number of contact surfaces in mechanical structures, and friction occurs when rough surfaces of materials come into contact with each other, and the tribological properties of the contact surfaces have an important impact on the performance of mechanical structures [1].

Fractal theory is an important tool for studying contact problems on rough surfaces. Majumdar and Bhushan [2] applied fractal theory to roughness characterization and surface contact mechanics as early as 1990 and found that when the surface treatment technique is isotropic and random error in the observed scale, the resulting surface conforms to fractal characteristics in that scale range, and the model they proposed is called the M-B model; Wang and Komvopoulos [3] proposed a more complete M-B model, which suggests that in the process of plane-to-plane friction contact, the contact interface undergoes complete plastic deformation and elasto-plastic deformation in turn, and the specific mechanism is as follows: at the beginning of contact, the top of the micro-convex body is the first to contact the rigid plane, and the radius of curvature of the top is small, so the stress is more concentrated and pure plastic deformation occurs; as the plane is continuously pressed down, the top of the micro-convex body becomes larger due to deformation and mutual fusion. In order to study the static frictional behavior of polymers and metals, a self-designed static friction coefficient measurement device was used by Benabdallah [4], focusing on the measurement of the exact moments during the sliding transition phase. The polynomial equation for calculating such static friction coefficient was finally obtained, revealing the law of the influence of two lubricants, water and paraffin, on the static friction coefficient between polymers and metals. With means of experiments and simulations, Zhang [5] established the static friction coefficient model associated with the bond surface scale based on the fractal model, linking the bond surface scale with the static friction coefficient. In recent years, there have been few theoretical studies on the coefficient of friction, most researchers have studied the effects of some specific surface treatment methods on the coefficient of friction based on experiments. For example, Sergei studied the use of lasers on diamond-like nanocomposite coatings under different environmental conditions. The surface is textured, and then the change law of the friction coefficient is studied [6]; Han studied the effect of ultraviolet radiation on the friction coefficient of the composite film structure [7].

The above-mentioned studies have their own focus on modeling the friction coefficient of rough surfaces of different materials, but none of them has seriously explored the influence of the elastic-plastic transition phase on the friction coefficient during frictional surface contact. In this paper, based on the basic model of fractal theory, the assumption that the truncated area of the asperity changes continuously is put forward, and then the distribution function and probability distribution density of the micro-convex body truncation area is derived by using mathematical and statistical means, and a new model of the critical cut-off area and the normal force during the fractal surface contact is proposed. Considering the difference between the actual contact area and the truncated area during plastic deformation of the micro-convex body, a correction factor is introduced, part of the mechanism of the elastic-plastic transition stage is discussed, and finally a friction coefficient model based on the fractal dimension, normal force and correction factor is derived and verified by simulation. In order to verify the above assumptions and the correctness of the final model for the prediction of the friction coefficient trend, simulation verification was carried out.

2. Methods

2.1. Micro-Convex Body Truncation Area Distribution Model

Fractal theory [2] holds that the number of micro-convex bodies with truncated area greater than or equal to s satisfies.

$$N(s)={\left(\frac{s_m}{s}\right)}^{(D-1)/2}$$

(1)

In the above equation, D is the fractal dimension which is used for describing the roughness of rough surfaces, and it indicates the degree of self-similarity of rough surfaces; s is the truncated area of micro-convex body; N(s) is the number of micro-convex bodies with truncated area greater than s, and s_m is the maximum truncated area of the micro-convex body. The truncated areas of all micro-convex bodies can be listed from smallest to largest: s₁, s₂, s₃, …, s_m. For a given plane, the maximum micro-convex truncation area at any height has been determined, and the number of micro-convexes at the microscopic level is known to be large, so it can be assumed that the micro-convex areas take continuous values. Let the minimum truncation area s₁ be taken as

$$s_1=\frac{1}{\alpha }{s}_m$$

(2)

Let s_i be the truncated area of the micro-convex body numbered i. α is a constant greater than 1, obviously s_i will take a value in the range of [s_m/α, s_m].

Let n be the number of micro-convex bodies and $n_i$ be the number of micro-convex bodies numbered i. Equation (1) can then be deformed as follows:

$$N(s_i)=n_i+n_{i+1}+\dots +n_m={\displaystyle \sum _{t=i}^m{n}_t}={\left(\frac{s_m}{s_i}\right)}^{(D-1)/2}$$

(3)

At this point, the total number N of all micro-convex bodies is

$$\begin{array}{ll}N& =n_1+n_2+\dots +n_m\\ & ={\displaystyle \sum _{t=1}^m{n}_i}\\ & ={\left(\frac{s_m}{s_1}\right)}^{(D-1)/2}\\ & ={\left(\frac{s_m}{\frac{1}{\alpha }{s}_m}\right)}^{(D-1)/2}\\ & ={\alpha }^{(D-1)/2}\end{array}$$

(4)

In this model, for a micro-convex body, the probability that its truncated area takes a certain value [8] is:

$$P\left(s_i\right)=\frac{n_i}{N}$$

(5)

This leads to the distribution function of the micro-convex body truncation area $F(s_i)$ as [9]:

$$\begin{array}{ll}F\left(s_i\right)=P\{s<s_i\}=1-P\left\{s\ge s_i\right\}& =1-\frac{n_i+n_{i+1}+\dots +n_m}{N}\\ & =1-\frac{{\displaystyle \sum _{t=i}^m{n}_t}}{{\alpha }^{(D-1)/2}}\\ & =1-\frac{{\left(\frac{s_m}{s_i}\right)}^{(D-1)/2}}{{\alpha }^{(D-1)/2}}\\ & =1-{\left(\frac{s_m}{\alpha s_i}\right)}^{(D-1)/2},\hspace{1em}\hspace{1em}\hspace{1em}{s}_i\in \left[\frac{s_m}{\alpha },s_m\right]\end{array}$$

(6)

This gives the probability distribution density of the truncated area $f\left(s_i\right)$ as

$$\begin{array}{ll}f\left(s_i\right)=\frac{dF\left(s_i\right)}{d{s}_i}& =\frac{d\left[1-{\left(\frac{s_m}{\alpha s_i}\right)}^{(D-1)/2}\right]}{d{s}_i}\\ & =\frac{D-1}{2}{\left(\frac{s_m}{\alpha }\right)}^{(D-1)/2}{s}_i^{(-1-D)/2}\end{array}$$

(7)

2.2. Actual Contact Area

According to the M-B model [10], the actual contact area of the two contact planes is:

When $s_m\le s_c$, the micro-convex bodies are in pure plastic contact. At this time:

$$S=S_p=\left(\frac{D-1}{3-D}\right)s_m$$

(8)

When $s_m>s_c$, both elastic and plastic contacts exist and:

$$S=\frac{D-1}{6-2D}\left[1+{\left(\frac{s_c}{s_m}\right)}^{\left(3-D\right)/2}\right]s_m$$

(9)

2.3. Elasticity Critical Point

2.3.1. Metal Material Stress Law

A diagram of the yield curve of a metallic material is shown in Figure 1.

As shown in the figure, the material has different stress-strain relationships in each of these stages. Among them, the “oa” segment is the elastic stage; the “ac” segment is the microplastic strain stage; the “cd” segment is the yield stage; the “de” segment is the strengthening stage; and the final “ef” segment is the necking stage [11].

From the yield curve of the material, it can be obtained that the stress is continuous at the cut-off point of elastic-plastic deformation, which is the yield point “c”. For the M-B model, the contact state changes and the calculation method also changes, but the stress should be continuous at the critical cut-off area.

2.3.2. Ideal Plastic Contact Load

According to Hertzian contact theory [12], the ideal plastic contact load for a micro-convex body with an actual contact area of $S_i$ is

$$P_p(S_i)=H{S}_i$$

(10)

The actual contact area S_i in plastic contact should be slightly larger than the truncated area $s_i$, and they are not equal [10]. Therefore

$$S_i=a{s}_i$$

(11)

where a is the correction factor and H is the material hardness.

At this point, the relationship between the ideal plastic contact load of the micro-convex body and the truncated area is

$$P_p(s_i)=aH{s}_i$$

(12)

2.3.3. Ideal Elastic Load

According to Hertzian contact theory [12], the ideal elastic contact load for a micro-convex body with an actual contact area of S_i is

$$P_e(S_i)=\frac{4\sqrt{\pi }{E}^{*}{G}^{D-2}{S}_i{}^{\frac{4-D}{2}}}{3}$$

(13)

In the ideal elastic contact state, the relationship between the actual contact area and the truncation area is [13]:

$$S_i=\frac{1}{2}{s}_i$$

(14)

This results in the relationship between the ideal plastic contact load of the micro-convex body and the cross-sectional area as

$$P_e(s_i)=\frac{2^{\frac{D}{2}}\sqrt{\pi }{E}^{*}{G}^{D-2}{s}_i{}^{\frac{4-D}{2}}}{3}$$

(15)

In the above equation, G is the fractal roughness and E* is the equivalent Young’s modulus.

2.3.4. Critical Truncation Area Model

For a certain micro-convex body with a truncation area equal to the critical truncation area s_c, the contact type changes from an ideal plastic contact state to an ideal elastic contact state. At this time, by the stress continuity theory, its internal stress should be continuous, so that

$$P_e(s_c)=P_p(s_c)$$

(16)

Combining Equations (12) and (15), it is obtained that

$$s_i={\left(\frac{2^{\frac{D}{2}}\sqrt{\pi }{E}^{*}{G}^{D-2}}{3aH}\right)}^{\frac{2}{D-2}}$$

(17)

2.4. Friction Coefficient

When s_m ≤ s_c, the micro-convex body is in full plastic contact state. At this point, combining Equations (4), (7) and (12), it can be introduced that

$$P=N{\displaystyle {\int }_{s_1}^{s_m}{P}_p(s_i)f(s_i)d{s}_i}=a\left(\frac{D-1}{3-D}\right)H{s}_m$$

(18)

At this point, combined with the Equation (18), the friction coefficient can be obtained [14]:

$$\begin{array}{ll}\mu & =\frac{\tau S}{P}\\ & =\frac{\tau \left(\frac{D-1}{D-3}\right)s_m}{aH\left(\frac{D-1}{D-3}\right)s_m}\\ & =\frac{\tau }{aH}\end{array}$$

(19)

where τ is the shear strength of the softer of the two materials in the friction sub.

When s_m > s_c, the micro-convex body is in mixed elastic-plastic contact. The total load is divided into elastic contact load and plastic contact load. Combining Equations (4), (7), (12) and (15), the total load of the micro-convex body P is obtained as

$$\begin{array}{ll}P& =P_p+P_e\\ & =[N-N(s_c)]{\displaystyle {\int }_{s_1}^{s_c}f(s_i)P_p(s_i)d{s}_i}+N(s_c){\displaystyle {\int }_{s_c}^{s_m}f(s_i)P_e(s_i)d{s}_i}\\ & =\frac{D-1}{3-D}aH{\left(\frac{s_c}{s_m}\right)}^{\left(3-D\right)/2}{s}_m+\frac{2^{\frac{D}{2}}\sqrt{\pi }{E}^{*}{G}^{D-2}\left(D-1\right)}{3\left(5-2D\right)}{s}_m^{\frac{D-1}{2}}(s_m^{\frac{5-2D}{2}}-s_c^{\frac{5-2D}{2}})\\ & =p(a,s_m)\end{array}$$

(20)

In this case, combining Equation (9), the contact surface friction coefficient can be obtained as

$$\mu =\frac{\tau S}{P}=\frac{\tau }{p(a,s_m)}\frac{D-1}{6-2D}\left[1+{\left(\frac{s_c}{s_m}\right)}^{\frac{3-D}{2}}\right]s_m$$

(21)

Among them, the correction factor is generally fixed and can be determined experimentally, while the maximum cut-off area s_m is related to the normal load and increases gradually with the increase of the normal load. From Equations (19) and (21), it can be seen that the friction coefficient is related to the correction factor a and the maximum cross-sectional area s_m with the same experimental object.

In summary, this friction coefficient model is based on the following assumptions:

(1)

The value of the cross-sectional area of the asperity is continuous, and the minimum value is close to 0;

(2)

There is a proportional relationship between the contact area and the cut-off area of the asperities in the pure plastic deformation stage, and the proportionality coefficient is a;

(3)

The stress of the asperity is continuous at the critical point of elastoplasticity.

Next, we need to verify that this model is correct.

3. Results and Discussion

3.1. Model Analysis

We took the data of No. 45 steel, which is more commonly used in engineering practice, as an example to illustrate the changing law of this model. The units of all physical quantities have been converted to international standard units, and the specific data are shown in

Table 1. Some physical values of the example materials.

Physical Quantities	Fractal Dimension	Fractal Roughness (m)	Hardness (N/m2)	Equivalent Young’s Modulus (Pa)	Critical Cut-Off Area (m2)
Symbol	D	G	H	E*	sc
Value	2.46	2.00 × 10−9	2.60 × 10−8	2.34 × 1011	4.46 × 10−6

[15]:

3.1.1. Maximum Cross-Sectional Area and Normal Force Relationship

When the two planes are in contact, if the experimental object remains unchanged, the maximum cross-sectional area must gradually increase with the increase of the normal force. Figure 2 gives a schematic diagram of the relationship between the normal force and the maximum cross-sectional area at the two stages of elastic contact and elasto-plastic contact.

In the figure above, some message can be obtained:

(1)

L1 shows the relationship between the maximum cross-sectional area and the normal force when s_m ≤ s_c without considering the critical cross-sectional area. The micro-convex body is in plastic contact at this time, and it can be seen from Figure 2 and Equation (18) that the normal force is linearly related to the maximum cut-off area.

(2)

L2 shows the relationship between the maximum cut-off area and the normal force when s_m > s_c without considering the critical cut-off area. At this time, the micro-convex body is in mixed elastic-plastic contact, and the relationship between the normal force and the maximum truncation area is more complicated, but the relationship between the normal force and the maximum truncation area is still largely linear, with a high linear correlation, but its slope is greater than that of the plastic contact stage.

(3)

L3 shows the relationship between the maximum truncation area and the normal force when the critical truncation area is considered. It can be seen from the graph that the normal force changes abruptly with a large change in the normal force when the contact type of the micro-convex body changes near the critical truncation area.

From the above analysis, it can be obtained that the relationship between the maximum cut-off area and the normal force is roughly linear in both phases, but the linear relationship is different in the plastic contact phase and the elastic-plastic mixing phase.

3.1.2. Maximum Cross-Sectional Area and Friction Coefficient Relationship

Figure 3 shows the graph of the friction coefficient as a function of the maximum cut-off area when the correction factor a is taken as 1.0, 1.1 and 1.2.

From Figure 3, the friction coefficient first remains constant during the gradual increase of the maximum cut-off area from 0. At this time, the friction state is pure plastic contact when s_m ≤ s_c, and the friction coefficient model corresponds to Equation (19); then, after the maximum cut-off area reaches the critical cut-off area, the friction turns into elastic-plastic contact, and the friction coefficient model corresponds to Equation (21). It can be seen that the friction coefficient model changes more obviously at the critical truncation area.

3.1.3. Relationship between Normal Force and Friction Coefficient

Both normal force and friction coefficient can be set or calculated by the simulation, so the normal force vs. friction coefficient relationship is the most effective way to validate the model. Figure 4 shows the normal force versus friction coefficient relationship constructed from the model.

Similar to Figure 3, the relationship between the normal force and the friction coefficient in Figure 4 also shows a general trend of constant force, then a sharp decline, and finally a leveling off. However, near the inflection point, the relationship between the normal force and the friction coefficient is disturbed, which may be related to the graph drawing method. However, in reality, the maximum cross-sectional area is not necessarily continuous and increasing in this process, and there may be more complex changes near the elastic-plastic critical point, which leads to the confusion near the critical point in Figure 4 [16]. However, the relationship between the normal force and the friction coefficient in this model must include the variable of the maximum cross-sectional area which cannot be measured, so it is not possible to directly plot the friction coefficient image with the normal force as the initial variable for the time being [17].

3.2. Simulation Material Simulation Analysis

3.2.1. Modeling

In this paper, we take the M-B model in fractal theory as the research object, so we need to establish a rough surface that meets the fractal characteristics first. The equation of the surface profile of the model is [10]:

$$z(x)={\displaystyle \sum _{n=0}^6{1.5}^{(D-2)n}[1-\cos ({1.5}^{n}x)]}$$

(22)

In the above equation, D is taken as 2.3, 2.4 and 2.5, and the obtained profile curves can be seen in Figure 5.

Next, according to this surface profile model to build a model, this simulation using MATLAB for modeling, and use the Python language to build a copy into ABAQUS for secondary development, to build a complete microscopic rough surface [18]. The complete surface model is shown in Figure 6.

The friction surface of the slider is an ideal rigid plane, and the sliding substrate is a metallic material with fractal surface characteristics whose physical parameters are set as follows: Young’s modulus 143,000 MPa, Poisson’s ratio 0.291, yield stress 590 MPa. In order to make the simulation more precise, the substrate uses a triangular grid, and the slider uses a quadrilateral grid for computational convenience [13]. The slider is a quadrilateral mesh for the same reason.

3.2.2. Analysis of Results

The calculation time is set to six seconds, the motion speed is 1 cm/s, and the tangential force and friction coefficient are calculated for each condition by taking different normal forces and fractal dimensions D, respectively. Figure 7 shows some of the stress concentrations during the calculation.

Summing up all the tangential stresses at the contact interface, the frictional force on the slider is obtained by dividing it by the corresponding normal force to obtain its friction coefficient [19]. The specific results of all simulations are shown in

Table 2. Simulation results of the friction coefficient for different normal forces and fractal dimensions.

Normal Force (N)	Friction Coefficient			Changing Rate of Friction Coefficient (1/N)
	D = 2.3	D = 2.4	D = 2.5	D = 2.3	D = 2.4	D = 2.5
10	0.98187	0.80004	0.76004	—	—	—
100	0.93519	0.77730	0.74301	−5.19 × 10−4	−2.53 × 10−4	−1.89 × 10−4
200	0.91853	0.49584	0.41757	−1.67 × 10−4	−2.81 × 10−3	−3.25 × 10−3
300	0.86853	0.42823	0.33837	−5.00 × 10−4	−6.76 × 10−4	−7.92 × 10−4
600	0.50164	0.29023	0.24023	−1.22 × 10−3	−4.60 × 10−4	−3.27 × 10−4
900	0.35007	0.22863	0.18675	−5.05 × 10−4	−2.05 × 10−4	−1.78 × 10−4
1200	0.2753	0.19942	0.16278	−2.49 × 10−4	−9.74 × 10−5	−7.99 × 10−5
1500	0.22905	0.18655	0.15655	−1.54 × 10−4	−4.29 × 10−5	−2.08 × 10−5
1800	0.19815	0.17387	0.15387	−1.03 × 10−4	−4.23 × 10−5	−8.93 × 10−6
2100	0.17606	0.16532	0.15032	−7.37 × 10−5	−2.85 × 10−5	−1.18 × 10−5
2400	0.15946	0.16166	0.14807	−5.53 × 10−5	−1.22 × 10−5	−7.52 × 10−6
2700	0.14652	0.15765	0.14765	−4.31 × 10−5	−1.33 × 10−5	−1.37 × 10−6
3000	0.13671	0.15434	0.14534	−3.27 × 10−5	−1.10 × 10−5	−7.71 × 10−6
3600	0.11727	0.14040	0.13840	−3.24 × 10−5	−2.32 × 10−5	−1.16 × 10−5

.

With the normal force as the horizontal coordinate and the friction coefficient as the vertical coordinate, the above table can be transformed into a graph, as shown in Figure 8, and the changing rate of the friction coefficient in the above table refers to the slope between the current node and the previous node.

From the above graphs it can be seen that:

(1)

The changing rate of friction coefficient represents the change trend of the friction coefficient between the current node and the previous node. It can be seen from Table 2 that all the rates of change are negative numbers, indicating that the friction coefficient is decreasing from beginning to end;

(2)

The friction coefficient does change slowly at the beginning of the gradual increase of the normal force from 0, which is different from the complete invariance expected in Figure 4, but the difference is not significant.

(3)

The change in the friction coefficient does have a clear inflection point, exactly as expected in Figure 4, which is particularly evident when D takes values 2.4 and 2.5. It can be seen from Table 2 that when the normal force changes from 10N to 100N, the slope is quite different from that near the elastoplastic critical point, and even there is an order of magnitude difference when D is 2.4 and 2.5;

(4)

The change trend of the friction coefficient after the inflection point is exactly the same as that of Figure 4, which both show a sharp decrease and then level off, and the different fractal dimensions D have little effect on the final friction coefficient.

It can be concluded that this model is generally accurate for the prediction of the change trend of the friction coefficient; the friction coefficient change trend is completely consistent with the model prediction, particularly after the elasto-plastic critical point, while the initial part before the elasto-plastic critical point is not exactly the same, but the difference is not huge.

4. Conclusions

Through the above analysis and derivation, the main work of this paper can be summarized as follows.

(1)

A more explicit and specific distribution model of the micro-convex body truncation area is proposed. The fractal contact model is analyzed and advanced, and the micro-convex truncated area distribution function and probability distribution density are obtained by applying mathematical and statistical methods.

(2)

A correction factor is proposed. The plastic contact area differs from the ideal plastic contact area when microscopic contact is considered, and a correction factor is thus proposed to correct the model.

(3)

A new friction coefficient model based on the fractal dimension, normal force and correction factor is proposed. Based on the principle of elastic-plastic stress continuity at the critical point, a new critical cut-off area model is proposed, which leads to a new friction coefficient model.

(4)

Simulation calculations were performed to verify the friction coefficient of the fractal surface, and the analysis and verification of the proposed model were compared with the simulation results to prove the correctness of the new model in predicting the trend of the friction coefficient change.

There are still some areas for improvement and further research. Although the new model can predict the trend of the friction coefficient, there is confusion at the elastic-plastic threshold due to the existence of unmeasurable variables in the model, which needs to be improved to eliminate such variables. In addition, the model is only simulated in this paper, and further experiments are needed to validate the model.