# A Revised Continuous Observation Length Model of Rough Contact without Adhesion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework

#### 2.1. The OLD Model

_{0}was regarded as a discrete surface composed of ideal subplanes with the observation length λ, λ

_{0}< λ < L

_{0}, as shown in Figure 1b. The ideal subplanes were regarded as asperities of the same curvature radius at a certain observation length when the contact force and the real contact area were calculated, as shown in Figure 1c. Therefore, the contact of a rigid plane and a deformable rough surface can be idealized as the contact between a rigid plane and deformable ideal subplanes. The height of the asperity is the average height of the asperity at the observation length division plane. Assuming that the heights of the ideal subplanes follow the Gaussian distribution in all observation lengths, the standard deviation of the ideal subplane height can be obtained. To obtain the relationship between the observation length and standard deviation, a linear relationship was assumed (as shown in Section 2.1.1).

#### 2.1.1. The Standard Deviation

^{n}determines the frequency spectrum of the surface roughness, and γ ≥ 1.0. For most typical surfaces, γ = 1.5. n

_{l}is the minimum frequency index, and n

_{l}= log

_{γ}(1/L

_{0}).

_{0}is the standard deviation of the surface profile at the minimum observation length.

#### 2.2. The Revised Continuous Observation Length Model

#### 2.2.1. The Standard Deviation

_{0}. The observation length is nondimensionalized to eliminate the influence of sampling length.

_{min}is the minimum observation length. According to the W-M function, D has a significant effect on the height of the rough surface profile [38].

^{(D−1)}, and z(x) becomes smaller as D becomes bigger. When the λ* is small, the change of z(x) is significant due to the increase in D. Therefore, the standard deviation σ also changes greatly, resulting in different rising rates.

_{0}+ B

_{1}λ*), quadratic polynomial fitting (σ = y

_{0}+ B

_{1}λ* + B

_{2}λ*

^{2}) and cubic polynomial fitting (σ = y

_{0}+ B

_{1}λ* + B

_{2}λ*

^{2}+ B

_{3}λ*

^{3}) are adopted, respectively. The fitting curves of λ* and σ are shown in Figure 3. It can be seen that the linear fitting curves have a large deviation from the data points compared to the polynomial fitting curves. The quadratic fitting curves and the cubic fitting curves are more consistent with the data points. Thus, the polynomial fitting is better than the linear fitting.

_{0}+ B

_{1}e

^{−x/A}

_{1}+ B

_{1}e

^{−x/A}

_{2}) is used for fitting. The fitting curves of λ* and σ are shown in Figure 4.

^{2}is important to evaluate the fitting effect. The closer the value of R

^{2}is to 1, the better the fit of the model. The relationship between R

^{2}and D in different fitting methods is shown in Figure 5. It can be concluded that: (a) when D ≤ 1.5, the power fitting is not suitable; (b) when D ≤ 1.5, the R

^{2}of all the three fitting methods is above 0.99, which is well fitted. Among the three methods, there is little difference between the two polynomial fitting methods, both of which are better than linear fitting, and the difference gradually decreases with the increase in D; (c) when D = 1.6, the R

^{2}of the three fitting methods is basically the same. However, from the figure, the quadratic polynomial fitting is more in line with the requirements; (d) when D ≥ 1.7, the R

^{2}of the linear fitting and polynomial fitting decreases with the increase in D. When D = 1.9, the R

^{2}of the linear fitting, quadratic fitting and cubic fitting is 0.82, 0.94, 0.96, respectively. This indicates that the fitting is poor. In addition, the R

^{2}of the ExpDec2 function fitting is bigger than 0.99, which is better than the other fitting methods.

#### 2.2.2. Establishment of Contact Mechanics Model

_{e}of the plastic contact is derived by Jackson based on Von Mises’ yield criterion, as shown below:

_{Y}, and σ

_{Y}is the yield strength. ϕ is the material property, ϕ = H/E. E is the equivalent modulus of Hertzian elasticity and equals

_{1}, E

_{2}, ν

_{1}and ν

_{2}are the Young’s modulus and Poisson’s ratios of two contacting surfaces, respectively.

#### 2.3. Contact Areas and Mechanics

_{e}, the asperity is at elastic deformation, and the ideal subplane height of elastic deformation should be d(λ) < h(x, λ) ≤ d(λ) + ω

_{e}. When ω

_{e}< ω < 6ω

_{e}, it is the first stage of elastoplastic deformation, and the height should be d(x, λ) + ω

_{e}< h(x, λ) ≤ d(x, λ) + 6ω

_{e}. When 6ω

_{e}< ω < 110ω

_{e}, it is the second stage of elastoplastic deformation, and the height should be d(x, λ) + 6ω

_{e}< h(x, λ)≤ d(x, λ) + 110ω

_{e}. In addition, when ω > 110 ω

_{e}, the contact is completely plastic.

#### 2.3.1. The Areas in Different Stages

_{r}of contact is

_{re}is

_{rep1}is

_{rep2}is

_{rp}is

_{0}is the nominal area, A

_{0}= L

_{0}

^{2}.

_{r}* is obtained by accumulating the different stages’ contributions. Hence, from Equations (14)–(17)

#### 2.3.2. The Relationship between F and ω_{e} of Single Asperity

_{e}is the critical contact depth; F

_{re}, F

_{rep1}, F

_{rep2}and F

_{rp}are the contact forces of a single asperity in four stages; F

_{ec}is the critical contact force of the elastic contact. The critical contact force is

#### 2.3.3. The Contact Force of the Surfaces in Different Stages

_{re}, P

_{rep}and P

_{rp}are the elastic, elastoplastic and plastic contact forces with the observation length λ, which can be calculated as the sum of forces in different stages of asperities.

_{0}is

_{0}is

_{N}and the nominal area A

_{0}are usually known for a certain contact system and will not change with the observation length λ. The separation d(λ) at any observation length can be solved by the numerical method. Consequently, the real contact area, elastic contact area and plastic contact area can be calculated with d(λ).

## 3. Numerical Results and Discussion

^{−12}m; the Poisson’s ratio ν = 0.3 and the yield strength σ

_{Y}= 345 MPa. The mean contact stress σ

_{0}is 10 MPa. By numerical analysis in Python, the changing rules between the contact area and λ* are obtained. The numerical analysis is based on the above equation through circular iterations and judgments.

#### 3.1. Surface Fractal Parameters

#### 3.1.1. The Influence of the Fractal Dimension

^{−12}m and D = 1.3, 1.4, 1.5 are plotted in Figure 6. After abundant and repetitive simulations, we found out that a high value of D results in more surface details.

_{r}* can be reduced by the decrease in λ* with the same D. Additionally, the observation scale decreases when A

_{r}* begins to reduce with the rising D. It is worth pointing out that A

_{r}* grows as D increases with the same λ*. This is because the surface with a smaller value of D reaches the incomplete contact.

^{−12}m and E = 100 GPa. When λ* decreases, the plastic contact dominates, and otherwise, the elastic contact dominates. The dimensionless elastic contact area A

_{re}* decreases as λ* reduces, and in the other three stages, the areas rise to the peak before decreasing to a small value. The maximum areas reduce with the increase in D. Moreover, the λ* where peaks occur reduce with the increase in D. For D = 1.3, when the peak occurs, λ* = 0.1, and A

_{rep1}* = 0.0142, λ* = 0.02 and A

_{rep2}* = 0.0034. For D = 1.4, when the peak occurs, λ* = 0.003, and A

_{rep1}* = 0.0036, λ* = 5 × 10

^{−4}and A

_{rep2}* = 0.0012, respectively. In general, the value of D has a large effect on the real contact area A

_{r}* and the area of each deformation stage.

_{re}* is 1, and there is only elastic deformation, as shown in Figure 8. The proportion gradually decreases when (a) D = 1.3 and λ* < 0.4; (b) D = 1.4 and λ* < 0.02; (c) D = 1.5 and λ* < 0.0006. This is because the elastic deformation transforms into the first elastoplastic deformation with the decrease in λ*. For the elastoplastic deformation stage, the trend is similar to that of their dimensionless areas, and the value of the peaks increases as D increases. For the plastic deformation stage, the proportion of the plastic dimensionless area is 0 when (a) D = 1.3 and λ* > 0.002; (b) D = 1.4 and λ* > 9 × 10

^{−5}; (c) D = 1.5 and λ* > 7 × 10

^{−6}. Additionally, when λ* is small enough, there is only plastic deformation.

#### 3.1.2. The Influence of the Equivalent Modulus

^{−12}m, ν = 0.3. Figure 9 is the dimensionless real contact area versus λ* with various E:E = 50 GPa, E = 75 GPa, E = 100 GPa.

_{re}* decreases with λ*, and a high value of E produces a big slope. As λ* reduces, the areas A

_{rep1}*, A

_{rep2}* and A

_{rp}* rise to the peak before decreasing to a small value. Additionally, the maximum λ* where the area begins to reduce is different with different E. For example, when the area A

_{rep2}* begins to reduce, E = 50 GPa and λ* = 7 × 10

^{−5}, E = 75 GPa and λ* = 2 × 10

^{−4}, E = 100 GPa and λ* = 1 × 10

^{−4}.

_{re}* decreases with λ*, and a high value of E produces a big slope. Similarly, as λ

^{*}reduces, the proportion of A

_{rep}* rises to the peak before decreasing to a small value, and the maximum λ*, where the peaks exist, becomes bigger when the value of E becomes higher. As E increases, the peak becomes lower in the first stage of elastoplastic contact, but it remains approximately unchanged in the other stage. The proportion of plastic contact area reduces as λ* increases, and when λ* > 0.001, the proportion is greater than 0.999.

#### 3.2. Model Comparison

_{Y}= 1 GPa, and ν. The equivalent parameters of our model are the surface fractal dimensions of 2D surface profile D = 1.2, the sample length L

_{0}= 0.01 m, the fractal roughness parameter G = 1.917 × 10

^{−27}, the mean contact stress σ

_{0}= 110 MPa. The relationship between the observation scale and the contact area can be obtained according to the multi-scale contact model, as shown in Figure 12. In order to compare with the Persson model, the magnification is defined [26]. The magnification ζ is defined as ζ = λ

_{0}/λ where λ

_{0}is the critical value of observation length. When ζ = λ

_{0}/λ, the two contact surfaces are in full contact, and A

_{r}* = 1. When ζ = λ

_{0}/λ, the two contact surfaces are in full contact, and A

_{r}* = 1. When λ

_{0}< λ, the contact is incomplete, and A

_{r}* < 1.

_{r}* decreases with the rise of ζ and increases with the reduction in λ. The three curves have an inflection point where the real contact area decreases with the observation length. According to the changing rules, the curves can be clearly divided into two parts. They have good consistency in the changing rule and trend. Therefore, the present model is consistent with Liu’s model and Persson’s model.

^{0.25}< ζ < 10

^{1.5}). This is because the present model replaces the contact areas of the asperities with the ideal subplane areas and obtains a relatively large result. The contact area of the present model is smaller compared to the Persson model at a smaller observation length (ζ > 10

^{1.5}). This is because Persson assumed that when the local contact is in a plastic state, the plastic contact area will no longer change as the observation length decreases. However, in the present model, after the contact enters into a plastic state, the local surface is still in an incomplete contact. It was suggested that the minimum observation length of the incomplete contact can reach the atomic level.

## 4. Conclusions

- When the fractal dimension D is different, the fitting relationship between the standard deviation and observation length is also different. For D ≤ 1.5, the polynomial fitting is the best. For D = 1.6, the quadratic polynomial fitting is more in line. For D ≥ 1.7, the ExpDec2 function fitting is better than the other fitting methods. The larger the D, the larger the error of the linear fitting.
- As λ* reduces, the areas A
_{r}* and A_{re}* decrease, and in the other three stages, the areas rise to the peaks before decreasing to a small value. The values of the peaks and the values of λ* where peaks occur are related to D and E. As λ* reduces, the proportion of A_{re}* decreases, the proportion of A_{rep}* rises to the peak before similarly decreasing to a small value, and the proportion of A_{rp}* increases. When λ^{*}reduces to a certain value, the proportion of A_{rp}* approaches 1, and the proportion of A_{rep1}* is larger than A_{rep2}*. - As D increases, the area A
_{r}* increases. The values of the peaks and λ* where peaks occur reduce with rising D. As E increases, the area A_{re}* drops quicker when λ* decreases. Additionally, the values of the peaks and λ* where peaks occur decrease with the increase in E. - The dimensionless real contact area of the present model is in good agreement with the Persson model and the Liu model in terms of variation patterns.
- The present model can accurately describe the contact characteristics of rough surfaces and the monotonicity and continuity in contact processes. The research results provide a theoretical basis for analyzing the contact characteristics on rough surfaces and designing a contact surface topography.

## 5. The Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

λ | the observation length, m |

λ* | the observation scale |

λ_{0} | the critical value of observation length |

x | the height of the ideal subplane |

σ | the standard deviation |

σ (λ) | the standard deviation of the ideal subplane |

h(λ) | the height of the ideal subplane |

h(λ) | the average height of the ideal subplane |

d(λ) | the separation at the observation length λ, m |

R^{2} | the coefficient of determination |

z(x) | the height of the surface profile |

L_{0} | the sample length, m |

n_{l} | the minimum frequency index |

γ^{n} | spatial frequency |

D | the surface fractal dimension |

G | the fractal roughness, m |

K | the hardness coefficient |

ν | the Poisson ratio |

H | the hardness of the softer material, MPa |

σ_{Y} | the yield strength, MPa |

R | the asperity curvature radius, m |

E_{1}, E_{2} | Young’s modulus of two contacting surfaces |

ν_{1}, ν_{2} | Poisson’s ratios of two contacting surfaces |

E | the equivalent modulus of Hertzian elasticity |

A_{r} | the real area of contact surface, m^{2} |

A_{re} | the elastic contact area, m^{2} |

A_{rep1} | the real area in the first stage of elastoplastic contact, m^{2} |

A_{rep2} | the real area in the second stage of elastoplastic contact, m^{2} |

A_{rp} | the real plastic contact area, m^{2} |

A_{0} | the nominal area, m^{2} |

A | the elastic dimensionless contact area |

A_{rep1}* | the dimensionless area in the first stage of elastoplastic contact |

A_{rep2}* | the dimensionless area in the second stage of elastoplastic contact |

A_{rp}* | the plastic dimensionless contact area |

A_{r}* | the total dimensionless contact area |

F_{ec} | the critical contact force, N |

F_{re} | the contact force of single asperity in elastic contact, N |

F_{rep1} | the contact force of single asperity in the first stage of elastoplastic contact, N |

F_{rep2} | the contact force of single asperity in the second stage of elastoplastic contact, N |

F_{rp} | the contact force of single asperity in plastic contact, N |

P_{r} | the total surface contact force |

P_{re} | the plastic contact forces, N |

P_{rep1}* | the dimensionless force in the first stage of elastoplastic contact |

P_{rep2}* | the dimensionless force in the second stage of elastoplastic contact |

P_{rp}* | the plastic dimensionless contact force |

P_{r}* | the total dimensionless contact force |

σ_{0} | the mean contact stress, MPa |

RMS | the root mean square of rough surface |

ζ | the magnification |

## References

- Bowden, F.P.; Tabor, D. The Hardness of Metals; Clarendon Press: Oxford, UK, 1951; pp. 279–281. [Google Scholar]
- Greenwood, J.A.; Williamson, J.P. Contact of nominally flat surfaces. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1966**, 295, 300–319. [Google Scholar] [CrossRef] - Chang, W.; Etsion, I.; Bogy, D.B. An Elastic-Plastic Model for the Contact of Rough Surfaces. J. Tribol.
**1987**, 109, 257–263. [Google Scholar] [CrossRef] - Xu, Y.; Jackson, R.L.; Marghitu, D.B. Statistical model of nearly complete elastic rough surface contact. Int. J. Solids Struct.
**2014**, 51, 1075–1088. [Google Scholar] [CrossRef] [Green Version] - Bhushan, B.; Majumdar, A. Role of Fractal Geometry in Roughness Characterization and Contact Mechanics of Surface. J. Tribol.
**1990**, 112, 205–216. [Google Scholar] [CrossRef] - Zhao, Y.; Maietta, D.M.; Chang, L. An Asperity Microcontact Model Incorporating the Transition from Elastic Deformation to Fully Plastic Flow. J. Tribol.
**2000**, 122, 86–93. [Google Scholar] [CrossRef] - Liou, J.L.; Lin, J.F. A modified fractal microcontact model developed for asperity heights with variable morphology parameters. Wear
**2010**, 268, 133–144. [Google Scholar] [CrossRef] - Kogut, L.; Etsion, I. Elastic-plastic contact analysis of a sphere and a rigid flat. J. Appl. Mech. Trans. ASME
**2002**, 69, 657–662. [Google Scholar] [CrossRef] [Green Version] - Morag, Y.; Etsion, I. Resolving the contradiction of asperities plastic to elastic mode transition in current contact models of fractal rough surfaces. Wear
**2007**, 262, 624–629. [Google Scholar] [CrossRef] - Yuan, Y.; Gan, L.; Liu, K.; Yang, X. Elastoplastic contact mechanics model of rough surface based on fractal theory. Chin. J. Mech. Eng.
**2017**, 30, 207–215. [Google Scholar] [CrossRef] - Hanaor, D.A.H.; Gan, Y.X.; Einav, I. Contact mechanics of fractal surfaces by spline assisted discretisation. Int. J. Solids Struct.
**2015**, 59, 121–131. [Google Scholar] [CrossRef] - Pan, W.; Song, C.; Ling, L.; Qu, H.; Wang, M. Unloading contact mechanics analysis of elastic–plastic fractal surface. Arch. Appl. Mech.
**2021**, 91, 2697–2712. [Google Scholar] [CrossRef] - Wen, Y.; Tang, J.; Zhou, W.; Li, L.; Zhu, C. New analytical model of elastic-plastic contact for three-dimensional rough surfaces considering interaction of asperities. Friction
**2021**, 10, 217–231. [Google Scholar] [CrossRef] - Zhang, W.; Jin, F.; Zhang, S.L.; Guo, X. Adhesive Contact on Randomly Rough Surfaces Based on the Double-Hertz Model. J. Appl. Mech. Trans. ASME
**2014**, 81, 051008. [Google Scholar] [CrossRef] - Song, H.; Van der Giessen, E.; Liu, X. Strain gradient plasticity analysis of elasto-plastic contact between rough surfaces. J. Mech. Phys. Solids
**2016**, 96, 18–28. [Google Scholar] [CrossRef] [Green Version] - Song, H.; Vakis, A.I.; Liu, X.; Van der Giessen, E. Statistical model of rough surface contact accounting for size-dependent plasticity and asperity interaction. J. Mech. Phys. Solids
**2017**, 106, 1–14. [Google Scholar] [CrossRef] [Green Version] - Kuzkin, V.A.; Kachanov, M. Contact of rough surfaces: Conductance-stiffness connection for contacting transversely isotropic half-spaces. Int. J. Eng. Sci.
**2015**, 97, 1–5. [Google Scholar] [CrossRef] - Jin, F.; Wan, Q.; Guo, X. Plane Contact and Partial Slip Behaviors of Elastic Layers With Randomly Rough Surfaces. J. Appl. Mech. Trans. ASME
**2015**, 82, 091006. [Google Scholar] [CrossRef] - Chen, Q.; Xu, F.; Liu, P.; Fan, H. Research on fractal model of normal contact stiffness between two spheroidal joint surfaces considering friction factor. Tribol. Int.
**2016**, 97, 253–264. [Google Scholar] [CrossRef] - Pan, W.; Li, X.; Wang, L.; Guo, N.; Mu, J. A normal contact stiffness fractal prediction model of dry-friction rough surface and experimental verification. Eur. J. Mech. A Solids
**2017**, 66, 94–102. [Google Scholar] [CrossRef] - Wang, R.; Zhu, L.; Zhu, C. Research on fractal model of normal contact stiffness for mechanical joint considering asperity interaction. Int. J. Mech. Sci.
**2017**, 134, 357–369. [Google Scholar] [CrossRef] - Zhai, C.; Hanaor, D.; Gan, Y. Contact stiffness of multiscale surfaces by truncation analysis. Int. J. Mech. Sci.
**2017**, 131, 305–316. [Google Scholar] [CrossRef] - Yuan, Y.; Chen, J.J.; Zhang, L.H. Loading-unloading contact model between three-dimensional fractal rough surfaces. AIP Adv.
**2018**, 8, 075017. [Google Scholar] [CrossRef] [Green Version] - Wang, H.H.; Jia, P.; Wang, L.Q.; Yun, F.H.; Wang, G.; Liu, M.; Wang, X.Y. Modeling of the Loading-Unloading Contact of Two Cylindrical Rough Surfaces with Friction. Appl. Sci.
**2020**, 10, 742. [Google Scholar] [CrossRef] [Green Version] - Persson, B.N.J. Contact Mechanics for Randomly Rough Surfaces. Surf. Sci. Rep.
**2006**, 61, 201–227. [Google Scholar] [CrossRef] [Green Version] - Afferrante, L.; Bottiglione, F.; Putignano, C.; Persson, B.N.J.; Carbone, G. Elastic Contact Mechanics of Randomly Rough Surfaces: An Assessment of Advanced Asperity Models and Persson’s Theory. Tribol. Lett.
**2018**, 66, 75. [Google Scholar] [CrossRef] - Ciavarella, M. An approximate JKR solution for a general contact, including rough contacts. J. Mech. Phys. Solids
**2018**, 114, 209–218. [Google Scholar] [CrossRef] [Green Version] - Guo, X.; Ma, B.B.; Zhu, Y.C. A magnification-based multi-asperity (MBMA) model of rough contact without adhesion. J. Mech. Phys. Solids
**2019**, 133, 103724. [Google Scholar] [CrossRef] - Liu, M.; Zhang, L.; Wang, L.Q.; Liu, H.X.; Sun, Y.Q.; Wang, Y.J. The leakage analysis of submarine pipeline connecter based on a new fractal porous media model. Desalin. Water Treat.
**2020**, 188, 390–399. [Google Scholar] [CrossRef] - Lorenz, B.; Persson, B.N.J. Leak rate of seals: Effective-medium theory and comparison with experiment. Eur. Phys. J. E
**2010**, 31, 159–167. [Google Scholar] [CrossRef] [Green Version] - Persson, B.N.J. Leakage of Metallic Seals: Role of Plastic Deformations. Tribol. Lett.
**2016**, 63, 42. [Google Scholar] [CrossRef] - Lorenz, B.; Rodriguez, N.; Mangiagalli, P.; Persson, B.N.J. Role of hydrophobicity on interfacial fluid flow: Theory and some applications. Eur. Phys. J. E
**2014**, 37, 57. [Google Scholar] [CrossRef] [PubMed] - Rodriguez, N.; Dorogin, L.; Chew, K.T.; Persson, B.N.J. Adhesion, friction and viscoelastic properties for non-aged and aged Styrene Butadiene rubber. Tribol. Int.
**2018**, 121, 78–83. [Google Scholar] [CrossRef] - Persson, B.N.J.; Scaraggi, M. Theory of adhesion: Role of surface roughness. J. Chem. Phys.
**2014**, 141, 124701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Persson, B.N.J. The dependency of adhesion and friction on electrostatic attraction. J. Chem. Phys.
**2018**, 148, 144701. [Google Scholar] [CrossRef] [PubMed] - Tiwari, A.; Dorogin, L.; Tahir, M.; Stockelhuber, K.W.; Heinrich, G.; Espallargas, N.; Persson, B.N.J. Rubber contact mechanics: Adhesion, friction and leakage of seals. Soft Matter
**2017**, 13, 9103–9121. [Google Scholar] [CrossRef] [PubMed] - Persson, B.N.J.; Scaraggi, M. Some Comments on Hydrogel and Cartilage Contact Mechanics and Friction. Tribol. Lett.
**2018**, 66, 23. [Google Scholar] [CrossRef] - Lorenz, B.; Persson, B.N.J. Leak rate of seals: Comparison of theory with experiment. Europhys. Lett.
**2009**, 86, 44006. [Google Scholar] [CrossRef]

**Figure 1.**The contact of two rough surfaces. (

**a**) The contact of two rough surfaces can be idealized as a rigid plane pressed on a deformable rough surface. (

**b**) The fractal surface can be regarded as a discrete surface composed of ideal subplanes. (

**c**) The fractal surface can be regarded as asperities of the same curvature radius.

**Figure 2.**The calculated standard deviation σ of the ideal subplane height with various fractal dimensions.

**Figure 3.**The different fitting curves between the dimensionless observation length and the standard deviation at different fractal dimensions (D ≤ 1.5).

**Figure 4.**Different fitting curves between the dimensionless observation length and the standard deviation at different fractal dimensions. (

**a**) Linear fitting, Quadratic fitting and Cubic fitting at D = 1.7. (

**d**) ExpDec2 fitting at D = 1.7, 1.8 and 1.9.

**Figure 7.**The real contact area A

_{r}* in different contact stages versus λ* with various D. (

**a**) Dimensionless elastic deformation area A

_{re}* versus λ*. (

**b**) Dimensionless elastoplastic deformation area A

_{rep1}* in the first stage versus λ*. (

**c**) Dimensionless elastoplastic deformation area A

_{rep2}* in the second stage versus λ*. (

**d**) Dimensionless plastic deformation area. A

_{rp}* versus λ

^{*}.

**Figure 8.**Influence of surface fractal dimension on multilength contact characteristics. (

**a**) Proportion of elastic contact area versus λ*. (

**b**) Proportion of elastoplastic contact area in the first stage versus λ*. (

**c**) Proportion of elastoplastic contact area in the second stage versus λ*. (

**d**) Proportion of plastic contact area versus λ*.

**Figure 10.**The real contact area A

_{r}* in different contact stages versus the λ* with various E. (

**a**) Dimensionless elastic contact area A

_{re}* versus λ*. (

**b**) Dimensionless elastoplastic contact area A

_{rep1}* in the first stage versus λ*. (

**c**) Dimensionless elastoplastic contact area A

_{rep2}* in the second stage versus λ*. (

**d**) Dimensionless plastic contact area A

_{rp}* versus λ*.

**Figure 11.**Influence of equivalent modulus of elasticity E on multilength contact characteristics. (

**a**) Proportion of elastic contact area versus λ*. (

**b**) Proportion of elastoplastic contact area in the first stage versus λ*. (

**c**) Proportion of elastoplastic contact area in the second stage versus λ*. (

**d**) Proportion of plastic contact area versus λ*.

Parameters | Values and Units |
---|---|

D | 1.1~1.9 |

G | 10^{−12} m |

L_{0} | 0.01 m |

λ_{min} | 1.0 × 10^{−9} m |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, L.; Wen, J.; Liu, M.; Xing, G.
A Revised Continuous Observation Length Model of Rough Contact without Adhesion. *Mathematics* **2022**, *10*, 3764.
https://doi.org/10.3390/math10203764

**AMA Style**

Zhang L, Wen J, Liu M, Xing G.
A Revised Continuous Observation Length Model of Rough Contact without Adhesion. *Mathematics*. 2022; 10(20):3764.
https://doi.org/10.3390/math10203764

**Chicago/Turabian Style**

Zhang, Lan, Jing Wen, Ming Liu, and Guangzhen Xing.
2022. "A Revised Continuous Observation Length Model of Rough Contact without Adhesion" *Mathematics* 10, no. 20: 3764.
https://doi.org/10.3390/math10203764