A Closure Contact Model of Self-Affine Rough Surfaces Considering Small-, Meso-, and Large-Scale Stage Without Adhesive

Abstract

Contact interface is essential for the dynamic response of the bolted structures. To accurately predict the dynamic characteristics of bolted joint structures, a fractal extension of the segmented scale model, i.e., the JK model, is proposed in this paper to comprehensively analyze the dynamic contact performance of engineering surfaces and revisit the multi-scale model based on the concept of asperities. The influence of asperity geometry, dimensionless material properties, and the elastic, elastoplastic, and full plastic mechanical models of a single asperity is established considering the asperity–substrate interaction. Then, a segmented scale contact model of rough surfaces is proposed based on the island distribution function in a strict sense. The mechanical contact process of determining rough surfaces is divided into small-scale, medium-scale, and large-scale stages. Moreover, cross-scale boundary conditions, i.e., a_l1′, a_l2′, and a_l3′, are provided through strict mathematical deduction. The results show that the real contact area and contact stiffness are positively correlated with fractal dimension and negatively correlated with fractal roughness. On a small scale, the contact damping decreases with an increase in load. In meso-scale and large-scale stages, the contact damping increases with the load. Finally, the reliability of the proposed model is verified by setting up three groups of modal vibration experiments.

1. Introduction

Contact is one of the most common ways of force transmission, whose mechanical response is extremely meaningful for performance and mechanical system integrity [1,2,3]. Among mechanical properties, contact stiffness and damping are the most common features in various engineering applications, such as biomedical, marine, civil, locomotive-railway, and aerospace engineering. The vast majority of mechanical systems normally operate by combining several independent parts. These interface contact studies span almost all scales, such as nanometer, micrometer, millimeter, and centimeter scales. Therefore, it is necessary to establish a contact mechanics model on a full scale to meet the requirements of academic research and engineering applications.

Long ago, people knew that the nominal smooth surface was rough and included many geometric asperities with random distribution and different sizes, which brought uncertainty and complexity to the research of contact mechanisms [4]. Archard was the first to propose the concept of “asperity on asperity”, which systematically explained that the load and the actual contact area are almost linearly dependent [5]. Greenwood and Williamson made a detailed analysis of non-adhesive rough surfaces based on useful pioneer insights [6]. The first statistical contact modeling method, i.e., the GW model, was proposed by assuming that the asperities’ height conforms to Gaussian distribution, the curvature radius of the asperities is constant, and there is no interaction between asperities. The GW model was later improved and expanded by many scholars to relax the strict assumptions of the GW model [7,8,9,10,11,12,13,14,15,16]. However, the statistical method cannot reflect the multi-scale properties of rough surfaces, which deviates from the original intention of Archard. Moreover, the statistical contact model depends on the sample length and the resolution of the measuring instrument, which challenges the interface contact research [17,18,19].

With the development of precision measurement technology on rough surfaces, people have realized that most engineering surfaces are multi-scale fractals [20]. In 1990, Majumdar and Bhushan proposed a scale-independent fractal contact model, i.e., the MB model [21]. Many fractal expansion models have emerged on this basis, such as the WK model [22], the YK model [23], the LL-2007 model [24], the Zhang model [25], the ZX model [26], the LS model [27], the LJ model [28], and the XZ model [29], etc. [30,31]. Although these models can explain some contact mechanics phenomena, they have obvious shortcomings, such as assuming that a single asperity is completely deformed and unrelated to the applied load, which leads to surprising results in popularization and application. Moreover, Persson [32,33] and Yang et al. [34] have developed a new modeling method for rough surfaces by employing the diffusion theory, which does not depend on fractal roughness or length scale like the GW model. Moreover, applying the developed method in engineering is difficult due to its complexity. In 2010, Morag and Etson [35] first proposed a scale-dependent contact model of single asperity, i.e., the ME model, which solved an important shortcoming of the MB model. Then, Lin et al. [36], Miao et al. [37], Afferrante et al. [38], and Goedecke and Jackson et al. [39,40] extended the ME model to rough surfaces and established the LL model, the Miao model, the AC model, the JS model, and the GJ model, respectively. Recently, Yu et al. [41,42], Chen et al. [43], and Guo et al. [44] proposed a multi-stage contact model of rough fractal surfaces according to the deformation mode of multi-scale asperities.

After much scientific research, the contact mechanics of rough surfaces have been further understood, and some results have been achieved. Standing on the shoulders of giants, we revisit the multi-scale contact theory by first putting forward the concept of subsection scale, such as the small-scale, medium-scale, and large-scale stages. A complete piecewise-scale fractal expansion model, i.e., the JK model, is proposed. The transformation process of asperities on the rough surface from full plastic to elastic-plastic and then to elastic behavior with an increase in the full-scale range load is explained. Compared with other multi-scale models, the model solves the problems of discontinuity in contact mechanics, and is simple to compile and easy to apply in engineering.

2. Single-Asperity Contact Model (JK Model with a Single Asperity)

The contact between rough surfaces is often equivalent to the contact between a rough surface and a rigid, smooth plane, as shown in Figure 1. From the microscopic point of view, the contact process involves many randomly distributed asperities with different geometric shapes, leading to intricate contact mechanisms. In this section, only the contact process of spherical asperities is considered, and the contact mechanism is clarified to standardize and normalize complex problems. Moreover, the complete mechanical process of an asperity, from elastic to elastic-plastic to full plastic deformation, is explored. Lastly, a reliable asperity contact model is established.

2.1. Problem Description

The bolted joints of the heavy-duty CNC machine tool are often subjected to a large pre-tightening force to ensure the reliability of the bolted structures. Only considering the asperity deformation on rough surfaces subjected to large loads is somewhat conservative. Due to Hertz’s stress distribution, it can be seen that the substrate is bound to bear large stress. When the surface is carburized, the asperities tend to be harder than the substrate. The single-asperity model is shown in Figure 2.

2.2. Asperity–Substrate Interaction

Yeo et al. [45] established an elastic contact model considering the interaction between the substrate and asperity. The relationship between δ_a and δ can be obtained as follows:

$${\delta }_a={\displaystyle \frac{\delta }{1+E_r\xi \sqrt{{\delta }_a}}}$$

(1)

where δ is the applied deformation; δ_a is the deformation of single asperity; δ_b is the interface deformation of the substrate; the geometric coefficient $\xi =1.5\sqrt{R}/r_a$; and the dimensionless material coefficient E_r = E_a/E_b, where E_a and E_b are the elastic moduli of the asperity and the substrate, respectively.

In Equation (1), δ_a involves the root formula, which can be conveniently applied to the analytical model. Thus, it is approximately discretized as follows:

$${\delta }_a\approx {\displaystyle \frac{\delta }{1+E_r\xi \sqrt{h_a^{ i}}}}=\lambda \delta$$

(2)

where h_aⁱ satisfies the linear distribution of [0, h_a] and i represents the number of iterations (0, 1, ..., N); $\lambda ={\displaystyle \frac{1}{1+E_r\xi \sqrt{h_a^{ i}}}}$.

The variation law of the applied deformation δ and asperity deformation δ_a are described in Figure 3, where there is a non-linear growth trend. The approximate solution agrees with the exact numerical solution. When ξ = 10⁴ m^−0.5, the harder the asperity is, the smaller its deformation, with an increase in the value of E_r. Therefore, the influence of the substrate is more significant. When E_r = 1, the asperity radius is larger, and its height is smaller with an increase in the value of ξ. In other words, the surface is smoother, and the asperity deformation is smaller. Therefore, substrate deformation cannot be ignored. In Figure 3, a relatively large range of E_r and ξ is provided, whose values represent the upper limit. The error of approximate and exact solutions is approximately 15%, which is too large. However, in the real physical field, the geometric coefficient ξ << 10⁴ m^−0.5 and the approximate solution is almost equal to the exact solution.

2.3. Asperity Contact Modeling in Elastic, Elastoplastic, and Full Plastic Deformation—JK Model

The asperity contact is established by cutting off the rough surface with a rigid plane used for various interferences. According to Figure 4, the rough surface profile can be written as follows [23].

$$z_0(x)=2^{(3-D)}{G}^{(D-2)}{(\ln \gamma )}^{1/2}{(r_a)}^{(3-D)}\cos \left({\displaystyle \frac{\pi x}{r_a}}\right)$$

(3)

Therefore, h_a is the height of an asperity, and can be expressed as follows:

$$h_a=2^{(3-D)}{G}^{(D-2)}{(\ln \gamma )}^{1/2}{(r_a)}^{(3-D)}$$

(4)

where D is the fractal dimension equal to D = <D_s> + 1, G is the fractal roughness obtained by the FPENN method (see ref. [46] for details), r_a is the truncated radius (or half wavelength), and γ is frequency index, generally 1.5.

Traditional fractal models include the MB model [21], YK model [23], LL model [24], LJ model [28], and ZX model [26]. These models assume complete deformation of the asperity, i.e., δ = h_a, which is false. Morag and Etsion [35] proved this point and thought that the deformation δ of an asperity depends on the applied load, which ranges from zero to complete deformation, i.e., 0 ≤ δ ≤ h_a. Then, the applied deformation δ can be written as follows.

$$\delta =h_a-z_0(r\prime )=2^{(3-D)}{G}^{(D-2)}{(\ln \gamma )}^{1/2}{(r_a)}^{(3-D)}\left[1-\cos \left({\displaystyle \frac{\pi r\prime }{r_a}}\right)\right]$$

(5)

The profile function of the cosine wave is approximately equivalent to a spherical asperity at half wavelength, as shown in Figure 4. The contact point must satisfy the geometric function $r_a\approx \sqrt{2R{h}_a}$. Thus, the geometric radius of the asperity can be expressed as follows.

$$R={\displaystyle \frac{r_a^{(D-1)}}{2^{(4-D)}{G}^{(D-2)}{(\ln \gamma )}^{0.5}}}$$

(6)

The asperity undergoes elastic, elastic-plastic, and full plastic deformation under external load. Therefore, the first critical deformation δ_1c, that is, when it comes into the elastic-plastic stage, can be proven as follows:

$${\delta }_{1c}=({\displaystyle \frac{\pi KH}{2E^{*}}}{)}^{2}R={\displaystyle \frac{{(\pi KH)}^2{r}_a^{(D-1)}}{2^{(6-D)}{E}^{*2}{G}^{(D-2)}{(\ln \gamma )}^{0.5}}}$$

(7)

where $K=0.454+0.41v$; H is the hardness of the softer material; and E* = E₁/(1 − v₁²). A relatively soft material, E₁ = E_b, is chosen if the material yields in a large area (δ > 0.1 h_a). If the material yields in a small area or does not yield (δ ≤ 0.1 h_a), E₁ = E_a.

Note: we assume that E₁ is E_a or E_b. However, for the better elastic modulus of coating materials or layered materials, please refer to [47].

The second critical deformation δ_2c is proven as follows.

$${\delta }_{2c}=76.8{\delta }_{1c}$$

(8)

If only half-wavelength peak contact is considered, Equation (5) is simplified as follows.

$$1 -\cos \left({\displaystyle \frac{\pi r\prime }{r_a}}\right)\approx 1.06{\left({\displaystyle \frac{r\prime }{r_a}}\right)}^2$$

(9)

If δ_max < δ_1c, the first critical truncated area a_1c′ of an asperity can be written as follows.

$$a_{1c}\prime ={\displaystyle \frac{{\pi }^3{(KH)}^2{r}_a^{ 2(D-1)}}{2^{(9-2D)}{G}^{2(D-2)}{E}^{*2}\ln \gamma }}$$

(10)

If δ_max < δ_2c, the second critical truncated area a_2c′ of an asperity can be written as follows.

$$a_{2c}\prime =76.8a_{1c}\prime$$

(11)

It should be noted that Equations (10) and (11) must satisfy $r\prime =\sqrt{2R\delta }$.

• Elastic deformation

If 0 < a′ < a_1c′, the normal load f_e, real contact area a_e, and average contact pressure ${\overline{p}}_e$ at elastic deformation of an asperity can be expressed according to the Hertz theory.

$$\begin{array}{l}{f}_e={\displaystyle \frac{4}{3}}{E}^{*}{R}^{1/2}{\delta }_a^{ 3/2}={\displaystyle \frac{2}{3}}K\pi R{\delta }_a{\left({\displaystyle \frac{{\delta }_a}{{\delta }_{1c}}}\right)}^{1/2}\\ ={\displaystyle \frac{{(1.06\lambda )}^{1.5}{2}^{(9-2D)/2}\ln {(\gamma )}^{0.5}{E}^{*}{G}^{(D-2)}{a}_l{\prime }^{(1-D)/2}a{\prime }^{3/2}}{3{\pi }^{(4-D)/2}}}\end{array}$$

(12)

$$a_e=\pi R{\delta }_a={\displaystyle \frac{1.06}{2}}\lambda a\prime$$

(13)

$$\begin{array}{l}{\overline{p}}_e={\displaystyle \frac{f_e}{a_e}}={\displaystyle \frac{4}{3}}{\displaystyle \frac{E^{*}}{\pi }}({\displaystyle \frac{{\delta }_a}{R}}{)}^{1/2}={\displaystyle \frac{2}{3}}KH({\displaystyle \frac{{\delta }_a}{{\delta }_{1c}}}{)}^{1/2}\\ ={\displaystyle \frac{\sqrt{1.06\lambda }}{3{\pi }^{(4-D)/2}}}{2}^{(11-2D)/2}{G}^{(D-2)}{E}^{*}\ln {(\gamma )}^{0.5}{a}_l{\prime }^{(1-D)/2}a{\prime }^{0.5}\end{array}$$

(14)

2.

Elastoplastic deformation

If a_1c′ < a′ < a_2c′, the normal load f_ep, real contact area a_ep, and average contact pressure ${\overline{p}}_{ep}$ at elastoplastic deformation of an asperity can be expressed as follows:

$$f_{ep}={\displaystyle \frac{2}{3}}KH\pi R{\delta }_{1c}{\left({\displaystyle \frac{{\delta }_a}{{\delta }_{1c}}}\right)}^{1.38}=H_{g2}{a}_l{\prime }^{0.38(1-D)}a{\prime }^{1.38}$$

(15)

$$a_{ep}=\pi R{\delta }_{1c}{\left({\displaystyle \frac{{\delta }_a}{{\delta }_{1c}}}\right)}^{1.16}=H_{g1}{a}_l{\prime }^{0.16(1-D)}a{\prime }^{1.16}$$

(16)

$$\begin{array}{l}{\overline{p}}_{ep}={\displaystyle \frac{H}{2.8}}1.08({\displaystyle \frac{{\delta }_a}{{\delta }_{1c}}}{)}^{0.22}\\ ={\displaystyle \frac{1.094{\lambda }^{0.22}{H}^{0.56}{E}^{*0.44}}{2.8{\pi }^{0.22(4-D)}{K}^{0.44}}}{2}^{0.22(9-2D)}{G}^{0.44(D-2)}\ln (\gamma {)}^{0.22}{a}_l{\prime }^{0.22(1-D)}a{\prime }^{0.22}\end{array}$$

(17)

where

$$H_{g1}={\displaystyle \frac{{1.06}^{1.16}{2}^{(0.44-0.32D)}{E}^{*0.32}{\lambda }^{1.16}\ln {\gamma }^{0.16}{G}^{0.32(D-2)}}{{\pi }^{(0.64-0.16D)}{(KH)}^{0.32}}};$$

$$H_{g2}={\displaystyle \frac{{1.06}^{1.38}{2}^{(3.42-0.76D)}{(KH)}^{0.24}\ln {\gamma }^{0.38}{E}^{*0.76}{\lambda }^{1.38}{G}^{0.76(D-2)}}{3{\pi }^{(1.52-0.38D)}}}.$$

3.

Full plastic deformation

If a_2c′ < a′ < a_l′, the normal load f_p, real contact area a_p and average contact pressure ${\overline{p}}_p$ at full plastic deformation of an asperity can be expressed as follows.

$$f_p=H{a}_p$$

(18)

$$a_p=2\pi R{\delta }_a=1.06\lambda a\prime$$

(19)

$${\overline{p}}_p=H$$

(20)

An accurate single-asperity model considering asperity–substrate interaction was established by introducing the geometric coefficient of the asperity and dimensionless material coefficient. The abnormal physical process of the contact based on the traditional fractal model was also corrected. Moreover, a comparative study of the classical contact model (KE model [48], LL-2005 model [49], and ZX model [26]) based on the contact research of single asperity will be conducted to verify the reliability and accuracy of the proposed model. Finally, the results will be compared with the finite element simulation. The detailed research is conducted in Section 2.4.

2.4. Analysis and Comparison of Different Single-Asperity Contact Models

Work with axisymmetric analysis involves a calculation process to ensure the unity of units (MPa, mm, N). The geometric parameters are set as the asperity radius R = 15 μm and height h_a = 0.1 μm, as shown in Figure 5. Material parameters are set as the elastic modulus E = 100 Gpa, Poisson’s ratio v = 0.3, and yield strength σ_s = 355 Mpa.

Parameters R = 15 μm and h_a = 0.1 μm are substituted into Equations (4) and (6) to obtain the fractal parameters D and G. Although the inverse solution of the equation is somewhat difficult, the “fsolve” function of the MATLAB toolbox is used to obtain D = 2.9961 and G = 1.5463 × 10⁻⁴ mm. The results conform to the fractal law, i.e., the smoother the surface, the fractal dimension is closer to three. Moreover, the fractal roughness is inversely proportional to the fractal dimension. Lastly, the values of R_v = 15 μm and h_av = 0.0997 μm are double-checked.

The following results are dimensionless to compare different models. KE and LL-2005 models are defined as follows.

$$\widehat{f}={\displaystyle \frac{f}{\pi R\delta H}};\widehat{a}={\displaystyle \frac{a}{\pi R\delta }};\widehat{p}={\displaystyle \frac{p}{H}};\widehat{\delta }={\displaystyle \frac{\delta }{{\delta }_{1c}}};$$

(21)

Furthermore, the dimensionless results of the JK and XU models are defined as follows:

$$\begin{gathered} \widehat{f}={\displaystyle \frac{f}{TH}};\widehat{a}={\displaystyle \frac{a}{T}} \\ \widehat{f}={\displaystyle \frac{f}{Ha\prime /2}};\widehat{a}={\displaystyle \frac{a}{a\prime /2}} \end{gathered}$$

(22)

where $T={\displaystyle \frac{1.06\lambda a\prime }{2}}$.

It should be noted that Equation (22) is derived by substituting the corresponding models R and δ into Equation (21).

The relationship between the dimensionless load, real contact area, average contact pressure, and applied displacement is shown in Figure 6. The JK model agrees with the LL-2005 model and the KE model. On the contrary, the trend gap of the ZX model is caused by the non-physical phenomenon of the traditional fractal contact model. Moreover, the KE and ZX models demonstrate obvious contact jumping. There are two contact jumping points in Figure 6a, both at the limit boundary of elasticity and full plasticity. However, two or three contact jumping points can be observed in Figure 6b, where the KE model appears in the first elastic limit, first elastic-plastic limit, and second elastic-plastic limit. On the other hand, the ZX model appears at the upper and lower boundaries of the elastic-plastic area. Lastly, since Figure 6c is the same as Figure 6a, it is not described here.

The ZX model is also a brave attempt based on the LL-2005 model. Specifically, there should be no contact jumping phenomena. The fundamental reason is that the ZX model violates Hertz’s theory during single-asperity contact, i.e., the radius of the asperity changes with force, which is also why the early fractal contact model is popular in the contact field. At the same time, the ZX model violates the geometric relationship as follows.

$$r\prime =\sqrt{2R\delta }(\delta <<R);$$

(23)

We rewrite Equation (23) as $a\prime =2\pi R\delta$ and replace δ with δ_1c and δ_2c to obtain the following.

$$\left\{\begin{array}{l}{a}_{1c}\prime =2\pi R{\delta }_{1c}\\ a_{2c}\prime =2\pi R{\delta }_{2c}\end{array}\right..$$

(24)

The critical truncated area increases with an increase in the applied deformation. This results in unprecedented troubles with the popularization and application of fractal contact theory. The statistical contact model represented by Muser et al. [19] and Greenwood et al. [17] criticizes the fractal contact model. A new fractal contact model (the JK model) with accuracy and reliability is proposed due to many doubts and difficulties. In addition, the real contact numerical solution is also presented and compared with the finite element simulation results in Figure 7.

According to Figure 7, the JK, KE, and LL-2005 models are highly consistent with the finite element results, which proves that the JK model is reliable and characterized by high accuracy. On the other hand, the ZX model error is very large, and there is an obvious contact paradox, which is the shortcoming of most fractal contact models (such as the MB and YK models). The earliest modified contact paradox was provided by Morag and Etsion [35], with Liou et al. [36] expanding their research. However, there are still some shortcomings in the rough surface contact model, especially in the full-scale process, while the mechanical model needs to be improved. Thus, while establishing a more comprehensive mechanical model, the JK model further considers the asperity–substrate interaction and the influence of the dimensionless material coefficient and asperity geometry. The advantages of the JK model are more evident in surface heat treatment or coating.

A comparison between the JK model and the finite element results for an asperity with different radii and dimensionless material coefficients is shown in

Table 1. Comparison of the JK model and finite element results under different parameters.

No	R [μm]	ha [nm]	ra [μm]	D	G [×10−5 mm]	ξ [mm−0.5]	Ea [Gpa]	Eb [Gpa]	Er	FEA		JK Model
										F [×10−3 N]	A [×10−6 mm2]	F [×10−3 N]	A [×10−6 mm2]
#1	15	100	1.729	2.996	15.46	106.24	100	100	1	0.41	0.589	0.405	0.646
	20	750	5.426	2.420	5.256	39.10	100	100	1	7.43	8.01	8.27	8.94
	40	3000	15.20	2.274	2.920	19.74	100	100	1	64.6	74.1	75.71	76.17
	80	6000	30.40	2.238	2.055	13.96	100	100	1	257.7	296.3	302.86	304.69
#2	15	100	1.729	2.996	15.46	106.24	150	100	1.5	0.43	0.589	0.402	0.629
	20	750	5.426	2.420	5.256	39.10	150	100	1.5	7.70	8.01	8.70	9.05
	40	3000	15.20	2.274	2.920	19.74	150	100	1.5	70.1	74.2	75.07	75.52
	80	6000	30.40	2.238	2.055	13.96	150	100	1.5	279.4	296.6	300.77	302.59
#3	15	100	1.729	2.996	15.46	106.24	200	100	2	0.45	0.590	0.393	0.611
	20	750	5.426	2.420	5.256	39.10	200	100	2	7.82	8.01	8.92	9.07
	40	3000	15.20	2.274	2.920	19.74	200	100	2	72.6	74.2	74.47	74.92
	80	6000	30.40	2.238	2.055	13.96	200	100	2	290.0	296.8	298.84	300.65

. Parameter R varies from 15 to 80 μm, and h_a varies from 100 to 6000 nm, corresponding to the extremely smooth surface to the rough surface. A high value of the geometric coefficient ξ represents the smooth nanoscale surface, while the low value of ξ represents the general rough surface, ranging from 106.24 mm^−0.5 to 13.96 mm^−0.5 (359.7 m^−0.5–441.4 m^−0.5).

When subjected to the external load, the asperity deformation must undergo a process of elasticity–elastic plasticity–full plasticity. The contact load and the actual contact area are significantly affected by the geometric size of an asperity, i.e., both increased with the geometric size. However, they are hardly affected by the elastic modulus on the rough surface. The reason is that the asperity and substrate yield plastically when the applied displacement is provided.

Some differences can be observed by comparing the two numerical results due to two reasons. First, the material parameter settings in the finite element analysis are simplified, and only the bilinear elastic-plastic material constitutive equation (σ_s = 355 MPa, k_g = 0 MPa) is considered. Second, the critical deformation of the JK model is affected by the coupling of the soft material substrate, thereby affecting the critical truncated area. Compared with the finite element method, the maximum error of the numerical results of this model is approximately 10%, demonstrating the JK model’s accuracy. The advantages of the JK model are demonstrated by comparing it with the LL-2005 and KE models. Thus, the details are omitted here for simplicity purposes.

Note: In 12 groups of cases, especially when the material properties of the asperity and substrate are different, if the materials yield, E^* in equations is referred to [47]. The treatment of the equivalent elastic modulus is simplified, similar to the contact mechanics of surface coating materials, which may be an important reason for the obtained error.

3. Contact Model of Rough Surfaces (JK Model with Rough Surfaces)

The contact modeling process of single asperity is analyzed in detail in Section 2. Mapping the mechanical properties of a single asperity to the nominally smooth joint surface requires introducing the “island distribution” function that approximately describes the truncated area distribution of asperities on the rough surface in contact. Then, the complete contact characteristic parameters on the nominal surface can be obtained by integration, including the real contact area, normal contact load, normal contact stiffness, tangential contact stiffness, normal contact damping, and tangential contact damping.

The proposed model is really difficult for engineers to apply in practice. In the future, we will start to develop interface application tools based on the ANSYS platform, which can be applied to a large number of CNC machine tools with joint surfaces, so as to complete the dynamic response research of the whole machine design.

3.1. Island Distribution Function

Mandelbrot used a plane to truncate the islands and found that the number of islands N with > a′ followed the power law [50]. Subsequently, Majumdar et al. introduced this phenomenon to rough fractal surfaces, and creatively proposed the fractal contact model, i.e., the MB model [21]. Then, considering that the contact area is affected by an increase in the contact temperature, Wang et al. [22] introduced the domain expansion coefficient ψ and revised the island distribution function, which was written as follows:

$$n(a\prime )={\displaystyle \frac{D-1}{2}}{\psi }^{{\displaystyle \frac{3-D}{2}}}{a}_l{\prime }^{{\displaystyle \frac{D-1}{2}}}a{\prime }^{{\displaystyle \frac{-(D+1)}{2}}}$$

(25)

where the maximum truncated area of asperities a_l′ equals a_l′ = πr_a²; the domain expansion coefficient ψ is shown in Reference [22].

It should be noted that obtaining the maximum truncated radius r_a of asperities is very important.

When the rough surface is determined (the fractal parameters are known), the change in a_l′ affects a_1c′ and a_2c′ in Equations (10) and (11). According to Figure 8, the contact between rough surfaces is characterized by a lengthy scale replacement in the range of the full-scale length a_l′. The range between 0 and a_l1′ is called the small-scale stage. As the load increases, the asperity undergoes elastic–elastoplastic–full plastic deformation, i.e., full plastic behavior. The range between a_l1′ and a_l2′ is called the meso-scale stage. Many asperities undergo full plastic behavior in the small-scale stag and embrace larger asperities. Then, asperities with a rough surface undergo elastic and elastoplastic deformation, i.e., elastoplastic behavior. The range between a_l2′ and a_l3′ is called the large-scale stage, where asperities undergo elastic deformation, i.e., elastic behavior.

When a_l1′ = a_2c′, it can be known from Equation (11).

$$a_{l1}\prime ={\left[{\displaystyle \frac{76.8{(KH)}^2}{2^{(9-2D)}{\pi }^{(D-4)}{E}^{*2}{G}^{2(D-2)}\ln \gamma }}\right]}^{1/(2-D)}$$

(26)

When a_l2′ = a_1c′, it can be known from Equation (10).

$$a_{l2}\prime ={76.8}^{1/(D-2)}{a}_{l1}\prime$$

(27)

Moreover, a_l3′ can be expressed as follows:

$$a_{l3}\prime ={\displaystyle \frac{A_0}{\left[\frac{D-1}{2(3-D)}{\psi }^{(3-D)/2}\right]}}$$

(28)

where A₀ is the nominal contact area A₀ = L₀² and L₀ is the sampling scanning length.

In Figure 9, the truncated area a_l′ is positively correlated with G and negatively correlated with D. When D and G are in a certain interval, a_l2′ is already higher than a_l3′. At this time, the upper integral limit is a_l3′, regardless of the load change. Therefore, the asperity deformation process of the complete scale on rough surfaces is closely related to the surface properties. A complete contact model of the joint surface in the full-scale process is provided by assuming that a_l1′, a_l2′, and a_l3′ satisfy the increasing relationship.

3.2. Real Contact Area

If 0 < a′ < a_1c′, the elastic contact area of rough surfaces, A_re, can be obtained by integrating the product of Equations (25) and (13) as follows.

$$A_{re}={\displaystyle {\int }_0^{a_{1c}\prime }{a}_{e}n(a\prime )d}a\prime ={\displaystyle \frac{1.06\lambda (D-1)}{2(3-D)}}{\psi }^{{\displaystyle \frac{3-D}{2}}}{a}_l{\prime }^{{\displaystyle \frac{D-1}{2}}}{a}_{1c}{\prime }^{{\displaystyle \frac{3-D}{2}}}$$

(29)

If a_1c′ < a′ < a_2c′, the elastic-plastic contact area of rough surfaces, A_rep, is as follows.

$$\begin{array}{l}{A}_{rep}={\displaystyle {\int }_{a_{1c}\prime }^{a_{2c}\prime }{a}_{ep}n(a\prime )d}a\prime =\\ {\displaystyle \frac{H_{g1}(D-1)}{2(1.66-0.5D)}}{\psi }^{{\displaystyle \frac{3-D}{2}}}{a}_l{\prime }^{0.34(D-1)}\left[a_{2c}{\prime }^{(1.66-0.5D)}-a_{1c}{\prime }^{(1.66-0.5D)}\right]\end{array}$$

(30)

If a_2c′ < a′ < a_l′, the plastic contact area of rough surfaces, A_rp, is as follows.

$$\begin{array}{l}{A}_{rp}={\displaystyle {\int }_{a_{2c}\prime }^{a_l\prime }{a}_{p}n(a\prime )d}a\prime =\\ {\displaystyle \frac{1.06\lambda (D-1)}{3-D}}{\psi }^{{\displaystyle \frac{3-D}{2}}}{a}_l{\prime }^{0.5(D-1)}\left[a_l{\prime }^{(1.5-0.5D)}-a_{2c}{\prime }^{(1.5-0.5D)}\right]\end{array}$$

(31)

1. In the small-scale stage (a_2c′ ≤ a_l1′), the real contact area A_r^s1 is as follows.

$$A_r^{ s1}=\left\{\begin{array}{l}{A}_{\mathit{re}}^{ s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<a\prime <a_{1c}\prime \\ A_{\mathit{re}}^{ s1}+A_{\mathit{rep}}^{\hspace{1em} s1}\hspace{1em}\hspace{1em}\hspace{1em} a_{1c}\prime <a\prime <a_{2c}\prime \\ A_{\mathit{re}}^{ s1}+A_{\mathit{rep}}^{\hspace{1em} s1}+A_{\mathit{rp}}^{ s1}\hspace{1em} a_{2c}\prime <a\prime <a_{l1}\prime \end{array}\right.$$

(32)

2. In the meso-scale stage (a_1c′ ≤ a_l2′), the real contact area A_r^s2 is as follows.

$$A_r^{ s2}=\left\{\begin{array}{l}{A}_{\mathit{re}}^{ s2}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{l1}\prime <a\prime <a_{1c}\prime \\ A_{re}^{ s2}+A_{rep}^{\hspace{1em} s2}\hspace{1em} a_{1c}\prime <a\prime <a_{2c}\prime \end{array}\right.$$

(33)

3. In the large-scale stage (a_1c′ > a_l2′), the real contact area A_r^s3 is as follows.

$$A_r^{ s3}=A_{re}^{ s3}$$

(34)

In the full-scale range, the real contact area of rough surfaces A_r can be expressed as follows.

$$A_r=A_r^{ s1}+A_r^{ s2}+A_r^{ s3}$$

(35)

It should be noted that Equation (35) is determined by a_l′, a_1c′, and a_2c′. The asperities with rough surfaces must undergo elastic, elastic-plastic, and full plastic deformation under different small scales of a_l′. Many asperities embrace larger asperities with an increase in the value of a_l′. Under the corresponding scale, asperities with rough surfaces undergo elastic and elastic-plastic deformation. With the further increase in a_l′ (limit: <30% of the nominal contact area), asperities with rough surfaces undergo elastic deformation in the large-scale stage. This observation is consistent with Gao’s et al. research [30], i.e., asperities exhibit fully plastic behavior on a small scale, elastic-plastic behavior on the meso-scale scale, and elastic behavior on a large scale. In addition, the upper and lower boundaries of piecewise functions may not be monotonic and continuous due to scale jumps between different scale stages. Therefore, continuous and strict monotony must be ensured in the numerical solution.

3.3. Normal Contact Load

1. In the small-scale stage (a_2c′ ≤ a_l1′), the normal contact load F^s1 is as follows.

$$F^{s1}=\left\{\begin{array}{l}{F}_e^{ s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<a\prime <a_{1c}\prime \\ F_e^{ s1}+F_{ep}^{\hspace{1em}s1}\hspace{1em}\hspace{1em}\hspace{1em} a_{1c}\prime <a\prime <a_{2c}\prime \\ F_e^{ s1}+F_{ep}^{\hspace{1em}s1}+F_p^{ s1}\hspace{1em} a_{2c}\prime <a\prime <a_{l1}\prime \end{array}\right.$$

(36)

2. In the meso-scale stage (a_1c′ ≤ a_l2′), the normal contact load F^s2 is as follows.

$$F^{s2}=\left\{\begin{array}{l}{F}_e^{ s2}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{l1}\prime <a\prime <a_{1c}\prime \\ F_e^{ s2}+F_{ep}^{\hspace{1em}s2}\hspace{1em}\hspace{1em} a_{1c}\prime <a\prime <a_{2c}\prime \end{array}\right.$$

(37)

3. In the large-scale stage (a_1c′ > a_l2′), the normal contact load F^s3 is as follows.

$$F^{s3}=F_e^{ s3}$$

(38)

Thus, in the full-scale range, the normal contact load of rough surfaces F is as follows.

$$F=F^{s1}+F^{s2}+F^{s3}$$

(39)

3.4. Normal Contact Stiffness

1. In the small-scale stage (a_2c′ ≤ a_l1′), the normal contact stiffness K_n^s1 is as follows.

$$K_n^{ s1}=\left\{\begin{array}{l}{K}_{ne}^{ s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<a\prime <a_{1c}\prime \\ K_{ne}^{ s1}+K_{nep}^{\hspace{1em} s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{a}_{1c}\prime <a\prime <a_{2c}\prime \\ K_{ne}^{ s1}+K_{nep}^{\hspace{1em} s1}+K_{np}^{ s1} a_{2c}\prime <a\prime <a_{l1}\prime \end{array}\right.$$

(40)

2. In the meso-scale stage (a_1c′ ≤ a_l2′), the normal contact stiffness K_n ^s2 is as follows.

$$K_n^{ s2}=\left\{\begin{array}{l}{K}_{ne}^{ s2}\hspace{1em}\hspace{1em}\hspace{1em} a_{l1}\prime <a\prime <a_{1c}\prime \\ K_{ne}^{ s2}+K_{nep}^{\hspace{1em} s2}\hspace{1em}\hspace{1em}{a}_{1c}\prime <a\prime <a_{2c}\prime \end{array}\right.$$

(41)

3. In the large-scale stage (a_1c′ > a_l2′), the normal contact stiffness K_n^s3 is as follows.

$$K_n^{ s3}=K_{ne}^{ s3}$$

(42)

Then, in the full-scale range, the normal contact stiffness of rough surfaces K_n is as follows.

$$K_n=K_n^{ s1}+K_n^{ s2}+K_n^{ s3}$$

(43)

3.5. Tangential Contact Stiffness

1. In the small-scale stage (a_2c′ ≤ a_l1′), the tangential contact stiffness K_t^s1 is as follows.

$$K_t^{ s1}=\left\{\begin{array}{l}{K}_{te}^{ s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<a\prime <a_{1c}\prime \\ K_{te}^{ s1}+K_{tep}^{\hspace{1em} s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{a}_{1c}\prime <a\prime <a_{2c}\prime \\ K_{te}^{ s1}+K_{tep}^{\hspace{1em} s1}+K_{tp}^{ s1}\hspace{1em} a_{2c}\prime <a\prime <a_{l1}\prime \end{array}\right.$$

(44)

2. In the meso-scale stage (a_1c′ ≤ a_l2′), the tangential contact stiffness K_t^s2 is as follows.

$$K_t^{ s2}=\left\{\begin{array}{l}{K}_{te}^{ s2}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{a}_{l1}\prime <a\prime <a_{1c}\prime \\ K_{te}^{ s2}+K_{tep}^{\hspace{1em} s2}\hspace{1em}\hspace{1em}\hspace{1em} a_{1c}\prime <a\prime <a_{2c}\prime \end{array}\right.$$

(45)

3. In the large-scale stage (a_1c′ > a_l2′), the tangential contact stiffness K_t^s3 is as follows.

$$K_t^{ s3}=K_{te}^{ s3}$$

(46)

Therefore, in the full-scale range, the tangential contact stiffness of rough surfaces K_t is as follows.

$$K_t=K_t^{ s1}+K_t^{ s2}+K_t^{ s3}$$

(47)

3.6. Normal Contact Damping

1. In the small-scale stage (a_2c′ ≤ a_l1′), the normal strain energy W_n^s1 is as follows.

$$W_n^{ s1}=\left\{\begin{array}{l}{W}_{ne}^{ s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} 0<a\prime <a_{1c}\prime \\ W_{ne}^{ s1}+W_{nep}^{\hspace{1em} s1}\hspace{1em}\hspace{1em}\hspace{1em} a_{1c}\prime <a\prime <a_{2c}\prime \\ W_{ne}^{ s1}+W_{nep}^{\hspace{1em} s1}+W_{np}^{ s1} a_{2c}\prime <a\prime <a_{l1}\prime \end{array}\right.$$

(48)

The normal damping loss factor, η_n^s1, can be expressed based on the energy dissipation principle.

$${\eta }_n^{ s1}=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<a\prime <a_{1c}\prime \\ {\displaystyle \frac{\alpha W_{nep}^{\hspace{1em}s1}}{W_{ne}^{ s1}+(1-\alpha )W_{nep}^{\hspace{1em}s1}}}\hspace{1em}\hspace{1em}{a}_{1c}\prime <a\prime <a_{2c}\prime \\ {\displaystyle \frac{\alpha W_{nep}^{\hspace{1em}s1}+W_{np}^{ s1}}{W_{ne}^{ s1}+(1-\alpha )W_{nep}^{\hspace{1em} s1}}}\hspace{1em}\hspace{1em}{a}_{2c}\prime <a\prime <a_{l1}\prime \end{array}\right.$$

(49)

2. In the meso-scale stage (a_1c′ ≤ a_l2′), the normal strain energy W_n^s2 is as follows.

$$W_n^{ s2}=\left\{\begin{array}{l}{W}_{ne}^{ s2}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{l1}\prime <a\prime <a_{1c}\prime \\ W_{ne}^{ s2}+W_{nep}^{\hspace{1em} s2}\hspace{1em}\hspace{1em}\hspace{1em}{a}_{1c}\prime <a\prime <a_{2c}\prime \end{array}\right.$$

(50)

The normal damping loss factor η_n^s2 is as follows.

$${\eta }_n^{ s2}=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{l1}\prime <a\prime <a_{1c}\prime \\ {\displaystyle \frac{\alpha W_{nep}^{\hspace{1em} s2}}{W_{ne}^{ s2}+(1-\alpha )W_{nep}^{\hspace{1em} s2}}}\hspace{1em}\hspace{1em}{a}_{1c}\prime <a\prime <a_{2c}\prime \end{array}\right.$$

(51)

3. In the large-scale stage (a_1c′ > a_l2′), the normal strain energy W_n ^s3 is as follows.

$$W_n^{ s3}=W_{ne}^{ s3}$$

(52)

The normal damping loss factor η_n^s3 is as follows.

$${\eta }_n^{ s3}=0$$

(53)

The normal strain energy W_n and normal damping loss factor η_n of rough surfaces in the full-scale range can be, respectively, expressed as Equations (54) and (55):

$$W_n=W_n^{ s1}+W_n^{ s2}+W_n^{ s3}$$

(54)

$${\eta }_n={\eta }_n^{ s1}+{\eta }_n^{ s2}+{\eta }_n^{ s3}$$

(55)

where α is the elastic-plastic load index under a loading–unloading cycle within the range of 0 ≤ α ≤ 1. The lower and upper limits correspond to pure elastic and full plastic load conditions, respectively. Details can be found in Etsion 2005 [51].

Hence, the normal contact damping of rough surfaces C_n can be expressed as follows:

$$C_n={\eta }_n\sqrt{M{K}_n}$$

(56)

where M is the structure’s mass.

3.7. Tangential Contact Damping

1. In the small-scale stage (a_2c′ ≤ a_l1′), the tangential strain energy W_t^s1 is as follows.

$$W_t^{ s1}=\left\{\begin{array}{l}{W}_{te}^{ s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<a\prime <a_{1c}\prime \\ W_{te}^{ s1}+W_{tep}^{\hspace{1em} s1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{1c}\prime <a\prime <a_{2c}\prime \\ W_{te}^{ s1}+W_{tep}^{\hspace{1em} s1}+W_{tp}^{ s1}\hspace{1em}\hspace{1em} a_{2c}\prime <a\prime <a_{l1}\prime \end{array}\right.$$

(57)

Similarly, the tangential damping loss factor η_t^s1 is as follows.

$${\eta }_t^{ s1}=\left\{\begin{array}{l}0 \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<a\prime <a_{1c}\prime \\ {\displaystyle \frac{\alpha W_{tep}^{\hspace{1em} s1}}{W_{te}^{ s1}+(1-\alpha )W_{tep}^{\hspace{1em} s1}}}\hspace{1em}{a}_{1c}\prime <a\prime <a_{2c}\prime \\ {\displaystyle \frac{\alpha W_{tep}^{\hspace{1em} s1}+W_{tp}^{ s1}\hspace{1em}}{W_{te}^{ s1}+(1-\alpha )W_{tep}^{\hspace{1em}s1}}}\hspace{1em} a_{2c}\prime <a\prime <a_{l1}\prime \end{array}\right.$$

(58)

2. In the meso-scale stage (a_1c′ ≤ a_l2′), the tangential strain energy W_t^s2 is as follows.

$$W_t^{ s2}=\left\{\begin{array}{l}{W}_{te}^{ s2}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} a_{l1}\prime <a\prime <a_{1c}\prime \\ W_{te}^{ s2}+W_{tep}^{\hspace{1em} s2}\hspace{1em}\hspace{1em}\hspace{1em}{a}_{1c}\prime <a\prime <a_{2c}\prime \end{array}\right.$$

(59)

The tangential damping loss factor η_t^s2 is as follows.

$${\eta }_t^{ s2}=\left\{\begin{array}{l}0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em} a_{l1}\prime <a\prime <a_{1c}\prime \\ {\displaystyle \frac{\alpha W_{tep}^{\hspace{1em} s2}}{W_{te}^{ s2}+(1-\alpha )W_{tep}^{\hspace{1em} s2}}}\hspace{1em} a_{1c}\prime <a\prime <a_{2c}\prime \end{array}\right.$$

(60)

3. In the large-scale stage (a_1c′ > a_l2′), the tangential strain energy W_t^s3 is as follows.

$$W_t^{ s3}=W_{te}^{ s3}$$

(61)

The tangential damping loss factor η_t^s3 is as follows.

$${\eta }_t^{ s3}=0$$

(62)

In the full-scale range, the tangential strain energy W_t and tangential damping loss factor η_t of rough surfaces can be, respectively, expressed by Equations (63) and (64).

$$W_t=W_t^{ s1}+W_t^{ s2}+W_t^{ s3}$$

(63)

$${\eta }_t={\eta }_t^{ s1}+{\eta }_t^{ s2}+{\eta }_t^{ s3}$$

(64)

It should be noted that the elastic-plastic load index α in Equations (55) and (64) is assumed to be the same. Therefore, a detailed study on it was not yet conducted. The index α can be simplified to a constant value (assuming α = 0.2) or satisfy the linear distribution between [0, 1] (related to the contact deformation state).

Thus, the tangential contact damping, C_t, can be expressed as follows.

$$C_t={\eta }_t\sqrt{M{K}_t}$$

(65)

In conclusion, it can be seen that fractal interface contact mechanics is the result of small-scale, meso-scale, and large-scale comprehensive action, which has a clear and reasonable physical basis. Furthermore, the length scale significantly affects the deformation of asperities, which is the key factor in rough surfaces. Important defects of the traditional fractal theory, such as asperity deformation discontinuity and cross-scale jump, are fundamentally solved in this paper according to proper theoretical derivation and analysis. The proposed model has high accuracy and provides theoretical support for engineering applications.

Additional explanation and the detailed derivation process can be found in the Appendices A–E.

3.8. Numerical Analysis of JK Model on Rough Surfaces

The following simulated physical parameters are set for the analysis: the elastic modulus E = 2 × 10⁵ Mpa, Poisson’s ratio v = 0.3, structural mass M = 1 kg, static friction coefficient μ = 0.25, yield strength σ_s = 355 Mpa, and sampling scanning length L₀ = 4 mm. The dimensionless Equations (35), (39), (43), (47), (56) and (65) are adopted as follows.

$$\begin{gathered} F^{*}={\displaystyle \frac{F}{E^{*}{A}_0}},A_r^{*}={\displaystyle \frac{A_r}{A_0}},K_n^{*}={\displaystyle \frac{K_n}{E^{*}\sqrt{A_0}}},K_t^{*}={\displaystyle \frac{K_t}{E^{*}\sqrt{A_0}}} \\ {C}_n^{*}={\displaystyle \frac{C_n}{A_0^{ 0.25}\sqrt{M{E}^{*}}}},C_t^{*}={\displaystyle \frac{C_t}{A_0^{ 0.25}\sqrt{M{E}^{*}}}} \end{gathered}$$

(66)

In engineering surfaces, Jiang et al. [46] have shown that the fractal dimension D is in the range of 2.3–2.6, and the fractal roughness G is in the order of magnitude of 10⁻¹²–10⁻¹⁰ m. The influence of fractal parameters on the change law of the real contact area is investigated for a given load determined by the maximum truncated area a_l′, as shown in Figure 10. The curve of the normal load–contact area agrees with a piecewise linear relationship in different scale stages. When the fractal roughness G is known, the real contact area increases with the fractal dimension D. In addition, the curve slope significantly changes in the meso-scale and large-scale stages, indicating that the elastic action dominates under a certain load in finer surfaces. On the contrary, when the fractal dimension D is known, the fractal roughness G negatively correlates with the contact area in a certain load range. The result is consistent with the common sense of physics because the surface quality is positively correlated with the fractal dimension and negatively correlated with the fractal roughness.

In Figure 11, the load–stiffness curve also conforms to the piecewise linear relationship at different scale stages, consistent with Persson’s research [52]. The contact stiffness is often determined by the longest wavelength roughness, which may be elastic rather than plastic deformation. Furthermore, the larger the fractal dimension D, the smaller the fractal roughness G, i.e., the finer and smoother the surface, and the greater the contact stiffness. In addition, if the contact stiffness is greater than the material’s bulk stiffness, the coefficient is related to the geometry when the load reaches a certain value ($K_b=1.129{\mathrm{E}}^{*}\sqrt{A_0}$ [53], as shown in the red line in the figure). Then, the contact stiffness obtained by the proposed model has no practical numerical value. The tangential contact stiffness is similar.

Mindlin provided the stiffness ratio for the axisymmetric Hertz contact as follows [54].

$${\displaystyle \frac{K_t}{K_n}}={\displaystyle \frac{2(1-v)}{2-v}},$$

(67)

According to Equation (67), the tangential–normal stiffness ratio is a function of Poisson’s ratio v. The stiffness ratio is constant for a predetermined material. Putignano et al. [55] further proved that the stiffness ratio depends on the contact zone for all possible contact areas in half space, providing a stiffness ratio limit.

$$1-v\le {\displaystyle \frac{K_t}{K_n}}\le 1$$

(68)

As shown in Figure 12, the stiffness ratio curve is similar to a Z shape. At different scales, the stiffness ratio varies with the load. In small-scale and large-scale stages, the stiffness ratio remains approximately constant. The stiffness ratio significantly decreases in the meso-scale stage. The value ratio accords with Equation (68), while Mindlin’s ratio is the upper and lower limits’ average. In addition, the stiffness ratio is negatively correlated with fractal dimension D and positively correlated with fractal roughness G. Compared with fractal roughness, the fractal dimension is the key parameter affecting the stiffness ratio.

The dimensionless contact damping–load curve is shown in Figure 13. On a small scale, the contact damping decreases with an increase in load. In meso-scale and large-scale stages, the contact damping increases with the load. This observation is consistent with Shi et al.’s research [56], proving that contact damping decreases with an increase in contact load under light load. The main reason is that the asperity has undergone complete mechanical deformation, and the damping loss factor decreases with an increase in the load, as shown in Equation (55).

Furthermore, the elastic-plastic load index is set as a constant (α = 0.2), which affects the result. However, in the meso-scale and large-scale stages, the damping loss factor shown by Equations (51) and (53) remains unchanged. Therefore, the contact damping is proportional to the mean root square of stiffness and increases monotonically with the load. In addition, when the fractal roughness G is determined, the contact damping decreases with an increase in the fractal dimension D in the small-scale stage. In the meso-scale and large-scale stages, the contact damping increases with the fractal dimension D. Once the fractal dimension D is determined, the contact damping increases with the fractal roughness G in the small-scale interval. An opposite effect occurs in the meso-scale and large-scale stages.

4. Analysis and Verification

Natural frequency and mode shape are important parameters in dynamic structural analysis. The dynamic performance of a discontinuous structure is determined by the contact state of the bolted joint surface. First, the surface topography parameters (fractal parameters D and G) can be estimated by the FPENN method used in previous research [46]. These parameters can be introduced into the full-scale and high-precision contact analytical model of rough surfaces. Then, the finite element method is employed to obtain the nodal forces of the nominal smooth surface and fit the nodal contact stiffness and damping values of the joint surface. Finally, the experimental solution is compared against the theoretical solution to complete the mechanical vibration analysis of the bolted structure.

The proposed contact model does not consider the influence of temperature, humidity, and corrosion. These complex environmental factors are coupled with each other, and their influence on interface contact mechanics is uncertain. Therefore, this study will not consider them for the time being.

4.1. Theoretical Analysis

A set of T-type specimens is designed, as shown in Figure 14. The material is 45# steel, whose material properties are shown in

Table 2. Material property.

Properties	Value	Unit
density ρ	7800	Kg/m3
elastic modulus E	2.0 × 105	MPa
yield strength σs	355	MPa
Poisson’s ratio v	0.3
hardness H	994	MPa
static friction coefficient μ	0.25

. The joint surface is milled with the nominal roughness R_a = 0.8 μm. Scanning blocks with a size of 20 × 20 × 30 mm are processed in the same way to obtain the fractal parameters of the surface morphology. In Figure 15, the measured surface is an area of 4 × 4 mm. Firstly, the data of this area are obtained by the optical profiler ST400. Then, the fractal parameters D_s = 1.39 and G = 1.53 × 10⁻¹⁰ m are identified by the FPENN method. Thus, the fractal dimension of the measured surface D = D_s + 1 = 2.39.

Firstly, with the help of the ANSYS (version R19.2), the force distribution of the contact nodes on the nominal smooth surface is obtained using static analysis, as shown in Figure 16. Then, each node force is imported into the MATLAB (version R2018) program. The normal and tangential contact stiffness and the normal and tangential contact damping of each node are obtained, respectively, by fitting the load–stiffness curve and the load–damping curve. Thirdly, the Matrix27 element is used to establish the stiffness and damping of the node–node contact. Finally, the frequency and vibration modes of the bolted structure are obtained by modal analysis.

It should be noted that the mesh nodes of the joint surface should correspond one by one during pre-processing. In addition, all nodes need to be renumbered by HYPERMESH (version R2017) to add the stiffness-damping element (Matrix27).

4.2. Experimental Validation

The test device includes the acceleration sensor, force hammer, and LMS vibration test analyzer. The T-type specimen is suspended on the elastic rope, and the PCB acceleration sensor is arranged on the structural surface (number 10 in Figure 17). Then, the force signal is applied by the force hammer to induce vibration. The LMS vibration test analyzer collects the vibration signal. Finally, the frequency and mode shapes of each structure order are obtained.

During bolt pre-tightening, the input torque is slowly applied by a digital torque wrench. The pre-tightening forces are obtained via ultrasonic detection technology as 4 kN, 8 kN, and 12 kN. The hammering test is repeated five times to avoid random operation errors. Note that the modal test is carried out indoors and its duration is limited, so the influence of temperature, humidity, and external interference can be ignored.

4.3. Comparison of the Results

The vibration frequency signal is shown in Figure 18. When hammering, the force signal induces a structural vibration response, while the acceleration sensor records the micro-vibration electrical signal. If the electrical signal has the maximum response, the resonance frequency of the structure can be determined.

The theoretical and experimental modal frequencies are compared in

Table 3. A comparison of theoretical values and test results.

Fp	Method	1st	2nd	3rd	4th	5th
4 kN	JK model	422.9	1153.2	2207.6	3307.7	4900.6
	Experiment 1	406	1126	2145	3249	4700
	Error 1	3.89%	2.35%	2.81%	1.76%	4.09%
8 kN	JK model	423.6	1154.1	2208.8	3311.7	4903.9
	Experiment 2	410	1135	2175	3285	4725
	Error 2	3.21%	1.65%	1.53%	0.81%	3.65%
12 kN	JK model	423.9	1154.6	2209.3	3313.78	4905.0
	Experiment 3	415	1140	2185	3265	4800
	Error 3	2.10%	1.26%	1.10%	1.47%	2.14%

. The frequencies of each order agree with each other, and the maximum errors are 4.09%, 3.65%, and 2.14% at the 5th mode. The constraint conditions of the flexible rope may greatly influence the high-order mode, which introduces certain errors. Moreover, all frequencies increase with an increase in the pre-tightening force F_p. However, the increase rate is relatively slow. In addition to the coincidence of the frequency, the corresponding vibration modes should also be matched to verify the accuracy of the joint surface’s contact stiffness and damping model. As shown in Figure 19, the vibration modes of each order are completely consistent. Therefore, the accuracy of the proposed contact model is proven.

5. Conclusions and Prospects

In this work, revisiting the multi-scale model, a fractal extension of the piecewise scale model on rough surfaces, i.e., the JK model, was proposed. Firstly, an accurate single-asperity model considering asperity–substrate interaction was established by introducing the geometric coefficient of the asperity and dimensionless material coefficient. Then, according to the island distribution function, we divided the contact mechanics process of rough surfaces into a small-scale stage, a meso-scale stage, and a large-scale stage, and originally deduced the precise scale limit. On this basis, a complete analytical solution was provided for contact mechanics with homogeneous and coating or layered materials by piecewise integration. In addition, we further explained the reasons why small asperities are plastically deformed in the small-scale stage, while medium asperities are elastic-plastically deformed in the meso-scale stage, and large asperities are elastically deformed in the large-scale stage. This paper is the most detailed contact model at present. Compared with most other contact models, it solves the discontinuous monotony problem of cross-scale contact mechanics, and comprehensively analyzes the relationships among load–real contact area, load–contact stiffness, load–stiffness ratio, and load-damping. Finally, we verified the accuracy of the proposed model using a dynamic modal experiment.

Based on the obtained results, hot issues, such as small-scale adhesion, friction, and wear, can be further investigated. Finally, a scientific basis for the popularization and application of the contact model can be provided.