A Novel Contact Resistance Model for the Spherical–Planar Joint Interface Based on Three Dimensional Fractal Theory

Abstract

In order to obtain the contact resistance of relay contacts more accurately, a novel contact resistance model for the spherical–planar joint interface is constructed based on the three-dimensional fractal theory. In this model, three-dimensional fractal theory is adopted to generate a rough surface at microscopic scale. Then, using contact mechanics theory, the deformation mechanism of asperities on rough surfaces is explored. Combined with the distribution of asperities, a contact resistance model for the planar joint interface is established. Furthermore, by introducing the surface contact coefficient, cross-scale coupling between the macro-geometric configuration and micro-surface topography is achieved, and a contact resistance model for the spherical–planar joint interface is constructed. After that, experiments are conducted to verify the accuracy of the proposed model, and the maximum relative error of the proposed model is 8.44%. Ultimately, combining numerical simulation analysis, the patterns of variation in contact resistance influenced by factors such as macroscopic configuration and microscopic topography are discussed, thereby revealing the influence mechanism of the contact resistance for the spherical–planar joint interface. The proposed model provides a solid theoretical foundation for the optimization of relay contact structures and improvements in manufacturing processes, which is of great significance for ensuring the safe and stable operation of power systems and electronic equipment.

1. Introduction

As an important carrier for connecting, disconnecting, and transmitting current in relays and other electromagnetic devices, contacts are key components that determine the electrical contact performance and reliability of electrical appliances. The instability of the contact resistance between contacts can cause circuit interruption or performance degradation. Therefore, through the in-depth study of contact resistance, potential connection problems can be predicted and prevented, thereby improving the reliability of the electrical system [1,2].

The research on contact resistance has always been a hot topic for relevant scholars [3,4]. Holm [5] derived the Laplace potential formula based on an ideal cylindrical conductor in a circular contraction region for the first time, thus obtaining an analytical expression of the contact resistance. On the basis of considering the actual surface roughness characteristics, Greenwood [6] proposed a formula for calculating the contact resistance. Without considering the size and distribution of asperities, this formula uses a single cluster of asperities to equivalently represent the influence of multiple internal asperities. Fractal theory has effectively solved the problem of describing random surface topography. In the 1970s, Mandelbrot and Bhushan [7] proposed the M-B two-dimensional fractal model, which can be applied to simulate the contact of rough surfaces, providing a new theoretical framework for describing random and irregular geometric phenomena. Kogut and Komvopoulos [8] established a functional relationship between contact resistance and the contact load for isotropic fractal surfaces and found that both the increase in fractal dimension and decrease in fractal roughness lead to a decrease in contact resistance. Subsequently, a calculation model for contact resistance that includes an insulating film was established [9], confirming that contact resistance is only related to current density and film thickness. Liu [10] established a finite element contact model for the electromechanical coupling of coated surfaces, explored the impact of rough surface topography on contact resistance, and compared the average contact resistance values at different locations with theoretical predictions. Using a rough surface generated by the two-dimensional Weierstrass–Mandbrot (W-M) function, Li [11] constructed an electrical contact model for rough surfaces that takes into account the interactions between asperities.

The above model adopts two-dimensional fractal theory to simulate the actual three-dimensional topography, which has limitations [12]. Two-dimensional fractal theory primarily focuses on the roughness characteristics within the contour section and is unable to comprehensively reflect the roughness characteristics outside the contour section. In contrast, based on fully considering the three-dimensional geometric topography of joint surfaces, three-dimensional fractal theory provides a better foundation for understanding and analyzing the anisotropy of roughness. By introducing multiple variables, Ausloos and Berman [13] extended the W-M function to a three-dimensional (3D) space and derived the generalized expression of the 3D W-M function, also known as the A-B function, which can be used to describe three-dimensional fractal topography. Yan [14] derived the A-B function and obtained the function for the three-dimensional fractal surface in a spatial Cartesian coordinate system. The equation is also known as the Y-K function, which can be used to simulate anisotropic or isotropic rough surfaces. The contact resistance model based on three-dimensional fractal theory is closer to the actual working conditions, and the calculation results are unique and deterministic [15]. Zhang [16] used the Y-K function to generate the rough surface and established a contact resistance model considering both basic thermal resistance and contraction thermal resistance. Zhang [17] established a simplified contact model for rough surfaces based on the fractal theory and the Monte Carlo methods, overcoming the difficulties of constructing resistance networks in traditional contact models and revealing the influence of fractal parameters D and G on surface topography and contact characteristics. Shen [18] used the Y-K function to generate fractal surfaces and developed a new fractal contact model to evaluate the contact resistance of rough surfaces.

The above models mainly focus on the contact resistance for the planar joint interface. However, in practical applications, spherical–planar joint interfaces appear to be more prevalent [19]. During the contact process, the geometric deformation of the macro sphere affects the contact state, leading to changes in the number of asperities involved in contact and the distribution of contact stress, which in turn affects the contact resistance [20]. Li [21] constructed a comprehensive contact model that encompasses various scenarios such as elastic, elastic–plastic, and plastic deformations in asperities. It is worth mentioning that Li’s model not only explores the plane–planar joint interface but also investigates the spherical–planar joint interface. Wang [22] employed the finite element method to conduct elastic–plastic deformation analysis on a deformable sphere and a rigid surface, with a focus on studying the impact of friction effects on elastic–plastic deformation while taking into account the material’s strain hardening characteristics. Megalingam [23] modeled the hemispherical asperity in contacts with a rigid plane and used the finite element method for analysis, extending the relationship of the obtained contact parameters to rough surface contacts. Zhao [24] studied the contact between an elastic–plastic sphere and a rigid plane during loading and unloading under viscous conditions. Song [25] investigated the mechanical and thermal responses of a rigid sphere sliding on an elastic–plastic sphere with a radius greater than its own, known as the plowing contact model. Zhao [26] researched the contact between a rigid plane and a coated asperity under frictionless and frictional conditions. Chen [27] applied finite element analysis to study the elastic–plastic contact behavior of a coated sphere compressed by a rigid plane. Zhang [28] studied the flattening contact behavior of an ideal elastic–plastic hemisphere squeezed by a rigid plane using the finite element method. In addition, a new elastic–plastic constitutive model was proposed to predict the contact area and contact pressure. Zwicker [29] investigated the influence of strain hardening and surface back rake angle on the flattening of deformed roughness under low normal pressure. Kono [30] proposed a global contact model that takes into account the deformation of the substrate and avoids assumptions about roughness features, which often introduce uncertainties. Meanwhile, some scholars have conducted research on contact models involving macroscopic configuration. Yu [31] proposed a contact correction coefficient to adjust the contact stiffness for contact problems between curved surfaces. Yalpanian [20] proposed a rapid correction method for the half-space theory in modeling contact between free-form surfaces.

The above models have made some progress in describing the contact behavior of rough surfaces involving macroscopic configuration. However, these models describe the effects of macroscopic configuration by introducing correction factors and do not consider the changing process of the number of micro-asperities and the actual contact area, thus failing to reveal the essence of how macro-geometric deformation affects contact behavior. In addition, there are many studies on interfacial force–electricity contact [31,32], which often equate interfacial conduction with the multi-range parallel circuit model. Some models consider the effect of contact load on contact resistance, but most are semi-empirical formulas that cannot reveal the force–electricity coupling mechanism [33,34]. Chen [35,36] established fractal contact models for two spherical surfaces and two cylindrical surfaces by constructing the surface contact coefficient, providing a new approach to studying the interfacial contact resistance under cross-scale coupling of macroscopic configuration and microscopic topography.

In summary, the two-dimensional fractal theory contact resistance model has deficiencies, and the contact resistance model for the spherical–planar joint interface is imperfect. Addressing the aforementioned issues, the contact resistance model for planar joint interface based on three-dimensional fractal theory is established in this paper. Then, by introducing the surface contact coefficient, a contact resistance model for the spherical-planar joint interface is established. Combined with the experimental tests, the accuracy of the proposed model is verified. Furthermore, the variation in contact resistance with influence factors such as macroscopic configuration and microscopic topography are explored, and the contact resistance influence mechanism for the spherical–planar joint interface is revealed. The proposed model holds important theoretical and practical value for improving electrical contact performance, optimizing electronic device design, and ensuring the stable operation of power systems.

2. Surface Generation Based on Three-Dimensional Fractal Theory

Surface generation is the critical prerequisite for constructing a contact resistance model. In this section, surface generation methods based on three-dimensional fractal theory are introduced. The micro-topography of the relay contacts was measured using the ZYGONexView (ZYGO Corporation, Middlefield, CT, USA), and the measurement results are shown in Figure S1. Observing the image, it can be seen that the microscopic topography of the surface is complex and disordered, and fractal theory provides an excellent simulation of this intricate and chaotic nature. Therefore, a surface based on fractal theory is generated in this section. Addressing the limitations of two-dimensional fractal theory, Ausloos and Berman [13] extended the W-M function to the A-B function by introducing multiple variables, making it possible to generate three-dimensional simulated surfaces. The expression of the A-B function is as follows:

$$W\left(\rho ,\theta \right)=\left({\left(\ln \gamma \right)}^{1/2}/M^{1/2}\right){\displaystyle \sum _{m=1}^M{A}_m}{\displaystyle \sum _{n=-\infty }^{\infty }\left[1-\exp \left(i{k}_0{\gamma }^n\rho \cos \left(\theta -{\alpha }_m\right)\right)\right]\times \exp \left(i{\phi }_{m,n}\right){\left(k_0{\gamma }^n\right)}^{D-3}}$$

(1)

where W(ρ, θ) is the vertical height of the fractal surface; ρ and θ are the coordinates of a polar coordinate system; D is the fractal dimension of the fractal surface, which determines the contribution of high- and low-frequency components in the surface function, and 2 < D < 3; G is the scale coefficient of the fractal surface, which represents the height scaling parameter independent of frequency; $\gamma$ is the frequency density parameter of the surface ( $\gamma$ = 1.5), and γⁿ refers to the spatial frequency on the surface. The symbol exp is an exponential function; k₀ is the wave number on the surface, which satisfies the relation k₀ = 2π/L, with L denoting sampling length; M is the number of overlapping wrinkles on the fractal surface. A_m is the height of the highest point on the fractal surface, and it satisfies the relation A_m = 2π(2π/G)^2−D; m and n are random phases, and the value range is [0, 2π]; α_m is used to represent the direction of the wave corresponding to m on the rough surface, which satisfies the relation α_m = πm/M; φ_m,n is the random phase, where the value range is [0, 2π]. N is the frequency index of the asperity, and its value range is (−∞, +∞).

For convenience of application, Yan [14] replaced ρ and θ in the A-B function with x and y, deriving the function for the three-dimensional fractal surface in a spatial Cartesian coordinate system, as shown in Formula (4).

$$\rho ={\left(x^2+y^2\right)}^{1/2}$$

(2)

$$\theta ={\tan }^{-1}\left(y/x\right)$$

(3)

$$z(x,y)=L{\left({\displaystyle \frac{G}{L}}\right)}^{(D-2)}{\left({\displaystyle \frac{\ln \gamma }{M}}\right)}^{{\displaystyle \frac{1}{2}}}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{n=0}^{n_{\max }}{\gamma }^{(D-3)n}}\left\{\cos {\phi }_{m,n}-\cos \left[\frac{2\pi {\gamma }^n\left(x^2+y^2\right)}{L}\cdot \cos \left({\tan }^{-1}\left(\frac{y}{x}\right)-\frac{\pi m}{M}\right)+{\phi }_{m,n}\right]\right\}}$$

(4)

where x, y, and z are the coordinates of the space rectangular coordinate system for constructing the surface; L is the sampling length; n_l is the minimum frequency, usually n_l = 0; n_max is the maximum frequency, where n_max = int[log(L/L_s)/log γ]; L_s is the cut-off length.

Based on Formula (4), the generation of a three-dimensional fractal surface can be achieved using MATLAB R2018b software [37]. The simulation process for the three-dimensional fractal surface can be divided into the following four steps:

① Set the initial values for the simulation. The initial values are set as follows: M =10; γ =1.5; L = 1 × 10⁻³ m; L_s = 5 × 10⁻⁹ m; D = 2.5; G = 10⁻⁸ m; φ_m,n =π/6.

② Define the simulation area, 2 mm × 2 mm, and interpolate the values within this area. The interpolated data from a 256 × 256 grid is used as x and y coordinate values.

③ Use m and n as loop variables to calculate the vertical profile height z of the fractal surface. Following the above process, the profile of the three-dimensional fractal surface can be obtained.

④ Display the profile heights at corresponding coordinates graphically. Figure 1a shows the three-dimensional fractal surface with D = 2.5 and G = 10⁻⁸ m. Combining the geometric formula of the macro spherical surface, the fractal surface under the macro spherical structure can be obtained, as shown in Figure 1b. The main code for three-dimensional fractal theory can be found in Main Code S1.

In this section, the three-dimensional fractal surfaces with planar structure and spherical structure are generated, achieving the generation of rough surfaces at the microscopic scale. Subsequently, combining the contact mechanics analysis of asperities, a contact resistance model based on three-dimensional fractal theory is constructed to explore the influence mechanism of surface micro-topography and macroscopic geometric configuration on contact resistance.

3. Contact Resistance Model for Planar Joint Interface

Based on the analysis in Section 2, the generation of rough surfaces is achieved. Then, the analytical model of contact resistance is constructed in this section. Before constructing the contact resistance for the spherical–planar joint interface, it is necessary to explore the contact resistance model for the planar joint interface. Subsequently, by incorporating the surface contact coefficient, the contact resistance model for the spherical–planar joint interface is constructed.

3.1. Contact Mechanics Analysis of Single Asperity

To construct a contact resistance model for the planar joint interface, it is essential to analyze the deformation mechanism of single asperity first. As contact load increases, the single asperity undergoes three deformation ranges: elastic, elastic–plastic, and plastic. A schematic diagram illustrating the deformation in asperity with a rigid plane is shown in Figure 2. In the diagram, R is the radius of asperity and δ represents the interference. Symbol r denotes the nominal contact radius of the undeformed asperity. Correspondingly, r’ represents the actual contact radius of the asperity.

Yan [14] derived the relationship between the interference δ of the asperity and the nominal contact radius r based on the Y-K function, as shown in Formula (5). It should be mentioned that the anisotropy in the stress field is ignored during the calculation of contact deformation.

$$\delta =2G^{(D-2)}{(\ln \gamma )}^{1/2}{(2r)}^{(3-D)}=2^{\left(4-D\right)}{G}^{D-2}{\left(\ln \gamma \right)}^{1/2}{\left({\displaystyle \frac{a}{\mathsf{\pi }}}\right)}^{\left(3-D\right)/2}$$

(5)

Meanwhile, based on geometric relations, the nominal contact area a between the asperity and the rigid plane can be obtained:

$$a=\mathsf{\pi }{r}^2=\mathsf{\pi }\left[R^2-{\left(R-\delta \right)}^2\right]\approx 2\mathsf{\pi }R\delta$$

(6)

Combining Formula (5), the relationship between the radius R and the nominal contact area a can be derived:

$$R={\displaystyle \frac{a^{\left(D-1\right)/2}}{2^{\left(5-D\right)}{\mathsf{\pi }}^{\left(D-1\right)/2}{G}^{\left(D-2\right)}{\left(\ln \gamma \right)}^{1/2}}}$$

(7)

With the increase in contact load, the asperity undergoes three deformation ranges: elastic, elastic–plastic, and plastic. The following is an analysis of the asperity under these three deformation ranges:

(1)

Elastic deformation range

As the contact load increases, the asperity first enters the elastic deformation range. The contact deformation of asperity in the elastic deformation range is analyzed by using Hertz contact theory. The interference δ of the asperity has the following relationship with the mean contact pressure p_me:

$$\delta ={\left({\displaystyle \frac{3\pi p_{me}}{4E}}\right)}^{2}R$$

(8)

where E′ is the equivalent elastic modulus, E = [(1 − ${\nu }_1^2$)/E₁ + (1 − ${\nu }_2^2$)/E₂]⁻¹. E₁ and E₂ and ν₁ and ν₂ are the elastic moduli and Poisson’s ratios of the two contact surfaces, respectively.

When the asperity undergoes elastic deformation range, the actual contact area a′ is half of the nominal contact area a:

$$a^{\prime }=\mathsf{\pi }{r}^{\prime 2}={\displaystyle \frac{1}{2}}a={\displaystyle \frac{1}{2}}\mathsf{\pi }{r}^2=\mathsf{\pi }R\delta$$

(9)

Meanwhile, the actual contact area a_e of the asperity under the elastic deformation range can be obtained:

$$a_e=a^{\prime }$$

(10)

According to Formulas (5), (7), (8), and (10), the normal contact load F_ne of a single asperity during the elastic deformation range can be derived:

$$F_{ne}=p_{me}\cdot a_{\mathrm{e}}={\displaystyle \frac{2^{\frac{11-2D}{2}}{(\ln \gamma )}^{\frac{1}{2}}E{G}^{D-2}{a}^{\frac{4-D}{2}}}{3{\pi }^{\frac{4-D}{2}}}}$$

(11)

When p_me = 1.1 k_μσ_y, the asperity begins to undergo plastic deformation. According to Formula (8), the critical interference δ_ec for elastic deformation can be determined:

$${\delta }_{\mathrm{ec}}={\left({\displaystyle \frac{33\pi k_{\mu }\varphi }{40}}\right)}^{2}R$$

(12)

where k_μ is the correction factor for the friction coefficient. When 0 ≤ μ ≤ 0.3, k_μ = 1 − 0.228μ; when 0.3 ≤ μ ≤ 0.9, k_μ = 0.932e^{−1.58(μ−0.3)}. Here, μ represents the friction coefficient of the joint interface. ϕ =σ_y/E, where σ_y is the yield strength of the material.

Combined with Formulas (5), (7), and (12), the critical nominal contact area a_ec can be derived:

$$a_{\mathrm{ec}}={\left[{\displaystyle \frac{2^{3D-11}{\pi }^{4-D}{G}^{4-2D}}{\ln \gamma }}\cdot {\left({\displaystyle \frac{33k_{\mu }\varphi }{40}}\right)}^2\right]}^{{\displaystyle \frac{1}{2-D}}}$$

(13)

According to Holm contact resistance theory, the contact resistance of a single asperity is as follows:

$$R_c=({\rho }_1+{\rho }_2)/4r_c={\displaystyle \frac{{\mathsf{\pi }}^{1/2}\left({\rho }_1+{\rho }_2\right)}{4a_c^{1/2}}}$$

(14)

where R_c represents the contact resistance; ρ₁ and ρ₂ are the resistivity of the materials (hereinafter assuming ρ₁ = ρ₂ = ρ); r_c is the radius of the conductive area, which is the actual contact area radius of the asperity.

Combining Formulas (10) and (14), the contact resistance R_ce of a single asperity in the elastic deformation range can be obtained:

$$R_{ce}={\displaystyle \frac{\rho {(\mathsf{\pi }/a_e)}^{1/2}}{2}}=\rho /2r_e={\displaystyle \frac{\rho {\mathsf{\pi }}^{1/2}}{2^{1/2}{a}^{1/2}}}$$

(15)

(2)

Plastic deformation range

With the further increase in contact load, the asperity enters the elastic–plastic deformation range. However, since there is no complete analytical formula for the elastic–plastic deformation range, the mathematical methods will be used for transition over that range. Therefore, the plastic deformation range of asperity is explored in this section. According to Reference [21], the actual contact area a_ep in the plastic deformation range is equal to the nominal contact area a:

$$a_p=a=\pi r^2$$

(16)

The mean contact pressure p_mp of a single asperity during the plastic deformation range is given by the following formula:

$$p_{mp}=2.79{\sigma }_{\mathrm{y}}$$

(17)

Based on Formula (16) and (17), the normal contact load F_np of a single asperity in the plastic deformation range can be derived:

$$F_{np}=p_{mp}\cdot a_p=2.79{\sigma }_{y}a$$

(18)

The empirical relationship for plastic deformation of a single asperity under normal contact load is as follows [21]:

$${\displaystyle \frac{p_{mp}}{{\sigma }_y}}=1.61{\left({\displaystyle \frac{{\delta }_{pc}}{{\delta }_e}}\right)}^{0.117}$$

(19)

where δ_pc is the critical deformation when the asperity enters the fully plastic deformation range.

Combined with Formulas (17) and (19), the critical plastic interference δ_pc of the asperity can be obtained:

$${\delta }_{pc}={\left({\displaystyle \frac{2.79}{1.61}}\right)}^{{\displaystyle \frac{1}{0.117}}}{\delta }_{ec}\approx 110{\delta }_{ec}$$

(20)

Further combining Formula (12), δ_pc can be expressed as follows:

$${\delta }_{pc}=110{\left({\displaystyle \frac{33\pi k_{\mu }\varphi }{40}}\right)}^{2}R$$

(21)

Thus, by substituting Formulas (5) and (7) into Formula (21), the nominal contact area a_pc of the asperity under the critical interference of full plastic deformation can be obtained, as shown in the following formula:

$$a_{pc}={\left[110\cdot {\displaystyle \frac{2^{2D-9}{\pi }^{4-D}{G}^{4-2D}}{\ln \gamma }}\cdot {\left({\displaystyle \frac{33k_{\mu }\varphi }{40}}\right)}^2\right]}^{{\displaystyle \frac{1}{2-D}}}$$

(22)

Based on Holm contact resistance theory, the contact resistance R_cp of a single asperity in the plastic range can be obtained:

$$R_{cp}={\displaystyle \frac{\rho {(\mathsf{\pi }/a_p)}^{1/2}}{2}}={\displaystyle \frac{\rho {\mathsf{\pi }}^{1/2}}{2a^{1/2}}}$$

(23)

(3)

Elastic–plastic deformation range

When the deformation is within the range of δ_ec ≤ δ ≤ δ_pc, the asperity enters the elastic–plastic deformation range. In this range, the material hardness H is not a fixed value but varies with the change in interference. In previous research [38], a transition method from the elastic range to the full plastic range was derived, which will not be repeated here. The mean contact pressure p_mep and contact load F_nep during the elastic–plastic deformation are defined as shown in Formulas (24) and (25):

$$p_{mep}=c{\sigma }_y{\left({\displaystyle \frac{a}{a_{ec}}}\right)}^{c_1}$$

(24)

$$F_{nep}=p_{mep}{a}_{ep}=c_2{\sigma }_y{a}_{ec}{\left({\displaystyle \frac{a}{a_{ec}}}\right)}^{c_3}$$

(25)

where, c, c₁, c₂, and c₃ are undetermined coefficients.

To solve for the mean contact pressure and contact load during elastic–plastic deformation, p_mep and F_nep should satisfy the following boundary conditions:

$$\begin{array}{l}{p}_{mep}\left(a_{ec}\right)=p_{me}\left(a_{ec}\right)\\ p_{mep}\left(a_{pc}\right)=p_{mp}\left(a_{pc}\right)\\ F_{nep}\left(a_{ec}\right)=F_{ne}\left(a_{ec}\right)\\ F_{nep}\left(a_{pc}\right)=F_{np}\left(a_{pc}\right)\end{array}$$

(26)

By substituting p_me(a_ec), p_mp(a_pc), F_nep(a_ec), and F_nep(a_pc) into Formula (26), the values of coefficients c, c₁, c₂, and c₃ can be obtained:

$$\begin{array}{l}c=2.79K=1.1k_{\mu }\\ c_1={\displaystyle \frac{\ln K}{\ln (a_{ec}/a_{pc})}}\\ c_2={\displaystyle \frac{1.1k_{\mu }}{2}}\\ c_3=1-{\displaystyle \frac{\ln \left(K/2\right)}{\ln (a_{ec}/a_{pc})}}\end{array}$$

(27)

where K = 0.4 k_μ.

From this, the mean contact pressure p_mep in the elastic–plastic deformation range can be obtained:

$$p_{mep}=1.1k_{\mu }{\sigma }_y{\left({\displaystyle \frac{a}{a_{ec}}}\right)}^{-{\displaystyle \frac{\ln K}{\ln \frac{a_{pc}}{a_{ec}}}}}$$

(28)

Similarly, the contact load F_nep in the elastic–plastic deformation range can be obtained:

$$F_{nep}={\displaystyle \frac{1.1k_{\mu }}{2}}{\sigma }_y{a}_{ec}{\left({\displaystyle \frac{a}{a_{ec}}}\right)}^{1-{\displaystyle \frac{\ln \frac{K}{2}}{\ln (\frac{a_{pc}}{a_{ec}})}}}$$

(29)

Then, combining Formulas (28) and (29), the actual contact area a_ep during the elastic–plastic deformation range can be derived:

$$a_{ep}=F_{nep}/P_{epa}={\displaystyle \frac{a_{ec}}{2}}{\left({\displaystyle \frac{a}{a_{ec}}}\right)}^{1+{\displaystyle \frac{\ln 2}{\ln \frac{a_{pc}}{a_{ec}}}}}={\displaystyle \frac{1}{2}}{a}^{1+{\displaystyle \frac{\ln 2}{\ln \frac{a_{pc}}{a_{ec}}}}}{a}_{ec}^{1-{\displaystyle \frac{\ln 2}{\ln \frac{a_{pc}}{a_{ec}}}}}$$

(30)

Based on Holm contact resistance theory, the contact resistance R_cep of a single asperity in the elastic–plastic range can be obtained:

$$R_{cep}={\displaystyle \frac{\rho {(\mathsf{\pi }/a_{ep})}^{1/2}}{2}}={\displaystyle \frac{\rho {(\mathsf{\pi }/\left(\frac{1}{2}{a}^{1+\frac{\ln 2}{\ln \frac{a_{pc}}{a_{ec}}}}{a}_{ec}^{1-\frac{\ln 2}{\ln \frac{a_{pc}}{a_{ec}}}}\right))}^{1/2}}{2}}$$

(31)

Based on the above analysis, the expressions of contact load and nominal contact area of a single asperity in the elastic, elastic–plastic, and plastic deformation ranges were obtained. Meanwhile, analytical expressions for contact resistance in different deformation ranges of a single asperity were also acquired. In the next section, the theoretical expression of contact resistance for the planar joint interface is derived by combining the distribution of asperities on the three-dimensional fractal surface.

3.2. Contact Resistance for Planar Joint Interface

Based on the analysis in Section 3.1, the deformation mechanism of the single asperity on rough surfaces was explored. In this section, combining the distribution of asperities on the three-dimensional fractal surface, the contact resistance model for the planar joint interface is constructed.

According to the literature [14], the formula for the nominal contact area distribution of asperities on the three-dimensional fractal surface is shown below:

$$n(a)={\displaystyle \frac{(D-1)}{2a_L}}{\left({\displaystyle \frac{a_L}{a}}\right)}^{{\displaystyle \frac{D+1}{2}}}={\displaystyle \frac{(D-1)}{2}}{a}_L{}^{{\displaystyle \frac{D-1}{2}}}{a}^{-{\displaystyle \frac{D+1}{2}}}$$

(32)

where a_L represents the maximum nominal contact area of the asperities.

Combining Formula (32), when D ≠ 2.5, the normal contact load F_n on the joint interface can be derived as follows:

$$\begin{array}{ll}{F}_n& ={\displaystyle {\int }_{a_{ec}}^{a_L}{F}_{ne}}\cdot n(a)da+{\displaystyle {\int }_{a_{pc}}^{a_{ec}}{F}_{nep}}\cdot n(a)da+{\displaystyle {\int }_0^{a_{pc}}{F}_{np}}\cdot n(a)da\\ & =2^{{\displaystyle \frac{11-2D}{2}}}{\displaystyle \frac{D-1}{5-2D}}\cdot {\displaystyle \frac{E{G}^{D-2}{(\ln \gamma )}^{1/2}{a}_L^{\frac{D-1}{2}}}{3{\pi }^{\frac{4-D}{2}}}}\cdot \left(a_L^{{\displaystyle \frac{5-2D}{2}}}-a_{ec}^{{\displaystyle \frac{5-2D}{2}}}\right)\\ & +{\displaystyle \frac{(D-1)c_2{\sigma }_y}{2c_3+2-D+1}}{a}_{ec}{}^{1-c_3}\left(a_{ec}^{c_3+1-{\displaystyle \frac{D+1}{2}}}-a_{pc}^{c_3+1-{\displaystyle \frac{D+1}{2}}}\right)a_L{}^{{\displaystyle \frac{D-1}{2}}}+{\displaystyle \frac{2.79(D-1){\sigma }_y}{3-D}}{a}_L^{{\displaystyle \frac{D-1}{2}}}{a}_{pc}^{{\displaystyle \frac{3-D}{2}}}\end{array}$$

(33)

Similarly, when D = 2.5, the normal contact load F_n on the joint interface can be derived as follows:

$$\begin{array}{ll}{F}_n& ={\displaystyle {\int }_{a_{ec}}^{a_L}{F}_{ne}}\cdot n(a)da+{\displaystyle {\int }_{a_{pc}}^{a_{ec}}{F}_{nep}}\cdot n(a)da+{\displaystyle {\int }_0^{a_{pc}}{F}_{np}}\cdot n(a)da\\ & ={\displaystyle \frac{2{(\ln \gamma )}^{0.5}E{G}^{0.5}}{{\pi }^{0.75}}}\times a_L{}^{0.75}\ln \left({\displaystyle \frac{a_L}{a_{ec}}}\right)+{\displaystyle \frac{1.1k_{\mu }(D-1)}{4\left(c_3-\frac{D+1}{2}+1\right)}}\times \\ & {\sigma }_y{a}_{ec}^{1-c_3}{a}_L{}^{{\displaystyle \frac{D-1}{2}}}\left(a_{ec}^{c_3-{\displaystyle \frac{D+1}{2}}+1}-a_{pc}^{c_3-{\displaystyle \frac{D+1}{2}}+1}\right)+{\displaystyle \frac{2.79(D-1){\sigma }_y}{3-D}}{a}_L^{{\displaystyle \frac{D-1}{2}}}{a}_{pc}^{{\displaystyle \frac{3-D}{2}}}\end{array}$$

(34)

Compared to the two-dimensional fractal model, the three-dimensional fractal model can more accurately simulate the three-dimensional topographical features of rough surfaces, including the actual contact area of micro-asperities and their random distribution in the contact interface. This allows for a more comprehensive representation of actual contact conditions. As a result, three-dimensional fractal models can provide more precise outcomes in contact resistance calculations, further enhancing the accuracy and reliability of the computations.

The asperities in the joint interface form a parallel relationship after contact, and the contact resistance of the entire joint interface is the parallel resistance R_c of all asperities on the interface, which has the following relationship:

$$\begin{array}{ll}{\displaystyle \frac{1}{R_c}}& ={\displaystyle {\int }_{a_{ec}}^{a_L}\frac{1}{R_{ce}}\cdot n(a)da}+{\displaystyle {\int }_{a_{pc}}^{a_{ec}}\frac{1}{R_{cep}}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}}\frac{1}{R_{cp}}\cdot n(a)da}\\ & ={\displaystyle \frac{2}{{\mathsf{\pi }}^{1/2}\rho }}\left({\displaystyle {\int }_{a_{ec}}^{a_L}{a}_e^{1/2}\cdot n(a)da}+{\displaystyle {\int }_{a_{ec}}^{a_L}{a}_{ep}^{1/2}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}}{a}_p^{1/2}\cdot n(a)da}\right)\end{array}$$

(35)

By combining the analytical expressions of the single asperity with the distribution expressions of asperities, the analytical expressions of the contact resistance for the entire planar joint interface can be derived:

In Formula (35):

$${\displaystyle {\int }_{a_{ec}}^{a_L}{a}_e^{1/2}\cdot n(a)da}={\displaystyle {\int }_{a_{ec}}^{a_L}{\left(\frac{1}{2}a\right)}^{1/2}\cdot \frac{(D-1)}{2}{a}_L{}^{\frac{D-1}{2}}{a}^{-\frac{D+1}{2}}da}={\displaystyle \frac{D-1}{2^{1/2}\left(2-D\right)}}\cdot a_L{}^{{\displaystyle \frac{D-1}{2}}}\cdot (a_L^{{\displaystyle \frac{2-D}{2}}}-a_{ec}^{{\displaystyle \frac{2-D}{2}}})$$

(36)

$$\begin{array}{ll}{\displaystyle {\int }_{a_{ec}}^{a_L}{a}_{ep}^{1/2}\cdot n(a)da}& ={\displaystyle {\int }_{a_{pc}}^{a_{ec}}\frac{2{\left(\frac{a_{ec}}{2}{\left(\frac{a}{a_{ec}}\right)}^{1+\frac{\ln 2}{\ln \frac{a_{pc}}{a_{ec}}}}\right)}^{1/2}}{{\mathsf{\pi }}^{1/2}\rho }\cdot \frac{(D-1)}{2}{a}_L{}^{\frac{D-1}{2}}{a}^{-\frac{D+1}{2}}da}\\ & ={\displaystyle \frac{D-1}{2^{1/2}{\mathsf{\pi }}^{1/2}\rho }}{a}_L{}^{{\displaystyle \frac{D-1}{2}}}{a}_{ec}^{{\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\ln 2}{\ln \frac{a_{pc}}{a_{ec}}}}\right)}{\displaystyle \frac{1}{1+\frac{1}{2}\left(\frac{\ln 2}{\ln \left(a_{pc}/a_{ec}\right)}-D\right)}}\left(a_{ec}^{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\ln 2}{\ln \left(a_{pc}/a_{ec}\right)}}-D\right)}-a_{pc}^{{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\ln 2}{\ln \left(a_{pc}/a_{ec}\right)}}-D\right)}\right)\end{array}$$

(37)

$${\displaystyle {\int }_0^{a_{pc}}{a}_p^{1/2}\cdot n(a)da}={\displaystyle {\int }_0^{a_{pc}}{a}^{1/2}\cdot \frac{(D-1)}{2}{a}_L{}^{\frac{D-1}{2}}{a}^{-\frac{D+1}{2}}da}={\displaystyle \frac{(D-1)}{2}}{a}_L{}^{{\displaystyle \frac{D-1}{2}}}{\displaystyle {\int }_0^{a_{pc}}{a}^{-\frac{D}{2}}da}={\displaystyle \frac{D-1}{2-D}}{a}_L{}^{{\displaystyle \frac{D-1}{2}}}{a}_{pc}{}^{1-{\displaystyle \frac{D}{2}}}$$

(38)

Based on the above analysis, the analytical expressions of the contact resistance for the planar joint interface at different deformation ranges were obtained. Then, the analytical model of contact resistance for the planar joint interface was established. In the next section, based on the contact resistance model for the planar joint interface, the contact resistance model for the spherical–planar joint interface is constructed by introducing the surface contact coefficient.

4. Contact Resistance Model for Spherical–Planar Joint Interface

The contact between a spherical surface and a plane is a commonly used connection method in engineering applications. The contact diagram of the spherical–planar joint interface is shown in Figure 3. The schematic diagram of the relay contact spring system is shown in Figure 3a, which essentially represents the mutual contact between a rough spherical surface and a plane, as illustrated in Figure 3b.

As can be seen from Figure 3, the construction of the contact resistance model for the spherical–planar joint interface needs to address the cross-scale coupling problem between macro-geometric configuration and micro-surface topography. To solve this problem, the surface contact coefficient is introduced to achieve the cross-scale coupling in this section.

4.1. Surface Contact Coefficient

To solve the contact problem between spherical and planar surfaces, the surface contact coefficient is introduced. Chen [35,36] analyzed the rationality of the contact coefficient λ in the literature. The surface contact coefficient of the two curved surfaces is defined as λ:

$$\lambda ={\left({\displaystyle \frac{S_h}{{\displaystyle \sum S}}}\right)}^{X_h}$$

(39)

where S_h is the theoretical contact area; ∑S is the sum of the areas of the two surfaces; X_h is the comprehensive curvature coefficient.

Based on Hertz contact theory, the contact area radius r_s during spherical–planar contact can be calculated using the following formula:

$$r_s{}^3={\displaystyle \frac{3}{4}}{\displaystyle \frac{F_{sn}{R}_1}{E}}$$

(40)

where R₁ represents the macroscopic spherical radius; F_sn denotes the normal contact load of spherical–planar contact.

The theoretical contact area S_h can be expressed by the following formula:

$$S_h=\mathsf{\pi }{r}_s{}^2$$

(41)

The comprehensive curvature coefficient X_h can be expressed by the following formula:

$$X_h={\displaystyle \frac{2}{R_1}}$$

(42)

The sum of the areas of the two surfaces ∑S can be obtained through the following formula:

$${\displaystyle \sum S}=S_1=4\mathsf{\pi }{R}_1{}^2$$

(43)

where S₁ represents the surface areas of the sphere.

By substituting the aforementioned parameters into Formula (39), the surface contact coefficient for the spherical–planar joint interface can be obtained:

$$\lambda ={\left({\displaystyle \frac{{\left(\frac{3}{4}\frac{F_{sn}{R}_1}{E}\right)}^{\frac{2}{3}}}{4R_1{}^2}}\right)}^{{\displaystyle \frac{2}{R_1}}}$$

(44)

To facilitate subsequent calculations, the following formula is defined:

$$\lambda ={\lambda }_0{F}_{sn}^{C_2{X}_h}$$

(45)

where ${\lambda }_0={\left({\left({\displaystyle \frac{3R_1}{4E}}\right)}^{C_2}/4R_1^2\right)}^{X_h}$, C₂ = 2/3, and X_h = 2/R₁.

4.2. Contact Resistance for Spherical–Planar Joint Interface

Combining the surface contact coefficient, the area distribution n(a) of asperities in planar contact is corrected by using the surface contact coefficient, and the area distribution n’(a) of asperities in spherical–planar contact can be obtained:

$$n^{\prime }\left(a\right)=\lambda \cdot n\left(a\right)$$

(46)

When the spherical surface is in contact with the plane surface, the contact load F_sn satisfies the following relationship:

$$\begin{array}{l}{F}_{sn}={\displaystyle {\int }_{a_{ec}}^{a_L}{F}_{ne}}\cdot n^{\prime }(a)da+{\displaystyle {\int }_{a_{pc}}^{a_{ec}}{F}_{nep}\cdot n^{\prime }(a)da}+{\displaystyle {\int }_0^{a_{pc}}{F}_{np}\cdot n^{\prime }(a)da}\\ ={\left({\displaystyle \frac{{\left(\frac{3}{4}\frac{F_{sn}{R}_1}{E}\right)}^{\frac{2}{3}}}{4R_1{}^2}}\right)}^{{\displaystyle \frac{2}{R_1}}}{\displaystyle {\int }_{a_{ec}}^{a_L}{F}_{ne}}\cdot n(a)da+{\displaystyle {\int }_{a_{pc}}^{a_{ec}}{F}_{nep}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}}{F}_{np}\cdot n(a)da}\end{array}$$

(47)

Combined with Formula (45), Formula (47) is simplified:

$$F_{sn}={\left({\lambda }_0{\displaystyle {\int }_{a_{ec}}^{a_L}{F}_{ne}}\cdot n(a)da+{\displaystyle {\int }_{a_{pc}}^{a_{ec}}{F}_{nep}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}}{F}_{np}\cdot n(a)da}\right)}^{{\displaystyle \frac{1}{1-\frac{2}{3}{X}_h}}}$$

(48)

Meanwhile, the actual contact area A_sa after introducing λ can be expressed in the following form:

$$\begin{array}{ll}{A}_{sa}& ={\displaystyle {\int }_{a_{ec}}^{a_L}{a}_e\cdot n^{\prime }(a)da}+{\displaystyle {\int }_{a_{pc}}^{a_{ec}}{a}_{ep}\cdot n^{\prime }(a)da}+{\displaystyle {\int }_0^{a_{pc}}{a}_p\cdot n^{\prime }(a)da}\\ & =\lambda {\displaystyle {\int }_{a_{ec}}^{a_L}{a}_e\cdot n(a)da}+{\displaystyle {\int }_{a_{pc}}^{a_{ec}}{a}_{ep}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}}{a}_p\cdot n(a)da}\end{array}$$

(49)

Referring to the derivation of contact resistance for the planar joint interface, the asperities can conduct electricity after contact and form a parallel relationship. The contact resistance of the entire spherical–planar joint interface is the parallel resistance R_sc of all the asperities in the interface:

$$\begin{array}{ll}{R}_{sc}& ={\displaystyle \frac{{\mathsf{\pi }}^{1/2}\rho }{\begin{array}{l}2\left({\displaystyle {\int }_{a_{ec}}^{a_L}{a}_e^{1/2}\cdot n^{\prime }(a)da}+{\displaystyle {\int }_{a_{ec}}^{a_L}{a}_{ep}^{1/2}\cdot n^{\prime }(a)da}+{\displaystyle {\int }_0^{a_{pc}}{a}_p^{1/2}\cdot n^{\prime }(a)da}\right)\\ \end{array}}}\\ & ={\displaystyle \frac{{\mathsf{\pi }}^{1/2}\rho }{\begin{array}{l}2\lambda \left({\displaystyle {\int }_{a_{ec}}^{a_L}{a}_e^{1/2}\cdot n(a)da}+{\displaystyle {\int }_{a_{ec}}^{a_L}{a}_{ep}^{1/2}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}}{a}_p^{1/2}\cdot n(a)da}\right)\\ \end{array}}}\end{array}$$

(50)

By substituting the formulas of ${\displaystyle {\int }_{a_{ec}}^{a_L}{a}_e^{1/2}\cdot n(a)da}$, ${\displaystyle {\int }_{a_{ec}}^{a_L}{a}_{ep}^{1/2}\cdot n(a)da}$, and ${\displaystyle {\int }_0^{a_{pc}}{a}_p^{1/2}\cdot n(a)da}$, the analytical expression of the contact resistance for spherical–planar joint interface in different deformation ranges can be derived.

For the convenience of comparison, the above parameters are dimensionless. The contact load F_sn is dimensionless as F_sn^*, as shown below:

$$F_{sn}{}^{*}={\left({\lambda }_0{\displaystyle {\int }_{a_{ec}{}^{*}}^{a_L{}^{*}}{F}_{ne}}\cdot n(a)da+{\displaystyle {\int }_{a_{pc}{}^{*}}^{a_{ec}{}^{*}}{F}_{nep}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}{}^{*}}{F}_{np}\cdot n(a)da}\right)}^{{\displaystyle \frac{1}{1-\frac{2}{3}{X}_h}}}$$

(51)

where F_sn^* = F_sn/(E∑S), a_L^* = a_L/∑S, a_ec^* = a_ec/∑S, and a_pc^* = a_pc/∑S.

The actual contact area A_sa is dimensionless into A_sa^*, as shown in the following formula:

$$\begin{array}{l}{A}_{sa}^{*}={\displaystyle {\int }_{a_{ec}{}^{*}}^{a_L{}^{*}}{a}_e\cdot n^{\prime }(a)da}+{\displaystyle {\int }_{a_{pc}{}^{*}}^{a_{ec}{}^{*}}{a}_{ep}\cdot n^{\prime }(a)da}+{\displaystyle {\int }_0^{a_{pc}{}^{*}}{a}_p\cdot n^{\prime }(a)da}\\ =\lambda {\displaystyle {\int }_{a_{ec}{}^{*}}^{a_L{}^{*}}{a}_e\cdot n(a)da}+{\displaystyle {\int }_{a_{pc}{}^{*}}^{a_{ec}{}^{*}}{a}_{ep}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}{}^{*}}{a}_p\cdot n(a)da}\end{array}$$

(52)

where A_sa^* = A_sa/∑S.

The contact resistance R_sc is dimensionless into R_sc^*, as shown in the following formula:

$$R_{sc}{}^{*}={\displaystyle \frac{{\mathsf{\pi }}^{1/2}\rho }{\begin{array}{l}2\lambda \left({\displaystyle {\int }_{a_{ec}{}^{*}}^{a_L{}^{*}}{a}_e^{1/2}\cdot n(a)da}+{\displaystyle {\int }_{a_{ec}{}^{*}}^{a_L{}^{*}}{a}_{ep}^{1/2}\cdot n(a)da}+{\displaystyle {\int }_0^{a_{pc}{}^{*}}{a}_p^{1/2}\cdot n(a)da}\right)\\ \end{array}}}$$

(53)

where R_sc^* = R_sc ∑S ^0.5/ρ.

Based on the above analysis, the analytical expressions of the contact resistance for the spherical–planar joint interface in different deformation ranges were obtained, and an analytical model of contact resistance for the spherical–planar joint interface was established. In the next section, contact resistance tests are carried out to verify the accuracy of the model proposed in this paper.

5. Experimental Verification and Simulation Analysis

Based on the analysis in Section 4, the analytical expression of contact resistance for the spherical–planar joint interface was obtained, and then a theoretical model of contact resistance was established. To verify the accuracy of the model proposed in this paper, a contact resistance experiment for the spherical–planar joint interface is carried out in this section. Subsequently, the simulation results of the proposed model are compared with the experimental results and the simulation results in the literature. Then, the numerical simulation and analysis of the proposed model is conducted, revealing the influence mechanism of macro-geometric configuration and micro-surface topography on contact resistance.

5.1. Parameters of the Test Samples

Before conducting the contact resistance experiment, the sample needs to be processed to obtain the relevant parameters for model simulation. The contact resistance between the copper sphere with a curvature radius of 0.02 m and the copper plane was experimentally tested. The micro-topography of the samples was measured using the ZYGONexView (ZYGO Corporation, Middlefield, CT, USA), and the measurement results are shown in Figure S1. Subsequently, the fractal characteristic parameters of these measured surfaces were derived through the application of the box counting method, as outlined in [39]. The fractal parameters of the sample surface are D = 2.512 and G = 1.479 × 10⁻⁸ m.

In addition, the spherical–planar joint interface was made of copper material, and the material parameters were tested by the universal testing machine (ZWICK Corporation, Bavaria, Germany). The material parameters of the sample are shown in

Table 1. The material parameters of the sample.

Material Parameter	Elastic Modulus	Yield Strength	Poisson’s Ratio	Friction Coefficient	Resistivity
Value	115 GPa	220 Mpa	0.34	0.19	1.78 × 10−8 Ω·m

. The elastic modulus of the sample was 115 GPa and the Poisson’s ratio of the sample was 0.34, so the equivalent elastic modulus E of the interface was 65 GPa; the yield strength was 220 Mpa, so the parameter ϕ was 0.0034; the friction coefficient μ was 0.19; the resistivity ρ was 1.78 × 10⁻⁸ Ω·m.

5.2. Experimental Device

After obtaining the relevant parameters of the test sample, the contact resistance test device is described in this section. The schematic diagram of the test device is shown in Figure 4.

As shown in Figure 4, the contact resistance test device mainly consists of two parts: the actuating mechanism and the data acquisition system. The actuating mechanism adopts a vertical structure, which includes components such as the loading system, the overall support frame, upper and lower insulating pads, contact clamping device, movable contact, static contact, and bottom support. Among them, the loading system uses a ZWICKZ010 electronic universal testing machine (ZWICK Corporation, Bavaria, Germany) for loading. The universal testing machine is equipped with a KAP-TC pressure sensor (Kewill GmbH, Hamburg, Germany) to measure the applied pressure during the loading process. The pressure sensor has a load range of 10,000 N and a pressure measurement accuracy of 0.5%FS.

The data acquisition system comprises two parts: the contact resistance tester and the data analysis system. The UC2518MX DC resistance tester (Youce Electronic Technology Corporation, Changzhou City, China) was used to test the contact resistance. It has a range of 3 mΩ, a test accuracy of 0.02%FS, and a sampling rate of 50 times per second.

Using this experimental setup, the contact load and contact resistance signals were collected, allowing for the acquisition of the contact resistance variation curve under different contact loads. It is important to note that to ensure the authenticity of the experimental data, three samples of each type were tested, and the average value was presented as the final result.

5.3. Numerical Simulation and Analysis

In this section, the simulation results of the proposed model are compared with the experimental results and the simulation results in the literature to verify the accuracy of the model in this paper. Then, a numerical simulation and analysis of the proposed model is conducted, revealing the influence mechanism of macro-geometric configuration and micro-surface topography on contact resistance.

The simulation results for comparison, hereinafter referred to as the statistical model and the YK model, respectively, are the statistical model from Reference [40] and the fractal model from Reference [14]—without considering the elastic–plastic deformation in asperities. By substituting the parameters of the sample into the three models, the numerical simulation results of the three models can be obtained. The comparison results of contact resistance are shown in Figure 5.

Combined with Figure 5, it can be observed that all three results exhibit similar trends, but there are numerical differences among them. Moreover, as the contact load increases, there is a gradual decrease in contact resistance. It is worth noting that the proposed model in this paper aligns more closely with experimental results. Since the values of the contact resistances in Figure 5 are relatively close, the relative errors between the simulation results of the three models and the experimental results were calculated, and the comparison of the results is shown in Figure 6.

As can be seen from Figure 6, the comparative results of relative errors are undoubtedly encouraging. Under different contact load conditions, the relative error of the model proposed in this paper is significantly lower than that of the other two models. The specific values of relative errors can be obtained in Table S1. The maximum relative error of the proposed model occurs at the dimensionless contact load F_n^* =1.22 × 10⁻⁵, and the maximum relative error is 8.44%. In comparison, the relative errors of the other two models are all around 20% to 30%. The model referenced in the literature [40] is rooted in statistical theory. However, statistical methods exhibit size dependence, meaning that statistical parameters of rough surfaces can vary depending on the measurement instrument used, directly affecting the initial values for model simulation. Deviations in these initial simulation values are the fundamental reason why the simulation results of that model diverge from experimental data. In contrast, the fractal parameters in fractal theory characterize geometric contours based on geometric self-similarity, independent of the measurement instrument used, which leads to the proposed model being closer to experimental results. In addition, the YK model in Reference [14] did not take into account the elastic–plastic deformation of asperity; therefore, the actual contact area calculated by the model is larger than the actual situation, and the calculated contact resistance result is smaller than that of the proposed model. Overall, the comparison results validate the accuracy of the proposed model in this paper.

After verifying the accuracy of the proposed model, the influence mechanism of macro-geometric configuration and micro-surface topography on contact resistance are explored. Factors such as macroscopic configuration and microscopic topography are specifically reflected in parameters like fractal dimension D, scale coefficient G, and spherical radius R₁. Therefore, the influence of the above parameters on contact resistance is discussed. The material parameters required for the simulation can be found in Section 5.1. Additionally, the value ranges for fractal dimension, scale coefficient, and spherical radius are specifically listed in each section.

(1)

The fractal dimension D

The fractal dimension D is a quantitative parameter that describes the complexity of rough surfaces. The fractal dimension more accurately reflects the self-similarity and irregularity of rough surfaces at different scales. A larger D value indicates that the surface is rougher, more complex, and contains more details and variations. Therefore, the numerical simulation on the influence of fractal dimension D for the interface contact resistance is conducted in this section. The value of the fractal dimension D ranges from 2.4 to 2.6. The scale coefficient is set as G = 1 × 10⁻⁸ m, and the spherical radius is set as R₁ = 0.02 m. By giving the specific a_L values, the simulation results of the contact resistance under different fractal dimensions for the spherical–planar joint interface can be obtained.

The variation curve of dimensionless contact resistance R_c^* with dimensionless contact load F_n^* under different fractal dimensions is shown in Figure 7. Overall, R_c^* follows the same variation pattern as F_n^* under different fractal dimensions. The contact resistance decreases as the contact load increases, and the rate of decrease gradually slows down. Analyzing with Formula (53), as the contact load increases, the conductive area of the asperities in the interface gradually increases, resulting in a gradual decrease in contact resistance. Meanwhile, due to the gradually decreasing rate of increase in conductive area, the rate of decrease in contact resistance gradually slows down.

It is worth noting that under the same contact load, the contact resistance varies with different fractal dimensions. A higher fractal dimension corresponds to lower contact resistance. The fractal dimension affects the density and amplitude of asperities on the rough surface. A larger fractal dimension is associated with smaller asperity amplitudes and a higher asperity density. Under the same contact load, smaller asperity amplitudes increase the actual contact area at the interface, resulting in a decrease in contact resistance. Meanwhile, a higher asperity density increases the number of asperities involved in contact. Since the contact resistances between different asperities are in parallel, this leads to a decrease in overall contact resistance.

(2)

The scale coefficient G

As the characteristic scale parameter of rough surfaces, the scale coefficient G can quantitatively describe the roughness of the surface. It reflects the height characteristics of the surface contour curve, which represents the degree of undulation in the surface micro-topography. By measuring or calculating the value of G, the roughness level of the surface can be objectively evaluated. Therefore, a numerical simulation of the influence of scale coefficient G for the interface contact resistance is conducted in this section. The value of the scale coefficient G ranges from 1 × 10⁻⁹ to 1 × 10⁻⁷ m. The fractal dimension is set as D = 2.5, and the spherical radius is set as R₁ = 0.02 m. By giving the specific a_L values, the simulation results of the contact resistance under different scale coefficients for the spherical–planar joint interface can be obtained.

The variation curve of dimensionless contact resistance R_c^* with dimensionless contact load F_n^* under different scale coefficients is shown in Figure 8. Overall, R_c^* follows the same variation pattern as F_n^* under different scale coefficients. The contact resistance decreases as the contact load increases, and the rate of decrease gradually slows down. Similar to the simulation results under different fractal dimensions, as the contact load increases, the conductive area of the asperities in the interface gradually increases, resulting in a gradual decrease in contact resistance. Meanwhile, due to the gradually decreasing rate of increase in conductive area, the rate of decrease in contact resistance gradually slows down.

Interestingly, under the same contact load, the contact resistance varies with different scale coefficients. A larger scale coefficient corresponds to greater contact resistance. The scale coefficient affects the amplitude of asperities on the rough surface, with a larger scale coefficient corresponding to greater asperity amplitude. Under the same contact load, a greater asperity amplitude will decrease the actual contact area at the interface, resulting in an increase in contact resistance.

(3)

The spherical radius R₁

A numerical simulation of the influence of spherical radius R₁ for the interface contact resistance is conducted in this section. The value of the spherical radius R₁ ranges from 0.01 to 0.03 m. The fractal dimension is set as D = 2.5, and the scale coefficient is set as G = 1 × 10⁻⁸ m. By giving the specific a_L values, the simulation results of the contact resistance under different spherical radii for the spherical–planar joint interface can be obtained.

The variation curve of dimensionless contact resistance R_c^* with dimensionless contact load F_n^* under different spherical radii is shown in Figure 9. Overall, R_c^* exhibits a consistent trend with F_n^* across various spherical radii. The contact resistance decreases as the contact load increases, with a gradually diminishing rate of decrease. As the contact load increases, the conductive area of the interface asperities progressively expands, leading to a gradual reduction in contact resistance. Simultaneously, the rate of decrease in contact resistance tapers off due to the slowing growth rate of the conductive area.

Excitingly, under the same contact load, there are discrepancies in contact resistance corresponding to different spherical radii. A larger spherical radius correlates with a smaller contact resistance. By examining the surface contact coefficient, it becomes evident that a larger contact area between the sphere and the plane results in a greater number of asperities participating in the contact. Since the contact resistances between different asperities are in parallel, it leads to a reduction in overall contact resistance.

6. Conclusions

Based on three-dimensional fractal theory, a novel contact resistance model for the spherical–planar joint interface was established. This model more accurately describes the evolution mechanism of contact resistance under loading conditions. The main conclusions are summarized as follows:

(1)

A contact resistance model based on three-dimensional fractal theory for the planar joint interface was proposed. In this model, based on three-dimensional fractal theory, the generation of rough surfaces at microscopic scale was achieved. Combining contact mechanics and asperity distribution, a three-dimensional fractal model of contact resistance considering the elastic, elastic–plastic, and plastic deformations in asperities was established.

(2)

Combining the contact resistance model for the planar joint interface with the surface contact coefficient, a novel contact resistance model for the spherical–planar joint interface was constructed. By introducing the surface contact coefficient, cross-scale coupling between the macro-geometric sphere and the micro-surface topography was achieved, thereby establishing the contact resistance model for the spherical–planar joint interface.

(3)

Experimental verification and simulation analysis was performed on the proposed model. It is worth noting that the model proposed in this paper aligned more closely with the experimental results. Furthermore, the influence mechanism of macro-geometric configuration and micro-surface topography on contact resistance was revealed. The proposed model provides important theoretical and practical value for improving electrical contact performance, optimizing electronic device design, and ensuring the stable operation of power systems.

However, the proposed model in this paper involves the exploration of the influence of factors such as macroscopic configuration and microscopic topography on contact resistance. In practical applications, the resistive heat generated when current passes through the contacts during relay contact has an impact on the contact resistance. The influence of temperature factors on spherical–planar joint interface contact resistance will be further explored in subsequent work.