Abstract
The purpose of this paper is to present a fractal model of normal contact stiffness of joint surfaces based on elliptical asperity. The effects of normal contact load, fractal dimension, fractal roughness and elliptical eccentricity on the normal contact stiffness of the joint surfaces were investigated by numerical simulations. This article equates the shape of the asperity on the joint surface to a more practical ellipse, including the elastic, elastic-plastic, and fully plastic deformation stages of the asperity. Additionally, a fractal model of the normal contact stiffness of the joint surface is established to investigate the influence of the relevant parameters of the joint surface on the normal stiffness. The results indicate that there is a nonlinear relationship between the dimensionless normal contact stiffness and the normal load when D < 2.4. When D > 2.4, the normal contact stiffness is approximately linearly related to the normal load, and as it increases, the linear relationship becomes increasingly apparent. When 2.1 ≤ D ≤ 2.6, the normal contact stiffness increases with increasing fractal dimension; when 2.6 ≤ D ≤ 2.9, it decreases as the fractal dimension increases. The normal contact stiffness of the joint surface increases with increasing contact load, decreases with increasing fractal roughness, and increases with the increase of eccentricity. Comparison of the simulation results between the theoretical model and the experimental model in this paper shows that the theoretical stiffness of the proposed model in this paper is greater than that of the experimental results, and the maximum error between them is 18.7%.This verifies the correctness and validity of the present model, which can better predict the normal contact stiffness of the fixed joint surface, and also demonstrates that the shape of the asperity on the rough surface has an important influence on the normal contact stiffness of the rough surfaces.
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Funding
The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (NSFC) (grant no. 52375113); Natural Science Foundation of Liaoning Province of China (grant no. 2022-MS-298); Shenyang Youth Science and Technology Innovation Talent Fund (grant no. RC230309); Scientific Research Fund of Liaoning Education Department (grant no. LJKMZ20220531).
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ABBREVIATIONS AND NOTATION
D fractal dimension of the rough surface |
G fractal roughness of the rough surface |
\(\gamma \) controllable frequency density |
L sample length |
M number of surface ridges |
\({{\phi }_{{mn}}}\) random phase |
n frequency index |
Ls minimum truncation length |
\({{n}_{{\max }}}\) upper limit of the frequency index |
F normal contact load applied to the asperity |
\(\delta \) deformation of the elliptical asperity |
la major radius of the elliptical asperity |
lb minor radius of the elliptical asperity |
l contact length |
Ra first effective curvature radius of the ellipse |
Rb second effective curvature radius of the ellipse |
Rm effective radius of curvature |
a contact area of the asperity |
e eccentricity of the ellipse |
ae contact surface area of the asperity in the elastic deformation stage |
\(\Delta {{F}_{e}}\) normal contact load of the asperity in the elastic deformation stage |
E effective modulus of elasticity |
\({{E}_{1}},{{E}_{2}}\) modulus of elasticity of the two rough surface contact materials |
\({{{v}}_{1}},{{{v}}_{2}}\) Poisson’s ratios of the two rough surface contact materials |
\({{f}_{1}}\left( e \right),{{f}_{2}}\left( e \right)\) functions of eccentricity |
\(K\left( e \right),E\left( e \right)\) elliptic integrals of the first and second kind |
\({{P}_{{\max }}}\) maximum contact pressure of the elliptical contact surface |
H hardness of the material |
\({{\sigma }_{y}}\) yield strength of softer materials |
\(\phi \) material property parameters |
\(K\) hardness coefficient |
\({v}\) Poisson’s ratio of softer materials |
\({{\delta }_{c}}\) critical deformation of plastic deformation of the elliptical asperity |
\({{a}_{c}}\) critical contact area of the elliptical asperity from elastic to elastoplastic deformation stage |
\({{a}_{{{\text{cp1}}}}},{{a}_{{{\text{cp2}}}}}\) critical elastic-plastic deformation areas in the first and second deformation stage |
\({{a}_{{ep1}}},{{a}_{{ep2}}}\) contact area in the first and second stage of elastic – plasticity |
\(\Delta {{F}_{{ec}}}\) critical contact load of the asperity from elastic to elastoplastic deformation stage |
\(\Delta {{F}_{{ep1}}},\Delta {{F}_{{ep2}}}\) contact load of the asperity in the first and second stage of elastic – plasticity |
\({{a}_{p}}\) contact area of the asperity in full plastic deformation stage |
\(\Delta {{F}_{p}}\) contact load of the asperity in full plastic deformation stage |
\({{a}_{L}}\) maximum contact area of the asperity |
\({{A}_{r}}\) total contact area of the joint surface |
\({{F}_{e}}\) normal contact load of the joint surface in the elastic deformation stage |
\({{F}_{{{\text{ep1}}}}},{{F}_{{{\text{ep2}}}}}\) normal contact load of the joint surface in the first and second elastic-plastic deformation stage |
\({{F}_{p}}\) normal contact load of the joint surface in full plastic deformation stage |
F total normal contact load of the joint surface |
\({{A}_{a}}\) nominal contact area of the joint surface |
\({{K}_{{ne}}}\) normal contact stiffness of the asperity in the elastic deformation stage |
\({{K}_{{{\text{nep1}}}}},{{K}_{{{\text{nep2}}}}}\) normal contact stiffness of the asperity in the first and second elastic – plastic stage |
\({{K}_{{np}}}\) normal contact stiffness of the asperity in fully plastic deformation stage |
\({{K}_{{re}}}\) normal contact stiffness of the joint surface in the elastic deformation stage |
\({{K}_{{{\text{rep1}}}}},{{K}_{{{\text{rep2}}}}}\) normal contact stiffness of the joint surface in the first and second stage of elastic-plasticity |
Kr overall normal contact stiffness of the joint surface |
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Pan, W., Guo, J., Li, X. et al. Contact Mechanics Analysis of Fractal Surfaces Considering Elliptical Asperity with Elliptical Geometry. Mech. Solids 59, 311–330 (2024). https://doi.org/10.1134/S002565442460274X
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DOI: https://doi.org/10.1134/S002565442460274X