Abstract
At present, there are many researches on the loading between contact surfaces, but it is also very important to predict the unloading process between rough surfaces. In this paper, based on the threedimensional fractal loading model we built earlier and the unloading finite element results of the single asperity by Etsion et al., the unloading fractal prediction model between rough contact surfaces is established. The model takes into account the friction factor between surfaces, the threedimensional fractal characteristics of rough surfaces and the elastic–plastic deformation mechanism of asperities. The model is compared with the unloading model based on the twodimensional fractal curve established by Miao et al. and the finite element results. The results show that the unloading process between rough surfaces depends on the final loading state of the surface; the loading and unloading curves form a hysteresis loop, which represents the energy dissipation in a contact cycle; both the friction factor and fractal dimension of the contact surface have important effects on the unloading model.
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Abbreviations
 \(f_{{{\text{eun}}}}\) :

Contact load of asperities in elastic deformation stage during unloading
 \(a_{{{\text{eun}}}}\) :

Contact area of asperities in elastic deformation stage during unloading
 \(R\) :

Radius of curvature at the top of equivalent asperity
 \(E\) :

Equivalent elastic modulus of two interface materials connected
 \(\omega_{{{\text{eun}}}}\) :

Interference of asperity during unloading
 \(\omega_{{{\text{max}}}}\) :

Maximum interference of asperity under loading
 \(\omega\) :

Height of maximum asperity
 \(d\) :

Interference of contact surfaces
 \(\omega_{{{\text{res}}}}\) :

Residual interference during unloading
 \(H\) :

Hardness of softer material
 \(\sigma_{{\text{y}}}\) :

Yield strength of softer material
 \(\nu\) :

Poisson’s ratio
 \(f_{{{\text{epun}}}}\) :

Contact load of asperities in elastic–plastic deformation stage during unloading
 \(a_{{{\text{epun}}}}\) :

Contact area of asperities in elastic–plastic deformation stage during unloading
 \(f_{{{\text{ep}}}}\) :

Contact load of asperities in elastic–plastic deformation stage during loading
 \(D\) :

Threedimensional fractal dimension of surface topography
 \(G\) :

Fractal roughness of surface topography
 \(a_{{\text{e}}}\) :

Elastic critical deformation area of asperity under loading
 \(a_{{\text{L}}}\) :

Maximum contact area of asperity in rough surface
 \(a_{{\text{p}}}\) :

Plastic critical deformation area of asperity
 \(\gamma\) :

Frequency density of rough surface
 \(n\left( a \right)\) :

Area distribution function of threedimensional surface topography
 \(A_{{{\text{eun}}}}\) :

Contact area of contact surfaces in elastic deformation stage during unloading
 \(F_{{{\text{eun}}}}\) :

Contact load of contact surfaces in elastic deformation stage during unloading
 \(A_{{{\text{epun}}}}\) :

Contact area of contact surfaces in elastic–plastic deformation stage during unloading
 \(F_{{{\text{epun}}}}\) :

Contact load of contact surfaces in elastic–plastic deformation stage during unloading
 \(A_{{{\text{un}}}}\) :

Total contact area during unloading
 \(F_{{{\text{un}}}}\) :

Total contact load during unloading
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Acknowledgements
The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (NSFC) (Grant No. 51905354); Natural Science Foundation of Liaoning Province of China (Grant No. 2019BS186); Scientific Research Fund of Liaoning Education Department (Grant No. JYT19030); Postdoctoral fund of Northeastern University (Grant No. 20200201); Scientific Research Foundation of Shenyang Aerospace University (Grant No. 18YB56).
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Appendix
Appendix
The previous research on the loading model of contact surfaces mentioned in this paper can be seen in reference [21], in which the contact load and contact area during loading are shown as follows.
In the loading stage, the load \(F_{{\text{e}}}\) of all asperities in the elastic stage between rough surfaces is
In the loading stage, the load \(F_{{{\text{ep}}}}\) of all asperities in the elastic–plastic stage between rough surfaces is
In the loading stage, the load \(F_{{\text{p}}}\) of all asperities in the plastic stage between rough surfaces is
Therefore, the total load of rough surfaces is the sum of the load borne by asperities in the three deformation stages, which can be expressed as the following equation.
In the loading stage, the contact area \(A_{{{\text{re}}}}\) of all asperities in the elastic stage between rough surfaces is
In the loading stage, the contact area \(A_{{{\text{rep}}}}\) of all asperities in the elastic–plastic stage between rough surfaces is
In the loading stage, the contact area \(A_{{{\text{rp}}}}\) of all asperities in the plastic stage between rough surfaces is
Therefore, the total contact area of rough surfaces is the sum of the contact area borne by asperities in the three deformation stages, which can be expressed as the following equation.
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Pan, W., Song, C., Ling, L. et al. Unloading contact mechanics analysis of elastic–plastic fractal surface. Arch Appl Mech (2021). https://doi.org/10.1007/s00419021019180
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Keywords
 Contact mechanics
 Elastic–plastic deformation
 Unloading analysis
 Fractal surface