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A modified elastic contact stiffness model considering the deformation of bulk substrate

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Establishing an accurate and effective stiffness model of the elastic contact surface is the basis for further modeling and analysis of machine tool dynamics. In this paper, a new elastic contact stiffness model is proposed that considers the bulk substrate deformation and modifies the Greenwood and Williamson microcontact model (GW model) of rough surfaces. Based on the Hertz contact theory and fixed-point iteration method, a single asperity contact model is created to analyze the effects of bulk substrate deformation and coating materials. To make the contact stiffness model more accurate and effective, two aspects are improved: One is to introduce the triangular distribution function to make the asperity heights distribution more consistent with the actual machined surface, and the other is to correct the defects in the micro-contact process. Comparing the finite element simulation results with the modal test data, the correctness of the proposed contact stiffness model is verified. The simulation results reveal the influence of distribution function, surface roughness and coating material on the contact characteristics of the joint surface.

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A0 :

Nominal contact area (mm2)

A :

Real contact area (mm2)

a 0 :

Contact area computed based on Hertz theory (mm2)

a :

The lower limit of the trigonometric distribution

b :

The upper limit of the trigonometric distribution

c :

The mode of the triangle distribution

D :

Fractal dimension

d :

Separation based on asperity heights (mm)

d n :

Dimensionless contact separation dlo)s

d :

Critical contact separation (mm)

d n :

Dimensionless critical contact separation d"lo)s

E :

Effective elastic modulus of joint surface (GPa)

E) a :

Elastic modulus of asperity (GPa)

E ae :

Effective elastic modulus of asperity (GPa)

E) b :

Elastic modulus of bulk substrate (GPa)

E) be :

Effective elastic modulus of bulk substrate (GPa)

F :

Normal load (N)

f :

Contact force computed based on Hertz theory (N)

G :

Fractal roughness factor (mm)

g(z)n :

Exponential distribution function

h(z) n :

Gaussian distribution function

h a :

Asperity height (urn)

K :

Overall contact stiffness (N/m)

K n :

Normal contact stiffness (N/m)

K j :

Tangential contact stiffness (N/m)

ka :

Stiffness of asperity (N/m)

kb :

Stiffness of bulk substrate (N/m)

kn :

Normal stiffness computed based on Hertz theory (N/m)

L :

Length of the sample (urn)

mo :

Zero-order moment spectra (jm)

m2 :

Second-order moment spectra

m4 :

Fourth-order moment spectra (urn2)

N :

Total number of asperities

n :

Number of asperities in contact

Po :

Maximum contact pressure (Pa)

prob :

Contact probability

R :

Curvature radius of asperity (urn)

r :

Radial distance from the center of contact (urn)

rb :

Radius of the contact area of substrate (urn)


Normal displacement of the contact surface


Normal applied displacement (urn)

δa :

Asperity deformation (urn)

δb :

Bulk substrate deformation (urn)


Asperity deformation in the range of 0<n<1 (urn)

v a :

Poisson’s ratio of asperity

V b :

Poisson’s ratio of bulk substrate

v :

Poisson’s ratio of joint surface

σs :

Standard deviation of the surface heights distribution (urn)

σ0 :

Standard deviation of the triangular distribution function


Asperity height measured from the mean of asperity heights (urn)

zn :

Dimensionless height of asperity zlo)s

k :

Ratio of the composite elastic modulus of the asperity to the bulk substrate


Geometrical parameter about the size of the asperity.


Density of asperity (urn-2)

μo :

Mean of the triangular distribution function


Triangle distribution function


Minimum sampling spacing of measuring instrument


Gamma function


Scale parameter of the spectral density


Power spectrum function

ωh :

The upper limit of the spectral bandwidth

ωi :

The lower limit of the spectral bandwidth


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This work was supported by National Natural Science Foundation of China Grant No. 51975449, Research Fund for Scientific Research Program Funded by Shaanxi Provincial Education Department Program No. 2018JM5066. The authors are grateful to other participants of the project for their cooperation.

Author information

Correspondence to Ling Li.

Additional information

Recommended by Editor Seungjae Min

Ling Li is a Professor of Mechanical and Electrical Engineering, Xi’an University of Architecture and Technology. He graduated from Beijing University of Technology with a Ph.D. in Mechanical Engineering. His research interests include machine dynamics, mechanical behavior of the bolted joints and interfaces of the critical structures, non-linear mechanics, and accurate design of machine tools.

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Li, L., Wang, J., Pei, X. et al. A modified elastic contact stiffness model considering the deformation of bulk substrate. J Mech Sci Technol 34, 777–790 (2020). https://doi.org/10.1007/s12206-020-0126-3

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  • Bulk substrate
  • Triangular distribution
  • Modified GW model
  • Contact stiffness