Investigation of the Sphere-Plane Contact Problem Under Micro-Scale Effects

MA Weiwei; WANG Yuxing; SHEN Huoming; LIU Juan

doi:10.21656/1000-0887.450260

Volume 46 Issue 12

Dec. 2025

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Article Navigation > Applied Mathematics and Mechanics > 2025 > 46(12): 1550-1559

MA Weiwei, WANG Yuxing, SHEN Huoming, LIU Juan. Investigation of the Sphere-Plane Contact Problem Under Micro-Scale Effects[J]. Applied Mathematics and Mechanics, 2025, 46(12): 1550-1559. doi: 10.21656/1000-0887.450260

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Investigation of the Sphere-Plane Contact Problem Under Micro-Scale Effects

doi: 10.21656/1000-0887.450260

MA Weiwei¹,
WANG Yuxing^{1,2
,
,},
SHEN Huoming^1,3,
LIU Juan¹

1. School of Mechanics and Aerospace Engineering, Southwest Jiaotong University, Chengdu 611756, P.R.China;
2. School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, P.R.China;
3. Chengdu University of Information Technology, Chengdu 610225, P.R.China

Funds:

The National Science Foundation of China(12402113)

Received Date: 2024-09-25
Rev Recd Date: 2024-12-06

Available Online: 2025-12-31

Abstract

Abstract

Based on the surface elasticity and couple stress theories, a 3D contact problem between a rigid spherical indenter and an elastic half-space at the micro and nano scale was studied. The size effects of materials were described with the characteristic material lengths of the surface and the material. Through combination of the surface elasticity theory and the couple stress theory, the frequency response function in the elastic half-space under normal loading was derived. Then, with the conjugate gradient method and the fast Fourier transform, a 3D semi-analytical contact model was established. The proposed model was then used to analyze the effects of surface residual stress, surface material constants, and the characteristic material lengths on the contact pressure, stress, and displacement on the contact surface. The results show that, compared with the classical contact solution, the contact pressure at the contact edge of the surface decrease sharply under the size effects. Additionally, the normal stress on the surface remains continuous and without abrupt changes at the contact edge, while the tangential stress is non-zero, and the normal displacement of the surface decreases. Furthermore, as the characteristic material length of the material increases, the direction of the tangential stress on the surface will reverse.
- contact problem,
- surface-elasticity theory,
- couple stress theory,
- size effect,
- conjugate gradient method

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References(21)

References

MA Q, CLARKE D R. Size dependent hardness of silver single crystals[J].Journal of Materials Research,1995,10(4): 853-863.

[2]GURTIN M E, MURDOCH A I. A continuum theory of elastic material surfaces[J].Archive for Rational Mechanics and Analysis,1975,57(4): 291-323.

[3]GURTIN M E, MURDOCH A I. Surface stress in solids[J].International Journal of Solids and Structures,1978,14(6): 431-440.

[4]COSSERAT E, COSSERAT F. Théorie des corps déformables[J].Nature,1909,81(2072): 67.

[5]MINDLIN R D. Influence of couple-stresses on stress concentrations[J].Experimental Mechanics,1963,3(1): 1-7.

[6]姜丽娟, 刘官厅, 高媛媛, 等. 磁电弹性材料含纳米尺度唇口次生两不对称裂纹的反平面问题[J]. 应用数学和力学, 2024,45(10): 1332-1344. (JIANG Lijuan, LIU Guanting, GAO Yuanyuan, et al. An antiplane problem of magnetoelectroelastic materials with nanoscale lip-shaped orifice with 2 asymmetric cracks[J].Applied Mathematics and Mechanics,2024,45(10): 1332-1344. (in Chinese))

[7]詹春晓, 李孝宝, 王美芹. 考虑表面效应的压电半导体梁的静态屈曲行为研究[J]. 应用数学和力学, 2024, 45(10): 1300-1312. （ZHAN Chunxiao, LI Xiaobao, WANG Meiqin. Static buckling behaviors of piezoelectric semiconductor beams with steigmann-ogden surface effects[J].Applied Mathematics and Mechanics,2024,45(10): 1300-1312. (in Chinese)）

[8]YUAN W, ZHANG J, NIU X, et al. Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity[J]. Applied Mathematical Modelling,2025,137: 115698.

[9]PINYOCHOTIWONG Y, RUNGAMORNRAT J, SENJUNTICHAI T. Rigid frictionless indentation on elastic half space with influence of surface stresses[J].International Journal of Engineering Science,2013,71: 15-35.

[10]SONG H X, KE L L, SU J, et al. Surface effect on the contact problem of a piezoelectric half-plane[J].International Journal of Solids and Structures,2020,185: 380-393.

[11]ZISIS T, GOURGIOTIS P A, BAXEVANAKIS K P, et al. Some basic contact problems in couple stress elasticity[J].International Journal of Solids and Structures,2014,51(11/12): 2084-2095.

[12]WANG Y, ZHANG X, SHEN H, et al. Three-dimensional contact analysis with couple stress elasticity[J].International Journal of Mechanical Sciences,2019,153: 369-379.

[13]WANG Y, SHEN H, LI J, et al. Three-dimensional frictional contact within the framework of couple stress elasticity[J].Applied Mathematical Modelling,2024,131: 288-305.

[14]LE T M, BUI T Q, LAWONGKERD J, et al. Frictionless contact on elastic half plane with influence of surface and couple stresses[J].Applied Mechanics and Materials,2020,897: 73-77.

[15]LE T M, WONGVIBOONSIN W, LAWONGKERD J, et al. Influence of surface and couple stresses on response of elastic substrate under tilted flat indenter[J].Applied Mathematical Modelling,2022,104: 644-65.

[16]WANG Y, ZHANG X, SHEN H, et al. Couple stress-based 3D contact of elastic films[J].International Journal of Solids and Structures,2020,191: 449-463.

[17]LIU S, WANG Q, LIU G. A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses[J].Wear,2000,243(1/2): 101-111.

[18]POLONSKY I A, KEER L M. A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques[J].Wear,1999,231(2): 206-219.

[19]杨阳, 王凯模, 沈火明, 等. 温差影响下的局部滑移接触行为的研究[J]. 应用数学和力学, 2023,44(2)：123-132. （YANG Yang, WANG Kaimo, SHEN Huoming, et al. Research on partial slip contact behaviors under temperature effects[J].Applied Mathematics and Mechanics,2023,44(2): 123-132. (in Chinese)）

[20]LAWONGKERD J, LE T M, KEAWSAWASVONG S, et al. Elastic solutions of axisymmetrically loaded half-space with surface and couple stress effects[J].Mechanics of Advanced Materials and Structures,2023,30(4): 835-55.

[21]JIA N, YAO Y, YANG Y, et al. Analysis of two-dimensional contact problems considering surface effect[J].International Journal of Solids and Structures, 2017,125: 172-183.

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