Skip to main content
SpringerLink
Log in

Surface effects on the indentation of a soft layer on a rigid substrate with an elliptical cylinder indenter

软材料薄层-刚性基底系统在椭圆柱压痕中的表面效应

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

Surface effects often play a significant role in the mechanical properties of soft materials such as hydrogels and biological tissues. In this paper, we investigate the plane-strain indentation of a soft elastic layer bonded to a rigid substrate. The surface effects on the indentation behavior of the elastic layer-substrate system are theoretically analyzed. Indentation tests using indenters with different elliptical shapes are compared. Analytical expressions are derived for the indentation force-displacement relation using the Kerr model with the effect of surface tension. The theoretical solution is verified by finite element simulations. The dependence of surface effects on the ratio of the indenter’s major and minor elliptical axes is also examined. This work helps understand the size effects on the indentation behaviors of soft materials and guides the design of corresponding measurement tests.

摘要

表面效应对软材料(如水凝胶和生物组织)的力学性能有重要影响. 本文研究了软材料薄层-刚性基底系统在椭圆柱压头下的 平面应变压痕问题,理论分析了表面效应对压痕过程的影响, 考虑了不同椭圆轴比参数下的压痕行为的差异. 采用具有表面张力 影响的弹性层的Kerr模型, 获得了压痕响应中力和位移的解析表达式, 并通过有限元模拟验证了理论解. 基于该理论解,分析了椭 圆压头横截面的长轴与短轴之比对表面效应的影响规律. 研究结果不仅有助于理解尺寸效应对软材料压痕行为的影响, 而且对相 关的力学性能测试具有指导价值.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Hay, Introduction to instrumented indentation testing, Exp. Tech. 33, 66 (2007).

    Article  Google Scholar 

  2. M. Khoeini, A. Kolahi, and S. Hesaraki, Mechanical properties, bioactivity and cell behavior of barium-containing calcium-phosphoalumino-borosilicate glass, Ceramics Int. 48, 7643 (2022).

    Article  Google Scholar 

  3. M. G. Zhang, Y. P. Cao, G. Y. Li, and X. Q. Feng, Spherical indentation method for determining the constitutive parameters of hyperelastic soft materials, Biomech. Model. Mechanobiol. 13, 1 (2014).

    Article  Google Scholar 

  4. H. Lu, S. Qu, and X. Feng, Preface: Mechanics of soft materials and flexible structures, Acta Mech. Sin. 37, 746 (2021).

    Article  Google Scholar 

  5. Y. Lai, and Y. Hu, The relation between adhesion properties and network properties of hydrogels: A study based on an indentation adhesion method, Mech. Mater. 159, 103877 (2021).

    Article  Google Scholar 

  6. W. Jiang, X. Zhu, J. Zhou, J. Liu, J. Liu, and W. Huang, The indentation responses of composite Y-type cores sandwich structure under various temperatures, Polym. Compos. 43, 533 (2022).

    Article  Google Scholar 

  7. Y. Zhang, X. Yang, and C. Xiong, Mechanical characterization of soft silicone gels via spherical nanoindentation for applications in mechanobiology, Acta Mech. Sin. 37, 554 (2021).

    Article  Google Scholar 

  8. M. G. Zhang, J. Chen, X. Q. Feng, and Y. Cao, On the applicability of Sneddon’s solution for interpreting the indentation of nonlinear elastic biopolymers, J. Appl. Mech. 81, 091011 (2014).

    Article  Google Scholar 

  9. E. Migliorini, E. A. Cavalcanti-Adam, A. E. Uva, M. Fiorentino, M. Gattullo, V. M. Manghisi, L. Vaiani, and A. Boccaccio, Nanoindentation of mesenchymal stem cells using atomic force microscopy: Effect of adhesive cell-substrate structures, Nanotechnology 32, 215706 (2021).

    Article  Google Scholar 

  10. S. Marcotti, G. C. Reilly, and D. Lacroix, Effect of cell sample size in atomic force microscopy nanoindentation, J. Mech. Behav. Biomed. Mater. 94, 259 (2019).

    Article  Google Scholar 

  11. Z. Wang, X. Zhong, L. Jiang, F. Qi, X. Ouyang, J. Wang, B. Liao, and J. Luo, Effect of interfacial delamination on coating crack in thick diamond-like carbon coatings under indentation, Acta Mech. Sin. 36, 524 (2020).

    Article  MathSciNet  Google Scholar 

  12. A. O. Olofinjana, J. M. Bell, and A. K. Jämting, Evaluation of the mechanical properties of sol-gel-deposited titania films using ultramicro-indentation method, Wear 241, 174 (2000).

    Article  Google Scholar 

  13. W. Wen, A. A. Becker, and W. Sun, Determination of material properties of thin films and coatings using indentation tests: A review, J. Mater. Sci. 52, 12553 (2017).

    Article  Google Scholar 

  14. H. Hertz, On the contact of elastic solids, J. Reine. Angew. Math. 92, 156 (1881).

    MATH  Google Scholar 

  15. M. J. Boussinesq, Application des potentiels à l’étude de l’éuilibre et du mouvement des solides élastiques (Gauthier-Villars, Paris, 1885).

    MATH  Google Scholar 

  16. I. N. Sneddon, The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile, Int. J. Eng. Sci. 3, 47 (1965).

    Article  MathSciNet  Google Scholar 

  17. X. Yin, and K. Komvopoulos, A discrete dislocation plasticity analysis of a single-crystal half-space indented by a rigid cylinder, J. Appl. Mech. 78, 041019 (2011).

    Article  Google Scholar 

  18. J. Su, L. L. Ke, and Y. S. Wang, Two-dimensional fretting contact analysis of piezoelectric materials, Int. J. Solids Struct. 73–74, 41 (2015).

    Article  Google Scholar 

  19. S. H. Chen, L. Liu, and T. C. Wang, Smallscale, grain size and substrate effects in nano-indentation experiment of film-substrate systems, Int. J. Solids Struct. 44, 4492 (2007).

    Article  Google Scholar 

  20. M. J. Jaffar, A numerical solution for axisymmetric contact problems involving rigid indenters on elastic layers, J. Mech. Phys. Solids 36, 401 (1988).

    Article  Google Scholar 

  21. P. Chen, S. Chen, and Z. Peng, Thermo-contact mechanics of a rigid cylindrical punch sliding on a finite graded layer, Acta Mech. 223, 2647 (2012).

    Article  MathSciNet  Google Scholar 

  22. P. Li, and T. J. Liu, The size effect in adhesive contact on a gradient nanostructured coating, Nanotechnology 32, 235704 (2021).

    Article  Google Scholar 

  23. R. W. Style, R. Boltyanskiy, Y. Che, J. S. Wettlaufer, L. A. Wilen, and E. R. Dufresne, Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses, Phys. Rev. Lett. 110, 066103 (2013).

    Article  Google Scholar 

  24. M. E. Gurtin, and A. Ian Murdoch, A continuum theory of elastic material surfaces, Arch. Rational Mech. Anal. 57, 291 (1975).

    Article  MathSciNet  Google Scholar 

  25. P. Shi, Imperfect interface effect for nano-composites accounting for fiber section shape under antiplane shear, Appl. Math. Model. 43, 393 (2017).

    Article  MathSciNet  Google Scholar 

  26. J. M. Long, G. F. Wang, X. Q. Feng, and S. W. Yu, Two-dimensional Hertzian contact problem with surface tension, Int. J. Solids Struct. 49, 1588 (2012).

    Article  Google Scholar 

  27. H. L. Duan, J. Wang, Z. P. Huang, and Z. Y. Luo, Stress concentration tensors of inhomogeneities with interface effects, Mech. Mater. 37, 723 (2005).

    Article  Google Scholar 

  28. G. F. Wang, and X. Q. Feng, Effects of surface stresses on contact problems at nanoscale, J. Appl. Phys. 101, 013510 (2007).

    Article  Google Scholar 

  29. D. X. Lei, L. Y. Wang, and Z. Y. Ou, Elastic analysis for nanocontact problem with surface stress effects under shear load, J. Nanomaterials 2012, 1 (2012).

    Article  Google Scholar 

  30. J. Long, Y. Ding, and G. Wang, Contact problems at micro/nano scale with surface tension, Procedia IUTAM 21, 40 (2017).

    Article  Google Scholar 

  31. A. D. Kerr, On the formal development of elastic foundation models, Ingenieur-Archiv 54, 455 (1984).

    Article  Google Scholar 

  32. M. Li, C. Q. Ru, and C. F. Gao, An alternative method for indentation of an elastic thin beam by a rigid indenter, Int. J. Mech. Sci. 149, 508 (2018).

    Article  Google Scholar 

  33. M. Li, C. F. Gao, and C. Q. Ru, Asymmetric indentation of an elastic beam by a rigid cylinder, Z. Angew. Math. Phys. 69, 93 (2018).

    Article  MathSciNet  Google Scholar 

  34. M. Li, C. Q. Ru, and C. F. Gao, Axisymmetric indentation of an elastic thin plate by a rigid sphere revisited, Z. Angew. Math. Mech. 98, 1436 (2018).

    Article  MathSciNet  Google Scholar 

  35. G. T. Wu, J. Wu, and L. Zhang, An analytical analysis of receding contact between a compressible elastic thin film and a rigid substrate due to spherical indentation, Z. Angew. Math. Phys. 70, 126 (2019).

    Article  MathSciNet  Google Scholar 

  36. M. Li, H. X. Zhang, Z. L. Zhao, and X. Q. Feng, Surface effects on cylindrical indentation of a soft layer on a rigid substrate, Acta Mech. Sin. 36, 422 (2020).

    Article  MathSciNet  Google Scholar 

  37. W. Yuan, and G. Wang, Cylindrical indentation of an elastic bonded layer with surface tension, Appl. Math. Model. 65, 597 (2019).

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11802225 and 11921002).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China

    Yan Li  (李艳), Xing Li  (李星), Pengpeng Shi  (时朋朋) & Shenghu Ding  (丁生虎)

  2. Institute of Biomechanics and Medical Engineering, Applied Mechanics Lab, Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

    Huanxin Zhang  (张焕新), Xiqiao Feng  (冯西桥) & Shenghu Ding  (丁生虎)

  3. School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Pengpeng Shi  (时朋朋)

Authors
  1. Yan Li  (李艳)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huanxin Zhang  (张焕新)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xing Li  (李星)
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Pengpeng Shi  (时朋朋)
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xiqiao Feng  (冯西桥)
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Shenghu Ding  (丁生虎)
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Pengpeng Shi and Xing Li organized the research. Xing Li and Xiqiao Feng provided resources. Pengpeng Shi and Shenghu Ding managed and coordinated the research. Pengpeng Shi and Xing Li designed the conceptualization. Yan Li and Pengpeng Shi developed the methodology, conducted the theoretical investigation, designed the supporting programs, and processed the research data. Huanxin Zhang completed the theoretical verification with finite element technology. Pengpeng Shi, Yan Li, Xiqiao Feng, and Shenghu Ding wrote the first draft of the manuscript. Pengpeng Shi, Yan Li, Xiqiao Feng, and Shenghu Ding revised and edited the final version. Pengpeng Shi and Xiqiao Feng acquired the financial support for the project leading to this publication.

Corresponding authors

Correspondence to Pengpeng Shi  (时朋朋) or Shenghu Ding  (丁生虎).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Zhang, H., Li, X. et al. Surface effects on the indentation of a soft layer on a rigid substrate with an elliptical cylinder indenter. Acta Mech. Sin. 38, 422098 (2022). https://doi.org/10.1007/s10409-022-22098-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10409-022-22098-x

Search

Navigation