Skip to main content
SpringerLink
Log in

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper studies the contact vibration problem of an elastic half-space coated with functionally graded materials (FGMs) subject to a rigid spherical punch. A static force superimposing a dynamic time-harmonic force acts on the rigid spherical punch. Firstly, we give the static contact problem of FGMs by a least-square fitting approach. Next, the dynamic contact pressure is solved by employing the perturbation method. Lastly, the dynamic contact stiffness with different dynamic contact displacement conditions is derived for the FGM coated half-space. The effects of the gradient index, coating thickness, internal friction, and punch radius on the dynamic contact stiffness factor are discussed in detail.

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. SURESH, S. and MORTENSEN, A. Fundamentals of Functionally Graded Materials: Processing and Thermomechanical Behavior of Graded Metals and Metal-ceramic Composites, IOM Communications Ltd., London (1998)

    Google Scholar 

  2. LI, W. and HAN, B. Research and application of functionally gradient materials. IOP Conference Series: Materials Science and Engineering, 394, 1–7 (2018)

    Google Scholar 

  3. SURESH, S. Graded materials for resistance to contact deformation and damage. Science, 292, 2447–2451 (2001)

    Article  Google Scholar 

  4. LIU, T. J. and WANG, Y. S. Axisymmetric frictionless contact problem of a functionally graded coating with exponentially varying modulus. Acta Mechanica, 199, 151–165 (2008)

    Article  MATH  Google Scholar 

  5. VASILIEV, A. S., VOLKOV, S. S., BELOV, A. A., LITVINCHUK, S. Y., and AIZIKOVICH, S. M. Indentation of a hard transversely isotropic functionally graded coating by a conical indenter. International Journal of Engineering Science, 112, 63–75 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. PATRA, R., BARIK, S. P., and CHAUDHURI, P. K. Frictionless contact between a rigid indentor and a transversely isotropic functionally graded layer. International Journal of Applied Mechanics and Engineering, 23, 655–671 (2018)

    Article  Google Scholar 

  7. GULER, M. A. and ERDOGAN, F. The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings. International Journal of Mechanical Sciences, 49, 161–182 (2007)

    Article  Google Scholar 

  8. CHOI, H. J. On the plane contact problem of a functionally graded elastic layer loaded by a frictional sliding flat punch. Journal of Mechanical Science and Technology, 23, 2703–2713 (2009)

    Article  Google Scholar 

  9. CHEN, P. J., CHEN, S. H., and PENG, J. Sliding contact between a cylindrical punch and a graded half-plane with an arbitrary gradient direction. Journal of Applied Mechanics, 82, 041008 (2015)

    Article  Google Scholar 

  10. HU, Y., ZHOU, L., DING, H. H., TAN, G. X., LEWIS, R., LIU, Q. Y., GUO, J., and WANG, W. J. Investigation on wear and rolling contact fatigue of wheel-rail materials under various wheel/rail hardness ratio and creepage conditions. Tribology International, 143, 106091 (2020)

    Article  Google Scholar 

  11. LORENZ, S. J., SADEGHI, F., TRIVEDI, H. K., ROSADO, L., KIRSCH, M. S., and WANG, C. A continuum damage mechanics finite element model for investigating effects of surface roughness on rolling contact fatigue. International Journal of Fatigue, 143, 105986 (2021)

    Article  Google Scholar 

  12. FU, P. L., ZHAO, J. Z., ZHANG, X., KANG, G. Z., WANG, P., and KAN, Q. H. Elastic shakedown analysis of two-dimensional thermo-elastic rolling/sliding contact for a functionally graded coating/substrate structure with arbitrarily varying thermo-elastic properties. Composite Structures, 280, 114891 (2022)

    Article  Google Scholar 

  13. WANG, Z., YU, C., and WANG, Q. An efficient method for solving three-dimensional fretting contact problems involving multilayered or functionally graded materials. International Journal of Solids and Structures, 66, 46–61 (2015)

    Article  Google Scholar 

  14. EL-BORGI, S., USMAN, S., and GÜLER, M. A. A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate. International Journal of Solids and Structures, 51, 4462–4476 (2014)

    Article  Google Scholar 

  15. YAN, J. and MI, C. W. A receding contact analysis for an elastic layer reinforced with a functionally graded coating and pressed against a half-plane. Journal of Mechanical Science and Technology, 33, 4331–4344 (2019)

    Article  Google Scholar 

  16. YAYLACI, M., ADIYAMAN, G., ONER, E., and BIRINCI, A. Investigation of continuous and discontinuous contact cases in the contact mechanics of graded materials using analytical method and FEM. Computers and Concrete, 27, 199–210 (2021)

    Google Scholar 

  17. JIN, F., GUO, X., and GAO, H. J. Adhesive contact on power-law graded elastic solids: the JKR-DMT transition using a double-Hertz model. Journal of the Mechanics and Physics of Solids, 61, 2473–2492 (2013)

    Article  MathSciNet  Google Scholar 

  18. CHIDLOW, S. J., CHONG, W. W. F., and TEODORESCU, M. On the two-dimensional solution of both adhesive and non-adhesive contact problems involving functionally graded materials. European Journal of Mechanics-A/Solids, 39, 86–103 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. KUDISH, I. I., VOLKOV, S. S., VASILIEV, A. S., and AIZIKOVICH, S. M. Lubricated point heavily loaded contacts of functionally graded materials, part 2, lubricated contacts. Mathematics and Mechanics of Solids, 23, 1081–1103 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. KANETA, M., MATSUDA, K., and NISHIKAWA, H. Effects of thermal properties of contact materials and slide-roll ratio in elastohydrodynamic lubrication. Journal of Tribology, 144, 061603 (2022)

    Article  Google Scholar 

  21. NAYAK, P. R. Contact vibrations. Journal of Sound and Vibration, 22, 297–322 (1972)

    Article  MATH  Google Scholar 

  22. SEIMOV, V. M. Application of the orthogonal polynomial method to dynamic contact problems. Soviet Applied Mechanics, 8, 52–58 (1972)

    Article  Google Scholar 

  23. BROCK, L. M. Frictionless indentation by an elastic punch: a dynamic Hertzian contact problem. Journal of Elasticity, 8, 381–392 (1978)

    Article  MATH  Google Scholar 

  24. BROCK, L. M. Dynamic analysis of non-symmetric problems for frictionless indentation and plane crack extension. Journal of Elasticity, 8, 273–283 (1978)

    Article  MATH  Google Scholar 

  25. BEDDING, R. J. and WILLIS, J. R. The dynamic indentation of an elastic half-space. Journal of Elasticity, 3, 289–309 (1973)

    Article  MathSciNet  Google Scholar 

  26. BEDDING, R. J. and WILLIS, J. R. High speed indentation of an elastic half-space by conical or wedge-shaped indentors. Journal of Elasticity, 6, 195–207 (1976)

    Article  MATH  Google Scholar 

  27. GEORGIADIS, H. G., BROCK, L. M., and RIGATOS, A. P. Dynamic indentation of an elastic half-plane by a rigid wedge: frictional and tangential-displacement effects. International Journal of Solids and Structures, 32, 3435–3450 (1995)

    Article  MATH  Google Scholar 

  28. SABOT, J., KREMPF, P., and JANOLIN, C. Non-linear vibrations of a sphere-plane contact excited by a normal load. Journal of Sound and Vibration, 214, 359–375 (1998)

    Article  Google Scholar 

  29. STREATOR, J. L. Dynamic contact of a rigid sphere with an elastic half-space: a numerical simulation. Journal of Tribology, 125, 25–32 (2003)

    Article  Google Scholar 

  30. MA, Q. L., KAHRAMAN, A., PERRET-LIAUDET, J., and RIGAUD, E. An investigation of steady-state dynamic response of a sphere-plane contact interface with contact loss. Journal of Applied Mechanics, 74, 249–255 (2007)

    Article  MATH  Google Scholar 

  31. ARGATOV, I. I. Slow vertical motions of an elliptic punch on an elastic half-space. International Journal of Engineering Science, 46, 711–724 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. ARGATOV, I. I. Slow vertical motions of a spherical indenter on an elastic half-space. The Quarterly Journal of Mechanics and Applied Mathematics, 65, 129–140 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. POPOV, M., POPOV, V. L., and POHRT, R. Relaxation damping in oscillating contacts. Scientific Reports, 5, 1–9 (2015)

    Article  Google Scholar 

  34. POPOV, M., POPOV, V. L., and POPOV, N. V. Reduction of friction by normal oscillations. I. Influence of contact stiffness. Friction, 5, 45–55 (2017)

    Article  Google Scholar 

  35. KORUK, H. Modelling small and large displacements of a sphere on an elastic half-space exposed to a dynamic force. European Journal of Physics, 42, 055006 (2021)

    Article  Google Scholar 

  36. GLUSHKOV, Y. V., GLUSHKOVA, N. V., and KIRILLOVA, Y. V. The dynamic contact problem for a circular punch adhering to an elastic layer. Journal of Applied Mathematics and Mechanics, 56, 675–679 (1992)

    Article  MATH  Google Scholar 

  37. YANG, J. and KOMVOPOULOS, K. Dynamic indentation of an elastic-plastic multi-layered medium by a rigid cylinder. Journal of Tribology, 126, 18–27 (2004)

    Article  Google Scholar 

  38. ESKANDARI-GHADI, M., PAK, R. Y. S., and ARDESHIR-BEHRESTAGHI, A. Transversely isotropic elastodynamic solution of a finite layer on an infinite subgrade under surface loads. Soil Dynamics and Earthquake Engineering, 28, 986–1003 (2008)

    Article  Google Scholar 

  39. ZHANG, P., LIU, J., LIN, G., and WANG, W. Y. Axisymmetric dynamic response of the multi-layered transversely isotropic medium. Soil Dynamics and Earthquake Engineering, 78, 1–18 (2015)

    Article  Google Scholar 

  40. WANG, X. M., KE, L. L., and WANG, Y. S. Dynamic response of a coated half-plane with hysteretic damping under a harmonic Hertz load. Acta Mechanica Solida Sinica, 33, 449–463 (2020)

    Article  Google Scholar 

  41. YU, H. Y. and LEE, S. Time-harmonic elastic singularities and oscillating indentation of layered solids. IMA Journal of Applied Mathematics, 85, 542–563 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  42. OGI, H., HIRAO, M., TADA, T., and TIAN, J. Y. Elastic-stiffness distribution on polycrystalline Cu studied by resonance ultrasound microscopy: Young’s modulus microscopy. Physical Review B, 73, 174107 (2006)

    Article  Google Scholar 

  43. TIAN, J. Y., OGI, H., TADA, T., and HIRAO, M. Young’s modulus mapping on SCS-6 SiCf/Ti-6Al-4V composite by electromagnetic-resonance-ultrasound microscopy. Journal of Applied Physics, 94, 6472–6476 (2003)

    Article  Google Scholar 

  44. TIAN, J. Y., OGI, H., TADA, T., and HIRAO, M. Vibration analysis on electromagnetic-resonance-ultrasound microscopy (ERUM) for determining localized elastic constants of solids. Journal of the Acoustical Society of America, 115, 630–636 (2004)

    Article  Google Scholar 

  45. LYU, X., KE, L. L., SU, J., and TIAN, J. Y. Axisymmetric contact vibration analysis of a rigid spherical punch on a piezoelectric half-space. International Journal of Solids and Structures, 210, 224–236 (2021)

    Google Scholar 

  46. LYU, X., SU, J., TIAN, J. Y., and KE, L. L. Dynamic contact response of an elastic sphere on a piezoelectric half-space. Applied Mathematical Modelling, 100, 16–32 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  47. O’SULLIVAN, T. C. and KING, R. B. Sliding contact stress field due to a spherical indenter on a layered elastic half-space. Journal of Tribology, 110, 235–240 (1988)

    Article  Google Scholar 

  48. OZTURK, M. and ERDOGAN, F. Axisymmetric crack problem in bonded materials with a graded interfacial region. International Journal of Solids and Structures, 33, 193–219 (1996)

    Article  MATH  Google Scholar 

  49. JOHNSON, K. L. Contact Mechanics, Cambridge University Press, Cambridge University (1985)

  50. ERDOGAN, F. and GUPTA, G. D. On the numerical solution of singular integral equations. Quarterly of Applied Mathematics, 29, 525–534 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  51. SUNG, T. Vibrations in semi-infinite solids due to periodic surface loading. Symposium on Dynamic Testing of Soils, 156, 35–68 (1954)

    Google Scholar 

  52. BYCROFT, G. Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on an elastic stratum. Philosophical Transactions of The Royal Society A: Mathematical Physical and Engineering Sciences, 248, 327–368 (1956)

    MathSciNet  MATH  Google Scholar 

  53. BARKAN, D. D. Dynamics of Bases and Foundations, McGraw-Hill Book Company, New York (1962)

    Google Scholar 

  54. TIAN, J. Y. Anisotropy influence of cubic solid on dynamic Hertzian contact stiffness for a vibrating rigid indenter. American Journal of Engineering and Applied Sciences, 3, 56–63 (2010)

    Article  Google Scholar 

Download references

Funding

Project supported by the National Natural Science Foundation of China (Nos. 11725207, 12021002, and 12072226)

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Tianjin University, Tianjin, 300350, China

    Xin Lyu, Liaoliang Ke & Jie Su

  2. National Institute of Natural Hazards, Ministry of Emergency Management of China, Beijing, 100085, China

    Jiayong Tian

Authors
  1. Xin Lyu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liaoliang Ke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiayong Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jie Su
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liaoliang Ke.

Additional information

Citation: LYU, X., KE, L. L., TIAN, J. Y., and SU, J. Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch. Applied Mathematics and Mechanics (English Edition), 43(8), 1187–1202 (2022) https://doi.org/10.1007/s10483-022-2885-7

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lyu, X., Ke, L., Tian, J. et al. Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch. Appl. Math. Mech.-Engl. Ed. 43, 1187–1202 (2022). https://doi.org/10.1007/s10483-022-2885-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-022-2885-7

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation