Skip to main content

Continuum Theory for Deformable Interfaces/Surfaces with Multi-field Coupling

  • Living reference work entry
  • First Online:
Handbook of Mechanics of Materials
  • 457 Accesses

Abstract

Continuum mechanics is a well-demonstrated powerful tool for dealing with comprehensive physical problems of macroscopic bodies involving deformation and flow processes. It has also been successfully and extensively applied to deformable bodies at micro- and nano-scales without or with appropriate modifications. One particular and interesting phenomenon of small-sized subjects is that they usually exhibit size-dependent characteristics, which cannot be well explained within the conventional framework of continuum mechanics. A modified version that accounts for the so-called surface elasticity has been developed. The rigorous derivation of surface elasticity is due to Gurtin and Murdoch as early in 1975 by creating a two-dimensional set of elasticity. This chapter aims to introduce a relatively traditional method to derive the interface/surface elasticity, which further incorporates the coupling among multiple physical fields (e.g., elastic, electric, and magnetic). The method is illustrated here by only considering the electroelastic coupling, but there is completely no difficulty to make a further step toward the situation where more fields are involved. The method first assumes a finite thickness of the interface/surface so that an interphase layer model is actually adopted, which is governed by the traditional three-dimensional theory of piezoelectricity. Then, the state-space formalism is derived, based on which a transfer relation between the state vectors at the upper and lower surfaces of the interphase layer can be easily obtained. By series expansion and truncation, various theories of interface/surface piezoelectricity of different orders can then be established. Comparison is made with those reported in the literature, which shows good agreement and hence validates the present approach.

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

Access this chapter

Institutional subscriptions

Similar content being viewed by others

References

  1. Mindlin RD. Polarization gradient in elastic dielectrics. Int J Solids Struct. 1968;4:637–42.

    Article  MATH  Google Scholar 

  2. Mindlin RD. Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Int J Solids Struct. 1969;5:1197–208.

    Article  Google Scholar 

  3. Cammarata RC. Surface and interface stress effects in thin films. Prog Surf Sci. 1994;46:1–38.

    Article  Google Scholar 

  4. Wang JX, Huang ZP, Duan HL, Yu SW, Feng XQ, Wang GF, Zhang WX, Wang TJ. Surface stress effect in mechanics of nanostructured materials. Acta Mech Solida Sin. 2011;24:52–82.

    Article  Google Scholar 

  5. Miller RE, Shenoy VB. Size-dependent elastic properties of nanosized structural elements. Nanotechnology. 2000;11:139–47.

    Article  Google Scholar 

  6. Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal. 1975;57:291–323.

    Article  MathSciNet  MATH  Google Scholar 

  7. Mindlin RD. High frequency vibrations of plated, crystal plates. In: Progress in applied mechanics. New York: MacMillan; 1963.

    Google Scholar 

  8. Tiersten HF. Elastic surface waves guided by thin films. J Appl Phys. 1969;40:770–89.

    Article  Google Scholar 

  9. Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct. 1978;14:431–40.

    Article  MATH  Google Scholar 

  10. Benveniste Y. A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J Mech Phys Solids. 2006;54:708–34.

    Article  MathSciNet  MATH  Google Scholar 

  11. Ting TCT. Mechanics of a thin anisotropic elastic layer and a layer that is bonded to an anisotropic elastic body or bodies. Proc R Soc A. 2007;463:2223–39.

    Article  MathSciNet  MATH  Google Scholar 

  12. Benveniste Y. An interface model for a three-dimensional curved thin piezoelectric interphase between two piezoelectric media. Math Mech Solids. 2009;14:102–22.

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen WQ. Surface effect on Bleustein–Gulyaev wave in a piezoelectric half-space. Theor Appl Mech Lett. 2011;1:041001.

    Article  Google Scholar 

  14. Chen WQ. Wave propagation in a piezoelectric plate with surface effect. In: Fang DN, Wang J, Chen WQ, editors. Analysis of piezoelectric structures and devices. Berlin/Boston: De Gruyter; 2103.

    Google Scholar 

  15. Chen WQ, Wu B, Zhang CL, Zhang C. On wave propagation in anisotropic elastic cylinders at nanoscale: surface elasticity and its effect. Acta Mech. 2014;225:2743–60.

    Article  MathSciNet  MATH  Google Scholar 

  16. Tiersten HF. Electroelastic interactions and the piezoelectric equations. J Acoust Soc Am. 1981;70:1567–76.

    Article  MATH  Google Scholar 

  17. Yang JS, Hu YT. Mechanics of electroelastic bodies under biasing fields. Appl Mech Rev. 2004;57:173–89.

    Article  Google Scholar 

  18. Yang JS. An introduction to the theory of piezoelectricity. New York: Springer; 2005.

    MATH  Google Scholar 

  19. Wu B, Zhang CL, Zhang C, Chen WQ. Theory of electroelasticity accounting for biasing fields: retrospect, comparison and perspective. Adv Mech. 2016;46:201604. (in Chinese)

    Google Scholar 

  20. Tiersten HF, Sinha BK, Meeker TR. Intrinsic stress in thin films deposited on anisotropic substrates and its influence on the natural frequencies of piezoelectric resonators. J Appl Phys. 1981;52:5614–24.

    Article  Google Scholar 

  21. Hoger A. On the determination of residual stress in an elastic body. J Elast. 1986;16:303–24.

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu B, Chen WQ, Zhang C. On free vibration of piezoelectric nanospheres with surface effect. Mech Adv Mater Struct. 2017. https://doi.org/10.1080/15376494.2017.1365986.

  23. Huang GY, Yu SW. Effect of surface piezoelectricity on the electromechanical behaviour of a piezoelectric ring. Phys Status Solidi B. 2006;243:22–4.

    Article  Google Scholar 

  24. Ding HJ, Chen WQ. Three dimensional problems of piezoelasticity. New York: Nova Science Publishers; 2001.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to W. Q. Chen .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Singapore Pte Ltd.

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Wu, B., Chen, W.Q. (2018). Continuum Theory for Deformable Interfaces/Surfaces with Multi-field Coupling. In: Schmauder, S., Chen, CS., Chawla, K., Chawla, N., Chen, W., Kagawa, Y. (eds) Handbook of Mechanics of Materials. Springer, Singapore. https://doi.org/10.1007/978-981-10-6855-3_27-1

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-6855-3_27-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-6855-3

  • Online ISBN: 978-981-10-6855-3

  • eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering

Publish with us

Policies and ethics