Skip to main content
SpringerLink
Log in

Simple geometrical explanation of Gurtin-Murdoch model of surface elasticity with clarification of its related versions

  • Research Paper
  • Published:
Science China Physics, Mechanics and Astronomy Aims and scope Submit manuscript

Abstract

It is showed that all equations of the linearized Gurtin-Murdoch model of surface elasticity can be derived, in a straightforward way, from a simple second-order expression for the ratio of deformed surface area to initial surface area. This elementary derivation offers a simple explanation for all unique features of the model and its simplified/modified versions, and helps to clarify some misunderstandings of the model already occurring in the literature. Finally, it is demonstrated that, because the Gurtin-Murdoch model is based on a hybrid formulation combining linearized deformation of bulk material with 2nd-order finite deformation of the surface, caution is needed when the original form of this model is applied to bending deformation of thin-walled elastic structures with surface stress.

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. Gurtin M E, Murdoch A I. A continuum theory of elastic material surfaces. Arch Ratl Mech Anal, 1975, 57: 291–323

    Article  MATH  MathSciNet  Google Scholar 

  2. Grutin M E, Murdoch A I. Effect of surface stress on wave propagation in solids. J Appl Phys, 1976, 47: 4414–4421

    Article  ADS  Google Scholar 

  3. Gurtin M E, Markenscoff X, Thurston R N. Effect of surface stress on the natural frequency of thin crystals. Appl Phys Lett, 1976, 29: 529–530

    Article  ADS  Google Scholar 

  4. Gurtin M E, Murdoch A I. Surface stress in solids. Int J Solids Struct, 1978, 14: 431–440

    Article  MATH  Google Scholar 

  5. Mogilevskaya S G, Crouch S L, Stolarski H K. Multiple interacting circular nano-inhomogeneities with surface/interface effects. J Mech Phys Solids, 2008, 56: 2298–2327

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Sharma P, Ganti S, Bhate N. Effect of surface on the size-dependent elastic state of nano-inhomogeneties. Appl Phys Lett, 2003, 82: 535–537

    Article  ADS  Google Scholar 

  7. Yang F Q. Size-dependent effective modulus of elastic composite materials. J Appl Phys, 2004, 95: 3516–3520

    Article  ADS  Google Scholar 

  8. Duan H L, Wang J, Huang Z P, et al. Size-dependenct effective elastic constants of solids containing nano-inhomogeneities with interface stress. J Mech Phys Solids, 2005, 53: 1574–1596

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Huang Z P, Wang J. A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mech, 2006, 182: 195–210

    Article  MATH  Google Scholar 

  10. He L H, Li Z R. Impact of surface stress on stress concentration. Int J Solids Struct, 2006, 43: 6208–6219

    Article  MATH  Google Scholar 

  11. Lim C W, Li Z R, He L H. Size dependent, non-uniform elastic field inside a nano-scale spherical inclusion due to interface stress. Int J Solids Struct, 2006, 43: 5055–5065

    Article  MATH  Google Scholar 

  12. Chen T, Chiu M S, Weng C N. Derivation of the generalized Young-Laplace equation of curved interface in nanoscaled solids. J Appl Phys, 2006, 100: 074308

    Article  ADS  Google Scholar 

  13. Wang G F, Feng X Q. Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl Phys Lett, 2007, 90: 231904

    Article  ADS  Google Scholar 

  14. Guo J G, Zhao Y P. The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnol, 2007, 18: 295701

    Article  Google Scholar 

  15. Tian L, Rajapakse R K N D. Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity. J Appl Mech, 2007, 74: 568–574

    Article  MATH  Google Scholar 

  16. Sharma P, Wheeler L T. Size-dependent elastic state of ellipsoical nano-inclusions incorporating surface/interface tension. J Appl Mech, 2007, 74: 447–454

    Article  MATH  MathSciNet  Google Scholar 

  17. Lachut M J, Sader J E. Effect of surface stress on the stiffness of cantilever plates. Phys Rev Lett, 2007, 99: 206102

    Article  ADS  Google Scholar 

  18. Quang H L, He Q C. Variational principles and bounds for elastic inhomogeneous materials with coherent imperfect interfaces. Mech Mater, 2008, 40: 865–884

    Article  Google Scholar 

  19. Li Q, Chen Y H. Surface effect and size dependence on the energy release due to a nanosized hole expansion in plane elastic materials. J Appl Mech, 2008, 75: 061008

    Article  Google Scholar 

  20. Kim C I, Schiavone P, Ru C Q. The effects of surface elasticity on an elastic solid with mode-III crack: Complete solution. J Appl Mech, 2010, 77: 021011

    Article  Google Scholar 

  21. Steigmann D J, Ogden R W. Plane deformation of elastic solids with intrinsic boundary elasticity. Proc R Soc London Ser A, 1997, 453: 853–877

    Article  MATH  MathSciNet  Google Scholar 

  22. Schiavone P, Ru C Q. Integral equation methods in plane strain elasticity with boundary reinforcement. Proc R Soc London Ser A, 1998, 454: 2223–2242

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Benveniste Y, Miloh T. Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech Mater, 2001, 33: 309–323

    Article  Google Scholar 

  24. Cahn J W, Larche F. Surface stress and the chemical equilibrium of small crystals. Acta Metal, 1982, 30: 51–56

    Article  Google Scholar 

  25. Nix W D, Gao H J. An atomistic interpretation of interface stress. Scripta Mater, 1998, 39: 1653–1661

    Article  Google Scholar 

  26. Cammarata R C, Sieradzki K, Spaepen F. Simple model for interface stress. J Appl Phys, 2000, 87: 1227–1234

    Article  ADS  Google Scholar 

  27. Benveniste Y. A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J Mech Phys Solids, 2006, 54: 708–734

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Van Bladel J G. Electromagnetic Fields. 2nd ed. New Jersey: John Wiley & Sons Inc., 2007

    Google Scholar 

  29. Ogden R W. Nonlinear Elastic Deformation. New York: Dover Publications, Inc., 1984

    Google Scholar 

  30. Wang Z Q, Zhao Y P, Huang Z P. The effects of surface tension on the elastic properties of nano structures. Int J Eng Sci, 2010, 48: 140–150

    Article  Google Scholar 

  31. Wang Z Q, Zhao Y P. Self-instability and bending behaviors of nano plates. Acta Mech Solida Sinica, 2009, 22: 630–643

    Google Scholar 

  32. He J, Lilley C M. Surface stress effect on bending resonance of nanowires with different boundary conditions. Appl Phys Lett, 2008, 93: 263108

    Article  ADS  Google Scholar 

  33. Park H S, Klein P A. Surface stress effects on resonant properties of metal nanowires: The importance of finite deformation kinematics and the impact of residual surface stress. J Mech Phys Solids, 2008, 56: 3144–3166

    Article  MATH  ADS  Google Scholar 

  34. Gavan K B, Westra H J R, et al. Size-dependent effective Young’s modulus of silicon nitride cantilevers. Appl Phys Lett, 2009, 94: 233108

    Article  ADS  Google Scholar 

  35. Wang G F, Feng X Q. Surface effects on buckling of nanowires under uniaxial compression. Appl Phys Lett, 2009, 94: 141913

    Article  ADS  Google Scholar 

  36. Ru C Q. Size effect of dissipative surface stress on quality factor of microbeams. Appl Phys Lett, 2009, 94: 051905

    Article  ADS  Google Scholar 

  37. Cuenot S, Fretigny C, Champagne S D, et al. Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy. Phys Rev B, 2004, 69: 165410

    Article  ADS  Google Scholar 

  38. Jing G Y, Duan H L, Sun X M, et al. Surface effects on elastic properties of silver nanowires: Contact atomic-force microscopy. Phys Rev B, 2006, 73: 235409

    Article  ADS  Google Scholar 

  39. Yun G, Park H S. Surface stress effects on the bending properties of fcc metal nanowires. Phys Rev B, 2009, 79: 195421

    Article  ADS  Google Scholar 

  40. Rao B N, Rao G V. Large amplitude vibration of clamped-free and free-free uniform beams. J Sound Vib, 1989, 134: 353–358

    Article  ADS  Google Scholar 

  41. Xie W C, Lee H P, Lim S P. Normal modes of a nonlinear clamped-clamped beam. J Sound Vib, 2002, 250: 339–349

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada, T6G 2G8

    C. Q. Ru

Authors
  1. C. Q. Ru
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Q. Ru.

Additional information

Contributed by Ru C. Q. (RU ChongQing)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ru, C.Q. Simple geometrical explanation of Gurtin-Murdoch model of surface elasticity with clarification of its related versions. Sci. China Phys. Mech. Astron. 53, 536–544 (2010). https://doi.org/10.1007/s11433-010-0144-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11433-010-0144-8

Keywords

Search

Navigation