Surface Energy and Nanoscale Mechanics

  • Kosar Mozaffari
  • Shengyou Yang
  • Pradeep SharmaEmail author
Living reference work entry


The mechanical response of nanostructures, or materials with characteristic features at the nanoscale, differs from their coarser counterparts. An important physical reason for this size-dependent phenomenology is that surface or interface properties are different than those of the bulk material and acquire significant prominence due to an increased surface-to-volume ratio at the nanoscale. In this chapter, we provide an introductory tutorial on the continuum approach to incorporate the effect of surface energy, stress, and elasticity and address the size-dependent elastic response at the nanoscale. We present some simple illustrative examples that underscore both the physics underpinning the capillary phenomenon in solids as well as a guide to the use of the continuum theory of surface energy.



Support from the University of Houston and the M. D. Anderson Professorship is gratefully acknowledged.


  1. Altenbach H, Eremeyev VA, Morozov NF (2013) Mechanical properties of materials considering surface effects. In: IUTAM symposium on surface effects in the mechanics of nanomaterials and heterostructures. Springer, pp 105–115Google Scholar
  2. Biria A, Maleki M, Fried E (2013) Continuum theory for the edge of an open lipid bilayer. Adv Appl Mech 21:1–78Google Scholar
  3. Cammarata RC (2009) Generalized thermodynamics of surfaces with applications to small solid systems. Solid State Phys 61:1–75CrossRefGoogle Scholar
  4. Cammarata R, Sieradzki K, Spaepen F (2000) Simple model for interface stresses with application to misfit dislocation generation in epitaxial thin films. J Appl Phys 87(3):1227–1234ADSCrossRefGoogle Scholar
  5. Chatzigeorgiou G, Meraghni F, Javili A (2017) Generalized interfacial energy and size effects in composites. J Mech Phys Solids 106:257–282ADSMathSciNetCrossRefGoogle Scholar
  6. Chhapadia P, Mohammadi P, Sharma P (2011) Curvature-dependent surface energy and implications for nanostructures. J Mech Phys Solids 59(10):2103–2115ADSMathSciNetCrossRefGoogle Scholar
  7. Courant R, Hilbert D (1953) Methods of mathematical physics, vol I (First English ed.). Interscience Publishers, Inc., New YorkzbMATHGoogle Scholar
  8. Duan H, Wang Jx, Huang Z, Karihaloo BL (2005) Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J Mech Phys Solids 53(7): 1574–1596ADSMathSciNetCrossRefGoogle Scholar
  9. Duan H, Wang J, Karihaloo BL (2009) Theory of elasticity at the nanoscale. In: Advances in applied mechanics, vol 42. Elsevier, Amsterdam, pp 1–68Google Scholar
  10. Fischer F, Waitz T, Vollath D, Simha N (2008) On the role of surface energy and surface stress in phase-transforming nanoparticles. Prog Mater Sci 53(3):481–527CrossRefGoogle Scholar
  11. Fried E, Todres RE (2005) Mind the gap: the shape of the free surface of a rubber-like material in proximity to a rigid contactor. J Elast 80(1–3):97–151MathSciNetCrossRefGoogle Scholar
  12. de Gennes PG, Brochard-Wyart F, Quere D (2004) Capillarity and wetting phenomenon. Springer, New YorkCrossRefGoogle Scholar
  13. Gurtin ME, Murdoch AI (1975a) Addenda to our paper a continuum theory of elastic material surfaces. Arch Ration Mech Anal 59(4):389–390CrossRefGoogle Scholar
  14. Gurtin ME, Murdoch AI (1975b) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323MathSciNetCrossRefGoogle Scholar
  15. Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14(6):431–440CrossRefGoogle Scholar
  16. Gurtin M, Weissmüller J, Larche F (1998) A general theory of curved deformable interfaces in solids at equilibrium. Philos Mag A 78(5):1093–1109ADSCrossRefGoogle Scholar
  17. Gurtin ME, Fried E, Anand L (2010) The mechanics and thermodynamics of continua. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  18. Haiss W (2001) Surface stress of clean and adsorbate-covered solids. Rep Prog Phys 64(5):591ADSCrossRefGoogle Scholar
  19. Henann DL, Bertoldi K (2014) Modeling of elasto-capillary phenomena. Soft Matter 10(5): 709–717ADSCrossRefGoogle Scholar
  20. Huang Z, Sun L (2007) Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis. Acta Mech 190(1–4):151–163CrossRefGoogle Scholar
  21. Huang Z, Wang Jx (2006) A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mech 182(3–4):195–210CrossRefGoogle Scholar
  22. Huang Z, Wang J (2013) Micromechanics of nanocomposites with interface energy effect. Handbook of micromechanics and nanomechanics. Pan Stanford Publishing, SingaporeGoogle Scholar
  23. Ibach H (1997) The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures. Surf Sci Rep 29(5–6):195–263ADSCrossRefGoogle Scholar
  24. Javili A, McBride A, Steinmann P (2013) Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Appl Mech Rev 65(1):010802ADSCrossRefGoogle Scholar
  25. Javili A, Ottosen NS, Ristinmaa M, Mosler J (2017) Aspects of interface elasticity theory. Math Mech Solids 23:1081286517699041MathSciNetzbMATHGoogle Scholar
  26. Li S, Wang G (2008) Introduction to micromechanics and nanomechanics. World Scientific Publishing Company, SingaporeCrossRefGoogle Scholar
  27. Liu L, Yu M, Lin H, Foty R (2017) Deformation and relaxation of an incompressible viscoelastic body with surface viscoelasticity. J Mech Phys Solids 98:309–329ADSMathSciNetCrossRefGoogle Scholar
  28. Miller RE, Shenoy VB (2000) Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3):139ADSCrossRefGoogle Scholar
  29. Müller P, Saúl A (2004) Elastic effects on surface physics. Surf Sci Rep 54(5–8):157–258ADSCrossRefGoogle Scholar
  30. Murdoch AI (2005) Some fundamental aspects of surface modelling. J Elas 80(1–3):33MathSciNetCrossRefGoogle Scholar
  31. Pala RGS, Liu F (2004) Determining the adsorptive and catalytic properties of strained metal surfaces using adsorption-induced stress. J Chem Phys 120(16):7720–7724ADSCrossRefGoogle Scholar
  32. Park HS (2008) Strain sensing through the resonant properties of deformed metal nanowires. J Appl Phys 104(1):013516ADSCrossRefGoogle Scholar
  33. Park HS, Klein PA, Wagner GJ (2006) A surface cauchy–born model for nanoscale materials. Int J Numer Methods Eng 68(10):1072–1095MathSciNetCrossRefGoogle Scholar
  34. Sharma P, Ganti S, Bhate N (2003) Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl Phys Lett 82(4):535–537ADSCrossRefGoogle Scholar
  35. Steigmann D, Ogden R (1997) Plane deformations of elastic solids with intrinsic boundary elasticity. In: Proceedings of the royal society of London A: mathematical, physical and engineering sciences, the royal society, vol 453, pp 853–877MathSciNetCrossRefGoogle Scholar
  36. Steigmann D, Ogden R (1999) Elastic surface substrate interactions. In: Proceedings of the royal society of London A: mathematical, physical and engineering sciences, the royal society, vol 455, pp 437–474ADSMathSciNetCrossRefGoogle Scholar
  37. Style RW, Hyland C, Boltyanskiy R, Wettlaufer JS, Dufresne ER (2013) Surface tension and contact with soft elastic solids. Nat Commun 4:2728ADSCrossRefGoogle Scholar
  38. Style RW, Jagota A, Hui CY, Dufresne ER (2017) Elastocapillarity: surface tension and the mechanics of soft solids. Ann Rev Conden Matter Phys 8:99–118ADSCrossRefGoogle Scholar
  39. Suo Z, Lu W (2000) Forces that drive nanoscale self-assembly on solid surfaces. J Nanopart Res 2(4):333–344CrossRefGoogle Scholar
  40. Wang J, Huang Z, Duan H, Yu S, Feng X, Wang G, Zhang W, Wang T (2011) Surface stress effect in mechanics of nanostructured materials. Acta Mechanica Solida Sinica 24(1):52–82CrossRefGoogle Scholar
  41. Wang ZQ, Zhao YP, Huang ZP (2010) The effects of surface tension on the elastic properties of nano structures. Int J Eng Sci 48(2):140–150CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Kosar Mozaffari
    • 1
  • Shengyou Yang
    • 1
  • Pradeep Sharma
    • 1
    Email author
  1. 1.Department of Mechanical Engineering, Cullen College of EngineeringUniversity of HoustonHoustonUSA

Section editors and affiliations

  • Ting Zhu
    • 1
  1. 1.Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations