Classification of Gradient Adhesion Theories Across Length Scale

  • Sergey LurieEmail author
  • Petr Belov
  • Holm Altenbach
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 42)


The sequence of continuum theories of adhesion is discussed. We give a brief analysis of the existing theories of adhesion and present a continuum theory of adhesion as a natural generalization of appropriate options for the theory of elasticity and gradient theories of elasticity. We offer a sequence of variational formulations of theories of adhesion and constitutive equations. In addition, the analysis of the structures of the tensors of adhesive elastic modules is presented. As a result, we propose a classification of theories of adhesion and gradient theories of elasticity in terms accounting for scale effects. The classification is based on the qualitative analysis of scale effects of different orders depending on the physical properties of the continuum.



This work was supported by the Russian Foundation for Basic Research project No. 15-01-03649-a.


  1. Altan BS, Aifantis EC (1992) On the structure of the mode iii crack-tip in gradient elasticity. Scripta Met 26:319–324CrossRefGoogle Scholar
  2. Altenbach H, Eremeyev VA, Lebedev LP (2011) On the spectrum and stiffness of an elastic body with surface stresses. ZAMM 91(9):699–710MathSciNetCrossRefzbMATHGoogle Scholar
  3. Belov PA, Lurie SA (2007) Theory of ideal adhesion interactions. J Compos Mech Des 14:545–561Google Scholar
  4. Belov PA, Lurie SA (2009) Continual theory of adhesion interactions of damaged media. J Compos Mech Des 15(4):610–629Google Scholar
  5. Belov PA, Lurie SA (2014) Mathematical theory of damaged media. Gradient theory of elasticity. Formulations hierarchy comparative analysis. Palmarium Academic Publishing, GermanyGoogle Scholar
  6. Duan HL, Wang J, Huang ZP, Karihaloo BL (2005) Size-dependent effective elastic constants of solids containing nanoinhomogeneities with interface stress. J Mech Phys Solids 53:1574–1596MathSciNetCrossRefzbMATHGoogle Scholar
  7. Duan HL, Wang J, Karihaloo BL (2008) Theory of elasticity at the nanoscale. In: Aref H, van der Giessen E (eds) Advances in applied mechanics, vol 42. Elsevier, Amsterdam, pp 1–68Google Scholar
  8. Eremeyev VA (2016) On effective properties of materials at the nano- and microscales considering surface effects. Acta Mech 227(1):29–42MathSciNetCrossRefzbMATHGoogle Scholar
  9. Eremeyev VA, Altenbach H, Morozov NF (2009) The influence of surface tension on the effective stiffness of nanosize plates. Dokl Phys 54(2):98–100CrossRefzbMATHGoogle Scholar
  10. Gao XL, Park SK (2007) Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int J Solids Struct 44:7486–7499CrossRefzbMATHGoogle Scholar
  11. Gurtin ME, Murdoch AI (1975a) Addenda to our paper a continuum theory of elastic material surfaces. Arch Ration Mech Anal 59(4):389–390MathSciNetCrossRefzbMATHGoogle Scholar
  12. Gurtin ME, Murdoch AI (1975b) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323MathSciNetCrossRefzbMATHGoogle Scholar
  13. Gurtin ME, Fried E, Anand L (2010) The mechanics and thermodynamics of Continua. Cambridge University Press, New YorkCrossRefGoogle Scholar
  14. Gusev AA, Lurie SA (2015) Symmetry conditions in strain gradient elasticity. Math Mech Solids Math pp 1–9Google Scholar
  15. Huang Z, Wang J (2013) Micromechanics of nanocomposites with interface energy effect. In: Li S, Gao XL (eds) Handbook on micromechanics and nanomechanics, vol 42. Pan Stanford Publishing, Stanford, pp 303–348Google Scholar
  16. Javili A, Steinmann P (2010) On thermomechanical solids with boundary structures. Int J Solids Struct 47(24):3245–3253CrossRefzbMATHGoogle Scholar
  17. Javili A, McBride A, Steinmann P (2012) 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(10):802–1–3Google Scholar
  18. Kim CI, Schiavone P, Ru CQ (2011) Effect of surface elasticity on an interface crack in plane deformations. Proc Roy Soc A 467(2136):3530–3549MathSciNetCrossRefzbMATHGoogle Scholar
  19. Liu IS (2009) Continuum mechanics. UNESCO Publications, OxdfordGoogle Scholar
  20. Lurie S, Tuchkova N (2009) A continuous adhesion model for deformed solid bodies and media with nanostructures. Kompoz Nanostruct 2(2):25–43Google Scholar
  21. Lurie S, Belov PA, Tuchkova NP (2010) Gradient theory of media with conserved dislocations: application to microstruc-tured materials. In: Maugin GA, Metrikine AV (eds) One hundred years after the Cosserats, advances in mechanics and mathematics, vol 21. Springer, Heidelberg, pp 223–234Google Scholar
  22. Lurie SA, Belov PA (2008) Cohesion field: Barenblatt’s hypothesis as formal corollary of theory of contonuous media with conserved dislocations. Int J Fract 50(1–2):181–194Google Scholar
  23. Lurie SA, Volkov-Bogorodsky DB, Zubov VI, Tuchkova NP (2009) Advanced theoretical and numerical multiscale modeling of cohesion/adhesion interactions in continuum mechanics and its applications for filled nanocomposites. Int J Compos Mater Sci 45(3):709–741CrossRefGoogle Scholar
  24. Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1(4):417–439CrossRefGoogle Scholar
  25. Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4:109–112CrossRefzbMATHGoogle Scholar
  26. Murdoch AI (2000) On objectivity and material symmetry for simple elastic solids. J Elast. 60:233–242MathSciNetCrossRefzbMATHGoogle Scholar
  27. Murdoch AI (2003) Objectivity in classical continuum physics: a rationale for discarding the principle of invariance under superposed rigid body motions in favor of purely objective considerations. Continuum Mech Thermodyn 15:309–320MathSciNetCrossRefzbMATHGoogle Scholar
  28. Murdoch AI (2005) On criticism of the nature of objectivity in classical continuum physics. Continuum Mech Thermodyn 17:135–148MathSciNetCrossRefzbMATHGoogle Scholar
  29. Ostoja-Starzewski M (2002) Lattice models in micromechanics. Appl Mech Rev 55(1):35–59MathSciNetCrossRefzbMATHGoogle Scholar
  30. Povstenko Y (2008) Mathematical modeling of phenomena caused by surface stresses in solids. In: Altenbach H, Morozov NF (eds) Surface effects in solid mechanics, vol 42. Springer, Heidelberg, pp 135–153Google Scholar
  31. Steigmann DJ, Ogden RW (1997a) Plane deformations of elastic solids with intrinsic boundary elasticity. Proc Roy Soc A 453(1959):853–877MathSciNetCrossRefzbMATHGoogle Scholar
  32. Steigmann DJ, Ogden RW (1997b) Plane deformations of elastic solids with intrinsic boundary elasticity. Proc Roy Soc A 453(1959):853–877MathSciNetCrossRefzbMATHGoogle Scholar
  33. Steigmann DJ, Ogden RW (1999a) Elastic surface-substrate interactions. Proc Roy Soc A 455(1999):437–474Google Scholar
  34. Steigmann DJ, Ogden RW (1999b) Elastic surface-substrate interactions. Proc Roy Soc A 455(1982):437–474MathSciNetCrossRefzbMATHGoogle Scholar
  35. Zhu HX, Wang JX, Karihaloo BL (2009) Effects of surface and initial stresses on the bending stiffness of trilayer plates and nanofilms. J Mech Mater Struct 4(3):589–604CrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute for Problem of Mechanics of RAS and Institute of Applied Mechanics of RASMoscowRussia
  2. 2.Institute of Research, Development and Technology TransferMoscowRussia
  3. 3.Lehrstuhl für Technische Mechanik, Institut für Mechanik, Fakultät für MaschinenbauOtto-von-Guericke-Universtät MagdeburgMagdeburgGermany

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