Antiplane Surface Wave Propagation Within the Stress Gradient Surface Elasticity

  • Victor A. EremeyevEmail author
Part of the Mathematics of Planet Earth book series (MPE, volume 6)


We discuss a new class of antiplane surface waves in an elastic half space with surface stresses. Here we consider a surface elasticity within stress gradient model , that is when the surface stresses relate to surface strains through an integral constitutive dependence. For antiplane motions the problem is reduced to the wave equation with nonclassical dynamic boundary condition. The dispersion relation is derived.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author gratefully thanks the Reviewer for the helpful constructive comments and recommendations.

The author acknowledges financial support from the Russian Science Foundation under the grant “Methods of microstructural nonlinear analysis, wave dynamics and mechanics of composites for research and design of modern metamaterials and elements of structures made on its base” (No 15-19-10008-P).


  1. Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  2. Askes, H., Aifantis, E.C.: Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results. Int. J. Solids Struct. 48(13), 1962–1990 (2011). CrossRefGoogle Scholar
  3. Berezovski, A., Engelbrecht, J., Berezovski, M.: Waves in microstructured solids: a unified viewpoint of modeling. Acta Mechanica 220(1-4), 349–363 (2011). CrossRefGoogle Scholar
  4. Berezovski, A., Giorgio, I., Corte, A.D.: Interfaces in micromorphic materials: wave transmission and reflection with numerical simulations. Math. Mech. Solids 21(1), 37–51 (2016). MathSciNetCrossRefGoogle Scholar
  5. dell’Isola, F., Madeo, A., Placidi, L.: Linear plane wave propagation and normal transmission and reflection at discontinuity surfaces in second gradient 3D continua. ZAMM 92(1), 52–71 (2012). MathSciNetCrossRefGoogle Scholar
  6. Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech., 42, 1–68 (2008). Google Scholar
  7. Engelbrecht, J., Berezovski, A.: Reflections on mathematical models of deformation waves in elastic microstructured solids. Math. Mech. Complex Systems 3(1), 43–82 (2015). MathSciNetCrossRefGoogle Scholar
  8. Engelbrecht, J., Berezovski, A., Pastrone, F., Braun, M.: Waves in microstructured materials and dispersion. Phil. Mag. 85(33-35), 4127–4141 (2005). CrossRefGoogle Scholar
  9. Engelbrecht, J., Pastrone, F., Braun, M., Berezovski, A.: Hierarchies of waves in nonclassical materials. In: Delsanto P.P. (ed.) Universality of Nonclassical Nonlinearity: Applications to Non-destructive Evaluations and Ultrasonic, pp. 29–47. Springer, New York (2006). CrossRefGoogle Scholar
  10. Eremeyev, V.A.: On effective properties of materials at the nano-and microscales considering surface effects. Acta Mechanica 227(1), 29–42 (2016). MathSciNetCrossRefGoogle Scholar
  11. Eremeyev, V.A., Rosi, G., Naili, S.: Surface/interfacial anti-plane waves in solids with surface energy. Mech. Res. Commun. 74, 8–13 (2016). CrossRefGoogle Scholar
  12. Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, New Jersey (2018a).
  13. Eremeyev, V.A., Lebedev, L.P., Cloud, M.J.: Acceleration waves in the nonlinear micromorphic continuum. Mech. Res. Commun. 93, 70–74 (2018b). CrossRefGoogle Scholar
  14. Eremeyev, V.A., Rosi, G., Naili, S.: Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses. Math. Mech. Solids 24(8), 2526–2535 (2019). MathSciNetCrossRefGoogle Scholar
  15. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002). zbMATHGoogle Scholar
  16. de Gennes, P.G.: Some effects of long range forces on interfacial phenomena. J. Phys. Lettr. 42(16), 377–379 (1981). CrossRefGoogle Scholar
  17. de Gennes, P.G., Brochard-Wyart, F., Quéré, D.: Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York (2004). CrossRefGoogle Scholar
  18. Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlin. Dyn. 88(1), 21–31 (2017). CrossRefGoogle Scholar
  19. Gourgiotis, P., Georgiadis, H.: Torsional and {SH} surface waves in an isotropic and homogenous elastic half-space characterized by the Toupin–Mindlin gradient theory. Int. J. Solids Struct. 62, 217–228 (2015). CrossRefGoogle Scholar
  20. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Anal. 57(4), 291–323 (1975). MathSciNetCrossRefGoogle Scholar
  21. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)CrossRefGoogle Scholar
  22. Israelachvili, J.N.: Intermolecular and Surface Forces, 3rd edn. Academic Press, Amsterdam (2011).
  23. Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010).
  24. Maugin, G.A.: Non-Classical Continuum Mechanics: A Dictionary. Springer, Singapore (2017). CrossRefGoogle Scholar
  25. Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Cont. Mech. Thermodyn. 28(1-2), 215–234 (2016). MathSciNetCrossRefGoogle Scholar
  26. Rosi, G., Auffray, N.: Anisotropic and dispersive wave propagation within strain-gradient framework. Wave Motion 63, 120–134 (2016). CrossRefGoogle Scholar
  27. Rosi, G., Nguyen, V.H., Naili, S.: Surface waves at the interface between an inviscid fluid and a dipolar gradient solid. Wave Motion 53, 51–65 (2015). MathSciNetCrossRefGoogle Scholar
  28. Rowlinson, J.S., Widom, B.: Molecular Theory of Capillarity. Dover, New York (2003)Google Scholar
  29. Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New York (1994)CrossRefGoogle Scholar
  30. Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. Roy. Soc. A 453(1959), 853–877 (1997). MathSciNetCrossRefGoogle Scholar
  31. Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. Roy. Soc. A 455(1982), 437–474 (1999). MathSciNetCrossRefGoogle Scholar
  32. Vardoulakis, I., Georgiadis, H.G.: SH surface waves in a homogeneous gradient-elastic half-space with surface energy. J. Elasticity 47(2), 147–165 (1997)MathSciNetCrossRefGoogle Scholar
  33. Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., Wang, T.: Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solida Sin. 24, 52–82 (2011). CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringGdańsk University of TechnologyGdańskPoland
  2. 2.Southern Federal UniversityRostov on DonRussia
  3. 3.Southern Scientific Center of RASRostov on DonRussia

Personalised recommendations