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On Dynamic Boundary Conditions Within the Linear Steigmann-Ogden Model of Surface Elasticity and Strain Gradient Elasticity

  • Victor A. EremeyevEmail author
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 103)

Abstract

Within the strain gradient elasticity we discuss the dynamic boundary conditions taking into account surface stresses described by the Steigmann–Ogden model. The variational approach is applied with the use of the least action functional. The functional is represented as a sum of surface and volume integrals. The surface strain and kinetic energy densities are introduced. The Toupin–Mindlin formulation of the strain gradient elasticity is considered. As a result, we derived the motion equations and the natural boundary conditions which include inertia terms.

Notes

Acknowledgements

Authors 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).

References

  1. 1.
    Abali, B.E., Müller, W.H., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87(9), 1495–1510 (2017)CrossRefGoogle Scholar
  2. 2.
    Aifantis, E.C.: Gradient material mechanics: perspectives and prospects. Acta Mech. 225(4–5), 999–1012 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Altenbach, H., Eremeyev, V.A., Lebedev, L.P.: On the spectrum and stiffness of an elastic body with surface stresses. ZAMM 91(9), 699–710 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Auffray, N., dell’Isola, F., Eremeyev, V.A., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bertram, A.: Compendium on Gradient Materials. Otto von Guericke University, Magdeburg (2017)Google Scholar
  6. 6.
    Cordero, N.M., Forest, S., Busso, E.P.: Second strain gradient elasticity of nano-objects. J. Mech. Phys. Solids 97, 92–124 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32(1), 33–52 (1997)MathSciNetCrossRefGoogle Scholar
  8. 8.
    dell’Isola, F., Sciarra, G., Vidoli, S.: Generalized Hooke’s law for isotropic second gradient materials. R. Soc. Lond. Proc. Ser. A 465(2107), 2177–2196 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. In: Advances in Applied Mechanics, vol. 42, pp. 1–68. Elsevier (2008)Google Scholar
  10. 10.
    Eremeyev, V.A.: On equilibrium of a second-gradient fluid near edges and corner points. In: Naumenko, K., Aßmus, M. (eds.) Advanced Methods of Continuum Mechanics for Materials and Structures, Advanced Structured Materials, vol. 60, pp. 547–556. Springer, Singapore (2016)Google Scholar
  11. 11.
    Eremeyev, V.A., Altenbach, H.: Equilibrium of a second-gradient fluid and an elastic solid with surface stresses. Meccanica 49(11), 2635–2643 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Eremeyev, V.A., Lebedev, L.P.: Mathematical study of boundary-value problems within the framework of Steigmann-Ogden model of surface elasticity. Cont. Mech. Thermodyn. 28(1–2), 407–422 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, New Jersey (2018)Google Scholar
  15. 15.
    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 (2018).  https://doi.org/10.1177/1081286518769960
  16. 16.
    Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)zbMATHGoogle Scholar
  17. 17.
    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
  18. 18.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)CrossRefGoogle Scholar
  20. 20.
    Han, Z., Mogilevskaya, S.G., Schillinger, D.: Local fields and overall transverse properties of unidirectional composite materials with multiple nanofibers and Steigmann-Ogden interfaces. Int. J. Solids Struct. 147, 166–182 (2018)CrossRefGoogle Scholar
  21. 21.
    Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61(12), 2381–2401 (2013a)MathSciNetCrossRefGoogle Scholar
  22. 22.
    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), 010802Google Scholar
  23. 23.
    Khakalo, S., Niiranen, J.: Form II of Mindlin’s second strain gradient theory of elasticity with a simplification: for materials and structures from nano-to macro-scales. Eur. J. Mech. A/Solids 71, 292–319 (2018)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kim, C.I., Schiavone, P., Ru, C.Q.: The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. Trans. ASME J. Appl. Mech. 77(2), 021011 (2010)Google Scholar
  25. 25.
    Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications in Mechanics. World Scientific, New Jersey (2010)CrossRefGoogle Scholar
  26. 26.
    Lurie, A.I.: Nonlinear Theory of Elasticity. North-Holland, Amsterdam (1990)zbMATHGoogle Scholar
  27. 27.
    Maugin, G.A.: Non-Classical Continuum Mechanics: A Dictionary. Springer, Singapore (2017)CrossRefGoogle Scholar
  28. 28.
    Mindlin, R.D.: Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16(1), 51–78 (1964)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1(4), 417–438 (1965)CrossRefGoogle Scholar
  30. 30.
    Mindlin, R.D., Eshel, N.N.: On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109–124 (1968)CrossRefGoogle Scholar
  31. 31.
    Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Schiavone, P., Ru, C.Q.: Solvability of boundary value problems in a theory of plane-strain elasticity with boundary reinforcement. Int. J. Eng. Sci. 47(11), 1331–1338 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New Yourk (1994)CrossRefGoogle Scholar
  34. 34.
    Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453(1959), 853–877 (1997)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. R. Soc. A 455(1982), 437–474 (1999)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11(1), 385–414 (1962)MathSciNetCrossRefGoogle Scholar
  37. 37.
    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 Sinica 24, 52–82 (2011)CrossRefGoogle Scholar
  38. 38.
    Zemlyanova, A.Y., Mogilevskaya, S.G.: Circular inhomogeneity with Steigmann-Ogden interface: Local fields, neutrality, and Maxwell’s type approximation formula. Int. J. Solids Struct. 135, 85–98 (2018)Google Scholar

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Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringGdańsk University of TechnologyGdańskPoland
  2. 2.Institute of Mathematics, Mechanics and Computer ScienceSouthern Federal UniversityRostov-on-DonRussia

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