Skip to main content

On Surface Kinetic Constitutive Relations

  • Chapter
  • First Online:
Nonlinear Wave Dynamics of Materials and Structures

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 122))

  • 364 Accesses

Abstract

In the framework of the strain gradient surface elasticity we discuss a consistent form of surface kinetic energy. This kinetic constitutive equation completes the statement of initial–boundary value problems. The proposed surface kinetic energy density is the most general function consistent with the constitutive relations in bulk. As the surface strain energy depends on the surface deformation gradient and its gradient, the kinetic energy is a quadratic function of the velocity and its surface gradient.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aifantis, E.: Chapter one - internal length gradient (ilg) material mechanics across scales and disciplines. In: Bordas, S.P.A., Balint, D.S. (eds.) Advances in Applied Mechanics, vol. 49, pp. 1–110. Elsevier (2016)

    Google Scholar 

  2. Aifantis, E.C.: Gradient deformation models at nano, micro, and macro scales. J. Eng. Mater. Technol. 121(2), 189–202 (1999)

    Article  Google Scholar 

  3. Altenbach, H., Eremeyev, V.A.: Eigen-vibrations of plates made of functionally graded material. CMC Comput. Mater. Continua 9(2), 153–178 (2009)

    MATH  Google Scholar 

  4. Bagdoev, A.G., Erofeyev, V.I., Shekoyan, A.V.: Wave Dynamics of Generalized Continua, Advanced Structured Materials, vol. 24. Springer, Berlin, Heidelberg (2016)

    Book  Google Scholar 

  5. Berdichevsky, V.: Variational Principles of Continuum Mechanics: I. Fundamentals. Springer, Heidelberg (2009)

    Book  Google Scholar 

  6. Chróścielewski, J., Makowski, J., Pietraszkiewicz, W.: Statyka i dynamika powłok wielopłatowych. Nieliniowa teoria i metoda elementów skończonych (in Polish), Wydawnictwo IPPT PAN, Warszawa (2004)

    Google Scholar 

  7. Cordero, N.M., Forest, S., Busso, E.P.: Second strain gradient elasticity of nano-objects. J. Mech. Phys. Solids 97, 92–124 (2016)

    Article  MathSciNet  Google Scholar 

  8. 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)

    Article  MathSciNet  Google Scholar 

  9. Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. In: Aref, H., van der Giessen, E. (eds.) Advances in Applied Mechanics, vol. 42, pp. 1–68. Elsevier (2009)

    Google Scholar 

  10. Eremeyev, V.A.: On effective properties of materials at the nano-and microscales considering surface effects. Acta Mech. 227(1), 29–42 (2016)

    Article  MathSciNet  Google Scholar 

  11. 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 (2016)

    Google Scholar 

  12. Eremeyev, V.A.: Strongly anisotropic surface elasticity and antiplane surface waves. Philos. Trans. R. Soc. A. 378(2162), 20190100 (2020). https://doi.org/10.1098/rsta.2019.0100

  13. Eremeyev, V.A., Pietraszkiewicz, W.: Local symmetry group in the general theory of elastic shells. J. Elast. 85(2), 125–152 (2006)

    Article  MathSciNet  Google Scholar 

  14. Eremeyev, V.A., Sharma, B.L.: Anti-plane surface waves in media with surface structure: discrete vs. continuum model. Int. J. Eng. Sci. 143, 33–38 (2019)

    Google Scholar 

  15. Eremeyev, V.A., Lebedev, L.P., Altenbach, H.: Foundations of Micropolar Mechanics. Springer-Briefs in Applied Sciences and Technologies, Springer, Heidelberg (2013)

    Book  Google Scholar 

  16. Eremeyev, V.A., Rosi, G., Naili, S.: Surface/interfacial anti-plane waves in solids with surface energy. Mech. Res. Commun. 74, 8–13 (2016)

    Article  Google Scholar 

  17. Eremeyev, V.A., Cloud, M.J., Lebedev, L.P.: Applications of Tensor Analysis in Continuum Mechanics. World Scientific, New Jersey (2018)

    Book  Google Scholar 

  18. 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, 2526–2535 (2019)

    Article  MathSciNet  Google Scholar 

  19. Eremeyev, V.A., Rosi, G., Naili, S.: Transverse surface waves on a cylindrical surface with coating. Int. J. Eng. Sci. 103188 (2020). https://doi.org/10.1016/j.ijengsci.2019.103188

  20. Erofeyev, V.I.: Wave Processes in Solids with Microstructure. World Scientific, Singapore (2003)

    Book  Google Scholar 

  21. Forest, S., Cordero, N.M., Busso, E.P.: First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales. Comput. Mater. Sci. 50(4), 1299–1304 (2011)

    Google Scholar 

  22. Georgiadis, H., Vardoulakis, I., Lykotrafitis, G.: Torsional surface waves in a gradient-elastic half-space. Wave Motion 31(4), 333–348 (2000)

    Article  MathSciNet  Google Scholar 

  23. Gorbushin, N., Eremeyev, V.A., Mishuris, G.: On stress singularity near the tip of a crack with surface stresses. Int. J. Eng. Sci. 146:103183 (2020)

    Google Scholar 

  24. 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)

    Article  Google Scholar 

  25. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)

    Google Scholar 

  26. Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)

    Article  Google Scholar 

  27. Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61(12), 2381–2401 (2013)

    Google Scholar 

  28. Javili, A., McBride, A., Steinmann, P.: 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), 010802 (2013)

    Google Scholar 

  29. Kim, C.I., Ru, C.Q., Schiavone, P.: A clarification of the role of crack-tip conditions in linear elasticity with surface effects. Math. Mech. Solids 18(1), 59–66 (2013)

    Article  MathSciNet  Google Scholar 

  30. Li, Y., Wei, P.J., Tang, Q.: Reflection and transmission of elastic waves at the interface between two gradient-elastic solids with surface energy. Eur. J. Mech. A Solids 52(C):54–71 (2015)

    Google Scholar 

  31. Libai, A., Simmonds, J.G.: The Nonlinear Theory of Elastic Shells, 2nd edn. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  32. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates. Trans. ASME J. Appl. Mech. 18, 31–38 (1951)

    MATH  Google Scholar 

  33. Murdoch, A.I., Cohen, H.: Symmetry considerations for material surfaces. Arch. Ration. Mech. Anal. 72(1), 61–98 (1979)

    Article  MathSciNet  Google Scholar 

  34. Pietraszkiewicz, W.: Refined resultant thermomechanics of shells. Int. J. Eng. Sci. 49(10), 1112–1124 (2011)

    Article  MathSciNet  Google Scholar 

  35. Placidi, L., Rosi, G., Giorgio, I., Madeo, A.: Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials. Math. Mech. Solids 19(5), 555–578 (2014)

    Article  MathSciNet  Google Scholar 

  36. Rosi, G., Auffray, N.: Anisotropic and dispersive wave propagation within strain-gradient framework. Wave Motion 63, 120–134 (2016)

    Article  Google Scholar 

  37. 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)

    Article  MathSciNet  Google Scholar 

  38. 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(3), 536–544 (2010)

    Article  MathSciNet  Google Scholar 

  39. Simmonds, J.G.: A Brief on Tensor Analysis, 2nd edn. Springer, New York (1994)

    Book  Google Scholar 

  40. Steigmann, D.J., Ogden, R.W.: Plane deformations of elastic solids with intrinsic boundary elasticity. Proc. R. Soc. A 453(1959), 853–877 (1997)

    Article  MathSciNet  Google Scholar 

  41. Steigmann, D.J., Ogden, R.W.: Elastic surface-substrate interactions. Proc. R. Soc. A 455(1982), 437–474 (1999)

    Article  MathSciNet  Google Scholar 

  42. Timoshenko, S.P.: LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41(245):744–746 (1921)

    Google Scholar 

  43. Truesdell, C., Noll, W.: The Non-linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004)

    Book  Google Scholar 

  44. Vardoulakis, I., Georgiadis, H.G.: SH surface waves in a homogeneous gradient-elastic half-space with surface energy. J. Elast. 47(2), 147–165 (1997)

    Article  MathSciNet  Google Scholar 

  45. 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)

    Article  Google Scholar 

  46. Wang, X., Schiavone, P.: A mode-III crack with variable surface effects. J. Theor. Appl. Mech. 54(4), 1319–1327 (2016)

    Article  Google Scholar 

  47. Xu, L., Wang, X., Fan, H.: Anti-plane waves near an interface between two piezoelectric half-spaces. Mech. Res. Commun. 67, 8–12 (2015)

    Article  Google Scholar 

  48. Yerofeyev, V.I., Sheshenina, O.A.: Waves in a gradient-elastic medium with surface energy. J. Appl. Math. Mech. 69(1), 57–69 (2005)

    Article  Google Scholar 

Download references

Acknowledgements

VAE acknowledges the support of the Ministry of Education and Science of the Russian Federation, Project No. 9.1001.2017/4.6.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Victor A. Eremeyev .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Eremeyev, V.A., Lebedev, L.P. (2020). On Surface Kinetic Constitutive Relations. In: Altenbach, H., Eremeyev, V., Pavlov, I., Porubov, A. (eds) Nonlinear Wave Dynamics of Materials and Structures. Advanced Structured Materials, vol 122. Springer, Cham. https://doi.org/10.1007/978-3-030-38708-2_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-38708-2_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-38707-5

  • Online ISBN: 978-3-030-38708-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics