Skip to main content

Stress Gradient Elasticity Theory: Existence and Uniqueness of Solution

Abstract

The objective of the present article is to assess the well-posedness of the stress gradient linear elastic problems recently introduced by Forest and Sab (Mech. Res. Commun. 40:16–25, 2012) and to formulate the corresponding existence and uniqueness theorems. In particular, we show that such theorems can be established in the case of the boundary value problems formulated in (Forest and Sab in Mech. Res. Commun. 40:16–25, 2012) with the corresponding boundary conditions.

This is a preview of subscription content, access via your institution.

References

  1. Adams, R.A.: Sobolev Spaces. Academic Press, San Diego, CA (1975)

    MATH  Google Scholar 

  2. Amrouche, C., Ciarlet, P.G., Gratie, L., Kesavan, S.: On the characterizations of matrix fields as linearized strain tensor fields. J. Math. Pures Appl. 86, 116–132 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  3. Auffray, N.: Analytical expressions for odd-order anisotropic tensor dimension. C. R., Méc. 342, 284–291 (2014)

    Article  Google Scholar 

  4. Auffray, N., Le Quang, H., He, Q.C.: Matrix representations for 3D strain-gradient elasticity. J. Mech. Phys. Solids 61, 1202–1223 (2013)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. Bleustein, J.L.: A note on the boundary conditions of Toupin’s strain-gradient theory. Int. J. Solids Struct. 3, 1053–1057 (1967)

    Article  Google Scholar 

  6. Chakravarthy, S.S., Curtin, W.A.: Stress-gradient plasticity. Proc. Natl. Acad. Sci. USA 108, 15716–15720 (2011)

    ADS  Article  Google Scholar 

  7. Dell’Isola, F., Seppecher, P.: The relationship between edge contact forces, double forces and interstitial working allowed by the principle of virtual power. C. R. Acad. Sci. Paris IIb 321, 303–308 (1995)

    MATH  Google Scholar 

  8. Dell’Isola, F., Seppecher, P.: Edge contact forces and quasi-balanced power. Meccanica 32, 33–52 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  9. Dell’Isola, F., Seppecher, P., Madeo, A.: How contact interactions may depend on the shape of Cauchy cuts in N-th gradient continua: approach “à la D’Alembert”. Z. Angew. Math. Phys. 63, 1119–1141 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  10. Eringen, A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)

    MATH  Google Scholar 

  11. Forest, S., Aifantis, E.C.: Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. Int. J. Solids Struct. 47, 3367–3376 (2010)

    Article  MATH  Google Scholar 

  12. Forest, S., Sab, K.: Stress gradient continuum theory. Mech. Res. Commun. 40, 16–25 (2012)

    Article  Google Scholar 

  13. Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus, première partie: théorie du second gradient. J. Méc. 12, 235–274 (1973)

    MathSciNet  MATH  Google Scholar 

  14. Iesan, D.: Thermoelasticity of bodies with microstructure and microtemperatures. Int. J. Solids Struct. 44, 8648–8662 (2007)

    Article  MATH  Google Scholar 

  15. Iesan, D., Quintanilla, R.: Existence and continuous dependence results in the theory of microstretch elastic bodies. Int. J. Eng. Sci. 32, 991–1001 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  16. Iesan, D., Quintanilla, R.: On a strain gradient theory of thermoviscoelasticity. Mech. Res. Commun. 48, 52–58 (2013)

    Article  Google Scholar 

  17. Jeong, J., Neff, P.: Existence, uniqueness and stability in linear Cosserat elasticity for weakest curvature conditions. Math. Mech. Solids 15, 78–95 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  18. Lebée, A., Sab, K.: A bending-gradient model for thick plates, Part I: Theory. Int. J. Solids Struct. 48, 2878–2888 (2011)

    Article  Google Scholar 

  19. Lebée, A., Sab, K.: A bending-gradient model for thick plates, Part II: Closed-form solutions for cylindrical bending of laminates. Int. J. Solids Struct. 48, 2889–2901 (2011)

    Article  Google Scholar 

  20. Mindlin, R.D.: Second gradient of strain and surface–tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965)

    Article  Google Scholar 

  21. Mindlin, R.D., Eshel, N.N.: On first strain gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968)

    Article  MATH  Google Scholar 

  22. Moreau, J.J.: Duality characterization of strain tensor distributions in an arbitrary open set. J. Math. Anal. Appl. 72, 760–770 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  23. Necas, J.: Les Méthodes Directes en Théorie des Equations Elliptiques. Masson, Paris (1967)

    Google Scholar 

  24. Neff, P., Forest, S.: A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure. Modelling, existence of minimizers, identification of moduli and computational results. J. Elast. 87, 239–276 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  25. Olive, M., Auffray, N.: Symmetry classes for even-order tensors. Math. Mech. Complex Syst. 1, 177–210 (2013)

    Article  MATH  Google Scholar 

  26. Olive, M., Auffray, N.: Symmetry classes for odd-order tensors. Z. Angew. Math. Mech. 94, 421–447 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  27. Polizzotto, C.: Stress gradient versus strain gradient constitutive models within elasticity. Int. J. Solids Struct. 51, 1809–1818 (2014)

    Article  Google Scholar 

  28. Polizzotto, C.: A unifying variational framework for stress gradient and strain gradient elasticity theories. Eur. J. Mech. A, Solids 49, 430–440 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  29. Ru, C.Q., Aifantis, E.C.: A simple approach to solve boundary value problems in gradient elasticity. Acta Mech. 101, 59–68 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  30. Toupin, R.A.: Elastic materials with couple stresses. Arch. Ration. Mech. Anal. 11, 385–414 (1962)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Karam Sab.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sab, K., Legoll, F. & Forest, S. Stress Gradient Elasticity Theory: Existence and Uniqueness of Solution. J Elast 123, 179–201 (2016). https://doi.org/10.1007/s10659-015-9554-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-015-9554-1

Keywords

  • Stress gradient elasticity
  • Higher order elasticity
  • Boundary conditions
  • Existence and uniqueness of solution

Mathematics Subject Classification (2000)

  • 35D30
  • 35J56
  • 35Q74