Skip to main content
SpringerLink
Log in

On the Ellipticity of Static Equations of Strain Gradient Elasticity and Infinitesimal Stability

  • MECHANICS
  • Published:
Vestnik St. Petersburg University, Mathematics Aims and scope Submit manuscript

Abstract

Conditions for the strong ellipticity of equilibrium equations are formulated within strain gradient elasticity under finite deformations. In this model, the strain energy density is a function of the first and second gradients of the position vector (deformation gradient). Ellipticity imposes certain constraints on the tangent elastic moduli. It is also closely related to infinitesimal stability, which is defined as the positive definiteness of the second variation of the potential-energy functional. The work considers the first boundary-value problem (with Dirichlet boundary conditions). For a 1D deformation, necessary and sufficient conditions for infinitesimal stability are determined, which are two inequalities for elastic moduli.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

REFERENCES

  1. M. Agranovich, “Elliptic boundary problems,” in Partial Differential Equations IX: Elliptic Boundary Problems, Ed. by M. Agranovich, Y. Egorov, and M. Shubin (Springer-Verlag, Berlin, 1997), in Ser.: Encyclopaedia of Mathematical Sciences, Vol. 79, pp. 1–144.

  2. G. Fichera, Linear Elliptic Differential Systems and Eigenvalue Problems (Springer-Verlag, Berlin, 1965), in Ser.: Lecture Notes in Mathematics, Vol. 8.

  3. S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,” Commun. Pure Appl. Math. 12, 623–727 (1959).

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II,” Commun. Pure Appl. Math. 17, 35–92 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  5. L. R. Volevich, “Solubility of boundary value problems for general elliptic systems,” Math. Sb. 68, 373–416 (1965).

    MathSciNet  Google Scholar 

  6. A. I. Lurie, Non-Linear Theory of Elasticity (North-Holland, Amsterdam, 1990).

    Google Scholar 

  7. R. W. Ogden, Non-Linear Elastic Deformations (Mineola, Dover, 1997).

    Google Scholar 

  8. V. A. Eremeyev, “Strong ellipticity conditions and infinitesimal stability within nonlinear strain gradient elasticity,” Mech. Res. Commun. 117, 103782 (2021). https://doi.org/10.1016/j.mechrescom.2021.103782

    Article  Google Scholar 

  9. V. A. Eremeyev and E. Reccia, “Strong ellipticity within the strain gradient elasticity: Elastic bar case,” in Theoretical Analyses, Computations, and Experiments of Multiscale Materials, Ed. by I. Giorgio, L. Placidi, E. Barchiesi, B. E. Abali, and H. Altenbach (Springer-Verlag, Cham, 2022), in Ser.: Advanced Structured Materials, Vol. 175, pp. 137–144.

  10. V. A. Eremeyev and E. Reccia, “Nonlinear strain gradient and micromorphic one-dimensional elastic continua: Comparison through strong ellipticity conditions,” Mech. Res. Commun. 124, 103909 (2022). https://doi.org/10.1016/j.mechrescom.2022.103909

    Article  Google Scholar 

  11. R. A. Toupin, “Elastic materials with couple-stresses,” Arch. Ration. Mech. Anal. 11 (1), 385–414 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  12. R. A. Toupin, “Theories of elasticity with couple-stress,” Arch. Ration. Mech. Anal. 17 (2), 85–112 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  13. R. D. Mindlin, “Micro-structure in linear elasticity,” Arch. Ration. Mech. Anal. 16 (1), 51–78 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  14. R. D. Mindlin and N. N. Eshel, “On first strain-gradient theories in linear elasticity,” Int. J. Solids Struct. 4 (1), 109–124 (1968).

    Article  MATH  Google Scholar 

  15. A. Mareno and T. J. Healey, “Global continuation in second-gradient nonlinear elasticity,” SIAM J. Math. Anal. 38 (1), 103–115 (2006).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  17. E. Aifantis, “Chapter One — Internal length gradient (ILG) material mechanics across scales and disciplines,” in Advances in Applied Mechanics, Ed. by S. P. A. Bordas and D. S. Balint (Elsevier, Amsterdam, 2016), Vol. 49, pp. 1–110. https://doi.org/10.1016/bs.aams.2016.08.001

    Book  Google Scholar 

  18. S. Forest, N. M. Cordero, and E. P. Busso, “First vs. second gradient of strain theory for capillarity effects in an elastic fluid at small length scales,” Comput. Mater. Sci. 50, 1299–1304 (2011). https://doi.org/10.1016/j.commatsci.2010.03.048

    Article  Google Scholar 

  19. N. M. Cordero, S. Forest, and E. P. Busso, “Second strain gradient elasticity of nano-objects,” J. Mech. Phys. Solids 97, 92–124 (2016). https://doi.org/10.1016/j.jmps.2015.07.012

    Article  MathSciNet  Google Scholar 

  20. M. Lazar, E. Agiasofitou, and T. Böhlke, “Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin-Mindlin anisotropic first strain gradient elasticity,” Continuum Mech. Thermodyn. 34, 107–136 (2022). https://doi.org/10.1007/s00161-021-01050-y

    Article  MathSciNet  Google Scholar 

  21. H. Abdoul-Anziz and P. Seppecher, “Strain gradient and generalized continua obtained by homogenizing frame lattices,” Math. Mech. Complex Syst. 6, 213–250 (2018). https://doi.org/10.2140/memocs.2018.6.213

    Article  MathSciNet  MATH  Google Scholar 

  22. F. dell’Isola and D. Steigmann, “A two-dimensional gradient-elasticity theory for woven fabrics,” J. Elasticity 118, 113–125 (2015). https://doi.org/10.1007/s10659-014-9478-1

    Article  MathSciNet  MATH  Google Scholar 

  23. F. dell’Isola and D. J. Steigmann, Discrete and Continuum Models for Complex Metamaterials (Cambridge University Press, Cambridge, 2020).

    Book  Google Scholar 

  24. Y. Rahali, I. Giorgio, J. F. Ganghoffer, and F. dell’Isola, “Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices,” Int. J. Eng. Sci. 97, 148–172 (2015). https://doi.org/10.1016/j.ijengsci.2015.10.003

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  26. C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, 3rd ed. (Springer-Verlag, Berlin, 2004).

    Book  MATH  Google Scholar 

  27. V. A. Eremeyev, “Local material symmetry group for first- and second-order strain gradient fluids,” Math. Mech. Solids 26, 1173–1190 (2021). https://doi.org/10.1177/10812865211021640

    Article  MathSciNet  MATH  Google Scholar 

  28. Mechanics of Strain Gradient Materials, Ed. by A. Bertram and S. Forest (Springer-Verlag, Cham, 2020).

    Google Scholar 

  29. A. Bertram, Compendium on Gradient Materials (Springer-Verlag, Cham, 2023).

    Google Scholar 

  30. N. Auffray, H. Le. Quang, and Q.-C. He, “Matrix representations for 3D strain-gradient elasticity,” J. Mech. Phys. Solids 61, 1202–1223 (2013). https://doi.org/10.1016/j.jmps.2013.01.003

    Article  MathSciNet  MATH  Google Scholar 

  31. N. Auffray, Q.-C. He, and H. Le. Quang, “Complete symmetry classification and compact matrix representations for 3D strain-gradient elasticity,” Int. J. Solids Struct. 159, 197–210 (2019). https://doi.org/10.1016/j.ijsolstr.2018.09.029

    Article  Google Scholar 

  32. O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Springer-Verlag, New York, 1985), in Ser.: Applied Mathematical Sciences (AMS), Vol. 49.

  33. S. P. Timoshenko, J. M. Gere, Theory of Elastic Stability, 2nd ed. (McGraw-Hill, Auckland, 1963).

    Google Scholar 

  34. H. L. Duan, J. Wang, and B. L. Karihaloo, Theory of Elasticity at the Nanoscale (Elsevier, 2008), in Ser.: Advances in Applied Mechanics, Vol. 42, pp. 1–68.

  35. J. Wang, Z. Huang, H. Duan, S. Yu, X. Feng, G. Wang, W. Zhang, and T. Wang, “Surface stress effect in mechanics of nanostructured materials,” Acta Mech. Solida Sin. 24, 52–82 (2011). https://doi.org/10.1016/S0894-9166(11)60009-8

    Article  Google Scholar 

  36. H. Altenbach and N. F. Morozov, Surface Effects in Solid Mechanics (Springer-Verlag, New York, 2013), in Ser.: Advanced Structured Materials, Vol. 30.

  37. H. Altenbach, V. A. Eremeyev, and N. F. Morozov, “Mechanical properties of materials considering surface effects,” in Proc. IUTAM Symp. on Surface Effects in the Mechanics of Nanomaterials and Heterostructures, Beijing, China, Aug. 8–12, 2010 (Springer-Verlag, Dordrecht, 2013), in Ser.: IUTAM Bookseries, Vol. 31, pp. 105–115.

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

    Article  Google Scholar 

  39. V. A. Eremeyev, G. Rosi, and S. Naili, “Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses,” Math. Mech. Solids 24, 2526–2535 (2019).

    Article  MathSciNet  MATH  Google Scholar 

Download references

ACKNOWLEDGMENTS

I thank Academician N.F. Morozov for drawing my attention to the problems of nanomechanics, in particular, to the problems of surface stress theory [34–37], which is also closely related to strain gradient elasticity (see, e.g., [38, 39]).

Funding

This work was supported by the Russian Foundation for Basic Research, project no. 20-08-00450.

Author information

Authors and Affiliations

  1. University of Cagliari, Cagliari, Italy

    V. A. Eremeyev

Authors
  1. V. A. Eremeyev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Eremeyev.

Ethics declarations

I declares that I have no conflicts of interest.

Additional information

Translated by O. Pismenov

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eremeyev, V.A. On the Ellipticity of Static Equations of Strain Gradient Elasticity and Infinitesimal Stability. Vestnik St.Petersb. Univ.Math. 56, 77–83 (2023). https://doi.org/10.1134/S1063454123010053

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063454123010053

Keywords:

Search

Navigation