Skip to main content
SpringerLink
Log in

Conditions for Strong Ellipticity of Anisotropic Elastic Materials

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

In this paper, we derive necessary and sufficient conditions for the strong ellipticity condition of anisotropic elastic materials. We first observe that the strong ellipticity condition holds if and only if a second order tensor function is positive definite for any unit vectors. Then we further link this condition to the rank-one positive definiteness of three second-order tensors, three fourth-order tensors and a sixth-order tensor. In particular, we consider conditions of strong ellipticity of the rhombic classes, for which we need to check the copositivity of three second-order tensors and the positive definiteness of a sixth-order tensor. A direct method is presented to verify our conditions.

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

Similar content being viewed by others

References

  1. Abeyaratne, R.C.: Discontinuous deformation gradients in plane finite elastostatic of imcompressible materials. J. Elast. 10, 255–293 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chiriţǎ, S., Ciarletta, M.: Spatial estimates for the constrained anisotropic elastic cylinder. J. Elast. 85, 189–213 (2006)

    Article  Google Scholar 

  3. Chiriţǎ, S., Danescu, A., Ciarletta, M.: On the strong ellipticity of the anisotropic linearly elastic materials. J. Elast. 87, 1–27 (2007)

    Article  Google Scholar 

  4. Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Springer, New York (1998)

    MATH  Google Scholar 

  5. Dacorogna, B.: Necessary and sufficient conditions for strong ellipticity for isotropic functions in any dimension. Discrete Contin. Dyn. Syst. Ser. B 1, 257–263 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gurtin, M.E.: The linear theory of elasticity. In: Truesdell, C. (ed.) Handbuch der Physik, vol. VIa/2. Springer, Berlin (1972)

    Google Scholar 

  7. Knowles, J.K., Sternberg, E.: On the ellipticity of the equations of non-linear elastostatics for a special material. J. Elast. 5, 341–361 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  8. Merodio, J., Ogden, R.W.: Instabilities and loss of ellipticity in fiber-reinforced compressible nonlinearly elastic solids under plane deformation. Int. J. Solids Struct. 40, 4707–4727 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ni, G., Qi, L., Wang, F., Wang, Y.: The degree of the E-characteristic polynomial of an even order tensor. J. Math. Anal. Appl. 329, 1218–1229 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Padovani, C.: Strong ellipticity of transversely isotropic elasticity tensors. Meccanica 37, 515–525 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Payton, R.G.: Elastic Wave Propagation in Transversely Isotropic Media. Nijhoff, Dordrecht (1983)

    MATH  Google Scholar 

  12. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  MATH  Google Scholar 

  13. Qi, L.: Rank and eigenvalues of a supersymmetric tensor, a multivariate homogeneous polynomial and an algebraic surface defined by them. J. Symb. Comput. 41, 1309–1327 (2006)

    Article  MATH  Google Scholar 

  14. Qi, L.: Eigenvalues and invariants of tensors. J. Math. Anal. Appl. 325, 1363–1377 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Qi, L., Wang, Y., Wang, F.: A global homogenous polynomial optimization problem over the unit sphere. Working Paper, Department of Applied Mathematics, The Hong Kong Polytechnic University (2007)

  16. Qi, L., Wang, F., Wang, Y.: Z-eigenvalue methods for a global polynomial optimization problem. Math. Program. 118, 301–316 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Rosakis, P.: Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109, 1–37 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Simpson, H.C., Spector, S.J.: On copositive matrices and strong ellipticity for isotropic elastic materials. Arch. Ration. Mech. Anal. 84, 55–68 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. Walton, J.R., Wilber, J.P.: Sufficient conditions for strong ellipticity for a class of anisotropic materials. Int. J. Non-Linear Mech. 38, 441–455 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Wang, Y., Aron, M.: A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media. J. Elast. 44, 89–96 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Zee, L., Sternberg, E.: Ordinary and strong ellipticity in the equilibrium theory of impressible hyperelastic solids. Arch. Ration. Mech. Anal. 83, 53–90 (1983)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing, Jiangsu, China

    Deren Han

  2. Department of Mathematics, The City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong

    H. H. Dai

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Liqun Qi

Authors
  1. Deren Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. H. Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liqun Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Liqun Qi.

Additional information

This work was supported by the Research Grant Council of Hong Kong.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Han, D., Dai, H.H. & Qi, L. Conditions for Strong Ellipticity of Anisotropic Elastic Materials. J Elasticity 97, 1–13 (2009). https://doi.org/10.1007/s10659-009-9205-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10659-009-9205-5

Keywords

Mathematics Subject Classification (2000)

Search

Navigation