Journal of Elasticity

, Volume 134, Issue 2, pp 219–234 | Cite as

Strong Ellipticity Conditions for Orthotropic Bodies in Finite Plane Strain

  • Adair R. AguiarEmail author


Necessary and sufficient conditions for strong ellipticity of the equilibrium equations governing finite plane deformations are established for the class of orthotropic and compressible hyperelastic materials. The conditions are then specialized to the case of an orthotropic material having the symmetry axes parallel to the principal directions of deformation. This case is relevant in experimental protocols employed in the characterization of constitutive relations of rubber-like materials and biological tissues. The classical conditions for infinitesimal linear elasticity and isotropic finite elasticity are recovered as particular cases.


Nonlinear elasticity Orthotropy Ellipticity condition Constitutive modeling 

Mathematics Subject Classification (2010)

74B20 74E10 



We would like to acknowledge the financial support provided by the National Council for Scientific and Technological Development (CNPq) and the State of São Paulo Research Foundation (FAPESP). We are also grateful to the referees, who have provided detailed comments that lead to an expansion of the scope of this work and improvements in the presentation.


  1. 1.
    Bose, N.K., Modarressi, A.R.: General procedure for multivariable polynomial positivity test with control applications. IEEE Trans. Autom. Control 21, 696–701 (1976) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, Y.: On strong ellipticity and the Legendre-Hadamard condition. Arch. Ration. Mech. Anal. 113, 165–175 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chirită, S., Danescu, A., Ciarletta, M.: On the strong ellipticity of the anisotropic linearly elastic materials. J. Elast. 87, 1–27 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gurtin, M.E., Fried, E., Anand, L.: The Mechanics and Thermodynamics of Continua. Cambridge University Press, Cambridge (2009) Google Scholar
  5. 5.
    Han, D., Dai, H.H., Qi, L.: Conditions for strong ellipticity of anisotropic elastic materials. J. Elast. 97, 1–13 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hasan, M.A., Hasan, A.A.: A procedure for the positive definiteness of forms of even order. IEEE Trans. Autom. Control 41, 615–617 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Holzapfel, G.A., Ogden, R.W.: On planar biaxial tests for anisotropic nonlinearly elastic solids. A continuum mechanical framework. Math. Mech. Solids 14, 474–489 (2009) CrossRefzbMATHGoogle Scholar
  8. 8.
    Humphrey, J.D., Strumpf, R.K., Yin, F.C.P.: Determination of a constitutive relation for passive myocardium: I. A new functional form. J. Biomech. Eng. 112, 333–339 (1990) CrossRefGoogle Scholar
  9. 9.
    Knowles, J.K., Sternberg, E.: On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elast. 5, 341–361 (1975) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Knowles, J.K., Sternberg, E.: On the failure of ellipticity of the equations for finite elastostatic plane strains. Arch. Ration. Mech. Anal. 63, 321–336 (1977) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ku, W.H.: Explicit criterion for the positive definiteness a general quartic form. IEEE Trans. Autom. Control 10, 372–373 (1965) CrossRefGoogle Scholar
  12. 12.
    Merodio, J., Ogden, R.W.: Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformations. Arch. Mech. 54, 525–552 (2002) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Merodio, J., Ogden, R.W.: Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int. J. Solids Struct. 40, 4707–4727 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Merodio, J., Ogden, R.W.: A note on strong ellipticity for transversely isotropic linearly elastic solids. Q. J. Mech. Appl. Math. 56, 589–591 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Merodio, J., Ogden, R.W.: Remarks on instabilities and ellipticity for a fiber reinforced compressible nonlinearly elastic solid under plane deformation. Q. Appl. Math. 63, 325–333 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Merodio, J., Ogden, R.W.: Tensile instabilities and ellipticity in fiber-reinforced compressible non-linearly elastic solids. Int. J. Eng. Sci. 43, 697–706 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Merodio, J., Ogden, R.W.: The influence of the invariant \(I_{8}\) on the stress–deformation and ellipticity characteristics of doubly fiber-reinforced non-linearly elastic solids. Int. J. Non-Linear Mech. 41, 556–563 (2006) ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rivlin, R.S., Saunders, D.W.: Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Philos. Trans. R. Soc. Lond. A 243, 251–288 (1951) ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Spencer, A.J.M.: Constitutive theory for strongly anisotropic solids. In: Spencer, A.J.M. (ed.) Continuum Theory of the Mechanics of Fiber Reinforced Composites. CISM Courses and Lectures, vol. 282, pp. 1–32. Springer, Wien (1984) CrossRefGoogle Scholar
  21. 21.
    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) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, F., Qi, L.: Comments on ‘Explicit criterion for the positive definiteness of a general quartic form’. IEEE Trans. Autom. Control 50, 416–418 (2005) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Structural EngineeringUniversity of São PauloSão CarlosBrazil

Personalised recommendations