Skip to main content
SpringerLink
Log in

Representations for transversely hemitropic and transversely isotropic stress-strain relations

  • Published:
Journal of Elasticity Aims and scope Submit manuscript

Abstract

The sets of polynomial stress-strain relations for elastic points which are transversely hemitropic and transversely isotropic are presented as projections of free algebraic modules having 20 and 10 generators, respectively. Complete sets of relations for the projections are presented which allow the sets of interest to be identified as free submodules having 12 and 6 generators, respectively. The results are established using the Cartan decomposition of the representation of the adjoint action of the two-dimensional rotation and orthogonal groups on the space of three-by-three symmetric matrices. The results are compared to known representations for nonlinear transversely isotropic stress-strain relations and for linear, transversely hemitropic and transversely isotropic ones.

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. J.E. Adkins, Symmetry relations for orthotropic and transversely isotropic materials. Arch. Rat. Mech. Anal. 4 (1960) 193–213.

    Google Scholar 

  2. F. Bethuel, H. Brezis, B.D. Coleman and F. Hélein, Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinder. Arch. Rat. Mech. Anal. 118 (1992) 149–168.

    Google Scholar 

  3. P. Biscari, G. Capriz and E. Virga, On Surface Biaxiality. Manuscript (1993).

  4. J.-P. Boehler, A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. ZAMM 59 (1979) 157–167.

    Google Scholar 

  5. Y.-C. Chen, Bifurcation and stability of homogeneous deformations of an elastic body under dead load tractions with Z 2 symmetry. J. Elasticity 25 (1991) 117–136.

    Google Scholar 

  6. A. Danescu, Bifurcation in the traction problem for a transversely isotropic material. Math. Proc. Camb. Phil. Soc. 110 (1991) 385–394.

    Google Scholar 

  7. J.L. Ericksen and R.S. Rivlin, Large elastic deformations. of homogeneous anisotropic materials. J. Rat. Mech. Anal. 3 (1954) 281–301.

    Google Scholar 

  8. M.J. Field and R.W. Richardson, Symmetry-breaking and branching patterns in equivariant bifurcation theory, I. Arch. Rat. Mech. Anal. 118 (1992) 297–348.

    Google Scholar 

  9. R.L. Fosdick and G.P. MacSithigh, Minimization in nonliner elasticity theory for bodies reinforced with inextensible cords. J. Elasticity 26 (1991) 239–289.

    Google Scholar 

  10. M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory, II. New York: Springer-Verlag (1985) 512 pp.

    Google Scholar 

  11. A.E. Green and J.E. Adkins, Large Elastic Deformations, 1st edn. Oxford: Clarendon Press (1960) 344 pp.

    Google Scholar 

  12. R. Lipton, Bounds and perturbation series for incompressible elastic composites with transverse isotropic symmetry. J. Elasticity 27 (1992) 193–225.

    Google Scholar 

  13. R.S. Marlow, On the stress in an internally constrained elastic material. J. Elasticity 27 (1992) 97–131.

    Google Scholar 

  14. A.N. Norris, Symmetry conditions for third order elastic moduli and implications in nonlinear wave theory. J. Elasticity 25 (1991) 247–257.

    Google Scholar 

  15. Peter J. Oliver, Canonical elastic moduli. J. Elasticity 19 (1988) 189–212.

    Google Scholar 

  16. P. Podio-Guidugli and M. Vianello, Internal constraints and linear constitutive relations for transversely isotropic materials. Rend. Mat. Acc. Lincei, S. 9 V. 2 (1991) 241–248.

  17. P. Podio-Guidugli and E. Virga, Transversely isotropic elasticity tensors. Proc. R. Soc. Lond. A 411 (1987) 85–93.

    Google Scholar 

  18. A.J.M. Spencer and R.S. Rivlin, Theory of matrix polynomials and its application to the mechanics of isotropic continua. Arch. Rat. Mech. Anal. 2 (1959) 309–336.

    Google Scholar 

  19. M. Stillman and D. Bayer, Macaulay User Manual. Preprint, NSF Regional Geometry Institute, Amherst College (1992).

  20. C. Truesdell and W. Noll, The non-linear theories of mechanics. In: S. Flügge (ed.), Handbuch der Physik, b. 3/3 (1965).

  21. C.-C. Wang, A new representation theorem for isotropic functions, Part 2. Arch. Rat. Mech. Anal. 36 (1970) 198–395.

    Google Scholar 

  22. H. Yong-Zhong and G.del Piero, On the completeness of the crystallographic symmetries in the description of the symmeties of the elastic tensor. J. Elasticity 25 (1991) 203–246.

    Google Scholar 

  23. A. Pipkin and A. Wineman, Material symmetry restrictions on non-polynomial constitutive relations. Arch. Rat. Mech. Anal. 12 (1963) 420–426.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, U.S. Naval Academy, 21402, Annapolis, MD, USA

    John F. Pierce

Authors
  1. John F. Pierce
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work supported in part by National Science Foundation Grant INT-9106519.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pierce, J.F. Representations for transversely hemitropic and transversely isotropic stress-strain relations. J Elasticity 37, 243–280 (1995). https://doi.org/10.1007/BF00041210

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00041210

AMS subject classification (1991)

Search

Navigation