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On the failure of ellipticity of the equations for finite elastostatic plane strain


In this paper we establish necessary and sufficient conditions, in terms of the local principal stretches, for ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. The material under consideration is assumed to be homogeneous and isotropic, but its strain-energy density is otherwise unrestricted. We also determine the directions of the characteristic curves appropriate to plane elastostatic deformations that are accompanied by a failure of ellipticity.

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  1. 1.

    J. K. Knowles & Eli Sternberg, On the ellipticity of the equations of nonlinear elastostatics for a special material, Journal of Elasticity, 5 (1975), 3–4, 341.

    Google Scholar 

  2. 2.

    P.J. Blatz & W.L. Ko, Application of finite elastic theory to the deformation of rubbery materials, Transactions of the Society of Rheology 6 (1962), 223.

    Google Scholar 

  3. 3.

    C. Truesdell & W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Vol. III/3, Springer, Berlin, 1965.

    Google Scholar 

  4. 4.

    R.J. Knops & L.E. Payne, Uniqueness theorems in linear elasticity, Springer Tracts in Natural Philosophy, Vol. 19, Springer, Berlin, 1971.

    Google Scholar 

  5. 5.

    M.E. Gurtin, The linear theory of elasticity, in Handbuch der Physik, Vol. VIa/2, Springer, Berlin, 1972.

    Google Scholar 

  6. 6.

    R.A. Toupin & B. Bernstein, Sound waves in deformed perfectly elastic materials. Acoustoelastic effect, The Journal of the Acoustical Society of America, 33 (1961), 2, 216.

    Google Scholar 

  7. 7.

    Fritz John, Plane strain problems for a perfectly elastic material of harmonic type, Communications on Pure and Applied Mathematics, XIII (1960), 239.

  8. 8.

    J.K. Knowles & Eli Sternberg, On the singularity induced by certain mixed boundary conditions in linearized and nonlinear elastostatics, International Journal of Solids and Structures, 11 (1975), 1173.

    Google Scholar 

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The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.

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Knowles, J.K., Sternberg, E. On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Rational Mech. Anal. 63, 321–336 (1976).

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