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Co-existent phases in the one-dimensional static theory of elastic bars

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References

  1. Gibbs, J. Willard, On the equilibrium of heterogeneous substances, in the Scientific Papers of J. Willard Gibbs, Vol. 1. Longmans, Green, & Co.: London-New York-Bombay, 1906.

    Google Scholar 

  2. Antman, Stuart S., Nonuniqueness of equilibrium states for bars in tension. J. Math. Anal. Appl. 44 (1973), 333–349.

    Google Scholar 

  3. Antman, Stuart S., & Ernest R. Carbone. Shear and necking instabilities in non-linear elasticity. J. of Elasticity, Vol. 7, No.2 (1977), 125–151.

    Google Scholar 

  4. Eriksen, J. L.. Equilibrium of bars. J. of Elasticity, Vol. 5, Nos. 3–4 (1975), 191–201.

    Google Scholar 

  5. Dafermos, Constantine M., The mixed initial-boundary value problem for the equations of nonlinear one-dimensional viscoelasticity. J. Diff. Equns. 6 (1969), 71–86.

    Google Scholar 

  6. Knowles, J. K., & Eli Sternberg. On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics. Technical Report # 37, Office of Naval Research, June, 1977.

  7. Knowles, J. K., & Eli Sternberg. On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Rational Mech. Anal. 63 (1977), 321.

    Google Scholar 

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

    Google Scholar 

  9. Bell, J. F., The experimental foundations of solid mechanics. Encyclopedia of Physics, vol. VI a/1, ed. c. Truesdell. Springer-Verlag: Berlin-Heidelberg-New York.

  10. Ward, I. M., & L. Holliday. A general introduction to the structure and properties of oriented polymers. Structure and Properties of Oriented Polymers, ed. I. M. Ward. John Wiley & Sons: New York-Toronto. 1–35.

  11. Keller, A., & J.G. Rider, On the tensile behavior of oriented polyethylene. J. Materials Science, 1 (1966), p. 389–398.

    Google Scholar 

  12. Sharp, William N., The Portevin-le Chatelier effect in aluminum single crystals and polycrystals. Ph. D. dissertation. The Johns Hopkins University. Baltimore, Maryland (1966).

    Google Scholar 

  13. Mc Reynolds, Adrew Wetherbee, Plastic deformation waves in aluminum. Trans. American Inst. Mining Metallurgical Engin., 185 (1949), 185.

    Google Scholar 

  14. Phillips, V. A., & A. J. Swain, Yield point phenomena and stretcher-strain markings in aluminum-magnesium alloys. J. Inst. Metals, 81 (1952–53), 625–647.

    Google Scholar 

  15. Young, L. C., Lectures on the Calculus of Variations and Optimal Control Theory. W. B. Saunders Company: Philadelphia-London-Toronto, 1969.

    Google Scholar 

  16. Hardy, G. H., J. E. Littlewood & G. Polya. Inequalities. Cambridge, 1934.

  17. Rockafellar, R. Tyrrell, Integral functionals, normal integrands and measurable selections, in Nonlinear Operators and the Calculus of Variations, ed. A. Dold & B. Eckmann. Lecture Notes in Mathematics. Springer-Verlag: Berlin-Heidelberg-New York, 1976.

    Google Scholar 

  18. Rudin, Walter. Real and Complex Analysis. Second Edition. McGraw-Hill: New York 1974.

    Google Scholar 

  19. Hadamard, J. Leçons sur le Calcul des Variations. Tome premier. Librarie Scientifique A. Hermann et Fils: Paris, 1910.

    Google Scholar 

  20. Hobson, E. W. Theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. II. Second edition. Harren Press: Washington, D.C., 1950.

    Google Scholar 

  21. Kearsley, E. A., L. J. Zapas & R. W. Penn, Private communication concerning their experiments on polyethylene. The National Bureau of Standards, Wash., D.C. 20234.

  22. Maxwell, James Clerk. On the dynamical evidence of the molecular constitution of bodies. Nature XI (1875), 357.

    MATH  Google Scholar 

  23. Thomson, James: Considerations on the abrupt change at boiling on condensation in reference to the continuity of the fluid state of matter. Collected Papers in Physics and Engineering, Cambridge (1912), 278.

  24. Kaczyński, H., & C. Olech, Existence of solutions of orientor fields with non-convex right-hand side. Annales Polonici Mathematics XXIX (1974), 61–66.

    Google Scholar 

  25. Hartman, Philip. Ordinary Differential Equations. John Wiley and Sons: New York, 1964.

    Google Scholar 

  26. Keller, Herbert B. Numerical Methods for Two-Point Boundary-Value Problems. Blaisdell Publishing Co.: Waltham-Toronto-London, 1968.

    Google Scholar 

  27. Gelfand, I. M., & S.V. Fomin, Calculus of Variations, trans. and ed. by R. A. Silverman. Prentice Hall: Englewood Cliffs, 1963.

    Google Scholar 

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  1. Department of Aerospace Engineering & Mechanics, University of Minnesota, Minneapolis

    Richard D. James

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  1. Richard D. James
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Communicated by C.-C. Wang

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James, R.D. Co-existent phases in the one-dimensional static theory of elastic bars. Arch. Rational Mech. Anal. 72, 99–140 (1979). https://doi.org/10.1007/BF00249360

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