Archive for Rational Mechanics and Analysis

, Volume 86, Issue 3, pp 213–231 | Cite as

On the rotated stress tensor and the material version of the Doyle-Ericksen formula

  • Juan C. Simo
  • Jerrold E. Marsden
Article

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Antman, S. S., & J. E. Osborne, 1979. “The Principle of Virtual Work and Integral Laws of Motion,” Arch. Rational Mech. Anal. 69, pp. 231–262.Google Scholar
  2. Arnold, V. 1966a. “Sur la Géometrie Différentielle des Groupes de Liede Dimension Infinie et ses Applications à l'hydrodynamique des Fluids Parfaits,” Ann. Inst. Fourier, Grenoble, 16, pp. 319–361.Google Scholar
  3. Arnold, V. 1966b. “An a priori Estimate in the Theory of Hydrodynamic Stability”, Trans. Amer. Math. Soc., 79 (1969), pp. 267–269.Google Scholar
  4. Arnold, V. 1978 Mathematical Methods of Classical Mechanics, New York: Springer-Verlag.Google Scholar
  5. Ball, J. M., & J. E. Marsden, 1984. Quasiconvexity at the boundary, second variations, and stability in nonlinear elasticity. Arch. Rational Mech. Anal. (to appear).Google Scholar
  6. Belinfante, F. 1939. “On the Current and the Density of the Electric Charge, the Energy, the Linear Momentum, and the Angular Momentum of Arbitrary Fields,” Physica, 6, p. 887 and 7, pp. 449–474.Google Scholar
  7. Coleman, B. D., & W. Noll, 1959. “On the Thermodynamics of Continuous Media,” Arch. Rational Mech. Anal., 4, pp. 97–128.Google Scholar
  8. Dienes, J. K., 1979. “On the Analysis of Rotation and Stress Rate in Deforming Bodies,” Acta Mechanica, 32, pp. 217–232.Google Scholar
  9. Doyle, T. C., & J. L. Ericksen, 1956. Nonlinear Elasticity, in Advances in Applied Mechanics IV. New York: Academic Press, Inc.Google Scholar
  10. Ericksen, J. L., 1961. “Conservation Laws for Liquid Crystals,” Tran. Soc. Rheol., 5, pp. 23–34.Google Scholar
  11. Green, A. E., & B. C. McInnis, 1967. “Generalized Hypo-Elasticity,” Proc. Roy. Soc. Edinburgh, A 57, p. 220.Google Scholar
  12. Green, A. E., & P. M. Naghdi, 1965. “A General Theory of an Elastic-Plastic Continuum,” Arch. Rational Mech. Anal., 18, p. 251.Google Scholar
  13. Green, A. E., & R. S. Rivlin, 1964a. “On Cauchy's Equations of Motion,” J. Appl. Math. and Physics (ZAMP), 15, pp. 290–292.Google Scholar
  14. Green, A. E., & R. S. Rivlin, 1964b. “Multipolar Continuum Mechanics,” Arch. Rational Mech. Anal., 17, pp. 113–147.Google Scholar
  15. Green, A. E., & R. S. Rivlin, 1964c. “Simple Force and Stress Multipoles,” Arch. Rational Mech. Anal., 16, pp. 327–353.Google Scholar
  16. Hawking, S., & G. Ellis, 1973. The Large Scale Structure of Spacetime, Cambridge, England: Cambridge Univ. Press.Google Scholar
  17. Holm, D., & B. Kuperchmidt, 1983. “Poisson Brackets and Clebch Representations for Magnetohydrodynamics, Multifluid Plasmas and Elasticity,” Physica D 6, 347–356.Google Scholar
  18. Holm, D. D., J. E. Marsden, T. Ratiu & A. Weinstein, 1983. “Nonlinear Stability Conditions and a priori Estimates for Barotropic Hydrodynamics” Physics Lett. 98 A 15–21.Google Scholar
  19. Knops, R., & E. Wilkes, 1973. Theory of Elastic Stability, in Handbuch der Physik, VIa/3, S. Flügge, ed., Berlin: Springer-Verlag.Google Scholar
  20. Lang, S., 1972. Differential Manifolds, Reading, MA: Addison-Wesley Publishing Co. Inc.Google Scholar
  21. Marsden, J. E., & T. J. R. Hughes, 1983. Mathematical Foundations of Elasticity, Englewood-Cliffs, N. Y. Prentice-Hall, Inc.Google Scholar
  22. Marsden, J. E., T. Ratiu & A. Weinstein, 1982. “Semidirect Products and Reduction in Mechanics,” Trans. Am. Math. Soc. (to appear).Google Scholar
  23. Misner, C., K. Thorne & J. Wheeler, 1982. Gravitation, S. Francisco: W. H. Freman & Company, Publishers, Inc.Google Scholar
  24. Naghdi, P. M., 1972. The Theory of Shells and Plates, in Handbuch der Physik, C. Truesdell ed., Vol. VIa/2, Berlin: Springer-Verlag.Google Scholar
  25. Noll, W., 1963. La Méthode Axiomatique dans les Mécaniques Classiques et Nouvelles, Paris, pp. 47–56.Google Scholar
  26. Rosenfeld, L., 1940. “Sur le Tenseur d'Impulsion-Energy,” Mem. Acad. R. Belg. Sci., 18, no.6, pp. 1–30.Google Scholar
  27. Sewell, M. J., 1966. “On Configuration-Dependent Loading,” Arch. Rational Mech. Anal., 23, pp. 327–351.Google Scholar
  28. Sokolnikoff, I. S., 1956. The Mathematical Theory of Elasticity (2d ed.), New York: McGraw-Hill Book Company.Google Scholar
  29. Toupin, R. A., 1964. “Theories of Elasticity with Couple Stress,” Arch. Rational Mech. Anal., 11, pp. 385–414.Google Scholar
  30. Truesdell, C., 1955. “Hypo-elasticity,” J. Rational Mech. Anal., 4, pp. 83–133.Google Scholar
  31. Truesdell, C., & R. A. Toupin, 1960. The Classical Field Theories, in Handbuch der Physik, Vol. III/1, S. Flügge, ed., Berlin: Springer-Verlag.Google Scholar
  32. Truesdell, C., & W. Noll, 1965. The Non-Linear Field Theories of Mechanics, in Handbuch der Physik, Vol. III/3, S. Flügge, ed., Berlin: Springer-Verlag.Google Scholar

Copyright information

© Springer-Verlag GmbH & Co 1984

Authors and Affiliations

  • Juan C. Simo
    • 1
    • 2
  • Jerrold E. Marsden
    • 1
    • 2
  1. 1.Structural Engineering and Structural MechanicsUniversity of CaliforniaBerkeley
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeley

Personalised recommendations