Abstract
Doyle & Ericksen [1956, p. 77] observed that the Cauchy stress tensor σ can be derived by varying the internal free energy ψ with respect to the Rie- mannian metric g on the ambient space: σ=2ϱ ∂ψ/∂g. Their formula has gone virtually unnoticed in the literature of continuum mechanics. In this paper we shall establish the material version of this formula: \( \sum { = 2\varrho \partial \bar{\Psi }} /\partial G \) for the rotated stress tensor Σ, and shall address some of the reasons why these formulae are of fundamental significance. Making use of these formulae one can derive elasticity tensors and establish rate forms of the hyperelastic constitutive equations for the Cauchy stress tensor and the rotated stress tensor, as discussed in Sections 4 and 5. The role of the rotated stress tensor in the formulation of continuum theories has been noted by Green & Naghdi [1965]. Generalizations of hypoelasticity based on the use of the rotated stress tensor have been considered by Green & McInnis [1967].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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.
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.
Arnold, V. 1966b. “An a priori Estimate in the Theory of Hydrodynamic Stability,” Trans. Amer. Math. Soc., 79 (1969), pp. 267–269.
Arnold, V. 1978 Mathematical Methods of Classical Mechanics, New York: Springer-Verlag.
Ball, J. M., & J. E. Marsden, 1984. Quasiconvexity at the boundary, second variations, and stability in nonlinear elasticity. Arch. Rational Mech. Anal, (to appear).
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.
Coleman, B. D., & W. Noll, 1959. “On the Thermodynamics of Continuous Media,” Arch. Rational Mech. Anal., 4, pp. 97–128.
Dienes, J. K., 1979. “On the Analysis of Rotation and Stress Rate in Deforming Bodies,” Acta Mechanica, 32, pp. 217–232.
Doyle, T. C., & J. L. Ericksen, 1956. Nonlinear Elasticity, in Advances in Applied Mechanics IV. New York: Academic Press, Inc.
Ericksen, J. L., 1961. “Conservation Laws for Liquid Crystals,” Tran. Soc. Rheol, 5, pp. 23–34.
Green, A. E., & B. C. Mcinnis, 1967. “Generalized Hypo-Elasticity,” Proc. Roy. Soc. Edinburgh, A 57, p. 220.
Green, A. E., & P. M. Naghdi, 1965. “A General Theory of an Elastic-Plastic Continuum,” Arch. Rational Mech. Anal., 18, p. 251.
Green, A. E., & R. S. Rivlin, 1964a. “On Cauchy’s Equations of Motion,” J. Appl. Math, and Physics (ZAMP), 15, pp. 290–292.
Green, A. E., & R. S. Rivlin, 1964b. “Multipolar Continuum Mechanics,” Arch. Rational Mech. Anal., 17, pp. 113–147.
Green, A. E., & R. S. Rivlin, 1964C. “Simple Force and Stress Multipoles,” Arch. Rational Mech. Anal., 16, pp. 327–353.
Hawking, S., & G. Ellis, 1973. The Large Scale Structure of Spacetime, Cambridge, England: Cambridge Univ. Press.
Holm, D., & B. Kuperchmidt, 1983. “Poisson Brackets and Clebch Representations for Magnetohydrodynamics, Multifluid Plasmas and Elasticity,” Physica D 6, 347–356.
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.
Knops, R., & E. Wilkes, 1973. Theory of Elastic Stability, in Handbuch der Physik, VIa/3, S. Flügge, ed., Belin: Springer-Verlag.
Lang, S., 1972. Differential Manifolds, Reading, MA: Addison-Wesley Publishing Co. Inc.
Marsden, J. E., & T. J. R. Hughes, 1983. Mathematical Foundations of Elasticity, Englewood-Clifïs, N. Y. Prentice-Hall, Inc.
Marsden, J. E., T. Ratiu & A. Weinstein, 1982. “Semidirect Products and Reduction in Mechanics,” Trans. Am. Math. Soc. (to appear).
Misner, C., K. Thorne & J. Wheeler, 1982. Gravitation, S. Francisco: W. H. Freman & Company, Publishers, Inc.
Naghdi, P. M., 1972. The Theory of Shells and Plates, in Handbuch der Physik, C. Truesdell ed., Vol. VIa/2,Berlin: Springer-Verlag.
Noll, W., 1963. La Méthode Axiomatique dans les Mécaniques Classiques et Nouvelles, Paris, pp. 47–56.
Rosenfeld, L., 1940. “Sur le Tenseur d’Impulsion-Energy,” Mem. Acad. R. Belg. Sei 18, no. 6, pp. 1–30.
Sewell, M. J., 1966. “On Configuration-Dependent Loading,” Arch. Rational Mech, Anal, 23, pp. 327–351.
Sokolnikoff, I. S., 1956. The Mathematical Theory of Elasticity (2d ed.), New York: McGraw-Hill Book Company.
Toupin, R. A., 1964. “Theories of Elasticity with Couple Stress,” Arch. Rational Mech. Anal., 11, pp. 385–414.
Truesdell, C, 1955. “Hypo-elasticity,” J. Rational Mech. Anal., 4, pp. 83–133.
Truesdell, C., & R. A. Toupin, 1960. The Classical Field Theories, in Handbuch der Physik, Vol. III/l, S. Flügge, ed., Berlin: Springer-Verlag.
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.
Author information
Authors and Affiliations
Additional information
Dedicated to J. L. Ericksen Contents
Rights and permissions
Copyright information
© 1986 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Simo, J.C., Marsden, J.E. (1986). On the Rotated Stress Tensor and the Material Version of the Doyle-Ericksen Formula. In: The Breadth and Depth of Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61634-1_20
Download citation
DOI: https://doi.org/10.1007/978-3-642-61634-1_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-16219-3
Online ISBN: 978-3-642-61634-1
eBook Packages: Springer Book Archive