Abstract
The purpose of these notes is to show that symmetry, in particular translational symmetry, is the base on which two notions in the theory of elasticity, associated with deformable bodies in ℝn, namely constitutive laws and a certain class of force densities, are in a one to one correspondence. This correspondence is given by a Neumann problem converting force densities into constitutive maps which in turn characterize constitutive laws.
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
R. Abraham, J.E. Marsden, T. Ratiu: Manifolds, Tensor Analysis,and Applications, Addison Wesley, Global Analysis, 1983
E. Binz: On the Notion of the Stress Tensor Associated with ℝn—invariant Constitutive Laws Admitting Integral Represent at ions, Reports on Mathematical Physics, Vol. 87, (1989), pp.49–57
E. Binz, G. Schwarz, D. Socolescu: On a Global Differential Geometric Description of the Motions of Deformable Media, to appear in Infinite Dimensional Manifolds, Groups, and Algebras, Vol. II, Ed. H.D. Doebner, J. Hennig, 1990
E. Binz, J. Sniatycki, H. Fischer: Geometry of Classical Fields, Mathematics Studies 154, North-Holland Verlag, Amsterdam, 1988
A. Frölicher, A. Kriegl: Linear Spaces and Differentiation Theory, John Wiley, Chichester, England, 1988
W. Greub, S. Halperin, J. Vanstone: Connections, Curvature and Cohomology, I, II, Acad. Press, New York, 1972–73
W. Greub: Lineare Algebra I, Graduate Texts in Mathematics, Vol.23, Springer Verlag, Berlin, Heidelberg, New York, 1981
M. Epstein, R. Segev: Differentiable Manifolds and the Principle of Virtual Work in Continuum Mechanics, Journal of Mathematical Physics, No.5, Vol.21, (1980), p.p. 1243–1245
E. Hellinger: Die allgemeinen Ansätze der Mechanik der Kontinua, Enzykl. Math. Wiss. 4/4, 1914
M. W. Hirsch: Differential Topology, Springer GTM, Berlin, 1976
L. Hörmander: Linear Partial Differential Operations, Grundlehren der mathematischen Wissenschaften, Vol.116, Springer Verlag, Berlin, Heidelberg, New York, 1976
F. John: Partial Differential Equations, Applied Mathematical Science, Vol.1, (1978)
L.D. Landau, E.M. Lifschitz: Lehrbuch der theoretischen Physik, Vol. VII, Elastizitätstheorie, 4. Auflage, Akademie Verlag, Berlin, 1975
Y. Matsushima: Vector Bundle Valued Canonic Forms, Osaka Journal of Mathematics, Vol.8, (1971), p.p. 309–328
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Science+Business Media New York
About this chapter
Cite this chapter
Binz, E. (1991). Symmetry, Constitutive Laws of Bounded Smoothly Deformable Media and Neumann Problems. In: Gruber, B., Biedenharn, L.C., Doebner, H.D. (eds) Symmetries in Science V. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-3696-3_3
Download citation
DOI: https://doi.org/10.1007/978-1-4615-3696-3_3
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6643-0
Online ISBN: 978-1-4615-3696-3
eBook Packages: Springer Book Archive