A Variational Framework for n-th Order Invariant Continuum Mechanics

  • R. Segev
Conference paper


Variational methods have found many applications in computational continuum mechanics as they suggest numerical algorithms for the solutions of concrete problems. However, in the modern axiomatic foundations of continuum mechanics as reflected in the works of Truesdell, Noll and others (e.g., [1,2]) the variational approach is nonexistent. The arguments against the application of variational principles in the formulation of the theory of continuum mechanics are given in [3]. These arguments may be summarized as follows. (1) Regarding the claim that variational principles are invariant it is argued that “now…the principles of tensor analysis offer us a simpler and direct method…for obtaining invariant statements. (2) “The boundary conditions emerging from a variational principle depend upon what boundary integrals… are included in the statement of the principle and the selection of the boundary integrals is not always dictated by the physical idea which the variational principle is assumed to express.” (3) Regarding the claim that variational principles are statements about the system as a whole and the elegance that variational principles sometimes exhibit, it is argued that unlike the “beautiful variational statements for systems of mass points” the variational principles in continuum mechanics are usually “awkward or unnatural”. (4) “No variational principle has ever been shown to yield Cauchy’s fundamental theorem in its basic sense as asserting that existence of the stress vector implies the existence of the stress tensor”.


Vector Bundle Variational Principle Virtual Work Cotangent Bundle Boundary Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Truesdell, C., A first course in rational continuum mechanics I. New York: Academic Press 1977.MATHGoogle Scholar
  2. 2.
    Truesdell C. and Noll, W., The nonlinear field theories of mechanics. Handbuch der Physik, III/3, ed. Flugge, S. Berlin: Springer Verlag 1965.Google Scholar
  3. 3.
    Truesdell, C. and Toupin R., The classical field theories. Handbuch der Physik, ed. Flugae, S. Berlin: Springer Verlag 1960.Google Scholar
  4. 4.
    Palais, R., Foundations of global non-linear analysis. New York: Benjamin 1968.MATHGoogle Scholar
  5. 5.
    Golubitsky, M. and Guillemin, V., Stable mappings and their singularities. New York: Springer Verlag 1973.MATHCrossRefGoogle Scholar
  6. 6.
    Segev, R., Forces and the existence of stresses in invariant continuum mechanics. J. Math. Phys., to appear.Google Scholar

Copyright information

© Springer Japan 1986

Authors and Affiliations

  • R. Segev
    • 1
  1. 1.The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical EngineeringBen-Gurion University of the NegevBeer ShevaIsrael

Personalised recommendations