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The Rayleigh-Ritz Technique and the Lagrange Equations in Continuum Mechanics: Formulations for Material and Non-Material Volumes

  • Hans Irschik
  • Helmut J. Holl
  • Franz Hammelmüller
Part of the International Centre for Mechanical Sciences book series (CISM, volume 444)

Abstract

In the present Lecture, we use the Rayleigh-Ritz technique in connection with the Lagrange equations in order to approximate the partial differential equations of continuum mechanics by means of a system of ordinary differential equations in time. We start from the local form of the equation of balance of momentum, from which we proceed to an extended Hamilton principle, eventually ending up with the Lagrange equations. We lay special emphasis upon the functional dependencies to be considered in the Rayleigh-Ritz technique with respect to the spatial and the material descriptions of continuum mechanics. In analogy to the notion of a material-time derivative, we define material variations and material partial derivatives of the quantities and entities that enter the Lagrange equations, and we relate these derivatives to local partial derivatives, the latter being particularly feasible with respect to the spatial formulation of continuum mechanics. Having formulated the Lagrange equations for the case of a material volume, we present a re-formulation for problems that are posed with respect to non-material volumes. As an example for such a problem, we shortly treat the coiling of a strip. The presented re-formulation of the Lagrange equations for non-material volumes represents an alternative derivation of recent results by Irschik and Holl (2002).

Keywords

Lagrange Equation Local Partial Derivative Virtual Displacement Spatial Formulation Dynamic Boundary Condition 
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.

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References

  1. Belyaev, A.K. (2004). Basics of Continuum Mechanics. In: Irschik, H. and Schlacher, K., eds., Advanced Dynamics and Control of Structures and Machines. CISM Courses and Lectures No. 444. Wien—New York: Springer-Verlag.Google Scholar
  2. Belyaev, A.K. (2004). Basics of Analytical Mechanics. In: Irschik, H. and Schlacher, K., eds., Advanced Dynamics and Control of Structures and Machines. CISM Courses and Lectures No. 444. Wien—New York: Springer-Verlag.Google Scholar
  3. Bremer, H.. and Pfeiffer, F. (1992). Elastische Mehrkörpersysteme. Stuttgart: B.G. Teubner.MATHGoogle Scholar
  4. Gershenfeld, N. (1999). The Nature of Mathematical Modelling. Cambridge: University Press.Google Scholar
  5. Gerstmayr, J., Irschik, H. and Dibold, M. (2004). Computational Dynamics of an Elasto-Plastic Structural Element With Rigid-Body Degrees-of-Freedom. In: Irschik, H. and Schlacher, K., eds., Advanced Dynamics and Control of Structures and Machines. CISM Courses and Lectures No. 444, Wien—New York: Springer-Verlag.Google Scholar
  6. Gurtin, M. E. (1972). The linear theory of elasticity. In: Flügge, S., ed., Handbuch der Physik, Vol. VIa/2, Berlin: Springer-Verlag.Google Scholar
  7. Gummert, P., Reckling, K.-A. (1987). Mechanik, 2nd Edition. Braunschweig: F. Vieweg Sohn.Google Scholar
  8. Hamel, G. (1967). Theoretische Mechanik. Berlin: Springer-Verlag.MATHGoogle Scholar
  9. Irschik, H. (2001). Zum Fingerschen Virial verformbarer Körper in der nichtlinearen Statik. Sitzungsberichte der Öster. Akademie der Wiss., Math.-Nat. Klasse Abt. II, 209: 47–66.Google Scholar
  10. Irschik, H. (2004). A Treatise on the Equations of Balance and on the Jump Relations in Continuum Mechanics. In: Irschik, H. and Schlacher, K., eds., Advanced Dynamics and Control of Structures and Machines. CISM Courses and Lectures No. 444, Wien—New York: Springer-Verlag.Google Scholar
  11. Irschik, H.. and Holl, H. J. (2002). The Equations of Lagrange Written for a Non-Material Volume, Acta Mechanica 153:.231–248.Google Scholar
  12. Irschik H., Holl H.J. (2004): Mechanics of Variable-Mass Systems: Part 1: Balance of Mass and Linear Momentum. ASME-Applied Mechanics Reviews, to appear.Google Scholar
  13. Parkus, H. (1988). Mechanik der festen Körper, Nachdruck der 2. Auflage. Wien-New York: Springer-Verlag.Google Scholar
  14. Schlacher K. and Kugi A. (2002): Symbolic Methods for Systems of Implicit Ordinary Differential Equations. Mechanics of Structures and Machines, 30: 411–429.CrossRefMathSciNetGoogle Scholar
  15. Truesdell, C. and Toupin, R. (1960). The Classical Field Theories. In: Flügge, S., ed., Handbuch der Physik, Vol. III/1, Berlin: Springer-Verlag.Google Scholar
  16. Sagan, H. (1992). Introduction to the Calculus of Variations, 2nd edition. New York: Dover Publ.Google Scholar
  17. Shabana, A.A. (1998). Dynamics of Multibody Systems, Second Ed. Cambridge: University Press.MATHGoogle Scholar
  18. Ziegler, F. (1998). Mechanics of Solids and Fluids, 2nd Engl. Edition, corrected 2nd printing. Berlin: Springer-Verlag.Google Scholar
  19. Warsi, Z. U. A. (1999). Fluid Dynamics, Theoretical and Computational Approaches,2nd Edition. CRC Press.Google Scholar

Copyright information

© Springer-Verlag Wien 2004

Authors and Affiliations

  • Hans Irschik
    • 1
  • Helmut J. Holl
    • 1
  • Franz Hammelmüller
    • 2
  1. 1.Johannes Kepler University of Linz and LCMLinzAustria
  2. 2.Linz Center of Excellence in Mechatronics (LCM)LinzAustria

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