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On Saint-Venant's problem for an elastic strip*

Published online by Cambridge University Press:  14 November 2011

Alexander Mielke
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D7000 Stuttgart 80, West Germany

Synopsis

The equilibrium equations for elastic deformations of an infinite strip are considered. Under the assumption of sufficiently small strains along the whole body, it is shown that all solutions lie on a six-dimensional manifold. This is achieved by rewriting the field equations as a differential equation in a function spaceover the cross-section, the axial variable taken as time. Then the theory of centre manifolds for elliptic systems applies. Thus the local Saint-Venant's problem is solved. Moreover, the structure of the finite-dimensional solution space is analysed to reveal exactly the two-dimensional rod equations of Kirchhoff. The constitutive relations for this rod model are calculated in a mathematically rigorous way out of the constitutive law of the material forming the strip.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1988

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