Symmetry and Bifurcation in Three Dimensional Elasticity. Part I.
Abstract
The purpose of this paper is to study the traction problem in three dimensional nonlinear elasticity using geometric techniques and singularity theory. The first two papers in the series deal with dead loads and configurations that are nearly stress free. As was shown by Signorini (1930) and Stoppelli (1958), this problem has nontrivial solutions. However, their analysis is incomplete for three reasons. First, their load was varied only by a scalar factor; in a full neighborhood in load space of a load which has an axis of equilibrium there are additional solutions missed by their analysis. Second, their analysis is only local in the rotation group, so additional nearly stress free solutions are missed by restricting to rotations near the identity. Third, some classes of loads with a degenerate axis of equilibrium were not considered. This paper completes their analysis by treating these questions as well as stability. The complexity of the answer is indicated by the fact that near certain types of loads, we find up to 40 distinct solutions that are nearly stress free. Our constitutive hypotheses on the stress tensor are 'generic'; for a degenerate stress tensor there can be even more solutions.
Document Details
- Document Type
- Technical Report
- Publication Date
- Apr 01, 1981
- Accession Number
- ADA100944
Entities
People
- D.r.j. Chillingworth
- J. E. Marsden
- Y. H. Wan
Organizations
- University of California, Berkeley