Symmetry and bifurcation in three dimensional elasticity

Chillingworth, D. R. J., J. E. Marsden and Y. H. Wan

Arch. Rational Mech. Anal., 80, (1982), 295-331; See also 83, (1983), 363-395, and 84, (1984), 203-233

Abstract:

This paper is the first of a series of three devoted to the study of the traction problem in three-dimensional nonlinear elasticity by means of geometric techniques and singularity theory. The first two papers in the series treat the traction problem with dead loads for configurations that are nearly stress-free. As was shown, this problem has nontrivial solutions. However, their analysis is incomplete for three reasons. First, their load is varied only by a scalar factor; in a full neighborhood in load space of a load that 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 the rotations to those near the identity. Third, some classes of loads with a degenerate axis of equilibrium are not considered. This series of papers completes their analysis by treating these questions as well as the stability of solutions, 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.