| Stability problems where nonlinear material behavior is seen are an important area of study in solid mechanics. Compressive buckling failure of beams and tensile necking in bars are two examples of stability problems where nonlinear material behavior is important. Sheet metal forming also offers a wide range of stability problems involving nonlinear material behavior.; In many cases where stability is an issue, a solid can be examined by looking at its relative dimensions. In this way many classical structural theories can be found. Here an asymptotic approach to stability problems finds a beam stability criterion based on the three-dimensional stability functional when two dimensions are much smaller than the third, and a shell stability criterion when one dimension is much smaller than the other two. Applications of these stability theories are investigated.; For one-dimensional problems a beam buckling theory is found that is valid when two dimensions are much smaller than the third. The validity of the asymptotic approach is examined for the case of thin walled beams, when a third small dimension is introduced. Comparisons are made with thin-walled linear elastic beam stability theory. Tensile loading of bars, resulting in a necking instability, is also examined.; For two-dimensional problems the shell stability theory is applied to sheet metal forming applications. Here large deformations and rotations, combined with nonlinear material behavior and contact with rigid surfaces, provide a highly nonlinear problem with a practical application. Instabilities that arise in sheet metal forming are examined for the axisymmetric conical cup test. Results from the asymptotic stability analysis are compared with experimental results. Wrinkling and localization are predicted and failure curves for the conical cup test are plotted. |
