In this talk, we discuss some issues on the bifurcation of equilibrium solutions in elasticity and the change of stability of the solutions.
As two research areas in mechanics and applied mathematics, the bifurcation theory and the stability theory deal with different phenomena: stability theory concerns the behavior of the solution of a physical system under disturbances, while bifurcation theory concerns the evolution of multiple solution branches. At present, there are considerable gaps and confusion on the relation of the two areas. On one hand, little study has been conducted to systematically identify possible connections between stability conditions and bifurcation conditions. On the other hand, researchers often use the notions and conditions in stability and bifurcation indistinguishably. It is not uncommon to find in the published work that a bifurcation is claimed from onset of instability, and that instability is concluded from the existence of a bifurcation.
Nevertheless, examples have suggested possible connections between stability conditions and bifurcation conditions. For instance, the buckling of an Euler column corresponds to a bifurcation point, and it has been shown that, at this point, the unbuckled deformation becomes unstable in the sense that it has greater energy than the buckled deformation does. Such examples, however, are too few to give us a complete understanding of the issue.
An effort has been made in Chen and Haughton [1] to compare the existence of bifurcation and the change of stability for the inflation and stretch of an elastic cylinder. A number of elastic materials were examined in a large range of deformations. It was found that the cylindrical deformations always change stability at a bifurcation point. This observation extends the result of classical problem of the Euler column to inhomogeneous deformations of nonlinear elastic materials.
As one of few known results on relationship between stability conditions and bifurcation conditions for general elastic materials, Ericksen and Toupin [2] and Hill [3] have shown, for a general elastic body of arbitrary geometry, that if a deformation is stable in the sense that the second variation of the total potential energy functional is positive definite for all admissible disturbances, then the linearized equilibrium equation has the trivial solution only, that is, there is no bifurcation. In this talk, we present the conditions under which the converse of their theorem holds, as well as a strengthened version of the theorem.
References:
[1] Y. C. Chen and D. M. Haughton: Stability and bifurcation of inflation of elastic
cylinders. Proceedings of the Royal Society London 459, 137-156, 2003.
[2] J. L. Ericksen and R. A. Toupin: Implications of Hadamard’s conditions for elastic
stability with respect to uniqueness theorems. Can. J. Math. 8, 432-436, 1956.
[3] R. Hill: On uniqueness and stability in the theory of finite elastic strain. J. Mech.
Phys. Solids 5, 229-241, 1957.