Stability and bifurcation are two research areas in mechanics. At present, there are considerable gaps and confusion on the relation of the two areas. On one hand, virtually no studies have 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 rare to find in the published work that a bifurcation is claimed from the onset of instability, and that instability is concluded from the existence of a bifurcation.
We give an analysis of the connections between bifurcation of equilibrium solutions in nonlinear elasticity and change of stability of the solutions. It establishes the relations between bifurcation conditions and stability conditions for general elastic materials of arbitrary geometry under general boundary conditions. The stability analysis is based on the energy criterion, which asserts that a deformation is stable if it renders the total potential energy functional a minimum. For a branch of equilibrium deformations under continuous loading, we show that if the undeformed body is stable in the sense that the second variation of the energy functional is positive definite, and if the linearized equilibrium equation has only the trivial solution along the branch (i.e., no bifurcation occurs), then all equilibrium deformations on the branch are stable. Furthermore, we show that if a certain non-vanishing third variation condition is satisfied, then the deformation becomes unstable at a bifurcation point.
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