Stability of relative equilibria II: Three dimensional elasticity

Simo, J. C., T. A. Posbergh and J. E. Marsden

Arch. Rational Mech. Anal., 115, (1991), 61-100

Abstract:

In this paper we consider the concrete application of the reduced energy-momentum method to an infinite-dimensional and highly non-trivial example: three-dimensional elasticity. Two main objectives motivate this work. First, we provide a detailed illustration of the abstract setting discussed in Part I of this paper in a concrete example which has a strong interest of its own. Second, we demonstrate that the application of the method to an infinite-dimensional example leads to explicit and readily tractable stability conditions, which can be implemented by means of numerical analysis techniques or modern symbolic computations. In particular, our analysis leads to the following results:

i. We provide a complete characterization of the possible relative equilibria of an anisotropic nonlinearly elastic body possessing a general form of stored energy function.

ii. We derive sufficient conditions for formal stability of the relative equilibria by exploiting in a crucial manner our block-diagonalization theorem for general simple mechanical systems with symmetry in the context of the reduced energy-momentum method.

iii. We provide a concrete mechanical interpretation for one set of stability conditions; namely, we show that stable stationary rotations are about the maximum axis of a certain locked inertia dyadic associated with the relative equilibrium. Furthermore, we give a constructive procedure for the remaining set of stability conditions in terms of a straightforward eigenvalue problem. This approach can be readily implemented in a numerical analysis context using a Galerkin finite element projection.

iv. We give a concrete mechanical interpretation of the block-diagonalization procedure and discuss in detail the structure of the symplectic two form in the context of elasticity. In particular, we show that the block diagonalization procedure also puts the linearized dynamics in normal form.