The Hamiltonian Structure of Nonlinear Elasticity: The Material and Convective Representations of Solids, Rods, and Plates

It is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation. For example, in the analysis of rotating structures, the coupling of the equations for geometrically inexact models, obtained by linearization or other approximations, with those for rotating rigid bodies can easily lead to misleading artificial “softening” effects that can significantly alter numerical results; see Simo and VuQuoc [1986c] (especially equations (3) and (5)). In this paper, we consider fully nonlinear geometrically exact models for rods, plates (and shells) which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models. These models can be obtained either from the three-dimensional theory by a systematic use of projection methods; see e.g., Antman [1972] and Naghdi [1972], or by a direct approach within the context of Cosserat continuum. Remarkably, the two approaches lead to essentially the same form of the governing field equations. In the present context, we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical Kirchhoff-Love model (see Love [1944]) to include shear deformations, as in