Geometrically exact, intrinsic theory for dynamics of curved and twisted anisotropic beams

A formulation is presented for the nonlinear dynamics of initially curved and twisted anisotropic beams. When the applied loads at the ends of, and distributed along, the beam are independent of the deformation, neither displacement nor rotation variables appear: an "intrinsic" formulation. Like well-known special cases of these equations governing nonlinear dynamics of rigid bodies and nonlinear statics of beams, the complete set of intrinsic equations has a maximum degree of nonlinearity equal to two. Advantages of such a formulation are demonstrated with a simple example. When the initial curvature and twist are constant along the beam, two space-time conservation laws are shown to exist, one being a work-energy relation and the other a generalized impulse-momentum relation. These laws can be used, for example, as benchmarks to check the accuracy of any proposed solution, including time-marching and finite element schemes. The structure of the intrinsic equations suggests parallel approaches to spatial and temporal discretization. A particularly simple spatial discretization scheme is presented for the special case of the nonlinear static behavior of end-loaded beams that, by virtue of the Kirchhoff analogy, leads to a time-marching scheme for the dynamics of a pivoted rigid body in a gravity field. This time-marching scheme conserves both the angular momentum about a vertical line passing through the pivot and total mechanical energy, whereas the analogous spatial discretization scheme for the nonlinear static behavior of end-loaded beams satisfies analogous integrals of deformation along the beam span. Remarkably, a straightforward generalization of these discretization schemes is shown to satisfy both space-time conservation laws for the nonlinear dynamics of beams when the applied loads are constant within a space-time element.