DYNAMICAL PROBLEMS FOR GEOMETRICALLY EXACT THEORIES OF NONLINEARLY VISCOELASTIC RODS

This paper surveys recent results and open problems for the equations of motion for geometrically exact theories of nonlinearly viscoelastic and elastic rods. These rods can deform in space by undergoing not on ly flexure and torsion, but also extension and shear. The paper begins with a derivation of the governing equations, which for viscoelastic rods form a quasilinear system of hyperbolic-parabolic partial differe ntial equations of high order. It then derives the energy equation and discusses difficulties that can arise in getting useful energy estima tes. The paper next treats constitutive assumptions precluding total c ompression. The paper then discusses the curious asymptotic problems t hat arise when the inertia of the rod is small relative to that of a r igid body attached to its end. The paper concludes with discussions of traveling waves and shock structure, Hopf bifurcation problems, and p roblems of control.