## Dynamical problems for geometrically exact theories of nonlinearly viscoelastic rods.(English)Zbl 0845.73034

Summary: 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 only 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 differential equations of high order. It then derives the energy equation and discusses difficulties that can arise in getting useful energy estimates. The paper next treats constitutive assumptions precluding total compression. The paper then discusses the curious asymptotic problems that arise when the inertia of the rod is small relative to that of a rigid body attached to its end. The paper concludes with discussions of traveling waves and shock structure, Hopf bifurcation problems, and problems of control.

### MSC:

 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74Hxx Dynamical problems in solid mechanics 74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
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