Dynamic Nonlinear Forcing of Elastic Rings

Equations of motion are derived that govern the large-deformational dynamics of thin-walled elastic rings due to arbitrary forcing. The derivation follows an `exact' nonlinear shell theory. Traditional small-order assumptions, usually associated with linear shell theory, are not made, nor is the ring material assumed to be circumferentially `inextensible'. Of particular note is the retention of rotary inertia, the rational inclusion of shear deformation and the identification of the undeformed shell. Conservation of mass and the first Piola-Kirchhoff identity are used to express the continuum mechanics in terms of the known undeformed geometry. At any time, the shell is the mass-weighted average of the continuum particle positions. Particle positions relative to the shell satisfy a dynamic consistency condition. Because the shell is the centre-of-mass axis, the equations of motion are dynamically uncoupled. This uncoupling of accelerations renders the equations amenable to explicit numerical integration in time using the method of lines.