Nonlinear viscoelastic damping in a Poynting oscillator

Abstract

This work considers a cylindrical rod, acting as a nonlinear viscoelastic spring, with an attached mass that can displace along and rotate about the rod’s axis, thereby inducing an extension and twist in the rod. The coupling between them produces the Poynting effect in the body. The material is modeled by the nonlinear single integral Pipkin–Rogers constitutive equation. The mass has been assumed to be at rest in a long-time equilibrium state and then given a small axial or rotational disturbance and released (plucked). The subsequent motion is governed by a linear Volterra integro-differential equation. For a particular choice of material parameters, analytical expressions for the time-dependent decay of the disturbance are obtained in terms of the extension/torsion coupling, material stiffness, the amount and rate of stress relaxation, and the inertia of the mass. For other choices of material parameters, the results provide insight into how these quantities affect the time-dependent decay. A number of plucking scenarios are treated such as extensional or rotational disturbances from the undeformed state, rotational plucking after a finite axial stretch, and extensional plucking after a finite rotation. Simultaneous rotational and extensional plucking leads to a system of Volterra integro-differential equations whose treatment is deferred to later work.