The Viscoelastic Coefficient of Restitution: Integrating the Wave Equation With Hereditary Physics

Abstract The viscoelastic coefficient of restitution (vCOR) is a functional of the entire constitutive law, including relaxation, retardation, memory kernels, impact velocity, and its duration. This work predicts the vCOR by embedding hereditary material behavior into a Zener-type wave model for sphere-plate impact. Impact generates both local deformation and stress waves, which attenuate rapidly in viscoelastic bodies due to intrinsic molecular dissipation. Various existing contact models typically relate force to deformation and deformation rate through forms of stiffness and damping parameters, but they neglect true hereditary viscoelasticity. In contrast, hereditary behavior requires that stresses depend on the full strain history, as described by Boltzmann's superposition principle and, for nonlinear intrinsics, the Volterra integral framework. Here, the Hertzian point contact force is reformulated to instill hereditary viscoelasticity into Zener's wave equation. A physically consistent separation condition is imposed to determine an effective coefficient of restitution, accounting also for the residual deformation present at separation in viscoelastic impacts. The associated hysteretic energy loss is quantified, and a parametric study is conducted. Two solution strategies are developed: a direct numerical integration of the nonlinear ordinary integro-differential wave equation, and an internal variable evolution formulation that converts the problem into a system of ordinary differential equations for rapid time integration. Both approaches yield independently identical numerical results.