Methods for Solving the Viscoelasticity Equations for Cylinder and Sphere Problems

Abstract

Abstract : This report considers techniques used to solve the Navier field equations of viscoelasticity in the Kelvin-Voigt or Maxwell models, for cylindrical of spherical geometries. Introducing scalar and vector potentials into the viscoelasticity equations formulation, ultimately yields telegraph-type partial differential equations governing those potentials. For harmonic time- dependence, these reduce to scalar and vector Helmholtz's equations with complex propagation constants. These constants are shown to be related to the viscoelastic material-constants in a more or less complicated fashion depending on the viscoelastic model used. The stresses, strains and displacements are then found from these potentials for a dozen cases of interest in those two coordinate systems. The formulation resembles that of electrodynamics in a Coulomb gauge. The above information is vital to set-up and solve various kinds of boundary-value-problems of dynamic viscoelasticity which appear when studying cases of acoustic scattering from sound-absorbing structures.