The Formulation and Solution of the Governing Equations of Motion for Three-Dimensional Linear Viscoelastodynamics
On the basis of the linearized Navier-Stokes equation of a viscous fluid and the displacement equation of motion of an elastic solid, the governing equations of motionsfor a class of three dimensional homogeneous, isotropic linearly viscoelastic continua are systematically constructed. The general solutions of these equation of motion are obtained by means of Lame-Helmholtz-Stokes potentials. Each equation of motion is then transformed into a scalar and a vector potential equation. The scalar potential equation is separable in eleven coordinate system, whereas the vector potential equation is separable in only six coordinate systems. Choosing the spherical coordinate system for illustration and introducing the Debye potentials to resolve the vector potential for the purpose of obtaining a separable solution, we can reduce each equation of motion into three independent scalar potential equations which are all separable in spherical coordinates. The field quantities are obtained in terms of these three scalar potentials from which the transient or harmonic solutions can be sought for mixed or nonmixed boundary value problems. Finally, applications of the general solution for physical problems in spherically symmetric, axially symmetric, torsional and nontorsional motion are given in thirteen examples. These examples can be used for the modeling studies of aero and hydro space vehicles, geological wave problems and macroscopic biomechanics.
KeywordsViscous Fluid Viscoelastic Material Potential Equation Spherical Coordinate System Separable Solution
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