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The Boltzmann constitutive representation is shown to provide a consistent means of incorporating dispersion effects into the mathematical modeling of wave behavior in layered elastic media. Attention is restricted to long wavelength conditions, with the waves propagating normal to the planes of layering. Special forms of a general Boltzmann law are derived for the special cases of periodic layering and one dimensionally random layering. Although there is no attenuation of harmonic waves in the periodic media case, an analytical representation is obtained for the attenuation measure in random media.

The effective attenuation of harmonic waves propagating through periodically layered elastic media is studied. The waves are taken to be propagating in the direction normal to that of the layering of the media, which has alternate layers of like material. The main restriction of the derivation is that the wavelength of the waves must be long compared with the periodic spacing of the layering. An explicit formula for the attenuation is derived by a perturbation method of analysis. The analysis reveals the basic cause of the attenuation effect in terms of the scattering properties of the medium. Specific examples are studied.

A derivation is given of a particular form of the isothermal nonlinear stress constitutive relation for materials with fading memory. This particular derivation results in a form for the constitutive relations that is well suited for application in solving boundary-value problems. The resulting forms are applied to obtain the exact quasi-static solution for the torsion of a right circular isotropic viscoelastic cylinder. Several effects due to the nonlinearity of the problem are discussed.

Analytical solutions are presented for two separate modes of deformation of an infinite elastic spherical wedge. In part, these solutions take the form of the stresses on the boundaries of the wedge caused by prescribed surface displacements. Full discussion is given to the means of utilizing these two solutions, along with corresponding tests, to determine the complete mechanical-properties characterization of a linear isotropic viscoelastic solid.

A solution is presented for the dynamic response of a solid sphere of linear viscoelastic material bonded to a mating, rigid spherical cavity which undergoes harmonic excitation. The dynamic response in the form of complex mechanical impedance is obtained in exact analytic form for two types of excitation of the rigid spherical boundary; rotation about a diameter and translation along a diameter. For the purpose of showing the usefulness of this system for experimentally determining the general stress-strain characteristics of the linear viscoelastic material, the sensitivity of the complex mechanical impedance of the sphere in each mode of excitation is examined relative to variations in the complex dynamic shear modulus and complex dynamic Poisson’s ratio of the material.