Analytic solution of ray-tracing equations for a linearly inhomogeneous and elliptically anisotropic velocity model

We study wave propagation in anisotropic inhomogeneous media. Specifically, we formulate and analytically solve the ray-tracing equations for the factorized model with wavefront velocity increasing linearly with depth and depending elliptically on direction. We obtain explicit expressions for traveltime, wavefront (phase) angle, and ray (group) velocity and angle, and study these seismological quantities for a model that successfully describes field measurements in the Western Canada Basin. By considering numerical examples, we also show that the difference between the wavefront and ray velocities depends only slightly on the anisotropy parameter, whereas the difference between the wavefront and ray angles is, in a first-order approximation, linear in the anisotropy parameter.

Stress-Strain Equation for Rubber in Tension

A stress-strain equation is derived for a homogeneous, isotropic material with the assumptions that for any given homogeneous simple tensile strain, the components of the stress tensor are to the first approximation linear, homogeneous functions of the components of the strain tensor and that no volume change occurs during the deformation. Utilizing the dependence of Poisson's ratio upon the extension referred to the initial coordinates, one elastic coefficient, C12, is found to be sufficient to roughly characterize the first stretch stress-strain curve. Although experimentally this elastic coefficient is found to be essentially constant for extensions greater than 250% its value increases rapidly as zero extension is approached. This behavior agrees qualitatively with data by Blanchard and Parkinson as to the distribution of secondary bond strengths, wherein they found a large number of relatively low-energy bonds which would be effective only at small extensions in contributing to modulus reinforcement. Various aspects of stress strain and cure behavior are examined with the derived equation as a basis.