Nonlinear elasticity and stress-induced anisotropy in rock

The elastic nonlinear behavior of rocks as evidenced by deviations from Hooke's law in stress-strain measurements, and attributable to the presence of mechanical defects (compliant features such as cracks, microfractures, grain joints), is a well-established observation. The purpose of this paper is to make the connection between the elastic nonlinearity and stress-induced effects on waves, in this case uniaxial-stress-induced transverse isotropy. The linear and nonlinear elastic coefficients constitute the most condensed manner in which to characterize the elastic behavior of the rock. We present both the second- and the third-order nonlinear elastic constants obtained from experimental data on rock samples assumed homogeneous and isotropic when unstressed. As is normally the case, the third-order (nonlinear) constants are found to be much larger than the second-order (linear) elastic constants. Contrary to results from intact homogeneous solids (materials without mechanical defects), rocks exhibit weak to strong nonlinearity and always in the same manner (i.e., an increase of the moduli with pressure). As a consequence the stress-induced P wave anisotropy and S wave birefringence can be large. The stress-induced P wave anisotropy appears to be much larger than the S wave birefringence. The fast direction is parallel to the stress direction, and the anisotropy goes as sin(2) theta, theta being the angle between the propagation direction and the stress direction. Experiments on rocks indicate that at low applied stresses, the proportionality of the stress and the induced S birefringence and P anisotropy predicted by theory is well corroborated.