Using leads to and . Energy restrictions usually require and hence we must have . Therefore, the material properties of a transversely isotropic material are described by the matrix In linear elasticity Condition for material symmetry In linear elasticity, the stress and strain are related by Hooke's law, i.e., or, using Voigt notation, The condition for material symmetry in linear elastic materials is. where Elasticity tensor Using the specific values of in matrix , it can be shown that the fourth-rank elasticity stiffness tensor may be written in 2-index Voigt notation as the matrix The elasticity stiffness matrix has 5 independent constants, which are related to well known engineering elastic moduli in the following way. These engineering moduli are experimentally determined. The compliance matrix (inverse of the elastic stiffness matrix) is where . In engineering notation, Comparing these two forms of the compliance matrix shows us that the longitudinal Young's modulus is given by Similarly, the transverse Young's modulus is The inplane shear modulus is and the Poisson's ratio for loading along the polar axis is . Here, L represents the longitudinal (polar) direction and T represents the transverse direction. In geophysics In geophysics, a common assumption is that the rock formations of the crust are locally polar anisotropic (transversely isotropic); this is the simplest case of geophysical interest. Backus upscaling is often used to determine the effective transversely isotropic elastic constants of layered media for long wavelength seismic waves. Assumptions that are made in the Backus approximation are: All materials are linearly elastic No sources of intrinsic energy dissipation (e.g. friction) Valid in the infinite wavelength limit, hence good results only if layer thickness is much smaller than wavelength The statistics of distribution of layer elastic properties are stationary, i.e., there is no correlated trend in these properties. For shorter wavelengths, the behavior of seismic waves is described using the superposition of plane waves. Transversely isotropic media support three types of elastic plane waves: a quasi-P wave (polarization direction almost equal to propagation direction) a quasi-S wave a S-wave (polarized orthogonal to the quasi-S wave, to the symmetry axis, and to the direction of propagation). Solutions to wave propagation problems in such media may be constructed from these plane waves, using Fourier synthesis. Backus upscaling (long wavelength approximation) A layered model of homogeneous and isotropic material, can be up-scaled to a transverse isotropic medium, proposed by Backus. Backus presented