Transverse isotropy
A transversely isotropic material is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy.
In geophysics, vertically transverse isotropy (VTI) is also known as radial anisotropy.
This type of material exhibits hexagonal symmetry (though technically this ceases to be true for tensors of rank 6 and higher), so the number of independent constants in the (fourth-rank) elasticity tensor are reduced to 5 (from a total of 21 independent constants in the case of a fully anisotropic solid).
The (second-rank) tensors of electrical resistivity, permeability, etc.
An example of a transversely isotropic material is the so-called on-axis unidirectional fiber composite lamina where the fibers are circular in cross section.
In a unidirectional composite, the plane normal to the fiber direction can be considered as the isotropic plane, at long wavelengths (low frequencies) of excitation.
In terms of effective properties, geological layers of rocks are often interpreted as being transversely isotropic.
Calculating the effective elastic properties of such layers in petrology has been coined Backus upscaling, which is described below.
The material matrix remains invariant under rotation by any angle
Linear material constitutive relations in physics can be expressed in the form where
In matrix form, Examples of physical problems that fit the above template are listed in the table below.
,[3] 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.
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, 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[4] 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: For shorter wavelengths, the behavior of seismic waves is described using the superposition of plane waves.
[5] Backus showed that layering on a scale much finer than the wavelength has an impact and that a number of isotropic layers can be replaced by a homogeneous transversely isotropic medium that behaves exactly in the same manner as the actual medium under static load in the infinite wavelength limit.
, specifying the matrix The elastic moduli for the effective medium will be where
denotes the volume weighted average over all layers.
Solutions to wave propagation problems in linear elastic transversely isotropic media can be constructed by superposing solutions for the quasi-P wave, the quasi S-wave, and a S-wave polarized orthogonal to the quasi S-wave.
is the angle between the axis of symmetry and the wave propagation direction,
The Thomsen parameters are used to simplify these expressions and make them easier to understand.
Thomsen parameters[8] are dimensionless combinations of elastic moduli that characterize transversely isotropic materials, which are encountered, for example, in geophysics.
In terms of the components of the elastic stiffness matrix, these parameters are defined as: where index 3 indicates the axis of symmetry (
Empirically, the Thomsen parameters for most layered rock formations are much lower than 1.
The name refers to Leon Thomsen, professor of geophysics at the University of Houston, who proposed these parameters in his 1986 paper "Weak Elastic Anisotropy".
In geophysics the anisotropy in elastic properties is usually weak, in which case
When the exact expressions for the wave velocities above are linearized in these small quantities, they simplify to where are the P and S wave velocities in the direction of the axis of symmetry (
The approximate expressions for the wave velocities are simple enough to be physically interpreted, and sufficiently accurate for most geophysical applications.
