Transverse isotropy
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A transversely isotropic (also known as polar anisotropic) material is one with physical properties that are
symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
about an axis that is normal to a plane of
isotropy In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also u ...
. This transverse plane has infinite planes of symmetry and thus, within this plane, the material properties are the same in all directions. 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 The elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness tensor. Common symbols include \mathbf and \mathbf. The defining equation can ...
are reduced to 5 (from a total of 21 independent constants in the case of a fully
anisotropic Anisotropy () is the structural property of non-uniformity in different directions, as opposed to isotropy. An anisotropic object or pattern has properties that differ according to direction of measurement. For example, many materials exhibit ver ...
solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...
). The (second-rank) tensors of electrical resistivity, permeability, etc. have two independent constants.


Example of transversely isotropic materials

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 the figure to the right, the fibers would be aligned with the x_2 axis, which is normal to the plane of isotropy. 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 Petrology () is the branch of geology that studies rocks, their mineralogy, composition, texture, structure and the conditions under which they form. Petrology has three subdivisions: igneous, metamorphic, and sedimentary petrology. Igneous ...
has been coined Backus upscaling, which is described below.


Material symmetry matrix

The material matrix \underline has a symmetry with respect to a given
orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we hav ...
(\boldsymbol) if it does not change when subjected to that transformation. For invariance of the material properties under such a transformation we require : \boldsymbol\cdot\mathbf = \boldsymbol\cdot(\boldsymbol\cdot\boldsymbol) \implies \mathbf = (\boldsymbol^\cdot\boldsymbol\cdot\boldsymbol)\cdot\boldsymbol Hence the condition for material symmetry is (using the definition of an orthogonal transformation) : \boldsymbol = \boldsymbol^\cdot\boldsymbol\cdot\boldsymbol = \boldsymbol^\cdot\boldsymbol\cdot\boldsymbol Orthogonal transformations can be represented in Cartesian coordinates by a 3\times 3 matrix \underline given by : \underline = \begin A_ & A_ & A_ \\ A_ & A_ & A_ \\ A_ & A_ & A_ \end~. Therefore, the symmetry condition can be written in matrix form as : \underline = \underline~\underline~\underline For a transversely isotropic material, the matrix \underline has the form : \underline = \begin \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end~. where the x_3-axis is the
axis of symmetry An axis (: axes) may refer to: Mathematics *A specific line (often a directed line) that plays an important role in some contexts. In particular: ** Coordinate axis of a coordinate system *** ''x''-axis, ''y''-axis, ''z''-axis, common names f ...
. The material matrix remains invariant under rotation by any angle \theta about the x_3-axis.


In physics

Linear material constitutive relations in physics can be expressed in the form : \mathbf = \boldsymbol\cdot\mathbf where \mathbf,\mathbf are two vectors representing physical quantities and \boldsymbol is a second-order material tensor. In matrix form, : \underline = \underline~\underline \implies \begin f_1\\f_2\\f_3 \end = \begin K_ & K_ & K_ \\ K_ & K_ & K_ \\ K_ & K_ & K_ \end \begin d_1\\d_2\\d_3 \end Examples of physical problems that fit the above template are listed in the table below. Using \theta=\pi in the \underline matrix implies that K_ = K_ = K_ = K_ = 0. Using \theta=\tfrac leads to K_ = K_ and K_ = -K_. Energy restrictions usually require K_, K_ \ge 0 and hence we must have K_ = K_ = 0. Therefore, the material properties of a transversely isotropic material are described by the matrix : \underline = \begin K_ & 0 & 0 \\ 0 & K_ & 0 \\ 0 & 0 & K_ \end


In linear elasticity


Condition for material symmetry

In
linear elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
, the stress and strain are related by
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
, i.e., : \underline = \underline~\underline or, using Voigt notation, : \begin \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end = \begin C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \\ C_ & C_ & C_ & C_ & C_ & C_ \end \begin \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end The condition for material symmetry in linear elastic materials is. : \underline = \underline^T~\underline~\underline where : \underline = \begin A_^2 & A_^2 & A_^2 & A_A_ & A_A_ & A_A_ \\ A_^2 & A_^2 & A_^2 & A_A_ & A_A_ & A_A_ \\ A_^2 & A_^2 & A_^2 & A_A_ & A_A_ & A_A_ \\ 2A_A_ & 2A_A_ & 2A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \\ 2A_A_ & 2A_A_ & 2A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \\ 2A_A_ & 2A_A_ & 2A_A_ & A_A_+A_A_ & A_A_+A_A_ & A_A_+A_A_ \end


Elasticity tensor

Using the specific values of \theta in matrix \underline,We can use the values \theta=\pi and \theta=\tfrac for a derivation of the stiffness matrix for transversely isotropic materials. Specific values are chosen to make the calculation easier. it can be shown that the fourth-rank elasticity stiffness tensor may be written in 2-index Voigt notation as the matrix : \underline = \begin C_&C_&C_&0&0&0\\ C_&C_&C_&0&0&0\\ C_&C_&C_&0&0&0\\ 0&0&0&C_&0&0\\ 0&0&0&0&C_&0\\ 0&0&0&0&0&(C_-C_)/2 \end =\begin C_ & C_-2C_ & C_ & 0 & 0 & 0 \\ C_-2C_ & C_ & C_ & 0 & 0 & 0 \\ C_ & C_ & C_ & 0 & 0 & 0 \\ 0 & 0 & 0 & C_ & 0 & 0 \\ 0 & 0 & 0 & 0 & C_ & 0 \\ 0 & 0 & 0 & 0 & 0 & C_ \end. The elasticity stiffness matrix C_ has 5 independent constants, which are related to well known engineering
elastic moduli An elastic modulus (also known as modulus of elasticity (MOE)) is a quantity that describes an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. Definition The elastic modu ...
in the following way. These engineering moduli are experimentally determined. The compliance matrix (inverse of the elastic stiffness matrix) is : \underline^ = \frac \begin C_ C_ - C_^2 & C_^2 - C_ C_ & (C_ - C_) C_ & 0 & 0 & 0 \\ C_^2 - C_ C_ & C_ C_ - C_^2 & (C_ - C_) C_ & 0 & 0 & 0 \\ (C_ - C_) C_ & (C_ - C_) C_ & C_^2 - C_^2 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac & 0 & 0 \\ 0& 0 & 0 & 0 & \frac & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac \end where \Delta := (C_ - C_) C_ + C_) C_ -2 C_C_/math>. In engineering notation, : \underline^ = \begin \tfrac & - \tfrac & - \tfrac & 0 & 0 & 0 \\ -\tfrac & \tfrac & - \tfrac & 0 & 0 & 0 \\ -\tfrac & - \tfrac & \tfrac & 0 & 0 & 0 \\ 0 & 0 & 0 & \tfrac & 0 & 0 \\ 0 & 0 & 0 & 0 & \tfrac & 0 \\ 0 & 0 & 0 & 0 & 0 & \tfrac \end Comparing these two forms of the compliance matrix shows us that the longitudinal
Young's modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
is given by :E_L = E_ = C_-2C_C_/(C_+C_) Similarly, the transverse
Young's modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
is :E_T= E_ = E_ = (C_-C_)(C_C_+C_C_-2C_C_)/(C_C_-C_C_) The inplane
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear s ...
is :G_=(C_-C_)/2=C_ and the
Poisson's ratio In materials science and solid mechanics, Poisson's ratio (symbol: ( nu)) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value ...
for loading along the polar axis is :\nu_=\nu_ = C_/(C_+C_). 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 wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
s. Transversely isotropic media support three types of elastic plane waves: * a quasi-
P wave A P wave (primary wave or pressure wave) is one of the two main types of elastic body waves, called seismic waves in seismology. P waves travel faster than other seismic waves and hence are the first signal from an earthquake to arrive at any ...
(
polarization Polarization or polarisation may refer to: Mathematics *Polarization of an Abelian variety, in the mathematics of complex manifolds *Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
direction almost equal to propagation direction) * a quasi-
S wave __NOTOC__ In seismology and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of elastic wave and are one of the two main types of elastic body waves, so named because t ...
* 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, G. E. (1962), Long-Wave Elastic Anisotropy Produced by Horizontal Layering, J. Geophys. Res., 67(11), 4427–4440 Backus presented an equivalent medium theory, a heterogeneous medium can be replaced by a homogeneous one that predicts wave propagation in the actual medium. 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. If each layer i is described by 5 transversely isotropic parameters (a_i, b_i, c_i, d_i, e_i), specifying the matrix : \underline =\begin a_i & a_i - 2e_i & b_i & 0 & 0 & 0 \\ a_i-2e_i & a_i & b_i & 0 & 0 & 0 \\ b_i & b_i & c_i & 0 & 0 & 0 \\ 0 & 0 & 0 & d_i & 0 & 0\\ 0 & 0 & 0 & 0 & d_i & 0\\ 0 & 0 & 0 & 0 & 0 & e_i\\ \end The elastic moduli for the effective medium will be : \underline = \begin A & A-2E & B & 0 & 0 & 0 \\ A-2E & A & B & 0 & 0 & 0 \\ B & B & C & 0 & 0 & 0 \\ 0 & 0 & 0 & D & 0 & 0 \\ 0 & 0 & 0 & 0 & D & 0 \\ 0 & 0 & 0 & 0 & 0 & E \end where : \begin A &= \langle a-b^2c^\rangle + \langle c^\rangle^ \langle bc^\rangle^2 \\ B &= \langle c^\rangle^ \langle bc^\rangle \\ C &= \langle c^\rangle^ \\ D &= \langle d^\rangle^ \\ E &= \langle e\rangle \\ \end \langle \cdot\rangle denotes the volume weighted average over all layers. This includes isotropic layers, as the layer is isotropic if b_i = a_i - 2e_i, a_i = c_i and d_i = e_i.


Short and medium wavelength approximation

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. However, the equations for the angular variation of velocity are algebraically complex and the plane-wave velocities are functions of the propagation angle \theta are. The direction dependent wave speeds for
elastic wave Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
s through the material can be found by using the Christoffel equation and are given by G. Mavko, T. Mukerji, J. Dvorkin. ''The Rock Physics Handbook''. Cambridge University Press 2003 (paperback). : \begin V_(\theta) &= \sqrt \\ V_(\theta) &= \sqrt \\ V_ &= \sqrt \\ M(\theta) &= \left left(C_-C_\right) \sin^2(\theta) - \left(C_-C_\right)\cos^2(\theta)\right2 + \left(C_ + C_\right)^2 \sin^2(2\theta) \\ \end where \begin\theta\end is the angle between the axis of symmetry and the wave propagation direction, \rho is mass density and the C_ are elements of the elastic stiffness matrix. The Thomsen parameters are used to simplify these expressions and make them easier to understand.


Thomsen parameters

Thomsen parameters are dimensionless combinations of
elastic moduli An elastic modulus (also known as modulus of elasticity (MOE)) is a quantity that describes an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. Definition The elastic modu ...
that characterize transversely isotropic materials, which are encountered, for example, in
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
. In terms of the components of the elastic
stiffness matrix In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution ...
, these parameters are defined as: : \begin \epsilon & = \frac \\ \delta & = \frac \\ \gamma & = \frac \end where index 3 indicates the axis of symmetry (\mathbf_3) . These parameters, in conjunction with the associated
P wave A P wave (primary wave or pressure wave) is one of the two main types of elastic body waves, called seismic waves in seismology. P waves travel faster than other seismic waves and hence are the first signal from an earthquake to arrive at any ...
and
S wave __NOTOC__ In seismology and other areas involving elastic waves, S waves, secondary waves, or shear waves (sometimes called elastic S waves) are a type of elastic wave and are one of the two main types of elastic body waves, so named because t ...
velocities, can be used to characterize wave propagation through weakly anisotropic, layered media. Empirically, the Thomsen parameters for most layered
rock formation A rock formation is an isolated, scenic, or spectacular surface rock (geology), rock outcrop. Rock formations are usually the result of weathering and erosion sculpting the existing rock. The term ''rock Geological formation, formation ...
s are much lower than 1. The name refers to Leon Thomsen, professor of geophysics at the
University of Houston The University of Houston (; ) is a Public university, public research university in Houston, Texas, United States. It was established in 1927 as Houston Junior College, a coeducational institution and one of multiple junior colleges formed in ...
, who proposed these parameters in his 1986 paper "Weak Elastic Anisotropy".


Simplified expressions for wave velocities

In geophysics the anisotropy in elastic properties is usually weak, in which case \delta, \gamma, \epsilon \ll 1. When the exact expressions for the wave velocities above are linearized in these small quantities, they simplify to : \begin V_(\theta) & \approx V_(1 + \delta \sin^2 \theta \cos^2 \theta + \epsilon \sin^4 \theta) \\ V_(\theta) & \approx V_\left + \left(\frac\right)^2(\epsilon-\delta) \sin^2 \theta \cos^2 \theta\right\\ V_(\theta) & \approx V_(1 + \gamma \sin^2 \theta ) \end where : V_= \sqrt ~;~~ V_= \sqrt are the P and S wave velocities in the direction of the axis of symmetry (\mathbf_3) (in geophysics, this is usually, but not always, the vertical direction). Note that \delta may be further linearized, but this does not lead to further simplification. The approximate expressions for the wave velocities are simple enough to be physically interpreted, and sufficiently accurate for most geophysical applications. These expressions are also useful in some contexts where the anisotropy is not weak.


See also

*
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
*
Linear elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechani ...
*
Orthotropic material In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength can ...
* Clinotropic material


References

{{Topics in continuum mechanics Crystallography Orientation (geometry) Elasticity (physics) Materials Continuum mechanics