A transversely isotropic (also known as polar anisotropic) material is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy. 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 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. 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 {\displaystyle 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 has been coined Backus upscaling, which is described below.
Material symmetry matrix The material matrix K _ _ {\displaystyle {\underline {\underline {\boldsymbol {K}}}}} has a symmetry with respect to a given orthogonal transformation ( A {\displaystyle {\boldsymbol {A}}} ) if it does not change when subjected to that transformation. For invariance of the material properties under such a transformation we require
A ⋅ f = K ⋅ ( A ⋅ d ) ⟹ f = ( A − 1 ⋅ K ⋅ A ) ⋅ d {\displaystyle {\boldsymbol {A}}\cdot \mathbf {f} ={\boldsymbol {K}}\cdot ({\boldsymbol {A}}\cdot {\boldsymbol {d}})\implies \mathbf {f} =({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {K}}\cdot {\boldsymbol {A}})\cdot {\boldsymbol {d}}}
Hence the condition for material symmetry is (using the definition of an orthogonal transformation)
K = A − 1 ⋅ K ⋅ A = A T ⋅ K ⋅ A {\displaystyle {\boldsymbol {K}}={\boldsymbol {A}}^{-1}\cdot {\boldsymbol {K}}\cdot {\boldsymbol {A}}={\boldsymbol {A}}^{T}\cdot {\boldsymbol {K}}\cdot {\boldsymbol {A}}}
Orthogonal transformations can be represented in Cartesian coordinates by a 3 × 3 {\displaystyle 3\times 3} matrix A _ _ {\displaystyle {\underline {\underline {\boldsymbol {A}}}}} given by
A _ _ = [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] . {\displaystyle {\underline {\underline {\boldsymbol {A}}}}={\begin{bmatrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{bmatrix}}~.}
Therefore, the symmetry condition can be written in matrix form as
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