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A transversely isotropic material is one with physical properties that are
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
about an axis that is normal to a plane of
isotropy
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
. This transverse plane has infinite planes of symmetry and thus, within this plane, the material properties are the same in all directions. Hence, such materials are also known as "polar anisotropic" materials. 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
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 th ...
are reduced to 5 (from a total of 21 independent constants in the case of a fully
anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
). 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
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
has a symmetry with respect to a given
orthogonal transformation (
) if it does not change when subjected to that transformation.
For invariance of the material properties under such a transformation we require
:
Hence the condition for material symmetry is (using the definition of an orthogonal transformation)
:
Orthogonal transformations can be represented in Cartesian coordinates by a
matrix
given by
:
Therefore, the symmetry condition can be written in matrix form as
:
For a transversely isotropic material, the matrix
has the form
:
where the
-axis is the
axis of symmetry
Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. . The material matrix remains invariant under rotation by any angle
about the
-axis.
In physics
Linear material
constitutive relations in physics can be expressed in the form
:
where
are two vectors representing physical quantities and
is a second-order material tensor. In matrix form,
:
Examples of physical problems that fit the above template are listed in the table below.
Using
in the
matrix implies that
. 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
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
, i.e.,
:
or, using
Voigt notation
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notat ...
,
:
The condition for material symmetry in linear elastic materials is.
:
where
:
Elasticity tensor
Using the specific values of
in matrix
,
[We can use the values and 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
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notat ...
as the matrix
:
The elasticity stiffness matrix
has 5 independent constants, which are related to well known engineering
elastic moduli
An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is ...
in the following way. These engineering moduli are experimentally determined.
The compliance matrix (inverse of the elastic stiffness matrix) is
:
where