Miguel Caro: Created page with "''Also check out the MattPy page.'' These are expressions for the projection of specific material tensors onto their closest tensor of specific symmetry. The original ten..."

2020-12-23T15:31:28Z

Created page with "''Also check out the MattPy page.'' These are expressions for the projection of specific material tensors onto their closest tensor of specific symmetry. The original ten..."

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''Also check out the [[MattPy]] page.''

These are expressions for the projection of specific material tensors onto their closest tensor of specific symmetry. The original tensor is assumed to have no symmetry (i.e. it has triclinic symmetry) for the sake of generality. See Ref. <ref name="moakher_2006" /> for details. Note that the expressions below do not consider the rotational degrees of freedom, see Ref. <ref name="caro_2014b" /> for a discussion. The projections below are representative of the tensor's underlying symmetry only if the latter is well "aligned" <ref name="caro_2014b" />.

== Elastic tensor ==

The elastic tensor has 21 independent components, expressed by <math>C_{ij}</math> in Voigt notation, where ''i'' and ''j'' run over the 6 Voigt indices:

<math>
C =
\left(
\begin{matrix}
C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\
C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\
C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\
C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\
C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\
C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66}
\end{matrix}
\right)
</math>

=== Cubic projection ===

The cubic elastic tensor has 3 independent components, <math>C_{11}^\text{cub}</math>, <math>C_{12}^\text{cub}</math> and <math>C_{44}^\text{cub}</math>, and the following form:

<math>
C^\text{cub} =
\left(
\begin{matrix}
C_{11}^\text{cub} & C_{12}^\text{cub} & C_{12}^\text{cub} & 0 & 0 & 0 \\
C_{12}^\text{cub} & C_{11}^\text{cub} & C_{12}^\text{cub} & 0 & 0 & 0 \\
C_{12}^\text{cub} & C_{12}^\text{cub} & C_{11}^\text{cub} & 0 & 0 & 0 \\
0 & 0 & 0 & C_{44}^\text{cub} & 0 & 0 \\
0 & 0 & 0 & 0 & C_{44}^\text{cub} & 0 \\
0 & 0 & 0 & 0 & 0 & C_{44}^\text{cub}
\end{matrix}
\right)
</math>

The projections are:

<math>C_{11}^\text{cub} = \frac{1}{3} \left( C_{11} + C_{22} + C_{33} \right)</math>

<math>C_{12}^\text{cub} = \frac{1}{3} \left( C_{12} + C_{13} + C_{23} \right)</math>

<math>C_{44}^\text{cub} = \frac{1}{3} \left( C_{44} + C_{55} + C_{66} \right)</math>

=== Hexagonal projection ===

The hexagonal elastic tensor has 5 independent components, <math>C_{11}^\text{hex}</math>, <math>C_{12}^\text{hex}</math>, <math>C_{13}^\text{hex}</math>, <math>C_{33}^\text{hex}</math> and <math>C_{44}^\text{hex}</math>, and the following form:

<math>
C^\text{hex} =
\left(
\begin{matrix}
C_{11}^\text{hex} & C_{12}^\text{hex} & C_{13}^\text{hex} & 0 & 0 & 0 \\
C_{12}^\text{hex} & C_{11}^\text{hex} & C_{13}^\text{hex} & 0 & 0 & 0 \\
C_{13}^\text{hex} & C_{13}^\text{hex} & C_{33}^\text{hex} & 0 & 0 & 0 \\
0 & 0 & 0 & C_{44}^\text{hex} & 0 & 0 \\
0 & 0 & 0 & 0 & C_{44}^\text{hex} & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{C_{11}^\text{hex} - C_{12}^\text{hex}}{2}
\end{matrix}
\right)
</math>

The projections are:

<math>C_{11}^\text{hex} = \frac{3}{8} \left( C_{11} + C_{22} \right) + \frac{1}{4} C_{12} + \frac{1}{2} C_{66}</math>

<math>C_{12}^\text{hex} = \frac{1}{8} \left( C_{11} + C_{22} \right) + \frac{3}{4} C_{12} - \frac{1}{2} C_{66}</math>

<math>C_{13}^\text{hex} = \frac{1}{2} \left( C_{13} + C_{23} \right)</math>

<math>C_{33}^\text{hex} = C_{33}</math>

<math>C_{44}^\text{hex} = \frac{1}{2} \left( C_{44} + C_{55} \right)</math>

{{Reference list}}

Projected tensors - Revision history

Miguel Caro: Created page with "''Also check out the MattPy page.'' These are expressions for the projection of specific material tensors onto their closest tensor of specific symmetry. The original ten..."