Projected tensors

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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. [1] for details. Note that the expressions below do not consider the rotational degrees of freedom, see Ref. [2] for a discussion. The projections below are representative of the tensor's underlying symmetry only if the latter is well "aligned" [2].

Contents

Elastic tensor

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


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)

Cubic projection

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


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)

The projections are:

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

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

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

Hexagonal projection

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


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)

The projections are:

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

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

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

C_{33}^\text{hex} = C_{33}

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

References

  1. M. Moakher and A. N. Norris. The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry. J. Elasticity 85, 215 (2006).
  2. 2.0 2.1 M. A. Caro. Extended scheme for the projection of material tensors of arbitrary symmetry onto a higher symmetry tensor. arXiv preprint arXiv:1408.1219 (2014).
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