Typology of Hooke matrices
===============================




Orthotropy
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This is a situation where the elastic material shows two symmetries with respect to two perpendicular planes (*orthorhombic* symmetry). In principle, the elasticity tensor has 9 independent coefficients, a consequence of the relationships obtained with these two symmetries between the 21 coefficients.

In principle, the thermal expansion tensor has 3 independent coefficients, a consequence of the relationships obtained with these two symmetries.

In the orthotropic axes:

:math:`\left[\mathrm{H}\right]=\left\{\begin{array}{cccccc}{H}_{\text{11}}& {H}_{\text{12}}& {H}_{\text{13}}& 0& 0& 0\\ & {H}_{\text{22}}& {H}_{\text{23}}& 0& 0& 0\\ & & {H}_{\text{33}}& 0& 0& 0\\ \text{SYM}& & & {H}_{\text{44}}& 0& 0\\ & & & & {H}_{\text{55}}& 0\\ & & & & & {H}_{\text{66}}\end{array}\right\}` :math:`\left\{\alpha \right\}=\left\{\begin{array}{c}{\alpha }_{\text{11}}\phantom{\rule{2em}{0ex}}\\ {\alpha }_{\text{22}}\phantom{\rule{2em}{0ex}}\\ {\alpha }_{\text{33}}\phantom{\rule{2em}{0ex}}\\ 0\\ 0\\ 0\end{array}\right\}`




Transverse isotropy
--------------------

Transverse isotropy (or revolution) is a restriction of orthotropy in which we have isotropy in one of the two orthogonal planes of elastic symmetry, following an invariance by rotation of :math:`2\pi /3` around the axis orthogonal to the plane of transverse isotropy for example :math:`{x}_{3}=0`. In principle, the elasticity tensor has 5 independent coefficients.

The :math:`\left[\mathrm{H}\right]` matrix will have the same shape as for orthotropy but with four additional relationships between the components. Thus, for the transverse isotropy in plane :math:`{x}_{3}=0`, we will have:

:math:`{H}_{11}={H}_{22}`; :math:`{H}_{13}={H}_{23}`; :math:`{H}_{44}={H}_{55}` and :math:`2{H}_{44}={H}_{11}-{H}_{12}` [:ref:`éq2.2-1 <éq2.2-1>`]

In principle, the thermal expansion tensor has 2 independent coefficients:

:math:`{\alpha }_{11}={\alpha }_{22}` eq2.2-2]




Isotropy
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The material is isotropic if :math:`\left[\mathrm{H}\right]` remains invariant in any change of frame of reference. In principle, the elasticity tensor has 2 independent coefficients. There is only one thermal expansion coefficient