2. Typology of Hooke matrices

2.1. Orthotropy

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:

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

2.2. 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 \(2\pi /3\) around the axis orthogonal to the plane of transverse isotropy for example \({x}_{3}=0\). In principle, the elasticity tensor has 5 independent coefficients.

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

\({H}_{11}={H}_{22}\); \({H}_{13}={H}_{23}\); \({H}_{44}={H}_{55}\) and \(2{H}_{44}={H}_{11}-{H}_{12}\) [éq2.2-1]

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

\({\alpha }_{11}={\alpha }_{22}\) eq2.2-2]

2.3. Isotropy

The material is isotropic if \(\left[\mathrm{H}\right]\) 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