Editing Poisson's ratio (section)

=== Orthotropic ===
{{Main|Orthotropic material}}
[[Orthotropic material]]s have three mutually perpendicular planes of symmetry in their material properties. An example is wood, which is most stiff (and strong) along the grain, and less so in the other directions.

Then [[Hooke's law]] can be expressed in [[Matrix (mathematics)|matrix]] form as<ref name=Boresi>{{cite book|last1=Boresi |first1=A. P |last2=Schmidt |first2=R. J. |last3=Sidebottom |first3=O. M. |date=1993 |title=Advanced Mechanics of Materials |publisher=Wiley}}{{page needed|date=April 2024}}</ref><ref name=Lekh>{{cite book |last=Lekhnitskii |first=S. G. |year=1981 |title=Theory of elasticity of an anisotropic elastic body |page=36 |url=https://archive.org/details/lekhnitskii-theory-of-elasticity-of-an-anisotropic-body-mir-1981/page/36/mode/2up |publisher=Mir Publishing}}</ref>
:<math>
  \begin{bmatrix} \epsilon_{xx} \\ \epsilon_{yy} \\ \epsilon_{zz} \\ 2\epsilon_{yz} \\ 2\epsilon_{zx} \\ 2\epsilon_{xy} \end{bmatrix}
  = \begin{bmatrix}
    \tfrac{1}{E_x} & - \tfrac{\nu_{yx}}{E_y} & - \tfrac{\nu_{zx}}{E_z} & 0 & 0 & 0 \\
    -\tfrac{\nu_{xy}}{E_x} & \tfrac{1}{E_y} & - \tfrac{\nu_{zy}}{E_z} & 0 & 0 & 0 \\
    -\tfrac{\nu_{xz}}{E_x} & - \tfrac{\nu_{yz}}{E_y} & \tfrac{1}{E_z} & 0 & 0 & 0 \\
    0 & 0 & 0 & \tfrac{1}{G_{yz}} & 0 & 0 \\
    0 & 0 & 0 & 0 & \tfrac{1}{G_{zx}} & 0 \\
    0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{xy}} \\
    \end{bmatrix}
  \begin{bmatrix}
    \sigma_{xx} \\ \sigma_{yy} \\ \sigma_{zz} \\ \sigma_{yz} \\ \sigma_{zx} \\ \sigma_{xy}
  \end{bmatrix}
</math>
where
*{{math|''E''<sub>''i''</sub>}} is the [[Young's modulus]] along axis {{mvar|i}}
*{{math|''G''<sub>''ij''</sub>}} is the [[shear modulus]] in direction {{mvar|j}} on the plane whose normal is in direction {{mvar|i}}
*{{math|''ν''<sub>''ij''</sub>}} is the Poisson ratio that corresponds to a contraction in direction {{mvar|j}} when an extension is applied in direction {{mvar|i}}.

The Poisson ratio of an orthotropic material is different in each direction ({{mvar|x}}, {{mvar|y}} and {{mvar|z}}).  However, the symmetry of the stress and strain tensors implies that not all the six Poisson's ratios in the equation are independent.  There are only nine independent material properties: three elastic moduli, three shear moduli, and three Poisson's ratios.  The remaining three Poisson's ratios can be obtained from the relations
:<math>\frac{\nu_{yx}}{E_y} = \frac{\nu_{xy}}{E_x}\,, \qquad
\frac{\nu_{zx}}{E_z} = \frac{\nu_{xz}}{E_x}\,, \qquad
\frac{\nu_{yz}}{E_y} = \frac{\nu_{zy}}{E_z} 
</math>

From the above relations we can see that if {{math|''E''<sub>''x''</sub> > ''E''<sub>''y''</sub>}} then {{math|''ν''<sub>''xy''</sub> > ''ν''<sub>''yx''</sub>}}.  The larger ratio (in this case {{math|''ν''<sub>''xy''</sub>}}) is called the '''major Poisson ratio''' while the smaller one (in this case {{math|''ν''<sub>''yx''</sub>}}) is called the '''minor Poisson ratio'''.  We can find similar relations between the other Poisson ratios.
<!--
The above stress-strain relation is also often written in the equivalent ([[transpose]]d) form
:<math>
  \begin{bmatrix}
    \epsilon_{11} \\ \epsilon_{22} \\ \epsilon_{33} \\ 2\epsilon_{23} \\ 2\epsilon_{31} \\ 2\epsilon_{12}
  \end{bmatrix}
  = \begin{bmatrix}
    \tfrac{1}{E_1} & - \tfrac{\nu_{21}}{E_2} & - \tfrac{\nu_{31}}{E_3} & 0 & 0 & 0 \\
    -\tfrac{\nu_{12}}{E_1} & \tfrac{1}{E_2} & - \tfrac{\nu_{32}}{E_3} & 0 & 0 & 0 \\
    -\tfrac{\nu_{13}}{E_1} & - \tfrac{\nu_{23}}{E_2} & \tfrac{1}{E_3} & 0 & 0 & 0 \\
    0 & 0 & 0 & \tfrac{1}{G_{23}} & 0 & 0 \\
    0 & 0 & 0 & 0 & \tfrac{1}{G_{13}} & 0 \\
    0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{12}}
    \end{bmatrix}
  \begin{bmatrix}
    \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12}
  \end{bmatrix}
 </math>
-->