Editing Poisson's ratio (section)

=== Isotropic ===

For a linear isotropic material subjected only to compressive (i.e. normal) forces, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axis in three dimensions. Thus it is possible to generalize [[Hooke's law]] (for compressive forces) into three dimensions:
:<math>\begin{align}
\varepsilon_{xx} &= \frac {1}{E} \left [ \sigma_{xx} - \nu \left ( \sigma_{yy} + \sigma_{zz} \right ) \right ] \\[6px]
\varepsilon_{yy} &= \frac {1}{E} \left [ \sigma_{yy} - \nu \left ( \sigma_{zz} + \sigma_{xx} \right ) \right ] \\[6px]
\varepsilon_{zz} &= \frac {1}{E} \left [ \sigma_{zz} - \nu \left ( \sigma_{xx} + \sigma_{yy} \right ) \right ] 
\end{align}</math>
{{Citation needed|date=August 2024}}
where:
*{{math|''ε''<sub>''xx''</sub>}}, {{math|''ε''<sub>''yy''</sub>}}, and {{math|''ε''<sub>''zz''</sub>}} are [[strain (materials science)|strain]] in the direction of {{mvar|x}}, {{mvar|y}} and {{mvar|z}}
*{{math|''σ''<sub>''xx''</sub>}}, {{math|''σ''<sub>''yy''</sub>}}, and {{math|''σ''<sub>''zz''</sub>}} are [[Stress (physics)|stress]] in the direction of {{mvar|x}}, {{mvar|y}} and {{mvar|z}}
*{{mvar|E}} is [[Young's modulus]] (the same in all directions for isotropic materials)
*{{mvar|ν}} is Poisson's ratio (the same in all directions for isotropic materials)

these equations can be all synthesized in the following:
:<math> \varepsilon_{ii} = \frac {1}{E} \left [ \sigma_{ii}(1+\nu) - \nu \sum_k \sigma_{kk} \right ] </math>

In the most general case, also [[shear stress]]es will hold as well as normal stresses, and the full generalization of Hooke's law is given by:
:<math> \varepsilon_{ij} = \frac {1}{E} \left [ \sigma_{ij}(1+\nu) - \nu \delta_{ij} \sum_k \sigma_{kk} \right ] </math>
where {{math|''δ''<sub>''ij''</sub>}} is the [[Kronecker delta]]. The [[Einstein notation]] is usually adopted:
:<math> \sigma_{kk} \equiv \sum_l \delta_{kl} \sigma_{kl} </math>
to write the equation simply as:
:<math> \varepsilon_{ij} = \frac {1}{E} \left [ \sigma_{ij}(1+\nu) - \nu \delta_{ij} \sigma_{kk} \right] </math>