Strain tensor and displacement field

The strain tensor $(\varepsilon _{ij})$ can be derived from the displacement field ${\vec {u}}$. The calculation is easy in two cases :

Linear deformation:

Pure shear strain:

The relative volume variation is evaluated by the trace of the strain tensor:

${\frac {\Delta V}{V_{0}}}=\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}$

The approximation done is described in the picture below:

• Top : volume before and after deformation;
• bottom: after deformation, the approximate volume is in green and the neglected part in orange.