 Strain

Strain measures local changes of length and angle relative to a chosen reference configuration. For a small displacement field $\mathbf{u}(\mathbf{X})$, the infinitesimal strain tensor isIts diagonal entries are fractional length changes along the coordinate axes. Off-diagonal entries describe changes of angle, with engineering shear strain equal to twice the corresponding tensor component.

A uniform translation has zero strain because its displacement gradient vanishes. An infinitesimal rigid rotation also has zero strain because its displacement gradient is antisymmetric. Finite rotations require finite strain theory, since the linear formula is then insufficient. In tensile testing, the simple ratio of extension to original gauge length is engineering strain; it coincides with the small axial component in the appropriate limit but is not a complete three-dimensional description.