Strain (source code)
= Strain
{title2=$\boldsymbol\varepsilon$}
{wiki}
<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> is
$$
\varepsilon=\frac12\left(\nabla\mathbf u+(\nabla\mathbf u)^{\mathsf T}\right).
$$
Its 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.
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