= Stretch tensor
{title2=$U, V$}
A <stretch tensor> isolates local changes of material length from the <rotation> contained in a <deformation gradient>. For an invertible orientation-preserving $F$, the <polar decomposition in continuum mechanics> is
$$
F=RU=VR,
$$
where $R$ is a <rotation matrix> and $U,V$ are <positive-definite symmetric matrices>. The <right stretch tensor> $U$ acts on directions in the <reference configuration>; the <left stretch tensor> $V$ acts on directions in the <current configuration>.
The <eigenvalues> of either <stretch tensor> are the <principal stretches>. A value two means a material line in the corresponding principal direction doubles its length, while a value one means unchanged length. For a pure <rotation>, both <stretch tensors> are $I$, even though $F$ need not equal $I$. This makes them useful when an experiment includes substantial specimen rotation that should not be mistaken for <strain>.
Back to article page
