 Stretch tensor

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 iswhere $R$ 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.