Stretch tensor ($U,V$)

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 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.

The principal stretches are the eigenvalues of a stretch tensor, or equivalently the singular values of the deformation gradient. They measure local length ratios along the principal reference directions.

The right stretch tensor is $U=(F^{\mathsf{T}}F{)}^{1/2}$, the positive matrix square root of the right Cauchy-Green deformation tensor. In the polar decomposition in continuum mechanics $F=RU$, it stretches material directions in the reference configuration before the rotation $R$ carries them to the final orientation.

For a unit reference direction $\mathbf{N}$, the squared length ratio is $|F\mathbf{N}{|}^2=\mathbf{N}\cdot U^2\mathbf{N}$. Along an eigenvector of $U$, that ratio is the square of its principal stretch. For $F=R\mathrm{diag}(2,1,1/2)$, one has $U=\mathrm{diag}(2,1,1/2)$ regardless of $R$: the local lengths double, stay fixed and halve along the three reference axes. Since the product is one, this example preserves volume. The Biot strain tensor $U-I$ expresses these extension ratios relative to unity.

The left stretch tensor is $V=(F{F}^{\mathsf{T}}{)}^{1/2}$, the positive matrix square root of the left Cauchy-Green deformation tensor. In the polar decomposition in continuum mechanics $F=VR$, it applies stretches along directions in the current configuration after the rotation $R$.

It is related to the right stretch tensor by $V=RU{R}^{\mathsf{T}}$. The two tensors therefore have the same eigenvalues, the principal stretches, but their eigenvectors belong to differently oriented configurations. For $F=R\mathrm{diag}(2,1,1/2)$, $V=R\mathrm{diag}(2,1,1/2)R^{\mathsf{T}}$. This distinction is useful when comparing a material-direction measurement before loading with an observed direction after the specimen has both stretched and turned. For a pure rotation, $V=I$, correctly reporting no local change in length.