Right stretch tensor ($U=(F^{T}F{)}^{1/2}$)

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.