Left Cauchy-Green deformation tensor ($B=F{F}^T$)

The left Cauchy-Green tensor measures stretch on the current configuration. It is a positive-definite symmetric matrix for an invertible deformation gradient, with eigenvalues equal to the squares of the principal stretches.

The first principal invariant of the left Cauchy-Green deformation tensor is the sum of the squares of the principal stretches. Since $B=F{F}^T$, it also equals ${\sum }_{i,I}{F}_{iI}^2$. Its derivative with respect to the deformation gradient is $2F$. For a strain energy density $W=w(I_1)$ with incompressibility, the Cauchy stress tensor is $T=2w^{\prime }(I_1)B-pI$.