Spectral strain measure ($E^f=f(U)$)

For a right stretch tensor $U={\sum }_a{\lambda }_a{n}_a\otimes n_a$, define $E^f=f(U)={\sum }_{a}f({\lambda }_a)n_a\otimes n_a$. A standard normalized choice has $f\in C^1(0,\infty )$, $f(1)=0$, $f^{\prime }(1)=1$ and $f$ strictly increasing; requiring $f^{\prime }>0$ gives a nonsingular stretch-to-strain parameterization. This strain vanishes exactly for a rigid rotation, is independent of superposed spatial rotations, and agrees to first order with the infinitesimal symmetric displacement gradient. Indeed $F=I+H$ gives $U=I+(H+H^T)/2+O(\Vert H{\Vert }^2)$, and Taylor expansion of $f$ proves the last statement.

The Biot strain tensor uses $f(\lambda )=\lambda -1$, so $E^{(1)}=U-I$. Its principal values are extension ratios minus one. It is a spectral strain measure, since $f(1)=0$, $f^{\prime }(1)=1$ and $f^{\prime }>0$. Its reference-volume work conjugate is the symmetric Biot stress.