= Spectral strain measure
{title2=$E^f=f(U)$}
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 $f\in C^1(0,\infty)$, $f(1)=0$, $f'(1)=1$ and $f$ strictly increasing; requiring $f'>0$ 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 $U=I+(H+H^T)/2+O(\|H\|^2)$, and Taylor expansion of $f$ proves the last statement.
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