Mean logarithmic stretching in an isotropic renovating shear ($\mu =\log (1+a^2/4)/(2T)$)

For independent uniform orientations in a renovating shear flow, the incident angle relative to each new shear is uniform even after earlier deformations. One interval stretches a unit material line by $\ell (\theta {)}^2=1+a\sin 2\theta +a^2{\sin }^2\theta$. Write this as $A+B\cos (2\theta -\delta )$, where $A=1+a^2/2$ and $A^2-B^2=1$. The angular logarithmic integral gives $⟨\log \ell ⟩=\frac{1}{2}\log [(A+1)/2]$. Add these independent logarithmic increments to get the displayed rate. At fixed shear rate, small $T$ gives $\mu \sim {\Lambda }^{2}T/8$ and large $T$ gives $\mu \sim \log (|\Lambda |T/2)/T$. The unique positive maximum is at $|\Lambda |T\approx 3.9605826$, obtained from $\log z=2(1-z^{-1})$ with $z=1+a^2/4$.