= Solution
For the specified affine linear <PTT fluid>, perturb about the relaxed state $A=0$, $\mathbf v=0$. Products of $A$ with the <velocity gradient> and the term $\alpha(\operatorname{tr}A)A$ are second order in the perturbation. The material-advection term is also second order. The linearized equation is therefore
$$
\partial_t A+\frac A\tau=2E.
$$
With vanishing stress in the remote past, an <integrating factor> gives
$$
A(t)=2\int_0^\infty e^{-s/\tau}E(t-s)\,ds.
$$
Since $\sigma' =G_0A$, comparison with the <linear viscoelastic fluid> convolution yields \b[$G(s)=G_0e^{-s/\tau}$] and <zero-shear viscosity> $G_0\tau$. The nonlinear relaxation parameter does not appear at linear order; the response is that of a <Maxwell fluid>.
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