Question #42215

A viscoelastic fiber exhibits an elastic after-effect modeled by the Kelvin-Voigt equation: $\sigma = Y \epsilon + \eta \frac{d\epsilon}{dt}$, where $Y$ is the Young's modulus and $\eta$ is the viscosity. The relaxation time $\tau = \eta/Y$ depends on the absolute temperature $T$ according to $\tau(T) = \tau_0 e^{E_a / k_B T}$, where $E_a$ is the activation energy and $k_B$ is the Boltzmann constant. Initially, a constant stress $\sigma_0$ is applied for a long time at temperature $T_0$, producing a steady strain $\epsilon_0$. At $t=0$, the stress is suddenly removed ($\sigma = 0$), and simultaneously, the fiber is heated such that its temperature increases linearly with time: $T(t) = T_0 + \alpha t$, where $\alpha > 0$ is a small constant. Assuming $\alpha t \ll T_0$ and $\frac{E_a \alpha t}{k_B T_0^2} \ll 1$, find the expression for the strain $\epsilon(t)$ during the early stages of recovery. Let $\tau_1 = \tau(T_0)$.