Materials Engineering (Non-Linear Fractional Material Strain)

A materials scientist is evaluating an advanced carbon-fiber reinforced polymer matrix designed for high-altitude aerodynamic radomes. Under intense aerodynamic friction and thermal load, the material's volumetric expansion $v$ responds to the normalized operating temperature $T$ according to an empirical power-tower equation:

1. Apply logarithmic differentiation to derive an expression for the volumetric sensitivity rate $\frac{dv}{dT}$.
2. Determine the fractional rate of thermal expansion (defined as $\frac{1}{v}\frac{dv}{dT}$, measuring percent expansion per degree temperature rise).
3. Evaluate the exact fractional rate of expansion $\frac{1}{v}\frac{dv}{dT}$ at the critical temperature baseline $T = e$.