Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section B / 21B / iv / Solution

Let the Maxwell branch contain spring modulus $G$ and series viscosity ${\eta }_M$, in parallel with a dashpot of viscosity ${\eta }_s$. The branch strains are equal and stresses add. Writing ${\sigma }_M$ for the Maxwell stress,For $\gamma =\mathrm{Re}({\gamma }_0{e}^{i\omega t})$ and $\tau ={\eta }_M/G$,Multiplying the Maxwell fraction by $1-i\omega \tau$ givesThe sign of the imaginary coefficient depends on the phasor convention; these expressions use $e^{i\omega t}$, so $G^{\prime \prime }>0$ represents dissipation. In the time domain, the equivalent constitutive relation isThe original mechanical schematic shows the series spring/dashpot branch separately from the parallel dashpot.