 Solution

The original diagram is a Kelvin-Voigt model: a spring and a dashpot act in parallel. The spring is an ideal elastic element, ${\sigma }_s=E{\epsilon }_s$; under its own sinusoidal stress its strain is in phase with stress and energy is stored reversibly. The dashpot is viscous, ${\sigma }_d=\eta {\dot{\epsilon }}_d$; its strain under an oscillating stress lags by90degrees and mechanical energy is dissipated.

In parallel the strain is common and the branch stresses add:For imposed harmonic stress, the strain amplitude and phase lag areThe short- and long-timescale limits follow under the printed roman labels below.

A sudden constant stress ${\sigma }_0$ gives delayed, bounded creep:The strain is initially zero because an instantaneous change would require an infinite dashpot stress. Initially the dashpot carries the load, with finite strain rate ${\sigma }_0/\eta$; the spring gradually carries more of it as the rate decays. On unloading, the strain recovers exponentially to zero. The network does not produce the instantaneous strain jump or unbounded constant-stress creep of a series Maxwell element.