= Solution
The original diagram is a <Kelvin-Voigt model>: a <spring> and a <dashpot> act in parallel. The <spring> is an ideal elastic element, $\sigma_s=E\varepsilon_s$; under its own sinusoidal <stress> its <strain> is in <phase> with <stress> and energy is stored reversibly. The <dashpot> is viscous, $\sigma_d=\eta\dot\varepsilon_d$; its <strain> under an oscillating <stress> lags by90degrees and mechanical energy is dissipated.
In parallel the <strain> is common and the branch <stresses> add:
$$
\sigma=E\varepsilon+\eta\dot\varepsilon,\qquad
\tau=\eta/E.
$$
For imposed harmonic <stress>, the <strain> amplitude and <phase> lag are
$$
|\varepsilon|=\frac{\sigma_0}{\sqrt{E^2+(\omega\eta)^2}},
\qquad
\delta=\tan^{-1}(\omega\tau).
$$
The short- and long-timescale limits follow under the printed roman labels below.
A sudden constant <stress> $\sigma_0$ gives delayed, bounded creep:
$$
\boxed{\varepsilon(t)=\frac{\sigma_0}{E}
(1-e^{-t/\tau}),\qquad t\geq0.}
$$
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.
\Image[/past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2015/paper-1-voigt-response.png]
{title=Original Kelvin-Voigt predictions for step-stress creep, frequency-dependent compliance and stress-strain phase lag}
Back to article page
