The Voigt model—M. The Voigt model for a viscoelastic fluid is shown in Fig. P11.18, and consists of a dashpot and spring in parallel, with constants η and G, respectively.
When a stress τ is applied at point P, the resulting strain is γ.
(a) By noting the resisting stresses offered by the dashpot and spring, derive the differential equation that relates τ , γ, η, and G.
(b) If a constant stress τ0 is applied, with an initial strain of zero, solve this differential equation for the strain as a function of time. Introduce the parameter λ = η/G if possible. Hint: If needed, try γ = c1 + c2e−c3t as a solution.
(c) If, after a long time, the stress is completely removed, what happens to the strain?
(d) Draw a diagram that shows both the stress and the strain as functions of time for both (b) and (c) above.