Solution (source code)

= Solution

For a rubber-like solid, a <spring> is a first model of recoverable <elasticity>, while a viscous element can represent rate-dependent loss and delayed recovery. An immediate extension followed by bounded delayed <material creep> is represented by <series spring and Voigt-branch creep>, a <standard linear solid model>:
$$
\varepsilon(t)=\frac{\sigma_0}{E_0}
+\frac{\sigma_0}{E_1}(1-e^{-E_1t/\eta}).
$$
The first term is immediate and recoverable; the second is delayed and also recoverable. For a load held until $t=t_h$, <recovery after a finite-duration stress load> predicts an immediate drop of $\sigma_0/E_0$ on unloading, followed by
$$
\boxed{\varepsilon(t)=\frac{\sigma_0}{E_1}(1-e^{-t_h/\tau})e^{-(t-t_h)/\tau},\qquad\tau=\frac\eta{E_1},\quad t>t_h.}
$$
This supplies a direct comparison with a loading-and-unloading observation: the model returns completely to its original length, with separate immediate and delayed recovery. A single <Kelvin-Voigt model> misses the immediate compliance, while a free series <dashpot> would predict unbounded flow that an intact crosslinked rubber need not show. A <spring> alone suffices over a timescale on which the measured response is effectively immediate and elastic.

These are local small-strain analogues. Large-strain <rubber elasticity> is nonlinear, nearly incompressible and sensitive to temperature, loading history and internal damage. A small set of linear <springs> and <dashpots> cannot reproduce every amplitude, timescale or <hysteresis> loop.

No numerical rubber trace or fitted constants are supplied, so these are explicit qualitative model predictions for comparison with the practical observations.