Assume that both grease and tailings exhibit decaying creep described by a Kelvin-Voigt model18:
$${\varepsilon }_{s,g}=\frac{\Delta h}{{h}_{0}}=\frac{q}{3{G}_{g}}\left(1-{e}^{-t/{\tau }_{g}}\right);\,{\varepsilon }_{s,t}=\frac{\Delta H}{{H}_{0}}=\frac{q}{3{G}_{t}}\left(1-{e}^{-t/{\tau }_{t}}\right)$$
(1)
where q is the constant deviatoric stress; \({h}_{0}\) and \({H}_{0}\) are the initial thicknesses of the grease and tailings, respectively; \({\varepsilon }_{s,g}\) and \({\varepsilon }_{s,t}\) are the shear strains, \({G}_{g}\) and \({G}_{t}\) are the shear moduli, \({\tau }_{g}\) and \({\tau }_{t}\) are relaxation times, subscripts “g” and “t” refer to the grease and tailings, respectively.
For \({h}_{0}\ll {H}_{0}\), the creep of the entire sample is given by
$${\varepsilon }_{s}=\frac{\Delta h+\Delta H}{{H}_{0}}=\frac{{h}_{0}}{{H}_{0}}\frac{q}{3{G}_{g}}\left(1-{e}^{-t/{\tau }_{g}}\right)+\frac{q}{3{G}_{t}}\left(1-{e}^{-t/{\tau }_{t}}\right)$$
(2)
Replacing the tailings by a rigid block with \({G}_{t}=\infty\) and changing the initial thickness of grease \({h}_{0}\) to \({h}_{g0}\), as carried out by the discussers, results in the creep rate of the entire sample (rigid block + grease):
$${\varepsilon }_{s}=\frac{\Delta h}{{H}_{0}}=\frac{{h}_{g0}}{{H}_{0}}\frac{q}{3{G}_{g}}\left(1-{e}^{-t/{\tau }_{g}}\right)$$
(3)
This can be adjusted to mimic exactly the response of the entire sample with grease and tailings in Eq, (2) in two cases only: (i) when \({h}_{g0}={h}_{0}\) and \({\tau }_{t}=0\), i.e., when the tailings are indeed rate independent, or (ii) when
$${h}_{g0}={h}_{0}+{H}_{0}\frac{{G}_{g}}{{G}_{t}}\:\:{and}\;{\tau }_{g}={\tau }_{t}$$
(4)
i.e., when the grease used by discussers is thicker than the one in the original Test 1, while the tailings and the grease are both rate-dependent with the same relaxation times, which would be ideal for any experiment aiming to correctly calibrate creep rates.
Similar conclusions are valid for a generalized Kelvin-Voigt model with n elements, where the creep responses of tailings and grease are given by Prony series:
$${\varepsilon }_{s,g}={\sum}_{i=1}^{n}\frac{q}{3{G}_{g,i}}\left(1-{e}^{-t/{\tau }_{g,i}}\right);\:\:\,{\varepsilon }_{s,t}={\sum}_{i=1}^{n}\frac{q}{3{G}_{t,i}}\left(1-{e}^{-t/{\tau }_{t,i}}\right)$$
(5)
allowing for simulating practically any experimental decaying creep curve. In this case, the perfect match between the tailings and the rigid block tests is achieved when
$${h}_{g0}={h}_{0}+{H}_{0}\frac{{G}_{g,i}}{{G}_{t,i}};\,\frac{{G}_{g,i}}{{G}_{t,i}}={const\;\: and}\: \,{\tau }_{g,i}={\tau }_{t,i}, \, {for}\;i=1,\ldots ,n$$
(6)
which is a generalized form of Eq. (4). It follows that even when the grease can mimic the entire sample response perfectly, this does not provide a unique proof of a rate independent behavior of tailings, as erroneously concluded in the Matters Arising.