The discussers’ main point of criticism refers to the fact that the equivalent granular void ratio e* and derived quantities like the equivalent relative density Dr* and equivalent state parameter Ψ* were not used by the authors in the analysis of the experimental data. It must be mentioned, however, that until date, the e* concept has been developed and limited to granular mixtures, meaning sands with a non-plastic fines content. In addition, the concept of e* was mainly developed by focusing exclusively on binary mixtures, i.e. mixtures of a coarse and a fine granular material ([1, 4, 7, 9,10,11,12], Chang and Deng 2019, [2, 6] and the references therein). Such purely granular mixtures, composed of Chlef sand and non-plastic Chlef silt, were also used in the own tests of the discussers. The relations addressed by the discussers are well known for mixtures of sand with non-plastic fines and have been confirmed by numerous experimental studies in the literature.

However, in the present study, the authors have tested Hostun sand mixed with plastic fines, i.e. kaolin and calcigel bentonite clay. The applicability of the equivalent granular void ratio concept to mixtures of sand with clay is a subject of controversial discussion, and there is not much research work on that aspect. On the one hand, the concept was originally developed for sand–silt mixtures only, and the parameter b describes the amount of fine particles contributing to the load transfer in the soil skeleton. On the other hand, one could interpret the parameter b in a similar way for sand–clay mixtures, although the micromechanical mechanisms of the contribution of clayey and silty fines to the mechanical behaviour of the mixtures will be different. Considering the doubts of the applicability of the e* concept to sand–clay mixtures, the authors have not used e* or the derived quantities Dr* and Ψ* in their analysis of the test data.

Although the application of the e* concept to the data of the tested sand–plastic fines mixtures was beyond the scope of the original paper, the authors present some respective analysis in the following. The b values, in the current case describing the amount of plastic fines contributing to the mechanical behaviour of the mixtures, were determined by two methods: the first set of b values was obtained applying Eq. (1) developed by Rahman and Lo [8] as shown below, using parameters of the grain size distribution curves of the fine and coarse constituents of the mixture. The second set of b values was obtained by back-analysis of the critical state data from the undrained monotonic triaxial tests performed on the mixtures, where b was chosen to achieve a unique line in the e*-p’cs space.

$$ {\text{b}} = \left\{ {1 - exp\left[ { - 0.30\frac{{\left( {f_{c} /f_{thr} } \right)}}{k}} \right]} \right\} \times \left( {r\frac{{f_{c} }}{{f_{thr} }}} \right)^{r} $$
(1)

where \(r = (\text{D}_{10}/\text{d}_{50})^{-1}\) and \(k = 1 - r^{0.25}\) with D10 = size of sand at 10% finer, d50 = size of fines at 50% finer.

As shown in Fig. 1a, there is a deviation between the b values obtained from the Rahman et al. [9] equation and the ones obtained via back-analysis. Equation (1) consistently yields larger values than those obtained via back-analysis. This is not surprising because Eq. (1) has been developed for granular mixtures, for which its prediction capacity has been well confirmed in former studies (e.g. [3, 5, 6, 9]). The data for the mixture of Hostun sand with 20% calcigel are not included in Fig. 1a since the slope of the steady-state line of this mixture considerably differed from the others. This is reasoned to be due to the fact that the threshold fines content fthr for the sand–calcigel mixtures is lower than 20%, and with an fc of 20%, the behaviour of the mixture is expected to be closer to that of pure calcigel (see original paper for further explanation). Replacing the global void ratio e with e*, the resulting data points for clean Hostun sand, sand with 10% calcigel or kaolin, and sand with 20% kaolin are shown in Fig. 1b, confirming a good agreement of the various data in the e*-p’cs space. The equation of the critical state line for clean Hostun sand is added as dashed line in the e*-p’cs diagram, also delivering a good description of most of the data for the various mixtures.

Fig. 1
figure 1

a b Parameters obtained from the equation of Rahman et al. [9] or via back-analysis, b critical state lines in e*-p’cs space for sand–fines mixtures below threshold fines content fthr

Full size image

The current analysis give hints that the e* concept could be also applied to mixtures of sand with plastic fines. However, the physical meaning of the b value in that case and its dependence on various parameters like grain size distribution curves of the involved coarse and fine materials or the plasticity of the latter needs further investigations.