1 Introduction

Sand–silt mixtures are abundant in the nature as well as in the man-made structures such as levees and embankments. Understanding the mechanical behavior of these materials is essential to the engineers to predict their deformation behavior and strength characteristics under different loading scenarios.

It has been shown by many researchers that the fines content has very important effects on different aspects of the mechanical behavior of sand–silt mixtures [1, 8, 9, 13, 14, 16,17,18,19, 21,22,23, 26,27,28,29,30, 32, 35, 37,38,39, 41,42,43]. Several experimental works have shown the effect of fines content on the compression behavior of sand–silt mixtures [3, 11, 12, 15, 20, 34, 44, 46].

A compression model for sand–silt mixture that accounts for the effect of fines content is needed in the practice of geotechnical engineering. For example, in studying the mechanics of internal erosion, as was shown by Chang and Meidani [4] and Hicher [14], it is essential to have a compression model for sand–silt mixtures that explicitly considers the amount of fines, to be able to simulate the effects of loss of fines on the overall deformation behavior and strength characteristics of the eroded soil.

To date, there is no compression model for sand–silt mixtures that explicitly considers the fines content as a variable in its formulation. The conventional compression models for granular materials consider factors such as initial void ratio, applied effective stress, and particles mineralogy [10, 24, 25, 31, 36]. The material constants in these conventional models need to be calibrated for each mixture with particular fines content.

In this paper, we develop a mathematical model to predict the compressibility of sand–silt mixtures by explicitly considering its fine content as an input variable. To do so, we introduce a novel concept of dividing the voids in a granular packing into two hypothetical fractions, named active and inactive voids. This postulate leads us to look at the compression test data from a non-traditional perspective; instead of treating compressibility as a function of stress level, we study the relationship between compressibility and density state. Based on the experimental evidence from compression tests on various sand–silt mixtures, we identify a linear relationship between the material compressibility and its density state. It is shown that this linear relationship is valid up to stress levels at which the effect of particle breakage is not prominent; however, there are limited particle crushing effects that are implicitly embedded in this linear relationship.

This linear relationship is expressed in the form of a differential equation. After integration, a mathematical model for the compressibility of sand–silt mixtures is derived. The two hypothetical fractions, i.e., active and inactive void ratios, are identified in the derived mathematical relationship. We show that active void ratio diminishes with an exponential rate with respect to the stress level. It is also shown that the inactive void ratio can be expressed as an explicit function of fines content. Finally, the proposed model is verified by comparing the predicted void ratios with the measured data for six different types of sand–silt mixtures with various amounts of fines and different mineralogy.

2 Active and inactive voids in granular materials

For convenience in the modeling process, we divide the voids in a granular packing into two hypothetical fractions: (1) active voids and (2) inactive voids. The traditional phase diagram for a granular material is shown in Fig. 1a. The active and inactive voids are separated in another phase diagram shown in Fig. 1b.

Fig. 1
figure 1

a Traditional phase diagram for a granular material; b separating the voids into two hypothetical fractions: active voids and inactive voids

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The compression of solid particles is negligible for the stress levels considered in this study. Therefore, we can divide the volumes of active and inactive voids by the total volume of solids \(V_{\text{s}}\) and define active void ratio \(e_{\text{active}}\) and inactive void ratio \(e_{\text{inactive}}\), respectively (Fig. 1). The total void ratio (\(e\)) of a soil at any stress level is the sum of \(e_{\text{active}}\) and \(e_{\text{inactive}}\).

We define active voids such as this hypothetical fraction of total voids is kinematically diminishable during compression due to particles rearrangement, i.e., particle sliding and rotation (Fig. 2). On the other hand, inactive voids are kinematically unavailable to be diminished by particles rearrangement. The volume of inactive voids is dependent on the particle size distribution (PSD) and particles shapes. These two factors dictate the densest kinematic limit of a granular packing as described in details by Meidani et al. [25]. PSD and particle shape can be changed by particles breakage, particles bonding, or particles loss. Varying the fines content in a sand–silt mixture changes the PSD of the soil (Fig. 2).

Fig. 2
figure 2

Causes and mechanisms that change the volumes of active and inactive voids in a sand–silt mixture

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A desirable compression model for sand–silt mixtures with a given fines content (\(f_{c}\)) can be written in the following general form as:

$$e\left( {p^{{\prime }} ,e_{0} ,f_{c} } \right) = e_{\text{active}} \left( {p^{{\prime }} ,e_{0} ,f_{c} } \right) + e_{\text{inactive}} \left( {p^{{\prime }} ,f_{c} } \right)$$
(1)

where \(e\) is the total void ratio of the mixture, \(p^{{\prime }}\) is the effective stress, \(e_{0}\) is the initial void ratio of the mixture. \(f_{c}\) is the fines content in the mixture. \(f_{c}\) is defined as the dry mass of fines in the mixture divided by the total dry mass of the mixture. e active and e inactive are the active and inactive void ratio components, respectively. They build up the total void ratio e. Equation (1) shows the simple postulate that we want to examine its applicability in this study.

In the next section, we derive a mathematical relationship for the compression of sand–silt mixtures. We show that active voids diminish with an exponential rate with respect to the effective stress and initial density of the mixture. We also show that the inactive void ratio is a function of fines content, which will be related to the minimum void ratio of the mixture. The inactive void ratio will be shown to be independent of the stress magnitude up to certain levels. We will show that the effects of particle crushing in the range of validity of the proposed model are implicitly embedded into the function that defines the evolution of active void ratio.

3 Derivation of the compression model

Meidani et al. [25] showed an approximate linear relationship can be identified between \(\frac{{{\text{d}}e}}{{{\text{d}}p^{{\prime }} }}\) and \(e\), based on the compression test data from 20 different types of granular soils. The result of compression test on a typical quartz sand is shown in Fig. 3. Determined from compression test data, we can observe that the approximate linear relationship is valid up to certain stress level, for example 40 MPa for most quartz sands and 10 MPa for most calcium-based sands [25]. It is clear that some particle breakage occurs in quartz sand prior to 40 MPa due to compression. Hence, the effects of particle breakage on the change of void ratio are implicitly included in the linear relationship between \(\frac{{{\text{d}}e}}{{{\text{d}}p^{{\prime }} }}\) and \(e\) [25].

Fig. 3
figure 3

\(\frac{{{\text{d}}e}}{{{\text{d}}p^{{\prime }} }}\) plotted versus void ratio \(e\). Data from compression tests on Cambria sand (after [25])

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We apply the similar concept to the compression of sand–silt mixtures. We select compression tests data on specimens made of Hukksond sand and Chengbei silt with various fines contents, ranging from 0 to 94% (Table 1), to check the validity of this concept in sand–silt mixtures. After normalizing all applied stresses to the reference atmospheric pressure and defining the normalized effective pressure as \(p_{*}^{{\prime }} = \frac{{p^{{\prime }} }}{{p_{\text{atm}} }}\), where \(p_{\text{atm}}\) = 0.101325 MPa, the values of \(\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{{\prime }} }}\) versus \(e\) are plotted in Fig. 4 for all fines contents.

Table 1 Properties of selected sand–silt materials used in this paper
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Fig. 4
figure 4

\(\frac{{{\text{d}}e}}{{{\text{d}}p^{{\prime }} }}\) plotted versus void ratio for mixtures of Hukksond sand and Chengbei silt with various fines contents. The solid lines are the best-fit linear approximation to the tests data with a given fines content. The slopes of three lines for the 30, 70, and 94% fines content specimens are shown as \(\alpha_{30\% }\), \(\alpha_{70\% }\), and \(\alpha_{94\% }\). The x-intercept of the trend line for the mixtures with 94% fines content is shown as \(e_{{r_{94\% } }}\)

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Linear correlations can be identified among the data points with a given fines content. For instance, the tests results from the mixtures with 70% fines content are shown with solid squares in Fig. 4. A line is fitted to these data points. The same procedure is done for all other fines contents. The trend lines have different slopes and different intercepts. In a general form, these lines can be expressed by the following relationship:

$$\left( {\frac{{{\text{d}}e}}{{{\text{d}}p_{ *}^{{\prime }} }}} \right)_{{f_{c} }} = - \alpha_{{f_{c} }} e + \beta_{{f_{c} }}$$
(2)

where \(f_{c}\) designates the fines content of the specimen. \(\alpha_{{f_{c} }}\) and \(\beta_{{f_{c} }}\) are the slopes and y-intercepts of the lines, respectively.

Equation (2) can also be written in the following form:

$$\left( {\frac{{{\text{d}}e}}{{{\text{d}}p_{ *}^{{\prime }} }}} \right)_{{f_{c} }} = - \alpha_{{f_{c} }} \left( {e - \frac{{\beta_{{f_{c} }} }}{{\alpha_{{f_{c} }} }}} \right) = - \alpha_{{f_{c} }} \left( {e - e_{{r_{{f_{c} }} }} } \right)$$
(3)

In this equation, \(e_{{r_{{f_{c} }} }}\) is equal to \(\frac{{\beta_{{f_{c} }} }}{{\alpha_{{f_{c} }} }}\), which is the x-intercept of the trend line for specimens with fines content \(f_{c}\), as shown in Fig. 4. As the void ratio decreases under compression, \(\left( {\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{{\prime }} }}} \right)_{{f_{c} }}\) decreases linearly. When the void ratio reaches \(e_{{r_{{f_{c} }} }}\), \(\left( {\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{{\prime }} }}} \right)_{{f_{c} }}\) becomes zero, which means that the void ratio will not decrease anymore. According to our postulate, this is when all kinematically available voids are diminished due to particles rearrangement. The remaining voids, \(e = e_{{r_{{f_{c} }} }}\), are no longer compressible, as they are kinematically unavailable to the compression process. We name \(e_{{r_{{f_{c} }} }}\), representative inactive void ratio. The reason we do not simply call this variable inactive void ratio is that the approximate linear correlation between \(\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{{\prime }} }}\) and \(e\) is only valid up to certain stress levels (Fig. 3). So, the x-intercept (\(e_{{r_{{f_{c} }} }}\)) only represents the compression behavior up to certain stress levels.

Taken from Fig. 4, we plot the slopes of the fitted lines \(\alpha_{{f_{c} }}\) versus the initial void ratios of various specimens made of Hukksond sand and Chengbei silt (Fig. 5). A correlation can be found between \(\alpha_{{f_{c} }}\) and \(e_{0}\), which is expressed as:

$$\alpha_{{f_{c} }} = \hat{a}_{{f_{c} }} e_{0}$$
(4)

where \(\hat{a}_{{f_{c} }}\) is a material parameter and \(e_{0}\) is the initial void ratio of each specimen. As shown in Fig. 5, the parameter \(\hat{a}_{{f_{c} }}\) is approximately 0.4 for all fines contents in this mixture. Therefore, we drop the subscript \(f_{c}\) from \(\hat{a}_{{f_{c} }}\) and rewrite Eq. 4 as:

$$\alpha_{{f_{c} }} = \hat{a}e_{0}$$
(5)

where \(\hat{a}\) is a material parameter. In the case of Hukksond sand and Chengbei silt mixtures, this parameter is 0.4. Even though the proposed linear relationship in Eq. 5 is approximate, we show later that it causes a very small error in void ratio predictions.

Fig. 5
figure 5

Linear correlation between the slopes of trend lines (shown in Fig. 4) and the initial void ratios of specimens for the mixtures of Hukksond sand and Chengbei silt with various fines contents

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Replacing Eq. 5 into Eq. 3 gives:

$$\left( {\frac{{{\text{d}}e}}{{{\text{d}}p_{ *}^{{\prime }} }}} \right)_{{f_{c} }} = - \hat{a}e_{0} \left( {e - e_{{r_{{f_{c} }} }} } \right)$$
(6)

or, if \(e_{0}\) is moved to the left side of Eq. 6, we have:

$$\frac{1}{{e_{0} }}\left( {\frac{{{\text{d}}e}}{{{\text{d}}p_{ *}^{ '} }}} \right)_{{f_{c} }} = - \hat{a}\left( {e - e_{{r_{{f_{c} }} }} } \right)$$
(7)

which means that if the values of \(\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{{\prime }} }}\) are divided by the initial void ratio of each specimen and plotted versus the void ratio, the result will be a group of parallel lines that has a slope equal to \(\hat{a}\) and different x-intercepts equal to \(e_{{r_{{f_{c} }} }}\). The intercepts vary with the fines content of the specimen, as shown in Fig. 6. The general relationship for these parallel lines is Eq. 7.

Fig. 6
figure 6

\(\frac{1}{{e_{0} }}\frac{{{\text{d}}e}}{{{\text{d}}p^{{\prime }} }}\) plotted versus void ratio \(e\) for mixtures of Hukksond sand and Chengbei silt with various fines contents. The slopes of three lines from 30, 70, and 94% fines content specimens are shown as \(\hat{a}\). The x-intercept of the trend line for the mixtures with 94% fines content is shown as \(e_{{r_{94\% } }}\)

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In order to derive the mathematical relationship for the compression model, we reorganize Eq. 7 as:

$$\frac{{{\text{d}}e}}{{e - e_{{r_{{f_{c} }} }} }} = - \frac{{\hat{a}e_{0} }}{{p_{\text{atm}} }}{\text{d}}p^{{\prime }}$$
(8)

Recall that \({\text{d}}p_{*}^{'} = \frac{{{\text{d}}p^{\prime}}}{{p_{\text{atm}} }}\).

This relationship can be integrated over the variables \(e\) and \(p^{{\prime }}\) for a constant \(e_{{r_{{f_{c} }} }}\) as follows:

$$\int \limits_{{e_{0} }}^{e} \left( {\frac{{{\text{d}}e}}{{e - e_{{r_{{f_{c} }} }} }}} \right) = - \frac{{\hat{a}e_{0} }}{{p_{\text{atm}} }} \int \limits_{0}^{{p^{{\prime }} }} {\text{d}}p^{{\prime }}$$
(9)

where the upper integral limit, \(e\), is the void ratio at the target stress \(p^{{\prime }}\). For a given fines content, \(e_{{r_{{f_{c} }} }}\) is constant (Fig. 6).

Integration of Eq. 9 leads to the following form, which expresses the void ratio as a function of applied stress \(p^{{\prime }}\), initial void ratio \(e_{0}\), and representative inactive void ratio \(e_{{r_{{f_{c} }} }}\):

$$e\left( {p^{{\prime }} ,e_{0} ,f_{c} } \right) = \left[ {e_{0} - e_{{r_{{f_{c} }} }} } \right]\exp \left( { - \frac{{\hat{a}e_{0} }}{{p_{\text{atm}} }}p^{{\prime }} } \right) + e_{{r_{{f_{c} }} }}$$
(10)

Through analogy of the terms between Eqs. 10 and 1, we can identify the active and inactive void ratios as follows:

$$e_{\text{active}} \left( {p^{{\prime }} ,e_{0} ,f_{c} } \right) = \left[ {e_{0} - e_{{r_{{f_{c} }} }} } \right]\exp \left( { - \frac{{\hat{a}e_{0} }}{{p_{\text{atm}} }}p^{{\prime }} } \right)$$
(11)
$$e_{\text{inactive}} \left( {p^{{\prime }} ,f_{c} } \right) = e_{{r_{{f_{c} }} }}$$
(12)

The graphical representation of Eq. (9) for a given fines content is shown in Fig. 7.

Fig. 7
figure 7

Graphical representation of Eq. 10, showing the evolution of void ratio during compression

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Figure 8 shows the evolution of volume of voids during compression in three phase diagrams for a mixture with a given fines content. Since the volume of the representative inactive voids does not change during loading (Eq. 12), and the volume of solids remains constant, all the volume change is attributed to the change in active voids (Eq. 11).

Fig. 8
figure 8

Three phase diagrams showing the evolution of active voids for a sand–silt mixture with a given fines content, while the volume of representative inactive voids and solids remains constant during loading

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The effect of variation of model parameters \(e_{{r_{{f_{c} }} }}\) and \(\hat{a}\) on the calculated compression lines is shown in Fig. 9. It can be seen that decreasing parameter \(e_{{r_{{f_{c} }} }}\) increases the slope of compression lines and slightly shifts them to the left, while increasing \(\hat{a}\) does not have a significant effect on the slope of compression lines, but shifts the calculated compression lines to the left, depicting a reduction in the yield stress of the material.

Fig. 9
figure 9

Effect of variation of model parameters a \(e_{{r_{{f_{c} }} }}\) and b \(\hat{a}\) on the calculated compression lines from Eq. 10

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In order to determine the model parameters \(\hat{a}\) and \(e_{{r_{{f_{c} }} }}\) for the selected sand–silt mixtures (Table 1), we construct \(\frac{1}{{e_{0} }}\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{'} }}\) versus \(e\) graphs similar to Fig. 6 for each type of mixture. Five of the mixtures are made from quartz sand and quartz silt with various fines contents [11, 12, 20, 44, 46]. The \(\frac{1}{{e_{0} }}\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{'} }}\) versus \(e\) graphs showed that for each type of mixture, all trend lines correlated with the compression tests data performed on specimens with various fines contents have the same slope (\(\hat{a}\)). The determined \(\hat{a}\) values for these five types of mixtures are reported in Table 1. The sixth type of mixture, tested by Cabalar [3], is made of quartz sand and mica fines. When \(\frac{1}{{e_{0} }}\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{{\prime }} }}\) versus \(e\) graph for Cabalar’s data was constructed, the trend lines did not seem to be parallel to each other, in contrast to the other five types of mixtures. We attribute this to the high crushability of mica particles compared with the quartz particles. The determined slopes of trend lines (i.e., \(\hat{a}\)) for these mixtures are shown in Fig. 10 for various fines contents.

Fig. 10
figure 10

Parameter \(\hat{a}\), i.e., slope of trend lines in \(\frac{1}{{e_{0} }}\frac{{{\text{d}}e}}{{{\text{d}}p^{{\prime }} }}\) versus \(e\) space for the mixture of quartz sand and mica fines

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For the six types of mixtures, the determined \(e_{{r_{{f_{c} }} }}\) values from \(\frac{1}{{e_{0} }}\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{{\prime }} }}\) versus \(e\) graphs are plotted in Fig. 11. For all mixtures and at each fines content, \(e_{0}\) is also presented in Fig. 11. The distance between \(e_{0}\) and \(e_{{r_{{f_{c} }} }}\) represents the range of active void ratio (i.e., compressible voids) for each mixture.

Fig. 11
figure 11

Representative inactive void ratios (\(e_{{r_{{f_{c} }} }}\)) versus fines content for six different types of sand–silt mixtures. The values of \(e_{{r_{{f_{c} }} }}\) are determined from the compression tests data by constructing \(\frac{{{\text{d}}e}}{{{\text{d}}p^{{\prime }} }}\) vs. \(e\) graphs similar to Fig. 6

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3.1 Predicting \(e_{{r_{{f_{c} }} }}\) as a function of fines content

The schematic trend of \(e_{{r_{{f_{c} }} }}\) versus fines content, which is observed in Fig. 11 for six different types of sand–silt mixtures, is drawn in Fig. 12. It can be seen that \(e_{{r_{{f_{c} }} }}\) (i.e., kinematically unavailable void ratio) is influenced by the amount of fines in the mixture. The next step is to find a mathematical relationship between \(e_{{r_{{f_{c} }} }}\) and the fines content, so that we can eliminate the need for compression tests at all fines contents and predict \(e_{{r_{{f_{c} }} }}\) from fewer experiments.

Fig. 12
figure 12

Schematics of variation of active and inactive void ratio for sand–silt mixtures with various amounts of fines

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As the fines content is increased from 0%, \(e_{{r_{{f_{c} }} }}\) decreases to a minimum value and then continues to increase after a reversal point. This trend is similar to that observed in the studies for the variation of minimum void ratio of sand–silt mixtures with various amounts of fines. For the minimum void ratio, the specific fines content corresponding to the reversal point is called threshold fines content (TFC) by Thevanayagam et al. [40], transitional fines content by Yang et al. [45], or limiting fines content by Polito and Martin [33]. This reversal point occurs at transient fine contents ranging from approximately 20–35% for most sand–silt mixtures [5].

Intuitively, the theory of mixtures [2] is the first choice for estimating \(e_{{r_{{f_{c} }} }}\) of a mixture from the properties of clean sand and clean silt specimens. We examine the validity of upper-bound and lower-bound solutions for predicting the \(e_{{r_{{f_{c} }} }}\) from \(e_{{r_{\text{sand}} }}\) and \(e_{{r_{\text{silt}} }}\) at various fines contents.

Voigt’s upper-bound solution for the representative inactive void ratio (\(e_{{r_{{f_{c} }} }}\)) of a mixture can be expressed as:

$$e_{{r_{{f_{c} }} }} = \left( {1 - f_{c} } \right) \cdot e_{{r_{\text{sand}} }} + f_{c} \cdot e_{{r_{\text{silt}} }}$$
(13)

where \(e_{{r_{\text{sand}} }}\) and \(e_{{r_{\text{silt}} }}\) are the properties of clean sand and clean silt, determined from \(\frac{1}{{e_{0} }}\frac{{{\text{d}}e}}{{{\text{d}}p_{*}^{{\prime }} }}\) versus \(e\) graphs.

The lower-bound solution for \(e_{{r_{{f_{c} }} }}\) of a mixture can be calculated from Ruess theory and is given by:

$$\frac{1}{{e_{{r_{{f_{c} }} }} }} = \frac{{\left( {1 - f_{c} } \right)}}{{e_{{r_{\text{sand}} }} }} + \frac{{f_{c} }}{{e_{{r_{\text{silt}} }} }}$$
(14)

Figure 13 shows typical measured values of \(e_{{r_{{f_{c} }} }}\) for sand–silt mixtures with various fines contents (circle symbols). Equations 13 and 14 are used to calculate the upper-bound (solid line) and lower-bound (dashed line) solutions of \(e_{{r_{fc} }}\) for mixtures with \(e_{{r_{\text{sand}} }} = 0.5\) and \(e_{{r_{\text{silt}} }} = 0.9\). As shown in Fig. 13, the upper and lower bounds of the mixture theory cannot cover the range of measured \(e_{{r_{fc} }}\) values. The typical measured \(e_{{r_{fc} }}\) values are significantly lower than the Reuss lower bound (Fig. 13), although the Voigt’s upper bound is not violated in this figure.

Fig. 13
figure 13

Comparison of the measured data with the upper bound and lower bound from the classic mixture theory

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Previously, Chang and Meidani [5] showed that the theory of mixtures is not applicable to the variation of minimum void ratio of sand–silt mixtures, because this theory does not account for the mixing mechanism of particles; for instance, it cannot consider the filling of fines into the voids of the coarse-grains skeleton, or in the other case, the embedment of coarse particles into the fine-grains skeleton. Chang and Meidani [5] proposed a model for the variation of minimum void ratio of sand–silt mixtures and showed that the same model can also predict the critical state void ratio of such mixture. Chang et al. [5, 7, 33] extended the previous model and proposed a new relationship that could predict the minimum and maximum void ratios of such mixtures with a unique set of parameters.

According to Chang et al. [6], minimum void ratio of sand–silt mixtures can be calculated from the following equations:

$$e_{{\min_{fc} }} = \hbox{max} \left[ {e_{{\min_{c} }} , e_{{\min_{f} }} } \right]$$
(15)

where

$$e_{{\min_{c} }} = e_{{\min_{\text{sand}} }} \left( {1 - f_{c} } \right) + e_{{\min_{\text{silt}} }} f_{c} - \alpha \left( {1 + e_{{\min_{\text{silt}} }} } \right)f_{c}$$
(16)
$$e_{{\min_{f} }} = e_{{\min_{\text{sand}} }} \left( {1 - f_{c} } \right) + e_{{\min_{\text{silt}} }} f_{c} - \beta e_{{\min_{\text{sand}} }} \left( {1 - f_{c} } \right)$$
(17)

The subscripts c and f on the left side of Eqs. 16 and 17 refer to the coarse-grain and fine-grain dominant structures, respectively (Fig. 14). The minimum void ratios of clean sand (\(e_{{\min_{\text{sand}} }}\)) and clean silt (\(e_{{\min_{\text{silt}} }}\)) are known a priori. Equations 16 and 17 represent the lines on the left and right sides of Fig. 14. Out of the two values of minimum void ratio calculated from Eqs. 15 and 16, the larger one is the easier outcome during mixing, because it consumes less energy. Based on Eq. 15, the solid line is the true solution for the minimum void ratio of a mixture.

Fig. 14
figure 14

Generic variation of \(e_{{\min_{fc} }}\) versus fines content calculated from Eqs. 15 and 16 for coarse-grain and fine-grain dominant mixtures

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The range of coefficients \(\alpha\) and \(\beta\) is between 0 and 1. These two coefficients are used to include the effects of inclusion and embedment of coarse and fine particles into the dominant grain structure. The special case of \(\alpha = \beta = 0\) corresponds to the Voigt solution (upper bound). The special case of \(\alpha = 1\) corresponds to a case where the change in voids volume is equal to the volume of fines particles added to or removed from the mixture. The special case of \(\beta = 1\) represents a case where the volume of voids remains the same after removing or adding coarse particles into the mixture. The coefficients \(\alpha\) and \(\beta\) are determined by back fitting Eqs. 15 and 16 to the minimum void ratio data obtained from experiments.

In what comes next, we examine the validity of Eq. 15 for prediction of \(e_{{r_{fc} }}\) for sand–silt mixtures with various fines contents.

Even though the two density state measures \(e_{{r_{fc} }}\) and \(e_{{\min_{fc} }}\) are determined from two different loading processes and different degrees of particle crushing occur in each process, they both bear the same meaning toward the inactive void ratio concept. The loading process to reach the minimum void ratio state includes vibration of the specimen under small vertical load. The loading process to reach \(e_{{r_{fc} }}\) is compression (isotropic or 1-D) where all active voids are diminished by particle rearrangement. Therefore, the values of \(e_{{r_{fc} }}\) and \(e_{{\min_{fc} }}\) are not expected to be the same as shown in Fig. 15, where experimentally measured values of \(e_{{r_{fc} }}\) and \(e_{{\min_{fc} }}\) for the six types of sand–silt mixtures are plotted versus fines content.

Fig. 15
figure 15

Predicted \(e_{{r_{{f_{c} }} }}\) (solid lines), and predicted e min (dot-dash lines) for all fines contents based on the same sets of \(\alpha\) and \(\beta\) coefficients for six different types of sand–silt mixtures. \(e_{{r_{{f_{c} }} }}\), e min values and specimen initial void ratios are shown with square, circle, and diamond symbols, respectively

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Despite their difference, the effect of varying fines content in a mixture seems to be similar on the variation of \(e_{{r_{fc} }}\) and \(e_{{\min_{fc} }}\) as shown in Fig. 15. We employ the forms of Eqs. 16 and 17 to predict \(e_{{r_{fc} }}\) for sand–silt mixtures by replacing \(e_{\hbox{min} }\) with \(e_{r}\):

$$e_{{r_{c} }} = e_{{r_{\text{sand}} }} \left( {1 - f_{c} } \right) + e_{{r_{\text{silt}} }} f_{c} - \alpha \left( {1 + e_{{r_{\text{silt}} }} } \right)f_{c}$$
(18)
$$e_{{r_{f} }} = e_{{r_{\text{sand}} }} \left( {1 - f_{c} } \right) + e_{{r_{\text{silt}} }} f_{c} - \beta e_{{r_{\text{sand}} }} \left( {1 - f_{c} } \right)$$
(19)

where \(e_{{r_{c} }}\) and \(e_{{r_{f} }}\) are the reference inactive void ratios for coarse-grain and fine-grain dominant mixtures, respectively. Similar to the case of minimum void ratio, out of the two calculated values from Eqs. 18 and 19, the higher one is the true solution:

$$e_{{r_{fc} }} = \hbox{max} \left[ {e_{{r_{c} }} , e_{{r_{f} }} } \right]$$
(20)

The coefficients \(\alpha\) and \(\beta\) used in Eqs. 18 and 19 are the same as those used for calculating \(e_{\hbox{min} }\) of the same type of mixture.

For six different types of mixtures selected for this study, the coefficients \(\alpha\) and \(\beta\) are determined for five types of mixtures from the minimum void ratio data. The sixth dataset from Konishi et al. [20] did not report the values of minimum void ratio, so these two coefficients are determined from \(e_{{r_{fc} }}\) values for this type of mixture. These values are shown in Table 1.

4 Calibration of the model parameters and instructions on using the proposed model

The proposed model can predict the compression lines of sand–silt mixtures with any fines content, using two main parameters, i.e., \(\hat{a}\) and \(e_{{r_{fc} }}\) (Eq. 10). Parameter \(\hat{a}\) is directly determined from a compression test on one mixture with an arbitrary fines content. Parameter \(e_{{r_{fc} }}\) is dependent on the fines content of the mixture and is predicted by an augmented model expressed by Eq. 20. In order to calculate \(e_{{r_{fc} }}\) from Eq. 20, four parameters are required: \(e_{{r_{\text{sand}} }}\), \(e_{{r_{\text{silt}} }}\), \(\alpha\), and \(\beta\). These four parameters are determined from compression tests on two specimens with different fines contents and maximum density (i.e. minimum void ratio) tests on four mixtures with different fines contents. Briefly, the model uses five parameters to predict the compression lines for sand–silt mixture with any amount of fines and with any initial void ratio. The total number of required experiments for calibration of the model parameters are: compression tests on two specimens and minimum void ratio tests on four mixtures.

The process of calibrating the model parameters is shown in Figs. 16 and 17 in detailed steps.

Fig. 16
figure 16

Flowchart showing the calibration process to determining the model parameters from compression tests and minimum void ratio data

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Fig. 17
figure 17

Steps toward calibration of the model parameters and implementing the model for prediction the compression lines of sand–silt mixtures with any amount of fines

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5 Performance of the proposed model

Based on the augmented model proposed for \(e_{{r_{{f_{c} }} }}\) (Eq. 20), and plugging the predicted values of \(e_{{r_{{f_{c} }} }}\) in the proposed model (Eq. 10), we predict compression lines for the selected sand–silt mixtures in the stress ranges shown in Table 1. The predicted compression lines are shown in Fig. 18 together with the measured void ratios from the experiments.

Fig. 18
figure 18

Calculated compression lines for six types of sand–silt mixtures using Eq. 10 with their representative inactive void ratios determined from Eq. 20

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To check the performance of the proposed model, the predicted void ratios are plotted versus the measured void ratios in Fig. 19 for the selected six types of mixtures. The coefficient of determination R 2 is equal to 0.97, which proves the suitability of the proposed model to predict the compression lines for different types of sand–silt mixtures with any amount of fines.

Fig. 19
figure 19

Correlation between the predicted and measured void ratios for the six types of sand–silt mixtures

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6 Conclusions

We introduced a postulate to divide the voids in a granular material into active and inactive fractions. Based on this postulation, we proposed a mathematical expression to predict the void ratio during 1-D and isotropic compression for sand–silt mixtures with various fines contents. This model has two main parameters \(\hat{a}\) and \(e_{{r_{fc} }}\). We also proposed a method to predict the model parameter \(e_{{r_{fc} }}\) as a function of fines content, which is based on four coefficients. Two of the coefficients (i.e., \(\alpha\) and \(\beta\)) can be determined from the minimum void ratio tests on four different sand–silt mixtures. The other two coefficients \(e_{{r_{\text{sand}} }}\) and \(e_{{r_{\text{silt}} }}\) are determined from compression tests on two specimens of sand–silt mixture with two different fines contents, one of which has fines content less than and the other has fines content higher than transitional fines content.

The performance of the proposed model is verified by comparing the measured and predicted compression lines for six types of sand–silt mixtures with various fines contents. The comparison shows a coefficient of determination R 2 = 0.97, thus proving the suitability of the proposed model for predicting compressibility of sand–silt mixtures with any amount of fines.