The tested sand-gravel mixture was obtained from Nanjing, China. The gravel grains of the mixture are prismatic. The mixture’s gravel content (G _c) is 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, and 100%. The particle size distribution curves of various sand-gravel mixtures are shown in Figure 1. The basic properties of the mixtures are listed in Table 1. The mixtures’ particle size distribution curves and basic properties were measured according to the ASTM D4254-14, 2006 and ASTM D4254-16, 2006.

As shown in Table 1, the e _max and e _min of the mixtures both decrease and then increase with the rise in G _c, which is consistent with the findings of Evans and Zhou., (1995) and Amini and Chakravrty, 2004. In addition, the e _max and e _min reach the minimum value at G _c equals 50%.

The measurement of shear wave velocity (V _s) and associated G _max was implemented using a pair of piezoceramic bender elements installed in the GCTS HCA-300 dynamic hollow cylinder-TSH testing system (Chen et al., 2019b). The test apparatus is shown in Figure 2. The confining and back pressure were measured using the standard pressure/volume controller. The axial static and dynamic force was controlled independently. Moreover, the maximum range of the dynamic force is 10 kN/5 Hz. The axial force and displacement sensors were placed at the top of the sample. Back pressure was applied at the top of the sample, and the excess pore water pressure was measured its bottom. Hardin and Black (1966), Goudarzy et al. (2016) detailed the testing principle of the bender element system.

The V _s is calculated via Eq. 1: V s = d t (1) where d is the effective distance of the shear wave propagation, and t is the time of the shear wave propagation.

The time domain method was used to determine t considering the simplicity and accuracy. Figure 3 shows the typical time histories of output signals from bender element tests, revealing that the received signals are always clear and efficient.

The cylindrical specimen has a diameter of 100 mm and a height of 150 mm. The specimen was prepared using a dry tamping method. This technique was adopted in several research works to test granular material. On the other hand, the well-mixed sand and grains were tamped into a cylindrical specimen creator for four layers in the apparatus using a dry tamping method. The pre-saturation was conducted after the specimen preparation. The pre-saturation consists of three steps: 1) permeating the specimen with CO₂ for 30 min; 2) flushing with de-aired water for 60 min; 3) flushing all water lines. After the pre-saturation, the back pressure saturation was initiated. Back pressure was gradually applied, and the Skempton B-value was checked until exceeding 0.95, which guaranteed the saturation of the tested sample. The saturated sample was consolidated under an effective target confining pressure until the strain was stable. After that, the bender element was conducted.

A series of bender element tests was conducted to study the V _s of the sand-gravel mixtures. The influence of relative density (D _r), effective confining pressure ( σ 0 ′ ), and G _c were considered. The D _r of the mixtures was taken as 30%, 45%, and 70%. Additionally, the σ 0 ′ of the mixtures was taken as 50, 100, 200, 300, and 400 kPa, and the G _c of the mixture was selected as 0%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, and 100%.

The relationship between the normalized effective confining pressure σ 0 ′ / P a and V _s of the sand-gravel mixtures is shown in Figure 4, where the atmospheric pressure (P _a) is approximately equal to 100 kPa. It can be seen that for a given G _c and D _r, the mixtures' vs. increases with the rise in σ 0 ′ / P a . The reason may be that the greater the pressure, the greater the contact force between the particles, and the more the granular materials converge to a whole, which leads to easier shear wave propagation and increased propagation speed. Moreover, the V _s increases with the increase in D _r when the G _c and σ 0 ′ / P a are given. The relationship between the V _s and σ 0 ′ / P a can be described by Eq. 2: V s = A σ 0 ′ / P a n (2) where A is the shear wave velocity value of the mixture when the σ 0 ′ is 100 kPa, and n is the best-fit coefficient, which reflects the influence of σ 0 ′ on the V _s.

The relationship between the fit coefficient n and coefficients of uniformity (C _u) for the mixtures is shown in Figure 5. It can be seen that the n increases with the increase in C _u. The relationship between the n and C _u can be described by Eq. 3: n = M C u N (3) where M and N are the best-fit coefficients, which for the sand-gravel mixture of this test are defined as M is 0.24, and N is 0.04. The goodness of fit (R ²) for this equation is 0.95.

The relationship between the normalized shear wave velocity, V s / σ 0 ′ / P a n , and G _c of the sand-gravel mixtures is depicted in Figure 6. It can be seen that the V s / σ 0 ′ / P a n increases and then decreases with the increase in G _c. The V s / σ 0 ′ / P a n reaches its peak when the G _c is 50%, meaning that the threshold gravel content value (G _cth) of the sand-gravel mixtures is 50%. The reason is that part of the force chain in sand particles is replaced by that of sand-gravel and gravel grains as the G _c increases. The contact area of the mixture grains increases, the V s / σ 0 ′ / P a n increases first. However, the sand particles fill the void of the gravel grains, and the force chain of sand particles is invalid when the increase in G _c exceeds the G _cth. As a result, the V s / σ 0 ′ / P a n decreases.

The relationship between the V s / σ 0 ′ / P a n and e of the sand-gravel mixtures is shown in Figure 7. Generally, the V s / σ 0 ′ / P a n decreases with the rise in e. The relationship between the V s / σ 0 ′ / P a n and e can be described using a power function when the G _c is given. However, the relationship between the V s / σ 0 ′ / P a n and e described by a power function varies with respect to G _c. Accordingly, the e is not a reasonable parameter to describe the dense state of the sand-gravel mixtures.

Sand-gravel mixture is composed of coarse gravel grain and fine sand particles, which is a fine-coarse grained mixture. Microstructural changes can affect the macro mechanical properties (Bai et al., 2019; Bai et al., 2021; Bai et al., 2022). The sand-gravel mixture’s force skeleton depends on the sand and gravel content, and the part that fills the void is invalid for the force skeleton. The force skeleton is composed of coarse gravel grain when the G _c is larger than G _cth. However, the force skeleton is composed of fine sand particles when the G _c is smaller than G _cth. The skeleton void ratio (e _sk) is defined as the volumetric ratio between the voids formed in the sand-gravel mixture skeleton and the volume of particles that make up the skeleton (Chang et al., 2014). This is used to describe the dense state of the fine-coarse-grained mixture. Thevanayagam (2007a, 2007b) proposed a binary intergranular contact theory of the fine-coarse-grained mixture and believed the particle contact state is divided into two types. The intergranular contact state of the sand-gravel mixture is shown in Figure 8.

The e _sk is calculated using Eqs 4, 5 when the intergranular contact state of the sand-gravel mixtures is in contact states 1 and 2, respectively. R _d is the average grain size ratio, which is the ratio of d _50-g and d _50-s. The d _50-g is the average size of the gravel, and d _50-s is the average size of the sand. b is the sand’s influence index, which ranges from 0 to 1. The sand particle is invalid for the force skeleton of the sand-gravel mixture when b is 0. Furthermore, the sand particles can be used in the force skeleton when b is 1. m is the gravel’s influence index that ranges from 0 to 1. The b and m can be determined using a back-fitting analysis (Thevanayagam, , ). e sk = e + 1 − b ⋅ 1 − G c 1 − 1 − b ⋅ 1 − G c (4) e sk = e 1 − G c + G c / R d m (5)

The relationship between the V s / σ 0 ′ / P a n and e _sk of the sand-gravel mixtures is shown in Figure 9. It can be seen that the V s / σ 0 ′ / P a n decreases with the increase in e _sk. The relationship between the V s / σ 0 ′ / P a n and e _sk can be fitted by two curves using Eq. 6, and the G _cth is the critical value. The mechanical behavior of the sand-gravel mixtures under the same e _sk is similar to that of pure gravel (G _c = 100%) when the G _c is larger than G _cth. Moreover, the mechanical behavior of the sand-gravel mixtures under the same e _sk is similar to that of pure sand (G _c = 0) when the G _c is smaller than G _cth. As a result, the relationship between the V s / σ 0 ′ / P a n and e _sk fitted by two curves using Eq. 6 is reasonable. V s = A 2 σ 0 ′ P a M × C u N (6) where A ₂ and B ₂ are the best-fit parameters determined as A ₂ is 210.29, B ₂ is -0.41 when the G _c is smaller than G _cth, and A ₂ is 216.48, and B ₂ is -0.53 when the G _c is larger than G _cth.

A series of bending element tests were conducted by Choo and Burns (2015) and Oka et al. (2018) to investigate the effects of fine granular content (FC) on V _s of coarse and fine granular mixtures. In this section, test data published in the previous literature were used to further verify the applicability of Eq. 6 for two types of coarse and fine granular mixtures. The V _s versus e _sk curves for two types of coarse and fine granular mixtures are shown in Figure 10. It can be clearly observed that e _sk can normalize V _s, indicating that it is reasonable for e _sk to V _s of coarse and fine granular mixtures.