Effect of Particle Shape on Stress-Dilatancy Responses of Medium-Dense Sands

Macroscopic mechanical behaviors of natural granular soils are significantly influenced by (1) internal factors, such as particle strength (Kuwajima et al. 2009; Shipton and Coop 2012; Wei and Yang 2014), particle size (Varadarajan et al. 2003; Frossard et al. 2012; Dai et al. 2016; Zhang et al. 2016; Zhou et al. 2016), particle size distribution (Kokusho et al. 2004; Li et al. 2013; Wang et al. 2013; Ovalle et al. 2014; Dai et al. 2016; Strahler et al. 2016; Xiao et al. 2018a), particle shape (Cho et al. 2006; Yang and Luo 2015; Altuhafi et al. 2016), and density (Been et al. 1991; Wan and Guo 1998; Xiao et al. 2014a); and (2) external factors, such as confining pressure (Charles and Watts 1980; Chiu and Fu 2008; Xiao et al. 2016a, 2017; Strahler et al. 2018), stress path (Vaid and Sasitharan 1992; Xu et al. 2012; Xiao et al. 2016b), loading mode (Chu and Wanatowski 2009), saturation (Oldecop and Alonso 2001; Yamamoto et al. 2009; Ovalle et al. 2015), and drainage conditions (Chu et al. 2012). Moreover, addition of fines (i.e., particle size finer than 74 μm) in different fractions to clean sands can greatly affect the strength, dilatancy, and critical-state behavior of these sand-fines mixtures (Thevanayagam 1998; Salgado et al. 2000; Polito and Martin 2001; Chiu and Fu 2008; Papadopoulou and Tika 2008; Carraro et al. 2009; Rahman and Lo 2014; Lashkari 2016; Simpson and Evans 2016). From a general point of view, a change in fines content is an internal factor (i.e., effect of particle size distribution) since the gradation of these mixtures is dependent on fines content. The plasticity of fines is also a crucial factor that strongly influences the undrained strength (Ni et al. 2004) and cyclic liquefaction resistance (Park and Kim 2013) of the mixture. Furthermore, Papadopoulou and Tika (2016) found from monotonic and cyclic triaxial tests on coarse-fine mixtures that increasing fines plasticity up to a threshold value led to a decrease in the undrained strength and liquefaction resistance, while continuously increasing fines plasticity over the threshold value caused an increase in the undrained strength and liquefaction resistance.

Yang and Wei (2012) conducted a series of undrained triaxial tests on four binary mixtures by adding distinct quantities of two nonplastic fines with different particle shapes (angular crushed silica and rounded glass beads) into two clean quartz sands. A significant finding is that adding rounded fines into clean sands caused an obvious decrease in the critical-state friction angle, while adding angular fines led to a slight increase in the critical-state friction angle. Moreover, the binary mixtures containing rounded fines were much more susceptible to collapse than the binary mixtures containing angular fines with the same content. Wei and Yang (2014) provided sufficient test data to confirm that the differences in shear behavior observed from Yang and Wei (2012) were determined by the particle shape rather than by the particle hardness. Yang and Luo (2015) also found that mixing sands with different particle shapes and different percentages could greatly influence the overall shear behavior. Furthermore, Yang and Luo (2017) emphasized that the critical-state friction angle of sands was affected by the particle shape rather than by the gradation. These findings illustrate that particle shape can have significant impacts on the susceptibility to liquefaction, undrained shear strength, and critical-state strength of sands (Rousé et al. 2008; Tsomokos and Georgiannou 2010; Suh et al. 2017) since the particle shape affects the assembly density (Cho et al. 2006; Shin and Santamarina 2013; Zheng and Hryciw 2016) and influences fabric stability at the contact scale (Yang and Wei 2012). Stress-dilatancy behavior is an intrinsic property of granular materials (Bolton 1986; Salgado et al. 2000; Zhao and Evans 2009; Strahler et al. 2016). However, the effects of particle shape on the strength, dilatancy, and stress-dilatancy response have not been fully studied. How the particle shape influences the peak-state strength, the maximum dilatancy, and their interrelationship needs to be systematically studied. Furthermore, implications of applying Bolton’s stress-dilatancy equation to sands with distinct particle shapes requires further discussion and investigation.

The current study aims to investigate the effect of particle shape on the stress-dilatancy behavior of sand mixtures through a series of drained triaxial compression tests at various confining pressures. Based on the particle-shape parameters measured with a binary image-based method, changes in the stress-strain relationship, peak-state friction angle, peak-state axial strain, maximum dilation angle, critical-state friction angle, and stress-dilatancy relationship as a function of a particle-shape parameter are presented.

Glass sands with rounded and angular particle shapes from the Mining in Xinyi Vanward Mining, Jiangsu Province, China, were used in the current study. Fig. 1(a) shows that the uniform gradations of the two glass sands were identical in order to eliminate the effects of particle size and gradation on the comparative behaviors of the materials. The minimum and maximum diameters of the particles were 0.6 and 0.8 mm, respectively. In addition, the silicon dioxide content of these glass sands was approximately 99%. The Mohs hardness was about 5.5 for typical glass including the current glass sands, as reported by Hagerty et al. (1993). The average specific gravity was 2.35. Following the work by Yang and Luo (2015) to compose sands with distinct particle shape distributions, five mixtures were prepared with different contents of rounded and angular glass beads by weight: 0% angular glass beads plus 100% rounded glass beads (denoted as A0R100), 25% angular glass beads plus 75% rounded glass beads (A25R75), 50% angular glass beads plus 50% rounded glass beads (A50R50), 75% angular glass beads plus 25% rounded glass beads (A75R25), and 100% angular glass beads plus 0% rounded glass beads (A100R0). Table 1 lists values of the minimum void ratio $e_{\min }$ and maximum void ratio $e_{\max }$ for the five mixtures, which were systematically measured in accordance with ASTM standards (ASTM 2016a, b). The initial relative density $I_D$ for all samples in this study was 0.6. As shown in Fig. 1(b), $e_{\min }$ and $e_{\max }$ both increase with increasing angular bead content due to the increased angularity in the mixtures (Cho et al. 2006; Shin and Santamarina 2013; Zheng and Hryciw 2016). In other words, the angular glass beads tend to form more loose structures in comparison with the rounded glass beads. In addition, the void ratio difference ($\mathrm{\Delta }e=e_{\max }-e_{\min }$) increases with increasing angular bead content up to 50% angular particles and then levels out. Fig. 2 presents scanning electron microscope (SEM) images of samples for each mixture conducted, illustrating that the overall average roundness in the mixtures varies over a wide range. Fig. 2 also shows that the surface roughness of rounded and angular glass beads was quite similar.

This study mainly aimed to investigate the effect of particle shape on the stress-strain-dilatancy behavior of granular materials. Of significant importance is how to define a representative shape parameter for describing the particle shape and how to establish an accurate method to quantify this shape parameter. Multiple definitions of shape parameters may be found in the literature (Jennings and Parslow 1988; Endoh et al. 1998; Fonseca et al. 2012; Yang and Wei 2012; Yang and Luo 2015; Lee et al. 2017). Wadell (1932) first proposed the shape parameter: sphericity ($S_C$), which is defined as the ratio of the surface area of the sphere with the same volume as the particle to its actual surface area. This shape parameter has been widely used to evaluate overall particle shape. Yang and Luo (2015) proposed four simplified shape parameters for sand mixtures: sphericity ($S_C$), aspect ratio ($A_R$), convexity ($C_X$) and overall regularity ($O_R$). Lee et al. (2017) not only redefined sphericity but also selected elongation, slenderness, and convexity to describe the irregularity of sand grains. The shape evaluation method proposed by Yang and Luo (2015) was adopted in this study since this method is more suitable for sand mixtures.

Two samples with clean angular glass beads (A100R0) and clean rounded glass beads (A0R100) were subjected to shape measurement. Figs. 3(a and b) present microscopic particle images for each sample, which were processed into binary images using ImageJ (Schneider et al. 2012). Fig. 4 shows a sequence of dimensions for an angular glass particle, from which the detailed definitions of sphericity, aspect ratio, and convexity are given as follows:

Yang and Luo (2015) proposed an overall regularity ($O_R$) to combine $S_C$, $A_R$, and $C_X$ for characterizing particle shape in an integrated manner

The binary image of each particle was accurately measured in Fig. 3 to compute $S_C$, $A_R$, $C_X$, and $O_R$. This approach is analogous to the imaging-based method utilized by Yang and Luo (2015). Fig. 5 presents the binary images of representative angular glass beads with different shape parameters. Comparison of these data shows that the shape of particles becomes more rounded with increasing magnitude of the overall regularity values. Particle images from a large sample of glass sands have been analyzed to obtain reliable shape data for each sample. The number of particles for the clean angular glass beads and clean rounded glass beads is 1,000. Based on these shape data, cumulative distributions of $S_C$, $A_R$, $C_X$, and $O_R$ are shown in Fig. 6. We selected the shape value at 50% of the cumulative distribution as the representative value for a given material. The representative values of $S_C$, $A_R$, $C_X$, and $O_R$ for A0R100 are 0.938, 0.977, 0.997, and 0.971, respectively; while the representative values of $S_C$, $A_R$, $C_X$, and $O_R$ for A100R0 are 0.846, 0.726, 0.961, and 0.844, respectively. Yang and Luo (2015) extended a concept of combined roundness proposed by Yang and Wei (2012) to generalized combined-shape parameters for binary mixtures. For instance, the combined $S_C$ for Mixture A75R25 can be calculated by

The $S_C$ value of 0.869 for Mixture A75R25 is reasonable when compared with 0.846 for pure angular glass beads and 0.938 for pure rounded glass beads. The concept of combined shape parameters is reasonable because the rounded and angular glass beads have the same gradation and specific gravity. All shape data for these mixtures in this paper are given in Table 2 for ease of reference. The combined $O_R$ as an integrated shape parameter of $S_C$, $A_R$, and $C_X$ are used to explore its relationships with the peak-state strength, peak-state strain, excess friction angle, and maximum dilation angle for these mixtures, which are presented in the following sections.

A strain-controlled digital triaxial apparatus was used in this study. All specimens had a nominal height of 80 mm and diameter of 39.1 mm as prepared. A rubber membrane with a thickness of 0.2 mm and length of 100 mm was placed inside a three-part split mold and secured to the top. The elongated particles due to the angular shape of angular glass beads have the potential to impact their orientation during specimen preparation and the corresponding initial fabric. It is well established that different specimen preparation methods result in different fabrics of mixtures (Ladd 1974; Vaid et al. 1999; Frost and Park 2003; Sadrekarimi and Olson 2012). The moist tamping method has been used in the current study to prepare specimens, which has been shown to produce an isotropic and uniform fabric that reduces the effect of particle shape on sample preparations compared with that prepared by the dry pluviation technique (Yang et al. 2008). The dry sand was mixed with 5% deaired water (relative to the weight of the dry sand) and divided into six equal parts prior to formation. Each part was carefully placed into the mold to form a layer using the undercompaction method proposed by Ladd (1978). The density of each layer in this undercompaction method should be compacted slightly more (about 1%) than that of the substratum layer in order to obtain a more uniform density throughout the specimen (Polito and Martin 2001; Chiu and Fu 2008; Belkhatir et al. 2012; Papadopoulou and Tika 2016). The membrane was fixed to the top cap when the last part of the mixture was compacted to the designed density. The mold was removed while a vacuum of approximately 20 kPa was applied to the sample. The triaxial cell was then placed around the specimen and the cell was filled with deaired water. The specimen was saturated for up to 1 h under a confining pressure of 30 kPa until the $B$-value reached 0.98 or higher. Finally, the specimen was isotropically consolidated at a given confining pressure (e.g., 0.05, 0.1, 0.2, 0.3, and 0.4 MPa) and then sheared under drained conditions with a constant vertical displacement rate of $0.21\mathrm{mm}/\min$. The stress and deformation results were recorded by a computerized data acquisition system.

Similar to the work conducted by Zhang et al. (2018), the specimen uniformity was assessed by checking the particle size distribution (PSD) and the cumulative distributions of overall regularity $O_R$ along the height of a specimen (A50R50). Fig. 7(a) shows that the specimen (A50R50) formed by the proposed moist tamping method was subjected to saturation and consolidated at ${\sigma }_3^{\prime }=0.1\mathrm{MPa}$. Then the specimen was divided into six slices (A, B, C, D, E, and F), which were subjected to particle size analysis and particle shape measurement. Fig. 7 shows that PSDs and cumulative distributions of $O_R$ for all six slices have few changes, illustrating that the specimens are largely uniform. Photos of the A50R50 sample before and after shearing at ${\sigma }_3^{\prime }=0.1\mathrm{MPa}$ in Fig. 7(b) show that the medium-dense sample could maintain an effectively homogeneous strain field even with the axial strain up to 30%.

In addition, it is known that particle breakage can alter the peak-state friction angle and dilation angle of granular soils (Indraratna et al. 1998; Tarantino and Hyde 2005; Xiao et al. 2014b). Specimens A75R25 and A25R75 at 0.4 MPa (i.e., the maximum confining pressure used in the current study) after tests were subjected to particle size analysis to evaluate the grain crushing. Fig. 8 shows that the grading after tests for both Specimens A75R25 and A25R75 uplifts slightly from the original grading. Particularly, the deviation of the grading after test to the original grading for Specimen A75R25 in Fig. 8(a) is a little larger than that for Specimen A25R75 in Fig. 8(b), indicating that the specimen with more angular-shape grains undergoes more crushing. This can be further validated by the relative breakage index $B_r$ (Hardin 1985) of 2.3% for Specimen A75R25 and 1.9% for Specimen A25R75. However, the particle breakage, as indicated by the small $B_r$ value, should not exert a significant influence on the mechanical behaviors of the mixtures. This finding is consistent with the high particle strength of glass sand used in this study, as mentioned in the previous section. In addition, microscopic images of sand particles after tests in Fig. 8 shows that only a small amount of sand fragments can be observed.

Fig. 9 shows the typical curves on the deviatoric stress versus axial strain versus volumetric strain for these mixtures at the medium-dense state in the drained triaxial compression tests. A peak state, which is defined as the state where the deviatoric stress is maximum, was observed. The axial strain at the peak state corresponds to the maximum rate of dilation, which is referred to as the state where the ratio of the volumetric-strain increment to the axial-strain increment is maximum. The corresponding definitions of friction angle and dilation angle at peak state are introduced in the next section.

Fig. 10 shows evolutions of the deviatoric stress and stress ratio with axial strain for different mixtures at various confining pressures (i.e., ${\sigma }_3^{\prime }=0.05$, 0.1, 0.2, 0.3, and 0.4 MPa). The deviatoric stress of all specimens at a given confining pressure increased to a maximum value at the peak state and then decreased slightly at a comparatively large axial-strain range, which illustrates that all specimens at an initially medium-dense state exhibited the strain-softening behavior, as presented in Figs. 10(a–e). Fig. 10 also clearly shows stick-slip behavior (Johnson et al. 2008; Cui et al. 2017) in the deviatoric stress. This is particularly evident in Fig. 10(e), where this stick-slip behavior becomes more obvious as confining pressure increases, which is consistent with results reported by Wu et al. (2017). In addition, Figs. 10(f–j) show that an increase in rounded glass bead content leads to an increase in stick-slip behavior. This behavior may be explained as follows: heterogeneity in the shape of angular particles results in heterogeneity in void space size and geometry, causing smoother evolution of fabric during shear. In assemblies of spherical particles, void space is topologically more uniform, which leads to greater frustration in particle reorganization, which manifests as stick-slip behavior at the macroscale. This is consistent with the observation that interlocking between rounded particles is more unstable than interlocking between angular particles (Yang and Wei 2012; Wei and Yang 2014). Comparing these stress-strain curves in Fig. 10 shows that increasing the fraction of rounded glass beads in a mixture at the same confining pressure reduces the deviatoric stress as well as stress ratio. Furthermore, the peak-state axial strain (i.e., the black vertical arrows in Fig. 10) increases with increasing the confining pressure or with increasing the percentage of angular glass beads. The detailed trends in the peak-state deviatoric stress and peak-state axial strain are introduced in the following section.

Fig. 11 compares the relationships between the volumetric strain and axial strain for different mixtures at various confining pressures (i.e., ${\sigma }_3^{\prime }=0.05$, 0.1, 0.2, 0.3, and 0.4 MPa). Sand mixtures shown in Figs. 11(a–e) exhibit a slight initial contraction that becomes more obvious with an increase in confining pressure and then begin to dilate from the phase-transformation state where the volumetric-strain increment is zero but the stress increment is not zero (Been and Jefferies 1985; Xiao et al. 2018b). Volumetric expansion decreases with increasing confining pressure for a given mixture. Furthermore, comparing the volumetric strain–axial strain curves in Figs. 11(f–j) indicates that the dilatancy behavior at a given confining pressure is also a function of the mean particle angularity in the mixtures. The initial contraction of pure angular glass beads (A100R0) is more pronounced than that of pure rounded glass beads (A0R100) before the phase-transformation state. This difference is particularly apparent at higher confining pressures. For example, the maximum contraction is about 0.447% for A100R0 but only 0.117% for A0R100 at the same effective confining pressure (${\sigma }_3^{\prime }=0.4\mathrm{MPa}$), as shown in Fig. 11(j). The volumetric expansion at the final state increases with increasing angular particle content. For example, Fig. 11(f) shows that Specimen A100R0 reaches the volumetric strain of $-11.29\%$ at an axial strain of 30%, whereas the volumetric strain of A0R100 is $-6.98\%$ at the same axial strain. However, the dilation rate of A100R0 at the peak state in Fig. 11(f) is lower than that of A0R100.

Fig. 12(a) shows the variation of the peak-state deviatoric stress with the overall regularity ($O_R$) and effective confining pressure. Consider, for example, that at an effective confining stress of 0.3 MPa, the peak-state deviatoric stress increases from 0.77 to 1.19 MPa with overall regularity decreasing from 0.971 to 0.844. The maximum differences in peak-state deviatoric stress due to changes in overall regularity are 0.10, 0.18, 0.33, 0.42, and 0.61 MPa for confining stresses of 0.05, 0.1, 0.2, 0.3, and 0.4 MPa, respectively. Fig. 12(b) presents the variation in the peak-state axial strain as a function of overall regularity and confining pressure. It is clear that an increase in ${\sigma }_3^{\prime }$ or a decrease in $O_R$ leads to an increase in the peak-state axial strain. For instance, increasing ${\sigma }_3^{\prime }$ from 0.05 to 0.4 MPa at $O_R=0.876$ results in an increase in the peak-state axial strain from 4.19% to 7.35%, whereas a decrease in $O_R$ from 0.971 to 0.844 at ${\sigma }_3^{\prime }=0.2\mathrm{MPa}$ gives an increase in the peak-state axial strain from 2.46% to 7.19%.

The peak-state friction angle ${\varphi }_{ps}^{\prime }$ (Bolton 1986) is defined as follows:where ${({\sigma }_1^{\prime })}_{ps}$ and ${({\sigma }_3^{\prime })}_{ps}$ = major and minor principle effective stresses at the peak state, respectively. Evolution of the peak-state friction angle ${\varphi }_{ps}^{\prime }$ with the overall regularity $O_R$ and confining pressure ${\sigma }_3^{\prime }$ is presented in Fig. 13(a), which shows that an increase in $O_R$ at a given ${\sigma }_3^{\prime }$ leads to a reduction in ${\varphi }_{ps}^{\prime }$. For example, ${\varphi }_{ps}^{\prime }$ decreases from 45.5° to 37.1° as $O_R$ increases from 0.844 to 0.971 at ${\sigma }_3^{\prime }=0.05\mathrm{MPa}$. In addition, an increase in ${\sigma }_3^{\prime }$ also gives a reduction in ${\varphi }_{ps}^{\prime }$, which, for example, decreases from 42.8° to 38.8° with increasing ${\sigma }_3^{\prime }$ from 0.05 to 0.4 MPa at $O_R=0.876$. Fig. 13(a) also shows that the peak-state friction angle reaches a maximum value of 45.5° at ${\sigma }_3^{\prime }=0.05\mathrm{MPa}$ and $O_R=0.844$ and a minimum value of 32.7° at ${\sigma }_3^{\prime }=0.4\mathrm{MPa}$ and $O_R=0.971$. The maximum differences in ${\varphi }_{ps}^{\prime }$ due to the changes of $O_R$ are 8.4°, 8.3°, 8.0°, 7.4°, and 8.6° at confining pressures of 0.05, 0.1, 0.2, 0.3, and 0.4 MPa, respectively.

The maximum dilation angle ${\psi }_{\max }$ can be expressed aswhere $d{ϵ}_a$ and $d{ϵ}_v$ = axial strain and volumetric strain increments, respectively; and ${(d{ϵ}_v/d{ϵ}_a)}_{\max }$ = maximum ratio of the volumetric strain increment to the axial strain increment. Fig. 13(b) presents the relationship between ${\psi }_{\max }$, $O_R$, and ${\sigma }_3^{\prime }$. An increase in ${\sigma }_3^{\prime }$ results in a decrease in ${\psi }_{\max }$, which, for example, decreases from 18.1° to 11.7° with increasing ${\sigma }_3^{\prime }$ from 0.05 to 0.4 MPa at $O_R=0.971$. Note that ${\psi }_{\max }$ at a given confining pressure in Fig. 13(b) increases slightly with increasing overall regularity $O_R$, which is in contrast with the variation in ${\varphi }_{ps}^{\prime }$ observed in Fig. 13(a). Specifically, ${\psi }_{\max }$ increases from 8.7° to 11.7° with increasing $O_R$ from 0.844 to 0.971 at ${\sigma }_3^{\prime }=0.4\mathrm{MPa}$. However, for a given confining pressure, the dilation at high strains in Figs. 11(f–j) increases with increasing $O_R$. Rounded particles exhibit a higher dilation rate at small strains than angular particles, while assemblies of rounded particles dilate less overall at critical state compared with angular particles. This behavior is intimately related to the stick-slip behavior observed in the stress-strain response of spherical particles. Specifically, spherical particles are prone to crystallization (i.e., degeneracy to regular packings) that requires significant dilation to overcome (e.g., consider the dilation required to overcome volumetric constraints in a tetrahedral packing). This behavior is influenced by particle contact orientations, particle rotations, and interlocking; discrete-element method simulations are used to investigate these mechanics and will be introduced in future work.

The relationship between peak-state friction angle and maximum dilation angle for the granular mixtures evaluated here is presented in Fig. 14. Bolton (1986) proposed an empirical stress-dilatancy equation, of which the intercept can be adopted as the critical-state friction angle (${\varphi }_{cs}^{\prime }$) in accordance with Vaid and Sasitharan (1992). The empirical equation can be given aswhere ${\chi }_d$ = dilatancy coefficient.

The best fit regressions in Fig. 14 show that a dilatancy coefficient ${\chi }_d$ is 0.6 for all of the tested mixtures, indicating that the coefficient is largely independent of the overall regularity. Yang and Luo (2017) showed that particle shape has a significant influence the critical-state friction angle ${\varphi }_{cs}^{\prime }$. They found that increasing the ratio of glass beads to Fujian sand in a mixture leads to a decrease in ${\varphi }_{cs}^{\prime }$, while increasing the ratio of angular glass beads to Fujian sand in a mixture leads to an increase in ${\varphi }_{cs}^{\prime }$. We observed similar behavior in the current study. It can be seen from Fig. 14 that the critical-state friction angles ${\varphi }_{cs}^{\prime }$ are 25.6°, 28.0°, 30.6°, 32.6°, and 35.7° for $O_R=0.971$, 0.939, 0.908, 0.876, and 0.844, respectively, indicating that ${\varphi }_{cs}^{\prime }$ increases substantially with a reduction in $O_R$. The difference between ${\varphi }_{ps}^{\prime }$ and ${\varphi }_{cs}^{\prime }$ can be described as the excess friction ${\varphi }_{ex}^{\prime }$, which is expressed as

Fig. 15 shows variations of ${\varphi }_{ex}^{\prime }$ with $O_R$ and ${\sigma }_3^{\prime }$. The maximum value of ${\varphi }_{ex}^{\prime }$ can be obtained at low confining stress and high overall regularity, i.e., ${\sigma }_3^{\prime }=0.05\mathrm{MPa}$ and $O_R=0.971$. This observation illustrates that a decrease in ${\sigma }_3^{\prime }$ or an increase in $O_R$ could lead to an increase in ${\varphi }_{ex}^{\prime }$. For example, ${\varphi }_{ex}^{\prime }$ at $O_R=0.908$ is found to increase by a total of 4.3° with a decrease in ${\sigma }_3^{\prime }$ from 0.4 to 0.05 MPa. For a given confining pressure of 0.2 MPa, an increase in $O_R$ from 0.844 to 0.971 results in an increase in ${\varphi }_{ex}^{\prime }$ by a total of 2.1°. Table 3 lists in detail the values of ${\varphi }_{ps}^{\prime }$, ${\varphi }_{ex}^{\prime }$, and ${\psi }_{\max }$. Also, variations of ${\varphi }_{ex}^{\prime }$ with $O_R$ and ${\sigma }_3^{\prime }$ in Fig. 15 are similar to variations of ${\psi }_{\max }$ with $O_R$ and ${\sigma }_3^{\prime }$ in Fig. 13(b). This suggests that an intrinsic relationship between ${\varphi }_{ex}^{\prime }$ and ${\psi }_{\max }$ exists. Combining Eqs. (8) and (9) gives

Fig. 16(a) shows that the best-fit surface relating ${\varphi }_{ex}^{\prime }$ and ${\psi }_{\max }$ at different overall regularities predicted by Eq. (10) ($R^2=0.956$) is in good agreement with test data. It may be seen from Fig. 16(a) that the fitted trends exhibit a zero intercept and are well captured by the predicted surface. In addition, the relationship between the test data, fitted trends, and fitting surface projected onto the ${\varphi }_{ex}^{\prime }\text{-}{\psi }_{\max }$ plane further indicates that the dilatancy coefficient ${\chi }_d$ is generally a constant for these mixtures with different overall regularities. This implies that the relationship between dilatancy and excess strength is independent of particle shape. However, the critical-state friction angle is dependent on the particle shape, as observed in Fig. 14. A linear regression can be employed to describe the relationship between ${\varphi }_{cs}^{\prime }$ and $O_R$ in Fig. 16(b)

Eq. (11) reasonably describes the relationship between critical-state friction angle and overall regularity with $R^2=0.996$. Finally, combining Eqs. (8) and (11) gives

Eq. (12) describes the quantitative contributions of $O_R$ and ${\psi }_{\max }$ to ${\varphi }_{ps}^{\prime }$. The peak-state strength is dependent not only on dilatancy but also overall regularity. Fig. 16(c) confirms that the prediction by Eq. (12) is in good agreement with the measured ${\varphi }_{ps}^{\prime }$ in relation to $O_R$ and ${\psi }_{\max }$ ($R^2=0.946$). The material constants in Eq. (12) vary for different types of sand; however, the basic trend in effect of particle shape on the stress-dilatancy relation should be general for other sands, though additional test data from other sands are needed to validate this inference.

This paper presented a systematic investigation of the effect of particle shape on the stress-strain-dilatancy behaviors of sand mixtures through a series of deliberately designed triaxial tests. The main findings can be summarized as follows:

The authors are grateful to Professor John McCartney for helpful comments provided during preparation of this manuscript. The authors would like to acknowledge the financial support from the 111 Project (Grant No. B13024), the National Science Foundation of China (Grant Nos. 51509024, 51678094, and 51578096), the Fundamental Research Funds for the Central Universities (Grant No. 106112017CDJQJ208848), and Special Financial Grant from the China Postdoctoral Science Foundation (Grant No. 2017T100681). T. Matthew Evans was supported by the National Science Foundation (NSF) (Grant No. CMMI-1538460) during the course of this work. This support is gratefully acknowledged.