Discrepancy in the Critical State Void Ratio of Poorly Graded Sand due to Shear Strain Localization

In engineering practice, the constitutive behavior of sand has been traditionally modeled using global-scale measurements of stress and strain at the boundaries of laboratory-size specimens. The development of the stress-dilatancy relationship (Roscoe et al. 1958) offered a major enhancement to the Mohr-Coulomb criterion and inspired the development of the critical state (CS) theory in soil mechanics. The CS theory was originally proposed for clays by Schofield and Wroth (1968), who postulated a material failure surface known as yield locus, and a unique postfailure state known as the CS. The unique postfailure CS is attributed to conditions of ultimate shear distortion without changes in the volume or effective stress and theoretically represented by the critical state line (CSL) tracing the top of yield locus in $p^{\prime }-q-e$ space, where $p^{\prime },q$, and $e$ are the mean effective principal stress, deviatoric stress, and void ratio, respectively. The CSL has conventionally been established in the stress plane ($p^{\prime }-q$) and compression plane ($p^{\prime }-e$) separately, since this procedure enhances the understanding of general triaxial stress and compression paths when superimposed in planes versus three-dimensional (3D) space representation (Poorooshasb et al. 1966; Wood 1990). Many one-dimensional (1D) compression experiments on sand have confirmed a similar response to clayey soil in which the CSL and normal compression lines (NCLs) are parallel when plotted in the compression plane (Been and Jefferies 1985; Coop 1990; McDowell and Bolton 1998), with a certain allowance for the influence of particle breakage (Coop and Lee 1993). However, the convergence of sand specimens to a unique CSL in the compression plane has been controversial in the current literature. To mention a few studies, Martins et al. (2001) examined the compression behavior of residual clayey sand specimens using oedometer tests, and the location of NCLs in the compression plane was found to be a function of the specimen’s initial $e$. Ferreira and Bica (2006) assessed the CS for sand-kaolin mixtures using triaxial compression experiments and reported a family of parallel CSLs in the compression plane depending on the specimen’s initial density state.

The nonconvergence of compression paths (NCLs and CSLs) has been acknowledged when describing the constitutive behavior of sand, and the published literature has introduced the term transitional to describe this mode of behavior (Nocilla et al. 2006). Several studies have demonstrated that the transitional behavior of sand is caused by particle breakage (Altuhafi and Coop 2011; Shipton and Coop 2012; Xiao et al. 2015), inherent effects of a specimen’s initial density state (Coop 2015; Shipton and Coop 2012; Xiao et al. 2016; Xu and Coop 2017), the complex morphology of natural sand particles (Alshibli and Cil 2018; Kandasami and Murthy 2017; Santamarina and Cho 2004; Yang and Luo 2015), influences of grain size distribution (Altuhafi et al. 2010; Altuhafi and Coop 2011), variation in the mineralogy of sand-fine mixtures (Ponzoni et al. 2017; Shipton and Coop 2015), and the percentage of fine content in sand specimens (Kwa and Airey 2016; Zuo and Baudet 2015). The transitional behavior of sand has also been attributed to the strong forms of microscale fabric that are difficult to break even at high applied strains. For instance, Todisco et al. (2018) used mercury intrusion porosimetry to investigate the evolution of pore structure within sand specimens subjected to conventional triaxial and uniaxial compression tests. Todisco et al. (2018) advocated for the complex evolution of pore void distribution, causing the transitional behavior of sand due to initial differences in the void structure that could not be erased during conventional testing. Although there are still active quests for integrating fabric evolution as an essential state parameter in CS theory (Dafalias and Manzari 2004; Fu and Dafalias 2011; Gao et al. 2014; Imseeh et al. 2017; Petalas et al. 2019; Theocharis et al. 2016; Wang et al. 2017), the development of these constitutive models remains incomplete because laboratory sand specimens tested in the conventional biaxial (Finno and Rechenmacher 2003) and triaxial (Alshibli and Cil 2018) apparatuses fail to deliver a unique CSL in the compression plane.

Most of the studies reported in the literature relied on global change in specimen volume to measure the evolution of $e$ toward the CS, with limited research investigating the accuracy of conventional triaxial or biaxial testing procedures (Garga and Zhang 1997; Jefferies and Been 2000). Mooney et al. (1998) conducted a series of biaxial compression tests in search of a unique CSL when locally assessed for the zone affected by a high shear distortion, which is also known as the shear band. Their tests were conducted using a special biaxial apparatus (Harris et al. 1995) that provided a local estimate of the volumetric strain within the shear band via several displacement sensors attached to the specimen surface at multiple vertical levels. The apparatus side walls and loading platens were also lubricated to minimize friction, and the bottom loading platen was connected to a free-sliding base that allowed for the onset and growth of the shear band by minimizing the influence of boundary constraints. Furthermore, one of the apparatus side walls was made of clear Plexiglas, which permitted monitoring of the shear strain localization from the specimen side using digital photographs. Mooney et al. (1998) reported a significantly higher local $e$ when measured near the shear band region compared to $e$ measured using the global volume change. Furthermore, they advocated for the notion of a CSL, but the CSL in the compression plane was found to depend on the initial $e$ and subsequent consolidation history of the specimens (Finno and Rechenmacher 2003). Localization of $e$ at higher values was also detected by optical microscopy near the center of sand specimens loaded in drained triaxial (Frost and Jang 2000) and biaxial (Evans and Frost 2010) compression using epoxy resin impregnation and two-dimensional (2D) section microscopy. Numerous discrete element method (DEM) studies have also demonstrated the localization of $e$ at higher values within the shear band that develops within specimens composed of 2D discs (Gu et al. 2014; Iwashita and Oda 1998, 2000; Jiang et al. 2011) and 3D clumped spheres (Lu and Frost 2010) loaded in biaxial compression.

In situ (i.e., in-position scanning during an experiment) X-ray computed tomography (CT) has recently offered a powerful nondestructive technique that imaged through the 3D internal structure (e.g., particles and voids) of sand specimens, and confirmed the localization of $e$ at high values within the developed shear band in sand specimens loaded under biaxial compression (Alshibli and Hasan 2008; Desrues and Viggiani 2004). However, sand specimens tested in biaxial compression exhibit much less volumetric strain at the CS in comparison to specimens loaded in triaxial compression (Alshibli et al. 2003). Tagliaferri et al. (2011) collected in situ X-ray scans for conventional triaxial compression (CTC) experiments on two sand specimens: biocemented and noncemented. Images of the noncemented specimen reaffirmed the localization of $e$ at high values within the shear band that gradually developed with axial compression. In recent advances, sets of in situ X-ray CT scans with excellent image quality were acquired for sand specimens subjected to CTC experiments (Alshibli et al. 2016; Andò et al. 2017). Compared to biaxial compression, the scans for CTC specimens exposed rather complex internal shearing patterns named micro shear bands (MSBs) that developed during the hardening stage of the experiments (Amirrahmat et al. 2018). Furthermore, in situ X-ray CT scans for conventional extension and compression triaxial experiments on sand specimens in Salvatore et al. (2017) revealed a unique trace of the CSL in the compression plane when $e$ measured from the scans was limited to the zones affected by the largest distortion (shear bands).

The main objective of this paper is to accurately establish the CSL and yield locus for four different types of poorly graded granular materials based on conventional laboratory experiments. The paper sheds light on a discrepancy in measuring the evolution of $e$ toward the CS using the conventional triaxial apparatus due to the development of internal shear bands within the specimens.

For each tested material, the CTC results showed a distinctive CSL in the stress plane as depicted in Fig. 3. Still, a trend can be noticed in Fig. 3 in which dense specimens tend to have a slightly higher slope ($M$). This trend was also reported by Alshibli and Cil (2018) and correlated well to the applied ${\sigma }_3$ as well as the specimen’s initial $D_r$. Overall, the linear regression models in Fig. 3 show significant statistical correlations: $R^2\sim 1$, $p$-value <0.05 for the F-statistics (the regression model is significant), and narrow 95% confidence limits for the estimate of $M$. However, the CSL appeared as a diffused stress-dependent zone in the compression plane (Fig. 4 and Table 3) rather than a linear representation (low $R^2$ values, $p\text{-}\text{value}>0.05$ for the F-statistics, a wide 95% confidence interval for the estimates $\mathrm{\Gamma }$ and $\lambda$. The extensive published literature has reported a similar pattern of response in the compression plane in which sand specimens with different initial $e$ do not approach a unique CSL when tested under general triaxial compression paths. For instance, Marschi et al. (1972) conducted drained CTC experiments on Pyramid Dam rockfill granular material and reported that the CTC paths in the compression plane of dense versus loose specimens did not approach the same CSL. Wood (1990) examined results of constant $p^{\prime }$ triaxial compression experiments on Chattahoochee River sand (Vesic and Clough 1968), and the results showed different $e$ for initially dense versus loose specimens at the CS. That is, Wood (1990) reported that dense specimens needed $\sim 17\%$ dilation by volume to attain the same $e$ as the loose specimens at the CS, which is unlikely to occur due to testing difficulties (e.g., rigid loading endplates, rubber membrane) hinder the ultimate dilation of the specimens. Although the testing difficulties were fairly alleviated by the special biaxial compression apparatus used by Mooney et al. (1998), they still failed to deliver a unique CSL in the compression plane. In fact, findings similar to those presented in Fig. 4 were reported by Mooney et al. (1998) and Finno and Rechenmacher (2003) based on the results of biaxial compression experiments. In summary, they characterized the CS zone by multiple CSLs depending on the initial $e$ of the specimen.

Visual observation of specimen failure at the CS manifested different modes for the 50 CTC experiments depending on the applied ${\sigma }_3$ and the specimens’ initial $D_r$ (Table 2). In summary, loose specimens exhibited slight bulging with no externally observed shear bands, whereas medium dense and dense specimens failed via apparent single or multiple shear bands. Characterizing the failure mode of specimens by visual observations on their surfaces can be misleading because they are just an external manifestation of more complex internal shearing patterns that cause a nonuniform distribution of $e$ within the sheared specimens. Therefore, the diffused CS zone in the compression plane can be attributed to the false reliance on the global volume change ($\delta v$) [Eq. (3)] to measure $e$ at failure. The following section evaluates the latter hypothesis that would explain the formation of the CS zone in the compression plane using accurate measurements of $e$ based on in situ synchrotron microcomputed tomography (SMT) scans that were collected for a series of CTC experiments on the tested materials.

Figs. 6 through 8 show clear evidence that $e$ within the scanned specimens was nonuniformly distributed, particularly when the specimens approached the CS (${\epsilon }_1>15\%$). Higher $e$ values can be seen in Figs. 6–8 at the center of the specimens, which is expected due to the effects of the rigid loading endplates and the flexible latex membrane surrounding the specimens. Moreover, the evolution of $e_{\mathrm{local}}$ versus ${\epsilon }_1$ in Fig. 9 is significantly different from the evolution of $e_{\mathrm{global}}$, similar to findings reported on sand specimens loaded in biaxial compression (Finno and Rechenmacher 2003; Mooney et al. 1998). In an attempt to assess the uniqueness of the CSL in the compression plane using $e_{\mathrm{local}}$, Fig. 10 shows boxplots of $e_{\mathrm{local}}$ for the scanned specimens at failure (last loading step), with the scattered points representing the variation in $e_{\mathrm{local}}$ versus the REV size (2–5 mm). Regarding the boxplot components, the green diamond represents the mean, the red line marks the median, the blue box represents the IQR, and the black whisker bounds the data points within $1.5\times \mathrm{IQR}$ of the upper and lower quartiles. Interestingly, equal $e_{\mathrm{local}}$ was assessed at failure for the F35_MD_15_SMT and F35_D_15_SMT specimens in Fig. 10(a) as well as the DG_MD_15_SMT and DG_D_15_SMT specimens in Fig. 10(b), which are the specimens with different initial $e$ tested at the same ${\sigma }_3$. The equality in $e_{\mathrm{local}}$ for these two pairs of specimens was assessed via the Wilcoxon Rank Sum (WRS) test, which tested the null hypothesis ($H_{\mathrm{o}}$) of equal median for the two specimens’ data of $e_{\mathrm{local}}$. In statistical hypothesis testing, the failure to reject $H_o$ is assessed by the test probability value ($p$-value) and a predefined significance limit (${\alpha }_s=0.05$). That is, a $p$-value greater than ${\alpha }_s$ indicates a failure to reject $H_0$, and one can conclude that the two specimens’ data of $e_{\mathrm{local}}$ have equal medians. The WRS test exhibited a $p\text{-}\text{value}>0.8$ for the F35_MD_15_SMT and F35_D_15_SMT boxplots in Fig. 10(a) and $p\text{-}\text{value}>0.1$ for the DG_MD_15_SMT and DG_D_15_SMT boxplots in Fig. 10(b). The equality between $e_{\mathrm{local}}$ for the medium dense and dense specimens in Figs. 10(a and b) supports the notion of a unique CSL in the compression plane, which confirms the findings reported in Salvatore et al. (2017), and goes a step beyond the conclusions reported in Mooney et al. (1998) and Finno and Rechenmacher (2003) of multiple CSLs, depending on the initial $e$ of the specimens tested in biaxial compression.

Amirrahmat et al. (2018) used the relative particle translation gradient (RPTG) concept proposed by Druckrey et al. (2018) to provide a thorough assessment of the internal shearing patterns that developed within the scanned specimens reported in this paper. Briefly, RPTG refers to the incremental displacement of each particle relative to its neighboring particles and normalized with respect to the global axial compression. When a shear band develops within a specimen, particles within the shear band rotate and translate as if the bulk specimen is divided into multiple frictionally sliding wedges, which produces higher RPTG values along the developed shear band and constant volume flow as postulated by CS theory. Figs. 11 and 12 presents side-by-side color maps for RPTG and $e$ distribution within the same central Y-Z vertical cut across the images acquired at the last loading step for the 10 scanned specimens. RPTG clearly exposed the development of a single shear band in the F35_D_400_SMT [Fig. 11(a)], GS40_D_400_SMT [Fig. 11(d)], and GB_D_400_SMT [Fig. 12(d)] specimens, while external surface bulging in the other specimens was internally manifested by the development of multiple shear bands in opposite directions. To quantify the influence of the failure mode (single versus multiple shear bands) on the discrepancy in $e$ at the CS, Fig. 13 depicts the evolution of $e_{\mathrm{local}}/e_{\mathrm{global}}$ versus ${\epsilon }_1$ for the specimens presented in Fig. 11. The discrepancy curves in Fig. 13 represent the solid curves/error bars ($e_{\mathrm{local}}$) from Figs. 9(a and c) normalized by their respective dashed curves ($e_{\mathrm{global}}$). Initially, $e_{\mathrm{local}}/e_{\mathrm{global}}$ was $\sim 1-1.05$ since isotropic normal compression under ${\sigma }_3$ produces no shear (no shear strain localization), then the difference between $e_{\mathrm{local}}$ and $e_{\mathrm{global}}$ gradually increased with ${\epsilon }_1$. At the CS (last loading step scans), $e_{\mathrm{local}}/e_{\mathrm{global}}$ was $\sim 1.25$ for the F35_D_400_SMT [Fig. 11(a)] and GS40_D_400_SMT [Fig. 11(d)] specimens, which failed via a single shear band. On the other hand, $e_{\mathrm{local}}/e_{\mathrm{global}}$ was less throughout compression for specimens that exhibited external bulging ($\sim 1.10–1.15$ at the CS). This discrepancy between $e_{\mathrm{global}}$ and $e_{\mathrm{local}}$ can actually explain the formation of the CS zone in the compression plane in Fig. 4. Furthermore, the slight bulging in the F35_MD_15_SMT specimen [Fig. 11(c)] produced the least severe discrepancy as the curve of $e_{\mathrm{local}}/e_{\mathrm{global}}$ was the closest to unity in Fig. 13. The lesser deviation between $e_{\mathrm{local}}$ and $e_{\mathrm{global}}$ for the F35_MD_15_SMT specimen can also be noticed in Fig. 11(c) as the distribution field of $e$ shows less variation, and RPTG reveals a dispersed development of multiple shear bands within the specimen. Therefore, the loose laboratory-size CTC specimens that manifested slight surface bulging at failure (Table 2) supposedly provided a more accurate (representative of the whole specimen) measurement of $e$ at the CS than the dense specimens.

The linear regression models in Fig. 16 established the CSL in the compression plane very well ($R^2\sim 1$, $p$-value $<0.05$ for the F-statistics, and narrow 95% confidence limits for the estimates $\mathrm{\Gamma }$ and $\lambda$). A trend can still be seen in Figs. 16(a–c) in which the data points of the tested sands show a higher slope ($\lambda$) for the loose, medium dense, and dense experiments, respectively. Again, this trend supports the transitional behavior of sand because of particle breakage, the effect of the specimens’ initial $D_r$, and fabric, for example. However, the linear models in Fig. 16 do not produce a diffused CS zone like the case in the CTC results (Fig. 4) due to the lack of shear band development in the oedometer tests.

To compare the oedometer with CTC results, the CSL in the compression plane was fitted with a linear model in Fig. 17 using the CS data points of the CTC experiments that were conducted on each initial $D_r$ state separately (dense, medium dense, and loose). The results of the linear fit among the CS data points of loose specimens in Fig. 17(a) closely agree with $\mathrm{\Gamma }$ and $\lambda$ determined by the oedometer tests (Table 7). This agreement supports the earlier conclusion of loose specimens providing more accurate CS assessment in the compression plane when tested in CTC conditions due to the slight bulging failure mode, which produced less discrepancy between $e_{\mathrm{global}}$ and $e_{\mathrm{local}}$ throughout the compression of the F35_MD_15_SMT specimen (Fig. 13).

This paper seeks to accurately measure the CSL and yield locus for poorly graded spherical glass beads and three types of silica sands with different particle morphologies. The CSL was quantified in the stress plane based on the results of CTC experiments that were conducted on specimens with different initial densities and multiple levels of ${\sigma }_3$. However, the CTC results revealed a diffused CS zone in the compression plane that was clearly dependent on the applied ${\sigma }_3$ and initial density of the specimen. Potential causes of the CS zone in the compression plane were investigated by analyzing high-resolution 3D images of in situ SMT scans collected for CTC experiments on the same tested materials. The SMT scans provided excellent 3D images that offered interesting insights into the discrepancy in measuring the evolution of $e$ within sand specimens subjected to CTC conditions. Processed 3D images affirmed a nonuniform distribution of $e$ within the scanned specimens as they approached the CS due to shear strain localization. $e_{\mathrm{local}}$ measured at the center of the specimen exhibited a very different evolution with ${\epsilon }_1$ in comparison to $e_{\mathrm{global}}$, particularly when a single shear band developed within the specimen at the CS. As proposed by CS theory, scanned specimens with initial medium dense and dense states attained equal $e_{\mathrm{local}}$ at the CS when tested at the same ${\sigma }_3$. Therefore, the CS zone in the compression plane was attributed to the reliance on measurements of global change in volume provided by the conventional triaxial apparatus to calculate the evolution of $e$ toward the CS. Alternatively, the CSL was quantified in the compression plane using the results of oedometer tests on specimens composed of the same granular materials. The oedometer specimens inhibit the development of internal shear bands, which leads to accurate quantification of the CS parameters in the compression plane compared to the CTC results. Accordingly, the yield locus and the CSL in $p^{\prime }-q-e$ space were established in Fig. 18 for the four granular material by synthesizing the results of the CTC and oedometer experiments. That is, the CS parameter $M$ was measured using the CTC results, while $\mathrm{\Gamma }$ and $\lambda$ were determined based on the oedometer tests, as summarized in Table 8.

This material was partially funded by the US National Science Foundation (NSF) under Grant CMMI-1266230. Any opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF. The SMT scans presented in this paper were collected using the X-Ray Operations and Research Beamline Station 13-BMD of the Advanced Photon Source (APS), a US Department of Energy (DOE) Office of Science User Facility operated by the Argonne National Laboratory (ANL) under Contract DE-AC02-06CH11357. We acknowledge the support of GeoSoilEnviroCARS (Sector 13), which is funded by the NSF Earth Sciences (EAR-1128799), and the DOE Geosciences (DE-FG02-94ER14466). We thank Dr. Mark Rivers for his guidance at APS.