End restraint effects in triaxial tests on H:D = 2:1 specimens of rockfill material Open Access

The mechanical characterisation of coarse soils is usually carried out based on tests in small-scaled samples, which may be affected by size effects (Marachi et al., 1972; Al-Hussaini, 1983; Hu et al., 2011; Girumugisha et al., 2026). Recent research has shown that the sources of size effects are mainly related to differences in grading (Muir Wood & Maeda, 2008; Girumugisha et al., 2024), particle shape (Linero et al., 2017; Ovalle & Dano, 2020; Carrasco et al., 2025), particle crushing (Frossard et al., 2012; Ovalle et al., 2014) and specimen size (Quiroz-Rojo et al., 2024; Cantor & Ovalle, 2025). However, test conditions can differ greatly between experimental setups, particularly end restraint effects in triaxial specimens. This topic was early reported in the literature for fine soils and sands (Taylor, 1941; Rowe & Barden, 1964; Bishop & Green, 1965; Duncan & Dunlop, 1968), but has rarely been studied for coarse soils.

In standard triaxial devices, the interface between the soil and the end platens is a rough porous stone, restricting free dilation. Such inherent kinematic constraints induce heterogeneous strain fields (Raju et al., 1972; Sheng et al., 1997; Frost & Yang, 2003; Liu et al., 2013; Peri et al., 2019), poor testing repeatability (Colliat-Dangus et al., 1988; Mozaffari et al., 2022) and uncertainty in critical state (CS) parameters (Lee & Vernese, 1978; Reid et al., 2021; Wightman et al., 2024).

Taylor (1941) suggested a minimum specimen slenderness to avoid end friction effects, defined as a height-to-diameter ratio H:D $≥$ 2. While systematic studies performed in fine soils (Shockley & Ahlvin, 1960; Olson et al., 1964; Duncan & Dunlop, 1968; Asaoka et al., 1994; Kodaka et al., 2007; Muraro & Jommi, 2019) and sands (Olson et al., 1964; Rowe & Barden, 1964; Bishop & Green, 1965; Roy & Lo, 1971; Raju et al., 1972; Colliat-Dangus et al., 1988) have shown consistent shear strength, nonhomogeneous strain fields are not fully overcome with H:D $≥$ 2 (Muraro & Jommi, 2019; Mozaffari et al., 2022; Yeh et al., 2024). Rowe & Barden (1964) proposed enlarged end platens covered with a greased latex rubber sheet. However, slender lubricated specimens often slide sideways and buckle at large strains (Olson et al., 1964; Rowe & Barden, 1964; Duncan & Dunlop, 1968), which can be avoided using H:D ≈ 1 (Hettler & Vardoulakis, 1984; Goto & Tatsuoka, 1988; Feda et al., 1993). The only study on the effects of triaxial end restraint in rockfill was reported by Al-Hussaini (1970). His results indicated that rough end platens significantly enhance the stiffness and dilatancy, while having a minor effect on shear strength. Although most testing standards have been drawn based on test results in fine soils and sands with CS friction angle lower than $ϕ′ ∼$35°, data in coarse soils are scarce. Provided that coarse angular soils typically exhibit high values of $ϕ′ ∼ 40°$ or more (Leps, 1970; Ovalle et al., 2020), it can be expected that frictional components at the specimen boundaries might have a great impact on the material response.

The main objective of this letter is to assess end restraint effects in triaxial tests on H:D = 2:1 specimens of coarse angular soil. Monotonic triaxial tests were performed on dry samples of D $=150$ mm, under consolidated drained conditions at confining stresses of $σ3=100$ and 400 kPa. The results with rough and enlarged lubricated platens are compared. In addition, the effects of grading were studied on scaled samples prepared by scalping and parallel grading techniques.

Well-graded rockfill material was sampled from a quarry in St-Eustache, Canada. The material consists of blasted and crushed dolomite rock, with specific gravity $Gs=2.75$ and subangular grain shapes. Field particle size distribution (PSD) shown in Fig. 1 indicates well-graded gravel without fines (GW) according to ASTM (2017). Maximum particle size is $dmax$ = 90 mm, $d50$ = 13·9 mm and uniformity and curvature coefficients are given by $Cu=d60/d10$ = 9 and $Cc=(d30)2/(d10d60)$ = 1·6, respectively.

Several samples were prepared using scalping (S) (Fig. 1(a)) and parallel grading (P) (Fig. 1(b)) techniques; $dmax$ of different scaled materials was set to 5, 6, 8, 12 and 25 mm. For S samples, particles coarser than a chosen $dmax$ were simply removed and a new PSD was generated. P samples were prepared by creating a parallel PSD curve between percents passing of 100% to 10%; for size fractions finer than $d10$⁠, the PSD curves were simply extended until the finest particle in the field material (⁠$dmin=0.08$ mm). The detailed information of all the materials is given in Table S1, included as supplementary information.

Specimens were prepared in a mould of H:D = 300/150 mm, covered by a 2-mm-thick latex membrane. D is in accordance with ASTM (2020) which stipulates a maximum ratio D$/dmax$ of 6. To minimise strain localisation promoted in dense soils, all the specimens were prepared in the loosest possible configuration, by pouring the material in layers without any compaction. Ten distinct layers of homogeneous dry material (≈950 g each) were gently placed in the mould. Due to different grading between samples, the attained dry densities varied slightly from $γd=16.14−17.64kN/m3$ in S and $γd=16.40−18.12kN/m3$ in P samples (see Table S1).

Two lubricated end configurations were evaluated, both with enlarged caps of 170 mm in diameter and made of 3-cm-thick lucite material (see Fig. 2). The aim was to evaluate the best approach that avoids buckling (Raju et al., 1972; Sheng et al., 1997). The first configuration consists of a thin film of silicone grease applied to the entire section of the enlarged caps, covered by a continuous sheet of latex membrane (1 mm thick). For the second lubricated configuration, a cross-shaped cut was introduced in the centre of the latex sheet, as proposed by Feda et al. (1993), to prevent the membrane from restricting radial expansion (see Fig. 2(b)).

Specimens were isotropically consolidated until volume stabilisation (≈45 min). Two effective confining pressures of $σ3′=100$ and 400 kPa were used. Since the specimens were dry, their volumetric strains (⁠$εv$⁠) were estimated through the volume changes of the confinement cell. Shearing was carried out at constant axial strain rate of 0·3%/min (ASTM, 2020). The deviatoric stress $q=σ1′−σ3′$ and the mean effective stress $p′=(σ1′+2σ3′)/3$ were monitored until reaching an axial strain of $εa=15%$⁠.

The following tests were carried out: (i) 30 tests on specimens with standard rough ends and (ii) 49 tests with enlarged platens and lubricated sheets (Fig. 2).

Figure 3 displays photos of the specimens after testing. Regarding specimens with rough ends (Fig. 3(a)), localised failure and specimen bulging are systematically observed. This is an unexpected result in such loose materials, and suggests that end restraint effects might be amplified in highly frictional soils, such as coarse crushed rock. Most of the cases with a continuous lubricated sheet exhibited buckling and sideways sliding, as qualitatively shown in Fig. 3(b). On the other hand, cross cutting the sheets helps to maintain the verticality of the specimens (see Fig. 3(c)), certainly because this setup facilitates radial movement of the particles in contact with the caps. Given these results, all tests with continuous sheets were repeated using cross-cut sheets, and the following analyses consider only the latter configuration.

Figure 4 presents typical stress–strain responses for selected S and P materials (⁠$dmax=$ 12·5 mm), tested with rough (R) and lubricated platens (L), where the stress ratio $q/p′$ and $εv$ are plotted against $εa$⁠.

Following the same trend of the specimens with $dmax=$12·5 mm shown in Fig. 4, all the tests exhibit stress hardening towards maximum $q/p′$ reached around $εa=15%$⁠. The stress–strain plots of the 79 tests are included as supplementary information: Figs. S1 and S2 for S and P samples, respectively. Tests with standard rough ends overestimate $(q/p′)max$ values and dilation in both sets of materials (S and P) and confining stresses. However, these differences are greatly reduced at higher $σ3′$⁠. To highlight these observations, Fig. 5 presents the variation of $ϕ′$ in all the tests, with $sinϕ′=3(q/p′)max/(6+(q/p′)max)$⁠; the results are plotted against $d50$⁠, displaying all the tests carried out on every PSD shown in Fig. 1 (detailed data in Table S1). Note that datapoints sharing the same marker for a given $d50$ correspond to test repetitions. Since the normalised critical strength does not depend on PSD (Muir Wood & Maeda, 2008; Li et al., 2013; Yang & Luo, 2018; Cantor & Ovalle, 2025), $ϕ′$ should be a stable value among all the samples tested, provided that boundary effects are negligible. For all tests with rough ends, $ϕ′$ is scattered between 39·6° and 46·4° at $σ3′=100$ kPa, and 38·8°–41·7° at 400 kPa, without a clear distinction between the SR and PR tests. On the other hand, lubricated tests exhibited slightly lower strength between $ϕ′=$39·2°–43·5° at $σ3′=100$ kPa and 37·3°–40·7° at 400 kPa, with less scattering compared to the cases with rough ends. These relatively high values of $ϕ′$ – compared with sands and fine soils – are consistent with enhanced end restraint effects, as hypothesised in the first section of this letter.

Figure 6 displays the maximum dilatancy rates (⁠$(dεv/dεa)max$⁠; negative values designate dilation) for all tests. As expected, the results reflect that rough boundaries promote dilatancy and increase dispersion of the data. The main source of scattering in R cases is probably related to strain localisation, as qualitatively shown in Fig. 3(a). In such heterogeneous strain fields, $εv$ does not necessarily represent the specimen strain, but a mean value between volume change during shear sliding within the localised shear band, and a relatively constant volume in the rest of the specimen.

Figure 7 presents the characteristic secant strain modulus (⁠$E50$⁠) variation across $d50$ for all tests; $E50$ is defined as the ratio of the 50% of the maximum deviatoric stress and the corresponding $εa$⁠. In SR and PR cases, $E50$ exhibits great scattering without a noticeable trend. Moreover, the values appear to increase with the particle size in tests at $σ3′=$ 100 kPa (from ≈8 to 19 MPa), while the inverse trend is observed at 400 kPa (from ≈23 to 17 MPa). Comparatively, $E50$ of specimens with lubricated platens exhibit more stable results in all materials S and P.

Due to the classical strain limitations of the triaxial test setup, the specimens did not fully reach CS. Nevertheless, Fig. 8 shows that the tendency displayed on Fig. 6 persists at the end of each test (⁠$εa=15%$⁠): the R configuration exhibits enhanced dilatancy rate and greater variability compared to L. While this study does not permit definitive conclusions regarding the influence of end restraint on CS, the results clearly indicate that boundary effects persist at large strains.

In order to assess the representativeness and test repeatability of small scaling methods, Fig. 9 summarises the results by displaying the mean values and their standard deviation for each testing condition (R and L) and scaling technique (S and P). In terms of shear strength (Fig. 9(a)), rough ends clearly give higher mean $ϕ′$⁠, particularly at $σ3′=$ 100 kPa. The difference in $ϕ′$ between 100 and 400 kPa is remarkably reduced in PL tests, indicating that lubrication and better graded materials give more stable results. The average $ϕ′$ and its standard deviation for SR and PL at $σ3′=100$ kPa are $ϕ′=44.9±1.5°$ and $40.8±0.9°$⁠, respectively. At $σ3′=400$ kPa, these differences are reduced to $ϕ′=40.3±0.3°$ and $39.1±0.8°$ for SR and PL, respectively. Regarding dilatancy (Fig. 9(b)), the differences due to end friction effects are noticeable only at $σ3′=$ 100 kPa, with enhanced $|(dεv/dεa)max|$ in R specimens. On the other hand, mean $E50$ values increase with rough ends, particularly in S samples (Fig. 9(c)). On average, $E50=14.9±3.2$ and $8.5±1.5$ MPa for SR and PL at $σ3′=$ 100 kPa, respectively, and $E50=19.4±1.6$ and $17.4±1.2$ MPa at 400 kPa.

Since this study did not include local strain measurements during shearing, the results are treated as a boundary-value problem, and only macro-mechanical values were quantified (i.e. at the specimen scale). Nevertheless, qualitative observations of strain heterogeneity at the end of the tests are consistent with the trends observed in data scatter and test repeatability. Future research should include meso- and micro-scale strain evaluations using image analysis techniques (Alshibli et al., 2003; Sachan & Penumadu, 2007). This would enable assessing the contribution of coarse angular particles to end restraint effects.

The relatively high shear strength of rockfill materials is largely attributed to the interlocking mechanism of angular grains (Charles & Watts, 1980; Barton & Kjærnsli, 1981). This phenomenon is fundamentally a kinematic constraint on grain rotation that arises from particle angularity, as demonstrated experimentally (Fonseca et al., 2013) and supported by numerical studies (Azéma & Radjaï, 2010). These constraints develop column-like force chains upon strain hardening (Kuhn & Bagi, 2004), which may buckle and lead to enhanced dilatancy (Iwashita & Oda, 1998). While such load-bearing structures are inherent to the mechanical behaviour of angular rockfill, the additional kinematic constraints imposed by end restraint may artificially promote their development, potentially resulting in an overestimation of their contribution to dilatancy. A perspective of this study could be to track and identify particle-scale mechanisms near the specimen ends using micro-CT scanning. These direct observations may shed light on the origin of the pronounced end restraint effects observed in highly angular materials.

The following conclusions are drawn:

• Unlike fine-grained soils and sands, end restraint effects in triaxial specimens of scaled rockfill materials are not fully overcome with slenderness H:D = 2:1.

• End restraint effects appear as higher and scattered shear strength, dilatancy and secant stiffness.

• Samples prepared by scalping and tested with standard rough platens revealed the strongest end friction effects, with higher values of $ϕ′$ and $E50$ compared with parallel graded materials under the same testing conditions.

• In general, parallel graded samples with lubricated caps displayed relatively consistent strain–strain curves in the whole set of tests, particularly at high confining pressure.

The results indicate that end lubrication and shredded sheets on lubricated caps should be systematically used in triaxial tests on H:D = 2:1 specimens of coarse rockfill materials. While these recommendations provide valuable guidance for practitioners, the findings of the study opens perspectives to wider research. Specifically, end restraint effects on nonuniform deformation of rockfill triaxial specimens should be quantified through advanced experimental methods. This might allow for evaluating how particle shape and size influence the material response in relation to boundary conditions.

This study was funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) [reference RGPIN-2019-06118] and the industrial partners of the Research Institute on Mines and the Environment (RIME) (Link to RIMELink to the cited article). The authors also thank Lafarge Canada for providing rockfill material and Patrick Bernèche for his support in the extensive experimental work.