Pore Water Pressure Generation and Energy Dissipation Characteristics of Sand–Gravel Mixtures Subjected to Cyclic Loading

Abstract

At least 32 case histories have shown that liquefaction can occur in gravelly soils (both natural deposits and manmade reclamations) during severe earthquakes, causing large ground deformations and severe damage to civil infrastructures. Gravelly soils, however, pose major challenges in geotechnical earthquake engineering in terms of assessing their deformation characteristics and potential for liquefaction. In this study, aimed at providing valuable insights into this important topic, a series of isotropically consolidated undrained cyclic triaxial tests were carried out on selected sand–gravel mixtures (SGMs) with varying degrees of gravel content (G_c) and relative density (D_r). The pore water pressure generation and liquefaction resistance were examined and then further scrutinized using an energy-based method (EBM) for liquefaction assessment. It is shown that the rate of pore water pressure development is influenced by the cyclic resistance ratio (CSR), G_c and D_r of SGMs. However, a unique correlation exists between the pore water pressure ratio and cumulative normalized dissipated energy during liquefaction. Furthermore, the cumulative normalized energy is a promising parameter to describe the cyclic resistance ratio (CRR) of gravelly soils at various post-liquefaction axial strain levels, considering the combined effects of G_c and D_r on the liquefaction resistance.

1. Introduction

Usually, gravelly soils are considered to be less susceptible to liquefaction than sandy soils due to their high permeability. Nonetheless, nowadays, there is notable evidence of gravelly soil liquefaction case histories, mostly in alluvial fan deposits and reclaimed soils, as a result of major earthquake events around the world [1]. The reported liquefied gravelly soils typically consist of well–graded mixtures of sand and gravel with gravel content (G_c) ranging from 5% to more than 85% [2].

Despite the large number of case histories, gravelly soil liquefaction studies are still very limited as compared to the large body of research focusing on the liquefaction resistance of sands. Consequently, the current engineering practice of evaluating the liquefaction performance of gravelly soils typically relies on the assessment procedures developed for sands. Yet, widely adopted sand-specific empirical correlations do not work for gravelly soils and could mislead engineering assessments; that is, because the micromechanical soil structure varies depending on the amount of sand and gravel present in the soils, the soil’s mechanical response, cyclic strain development and liquefaction resistance also vary. Therefore, research focusing on better understanding the liquefaction characteristics of gravelly soils and developing gravel-specific liquefaction triggering analyses is much needed.

From systematic laboratory investigations on undisturbed [3,4,5], reconstituted well-graded [6,7,8,9,10] and gap-graded (i.e., sand–gravel mixtures, SGMs) [11,12,13,14,15,16,17,18] specimens of gravelly soils, the liquefaction resistance of a few gravelly soils has been evaluated and key factors affecting it have been identified. There is a general agreement that the liquefaction resistance of gravelly soils can be as low as that of clean sands; moreover, previous studies also suggested that it is significantly affected by two key factors, namely, relative density (D_r) and gravel content (G_c). While it is recognized that the liquefaction of gravels increases with increasing D_r, there are inconclusive and/or contradictory findings regarding the effect of G_c on the liquefaction resistance of gravelly soils [19]. The conflicting finding regarding the effects of G_C on the liquefaction behavior is probably due to the inactive and active participation of gravel particles in the force chain network of the sand matrix; thus, the soil’s mechanical response of soils remains the same or it may differ. Due to the difference in such behavior, adopting D_r (or global void ratio) for the liquefaction assessment of such SGMs becomes problematic, in a similar manner to that found in other studies carried out on sand–silt mixtures [20]. Therefore, intergranular state frameworks such as the equivalent fraction density model [11] and skeleton void ratio [13] have been used to capture the collective effect of G_C and D_r on the liquefaction resistance of SGMs. The equivalent void ratio proposed by Thevanayagam et al. [21] is also applicable to uniquely describe the liquefaction resistance of gravelly soils, as it captures the combined effects of both G_C and D_r [2]. For the application of those state parameters to in situ (mix) soils, not only is detailed information on index properties and the amount of gravel particles present in such soils needed, but it is also necessary to quantify the active and inactive participation of gravel particles in such a soil matrix. Therefore, applying the intergranular state concept is not a straightforward procedure.

In the liquefaction assessment, the generation of pore water pressure during undrained cyclic loading is of particular interest. For gravelly soils, a few studies [7,8,13,22,23,24] analyzed the pore pressure generation pattern and compared it with sand-based results obtained by Lee and Albaisa [25]. Those experimental investigations show contradicting results; for example, the rate of pore water pressure generation in gravel is significantly higher than that in sand [8], but similar to that of sand if the SGMs have G_C less than 60% [13]. Later, Hubler et al. [24] also indicated that the rate of pore pressure generation is close to the upper bound for sand, more dependent on G_C, D_r and gradation than particle size for SGMs, as well as cyclic stress ratio (CSR) and particle angularity. Evans and Seed [22] and Haeri and Shakeri [23] both acknowledged that the pore water pressure development in gravel specimens is significantly affected by the membrane penetration effect. These conflicting results can be explained by the effect of particle gradation, D_r, G_C and CSR, in a similar way to that found for sand–silt mixtures [26,27].

The energy-based liquefaction evaluation method (EBM), introduced by Davis and Berrill [28], followed by the semi-theoretical work of Nemat-Nasser and Shokooh [29], has proven to be a promising method for evaluating pore water pressure generation [30,31,32,33,34] and liquefaction potential [33,35,36,37,38,39] even for mixed types of soils (i.e., sand–silt mixtures) [26,40]. The level of excess pore water pressure generated under undrained cyclic loading is proportional to the amount of energy dissipation in the soil [41]. The normalized dissipated energy can be uniquely correlated with the pore water pressure generation irrespective of D_r, CSR and consolidation stress ratio [42], as well as fine content [40]. Similarly, the EBM approach shows the potential applicability to assess the liquefaction resistance of mixed types of soils by simultaneously capturing the effect of fine content and D_r [36,38,39,40]. Furthermore, the energy-based method has shown suitability in predicting pore pressure generation across different loading conditions [38], suggesting that EBM can effectively predict pore pressure in real-world scenarios, including those involving non-sinusoidal loading patterns typical of seismic events. Regardless of the amount of evidence that exists for sand and sandy soils, the EBM approach has yet to be applied for the assessment of pore water pressure generation and liquefaction resistance of gravelly soils.

Therefore, given the importance of finding a suitable approach to assess the liquefaction resistance and pore water pressure generation characteristics of gravelly soils by capturing the combined effect of G_C and D_r, a series of isotropically consolidated undrained cyclic triaxial tests on SGM specimens with varying G_C and D_r were conducted by the authors. The liquefaction resistance characteristics of the investigated SGMs have been described in detail in the work of Pokhrel et al. [2]. This paper mainly focuses on the pore water pressure generation characteristics (using the boundary curves obtained by Lee and Albaisa [25] for sand as well as the mathematical model proposed by Booker et al. [43] for sand as a comparison) and further examines the liquefaction resistance using the energy-based method (EBM) for liquefaction assessment developed by Kokusho et al. [36].

2. Test Materials and Procedure

2.1. Materials

New Brighton sand (NB sand; mean diameter D₅₀ = 0.2 mm; maximum diameter D_max = 0.425 mm; specific gravity G_s = 2.66), Dalton River Washed sand (DRW sand, D₅₀ = 0.75 mm; D_max = 3.5 mm; G_s = 2.66), and rounded pea gravel (D₅₀ = 5.5 mm; D_max = 8 mm; G_s = 2.66) were the three materials used to undertake this study. The NB sand was collected from New Brighton Beach, Christchurch, New Zealand. Alternatively, DRW sand and pea gravel were commercially sourced. To ensure the testing materials were free from organic matter and unwanted chemicals, they were carefully washed with water and oven-dried.

The NB sand and DRW sand were mixed in equal proportion by mass (50:50) to produce a fine-to-medium reference sand. Then, the pea gravel was added to generate the desired SGM with a gravel content (G_C) of 0, 10, 25, and 40% by mass. Figure 1 shows the particle size distribution curves of the parents’ materials and selected the SGM tested in this study.

The index properties of principal materials are listed in

Table 1. Index properties of the tested materials.

Materials	Gc (%)	D50 (mm)	Gs	emax	emin	Cu	Cc
GC0	0	0.26	2.66	0.889	0.538	2.50	0.90
GC10	10	0.29	2.66	0.739	0.494	2.77	0.66
GC25	25	0.41	2.66	0.632	0.415	4.50	0.42
GC40	40	0.90	2.66	0.520	0.343	11.76	0.47

Note: D₅₀ = mean diameter; G_s = specific gravity; e_max and e_min = maximum and minimum void ratio, respectively; C_u and C_c = coefficient of uniformity and gradation, respectively.

. The G_s of all the materials were obtained experimentally following JGS0111–2009 [44]. The minimum and maximum void ratios of the SGM were obtained experimentally utilizing a mold with a diameter of 200 mm and a depth of 200 mm following the JGS 0162–2009 [45]. Details are available in the work of Pokhrel et al. [2].

2.2. Test Procedure

In this study, reconstituted SGM specimens with a height of 130 mm and a diameter of 61 mm were prepared in 5 layers by adopting the wet tamping (WT) method. The required mass of mixed materials was calculated based on the dry density of the mix and the specimen size/volume. The materials were further divided into 5 equal-mass portions to obtain homogeneous specimens with uniform particle size distribution and required G_C. The pre-weighed oven-dried SGMs were mixed with deaired water at a water content of less than 5%, as recommended by Ishihara [46]. Then, the moist material was placed inside a split mold by spoon and compacted at a predetermined D_r.

Carbon dioxide was circulated through the specimens, followed by water percolation and double vacuum saturation. During the saturation process, the cell and back pressures were slowly increased simultaneously with an increment of 5 kPa/min, keeping the effective stress at a constant level of 20 kPa. After confirming B values greater than 0.95, fully saturated specimens were then isotropically consolidated at a 100 kPa confining pressure under 200 kPa back pressure. Thereafter, stress-controlled undrained cyclic triaxial tests were performed at the frequency of 0.05 Hz using a pneumatic cyclic loading system. Specimens were sheared under cyclic stress ratios (CSR) ranging between 0.145 and 0.481.

Past studies on gravelly soils strongly recommended that the specimen’s diameter be around 6 to 8 times larger than the D_max [11]. Considering the 61 mm diameter of the specimens and the 8 mm D_max of tested materials, the ratio is close to 8 in this study. The list of 44 undrained cyclic triaxial tests conducted in this study is reported in

Table 2. List of undrained cyclic triaxial tests performed in this study.

Test	GC (%)	Dr* (%)	Dr (%)	Dr,ic (%)	CSR	ru ≥ 0.95				εSA = 5%				N/NL		ru = f(WF)
						NL	$\mathit{\Sigma }$WL	CRRL	$\mathit{\Sigma }$WL*	NF	$\mathit{\Sigma }$WF	CRRF	$\mathit{\Sigma }$WF*	α	R2	β	R2
1	0	25	29.7	31.5	0.202	6.7	0.0141	0.178	0.014	7.0	0.0189	0.178	0.018	0.93	0.99	360	0.96
2			28.6	32.4	0.170	17.5	0.0141			17.6	0.0175			0.59	0.96	269	0.93
3			31.1	33.1	0.145	56.3	0.0153			56.6	0.0182			0.39	0.85	147	0.98
4		45	46.3	47.2	0.348	4.6	0.0385	0.279	0.041	4.8	0.0558	0.279	0.057	0.67	0.91	104	0.95
5			46.6	50.8	0.255	24.0	0.0428			24.6	0.0643			0.68	0.95	75	0.95
6			45.1	48.0	0.200	84.7	0.0424			85.6	0.0490			0.73	0.92	86	0.93
7		55	55.0	56.9	0.340	8.7	0.0443	0.309	0.045	9.8	0.0582	0.315	0.061	1.98	0.93	133	0.95
8			56.0	59.5	0.316	13.0	0.0428			13.7	0.0674			0.60	0.96	73	0.98
9			57.0	60.4	0.285	23.6	0.0473			24.7	0.0576			0.54	0.94	78	0.98
10	10	25	28.6	31.5	0.200	11.0	0.0169	0.193	0.017	11.6	0.0251	0.194	0.026	0.80	0.98	336	0.96
11			25.9	28.9	0.182	28.0	0.0201			28.6	0.0309			0.75	0.93	186	0.92
12			25.9	30.5	0.176	24.0	0.0156			24.6	0.0248			0.49	0.95	224	0.97
13			26.7	28.9	0.152	96.0	0.0197			97.6	0.0335			0.60	0.89	144	0.95
14		35	35.5	38.8	0.260	8.7	0.0239	0.236	0.024	9.7	0.0365	0.239	0.039	1.19	0.97	217	0.95
15			34.9	36.7	0.227	16.0	0.0212			17.6	0.0403			0.84	0.98	206	0.96
16			34.8	36.7	0.180	94.0	0.0306			95.6	0.0518			0.59	0.86	79	0.95
17		45	47.9	49.5	0.317	7.8	0.0370	0.288	0.039	8.8	0.0513	0.292	0.056	1.75	0.94	149	0.96
18			45.0	47.4	0.286	15.6	0.0379			16.6	0.0563			0.72	0.99	107	0.98
19			50.4	50.9	0.257	30.0	0.0415			32.7	0.0680			0.95	0.98	99	0.94
20	25	25	25.8	30.0	0.227	6.0	0.0151	0.207	0.022	6.1	0.0250	0.208	0.035	0.73	0.99	360	0.98
21			23.9	26.7	0.206	46.0	0.0265			47.6	0.0442			0.48	0.94	129	0.95
22			26.2	30.2	0.203	39.8	0.0286			41.7	0.0448			0.70	0.97	147	0.91
23			24.1	30.2	0.182	22.0	0.0175			22.1	0.0226			0.64	0.96	302	0.96
24			26.4	30.0	0.158	78.0	0.0255			78.7	0.0431			0.40	0.89	108	0.97
25		35	35.2	37.8	0.265	13.0	0.0330	0.258	0.031	14.2	0.0661	0.262	0.062	1.00	0.99	146	0.93
26			32.1	37.4	0.240	23.0	0.0258			23.7	0.0359			0.69	0.97	157	0.97
27		40	40.7	46.0	0.305	11.0	0.0310	0.292	0.032	13.6	0.0663	0.298	0.066	1.22	0.98	131	0.95
28			39.4	45.3	0.262	37.0	0.0349			39.7	0.0654			0.38	0.98	67	0.99
29			40.3	47.3	0.255	36.0	0.0423			38.7	0.0921			0.64	0.45	62	0.95
30		45	44.7	52.1	0.397	5.0	0.0337	0.326	0.040	7.7	0.0777	0.346	0.091	2.14	0.93	147	0.97
31			44.7	52.5	0.351	11.0	0.0435			15.7	0.0976			2.17	0.89	134	0.94
32			46.5	52.4	0.300	22.0	0.0369			27.7	0.0982			0.91	0.97	88	0.94
33			48.1	54.1	0.245	73.0	0.0517			78.7	0.1101			0.44	0.96	54	0.99
34		55	54.4	61.4	0.481	12.0	0.1031	0.425	0.099	15.7	0.1720	0.470	0.156	3.38	0.78	83	0.93
35			55.5	59.8	0.478	10.0	0.0820			14.7	0.1575			3.56	0.77	94	0.95
36			55.3	60.0	0.366	20.0	0.1119			20.7	0.1361			0.77	0.98	57	0.99
37			56.5	61.1	0.274	54.0	0.1110			60.8	0.1992			0.79	0.99	67	0.96
38	40	25	24.4	25.6	0.251	10.0	0.0167	0.241	0.019	11.2	0.0417	0.243	0.044	0.92	0.99	243	0.97
39			25.7	31.5	0.247	15.0	0.0247			16.6	0.0480			0.88	0.98	167	0.95
40			25.6	29.4	0.233	16.0	0.0167			17.6	0.0414			0.79	0.98	199	0.96
41			25.8	29.4	0.208	53.0	0.0187			74.8	0.0387			1.00	0.99	163	0.97
42		45	47.8	51.5	0.421	13.0	0.0867	0.391	0.086	22.7	0.2523	0.472	0.226	2.23	0.9	68	0.96
43			46.3	50.3	0.341	20.0	0.0851			26.8	0.1888			1.49	0.91	104	0.95
44			45.8	48.4	0.277	47.7	0.0898			59.8	0.2627			0.96	0.92	60	0.99

Note: D_r* = nominal relative density; D_r = measured relative density; D_r,ic = consolidated relative density; CSR = cyclic stress ratio; N_L = number of cycles to liquefaction (r_u ≥ 0.95); $\Sigma$W_L = cumulative dissipated energy to liquefaction; CRR_L = cyclic resistance ratio for r_u ≥ 0.95; $\Sigma$W_L* = cumulative dissipated energy with respect to CRR_L; N_F = number of cycles to cyclic failure (ε_SA = 5%);$\Sigma$W_F = cumulative dissipated energy to cyclic failure; CRR_F = cyclic resistance ratio for cyclic failure; $\Sigma$W_F* = cumulative dissipated energy with respect to CRR_F; α = pore water pressure parameters for stress-based method (Equation (2)); and β = pore water pressure parameters for energy-based method (Equation (8)).

. Full details of the test procedure, including the evaluation and correction for membrane force and penetration effects, are reported in the work of Pokhrel et al. [2].

3. Test Results and Discussions

3.1. Cyclic Resistance

Typical triaxial test results have been reported by Pokhrel et al. [2], in terms of effective stress paths and stress–strain relationships, and thus are not reported here. Alternatively, the liquefaction resistance characteristics of the SGMs are presented in Figure 2 for completeness and to better support subsequent pore water pressure generation and energy dissipation analyses.

The CSR against the number of loading cycles to achieve the initial liquefaction (N_L; based on pore water pressure ratio, r_u ≥ 0.95) and to achieve the cyclic failure (N_F; at 5% single amplitude axial strain, ε_SA = 5%, i.e., specimen failure condition assumed in this study) for the SGMs is presented in Figure 2, where Figure 2d correspond to G_C = 0%, 10%, 25% and 40%, respectively. The CSR–N_L and CSR–N_F data points generally overlap each other for the lower G_C and D_r values, as shown in Figure 2a,b. However, for higher G_C and D_r values (Figure 2c,d), N_F data points are found to be higher than N_L, indicating that in those conditions, the specimens further developed axial strain even after the liquefaction state is achieved.

The CSR–N_L or CSR–N_F can be approximated by the following power-form function:

$$CSR=a{ \left(N_{\mathrm{L}}\right)}^b$$

(1a)

$$CSR=a {\left(N_{\mathrm{F}}\right)}^b$$

(1b)

where a and b are the fitting parameters (for the tested SGMs, the values of a and b can be found in Pokhrel et al. [2]).

In the reported log–log plots, the trends for CSR–N_L and CSR–N_F given by Equation (1a,b) are linear and distinct for the different testing conditions. Such trends shift upwards with increasing G_C and D_r, indicating that the position of the liquefaction resistance curve of an SGM is significantly influenced by both G_C and D_r. Considering that 15 loading cycles are frequently employed in the literature to estimate the cyclic stress ratio (CRR₁₅) in stress-based approaches [47], the CRR₁₅ values for N_L and N_F (i.e., CRR_L and CRR_F) have been estimated for the SGMs and are reported in Table 2.

3.2. Pore Water Pressure Generation

In this study, the pore water pressure generation of all the SGM specimens was analyzed, and comparisons were made to assess the effect of G_C, D_r and CSR.

The typical development of the pore water pressure for loose SGMs (D_r = 26–30%) with various G_C is presented in Figure 3. The transient pore water pressure (r_u,T, which represents the real-time change in r_u during cycling loading) is plotted in solid lines, and the residual pore water pressure (r_u,R, which is measured at the end of each loading cycle when the deviatoric stress goes back to zero) is indicated by symbols. While the r_u,T varies in the range of 0.5 to 1.0 for all the SGMs, the range of r_u,_R is narrowed to 0.93–1.0.

For specimens with G_C = 0 and 10%, no noticeable axial strain developed is observed until r_u becomes equal to 0.80 or more, even though the r_u develops uniformly. Then, the axial strain suddenly increases, and the specimen fails in just one to two additional loading cycles. ε_SA = 5% and r_u ≥ 0.95 are reached almost simultaneously, showing a liquefaction behavior typical of loose sand.

In contrast, the SGMs with G_C = 25 and 40% show different axial strain and r_u responses. Both the axial strain and r_u increase gradually until r_u reaches 0.95 or more. Following, the axial strain continues to increase gradually before the specimen fails in an additional 16 loading cycles for the SGMs with G_C = 40%. Despite the specimens being loose, their cyclic response is similar to that of dense sand.

The variation in r_u for loose SGM specimens is presented in Figure 4 together with the boundary curves proposed by Lee and Albaisa [25] for Monterey sand.

In laboratory stress-controlled undrained cyclic triaxial tests, r_u is usually related to the number of loading cycles to achieve initial liquefaction, N_L [48]. The r_u pattern for SGMs can be approximated by using the mathematical model proposed by Booker et al. [43] shown in Equation (2):

$$r_{\mathrm{u}}={\displaystyle \frac{2}{\pi }}{\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}{\left({\displaystyle \frac{N}{N_{\mathrm{L}}}}\right)}^{1∕2\alpha }$$

(2)

where N is the number of loading cycles, and α is a parameter accounting for the soil properties and test conditions.

In Figure 4, the solid lines represent the r_u,R of the tested SGMs, while the fitting curves are shown as dotted lines. The experimentally obtained rate of r_u generation is found to be higher, at the initial and final stages of the tests, than predicted. It can be also noticed that the test results fall inside the boundary curves reported by Lee and Albaisa [25] for clean sand. The values of α obtained for G_C = 0, 10, 25 and 40% are 0.9, 0.8, 0.7 and 1.0, respectively, and, thus, close to the value of α = 0.7 recommended by Seed et al. [48] for clean sand.

Figure 5 and Figure 6 show the r_u generation at different CSR conditions for SGMs prepared at D_r = 26–33% and 47–54%, respectively. The fitting curves are also reported, and the corresponding α values are specified. It is observed that the Booker et al. [43] model conforms well with the variation in r_u with the normalized number of stress cycles (N/N_L) for all the tested SGMs and both D_r conditions, yielding coefficient of determination (R²) values exceeding 0.85 in all cases. The values of α obtained by the curve fitting method are generally higher for higher CSR values, indicating the dependency of r_u generation on CSR. The values of α for D_r = 27–33% fall in the range of 0.4–1.0, as shown in Figure 5a–d, the upper bound value of 1.0 being for G_C = 40% (Figure 5d) and the lower bound value of 0.4 being for G_C = 0 and 25% (Figure 5a,c). For G_C = 0%, the average value of α is close to 0.7 for both D_r conditions, which is consistent with the recommended value of 0.7 for clean sand [48]. However, for other G_C configurations, the value of α is much higher for D_r = 47–54%. The maximum value of α is 2.23 for G_C = 40% (Figure 6d), and 0.7 is the minimum for G_C = 0 and 10% (Figure 6a,b). However, in the case of D_r = 47–54%, the fitting lines show a lower level of consistency with the experimental results. This is consistent with Haeri and Shakeri’s [23] study. Overall, the mathematical model proposed by Booker et al. [43] can be conveniently used to describe the pore water pressure generation for various SGMs with different G_C and D_r as it yields R² values greater than 0.85 (41 out of 44 tests), with the majority being over 0.9 (36 out of 44 tests) as reported in Table 2.

Figure 7 compares the upper and lower r_u generation curves from this study with those reported in the literature for gravel and SGMs [8,23,49]. It also includes the curves for Monterey sand by Lee and Albaisa [25]. The lower bound curve from this study is similar to that for clean sand by Lee and Albaisa [25]. Instead, the upper bound curve is closer to the results of Haeri and Shakeri [23]; this curve shows that the r_u generation increases rapidly in the first few cycles of loading, and then flattens after reaching an r_u value greater than 0.8. From the plot, it is evident that the r_u generation of gravel and gravelly soil mixtures may vary over a wide range of values. The differences can be partially explained by the different gradation characteristics of the tested material [24]. It strongly depends on the coefficient of uniformity (C_u) of the gravel and SGMs, and its location shifts up with increasing C_u: i.e., C_u = 1.6 [49], C_u = 2.5 to 11.8 (this study), C_u = 28 [23], and C_u = 47 [8]. By further scrutinizing the upper bound curves of the various materials, it can be said that naturally deposited gravelly soils, which have higher C_u compared to the clean sands, would have a higher rate of r_u generation in the first few cycles of seismic loading.

4. Energy Dissipation Approach

4.1. Theoretical Framework

In the case of saturated granular soils, under undrained cyclic loading, the tendency of the soil to densify controls the amount of energy dissipated during the rearrangement of soil particles and causes pore water pressure generation. The build-up of pore water pressure induces interparticle forces to decrease, and thus the ability of the soil to resist shear deformation decreases, resulting in significant strain development.

The dissipated energy during each undrained loading cycle is given by the hysteresis area of a stress–strain loop [36,50] and the cumulative dissipated energy can be calculated using the following equation [31]:

$$\mathsf{\Sigma }w={\displaystyle \frac{1}{2}}\sum _{i=1}^{n-1}\left({\sigma }_{d,i+1}+{\sigma }_{d,i}\right)\left({\epsilon }_{a,i+1}-{\epsilon }_{a,i}\right)$$

(3)

where n is the number of loading increments; σ_(d,i) and σ_(d,i+1) are the deviator stress at load increment i and i + 1, respectively; ε_(a,i) and ε_(a,i+1) are the axial strains at load increments i and i + 1, respectively.

According to Kokusho and Tanimoto [39], the cumulative dissipated energy can be calculated by integrating the deviatoric stress into axial strain as follows:

$$\mathsf{\Sigma }w=\int qd\epsilon$$

(4)

where q is the deviatoric stress and ε is the axial strain.

To remove the effect of the initial effective mean stress (σ₀′), the cumulative dissipated energy is typically normalized or nondimensionalized:

$$\mathsf{\Sigma }W=\Sigma w/{\sigma }_0^{\prime }$$

(5)

A typical deviator stress-axial strain plot is shown in Figure 8a for a medium-dense specimen of SGM (D_r = 51%) with G_C = 10% subjected to CSR = 0.26. The corresponding cumulative normalized dissipated energy ΣW (hereinafter referred to as “dissipated energy”) is calculated by using the above Equations (3)–(5); its variation is plotted in Figure 8 against the pore water pressure ratio (Figure 8b), the number of loading cycles (Figure 8c), the double amplitude axial strain (Figure 8d) and the axial strain (Figure 8e) for completeness.

In the early stages of the cyclic loading, ΣW remains minimal, but it starts to increase significantly with increasing N and the effective stress reduces by half. Specifically, in Figure 8b,c, in the first 20 loading cycles, ΣW is almost 0.01 and r_u is more than 0.6, but in the next 13 cycles, Σ_W increased by more than 600%. Moreover, only ε_DA = 1% is developed in the first 26 loading cycles (Figure 8d), but then ε_DA starts to increase dramatically afterward, reaching 5% within the next 5 loading cycles (Figure 8e).

The ΣW required to achieve initial liquefaction is here defined as ΣW_L and is based on the r_u ≥ 0.95 criterion; the ΣW required to achieve cyclic failure is here defined as ΣW_F and is based on the axial strain level ε_SA = 5% approach. Their variation with the number of loading cycles is plotted in Figure 9 for SGMs with varying G_C and D_r, where Figure 9a–d represents G_C = 0%, 10%, 25% and 40%, respectively. For the easy identification of data grouping, the data are presented with different symbols using hollow blocks connected by a dashed line for ΣW_L, and full blocks connected by a solid line for ΣW_F-based results. It can be seen that, for all the G_C and D_r conditions, ΣW_L is generally lower than ΣW_F. This is due mainly to the fact that the r_u,R reaches more than 0.95 before the specimen fails at ε_SA = 5%. Moreover, at G_C = 0 and 10% (Figure 9a,b), the ΣW is relatively small and unchanged with the number of loading cycles, but a slightly to significantly increasing trend can be observed with increasing the number of loading cycles for G_C = 25 and 40% (Figure 9c,d).

The above can be further discussed by looking at the relationships between ΣW and CSR, as shown in Figure 10. In fact, the number of loading cycles presented in Figure 9 is directly related to the CSR; that is, the higher the CSR is, the lower the N is, and vice versa. The ΣW–CSR is relatively flat for G_C = 0% (Figure 10a) under various D_r values, indicating the independency of ΣW on CSR. However, for other G_C and D_r conditions (Figure 10b,c), ΣW mostly decreases with increasing CSR. The marginal dependence of ΣW on CSR is consistent with the findings of Kokusho and Tanimoto [39].

The ΣW during liquefaction can be directly correlated with the CRR [36]. From undrained cyclic triaxial tests carried out on reconstituted specimens over a wide range of fine content, relative density and effective stress, Kokusho [36] found that the ΣW and CRR have a unique relationship that can be described using a parabolic function as follows:

$$\mathsf{\Sigma }W=A·CR{R}^2+B·CRR+C$$

(6)

where A, B and C are fitting constants obtained from the experimental results. In this study, the CRR values for N = 15 loading cycles (CRR₁₅) can be determined from Figure 2.

According to Kokusho and Tanimoto [39], the ΣW value can be interpolated using the log ΣW–log CSR plot (e.g., Figure 10) by using the following function with positive fitting constants c and d:

$$\mathsf{\Sigma }W=c{\left(CSR\right)}^{-d}$$

(7)

Thus, the ΣW corresponding to a given CRR₁₅ value can be estimated from Equation (7) by setting CSR = CRR₁₅.

The relationships between ΣW and CRR₁₅ obtained using the above-described approach are presented in Figure 11a for liquefaction criteria adopted in this study (i.e., ε_SA = 5% and r_u ≥ 0.95) and Figure 11b for other criteria (i.e., ε_DA = 1, 3 and 5%). The CRR₁₅ is found to be uniquely correlated with the respective ΣW in all cases considered.

For a comparison, the relationships obtained by Kokusho and Tanimoto [39] and Zhou et al. [40] are also included in Figure 11a. The deviation between the trend found in this study and the two previous studies is due to the differences in test materials, test conditions and liquefaction evaluation criteria. However, they all follow a similar trend, confirming that ΣW monotonically increases with increasing CRR. The fitting parameters A, B and C from this and previous studies are listed in

Table 3. Summary of the fitting parameters for $\mathsf{\Sigma }W-CRR$ relations based on the experimental finding from this study on SGMs and previous studies on gravelly soils and SGMs.

Reference	CRR Criterion	A	B	C	R2
This study	${\epsilon }_{DA}$ = 1%, NL = 15	0.054	0.087	−0.004	0.97
	${\epsilon }_{DA}$ = 3%, NL = 15	0.497	−0.049	0.006	0.99
	${\epsilon }_{DA}$ = 5%, NL = 15	0.623	−0.046	0.006	0.99
	$r_u$ = 1, NL = 15	1.177	−0.367	0.044	0.97
	${\epsilon }_{SA}$ = 5%, NL = 15	1.383	−0.331	0.040	0.93
Kokusho and Tanimoto [39]	${\epsilon }_{DA}$ = 5%, NL = 15	2.7	−0.54	0.035	0.92
Zhou et al. [40]	${\epsilon }_{SA}$ = 5%, NL = 20	1.27	−0.272	0.0189	-

.

The relationship between ΣW and CRR found in this study confirms that the energy dissipation approach can uniquely describe the liquefaction resistance of SGMs (i.e., gravelly soils) by simultaneously capturing the effects of G_C and D_r. Considering the output of this and previous studies, the energy-based method is a promising technique to assess the liquefaction resistance of different types of soils, including sand–silt mixtures and SGMs of varying densities. Additionally, this method can also be applied to assess the liquefaction behavior of in situ soils [40]. Regardless, further laboratory and in situ investigation, specifically focusing on the different gradations of gravel, fabric, and stress state, is recommended to improve the suitability of this method for gravelly soils.

4.2. Relationship Between Energy Dissipation and Double Amplitude Axial Strain

In Figure 12, the ΣW−ε_DA relationships obtained for the SGMs with D_r = 26–33% and 47–60% and for different G_C conditions are plotted. Each data set includes 3 to 5 test results corresponding to different values of CSR. The effect of CSR on the ΣW−ε_DA relation is noticeable, but the data can still be grouped according to their D_r. The ΣW values are found to be lower for D_r = 26–33% than D_r = 47–60% for any G_C configuration, clearly demonstrating the effect of D_r on the ΣW−ε_DA relation. Specifically, for SGMs with G_C = 0% (Figure 12a), to attain the ε_DA = 5%, ΣW = 0.015 shall be dissipated for D_r = 26–33%, while ΣW = 0.042 is needed for D_r = 47–60%. The difference in ΣW required to attain similar ε_DA for D_r = 26–33% and D_r = 47–60% is almost similar to that for G_C = 0 and 10% but increases in magnitude with increasing G_C from 10% to 25% and 40% (Figure 12b–d).

Moreover, considering different G_C but the same D_r, the curves tend to shift systematically rightwards, demanding higher energy dissipation to reach the same ε_DA. This is, for D_r = 47–60%, ΣW = 0.042 is required to reach the ε_DA = 5% for G_C = 0% and ΣW = 0.048, 0.068 and 0.1 for G_C = 10, 25 and 40%, respectively. Similarly, for D_r = 26–33%, for G_C = 0, 10, 25 and 40%, the ΣW needed to attain ε_DA = 5% is 0.015, 0.017, 0.023 and 0.03, respectively. The marginal difference between G_C = 0 and 10%, as well as the significant difference for the higher G_C (25 and 40%), can be related to the effect of G_C on the liquefaction resistance of SGMs. Therefore, it can be concluded that the ΣW−ε_DA relation is significantly affected by both the G_C and D_r. This conclusion is consistent with the results obtained for sand–silt mixtures by Kokusho and Kaneko [38].

4.3. Relationship Between Energy Dissipation and Pore Water Pressure

The stress-based model developed by Seed et al. [48] is widely used for estimating pore water pressure build-up under specific cyclic loading conditions or earthquake intensity. This model requires selecting an equivalent number of uniform stress cycles or loading cycles to describe the actual in situ pore water pressure response to earthquake loading [51]. The application of an equivalent number of loading cycles is limited to liquefiable soils. However, even non-liquefiable soils containing plastic fines and dense sands can undergo significant pore water pressure increases and cyclic deformation due to cyclic softening [52]. Under more general loading conditions and different types of soil, a second alternative method based on ΣW has been used in cyclic liquefaction assessment. This theory is based on the observation that the expenditure of energy during particle rearrangement in undrained cyclic tests causes an increase in pore water pressure. Originally, Nemat-Nasser and Shokooh [29] suggested a relationship between ΣW and r_u. This concept was further validated by Davis and Berrill [28]. Initially, a linear relationship between r_u and ΣW was assumed; however, Simcock et al. [53] suggested a non-linear relationship. Based on undrained cyclic triaxial testing on New Brighton sand conducted by Simcock et al. [53], Berrill and Davis [54] proposed a nonlinear relationship between r_u and ΣW. Following the same hypothesis, numerous energy-based pore pressure models based on laboratory undrained cyclic tests have been developed [32,55,56,57,58,59,60,61]. Similarly, the variation in r_u with ΣW for different types of soils (sand and its mixtures) under different loading conditions (harmonic and irregular) has been investigated [34,36,38,40,41,50]. Such studies concluded that the energy-based approach reveals an intrinsic correlation between the ΣW and r_u irrespective of the D_r, CSR, and consolidation ratio for various sands and silts under different loading conditions.

Considering that the ΣW–r_u relation is independent of CSR, Amini et al. [34] proposed that the variation in ΣW with r_u can be represented by the following simple model developed by Davis and Berrill [61]:

$$r_{\mathrm{u}}=1-\exp \left(-\beta \mathsf{\Sigma }W\right)$$

(8)

where β is a dimensionless constant relating the dynamic excess pore water pressure determined from cyclic undrained tests.

The model was checked against the data points obtained from all 44 tests tested in this study. The estimated β and the fitting coefficient R² value for each condition are presented in

Table 4. Fitting parameter β and R² value.

GC (%)	0			10			25					40
Dr (%)	31–33	47–51	57–60	29–32	37–39	47–91	27–30	37–38	45–47	52–54	60–61	26–32	48–52
$\beta$	199.8	82.6	77.85	186.8	110.9	106.9	177.8	153.3	70.74	69.19	67.78	178.6	70.21
R2	0.88	0.94	0.98	0.91	0.84	0.95	0.85	0.95	0.92	0.91	0.95	0.95	0.94

. The R² value over 0.91 indicated that the model accurately captured the ΣW–r_u relationship of SGMs. The constant parameter β is found to vary from 54 to 360 depending on the G_C, D_r and CSR. For any given G_C and D_r combination, β is found to generally decrease with decreasing CSR.

Figure 13 shows the variation in ΣW with r_u for SGMs with D_r = 26–33%, where Figure 13a–d corresponds to G_C = 0%, 10%, 25% and 40%, respectively. The pore water pressure is found to be well correlated with ΣW for each G_C condition and different CSR, even though the β value is different for a different CSR. The ΣW required to reach r_u ≥ 0.95 is also similar for any given G_C and D_r but with a different CSR. For G_C = 0 and 10% (Figure 13a,b), the r_u value is greater than 0.95 at almost similar ΣW = 0.018. But, for G_C = 25 and 40% (Figure 13c,d), the r_u is slightly delayed and requires a higher ΣW than SGMs with G_C = 0 and 10% to reach the same r_u level. The grey shaded zone in Figure 13 represents the range of results obtained by Kokusho and Kaneko [38] for liquefaction tests conducted on sandy soil mixed with different fine content, densities and confining pressures. The ΣW–r_u obtained in this study for SGMs is within the range found by Kokusho and Kaneko [38], indicating that the pore water pressure generation response of SGMs with different G_C and low D_r could be the same as that of sandy soils.

The relationship between ΣW–r_u for SGMs with D_r = 47–54% is presented in Figure 14. The requirement of ΣW for r_u to reach 0.95 increases from 0.037 to 0.057 when G_C increases from 0 to 40%. Yet again, the r_u for SGMs with G_C = 0 and 10% (Figure 14a,b) attains its maximum value at a smaller ΣW than SGMs with G_C = 25 and 40% (Figure 14c,d). The requirement of lower ΣW for lower G_C can be related to the effect of G_C on the liquefaction behavior of gravelly soils. The ΣW–r_u range is also shifted rightwards as compared to that of sandy soils reported by Kokusho and Kaneko [38]. The effect of D_r on ΣW–r_u relation can be better understood by comparing Figure 13 and Figure 14. The development of r_u is slightly more rapid for SGMs with D_r = 26–33% (Figure 13) than for those with D_r = 47–54% (Figure 14) for all the G_C conditions. Similarly, by looking at the ΣW–r_u data points for any given testing conditions (i.e., same G_C and D_r), the test results fall in a small range and r_u also reaches its maximum value at a similar ΣW for different CSR. Therefore, the ΣW–r_u relation can be considered to be somehow independent of CSR.

By assuming that the ΣW–r_u relation is actually independent of CSR, β could then be estimated by using Equation 8 for each testing condition, as tabulated in Table 4. The higher values of β indicate that the growth of pore water pressure would be higher for any given ΣW. The obtained fitting lines are also indicated in the plots. The fitting constant β is in a narrow range for each G_C except for D_r = 26–33%. For the same D_r, but with different G_C, β decreases with increasing G_C but is in a very narrow range. The small effect of G_C is similar to the effect of fines observed by Kokusho and Kaneko [38] and Zhou et al. [40] for silty sand. Similarly, for the same G_C but with different D_r, β decreases with increasing D_r, indicating the effect of D_r on the ΣW–r_u relation. The small effect of D_r on ΣW–r_u relation is consistent with the findings of Kokusho and Kaneko [38].

Figure 15 shows the variations in r_u with ΣW for all the cyclic tests tested in this study. Despite the marginal effect of G_C, D_r and CSR as observed, all the test data points are found to be falling within a narrow band. This outcome is well in agreement with the previous experimental studies on Toyoura sand conducted at different D_r, CSR and consolidation stress ratios by Yang and Pan [42] and on mixtures of Fujian silica sand and its fines (fine content up to 20%) under different consolidation stress ratios [40].

5. Conclusions

A series of 44 isotropically consolidated undrained cyclic triaxial tests were conducted on selected sand–gravel mixtures (SGMs), with varying gravel content (G_C = 0, 10, 25 and 40%) and prepared at different relative densities (D_r,ic = 30–60%), to characterize the undrained cyclic responses of gravelly soils. The pore water pressure generation (r_u) response and the liquefaction resistance were examined. The r_u results were compared with the experimentally derived boundary curves obtained by Lee and Albaisa [25] for sand and Banerjee et al. [8], Haeri and Shakeri [23] and Hubler [49] for gravelly soils, as well as with the mathematical model proposed by Booker et al. [43] for sand. The combined effect of G_C and D_r on the liquefaction resistance of gravelly soils was also scrutinized using the energy–based method (EBM) for liquefaction assessment proposed by Kokusho [36].

The following main conclusions can be drawn from this experimental study:

• It is confirmed that the liquefaction resistance (CRR) and pore water pressure generation (r_u) of SGMs are influenced by the CSR, G_C, and D_r; however, this study shows that it is critical to consider the combined effects of G_C and D_r to properly characterize the response of SGMs subjected to undrained cyclic loading conditions;
• For looser (D_r = 26–33%) SGMs, the influence of G_C on the r_u is marginal irrespective of the CSR level; alternatively, for denser (D_r = 47–60%) SGMs, the influence of G_C on the r_u becomes significant with an increasing CSR level;
• The boundary curves for r_u proposed by Lee and Albaisa [25] for sand describe well the response of loose SGMs; alternatively, the response of the denser SGMs is better described by the boundary curves proposed by Haeri and Shakeri [23] for gravels. The mathematical model proposed by Booker et al. [43] is fairly applicable to all SGMs cases examined in this study (G_C = 0–40%; D_r = 26–60%; CSR = 0.145–0.481);
• For SGMs with G_C = 0 and 10%, the normalized cumulative energy dissipation (ΣW) slightly increases with increasing D_r; in contrast, for SGMs with G_C = 25 and 40%, ΣW significantly increases with increasing D_r. Irrespective of the G_C and D_r combinations, the effects of CSR on ΣW are mostly marginal;
• EBM is found to be an effective approach for uniquely describing the liquefaction potential of SGMs while simultaneously capturing the effects of G_C and D_r. This highlights that EBM is a promising alternative to the CSR approach, which requires additional parameters, such as the skeleton void ratio and/or equivalent void ratio, to capture the effect of G_C and D_r as described and correlated with CRR in the work of Pokhrel et al. [2]. This contribution adds significant value to the existing body of literature, which primarily focuses on sand and sand–silt mixtures. Based on the existing knowledge and the findings of this study, the EBM approach has demonstrated its robustness in predicting liquefaction potential across various soil types and loading conditions.
• Both the CSR and EBM approaches demonstrate significant effectiveness in predicting the pore pressure generation of SGMs tested in this study. However, the EBM approach appears to be better suited due to the narrow band observed in the ΣW–r_u. In contrast, the CSR approach shows a wider variability in the relationship between N/N_L and the r_u.