A Constitutive Model of Sandy Gravel Soil under Large-Sized Loading/Unloading Triaxial Tests

Based on large-scale triaxial tests of sandy gravel materials, the strength and deformation characteristics under loading/unloading conditions are analyzed. At the same time, the applicability of the hyperbolic constitutive model to sandy gravel is studied using experimental data. +e results indicate that sandy gravel under low confining pressures (0.2 and 0.4MPa) shows a weak softening trend; the higher the confining pressure, the more obvious the hardening tendency (0.6 and 0.8MPa) and the greater the peak strength. During unloading tests, strain softening occurs, and the peak strength increases with increasing confining pressure. During loading tests, dilatancy appears when the confining pressure is low (0.2MPa).With increasing confining pressure (0.4, 0.6, and 0.8MPa), the dilatancy trend gradually weakens, and the cumulative volume tric strain increases, which reflects the relevance of the stress paths.+rough research, it is found that the hyperbolic constitutive model has good applicability to sandy gravel soils, and the corresponding model parameters are obtained.


Introduction
With the rapid development of rail transit, shield tunnel construction has encountered many severe challenges in sandy gravel strata in China. For example, the first phase of the Luoyang Urban Rail Transit Line 2 project under construction uses the shield method to cross the sandy gravel stratum, which is particularly important for the ground settlement, seepage damage, and the stability of the excavation surface [1][2][3]. In addition to a large number of field tests, the mechanical characteristics and description of the stress-strain relationship of coarse-grained soil are hot topics [4,5], which can provide necessary conditions for numerical calculations in practical engineering. e mechanical properties of soils are determined by many factors, including externally applied stress, soil type, microstructure damage [6], density, matrix suction caused by surface tension passing through the air in unsaturated environments [7], and temperature [8][9][10]. To date, many constitutive models of saturated/unsaturated soil have been proposed, but there are still many controversies regarding some basic problems [9,11], such as the selection of state variables of stress and strain and the theoretical framework of hydraulic coupling processes [12,13]. Alonso et al. [14] established a constitutive model of unsaturated soil materials by utilizing the double-stress variables of net stress and matrix suction. Many scholars [15,16] have discussed the hydraulic coupling characteristics of saturated/unsaturated soil and even some basic thermo-hydromechanical coupling characteristics [17,18]. ese theoretical studies considered the influence of the volume water content on mechanical properties, such as irreversible compression during drying and changes in the water content during loading/unloading. Based on the principle of thermodynamics, some researchers [19][20][21] established the hydraulic coupling effects of threephase unsaturated soil by analyzing the thermodynamic potential of the solid-liquid-gas phase and dissipated energy. Recently, Bai et al. [22] proposed a thermo-hydromechanical constitutive model for geomaterials in view of the concept of particle rearrangement during the thermodynamic process of porous granular materials, which is contributed to understand the mechanical properties of the particle movement of sandy gravel soils from the microscopic point of view.
For subway tunnel projects in sandy gravel strata [23][24][25], when discussing settlement control and deformation characteristics, the constitutive relationship of sandy gravel is rarely mentioned, and the commonly used Mohr-Coulomb relationship is still utilized [26]. e actual sandy gravel is a kind of coarse-grained soil; however, it is not reasonable to use the Mohr-Coulomb relationship. erefore, it is particularly important to study the deformation characteristics of shield tunneling in sandy gravel formations and to determine the appropriate constitutive model and parameters [27,28]. It has important engineering significance in stratum settlement control and station foundation pit excavation [29].
In this paper, a large-sized laboratory triaxial shear test is carried out on sandy gravel soil taken from the Luoyang subway site, and then triaxial shear tests on sandy gravel under different confining pressures are carried out to determine the appropriate sandy gravel constitutive model as well as the model parameters accordingly. is work can help with engineering design and practice. e established constitutive model can well reflect the loading process and unloading process of sandy gravel soil, which is also used for the numerical calculation during the unloading process in the shield tunnel excavation process.

Test Schemes.
is paper selects sandy gravel soil samples from the first phase of the project of Luoyang Urban Rail Transit Line 2 to conduct a large-scale triaxial test. e purpose is to investigate two typical stress state change processes (namely, loading and unloading stress paths) and then determine the stress-strain relationship characteristics and volumetric strain-axial strain characteristics, as well as the seepage effect [30,31] and damage mechanism of the soil layer [24,32,33].
Actually, the stress and deformation state changes of the surrounding stratum are very complicated during the shield tunnel excavation process [34,35], even related to the effect of environmental temperature [36,37] and the thermalmechanical behavior of tunnel lining segments [38,39]. For simplicity, this test simulates two extreme stress path change processes, namely, active failure and passive failure of the excavation surface. e specific test scheme is shown in Table 1, including 2 groups of tests. Among them, the sample size is 30 cm in diameter and 70 cm in height, and the test confining pressure is determined by the actual stress state at the site to be 200, 400, 600, and 800 kPa. e loading/ unloading tests were accomplished by the stress-controlled method, and the shear rate was 0.2 kPa/s. is test primarily studies the strength and deformation characteristics of sandy gravel under triaxial compression and unloading conditions.

Preparation of Soil Samples.
e soil layer in the project area is composed of silty clay, silt soil, a sand layer, and a gravel soil layer, which is a multilayer structure. According to the actual situation of the project, samples were taken at a depth of 10 m, and the dry density was controlled to be 2.27 g/cm 3 . rough particle analysis, the maximum particle size of sandy gravel tested was 80 mm, and its characteristic particle size is shown in Table 2. e inhomogeneity coefficient of the soil is greater than 10, and the curvature coefficient is between 1 and 5. It belongs to coarse-grained soil with good gradation. e test equipment was an SJ70 large-scale high-pressure triaxial shear tester.
e test specimen has a diameter of 300 mm and a height of 700 mm. According to the sampling and sieving situation, the gravels that exceed the particle size limit of 60 mm account for only approximately 5%, so a particle size of 40−60 mm was used instead. e sample was prepared by splitting the soil into 5 layers. e sample preparation method uses a vibrator to vibrate. e static pressure of the vibrator bottom plate is 14 kPa, and the vibration frequency is 40 Hz. e vibration time is determined according to the dry density of the sample.
According to the actual engineering conditions and the high water permeability of sandy gravel, this test adopts the consolidated drained shear (i.e., CD) test. e sample saturation method adopts the suction saturation method.

Stress-Strain-Strength Characteristics.
From the stressstrain relationship curves of sandy gravel in Figures 1 and 2, under the condition of the same shear rate, sandy gravel is loaded under low confining pressure (0.2 and 0.4 MPa), showing a weak softening trend. e higher the confining pressure, the more obvious the hardening tendency (0.6 and 0.8 MPa) and the greater the peak strength. For the unloading test, there is a more obvious softening phenomenon, and as the confining pressure increases, the peak value becomes more obvious. e deviator stress-strain curves of the two groups of sandy gravels show a strong nonlinear relationship when the stress and strain reach a certain value.

Volume Deformation Characteristics. Figures 3 and 4
give the ε v − ε a relationship of sandy gravel when loading and unloading under different confining pressures. From Figure 3, under conditions of low confining pressure (0.2 MPa), sandy gravels exhibit shear shrinkage first and then dilatancy. As the confining pressure increases (0.4, 0.6, and 0.8 MPa), the dilatancy trend gradually weakens, and the cumulative body strain increases. Under unloading conditions (Figure 4), the main manifestation is dilatancy, and with increasing confining pressure, the dilatancy phenomenon becomes more obvious, which fully reflects the relevance of the stress path.

Constitutive Model of Sandy Gravel Considering the Loading/ Unloading Process
Kondner [40] thought that the hyperbola can be used to fit the general triaxial test (σ 1 − σ 3 )-ε 1 curve, namely, where a and b are test constants. For the conventional triaxial shear test, ε a � ε 1 . Later, Duncan and Chang [41] proposed a widely used nonlinear elastic model, namely, the Duncan-Chang model, which was developed widely [42].

Conventional Triaxial Loading Test.
Under conventional triaxial loading conditions, (1) can be written as where a is the intercept of the line and b is the slope of the line.
Obviously, there is an approximately linear relationship between ε 1 /(σ 1 − σ 3 ) and ε 1 . In the conventional triaxial shear test, the tangent elastic modulus can be expressed as When ε 1 � 0 and E t � E i , the combination of (3) leads to When ε 1 ⟶∞, from (1), we can get In (4) and (5), a represents the reciprocal of the initial elastic modulus E i , and b represents the reciprocal of the ultimate deviator stress (σ 1 − σ 3 ) ult corresponding to the asymptote of the hyperbola.
In the conventional triaxial test, the shear strength (σ 1 − σ 3 ) f is determined according to the stress value corresponding to the strain value ε 1 � 15%; for the stress-strain curve with a peak value, it generally takes its strength, that is, In this way, the damage ratio R f can be defined as Considering (5), one has b � 1 σ 1 − σ 3 ult , Consider E t as a function of stress, which can be obtained by transforming (2): Substituting (8) into (3), we get Substituting (4) and (7) into (9), we can get According to the Mohr-Coulomb strength criterion, there is e expression of the initial elastic modulus E i with consolidation pressure σ 3 can be described as where p a is the atmospheric pressure (p a � 101.4 kPa) and K and n are experimental parameters, representing the intercept and slope of the straight line between log (E i /p a ) and log (σ 3 /p a ). Substituting (11) and (12) into (10), we can obtain In (13), the tangent elastic modulus E t contains five material parameters (K, n, c, φ, and R f ). ere is a hyperbolic relationship between the axial strain ε 1 and the confining strain −ε 3 based on the conventional triaxial loading test data of the soil ( Figure 5). e hyperbolic equation shown in Figure 5(a) can be written as e linear equation in Figure 5(b) can be written as When ε 3 � 0, −ε 3 /ε 1 � f � ] i is initial Poisson's ratio; when −ε 3 ⟶∞, D � 1/(ε 1 ) ult . e triaxial shear test shows that 4 Advances in Civil Engineering initial Poisson's ratio is related to the confining pressure [9,12]. As shown in Figure 5(c), it is assumed that where G and F are test constants.

Conventional Triaxial Unloading Test.
Assuming that the soil is an isotropic medium, according to the generalized Hooke's law, the stress-strain relationship can be written as where Δσ a � Δσ 1 is the axial stress increment, Δσ r � Δσ 3 is the lateral stress increment, Δε a � Δε 1 is the axial strain increment, and Δε r � Δε 3 is the lateral strain increment. When Δσ a � 0 and Δσ r ≠ 0, the tangent elastic modulus E t can be written as Referring to the derivation process of the triaxial loading test, under the condition of lateral unloading, the relationship curve of 2(σ rc − σ r ) − (ε a − 2ε r ) is still satisfied with a hyperbolic relationship, namely, (ε a − 2ε r )/ 2(σ rc − σ r ) − (ε a − ε r ).
Use (12) to establish the relationship between the initial tangent elastic modulus E i and the axial consolidation pressure σ ac . Here, E i is actually the slope of the curve between 2(σ rc − σ r ) and (ε a − 2ε r ) at the origin of the curve. Hence, During lateral unloading, the failure deviator stress can be deduced: Because σ a � σ ac is a constant during lateral unloading, it can be obtained from (20):  Advances in Civil Engineering 5 . (23) e tangential elastic modulus E t of the triaxial unloading test still contains 5 material constants (i.e., K, n, c, φ, and R f ). Comparing the unloading and loading paths, it can be seen that the axial strain ε 1 increases and ε 3 decreases.
at is, the two effects on the test results are the same. From this point of view, the calculation formula of Poisson's ratio for loading and unloading tests is the same. erefore, the final expression of tangent Poisson's ratio in the unloading test is

Parameters of the Triaxial Loading.
According to the stress-strain relationship shown in Figure 2, three groups of confining pressures and corresponding failure stresses (σ 1 − σ 3 ) f are obtained, as shown in Table 3. us, we can obtain the cohesive force c � 19.2 kPa and the friction angle φ � 40°. In the coordinates of ε 1 and ε 1 /(σ 1 − σ 3 ), the relationship between the two is approximately linear (Figure 6). In Figure 6, a and b are the intercept and slope obtained by fitting a straight line. Combining equation (24), the E i value can be obtained as shown in Table 4. Figure 6 indicates that ε 1 and ε 1 /(σ 1 − σ 3 ) of sandy gravel have a good linear relationship, which means a hyperbolic relationship between shear stress and strain. e determination parameter R 2 values are all greater than 0.98, and the hyperbolic model has better applicability. Combining (7) can calculate the values of R f to be 0.91, 0.93, and 0.92, and the average value is 0.92.
According to (12), the following expression can be obtained: From the results in Table 4, three sets of E i and σ 3 values can be obtained. According to the fitting results, we can obtain the intercept K � 97 and the slope n � 0.538. According to the relationship between the axial strain of sandy gravel and the volume tric strain, the relationship of −ε 3 /ε 1 and −ε 3 can be obtained as shown in Figure 7. Figure 7 indicates that there is a good linear relationship between −ε 3 / ε 1 and −ε 3 . By fitting the result to a straight line, the average value of the slope D is 0.07.
According to the three groups of values of v i and σ 3 in Figure 7, the relationship between v i and log(σ 3 /p a ) is shown in Figure 8. Due to v i � G − Flg(σ 3 /p a ), the parameters G � 0.49 and F � 0.186 can be obtained.

Parameters for Triaxial Unloading.
A large number of experimental studies [13][14][15] show that the cohesion c and the friction angle φ in the unloading test are equal to those in the load test, that is, the cohesion c is 19.2 kPa and the friction angle φ is 40°. e damage ratio R f of the unloading test is consistent with (26). In the coordinates (ε 1 − 2ε 3 )/ 2(σ 1 − σ 3 ) − (ε 1 − 2ε 3 ) (Figure 9), the linear fitting results are poor. is is because high or low stress levels will cause the test point to deviate from the straight line.
e relevant values obtained according to this method are shown in Table 5, and the fitted straight line relationship is shown in Figure 10. Combining the stress paths of the unloading test, there are where a and b are the intercepts and slopes obtained by fitting in Figure 4. Combining the results in Figure 4, the relevant parameters of the unloading test and E i and R f can be obtained (Table 6). Finally, the average value of R f in the sandy gravel unloading test can be obtained from Table 6, namely, R f � 0.43. Using (21), the following equation can be obtained: where K represents the intercept and n represents the slope. According to Table 6, three groups of E i and σ 1 values can be obtained to fit a straight line, and the expression is Comparing (28) and (29), we can obtain that intercept K � 43 and slope n � 0.759. Finally, the parameters G, F, and D related to tangent Poisson's ratio are determined. Figure 5 shows the relationship between the axial strain and the body strain and the relationship between −ε 3 /ε 1 and −ε 3 ( Figure 11). 6 Advances in Civil Engineering Figure 11 indicates that there is an obvious linear relationship between −ε 3 /ε 1 and −ε 3 , indicating that the assumption that the axial strain ε 1 and the lateral strain ε 3 are hyperbolic in the unloading test is applicable. e average value of the slope D of the fitted straight line is −0.015. According to Figure 11, three groups of values of v i and σ 3            Figure 11: Relationship between −ε 3 /ε 1 and −ε 3 .

Advances in Civil Engineering
can be obtained, and the corresponding relationship between v i and log(σ 3 /p a ) can be drawn, as shown in Figure 12.
the equation of the fitted straight line in Figure 12 is Comparing (30) and (31), the value of G is 0.47, and the value of F is −0.145.

e Tangent Elastic Modulus and Tangent Poisson's Ratio.
For triaxial loading, according to the relevant parameters determined above, the tangent elastic modulus E t and tangent Poisson's ratio v t of sandy gravel under triaxial loading can be obtained as For triaxial unloading, E t and v t during the unloading test are

Conclusions
Consolidation and drainage shear tests of sandy gravel under different confining pressures under loading/unloading conditions are carried out. e differences in the strength and deformation of sandy gravel are analyzed. In using the proposed model, the parameters are few and can be obtained from normal geotechnical tests, which has high application value in practical engineering. e main conclusions drawn are as follows: (1) e relationships between σ 1 − σ 3 and ε 1 and ε v and ε a of sandy gravel soil under loading conditions are all nonlinear. e smaller the confining pressure, the Advances in Civil Engineering weaker the stress-strain relationship.
e stressstrain relationship exhibits a softening phenomenon, and the bulk strain undergoes dilatancy deformation. In contrast, it shows strain hardening and shear shrinkage deformation.
(2) Under unloading conditions, the sandy gravel soil softens significantly with increasing confining pressure, and the peak strength increases. In the meantime, the tric strain response exhibits dilatancy characteristics. e strength and deformation characteristics of sandy gravel soil have a high dependence on the stress path.
(3) e Duncan-Chang hyperbolic model has good applicability for simulating sandy gravel soils, and the model parameters are calculated according to the test results, which can provide help for engineering design and practice.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.