The basic principle of the large-scale triaxial test for sandy pebble soil is the same as that of triaxial shear test for fine-grained soil, except that the scope of the study object is wider, the particle size is larger as well as the scale and size of the instrument are larger. This pertains to the coarse-grained and discrete sandy gravel soil suitable for the research presented in this paper. During the triaxial test, the vertical stress is first applied to the specimen, so that the vertical principal stress becomes greater than the horizontal one. When the horizontal principal stress remains unchanged and the vertical principal stress is gradually increased, the specimen finally damaged by shear stress. Provided that shear failure occurs in the specimen, the increment of the vertical compressive stress on the specimen is Δσ ₁, the major principal stress in the specimen is expressed as σ ₁=σ ₃+Δσ ₁, and then the small primary stress is σ ₃, so Mohr’s limit stress circle is obtained. Several specimens at least the three same specimens under different confining pressuresσ ₃are used to conduct the compressive test as per the above-stated method respectively, then large principal stress σ ₁ at shear failure is obtained. In combination with these results, a set of Mohr’s limit stress circles are obtained, including a common tangent line. This line indicates the shear strength envelope of sandy pebble soil under triaxial compression. The angle included between the tangent line and the x-axis denotes the internal friction angle φ of sandy pebble soil, while the intercept between the tangent line and y-axis illustrates the cohesion c of sandy pebble soil as shown in Figure 1.

To conduct triaxial compressive test for sandy pebble soil specimen, YLSZ30-3 stress-typed large triaxial testing machine is used (as shown in Figure 2). This set of apparatus is used to measure such properties as the total shear strength parameters, effective shear strength parameters and pore pressure coefficient of sandy pebble soil. Triaxial compressive tests for sandy pebble soil subject to constant confining pressure have been carried out according to the properties and engineering conditions of sandy pebble soil under different situations. The main performance indexes of the apparatus are listed in Table1.

The soil samples selected for this experiment were from a metro project in a city in southwest China. Since most of the tunnel depths are within the sandy pebble soil layer, this experiment primarily focused on the study of sandy pebble soil. The composition of the pebbles mainly consists of igneous and metamorphic rocks. They are predominantly sub-rounded to sub-angular, with some rounded pebbles, poor sorting, and a pebble content ranging from 60% to 85%. The particle size mainly falls in the range of 20–80 mm, with some particles exceeding 100 mm, and the largest pebble has a diameter of 180 mm. The interstitial material is fine sand, and occasional floaters are present, as shown in Figure 3. The arithmetic average method of particle grading was used to obtain the grading of sand and pebble soil particles used in the test, as shown in Table 2. Field measurements determined that the density of the soil samples ranged between 0.4 and 0.6, with a natural moisture content of 4%. Consequently, the density of the specimens during the experiment was set at 0.4, 0.5, and 0.6. Furthermore, based on the depth from which the samples were extracted, the confining pressures for the laboratory tests were calculated to be 100 kPa, 200 kPa, and 300 kPa, respectively.

In the triaxial shear test, it is essential first to prepare the testing apparatus, which involves filling the organic glass tube in the volume change measurement cabinet with water and removing any air. Next, the soil sample is prepared and placed step by step, including laying the permeable plate, filter cloth, and rubber mold, and layering the soil sample with compaction at each layer. Subsequently, a vacuum pump is used to create a negative pressure inside the sample to help remove air. Then, the pressure chamber shell is installed and ensured to be sealed, followed by water injection into the pressure chamber. Finally, the axial loading and the pressure chamber are synchronously lifted using the oil pump, and the test parameters are set on the control panel, marking the commencement of the test.

With the increase in confining pressure, the tangent slope and linearity of the stress-strain curve for the sand-pebble soil specimens improved (as shown in Figure 4). Additionally, the initial tangent modulus gradually increased, suggesting an enhancement in the compression modulus due to the interplay between hydrostatic pressure and shear deformation. This indicates a strengthening effect resulting from the coupling of these factors. Moreover, the peak stress of the sandy pebble soil specimen gradually increases, along with its initial elastic modulus. The shear failure in the sandy pebble soil specimen occurs as its axial strain approaches 3.5%. This behavior contrasts with fine-grained soil, where the axial strain reaches approximately 15% before specimen damage occurs. The higher density of sandy pebble soil prevents obvious strain softening, enabling it to withstand higher loads post its peak strength. The presence of large pebble particles in the sand and pebble soil induces a crushing effect during the compression process, resulting in non-ideal test curves with fluctuations observed during experimentation.

The difference in principal stresses obtained at various circumferential pressures increased with the rising Dr. At Dr = 0.6, the range of principal stress differences at different perimeter pressures reached 800–1,500 kPa.

The Mohr stress circle was constructed based on the results of triaxial shear tests, yielding the strength parameters of sandy pebble soil under varying relative densities, as displayed in Table 3. These results reveal a significant correlation between specimen density and the shear strength parameters of sandy pebble soil. Figure 5 demonstrates that the internal friction angle of sandy pebble soil decreases as relative density increases, showing a maximum reduction of 31.7%. In contrast, cohesion initially rises and subsequently diminishes, exhibiting a more notable decline when relative density surpasses 0.5. The correlation between the mechanical parameters of sandy pebble surrounding rock and relative density is shown as Eqs. 1, 2 (These formulas are only applicable to sandy pebbly soils with a densification of 0.4–0.6): Sin ⁡ φ = − 24.8 D r 2 + 21.4 D r − 4.5 (1) c = − 7.9 D r 2 + 7.1 D r − 1.6 (2)

The Eq. 8 reveals that (σ₁-σ₃)_ult is expressed as a function of R_f and (σ₁-σ₃)_f. Furthermore, as indicated in Table 2, it becomes apparent that Dr exerts a substantial influence on (σ₁-σ₃)_f, while its impact on R_f remains relatively insignificant. The effect of Dr on parameter b is considered in the context of (σ₁-σ₃)_f. Taking into account the relationship between (σ₁-σ₃)_f and the confining pressure σ₃, Eq. 9 is formulated as follows. σ 1 − σ 3 f = O p a σ 3 p a P (9)

Taking the logarithm of both sides of Eq. 9. ln σ 1 − σ 3 f / p a = ⁡ ln ⁡ O + P ⁡ ln σ 3 / p a (10)

Where parameter lnO represents the intercept of the line, P is the slope of the line. If the test data in Table 4 are substituted into Eq. 10, then the values of O and P under different relative compactness are derived and listed in Table 5.

From Table 5, it can be observed that, under different relative soil compaction levels, the value of P remains relatively constant, while O exhibits a relatively significant variation with respect to Dr. Figure 6 illustrates the relationship between lnO and Dr, revealing a roughly linear association between the lnO parameter and Dr. Consequently, Eq. 11 can be employed to perform a linear fit on lnO and Dr. ln ⁡ O = o + p D r (11)

Figure 6 reveals a compelling linear association between lnO and the relative compactness parameter Dr. In the context of sandy pebble soil, the dimensionless constants o and p within Eq. 11 were determined through experimental investigations, resulting in values of o = 0.18 and p = 1.331, accompanied by a remarkably high correlation coefficient of R ² = 0.999. These findings provide strong evidence supporting the appropriateness and efficacy of the linear regression model applied to lnO and Dr.

A great deal of research has been done to show that the initial shear modulus Ei of the soil is an exponential function of the lateral limit pressure, expressed as follows E i = M p a σ 3 p a N (12)

A certain deformation of the formula can be obtained: ln E i p a = ⁡ ln ⁡ M + N ⁡ ln σ 3 p a (13)

By substituting the parameters in Table 4 into Eq. 13, the values of lnM and N under different relative compactness are obtained and listed in Table 6.

It can be seen from Table 6 that the value of N basically remains unchanged, and lnM changes with Dr in a large range under different relative compactness. Figure 7 shows the linear relationship between lnM and Dr. As a result, Eq. 14 is obtained to express the linear relationship between lnM and Dr. ln ⁡ M = m + n D r (14)

Additionally, Figure 7 presents a clear and apparent linear correlation between lnM and the relative compactness parameter, Dr. Notably, for sandy pebble soil, the dimensionless constants m and n within Eq. 14 were derived from experimental data, resulting in the values m = 3.288 and n = 1.115. (accompanied by a high correlation coefficient of R ²=0.996). This further affirms the logical and efficient nature of the linear regression model applied to lnM and Dr.

According to the theory of disturbed state concept (DSC), the microstructure of sandy pebble soil is disturbed by external force, resulting in the damage of micro-cracks or the reinforcement of particles due to the relative movement of particles. The disturbance process is described with function D, namely, disturbance function, and its magnitude is determined with disturbance degree. The disturbance evolution is expatiated by macroscopic measurement so as to simulate the behavior of sandy pebble soil. The disturbance function D of soil is established according to test results D = D u 1 − ⁡ exp − A ε z (15)

It is apparent from Eq. 15 that the disturbance degrees introduced by Desai all possess positive values and are confined to the [0,1] interval, akin to the traditional interpretation of a damage variable. However, it's essential to recognize that disturbance doesn’t solely lead to the micro-crack damage within the material; it also has the capacity to enhance material properties, such as the compaction of sandy pebble soil. In this context, D assumes negative values, signifying an improvement in material performance following a disturbance. For the sake of clarity, the term “positive disturbance” is used to refer to a disturbance that decreases Dr, while “negative disturbance” characterizes a disturbance that increases Dr. Based on the fundamentals of the Disturbed State Concept (DSC), Dr is used as the perturbation parameter. The final perturbation function can be expressed as follows.

(a) Positive disturbance, D r 0 ≥ D r ,

The functional relationship of Eqs 16, 17 is drawn and shown in Figure 8. It can be seen that the disturbance degree falls in to interval [−1, 1], it indicates the whole process of the change of sandy pebble soil from the initial state to the densest or the loosest state.

The physical and mechanical parameters of sandy pebble soil are prone to change under the influence of soil disturbance, resulting in significant alterations in the soil’s deformation and strength characteristics. The degree of disturbance in sandy pebble soil is primarily characterized by a disturbance equation. To effectively depict the changes in the soil’s mechanical properties under any disturbance scenario, it is necessary to establish a functional relationship between the soil’s mechanical parameters and D. Within the Duncan-Chang model, the initial modulus E_i and peak strength (σ₁-σ₃)_f are most significantly affected by the relative compactness of the soil.

Under the condition of positive disturbance, D r 0 ≥ D r , D _rmin =0, Combining Eq. 16 leads to Eq. 18. D = 2 π arctan D r 0 − D r D r − D r ⁡ min 4 (18)

Morphing Eq. 18 gives D r = D r 0 tan π D 4 2 + 1 (19)

Substituting Eqs 9, 11 into Eq. 19 yields Eqs 20, 21. O = e o 1 + ⁡ tan π 2 D 4 + p D r 0 1 + ⁡ tan π 2 D 4 (20) O = e o + p D r 0 e o D r ⁡ tan π 2 D 4 (21)

If let O 0 = e o + p D r 0 , then it can be illustrated in Eq. 22. O = O 0 e p D r ⁡ tan π 2 D 4 (22)

Combining Eqs 12, 14 with Eqs 19, 3 can be obtained M = e m 1 + ⁡ tan π 2 D 4 + n D r 0 1 + ⁡ tan π 2 D 4 (23)

Simplifying Eq. 23 gives M = e m + n D r 0 e n D r ⁡ tan π 2 D 4 (24)

Substituting M₀ ( M 0 = e m + n D r 0 ) into Eq. 24 gives Eq. 25 M = M 0 e n D r ⁡ tan π 2 D 4 (25)

(2) Under the condition of Negative disturbance, D _r >D _r0, D _rmax =1.

Morphing Eq. 26 gives D r = D r 0 − ⁡ tan π D 4 2 1 − ⁡ tan π D 4 2 (27)

Combining Eqs 9, 11 with Eq. 27 gives O = e o + p tan π 2 D 4 − D r 0 tan π 2 D 4 − 1 (28)

Simplifying the formula Eq. 28 gives O = e o + p D r 0 e p 1 − D r tan π 2 D 4 (29)

Substituting O₀ ( O 0 = e o + p D r 0 ) into Eq. 29 gives O = O 0 e p 1 − D r tan π 2 D 4 (30)

Combining Eqs 12, 14 with Eq. 30 gives M = e m + n tan π 2 D 4 − D r 0 tan π 2 D 4 − 1 (31)

Simplification of Eq. 31 gives M = e m + n D r 0 e n 1 − D r tan π 2 D 4 (32)

Substituting M₀ ( M 0 = e m + n D r 0 ) into Eq. 32 gives Eq. 33. M = M 0 e n 1 − D r tan π 2 D 4 (33)

In summary, it can be concluded that when sandy pebbly soils are undisturbed:

D _r =D _r0, M = e m + n D r 0 , O = e o + p D r 0

When the soil reaches its loosest state as a result of disturbance:

D _r =D _rmin =0, D=1, M = M 0 e n D r 0 , O = O 0 e p D r 0

When the soil reaches its densest state as a result of disturbance:

D _r =D _rmax =1, D=−1, M = M 0 e n D r 0 − 1 , O = O 0 e p D r 0 − 1 Substituting the theoretical formulas obtained in this study into the Duncan-chang model, the modified Duncan-Chang model can be obtained as follows:

(1) Under the condition of positive disturbance, D r 0 ≥ D r ,

In Eq. 34, E i = M p a σ 3 p a N , R f = σ 1 − σ 3 f σ 1 − σ 3 u l t , σ 1 − σ 3 u l t = O 0 σ 3 , then σ 1 − σ 3 = ε 1 e n D r ⁡ tan π D 4 / 2 M 0 p a σ 3 p a N 0 + R f 0 ε 1 O 0 e p D r ⁡ tan π 2 D 4 σ 3 (35)

(2) Under the condition of Negative disturbance, D _r >D _r0, the following can be obtained:

Eqs 35, 37 stand for the modified Duncan-Chang model considering the influence of the disturbance of sandy pebble soil under different compactness.

To validate the rationality and accuracy of the modified Duncan-Chang model considering the influence of disturbance, based on the test data in case D _r = 0.6, Both Eqs 35, 37 are used to predict the stress-strain curve of sandy pebble soil with different disturbance degrees at confining pressure σ ₃ = 100 kPa, and are compared with the Duncan-Chang model and the measured data. Moreover, the deviator stress (σ ₁-σ ₃) _f is calculated by using Eq. 10, The obtained results are shown in Figure 9.

Upon comparing predicted and experimental results in Figure 9, it is evident that, at D = 0, the predicted values of the modified Duncan-Chang model align closely with the experimental results in the elastic phase. Predicted stress under the positive disturbance state (D>0) is notably higher, whereas under the negative disturbance state (D<0), it is significantly lower. Consequently, the Duncan-Chang model proves unsuitable for predicting the behavior of sandy pebble soil under disturbance states. Utilizing this modified Duncan-Chang model to forecast stress-strain curves under disturbance states (D>0), the predicted stress closely aligns with experimental values, akin to situations of negative disturbance (D<0) and D = 0, where predicted stress in the elastic stage and stress peak values closely match experimental data. Hence, the modified Duncan-Chang model demonstrates wider applicability.

The modified constitutive model proposed in this paper allows for the calculation of stress values of gravelly soil under various strain conditions. Analyzing experimental data from both the elastic and peak stress sections for comparative analysis of the fitting results. Table 7 presents the comparative results between initial model calculations and modified model calculations alongside experimental values. Stress calculation errors of the initial model surpass 35% when compaction densities range between 0.4 and 0.6, failing to meet engineering precision standards. Conversely, stress calculation errors of the modified model consistently range between 0% and 8%, aligning more closely with engineering application requirements. Comparative analysis with experimental data indicates that the calculations of the modified Duncan-Chang model closely align with experimental results. These findings affirm that the established modified constitutive model effectively represents the stress-strain relationship of gravelly soil under disturbance conditions.

In this article, a unified disturbance degree function expression was established using the relative density of sandy gravel soil as the disturbance parameter. Subsequently, a modified Duncan-Chang model, which accounts for the effects of disturbances, was proposed. This model incorporates the influence of the gravelly soil’s physical parameter, Dr, on the characteristics of sandy gravel soil, thereby reflecting, to a certain extent, the nature of soil strength and deformation. However, the study of the modified constitutive model was limited to the elastic phase of sandy gravel soil, neglecting the evolution of the plastic and softening phases. Furthermore, while numerous physical parameters can indicate soil disturbances, and relative density (Dr) is a crucial factor, it is not the sole factor. Therefore, future research should focus on developing a disturbance theory for soils that correlates with coarse grain content, moisture content, and other physical parameters.