Introduction

Red clay is widely distributed in China, such as Guangxi, Guizhou, and Yunnan1,2. Red clay as a special soil has a high liquid-plastic limit, water swelling and water loss shrinkage3. The main mineral composition of red clay includes limestone, dolomite, marl, montmorillonite, illite and other minerals4. With the development of transportation facilities such as railways and highways in the western region of China, red clay as a roadbed filler reduces costs5. However, the poor engineering properties cause many hidden dangers for the implementation of the project. Many scholars have studied the treated soil with different materials6,7,8,9,10,11,12. Stabilizing peat soils using demolished concrete and scrap tyres is a sustainable method13,14. The crushed stone mixture improves the performance15. Bituminous stabilizer as a non-traditional soil additive has a positive effect on increasing strength16. Traditional soil stabilizers are similar to Portland cement, which increases strength and improves the microscopic pore structure after hydration17,18,19,20,21. Phosphoric acid and lime treatment enhance the strength22,23. Calcium carbonate is a typical carbonate rock widely distributed in the red clay area. Hezhou is one of the main producing regions in China. Calcium carbonate is widely used as a modifying agent. Calcium carbonate improves the performance of asphalt24. Nano calcium carbonate is used for coating modification25. The calcium carbonate incorporated into red clay will change its mechanical properties26.

Currently, the modification of soil with calcium carbonate in engineering is mainly focused on mechanical properties. Although the strength of the subgrade filler is important, the influence of other aspects on the service life also needs to be studied, such as the liquid-plastic limit, water loss shrinkage, and constitutive model. The soil constitutive model provides an effective theoretical calculation method for engineering construction and captures geotechnical characteristics well27. The existing constitutive models used in practical engineering include the linear-elastic and the elastic-plastic28. The Duncan-Chang based on the hyperbolic hypothesis has a good reflection of the nonlinear characteristics29. Liu et al. studied the Duncan-Chang parameters of lean clay considering the number of cycles and freezing temperature30. Ni et al. established a modified Duncan-Chang for the variation of shear strength with water content31. The Duncan-Chang describes the stress-strain relationship of soil, but it needs according to the actual situation consider the application.

In this study calcium carbonate is used as an admixture to treat red clay. The soil with different contents of calcium carbonate was subjected to liquid-plastic limit, water loss shrinkage, NMR and the triaxial consolidation drained shear test. The experiment revealed the influence of calcium carbonate content changes on the microscopic pores and physico-mechanical properties of modified soil. The correction coefficient of the Duncan-Zhang is introduced according to the change of confining pressure. The applicability of the model to modified soil is compared before and after correction.

Materials and methods

Materials

The soil was taken from 5 m underground in Guilin, Guangxi, China (25.29 degrees north latitude, 110.28 degrees east longitude). The main mineral content is shown in Table 1. The basic physical and mechanical indexes are shown in Table 2. Figure 1(a) shows the SEM image of air-dried soil. The microstructure of “agglomerates” can be observed. “Agglomerates” are formed by the stacking and aggregation of flat mineral particles. The red clay mineral particles are irregular in shape and the micropores after cementation are very developed, which leads to a large specific surface area. Calcium carbonate is produced in Hezhou. Figure 1(b) shows the SEM image. The particle size distribution range is 0.237–16.44 μm. The shape of calcium carbonate particles is irregular cubes with obvious edges and corners and the surface is rough. The specific surface area of calcium carbonate is 0.4596 m2/g.

Table 1 The main mineral composition and content of red clay.
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Table 2 The basic physical and mechanical indicators.
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Fig. 1
Fig. 1The alternative text for this image may have been generated using AI.
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SEM images: (a) Red clay and (b) Calcium carbonate.

Methods

The triaxial shear test was prepared according to the " Standard for Soil Test Method " (GB/T50123-1999). The soil was air-dried, crushed and passed through a 2 mm sieve. Calcium carbonate was added at a mass fraction of 5%. Then prepared into a cylindrical sample with a diameter of 39.1 mm, a height of 80 mm and a density of 1.5 g/cm3 by static pressure method. The confining pressure of the triaxial shear test is 100, 200 and 300 kPa, respectively. The triaxial shear test is consolidated drained shear with a rate of 0.08 mm/min. The steps of the triaxial shear test are as follows: Firstly, the soil sample is placed on the base and then covered with a rubber membrane to remove the bubbles between the rubber membrane and the soil sample. Secondly, the pressure chamber is sealed and water is injected to remove the air in the pressure chamber. Finally, when the confining pressure reaches the target value the shear test is started.

In the water loss shrinkage test, the contents of calcium carbonate are 0%, 5%, 10%, 15% and 20%. For the water loss shrinkage test, take a particle size of less than 2 mm of air-dried red clay and mix it with the calcium carbonate, spray pure water and stir evenly. The moisture content of the soil reaches the optimal and then stands for 24 h. The soil sample is placed on a porous plate of a constant temperature box, the temperature is set to 30 °C and its vertical water loss shrinkage is measured with a dial indicator. The dial indicator reading and soil sample mass are recorded every 4 h. When the dial indicator readings remain unchanged for two times take out the soil sample and measure its moisture content.

The MacroMR12 microscopic pore structure analysis and imaging system of Suzhou Newmai Analytical Instrument Co., Ltd. was used to test the soil sample. The relevant parameters during the test are set as follows: the magnetic field strength of the magnet unit is 0.3 Tesla, the polarization time (TW) is 1000 ms, the echo interval (TE) is 0.5 ms, the number of repetitions (NS) is 16 times, and the number of 180° pulses (NECH) is 100032,33. The T2 spectrum test signal is obtained by 10,000 stable iterative fittings through inversion analysis. During the test to prevent water evaporation and ensure the integrity of the triaxial soil sample is wrapped with a polytetrafluoroethylene film. The wrapped sample was placed in the NMR sample tube26. The contents of calcium carbonate are 0%, 5%, 10%, 15% and 20% added into the soil for the NMR test.

The Hitachi High-Tech’s S-4800 field emission scanning electron microscope from Oxford, UK was used to take SEM images of the gold-sprayed modified soils. The magnification range is between 120,000 and 1 million times.

In the XRD test, the soil is crushed to a particle size of less than 1 mm and then soaked in distilled water. Ultrasonic vibration is used to promote its dispersion and the sample is left to stand for 8 h. The upper suspension with a particle size of less than 2 μm is transferred into a test tube. After centrifugation, the upper clear liquid of the test tube is poured out. The clay minerals in the lower part are evenly dispersed on a 2 × 3 cm2 frosted glass slide and left to stand. The sample that has been naturally air-dried is called a naturally oriented sheet (N sheet). After the N sheet is subjected to the X-ray diffraction test, it is placed in a constant temperature 60 °C ethylene glycol atmosphere for saturation treatment for no less than 8 h. The obtained ethylene glycol saturated sheet (EG sheet). Finally, the EG sheet is heated to 550 °C and the temperature is maintained for 2.5 h. Then naturally cooled to room temperature to obtain the T sheet. The diffraction data measured by the N, EG, and T sheets are comprehensively analyzed to obtain the relative content of various clay minerals.

Results and discussion

The content of calcium carbonate effect on liquid-plastic limits and water loss shrinkage

Red clay is a special soil with a high liquid-plastic limit and water loss shrinkage. Table 3 presents the influence of calcium carbonate content on liquid-plastic limits. The calcium carbonate reduces the liquid limit, especially when the content is 5%. When the content is from 5 to 15% the liquid limit increases. The plastic limit decreases when the content is 5%. The content exceeding 5% has little impact on the plastic limit. The content improves the plasticity index to varying degrees. The content of calcium carbonate changes leading to the content of Montmorillonite change. The Montmorillonite has a significant impact on the physical properties34. The presence of Montmorillonite will lead to an increase in the liquid-plastic limits35. Table 3 shows the change of Montmorillonite in soil with the increase of calcium carbonate content, which is the main factor in reducing the liquid-plastic limit.

Table 3 The plasticity index under different content of calcium carbonate.
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The water loss shrinkage is tested by the vertical linear shrinkage. Figure 2(a) shows the linear shrinkage curve with different contents of calcium carbonate. The degree of compaction in the linear shrinkage test is 95%. The water content-linear shrinkage curve is divided into three stages: (I) the initial water loss shrinkage stage of intergranular, in which pores are filled with free water. (II) the rapid shrinkage stage of all intergranular pores; (III) the stage of shrinkage and stagnation of intragranular pore water loss. The initial moisture content under different calcium carbonate content is about 28%. In the first stage, the moisture content will be about 26% resulting in less shrinkage. In the second stage, the moisture content will be about 17%. The linear shrinkage rate will increase and the water is not filled with intergranular pores. There is a large pressure difference between the soil boundary and the liquid surface of the pore water. The volume of intergranular pores at this stage is significantly reduced. When the moisture content is close to 18% the soil particles have little pore water. The linear shrinkage rate slows down. In the third stage, the moisture content is less than 17% the intergranular pore water has completely disappeared. The intergranular pore water is gradually decreasing.

During the process of water loss shrinkage, pore water evaporates first from large pores and then from small pores. When the drying continues to increase the adsorbed water on the surface of the soil particles begins to disappear. In the first stage, most of the loss is free water and surface water in the pores (Fig. 2(a)). In the second stage, a large amount of pore water disappears. The remaining small pore water and particle surface water gradually disappear. The soil particles are close to each other resulting in rapid shrinkage (Fig. 2(a)). In the third stage, the water content continues to decrease, but the volume of the soil is not shrinking (Fig. 2(a)).

Figure 2(b) shows the relationship between different calcium carbonate contents and shrinkage coefficients. The shrinkage coefficients into a downward trend with the increase in the content. The content of calcium carbonate improves the shrinkage characteristics. This is due to the low internal porosity of calcium carbonate particles, which reduces the content of pore water. Calcium carbonate replaces part of the pores in the soil and acts as part of the soil skeleton, which is also a factor that reduces shrinkage (Fig. 2(c)).

Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
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The content effect on the water loss shrinkage: (a) Moisture content-linear shrinkage curve, (b) curves of different calcium carbonate contents versus water loss shrinkage coefficients and (c) SEM of calcium carbonate modified soil.

The content of calcium carbonate effect on microscopic pores

NMR testing with different contents of calcium carbonate at the same degree of compaction. The total peak area obtained from NMR reflects the porosity. The amplitude represents the content of pores. The T2 represents the size of the pore diameter. The modified soil adopts the degree of compaction as the evaluation standard in the design of the way subgrade. Figure 3(a) shows the microscopic pore distribution. The curves have two peaks indicating that the larger pores are completely compacted. When the degree of compaction is 95%, the density is close to the maximum dry density, the compactness is high, and the pores are greatly compressed. The calcium carbonate particles are in contact with the soil particles, and which content change will impact the soil. The content increased from 0 to 5%, the area of the I peak increased and the area of the second peak decreased. The content would lead to an increase in the smaller pores and the larger pores would be filled. The content is 15% and 20% will lead to a blurring of the bimodal boundary. The content increase leads to the peak area of I decrease and the peak area of peak II increase. Figure 3(b) shows the trend of porosity increasing with the content. The more calcium carbonate added, the more pores there are. Figure 2(c) shows that calcium carbonate fills the soil pores, but there are a lot of gaps where the edges of calcium carbonate contact the soil particles, which is the reason for the increase in the porosity of the mentor.

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
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The contents of calcium carbonate effect on microscopic pores: (a) T2 of soil under 95% degree of compaction and (b) the content effect on porosity.

Nonlinear analysis for the modified soil

The stress-strain reflects the process of plasticity, brittleness and compaction deformation. Also, reflects the deformation and strength characteristics. Duncan and Chang et al. proposed a simplified and practical nonlinear stress-strain relationship based on the stress-strain relationship of the Kondner hyperbola36. The Duncan-Zhang is a typical nonlinear elastic model, which reflects the nonlinear characteristics to a certain extent. Based on the consolidated drainage shear test the Duncan-Zhang is established to analyze the influence of confining pressure change.

The soil has nonlinear deformation characteristics and the deformation is described by a nonlinear elastic. Assuming that the elastic parameters change with the stress state. The law of elastic parameters changing with stress is obtained by the triaxial shear test. Based on the generalized Hooke’s law the increment of the nonlinear relationship is reflected. Due to the triaxial test, the confining pressure remains constant during the shear process, i.e. \(\:{d\sigma\:}_{1}={d\sigma\:}_{3}=0\), therefore

$$\:{E}_{t}=\frac{d\sigma\:}{d{\epsilon\:}_{a}}$$
(1)

Since the confining pressure remains constant, the modulus of elasticity formula can be transformed as follows30:

$$\:{E}_{t}=\frac{d\sigma\:}{d{\epsilon\:}_{1}}$$
(2)

The stress-strain relationship conformed to the hyperbolic model under the double parameters, as follows:

$$\:{\sigma\:}_{1}-{\sigma\:}_{3}=\frac{{\epsilon\:}_{a}}{a+b{\epsilon\:}_{a}}$$
(3)

where a and b are the test constants. In the triaxial shear test the shear strain \(\:{\epsilon\:}_{a}={\epsilon\:}_{1}\).

Based on the initial stress-strain the hyperbolic relationship is converted into the following formula:

$$\:\frac{{{\upepsilon\:}}_{\text{a}}}{{{\upsigma\:}}_{1}-{{\upsigma\:}}_{3}}=\text{a}+\text{b}{{\upepsilon\:}}_{\text{a}}$$
(4)

The model is derived from Eq. (4) to approximate the straight-line model. Where a is the intercept of the line and b is the slope of the line.

The Duncan-Chang derives the modulus of elasticity formula from the above relationship, as follows:

$${\text{E}}_{\text{t}}=\frac{{\text{d}{\upsigma}}_{1}}{{\text{d}{\upepsilon}}_{1}}=\frac{\text{d}({{\upsigma}}_{1}-{{\upsigma}}_{3})}{{\text{d}{\upepsilon}}_{\text{a}}}$$
(5)

The elastic model formula with parameters a and b is derived from the Eq. (3) to (5) as follows:

$$\:{\text{E}}_{\text{t}}=\frac{1}{\text{a}}{[1-\text{b}\left({{\upsigma\:}}_{1}-{{\upsigma\:}}_{3}\right)]}^{2}$$
(6)

When \(\:{{\upepsilon\:}}_{\text{a}}\to\:0\), \(\:\text{a}={\left(\frac{{{\upepsilon\:}}_{\text{a}}}{{{\upsigma\:}}_{1}-{{\upsigma\:}}_{3}}\right)}_{{{\upepsilon\:}}_{\text{a}\to\:0}}\), the initial tangent modulus of the curve (σ1-σ3) ~ εa, expressed in Ei. When ε1 = 0, Et=Ei, obtained:

$$\:\text{a}={\left(\frac{{{\upepsilon\:}}_{\text{a}}}{{{\upsigma\:}}_{1}-{{\upsigma\:}}_{3}}\right)}_{{{\upepsilon\:}}_{\text{a}\to\:0}}=\frac{1}{{\text{E}}_{\text{i}}}$$
(7)

where a is the reciprocal of the initial deformation modulus Ei in the triaxial shear test.

When \(\:{{\upepsilon\:}}_{1}\to\:{\infty\:}\), \(b=\frac{1}{{({\sigma\:}_{1}-{\sigma}_{3})}_{{\epsilon}_{a}\to\infty}}=\frac{1}{{({\sigma}_{1}-{\sigma}_{3})}_{ult}}\). Where \(\:{({\sigma\:}_{1}-{\sigma\:}_{3})}_{ult}\) is the limit value of \(\:({\sigma\:}_{1}-{\sigma\:}_{3})\). In practice, \(\:{\epsilon\:}_{1}\) can not tend to infinity. When \(\:{\epsilon\:}_{1}\) reaches a certain level the soil is damaged the difference in principal stress is \(\:{({\sigma\:}_{1}-{\sigma\:}_{3})}_{f}\). The value is always less than \(\:{({\sigma\:}_{1}-{\sigma\:}_{3})}_{u}\).

Parameters a and b are determined by the stress-strain in the triaxial shear test. According to the provisions of the " Standard for Soil Test Method " (GB/T50123-1999), if there is a peak of the principal stress difference in the conventional triaxial shear test, the peak point should be taken as the \(\:{({{\upsigma\:}}_{1}-{{\upsigma\:}}_{3})}_{\text{f}}\). The principal stress difference does not peak the corresponding principal stress difference at ε1 = 15% is taken as the peak strength \(\:{({{\upsigma\:}}_{1}-{{\upsigma\:}}_{3})}_{\text{f}}\). Thus, the damage ratio Rf is obtained as follows:

$$\:{\text{R}}_{\text{f}}=\frac{{({{\upsigma\:}}_{1}-{{\upsigma\:}}_{3})}_{\text{f}}}{{({{\upsigma\:}}_{1}-{{\upsigma\:}}_{3})}_{\text{u}\text{l}\text{t}}}$$
(8)

where Rf has a value range between 0.75 and 1. Rf is determined based on the average value of different confining pressures.

According to the Eq. (6) to (8), the relationship between Et and Ei is as follows:

$$\:{E}_{t}=\frac{d({\sigma\:}_{1}-{\sigma\:}_{3})}{d{\epsilon\:}_{1}}={E}_{i}{[1-\frac{{R}_{f}({\sigma\:}_{1}-{\sigma\:}_{3})}{{({\sigma\:}_{1}-{\sigma\:}_{3})}_{f}}]}^{2}$$
(9)

where Ei is the initial tangent modulus, which is related to the confining pressure. \(\:{({\sigma\:}_{1}-{\sigma\:}_{3})}_{f}\) is the strength of the destruction also related to the confining pressure.

Introducing standard atmospheric pressure (Pa = 101 kPa) into the equation, the link to the initial tangent modulus is established as follows:

$$\:{E}_{i}=K{p}_{a}{({\sigma\:}_{3}/{p}_{a})}^{n}$$
(10)

The initial tangent modulus changes with the change of confining pressure.

Introducing the cohesion force and internal friction angle into the binding formulas (9) and (10) yields the modified model as follows:

$$\:{E}_{t}=K{p}_{a}{\left(\frac{{\sigma\:}_{3}}{{p}_{a}}\right)}^{n}{\left[1-\frac{{R}_{f}\left({\sigma\:}_{1}-{\sigma\:}_{3}\right)(1-\text{s}\text{i}\text{n}\phi\:)}{2c\text{c}\text{o}\text{s}\phi\:+2{\sigma\:}_{3}\text{s}\text{i}\text{n}\phi\:}\right]}^{2}$$
(11)
$$\:{B}_{t}={K}_{b}{p}_{a}{\left(\frac{{\sigma\:}_{3}}{{p}_{a}}\right)}^{m}$$
(12)

where Et is the tangent modulus, Bt is the tangent volume modulus, Pa is atmospheric pressure, cohesion c, internal friction angle φ, and Rf, K, Kb, m are parameters.

The effect of confining pressure and soil with a density of 1.5 g/cm3 at a calcium carbonate content of 5% is selected for nonlinear analysis. According to the stress-strain in the shear test, the parameters are shown in Tables 4 and 5.

Table 4 Ei and Rf parameters of the model.
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Table 5 Model parameters at calcium carbonate content of 5%.
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The parameters in Tables 4 and 5 are substituted into the model to obtain the fitting and experimental results, as shown in Fig. 4. The data of the fitting curves at the peak point fit well, but the initial stress-strain performance is not completely consistent with the actual situation. The model with small confining pressure (100 kPa) has a good fitting effect. The fitting effect is poor with increasing confining pressure (Fig. 4). Table 4 shows that the damage ratio decreases with increasing confining pressure. When the confinement pressure is low, the damage ratio is close to 1. Therefore, the fitting accuracy fluctuates.

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
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The model fitting under different confining pressures are compared with the experiments.

The fitting curve in Fig. 4 is compared with the measured stress strain, which influence of confining pressures on the deformation characteristics is not well-reflected. Therefore, it needs to make certain corrections to the model parameters to achieve the purpose of improving the fitting curve. By introducing the coefficient k to correct the model, the modified stress strain is as follows:

$$\:{{\upsigma\:}}_{1}-{{\upsigma\:}}_{3}=\frac{{{\upepsilon\:}}_{1}+\text{k}\cdot\:{{\upepsilon\:}}_{1}^{2}}{\text{a}+\text{b}\cdot\:({{\upepsilon\:}}_{1}+\text{k}\cdot\:{{\upepsilon\:}}_{1}^{2})}$$
(13)

Substituting equations (13) into (1) to (12) yields the following expression for the modified model:

$$\:\begin{array}{c}{E}_{t}=K{p}_{a}{\left(\frac{{\sigma\:}_{3}}{{p}_{e}}\right)}^{n}\times\:{\left[1-R,\frac{\left({\sigma\:}_{1}-{\sigma\:}_{3}\right)(1-\text{s}\text{i}\text{n}\phi\:)}{2c\cdot\:\text{c}\text{o}\text{s}\phi\:+2{\sigma\:}_{3}\cdot\:\text{s}\text{i}\text{n}\phi\:}\right]}^{\frac{3}{2}}\\\:\times\:\sqrt{\left[1-{R}_{f}\frac{\left({\sigma\:}_{1}-{\sigma\:}_{3}\right)(1-\text{s}\text{i}\text{n}\phi\:)}{2c\cdot\:\text{c}\text{o}\text{s}\phi\:+2{\sigma\:}_{3}\cdot\:\text{s}\text{i}\text{n}\phi\:}+\frac{4k\cdot\:\left({\sigma\:}_{1}-{\sigma\:}_{3}\right)}{K{p}_{a}}\times\:{\left(\frac{{\sigma\:}_{3}}{{p}_{n}}\right)}^{(-m)}\right]}\end{array}$$
(14)

The coefficient k determines the fitting effect of the \(({{\upsigma}}_{1}-{{\upsigma}}_{3})\sim{{\upepsilon}}_{\text{a}}\) curve. A reasonable value is selected by analyzing the coefficient k. When the values of parameters a and b remain unchanged, adding coefficient k improves the characteristics and corrects the peak stress to a certain extent (Fig. 5). The value of k is larger the fitted curve is closer to the ideal plasticity. The fitted curve reflects a certain hardness when the k value decreases. The selection of k values for different confining pressures is shown in Table 6.

Fig. 5
Fig. 5The alternative text for this image may have been generated using AI.
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The effect of coefficient k on the fitting effect.

Table 6 Corrects the value of the model coefficient k.
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The model parameters in Tables 5 and 6 are substituted into the correction model formula (14). The comparison results between the before and after correction and the actual test are obtained, as shown in Fig. 6. Although the Duncan-Zhang reflects the stress strain under different confining pressures, the prediction effect on the peak of the principal stress difference is poor and the actual results are different. The coefficient k is introduced to correct it the modified model is in good agreement with the actual results under different confining pressures. Under the action of small confining pressure, the modified soil exhibits weak brittle deformation characteristics. The strain reaches the turning point of the peak the stress is slightly reduced. The Duncan-Zhang has a relatively poor fitting effect for this deformation characteristic. The corrected model has better improvement. The confinement pressure is 200 or 300 kPa the modified soil shows obvious stress-strain hardening characteristics. Although the Duncan-Zhang fits it well, the fitting degree of the modified model is further improved. Figure 6(d) shows the fitting relationship between confining pressure and correction coefficient k. The correction coefficient k at different confining pressures is obtained by fitting the curve, making the corrected model more applicable.

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
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Comparative analysis plot of model fit and measured results: (a) 100 kPa confining pressure, (b) 200 kPa confining pressure, (c) 300 kPa confining pressure and (d) the coefficient k and confining pressure fitting curve.

Conclusions

In the study, the liquid-plastic limit, water loss shrinkage and microscopic pore change were performed under different calcium carbonate contents. The consolidated drained triaxial shear test with 5% calcium carbonate content of the modified soil to obtain the stress-strain describing the deformation characteristics. The shortcomings of the Duncan-Zhang in describing the deformation of its stress-strain, a correction model with a correction coefficient k is introduced. The main conclusions are as follows:

The addition of calcium carbonate is beneficial for reducing the liquid-plastic limit and water loss shrinkage properties. The content of calcium carbonate exceeding 5% has little effect on the liquid-plastic limit. With the content increase the water loss shrinkage is greatly inhibited. Calcium carbonate content has a great influence on the pore distribution in modified soil. If the content exceeds 10%, the variations of large and small pores opposite phenomenon occur. According to the change of liquid-plastic limit, water loss shrinkage performance test and economic applicability, the calcium carbonate content of 5% is optimal. The Duncan-Zhang has a poor fitting in the peak load of the principal stress difference of modified soil, it needs to be corrected. A better fitting effect is achieved by introducing a correction coefficient related to the confining pressure. Although the modified Duncan-Zhang has a good prediction effect on modified soils, further research on density sensitivity and durability is needed to make the model more widely applicable.