Constitutive model of lime-stabilized laterite considering curing time and softening characteristics

Abstract

This study presents a comprehensive investigation into the mechanical properties of lime-stabilized lateritic soil, with a focus on developing an improved constitutive model that incorporates both curing time and strain-softening effects. Current constitutive models fail to accurately capture the stress–strain behavior of lime-stabilized soils, particularly over extended curing periods. To address this, unconfined compressive strength (UCS) tests were conducted using lime contents of 0%, 1%, 3%, 5%, 7%, 9%, and 11% revealing that 7% lime content optimally enhances the compressive strength of the soil by 1202.66% compared to untreated soil. Triaxial consolidated-drained tests were then performed with the optimal 7% lime content, considering curing times of 3, 7, 14, and 28 days under confining pressures of 100 kPa, 200 kPa, 300 kPa, and 400 kPa. The results demonstrated that the shear strength, cohesion, internal friction angle, and initial tangent modulus of lime-stabilized lateritic soil increased with longer curing times and higher confining pressures. These findings were integrated into a re-modified Duncan-Chang model, which incorporates both strain softening and curing time as key factors. The revised model was validated through comparisons with experimental data, achieving an average relative error of 2.12% at 7 days, 1.46% at 14 days, and 17.55% at 28 days. This validation demonstrates the model's ability to accurately predict the stress–strain behavior of lime-stabilized lateritic soil under different curing conditions. The novelty of this research lies in the successful integration of curing time and strain-softening effects into the Duncan-Chang model, providing a more accurate tool for predicting the long-term mechanical performance of stabilized soils. The findings have significant implications for engineering applications, particularly in the context of soil stabilization for infrastructure projects in tropical and subtropical regions.

Introduction

Laterite is a widely distributed soil characterized by unique engineering properties. In China, it is predominantly found in southern provinces such as Yunnan, Guangxi, Guizhou, Fujian, Hunan, and Jiangxi. Renowned for its relatively high natural strength, laterite is often regarded as an ideal material for subgrade construction in road and hydraulic engineering projects (Rashid et al. 2014; Razali et al. 2023). However, the climatic conditions in Yunnan, marked by distinct wet-dry cycles and concentrated rainfall, pose significant challenges. These conditions intensify the deterioration of laterite-based infrastructure, leading to subgrade swelling and shrinkage deformations, strength degradation, and seepage issues in dam structures (Fan et al. 2018).

To address the aforementioned engineering challenges, researchers have developed a range of soil stabilization techniques, including physical, chemical, and microbial methods (Qu et al. 2021; Moslemi et al. 2022; Ji et al. 2024). Among these, chemical and physical stabilization methods are widely adopted due to their effectiveness in significantly enhancing soil strength. Commonly used materials, such as cement, lime, fly ash, and fibers, have been extensively applied to improve the engineering properties of soils (Poltue et al. 2020; Wahab et al. 2021; Yanou et al. 2021; Tamassoki et al. 2022). Lime stabilization, in particular, stands out as one of the most widely used chemical methods due to its high efficiency and broad applicability. This technique enhances soil strength through cation exchange and pozzolanic reactions, which generate cementitious compounds that significantly improve both compressive and shear strength (Consoli et al. 2016; Bouras et al. 2021; Kan and François 2023; Shi et al. 2024). Additionally, compared to traditional cement stabilization methods, lime stabilization is associated with substantially lower carbon emissions, making it a more sustainable solution aligned with low-carbon development goals (Li et al. 2022; Razali et al. 2023; Abdelbaset et al. 2024).

Lime stabilization techniques have made significant advancements in enhancing the mechanical properties of soils. However, a critical challenge for researchers lies in systematizing and theorizing the patterns observed in macroscopic mechanical experiments, as well as developing constitutive models capable of accurately describing and predicting the mechanical behavior of stabilized soils (Bahmed et al. 2024). Among the various constitutive models available for soils—such as the Modified Cam-Clay model, the Duncan-Chang model, and the Mohr–Coulomb model—the Duncan-Chang model is particularly favored in geotechnical engineering due to its computational simplicity, clear physical interpretation of parameters, and ease of parameter determination (Xiong et al. 2012; Zhou et al. 2018; Liu et al. 2019). The Duncan-Chang model (hereafter referred to as the traditional Duncan-Chang model) is an incremental elastic model proposed by Duncan based on Konder's foundational work (Konder and Zelasko 1963; Duncan James and Chang 1970). This model is commonly applied to describe and predict the stress–strain behavior of soils under conventional drained conditions. Nevertheless, its application to stabilized soils is limited by two significant shortcomings: (1) Inability to capture strain-softening behavior. The model assumes that soil continues to exhibit a hardening trend after reaching peak strength. However, most stabilized soils display strain-softening behavior beyond peak strength, where stress gradually decreases as strain increases. (2) Neglect of curing time as a critical factor. Curing time plays a pivotal role in the strength development of stabilized soils. The traditional Duncan-Chang model fails to incorporate curing time as a variable, which undermines its accuracy in simulating the long-term mechanical behavior of stabilized soils. Extensive studies have demonstrated that curing time significantly influences the strength and deformation characteristics of various chemically stabilized soils. For instance, curing time directly affects hydration and pozzolanic reactions in cement-stabilized clays and silts (Liu et al. 2023; Zhou et al. 2024), as well as in lime-treated loess and saprolitic soils (Jia et al. 2019; Ma et al. 2024). These findings highlight that curing time is not only essential for strength development but also critical for accurately predicting the long-term mechanical behavior of stabilized soils.

To address the aforementioned challenges, researchers have developed various modified Duncan-Chang models in recent years (Tang et al. 2022; Jiang et al. 2023; Yang et al. 2023; Xie et al. 2024). Among these, He et al. (2024) proposed a modified Duncan-Chang model incorporating curing time to predict the stress–strain behavior of lime-modified dispersive soils. While this model accurately predicts the stress–strain behavior before peak strength during long-term curing beyond 28 days, it fails to describe the complete stress–strain curve, particularly the strain-softening behavior of stabilized soils. Lai et al. (2007) introduced another modified Duncan-Chang model designed to account for the strain-softening behavior of stabilized soils. This model, applied to nickel slag-clay-cement modified soils, demonstrated good agreement between experimental and simulated stress–strain curves (Yin et al. 2022). However, despite addressing strain-softening, it overlooks the influence of curing time on strength development and the associated strain-softening effects. Weng et al. (2023) further modified the traditional Duncan-Chang model to describe the stress–strain behavior of xanthan gum-stabilized red clay. However, this model primarily focuses on mechanical behavior under varying loading conditions within a single curing period, without incorporating curing time as a dynamic variable. Additionally, it does not reflect the strain-softening characteristics of stabilized soils. Chemical stabilizers, such as lime, exhibit strong sensitivity to curing time. With prolonged curing, lime-stabilized soils increasingly display brittle behavior, with stress–strain curves prominently characterized by strain-softening.

In summary, existing modified Duncan-Chang models focus primarily on curing time or strain-softening effects but lack a systematic investigation of their coupled influence. This limitation restricts their ability to accurately predict the long-term mechanical behavior of lime-stabilized soils. To tackle this issue, this study introduces a re-modified Duncan-Chang constitutive model. The model incorporates curing time as a dynamic variable and integrates strain-softening effects, emphasizing the coupled influence of these factors. This innovation aims to provide a comprehensive prediction of the stress–strain behavior of lime-stabilized soils over extended curing periods. The development of this model not only enables more accurate predictions of the long-term mechanical behavior of lateritic soils but also serves as a reference for the study of other chemically stabilized soils. To validate the model, unconfined compressive strength (UCS) tests and triaxial consolidated-drained tests were performed. The experiments investigated the mechanical properties of lime-stabilized laterite with varying lime contents (0%, 1%, 3%, 5%, 7%, and 9%) and curing times (3 days, 7 days, 14 days, and 28 days). The experimental results demonstrate the significant impact of curing time and confining pressure on soil strength and deformation characteristics. These findings not only enhance the understanding of lime-stabilized soils but also provide critical data to refine the proposed constitutive model further.

Materials and methodologies

Soil and lime properties

The lateritic soil used in this study was sourced from Kunming, Yunnan, China, as shown in Fig. 1, with a sampling depth of 2 m below the surface. The basic physical properties of the lateritic soil are presented in Table 1. The grain size distribution of the lateritic soil, as shown in Fig. 2, was determined using dry sieving and the hydrometer method. The results indicate that the soil consists of 81.7% sand, 11.5% silt, and 6.8% clay. The lime used in the experiments is also shown in Fig. 1, with its primary chemical components detailed in Table 2.

Fig. 1 [Images not available. See PDF.]

Geographic location and test materials

Table 1. Basic physical properties of laterite

Natural moisture content (%)	Optimum moisture content (%)	Maximum dry density (g⋅cm−3)	Liquid limit (%)	Plastic limit (%)	Specific gravity
45.5	38.5	1.32	65.12	42.39	2.84

Fig. 2 [Images not available. See PDF.]

Grain size distribution curve of the lateritic soil

Table 2. Chemical composition of lime

Name of ingredient	CaO	SiO2	Al2O3	Fe2O3	MgO	SO3
Content (%)	86.72	0.90	0.37	0.19	1.87	0.16

Sample preparations and test scheme

1. Representative lateritic soil was selected, air-dried, crushed, and sieved through a 2 mm standard sieve (Liang et al. 2023). According to the experimental design shown in Table 3, the required quantities of water, lime, and air-dried lateritic soil were calculated and thoroughly mixed. The mixture was then cured for 12 h under standard conditions (temperature 20 ± 2℃, relative humidity 95%) to ensure uniform moisture distribution. Cylindrical specimens were subsequently prepared with a dry density of 1.31 (g⋅cm−3), a diameter of 39.1 mm, and a height of 80 mm. The specimens were placed in a humidity chamber and cured under standard conditions for 3, 7, 14, and 28 days prior to testing.

2. To determine the optimal lime, unconfined compressive strength tests were conducted on lime-treated soil with varying lime contents using an electronic universal material testing machine from Shenzhen Suns Technology Stock Co., Ltd. with a loading rate controlled at 1 mm/min, as shown in Fig. 3(a).

   Fig. 3 [Images not available. See PDF.]

   Testing instruments: a Electronic universal material testing machine; b Automatic triaxial apparatus

3. To study the effects of different curing durations and confining pressures on the strength of lime-stabilized lateritic soil, lime content was fixed at 7%, and samples were cured for 3, 7, 14, and 28 days before conducting triaxial consolidation drained tests. The tests used an automatic triaxial apparatus from Nanjing TKA Technology Co., Ltd., as depicted in Fig. 3(b). The experimental design is provided in Table 3. Each sample set was consolidated under confining pressures of 100 kPa, 200 kPa, 300 kPa, and 400 kPa for 24 h, followed by drainage shearing at a rate of 0.008 mm/min.

Table 3. Experimental design options

Number	Lime dosing (%)	Curing time (d)	Name of the test
1 ~ 7	0, 1, 3, 5, 7, 9, 11	3	Unconfined compressive strength
8 ~ 11	7	3	Consolidated drained triaxial test
12 ~ 15		7
16 ~ 19		14
20 ~ 23		28

Test results and discussion

Unconfined compressive strength

As shown in Fig. 4, the unconfined compress strength of soil with varying lime content exhibits an initial increase followed by decrease. The peak strength is reached at a 7% lime content, showing a 1202.66% improvement over the unconfined compressive of the native soil. Beyond this percentage, a decreasing trend in strength is observed. Lime, lacking cohesion and possessing lubricating properties, reduces the strength when overly abundant (Kumar et al. 2007).

Fig. 4 [Images not available. See PDF.]

Relationship between unconfined compressive strength and lime dosage

When the lime content is insufficient, the strength requirements for practical engineering applications cannot be met; conversely, while excessive lime content can meet the strength criteria, is economically undesirable due to its high proportion. Therefore, the optimal lime content for stabilizing laterite is determined to be 7%.

Consolidated drained triaxial test

Stress–strain relationship considering curing time

Consolidated drained triaxial tests were conducted on lime-stabilized laterite with a 7% lime content, with curing times of 3, 7, 14, and 28 days. The resulting deviatoric stress-axial strain curves are presented in Fig. 5.

Fig. 5 [Images not available. See PDF.]

Stress–strain curves of lime-cured lateritic soils under different conservation times and different confining pressures: a 3d, b 7d, c 14d, d 28d

As shown in Fig. 5, under the combined effects of confining pressure and curing time, the stress–strain curves of lime-stabilized laterite exhibit various forms. The deviatoric stress of the lime-stabilized laterite increases with both confining pressure and curing time. Moreover, as the curing time increases, the stress–strain behavior of the lime-stabilized laterite gradually transitions from strain hardening to strain softening.

Internal friction angle and cohesion

Based on the Mohr–Coulomb criterion the triaxial test data were analyzed, and the Mohr's stress circles along with the strength envelope are illustrated in Fig. 6. As the test was a consolidated drained test, the pore water pressure was zero, resulting in the determination of the soil's effective cohesion ($c_d$) and effective internal friction angle (${\phi }_d$).

Fig. 6 [Images not available. See PDF.]

Mohr–Coulomb strength envelope of lime-stabilized laterite at different curing times: a 3d, b 7d, c 14d, d 28d

Figure 7 illustrates that as the curing time increases, both the effective cohesion and effective internal friction angle of lime-stabilized laterite linearly increase. Moreover, there is a linear relationship between these properties and curing time, as described by the fitting functions in Eqs. (1) and (2), with R² values of 0.999 and 0.982, respectively.

Fig. 7 [Images not available. See PDF.]

Effective cohesion and effective angle of internal friction of lime-cured laterite at different curing times

1

$$c_d=88.942+2.698t$$

2

$${\phi }_d=31.073+0.018t$$

The original Duncan-Chang model

Establishment of the model

As indicated in Fig. 5, the stress–strain curve of lime-stabilized lateritic soil approximates a hyperbola. This relationship can be described using the original Duncan-Chang model. Therefore, based on this model and stress–strain data obtained from consolidated drained triaxial test, the parameters of the original Duncan-Chang model for lime-stabilized lateritic soil were determined. The derivation process of the model parameters is outlined below:

According to the original Duncan-Chang model, the stress–strain relationship of the soil during shearing can be expressed as Eq. (3).

3

$${\sigma }_1-{\sigma }_3=\frac{{\epsilon }_1}{a+b_{\epsilon 1}}$$

In Eq. (3): ${\sigma }_1$ represents axial stress; ${\mathrm{\sigma }}_3$ represents confining pressure; ${\epsilon }_1$ is axial strain; $a$, $b$ are the test constants. Equation (3) can be transformed into Eq. (4).

4

$$\frac{{\epsilon }_1}{{\sigma }_1-{\sigma }_3}=a+b_{\epsilon 1}$$

The triaxial test results are organized according to Eq. (4) and plotted in the $\frac{{\epsilon }_1}{{\sigma }_1-{\sigma }_3}-{\epsilon }_1$ coordinate system. The experimental curve is approximately linear, with 'a' representing the y-intercept and 'b' the slope of the line.

Taking the derivative of Eq. (3) with respect to ${\epsilon }_1$ yields Eq. (5); setting ${\epsilon }_1\to \infty$ in Eq. (3) gives Eq. (6).

5

$$E_i={[\frac{\mathrm{d}({\sigma }_1-{\sigma }_3)}{d_{\epsilon 1}}]}_{{\epsilon }_1\to 0}=\frac{1}{a}$$

6

$${({\sigma }_1-{\sigma }_3)}_{\mathit{ult}}={({\sigma }_1-{\sigma }_3)}_{{\epsilon }_1\to \infty }=\frac{1}{b}$$

In the equation: $E_i$ represents the initial tangent modulus; ${({\sigma }_1-{\sigma }_3)}_{\mathit{ult}}$ stands for ultimate shear stress; $a$ denotes the reciprocal of the initial tangent modulus $E_i$; $b$ represents the reciprocal of the ultimate shear stress corresponding to the asymptote of the stress–strain curve.

Based on the above, the initial secant modulus of the samples under different curing times and confining pressures is obtained as shown in Table 4.

Table 4. Initial tangent modulus $E_i$

Curing time (d)	$E_i$(kPa)
	100 kPa	200 kPa	300 kPa	400 kPa
3	108.00	236.00	582.00	668.00
7	436.00	520.00	696.00	796.00
14	472.00	628.00	742.00	825.33
28	700.00	790.67	964.00	1060.00

When obtaining parameter $b$, it's impossible to make ${\epsilon }_1$ infinitely large in the experiment to calculate ${({\sigma }_1-{\sigma }_3)}_{\mathit{ult}}$. Thus, it's not feasible to directly obtain b from the experiment (Luo et al. 2010). When stress–strain curves have no peak points, ${({\sigma }_1-{\sigma }_3)}_{\mathit{ult}}$ is often taken as ${({\sigma }_1-{\sigma }_3)}_{{\epsilon }_1=15\%}$, where ${({\sigma }_1-{\sigma }_3)}_{{\epsilon }_1=15\%}$ represents the deviatoric stress corresponding to an axial strain of 15%. For cases with peak points, ${({\sigma }_1-{\sigma }_3)}_{\mathit{ult}}$ is taken as ${({\sigma }_1-{\sigma }_3)}_f$, where ${({\sigma }_1-{\sigma }_3)}_f<{({\sigma }_1-{\sigma }_3)}_{\mathit{ult}}$. Therefore, the failure ratiof is defined as (Li 2016):

7

$$R_f=\frac{{\left({\sigma }_1,-,{\sigma }_3\right)}_f}{{\left({\sigma }_1,-,{\sigma }_3\right)}_{\mathit{ult}}}=\frac{{\left({\sigma }_1,-,{\sigma }_3\right)}_f}{{\displaystyle \frac{1}{b}}}$$

In the formula:${({\sigma }_1-{\sigma }_3)}_f$ represents the peak stress of the stress–strain curve. According to the Mohr–Coulomb strength criterion, the peak strength can be expressed by Eq. (8).

8

$${\left({\sigma }_1,-,{\sigma }_3\right)}_f=\frac{{2c}_d{cos\phi }_d+{2\sigma }_3{sin\phi }_d}{1-{sin\phi }_d}$$

In the formula: $c_d$ denotes cohesion; ${\phi }_d$ represents the angle of internal friction; substituting Eq. (8) into Eq. (7) and simplifying yields Eq. (9).

9

$$b=\frac{R_f(1-sin{\phi }_d)}{2c_{d}cos{\phi }_d+2{\sigma }_{3}sin{\phi }_d}$$

In 1963, Janbu discovered that the initial modulus of soil, $\left(E_i\right)$, in triaxial tests is related to the confining pressure, as represented by Eq. (10).

10

$$E_i={\mathit{KP}}_a{\left(\frac{{\sigma }_3}{P_a}\right)}^n$$

In the formula: $K$ and $n$ are experimental constants, $P_a$ represents atmospheric pressure, $P_a$=101 kPa. Taking logarithm of both sides of Eq. (10) and rearranging gives Eq. (11).

11

$$lg\left(\frac{E_i}{P_a}\right)=lgK+nlg\left(\frac{{\sigma }_3}{P_a}\right)$$

At this stage, using $\lg \left(\frac{E_i}{P_a}\right)$ as independent variable and $\lg \left(\frac{{\sigma }_3}{P_a}\right)$ as the dependent variable, Eq. (11) is further rearranged into the form of a linear function, resulting in Eq. (12).

12

$$y=a+nx$$

Through multiple experiments under different confining pressures, the intercept $a=logK$ and the slope $n$ of the linear equation are determined using linear fitting. Consequently, the parameters $K$ and $n$ are obtained. Additionally, substituting Eq. (10) into Eq. (5) yields Eq. (13).

13

$$a=\frac{1}{{\mathit{KP}}_a{\left(\frac{{\sigma }_3}{P_a}\right)}^n}$$

By substituting Eqs. (9) and (13) into Eq. (3), Eq. (14) is derived, representing the expression for the stress–strain curve in a consolidated drained triaxial test.

14

$$({\sigma }_1-{\sigma }_3)=\frac{{\epsilon }_1}{{\mathit{KP}}_a{\left(\frac{{\sigma }_3}{P_a}\right)}^n+\frac{R_f(1-\sin {\phi }_d)}{2c_d\cos {\phi }_d+2{\sigma }_3\sin {\phi }_d}{\epsilon }_1}$$

In summary, employing the aforementioned method yields the$a$,$b$,$R_f$, $K$ and $n$ parameters of the traditional Duncan-Chang model, as depicted in Table 5. When determining parameters a and b, it is common for experimental points at both low and high stress levels to deviate from the straight line, leading to inaccurate fitting results. Therefore, a correction is necessary for parameter a, with the correction coefficient also listed in Table 5.

Table 5. Parameters of the original Duncan-Chang model $a$, $b$,$R_f$, $K$ and $n$

Curing time (d)	Confining pressure(kPa)	$a$	$b$	$R_f$	Correction coefficient	$K$	$n$
3	100	0.00959	0.00173	0.97	0.07	1.046	1.394
	200	0.00365	0.00123	0.95	0.11
	300	0.00207	0.00096	0.95	0.14
	400	0.00139	0.00079	0.95	0.36
7	100	0.000550	0.00164	0.98	0.23	4.159	0.443
	200	0.000450	0.00118	0.96	0.26
	300	0.000400	0.00094	0.96	0.27
	400	0.000360	0.00075	0.93	0.28
14	100	0.00030	0.00148	0.97	0.142	4.699	0.405
	200	0.00025	0.00106	0.94	0.156
	300	0.0002	0.00085	0.95	0.148
	400	0.00015	0.00070	0.95	0.124
28	100	0.00080	0.001188	0.98	0.545	6.761	0.305
	200	0.00030	0.000961	0.98	0.252
	300	0.00025	0.000799	0.97	0.238
	400	0.00020	0.000660	0.93	0.208

Duncan-Chang model parameters ${\mathit{E}}_{\mathit{i}}$, $\mathit{K}$ and $\mathit{n}$ considering curing time

From Table 5, it is observed that $K$ and $n$ exhibit nonlinear relationship with curing time ($t$), as depicted by fitting curve in Fig. 8. The fitting functions are shown in Eqs. (15) and (16).

Fig. 8 [Images not available. See PDF.]

Relationship between parameters $K$,$n$ and curing time: a parameter K, b parameter n

15

$$K=-8.01\exp \left(\frac{-t}{7.61}\right)+6.68$$

16

$$n=6.44\exp \left(\frac{-t}{1.65}\right)+0.35$$

Inserting Eqs. (15) and (16) into Eq. (10) results in Eq. (17), indicating that the initial tangent modulus ($E_i$) of the soil is a function of curing time ($t$) and confining pressure.

17

$$E_i=[-8.01\exp \left(\frac{-t}{7.61}\right)+6.68]P_a{\left(\frac{\sigma 3}{\mathit{Pa}}\right)}^{6.44\exp (\frac{-t}{1.65})+0.35}$$

Additionally, using the experimental data from Table 4, nonlinear surface fitting was performed on the data using OriginPro 2021. An empirical formula relating the initial tangent modulus ($E_i$), curing time ($t$), and confining pressure (${\sigma }_3$) was established. The fitted surface is shown in Fig. 9, and the function relationship is expressed in Eq. (18), with an R² of 0.913. Equation (18) is more concise than Eq. (17) in its expression form.

Fig. 9 [Images not available. See PDF.]

Fitted surface for the parameter $E_i$

18

$$E_i=84.962+19.177t+1.513{\sigma }_3-0.187t^2-2.676\mathrm{E}-4{\sigma }_3^2$$

To verify the accuracy of the simulation results of Eqs. (17) and (18), a comparison between the simulated and actual was conducted, as shown in Fig. 10. Figure 10(a) reveals that the simulation of Eq. (17) exhibit non-linear characteristics, while Eq. (18) demonstrates linear characteristics as shown in Fig. 10(b). Both methods simulate experimental values well. However, limitations include: at a curing time of 3 days and confining pressure of 400 kPa, the results of Eq. (17) are slightly higher than the actual measurements; at confining pressures of 100 and 200 kPa, the results of Eq. (18) are also slightly above the experimental values. Therefore, it is recommended to combine both methods for more accurate simulation results.

Fig. 10 [Images not available. See PDF.]

Comparison of simulated and measured values of the two $E_i$ calculation methods: a Eq. (17), b Eq. (18)

Verification of the original Duncan-Chang model

By incorporating the parameters a and b from Table 5 into Eq. (3), the stress–strain curves for lime-stabilized red soil under various curing durations and confining pressures were simulated using spreadsheet software. The simulated values were then compared with the actual results from consolidated drained triaxial tests to verify the accuracy of the simulations, as shown in Fig. 11.

Fig. 11 [Images not available. See PDF.]

Simulation and experimental results of specimens under different curing times: a 3d, b 7d, c 14d, d 28d

Relative error analysis was conducted on the simulated values mentioned above, with results detailed in Table 6. It's observed from Table 6 that at a curing period of 3 days, the relative mean error of deviatoric stress under various confining pressures is minimal, indicating more accurate simulation results. However, as the curing time increases to 7, 14, and 28 days, the relative mean error of deviatoric stress under each confining pressure gradually increases, with a maximum increase of 1370.18%. This discrepancy between simulation and actual measurements suggests an inability to accurately represent the stress–strain relationship of lime-stabilized laterite. The findings of the relative error analysis once again confirm that the original Duncan-Chang model cannot accurately describe the stress–strain relationship of lime-stabilized laterite considering curing time.

Table 6. Mean relative error analysis of simulated values

Curing time (d)	Average relative error (%)
	100 kPa	200 kPa	300 kPa	400 kPa
3	3.32	4.59	4.23	2.60
7	21.71	10.15	8.39	5.17
14	16.65	16.55	15.32	18.96
28	48.81	37.11	39.80	20.88

Modified Duncan-Chang model considering strain softening

Analysis reveals that the original Duncan-Chang model effectively predicts the stress–strain curves of lime-stabilized laterite for ages up to 3 days with high accuracy between simulation and experimental values. However, with increasing curing time, reactions such as cation exchange, pozzolanic reactions, carbonation, and flocculation occur within the stabilized soil (Consoli et al. 2011; Kamaruddin et al. 2020), producing compounds that enhance the mechanical properties of the soil (Bell 1996; Okyay and Dias 2010; Jha and Sivapullaiah 2015). Consequently, during triaxial testing, the stress–strain curves of lime-stabilized laterite exhibit strain-softening behavior as curing time extends, leading to increased discrepancies between simulated and experimental values and reduced model precision.

To address the strain-softening characteristics of stabilized soil, Lai et al. (2007) developed a modified Duncan-Chang model based on the original Duncan-Chang model, capable of describing soil strain-softening. The expressions for this model are provided in Eqs. (19), (20), (21), and (22). The modified Duncan-Chang model introduces an adjustment term $l{{\epsilon }_1}^2$. Compared to the original model, the additional term in the denominator changes the shape of the curve, especially under large strain conditions. As the denominator increases rapidly, the principal stress difference decreases, which leads to the manifestation of strain-softening behavior.

19

$${\sigma }_1-{\sigma }_3=\frac{{\epsilon }_1}{m+n{\epsilon }_1+l{\epsilon }_1^2}$$

20

$$m=\frac{1}{E_i}$$

21

$$n=\frac{1}{{({\sigma }_1-{\sigma }_3)}_f}-\frac{2}{Ei{\epsilon }_f}$$

22

$$l=\frac{1}{Ei{\epsilon }_f^2}$$

In the formula: $m$, $n$, and $l$ are model parameters; ${\epsilon }_f$ represents the axial strain corresponding to ${({\sigma }_1-{\sigma }_3)}_f$.

The re-modified Duncan-Chang model

Although the modified Duncan-Chang model mentioned above can describe the strain softening phenomenon of stabilized soil, unfortunately, the curing time is not considered as an independent variable in the model. Therefore, the modified model does not reflect the stress–strain relationship of stabilized soil under different curing times.

The experiments revealed that the modified Duncan-Chang model in Eq. (19) does not adequately describe the stress–strain behavior of lime-stabilized soil with curing periods of 7, 14, and 28 days. To more accurately capture the stress–strain relationship of lime-stabilized lateritic soil at curing periods of 7, 14, and 28 days, this study introduces an initial strength correction term ${i{(\sigma }_1-{\sigma }_3)}_f$ to Eq. (19), as shown in Eq. (23). This revised version is referred to as the re-modified Duncan-Chang model.

23

$${\sigma }_1-{\sigma }_3=\frac{{\epsilon }_1}{m^{\prime }+n^{\prime }{\epsilon }_1+l^{\prime }{\epsilon }_1^2}+i{\left({\sigma }_1,-,{\sigma }_3\right)}_f$$

In the formula, $m^{\prime }$, $n^{\prime }$, $l^{\prime }$, and i are experimental parameters, with values for i as shown in Table 7. Substituting Eq. (8) into Eq. (23) yields Eq. (24).

24

$${\sigma }_1-{\sigma }_3=\frac{{\epsilon }_1}{m_{}^{_{}^{\prime }}+n_{}^{_{}^{\prime }}{\epsilon }_1+l_{}^{_{}^{\prime }}{\epsilon }_1^2}+i\frac{2c_{d}cos\phi +2{\sigma }_{3}sin{\phi }_d}{1-sin{\phi }_d}$$

Table 7. Parameters of the re-modified Duncan-Chang model i

Curing time (d)	i
	100 kPa	200 kPa	300 kPa	400 kPa
7	0.28	0.30	0.32	0.40
14	0.21	0.22	0.23	0.24
28	0.30	0.36	0.40	0.48

The text mentions that the effective cohesion and internal angle of lime-stabilized laterite change with curing time., substituting Eqs. (1) and (2) into Eq. (24) results in Eq. (25).

25

$${\sigma }_1-{\sigma }_3=\frac{{\epsilon }_1}{m^{\prime }+n^{\prime }{\epsilon }_1+l^{\prime }{\epsilon }_1^2}+i\frac{(177.884+5.396t)\cos (31.073+0.018t)+2{\sigma }_{3}sin(31.073+0.018t)}{1-\sin (31.073+0.018t)}$$

The research indicates that parameters $m^{\prime }$, $n^{\prime }$ and $l^{\prime }$ are simultaneously affected by curing time and conf pressure. Furthermore, as previously mentioned, the initial tangent modulus $\left(E_i\right)$ should be calculated using Eqs. (17) and (18). Here, using Eq. (18) an example, substituting it into Eq. (20) yields Eq. (26).

26

$$m^{\prime }=\frac{1}{84.962+19.177t+1.513{\sigma }_3-0.187t^2-2.676\mathrm{E}-4{\sigma }_3^2}$$

To better describe the strain-softening characteristics of lime-stabilized red soil, the strain-softening parameter F is introduced. Equation (21) is modified to Eq. (27).

27

$$n^{\prime }=\frac{1}{{({\sigma }_1-{\sigma }_3)}_f}-\frac{F(t,{\sigma }_3)}{{Ei\epsilon }_f}$$

In Eq. (27), ${F(t,\sigma }_3)$ represents the strain-softening parameter for lime-stabilized red soil, where a higher ${F(t,\sigma }_3)$ value indicates a more significant softening effect. Equation (22) remains unchanged. By setting $l^{\prime }=l$, we obtain Eq. (28).

28

$$l^{\prime }=l=\frac{1}{Ei{\epsilon }_f^2}$$

In the equation, $l^{\prime }$ represents the residual stress parameter. The smaller the value of $l^{\prime }$, the higher the residual stress.

The study revealed that parameters ${F(t,\sigma }_3$) and $l^{\prime }(t,{\sigma }_3)$ are influenced by curing time and confining pressure. Non-linear surface fitting of the triaxial test data was conducted using OriginPro 2021. Empirical formulas for parameters ${F(t,\sigma }_3)$ and $l^{\prime }(t,{\sigma }_3)$, considering the combined effects of curing time and confining pressure, were established. The fitted surfaces are shown in Fig. 12, with the fitting functions presented in Eqs. (29) and (30), yielding R² values of 0.952 and 0.812, respectively. Additionally, from Figs. 12(a) and (b), it can be observed that as the curing time and confining pressure increase, the value of ${F(t,\sigma }_3)$ gradually increases, while the value of $l^{\prime }(t,{\sigma }_3)$ decreases. This indicates that the softening effect of lime-stabilized laterite becomes more pronounced with extended curing time and increased confining pressure, and the residual stress progressively increases. This trend is consistent with the stress–strain curve of lime-stabilized laterite shown in Fig. 5.

Fig. 12 [Images not available. See PDF.]

Plot of the fitted surface for the parameters: a${F(t,\sigma }_3)$, b$l^{\prime }(t,{\sigma }_3)$

29

$$F(t,{\sigma }_3)=<spanclass{=}^{\prime }convertEndas{h}^{\prime }>0.935-0</span>.00299t+8.951\mathrm{E}-4{\sigma }_3+5.946\mathrm{E}-4t^2+9.519\mathrm{E}-7{\sigma }_3^2+6.851\mathrm{E}-6t{\sigma }_3$$

30

$$l^{\prime }=4.252\mathrm{E}-5+8.373\mathrm{E}-4\exp (-\frac{t}{4.383E22}-\frac{\sigma 3}{125.937})$$

Validation of the re-modified Duncan-Chang model

To test the accuracy of the re-modified Duncan-Chang model, parameters $m^{\prime }$, $n^{\prime }$ and $l^{\prime }$ were first determined for curing times of 7, 14, and 28 days using Eqs. (26), (27), and (29). These parameters were then used in Eq. (23) to simulate the stress–strain curves of lime-stabilized lateritic soil at different curing ages. Finally, the simulated curves from the re-modified Duncan-Chang model were compared with the experimental curves, as illustrated in Fig. 13.

Fig. 13 [Images not available. See PDF.]

Comparison of model predictions with experimental measurements: a 7 d, b 14 d, c 28 d

From Fig. 13, it is evident that the re-modified Duncan-Chang model, which considers curing time and strain-softening effects, can accurately describe the stress–strain relationship of lime-stabilized lateritic soil compared to Fig. 11. This indicates that using the re-modified Duncan-Chang model to simulate the stress–strain relationship of lime-stabilized lateritic soil is reasonable.

Additionally, Table 8 presents the relative errors between the model predictions and experimental measurements shown in Fig. 13. The results from Table 8 indicate that the model's average relative errors under the combined effects of various curing time and confining pressures are within an acceptable range. This accuracy surpasses that of the original Duncan-Chang model, which neither considers curing time nor softening effects, in depicting the stress–strain relationship of lime-stabilized soil. This reaffirms the feasibility of using the re-modified Duncan-Chang model proposed in this paper, which accounts for both curing time and strain-softening effects, to describe the stress–strain relationship of lime-stabilized lateritic soil.

Table 8. Mean relative error of model prediction results

Curing time (d)	Average relative error (%)
	100 kPa	200 kPa	300 kPa	400 kPa
7	1.22	2.75	6.34	5.34
14	1.46	4.64	0.29	1.71
28	2.12	4.61	17.55	4.82

Conclusions

1. Unconfined compressive strength (UCS) tests show that 7% is the optimal lime content for stabilizing lateritic soil, increasing compressive strength by 1202.66% compared to untreated soil. However, further increases in lime content reduce strength due to the lubricating effect of excessive lime. This study provides clear guidance on optimizing lime content for large-scale lateritic soil stabilization, with significant practical value.

2. The mechanical properties of lime-stabilized lateritic soil significantly improve with curing time and confining pressure. Using equations for initial tangent modulus, cohesion, and internal friction angle, we revealed relationships between these properties and curing times (3, 7, 14, and 28 days) and confining pressures (100, 200, 300, and 400 kPa). The fitting accuracies for cohesion (R² = 0.999) and internal friction angle (R² = 0.982) confirm that longer curing significantly enhances mechanical performance. This finding offers practical design guidance, especially for projects where curing time is critical.

3. The re-modified Duncan-Chang model significantly improves stress–strain prediction by incorporating both strain-softening and curing time, better reflecting the behavior of lime-stabilized lateritic soil under different curing periods and stress conditions. At 100 kPa confining pressure, the model's average relative errors for 7, 14, and 28 days were 1.22%, 1.46%, and 2.12%, respectively, demonstrating its predictive accuracy.

4. By introducing both a strain-softening parameter and curing time as key factors, the re-modified model not only improves the prediction of post-peak softening behavior in lime-stabilized lateritic soil, but also accurately captures strength variations during long-term curing. This enhancement provides a more robust tool for future designs of lateritic soil reinforcement, effectively addressing the original model's shortcomings in representing strain-softening behavior and accurately predicting long-term performance.

5. The model's potential in engineering applications is particularly significant for soil stabilization projects in tropical and subtropical lateritic regions. By considering the optimal lime content (7%) and accounting for curing time and strain-softening effects, the model offers reliable performance predictions for lime-stabilized lateritic soil in projects such as road construction and foundation reinforcement.

Authors contribution

Guiyuan Xiao: Conceptualization, Methodology, Validation, Resources, Writing-review and editing, Funding acquisition. Xing Liu: Data curation, Writing-original draft, Formal analysis, Visualization. Dunhan Yang: Investigation, Data curation, Writing-review and editing. Yipeng Wang: Investigation, Supervision, Writing-review and editing. Ji Zhang: Investigation, Funding acquisition。All authors participated in the final review and approval of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (No.52169022) and the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities. (No.202101BA070001-278).

Data availability

The data supporting the findings of this study are available upon request.

Declarations

Competing interest

The authors declare that they have no relevant conflicts of interest that could influence the outcomes of this study.

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