https://orcid.org/0000-0003-2273-4489
Figures
Abstract
The subgrade resilient modulus is a crucial parameter in pavement structure design, directly related to the safety and economy of the pavement structure. Currently, the subgrade resilient modulus used in pavement structure design is the static resilient modulus. To address this issue, this paper employs dynamic triaxial tests to analyze the effects of confining stress (), cyclic dynamic stress (
), wet-dry cycles (N), and cement dosage (C) on the resilient modulus (MR). It explores the relationship among the MR, static resilient modulus, and 7-day unconfined compressive strength (
), and proposes a predictive model for MR considering C and N. SEM and XRD methods are used to analyze the mechanism of the effect of wet-dry cycles on MR at the microstructural level. The results show that; (1) The MR of the PG mixture increases with the rise of
and decreases with the increase of
, is positively correlated with C, and negatively correlated with the number of N, with the decay of MR mainly concentrated in the first four wet-dry cycles and gradually stabilizing afterwards, which is explained through SEM and XRD analysis. (2) By comparing MR and the static resilient modulus, it is found that the MR of PG specimens under wet-dry cycles is 1.25–1.30 times that of the static resilient modulus, this ratio can be used to make a preliminary estimation of the MR of C-stabilized PG materials. (3) The model relating
and MR follows a linear relationship, with correlation coefficients for the fit all exceeding 0.94, indicating a good fit. (4) A model for predicting MR that incorporates N and C was proposed, and the reliability of this model was verified by comparing predicted values with experimental data. The predicted MR provides a reliable basis for pavement design and holds significant engineering value.
Citation: Gan W, Chen K, Chen J, Chen Y, Deng S (2026) Resilient modulus characteristics and model prediction of cement phosphogypsum under dry-wet cycling conditions. PLoS One 21(9): e0341539. https://doi.org/10.1371/journal.pone.0341539
Editor: Pratheep K. Annamalai, University of Queensland, AUSTRALIA
Received: January 7, 2026; Accepted: August 11, 2026; Published: September 8, 2026
Copyright: © 2026 Gan et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the manuscript and its Supporting Information files.
Funding: This research was supported by the Guizhou Provincial Science and Technology Programme (Qian Ke He Ji Chu-ZK[2023] Key Project 016); the Guiyang Municipal Science and Technology Programme (Zhu Ke He Tong [2024]-1-12); the Guizhou Provincial Basic Research Programme (Natural Science) for Youth Guidance (No. QN[2025] 246); High-level Innovative Talents in Guizhou Province (Grant No. [2025]202304). All four of the above funders were involved in the study design, data collection and analysis, and decisions regarding publication or manuscript preparation, and provided important guidance on this study.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
PG is a solid waste derivative produced in the wet-process phosphoric acid industry [1]. Statistics show that five tons of PG are generated for every ton of phosphoric acid produced [2]. With the increasing agricultural demand and continuous growth of the phosphate fertilizer industry, the production of PG shows an upward trend, resulting in a large accumulation of PG. Taking Guizhou as an example, the output of PG in Guizhou Province was about 18.5 million tons in 2024, with a historical stockpile as high as 200 million tons. PG contains trace amounts of silicon, fluorides, phosphorus pentoxide, organic compounds, rare earth elements, and radioactive substances [3–6]. For the environment, the large amount of fluorides in PG can contaminate groundwater and surface water. Such polluted water is also rich in phosphorus, which is prone to cause water eutrophication and damage aquatic ecosystems. Long-term intake of high-fluoride water by humans can not only cause dental fluorosis but also lead to bone deformation and skeletal fluorosis. In addition, since PG contains a large amount of heavy metals and radioactive substances, inhalation of PG dust or consumption of contaminated water can damage human liver and kidney functions as well as the nervous system. It is also accompanied by risks such as anemia and carcinogenicity. Therefore, the development and utilization of PG has become a research focus and has attracted extensive attention from many scholars. At present, PG is applied in construction, agriculture, chemical industry and other fields through various approaches [6]. In the construction industry, PG can be used to manufacture innovative bricks and cement-based materials [7,8]. In the chemical industry, it is mainly used to enhance the properties of cementitious materials and produce cement-based subgrade composites. In agriculture, PG is used as a soil stabilization material to prepare alkaline soil conditioners [9,10]. PG can also be used in the field of filling, serving as a basic aggregate to restore stability in mining areas, promote resource recycling, and reduce the risk of geological disasters. The comprehensive utilization of PG allows its application as a new type of phase change material [11], uranium adsorption material, and porous sound-absorbing material [12,13].
Besides the uses of phosphate gypsum (PG) mentioned above, many researchers have been exploring the feasibility of using PG in highway engineering. Using PG in road base materials can not only help absorb industrial by-products on a large scale and ease the pressure of open-air PG stacking, but also partially replace natural sand and gravel aggregates, reducing the mining of mineral resources and energy consumption, cutting construction costs, and supporting the development of a circular economy. This brings notable economic and social benefits. C-PG stabilized macadam materials were prepared by partially replacing fine aggregates with PG, and the influencing factors of water stability and crack resistance were analyzed, and the feasibility of the mix ratio was verified through experiments [14]. Combined with the special engineering properties of PG and red clay, cement (lime) was used as a curing agent to establish a model to reflect the dynamic strength index of the mixture [15]. The mechanical properties and durability of the new PG material were studied. By measuring indexes such as unconfined compressive strength, splitting strength, dry-wet cycles and freeze-thaw cycles, it was verified that the mixture meets the design standards for base materials of highways and first-class highways [16]. Liu et al. used PG as a base course material for sub-base paving [17,18], and Lu G et al. used PG as a surface course material to produce artificial aggregates and asphalt modifiers [19,20]. PG used as subgrade material is also regarded as the optimal choice for large PG consumption with low pavement safety risks [21,22]. However, the application market still faces challenges such as limited application fields, low market acceptance, low utilization rate, high production cost and regional restrictions.
Resilient modulus (MR) is a key parameter for evaluating subgrade stiffness and characterizing the deformation resistance of soil under repeated traffic loads. Since MR can quantitatively reflect the mechanical properties of subgrade materials, it has become a focus in geotechnical engineering research. Numerous studies have discussed the effects of various internal and external factors on MR through laboratory tests. Among internal factors, moisture content has consistently been identified as the primary variable; many studies show that MR decreases with increasing moisture content, highlighting the moisture sensitivity of subgrade materials [23–25] Zhang et al. further demonstrated that the degree of compaction and matrix suction exert a significant influence on MR [26]. Similarly, other researchers have emphasized the influence of initial dry density and particle size distribution, which modulate soil structure and thereby regulate its elastic response [27,28]. The results of Nazerke S et al. show that MR exhibits a decreasing trend as the number of freeze-thaw cycles increases, with the rate of MR decline slowing as the freeze-thaw process progresses [29,30].
Based on the aforementioned research, the effect of PG as a subgrade material on MR under different N and C conditions remains to be investigated. Therefore, this study examines the MR of PG under heavy traffic conditions in regions with seasonal rainfall, with particular attention to its material properties and mechanical evolution. This study utilized dynamic triaxial tests to analyze the effects of confining stress (), cyclic dynamic stress
, the number of wet-dry cycles (N) and cement dosage (C) on the resilient modulus (MR). The relationship between MR and the static resilient modulus, as well as the 7-day unconfined compressive strength (
), was examined. A predictive model for MR that accounts for C and N was proposed, and SEM and XRD methods were employed to analyze the microstructure and elucidate the mechanism by which wet-dry cycles affect MR. This research fills a knowledge gap regarding the influence of C and N on MR and holds significant engineering value.
2. Materials and methods
2.1. Materials
The PG used in this experiment was obtained from Wengfu Phosphate Mine in Qiannan, Guizhou Province. It appears as an off-white powder in a dry state, as shown in Fig 1. Since PG is a fine-grained soil, a hydrometer method was adopted for supplementary sieve analysis to obtain more accurate particle gradation characteristics. A Type-A hydrometer was used in the test, and the PG particle gradation curve was plotted based on the results, as shown in Fig 2. The characteristic particle sizes of the PG sample are as follows: D60 (control particle size) = 0.0144, D30 (median particle size) = 0.0074, D10 (effective particle size) = 0.0048, CU (coefficient of uniformity) = D60 × D10−1 = 0.0144/0.0048 = 3.00, CC (coefficient of curvature) = D30 × D30 × D60−1 × D10−1 = 0.00774 × 0.0074 × 0.0144−1 × 0.0048−1 = 0.79, CU = 3.00 is less than 5, CC = 0.79, the PG gradation is uniform but poorly graded. The proportion of PG particles <0.075mm is 57.65%, exceeding 50%, making it fine-grained soil.
The main components of the raw material are CaO, SO3 and SiO2, and its main chemical compositions are listed in Table 1. The basic physical indexes of PG are shown in Table 2. Since PG contains potentially harmful substances like phosphorus and fluorine, it should be tested for heavy metal content and radioactivity before the experiment. The test results are shown in Tables 3 and 4. The cement used in the experiment is P·C 32.5R Portland cement of a brand from Southwest China, which is dry and free of agglomeration. The basic parameters and chemical compositions are presented in Tables 5 and 6. Tests show that all indexes of the cement satisfy the technical specifications of Common Portland Cement (GB 175–2007).
2.2. Mixture proportion design and sample preparation
The overall experimental procedure is shown in Fig 3 below.
2.2.1. Mixture proportion design.
Extensive studies have been conducted on the mix proportion of C-PG materials worldwide. According to the Technical Specifications for Construction of Highway Pavement Base Courses (JTG/T F20-2015), when cement-stabilized soil is used as the subbase and the plasticity index of the soil is greater than 12, the recommended cement dosage is 6%–14%. Scholars such as Mansheng Dong et al. conducted in-depth research on cement-stabilized macadam bases and found that of the mixture reaches the maximum when the cement dosage is 5% and the C:PG ratio is 1:3, which can meet the strength requirements of pavement bases [31,32]. Mingkai Zhou et al. carried out unconfined compressive strength tests on C-PG-stabilized macadam and found that the compressive strength peaks at a cement dosage of 5% [33]. Zhengfu Qian used 10% cement to modify PG, and unconfined compressive strength tests showed that C-PG-stabilized materials improved by the curing agent exhibit good long-term strength performance [34]. Based on previous research, C-PG-stabilized materials with cement dosage of 7%, 9%, and 11% can meet the specification requirements for pavement bases and subbases in most cases. Therefore, these three cement dosage levels were adopted for sample preparation in this experiment. The cement dosage was calculated in accordance with Technical Specifications for Construction of Highway Pavement Base Courses (JTG/T F20-2015), expressed as the percentage of cement mass to the total mass of dry stabilized materials. Compaction tests were then performed on the mixture following the Standard for Highway Geotechnical Tests (JTG 3430−2020). The compaction results are shown in Table 7. The compaction results at 9% cement dosage are presented in Fig 4. Specimens were prepared according to the compaction test results. All samples were statically compacted at the optimum moisture content with a compaction degree of 98%. The specimens were cylindrical with a height of 80 mm and a diameter of 39.1 mm. The sample preparation scheme and strength tests are shown in Fig 5.
2.2.2. Preparation of samples.
The sample preparation method follows the static pressure method specified in the Test Procedures for Inorganic Binder Stabilized Materials in Highway Engineering (JTG_3441–2024) [35]. Before specimen preparation, the PG used in the test was placed in an oven at 40°C and dried for 24 hours to evaporate most of its internal moisture. The moisture content before and after must be measured by weighing, and it must be less than 1% to be used for the experiment. The bulk PG was crushed with a hammer or other tools and then sieved through a 2 mm sieve. Subsequently, the first mixing was conducted. According to the designed compaction degree and optimum moisture content, the required masses of PG and water were calculated. All the weighed PG and 90% of the required water were poured into a bucket and mixed uniformly with a hand-held mixer. The well-mixed PG was sealed and stored for 24 hours to allow sufficient water absorption. Then the second mixing was performed. After 24 hours of standing, the PG was mixed with the remaining 10% of the required water, and the corresponding mass of cement was added simultaneously, followed by thorough mixing again with the mixer. The mixture after the second mixing was poured into the mold and manually compacted and demolded using a hydraulic jack. The duration from the end of the second mixing to the completion of specimen preparation should not exceed 1 hour, since excessive time may lead to premature hardening of cement and thus affect the properties of the specimens. Finally, standard curing was carried out. The prepared specimens were placed in a pre-prepared tray with permeable stones underneath, sealed with plastic wrap, and then placed in a curing chamber at a temperature of 18–22°C and a humidity of no less than 95%. The specimen preparation process is shown in Fig 6.
2.3. Test methods
2.3.1. Wet-dry cycle program.
When PG is used as subgrade filling material, it will inevitably be affected by factors such as groundwater level rise, rainfall infiltration, and solar radiation under open-air conditions. The material will undergo dry-wet cycles. Under repeated dry-wet cycles, the internal structure of the subgrade will be damaged, eventually leading to problems such as reduced subgrade strength and subgrade cracking, thus resulting in potential safety hazards. Harianto Rahardjo et al. utilized analyses of slope stability based on the temporal variations in pore water pressure and soil moisture during wet and dry periods [36]. When calculating the shear strength of unsaturated soils using water-soil characteristic curves, Qian et al. took into account the inherent uncertainty of these curves and proposed correcting the SWCC values obtained from conventional Tempé cells or pressure plates by considering different stress levels, in order to assess the shear strength of unsaturated soils [37]. Ye et al. converted sampled soil data into continuous field-scale outputs of unsaturated soil properties for incorporation into slope stability analyses [38]. Therefore, studying the mechanism and influence law of moisture content of road materials on subgrade performance is of great significance for optimizing subgrade design and ensuring engineering safety. To simulate real field conditions as closely as possible, this paper simplifies the influence of material moisture content on roads through dry-wet cycles. Studies have shown that the dry-wet cycle path has a significant impact on the strength of base materials. In general, the strength degradation caused by wetting first and then drying is much greater than that caused by drying first and then wetting. Therefore, the wetting-first and drying-later path was adopted in this study [39,40].The amplitude of dry-wet cycles (the upper limit moisture content for wetting and the lower limit moisture content for drying) also affects the strength of the mixture to a certain extent. According to the research results in references [41,42], After the sub-base material has been compacted at its optimal moisture content, exposure to natural climatic conditions for a period of time will cause the moisture content to increase, fluctuating periodically or non-periodically within a range of ±5% of the ‘optimal moisture content’ (OMC). To simulate this real-world engineering scenario, and taking into account the liquid and plastic limit moisture contents of PG, a moisture variation range of 10% has been established; that is, during the wetting process, the moisture content of the test specimen is adjusted to 5% above the optimal moisture content, and during the drying process, the moisture content of the specimen is reduced to 5% below the initial moisture content. The main steps of a complete wet-dry cycle test are as follows: (1) Wetting stage: Remove the plastic wrap from the cured specimens and place them in a curing box. Turn on the humidifier to wet the specimens by spraying. Take out the specimens at intervals and monitor their moisture content by weighing. When the moisture content reaches the upper limit (OMC + 5%), stop wetting and seal the specimens for 24 hours. To ensure uniform moisture distribution inside the specimens, a pre-experiment was conducted in this study. The specimens were first placed in a dry-wet cycle test chamber for cycles, and parts of the specimens from the upper, middle and lower layers were taken to measure the moisture content by the drying method. Uniform moisture distribution was confirmed when the difference in moisture content among the three layers was less than 1%. The formal test was started only after calibrating the instrument. (2) Drying stage: The wetted specimens were dried in an oven at 40°C. Similarly, the moisture content of the specimens was measured by weighing at intervals. When the moisture content reached the drying target (OMC – 5%), the specimens were taken out and drying was stopped. After standing for 24 hours, the specimens were rewetted to the optimum moisture content, completing one dry-wet cycle. Subsequent cycles were performed in the same manner.
2.3.2. Dynamic triaxial test.
The resilient modulus test was carried out using an SDT-20 microcomputer-controlled dynamic triaxial test machine for soil. According to the Standard for Highway Geotechnical Tests (JTG 3430−2020), the stress state experienced by different subgrade regions under vehicle load is governed by the coupling effect of and
[43]. Given the strong stress dependence of the subgrade soil MR [44], the applied test conditions must be as close as possible to the in-situ stress environment. The selection of triaxial test parameters was guided by empirical observations and field data. Specifically,
was set to 15, 30, 45, and 60 kPa, which are consistent with the ranges commonly specified in various pavement layers. Field investigations have shown that
in subgrade soils typically spans 0–60 kPa [45]. To account for heavy traffic effects, the load response at different depths within the subgrade was examined. The results indicate that the amplitude of
can reach 150 kPa at a depth of 0.5 m below the subgrade surface, while a smaller
is still observed at 3.5 m depth [46]. Therefore,
values of 30, 55, 75, and 105 kPa were adopted to reflect a gradient of stress conditions, including those aggravated by traffic growth and related stress amplification effects.
During the test, the confining stress supply line and the triaxial cell were connected first. The standard loading sequence for fine-grained soil specimens is listed in Table 8. A pre-loading confining stress of 30 kPa was applied to the specimen, where . A haversine pulse load was used as the test load, and at least 1000 cycles of haversine pulse load were applied to the specimen at a frequency of 10 Hz. The loading duration and rest period were 0.1 s and 0.9 s, respectively. The maximum axial stress of the specimen could be taken as 61.0 kPa. For each loading sequence, after applying the corresponding number of pulse loads, the dynamic resilient deformations of the last 5 cycles in the sequence were selected. If these 5 deformation values showed no obvious fluctuation and tended to be constant, the specimen was judged to have reached a stable elastic deformation response. The MR under that sequence was calculated using the average value of these 5 deformations. The formula for calculating the MR is given in Equation (1).
In the formula: MR denotes the resilient modulus (MPa); denotes the cyclic dynamic stress (MPa);
denotes the axial resilient strain.
2.3.3. Unconfined compressive strength test.
Place the properly cured specimens onto the unconfined compressive strength testing machine for compression testing. In accordance with the “Test Procedures for Inorganic Binder Stabilized Materials in Highway Engineering” (JTG3441−2024), maintain a loading rate of 1 mm/min throughout the test. Record the maximum pressure at specimen failure during testing. Then calculate the compressive strength using the unconfined compressive strength formula (2).
In the formula: RC denotes the unconfined compressive strength of the specimen (MPa); P denotes the maximum pressure at specimen failure (kN); A denotes the cross-sectional area of the specimen (mm²).
2.3.4. Microtesting program.
SEM and XRD tests were conducted to analyze the strength development mechanism under different dry-wet cycles from the perspectives of particle morphology, arrangement characteristics (pores) and chemical composition of the specimens. The influence of each factor could be obtained by comparing the micro results of different groups. The mixture was prepared by the static pressure method and taken out after standard curing. Then the specimens were immersed in anhydrous ethanol to stop the hydration reaction. After that, they were placed in an oven and dried at 40°C to remove free water until the weight remained constant. The specimens were taken out, crushed and processed into particles with a length and width of 1–2 cm (generally not less than 1 cm) and a thickness of no more than 10 mm. The particles were placed in a German ZEISS GeminiSEM 300 scanning electron microscope to observe the morphology characteristics of the specimen section. In addition, a part of the crushed specimens was ground into powder with a particle size of about 0.05 mm for X-ray diffraction test to analyze the composition under different conditions.
3. Results and discussion
3.1. Effects of confining stress on resilient modulus
Fig 7 shows the relationship between MR and for specimens tested at C = 7% and N = 6. similar patterns are observed for other wet-dry cycles, due to space constraints, these are not shown in detail here. The figure indicates that, for a given
. The MR of the test specimen is positively correlated with
and the relationship is approximately linear. Taking
=55 kPa as an example, when
is 15 kPa, 30 kPa, 45 kPa and 60 kPa, the corresponding MR values are 152.84 MPa, 176.48 MPa, 198.24 MPa and 215.64 MPa, representing increases of 12.47%, 12.33% and 8.76% respectively. Analyzing the underlying causes, the presence of confining stress subjects the specimen to lateral pressure, restricting its lateral deformation during loading. This enhances the interlocking and friction between the particles within the specimen, reduces its porosity, and thereby improves its resistance to deformation and failure, consequently increasing its strength.
3.2. Effects of Cyclic dynamic Stress on resilient modulus
Fig 8 shows the relationship between MR and for C = 7% and N = 3. Similar patterns are observed for other values of N, due to space constraints, these are not shown here. The MR of a specimen is influenced not only by the material itself but is also closely related to the stress conditions under which the test is conducted. As can be seen from the figure, when all other conditions are held constant, the MR decreases continuously as
increases, and the rate of decrease is significant. Taking N = 3 and
=30 kPa as an example, the values of
at 30 kPa, 55 kPa and 75 kPa are 220.04 MPa, 197.87 MPa and 178.24 MPa respectively, representing decreases of 11.2% and 11.0% respectively, with the rates of decline being relatively similar. This is because cyclic dynamic stress causes changes in the intergranular structure within the specimen, leading to fatigue damage. As a result, the stress in certain areas exceeds the material’s strength limit, causing microcracks to form and propagate within the material. This reduces the specimen’s effective bearing area, thereby diminishing its strength.
3.3. Effects of the Dry-Wet Cycle on resilient modulus
The relationship between MR and N at C = 9% and =15 kPa is shown in Fig 9. Similar patterns were observed for other C; therefore, these are not shown here. The data indicate that prior to the wet-dry cycling, the internal structure of the specimens with the optimal soil conditions remained undisturbed. This happens because when cement-stabilized PG is mixed with water, the cement clinker minerals quickly hydrate to form calcium silicate hydrate (C-S-H gel) and create an alkaline environment; the PG dissolves and releases sulfate ions, which combine with calcium hydroxide and calcium aluminate to form ettringite (AFt). The C-S-H gel and AFt intertwine, not only filling the gaps between particles but also hardening the slurry and forming a three-dimensional network structure, which greatly boosts the strength of the mixture [47].Taking
= 55 kPa and N = 0 as an example, the MR is 266.59 MPa; when N = 1, the MR is 240.87 MPa, with a rate of decrease of approximately 9.65%; when N = 4, the MR is 198.12 MPa, and the average rate of decrease over the four trials is 8.81%. When N = 6, MR is 168.89 MPa, and the average rate of decrease for the last two cycles is 4.26%. It can be seen that after one wet-dry cycle, the curve flattens out, the hardening effect gradually weakens, and the soil is damaged under the wet-dry cycles. After 2–4 wet-dry cycles, the damage continues to increase, and internal cracks in the soil develop further, leading to a reduction of MR. The decline in MR is mainly concentrated in the first four wet-dry cycles, after which it gradually stabilizes. An analysis of the experimental process suggests that during drying in the oven, temperature differences between the specimen’s surface and interior caused the moisture content to increase gradually from the outside inwards. This led to the generation of residual stresses, inducing tensile stress within the specimen. When the residual stress reached a critical value, cracks formed in the specimen, and subsequent wet-dry cycles exacerbated this phenomenon. However, as the N continues to increase, the development of cracks within the soil stabilizes and the soil reaches a state of equilibrium. Consequently, after four wet-dry cycles, the MR stabilizes.
3.4. Effect of Cement dosage on resilient modulus
Fig 10 shows the relationship curve of MR and C when = 30 kPa and N = 0. Experimental data reveal a positive correlation between the MR of stabilized materials and C, clearly demonstrating that increased cement dosage effectively enhances the MR of PG. This happens because when cement-stabilized PG is mixed with water, the cement clinker minerals quickly hydrate to form calcium silicate hydrate (C-S-H gel) and create an alkaline environment; the PG dissolves and releases sulfate ions, which combine with calcium hydroxide and calcium aluminate to form ettringite (AFt). The C-S-H gel and AFt intertwine, not only filling the gaps between particles but also hardening the slurry and forming a three-dimensional network structure, which greatly boosts the strength of the mixture [47], as shown in equations (3)-(4) below:
4. Prediction model of resilient modulus
4.1. Comparison of resilient modulus and Static resilient Modulus under Dry-Wet Cycling
Since the measurement of MR has high requirements for test conditions, while the static resilient modulus test is relatively simple, if the proportional relationship between them can be found, MR of the material can be indirectly obtained by measuring the static resilient modulus when MR testing is unavailable. Therefore, this section compares MR and static resilient modulus under dry-wet cycles to provide a reference for obtaining the MR of C-stabilized PG materials. In the test, MR under the most unfavorable conditions of =105 kPa and
=15 kPa was taken as the representative value of C-stabilized PG materials, and the comparison results are shown in Fig 11.
Fig 11 shows the relationship between MR and static resilient modulus at C = 9%. Similar trends are observed at C = 7% and C = 11%, which are not presented here. It can be seen from Fig 11 that both MR and static resilient modulus of the specimens decrease with repeated dry-wet cycles and gradually tend to be stable, and the MR is always much larger than the static value. The variation amplitude of resilient modulus of the specimens becomes very small after 4–6 dry-wet cycles. Therefore, this paper adopts the average values of the last three dry-wet cycles to convert the ratio between them. The representative values of MR and static resilient modulus of specimens with different cement dosage under dry-wet cycles are listed in Table 9. It can be seen from Table 9 that the ratio of MR to static resilient modulus under dry-wet cycles is approximately between 1.25 ~ 1.30. This ratio can be used to make a preliminary estimate of the MR of C-stabilized PG materials, providing a reference for pavement design; however, as the characteristics of PG and local environmental and climatic conditions vary from place to place, further field data is required to verify whether this approach can be widely applied.
4.2. Establishing the Relationship Between resilient modulus and 7-Day Unconfined Compressive Strength
This paper attempts to establish the relationship between MR and for C-stabilized PG materials subjected to wet-dry cycling. The experimental data for MR and
were fitted using Origin 2018 software; the fitting equation is shown in Equation (5). The results of the fitting curve under the condition of
= 15 kPa are shown in Fig 12. Fig 12 clearly illustrates the degree of deviation between the predicted strength values and the experimental values for specimens with different cement dosage. The fitting results for cement dosage of 7%, 9% and 11% are all satisfactory. Table 10 presents the relevant fitting parameters. As shown in Table 10, the correlation coefficients (R²) are all greater than 0.94, which further demonstrates the high quality of the fitting.
This model establishes a mathematical relationship between the MR and of C-stabilized PG materials under wet-dry cycling. As MR testing imposes high demands on experimental equipment and environmental conditions, this model can indirectly estimate the MR of C-stabilized PG by testing
under wet-dry cycling, thereby providing a reference for MR prediction and holding practical significance. However, given the variability of on-site conditions, the complexity of engineering scenarios, and the limited sample size of the tests, the application of this model in engineering practice must be supplemented by in-situ testing, long-term monitoring data, and adaptive parameter calibration to further enhance its reliability and engineering applicability under real-world, complex conditions.
In the equation: a and b are fitting parameters; is the 7-day unconfined compressive strength.
4.3. Significance analysis
The analyses presented in Sections 3.1 to 3.4 indicate that ,
, C and N have varying degrees of influence on the MR of PG. However, the relative importance of each factor remains unclear. Therefore, it is essential to conduct a sensitivity analysis of these influencing factors before developing an MR prediction model. This analysis will provide valuable insights into the selection of appropriate model parameters, thereby ensuring the accuracy and reliability of the predictive model. A three-factor analysis of variance was conducted using SPSS to evaluate the effects of
, C and N on the PG mixture, and to analyse the extent to which the three factors
,C and N influence MR when
=15 kPa. The factor levels and the results of the sensitivity analysis are shown in Tables 11 and 12.
In Table 12, the smaller the p-value and the larger the F-value, the more significant the influence of that factor on the test results. It can therefore be concluded that there are differences in the extent to which each factor influences the material’s MR: The influence intensity of the three factors on the response indicator follows the order of C > N>. Factor C exerts a significant effect (P < 0.05), whereas factors N and
exhibit marginally significant effects.
4.4. Classical prediction mode
In order to quantitatively assess the strength characteristics of subgrade materials under long-term loading, researchers both domestically and internationally have developed a series of MR prediction models, as shown in Table 13. Model 1 considers only , Model 2 considers only
, and Model 3 considers only the confining effect of the volumetric stress θ. Model 4 comprehensively considers both the confining effect of the volumetric stress θ and the influence of the octahedral shear stress
, providing a more comprehensive description of soil behaviour. The experimental results of this study were fitted using the parameters of typical MR prediction models. However, the overall fitting accuracy was relatively low. This discrepancy arises because classical models contain fewer parameters, making them relatively simple and insufficient to capture the effects of multiple factors.
In the given formula. MR denotes the resilient modulus (MPa); represents confining stress (kPa);
refers to cyclic dynamic stress (kPa); θ is the volumetric stress (kPa); τoct indicates octahedral shear stress (kPa); pa is typically set to 100 kPa for standard atmospheric pressure; k1, k2, and k3 are fitting parameters that need to be determined.
4.5. Estimation model based on wet and dry cycle and cement dosage
Sub-base materials are inevitably subject to wet-dry cycles, which consequently affect their MR values; The traditional prediction models mentioned above do not take this factor into account. Combining the stress conditions and typical stress levels of Chinese subgrades, this section builds upon the NCHRP 1-28A model and, based on experimental data, further incorporates C and N to propose an MR prediction model based on the effects of wet-dry cycling. This model comprehensively considers four factors—C, ,
and N—to provide a reference for estimating the MR values of test specimens. By introducing C in exponential form (1 + C)k and N in logarithmic form [ln(e + N)]k into the NCHRP 1-28A model, an improved model is obtained. The use of an exponential function reflects the non-linear enhancing effect of C on the mixture’s bonding properties and overall strength, consistent with the macroscopic mechanical evolution patterns of cement-stabilized materials; The use of the form [ln(e + N)]k reasonably describes the non-linear characteristic whereby material performance gradually decreases and stabilizes under wet-dry cycling. This expression is concise and effectively reflects the actual damage development process of the material. The model is suitable for low-stress conditions; it is not only structurally simple and intuitive but also comprehensively integrates key influencing factors. Furthermore, the parameters of this model have clear physical significance and can be readily obtained through experimental testing. Specifically, as follows:
In the given formula, C denotes cement dosage, N represents the number of wet-dry cycles; k4, and k5 are fitting parameters to be determined. Parameter k4 reflects the exponential enhancement effect of cement dosage on MR, and k5 reflects the extent of strength reduction in the mixture due to wet-dry cycles. ,
, and
are principal stresses, where
, and
, yielding Equation (10).
By substituting the experimental data for C-stabilized PG materials at N = 0, 3 and 6 into Equation (6), and using a MATLAB programming loop to iterate, the values of k1 to k5 were obtained, where k2 and k4 are greater than 0, and k3 and k5 are less than 0; this is consistent with the conclusions in Sections 3.1 to 3.4. Based on this model, it can be seen that MR increases with an increase in θ () and decreases with an increase in τoct
. This is because the weak bonding bonds within the specimen are continuously breaking down, leading to the gradual initiation, propagation and connection of microcracks within the specimen; under repeated loading, the aforementioned damage accumulates continuously, ultimately resulting in a deterioration of the material’s overall stiffness, which manifests macroscopically as a continuous decrease in MR. It is noted that the value of k4 = 6.8024 is abnormally high, indicating that the test soil is highly sensitive to C. This elevated value is attributed to the combined effect of the selected model form and the material’s inherent non-linearity. Fig 13 was plotted based on the prediction results and experimental data. The MAE1 of the prediction results is 13.3, the RMSE1 is 17.50, and RD1 = 5.6%; the prediction results are satisfactory. To further verify the reliability of the improved model, the MAE1, RMSE1, and RD1 of the MR prediction results under the most unfavorable conditions (C = 7%,
=15 kPa,
=105 kPa) for N = 1, 2, 4, and 5 are presented. Table 14 provides the fitted parameter values, the model’s fitting accuracy, and the MAE2, RMSE2 and RD2 of the validation results, where the units for MAE and RMSE are MPa.
4.6. Microscopic mechanism analysis of the mixture under wet-dry cycling
To elucidate the mechanisms by which wet-dry cycling affects the properties of the mixture from a microscopic perspective, Figs 14 and 15 present the SEM and XRD micrographs of the C = 9% specimen after 0, 4 and 6 wet-dry cycles, respectively. Observing Fig 14, When N = 0, it can be seen that there is more columnar gel substance between the particles. compared with the specimen that had not undergone wet-dry cycling, after 4 cycles, the content of columnar cementitious material between the lumpy particles was lower, and the inter-particle bonding strength was weaker; when the number of cycles was increased to 6, the pores in the specimens tended to enlarge, and the content of columnar cementitious material between the particles was lower than that observed after 4 cycles. These characteristics of microstructural evolution are consistent with the trend in specimen strength, which initially decreases rapidly and then gradually stabilizes as N increases.
Fig 15 shows that disappears in the specimen after 4 ~ 6 dry and wet cycles, the AFt content decreases, and the
content increases, because part of the AFt decomposes into calcium oxide (CaO), sulfur trioxide (SO3) and other substances during the drying process, and SO3 dissolved in water to form sulfuric acid (H2 SO4) during the humidification process, and PG also contains a large amount of
, which makes the system acidic. It accelerates the degradation of AFt, which is one of the main reasons for the decrease in AFt in specimens after wet and dry cycling [52,53].
content increases and
disappears, and it is speculated that the possible reason is that the SO3 obtained by AFt decomposition reacts with
after dissolving in water to form
, and the reaction process is as follows:
The main mechanisms of strength attenuation of cement PG stabilized materials under the above dry and wet cycles are as follows: (1) The dry and wet cycle increases the porosity inside the specimen, reduces the cementitious matter between the plates, weakens the connection effect, and reduces the integrity of the specimen. (2) The dry and wet cycle partially decomposes AFt, and the reinforcement effect of AFt is weakened.
5. Conclusions
This study investigates the variation patterns of the MR of cement-stabilized PG materials under different conditions through dynamic triaxial tests, analyzes the mechanism behind MR reduction, and focuses on the mechanical properties of PG mixtures in regions with seasonal rainfall. Through a series of dynamic triaxial experiments and microscopic analyses, the microstructural characteristics of specimens were examined, and an MR prediction model was established for different cement dosages under wet-dry cycling conditions. The main conclusions are as follows:
- (1) The MR of stabilized materials increases with higher confining stress and cement dosage, while gradually decreasing with increasing cyclic dynamic stress. Under wet-dry cycles, the MR decline of PG mixtures undergoes three stages: rapid decay, moderate decay, and stable stage. The most pronounced MR decline occurs during the first cycle. As the number of wet-dry cycles increases, MR continues to decrease but at a slower rate, stabilizing after approximately four cycles. And through a microscopic explanation of its causes, the wet-dry cycles increase the porosity inside the specimen, reduce the cementitious material between the plates, weaken the bonding effect, and decrease the overall integrity of the specimen; the wet-dry cycles cause partial decomposition of AFt, weakening the reinforcing effect of AFt.
- (2) Under dry-wet cycling, the relationship between MR of stabilized materials and their 7-day unconfined compressive strength can be described by MR = a + b
. The correlation coefficients for fitting exceed 0.94, indicating good fitting quality. This validates the reliability of estimating the MR of stabilized materials using 7-day unconfined compressive strength, providing a reference for studying the strength characteristics of cement-stabilized PG materials.
- (3) Comparing dynamic and static resilient modulus reveals that the MR of PG specimens under wet-dry cycling is 1.25–1.30 times the static resilient modulus. This ratio can be used to estimate the MR of cement-stabilized PG materials. Of course, whether it can be rolled out for wider use will require further verification through field data.
- (4) Multi-factor sensitivity analysis indicates that C and N influence the mixture’s MR. Therefore, a MR prediction model incorporating wet-dry cycling effects and cement dosage is proposed:
. The model’s reliability was validated by comparing predicted values with experimental data. This model is no longer limited to simple statements at the phenomenological level, but rather, through function fitting and pattern analysis, it extracts the nonlinear evolution mechanism of modulus changes with influencing factors, thereby deepening the understanding of the intrinsic mechanical response of cement-stabilized PG materials under complex working conditions
- (5) This experiment fully utilizes the readily available
and static resilient modulus, and establishes predictive models of C and N with respect to MR. By comparing measured values with model predictions, the reliability of the models is verified, providing a new approach for predicting MR. Additionally, microscopic experiments are used to explain the reasons for MR reduction under wet-dry cycles. Although actual road engineering conditions are relatively complex, with multi-factor coupling phenomena, this experiment is a reasonable simplification of field complexity under controlled conditions. The conclusions obtained have certain engineering value.
- (6) This study creatively proposes a MR prediction model that considers the effects of wet-dry cycles and cement dosage, and validates the reliability of the model by comparing predicted values with experimental results. The predicted MR provides a reliable basis for pavement design and holds significant engineering guidance value. Because of the variability of onsite conditions and the complexity of engineering situations, and given that our test sample size is limited, this model, when applied to actual projects, still needs to be combined with on-site testing, long-term monitoring data, and adaptive parameter calibration to further improve its reliability and suitability under real, complex conditions. We hope more researchers will carry out in-depth analyses on the environmental safety of PG, testing the leachate of cement-stabilized PG samples under wet-dry cycles for radioactivity, heavy metal content, and pH value, to ensure the environmental safety of PG when exposed onsite.
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