# Fatigue behavior of lime- and/or cement-treated soils: A database analysis and empirical modeling approach

Vincent Krzewinski, Nathalie Dufour, Christophe Chevalier, Julien Borderon,
Muriel Gasc-Barbier

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Vincent Krzewinski, Nathalie Dufour, Christophe Chevalier, Julien Borderon, Muriel Gasc-Barbier. Fatigue behavior of lime- and/or cement-treated soils: A database analysis and empirical modeling approach. Results in engineering, 2025, 28, pp.107682. 10.1016/j.rineng.2025.107682. hal-05464683

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# Fatigue behavior of lime- and/or cement-treated soils: A database analysis and empirical modeling approach

Vincent Krzewinski<sup>a,b</sup>, Nathalie Dufour<sup>a,*</sup>, Christophe Chevalier<sup>b</sup>, Julien Borderon<sup>a</sup>, Muriel Gasc-Barbier<sup>a,●</sup>

## ARTICLE INFO

### Keywords:

- Fatigue

- Treated soil

- Cyclic loading

- Stress ratio

- Cement treatment

- Curing duration

- Geotechnical engineering

## ABSTRACT

Soil improvement using lime and/or hydraulic binders is a well-established technique in geotechnical engineering, particularly in road and railway infrastructures. However, treated soils are subjected to cyclic loading, leading to a fatigue phenomenon that progressively degrade their mechanical properties. This study is based on a database of 410 laboratory tests extracted from 12 published articles (1984–2025), covering various soil types and experimental conditions. A detailed analysis of the influence of parameters on fatigue is conducted using correlation matrices, Principal Component Analysis, and Wöhler diagrams. The results highlight the significance of curing duration, with long-term curing (>120 days) giving more consistent fatigue behavior. The influence of cement classification used to treat soil and soil composition is also demonstrated, showing that CEM II treated specimens exhibit higher fatigue resistance. Finally, an empirical fatigue model is developed to predict the fatigue resistance of treated soils.

## 1. Introduction

Treated soil is natural soil that has been improved by adding stabilizing agents [1–4]. In geotechnical engineering, using lime or hydraulic binders to treat soil is a common technique to improve its mechanical behavior [5–7]. These treated materials are used in many infrastructures, like road subgrades, railway platforms, and embankments [8–10]. Treated soils undergo significant changes in their physical structure and mechanical properties due to compaction and curing processes. Compaction densifies the soil matrix, reducing voids and enhancing particle contact, which directly impacts the soil’s morphology and strength [11,12]. The curing conditions, often standardized [13] in laboratory settings, play a crucial role in the development of these properties over time by allowing chemical reactions between the binder and soil to progress. Under static loading, treated soils generally exhibit improved stiffness, increased shear strength, and reduced plasticity compared to untreated soils. These enhancements contribute to greater load-bearing capacity and better deformation resistance, making treated soils a reliable choice for foundational applications. This process has the advantage of minimizing environmental impact by reducing the need for imported materials and promoting the

reuse of local soils [14]. In this article, we focus specifically on soil treatment using hydraulic binders and/or lime, as these are among the most widely adopted and documented methods in current geotechnical practice. However, it is important to note that alternative soil improvement techniques, such as enzymatic treatments, bio-cementation, or the use of geopolymers [15–18], are also being actively studied.

Under static loading, treated soils generally show better load-bearing capacity. They tend to have higher shear strength and lower plasticity, which helps reduce deformation when loads are applied [19]. This is why soil stabilization is often considered a useful and affordable solution, especially where local soils don’t meet performance requirements for construction. Chemically, the treatment process involves several stages. One of the first effects is a decrease in the natural water content, due to the reaction between the soil and the binder. Then, a flocculation step takes place, where fine particles in suspension group together into larger clusters because of electrostatic forces and Van der Waals attractions. The main chemical reaction involves the hydration of tricalcium silicate (C₂S) and dicalcium silicate (C₂S) with water, forming calcium silicate hydrate (C-S-H) and calcium hydroxide (Ca(OH)₂). Later, as the process continues, a pozzolanic reaction starts between the

<footnote><sup>a</sup> GéoCoD, Cerema, Aix-en-Provence, France</footnote>

<footnote><sup>b</sup> Univ Gustave Eiffel, GERS-SRO, Marne-la-Vallée, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: nathalie.dufour@cerema.fr (N. Dufour).</footnote>
calcium hydroxide and other components in the soil, producing more C-S-H [20,21]. These chemical changes help strengthen the soil over time [20].

Fatigue is a progressive process in which a material gradually loses strength and its internal structure deteriorates under repeated stress [22, 23]. Treated soils are not only exposed to static loads as, in practice, they often face repeated or cyclic loading, caused by traffic or changing water levels in hydraulic structures. Over time, these repeated loads induce fatigue. The fatigue behavior of treated soils depends on many variables, including load magnitude and frequency, soil type, binder characteristics, curing conditions, and environmental factors. This complexity, combined with the inherent variability of soils, makes it difficult to compare results across studies or to establish general design rules. Fatigue phenomena are well recognized in other construction materials, such as concrete [24,25], which share some similarities with treated soils (grains, cementitious matrix, curing duration).

Unlike static loading, cyclic loading generates progressive damage at the microstructural level, eventually resulting in crack initiation, propagation, and ultimate failure [22]. Understanding the fatigue behavior of treated soils is therefore critical for designing more resilient infrastructures and mitigating the risks of premature failure. One concept is crucial for understanding the performance of materials under cyclic loading, where damage accumulates gradually over time: the fatigue life. It refers to the number of cycles a material or structure can endure under loads before failure occurs. In the case of materials, localized permanent damage results in the formation of cracks or sudden fractures. The fatigue life prediction in this context is focused on the number of cycles until materials or structures fail to resist the applied cyclic load (such as airport pavements [26]). While this study focuses on mechanical loading fatigue, it is important to note that other types of fatigue exist, such as fatigue induced by thermal cycles [26,27] or hydric cycles [13,28,29].

Research in this field has demonstrated that different loading conditions lead to varying fatigue responses. There are three categories of fatigue based on the number of cycles to failure. According to [30], fatigue can be categorized into *very high cycles fatigue* (number of cycles > 10⁷), *high cycles fatigue* (10⁵-10⁷ cycles), and *low cycles fatigue* (< 10⁵ cycles). Low-cycle fatigue involves plastic deformations that accelerate degradation.

The early 1980s marked the beginning of laboratory investigations into the fatigue phenomenon in treated soils. Initial research included two articles and a report from France [30–32]. This early work involved extensive research and parametric analysis of different soil types, focusing on the application of treated soils in road construction. However, there is a noticeable gap in publications between these early studies and more recent work, with four articles published before 1990 and the remainder after 2007. This gap may be partly due to the fact that early research on fatigue in treated soils focused primarily on road and railways applications. After conducting thorough analyses and setting standards, new advancements in the use of treated soils began to surface in the 2000s with exploring the effects of additional factors, such as incorporating extra binders. The shift indicates a rising interest in enhancing soil stabilization methods for a wider array of uses, extending beyond just transportation infrastructure. More recently, fatigue behavior has also been investigated using cyclic triaxial testing, providing deeper insights into material response under specific stress conditions [22,33].

Fatigue models in treated soils have long been recognized as a key tool for predicting material performance under cyclic loading. To date, most studies have relied on empirical approaches, providing practical predictions and design guidance. A wide range of models has been proposed, including linear relationships in a semi-logarithmic space (Stress-Number) [3,34–38], power-law scaling [39,40], and other stress-based formulations, reflecting the diversity of soil types, binders, and testing methods considered. Several types of fatigue curves have been observed, including S-shaped curves [38]. These empirical models

serve as valuable tools for engineers, allowing estimation of short-term and long-term behavior. This reliance on empirical predictions remains prevalent even nowadays [17], highlighting the ongoing need for comprehensive frameworks that integrate multiple influencing factors and capture the full spectrum of observed material responses.

The challenges related to the fatigue phenomenon have thus been recognized since the 1980s. To tackle these issues, the authors have compiled a comprehensive experimental database that takes into account various parameters influencing the fatigue performance of treated soils. This database is essential for understanding the complex factors that contribute to fatigue degradation. Fatigue life is often predicted using both statistical methods and empirical modeling, to enhance fatigue life predictions. This ongoing work helps in developing optimized material formulations that are specifically tailored to different engineering applications. In this study, we first provide a description of the database and present statistical analysis. Then an analysis based on parameters is conducted. Finally, a global empirical model is proposed.

## 2. Database

### 2.1. Database introduction

The database [41] compiles results from 410 laboratory tests conducted in the context of linear infrastructure projects (roads, railway, embankment), reported across 12 articles published between 1984 and 2025. These tests were performed on clayey, silty, sandy, and gravelly soil specimens, each treated with at least one type of hydraulic binder. To study the fatigue phenomenon cyclic bending tests and occasionally cyclic triaxial tests were used; however, the cyclic triaxial tests were too different from the others tests and were therefore excluded from the analysis. All 12 studies included in the database focus on stress-controlled fatigue testing procedures. Very few usable data are available [42] from strain-controlled fatigue tests, and even fewer studies exist that report such results in sufficient detail for integration into a structured database.

Different cyclic tests are presented, including two-point bending, three-point bending, four-point bending, flexural tensile, indirect tensile, splitting fatigue, and uniaxial tests. Table 1 shows a summary of tests done in articles and the goals of articles including schemas of tests with dark arrows representing loadings and blue ones showing support. Two-point bending tests involve applying a load at two points on a beam from a soil specimen to create a bending moment. In the case of fatigue, the load is cyclical until failure. Three-point bending tests are similar to two-point bending tests, but the load is applied at the beam's center and supported at two ends. For four-point bending fatigue tests two loads are applied symmetrically on the beam cyclically until failure. Flexural tensile tests use a beam specimen and create bending cyclic flexural stress until failure. Indirect tensile fatigue tests use cylindrical specimens. Cyclic compressive loads are applied along the diameter to create repeated tensile stresses to evaluate fatigue life. Splitting fatigue tests also use cylindrical specimens and a load on the diameter. Specimens are subjected to splitting tensile stresses perpendicular to the direction of the force applied. Finally, during a uniaxial fatigue test, cyclic loading is applied in a single direction to a cylindrical specimen until failure.

The studies investigated a range of parameters related to fatigue behavior in treated soils [37] examined the influence of compaction on the fatigue resistance of four clayey soils treated exclusively with lime. In addition to lime content and stress ratio, they also varied the compaction method [36] compared fatigue resistance between two gravelly soils with different cement contents and stress ratios [31,32] reports trends [32] focused on silty and clayey soils treated with cement and occasionally lime, with curing periods between 90 and 360 days. They also included extra curing conditions and analyzed the effects of cement content, lime content [2] studied both sandy and clayey soils treated with hydraulic binders, varying curing duration, lime content, and stress ratio [44] extended this work by adding fibers as an extra
<table><caption>**Table 1**
Summary of studies on fatigue tests of treated soils.</caption><thead><tr><th>Article</th><th>Soil</th><th>Fatigue tests</th><th>Goal of research</th><th>Parameters analyses</th></tr></thead><tbody><tr><td>Festugato et al. [43]</td><td>Sand</td><td>Cyclic splitting fatigue test</td><td>Analyze the cyclic response and fatigue behavior of cement-treated sand using the porosity/cement index.</td><td>Stress ratio Cement content</td></tr><tr><td>Jiang et al., [39]</td><td>Gravel</td><td>Cyclic splitting fatigue test</td><td>Investigate the fatigue performance of cement-stabilized crushed gravel and establish a Weibull-based equation.</td><td>Type of soil Cement content<br>Compaction methods Stress ratio</td></tr><tr><td>Biswal et al., [34]</td><td>Sand</td><td>Cyclic four-point bending</td><td>Study the fatigue properties and develop prediction models for cement-stabilized lateritic soils.</td><td>Stress ratio Type of soil Cement content</td></tr><tr><td>Xie et al., [3]</td><td>Gravel</td><td>Cyclic indirect tensile<br>Cyclic uniaxial compression</td><td>Examine fatigue behavior and establish a standardized fatigue equation for cement-treated aggregate base materials under varying stress states.</td><td>Cure duration Cement content<br>Stress ratio Field core Type of test</td></tr><tr><td>Lv et al., [40]</td><td>Gravel</td><td>Cyclic flexural tensile</td><td>Evaluate the fatigue behavior of cement-treated aggregates.</td><td>Stress ratio Cement content on gravelly soils</td></tr><tr><td>Lenoir et al., [44]</td><td>Sand</td><td>Cyclic four-point bending<br>Cyclic two-point bending</td><td>Understand the effect of fibers on the mechanical behavior of cement-modified soils.</td><td>Extra treatment Stress ratio</td></tr></tbody></table>

(continued on next page)
<table><caption>**Table 1 (continued)**</caption><thead><tr><th>Article</th><th>Soil</th><th>Fatigue tests</th><th>Goal of research</th><th>Parameters analyses</th></tr></thead><tbody><tr><td>Pretesteille and Lenoir, [2]</td><td>Sand Clay</td><td>Cyclic two-point bending</td><td>Evaluate mechanical fatigue performance of treated soils</td><td>Type of soil Stress ratio Lime content Cure duration</td></tr><tr><td>Pretesteille [35]</td><td>Sand</td><td>Two-point bending</td><td>Fatigue Behavior of Soils Treated with Hydraulic Binders in railway structure platforms</td><td>Type of soils Cure duration Stress ratio</td></tr><tr><td>Sobhan and Das [36]</td><td>Gravel</td><td>Three-point bending</td><td>Assess the durability and damage behavior of cement-fly ash stabilized recycled aggregates under repeated stresses.</td><td>Type of soil Stress ratio</td></tr><tr><td>Bhattacharya and Pandey [37]</td><td>Clay</td><td>Three-point bending</td><td>Examine flexural fatigue life and properties of lime-laterite soil mixtures.</td><td>Type of soil Stress ratio Lime content Compaction methods</td></tr><tr><td>Nguyen and Mulders [38]</td><td>Silt Clay</td><td>Two-point bending</td><td>Understand the mechanical performance of treated silty soil.</td><td>Type of soil Stress ratio Extra cure condition Cure duration Cement content Lime content</td></tr></tbody></table>

treatment and investigating their effect on stress response under cyclic loading [43] tested sandy soils with cement contents ranging from 1 % to 9 %, focusing on the relationship between cement content and stress ratio [35] analyzed three sand types under different curing durations and stress ratios, the role of soil type and binder performance over time [39] investigated gravelly soils with various cement dosages and compaction methods, studying the effect of stress ratio [34] explored three lateritic soils stabilized with different cement contents, considering stress ratio in their analysis [40] evaluated the fatigue behavior of gravelly soils, varying cement content and stress ratio [3] conducted an investigation that included several test types (indirect tensile, uniaxial compression, and flexural tensile), using both laboratory and field core samples. They also examined the combined effects of curing duration, cement content, stress ratio, and testing configuration.

A total of 26 parameters were extracted for each test. These parameters were selected to best describe the test conditions. They cover characteristics of the specimens, details of the treatment applied, and information about the test methods used. The 26 parameters constitute the architecture of the database [41]. This structured categorization facilitates the classification of the tests and supports a more effective analysis of how these parameters influence fatigue behavior.

* Bibliographic identification

* Title of the article

* Publication date

* Physical parameters

* UCS [45]: The Unified Soil Classification System uses the following notations, G for gravel, S for sand, M for silt and C for clay.

* Simplified UCS, (Only the first letter of UCS is selected, e.g., SC becomes S)

* Gravel content (%): percentage of gravel particles (Φ>4.75 mm) with Φ the diameter of the grain

* Sand content (%): percentage of sand particles (0.075 mm < Φ < 4.75 mm)

* Fine particles content (%): percentage of silt and clay particles (Φ < 0.075 mm)

* Plasticity index (%) (PI) [46]

* Liquid limit (%) (LL) [46]

* Plastic limit (%) (PL) [46]

* Soil treatment

* Cement content (%): percentage of mass of cement in a specimen

* Cement classification (CEM I, CEM II, CEM III) [47]: CEM I corresponds to a hydraulic binder with at least 95 % clinker and <5 % secondary component. CEM II has at least 65 % clinker and a maximum of 35 % of other components like blast furnace slag, and fly ash. CEM III has between 36 and 80 % blast-furnace slag, and 20 to 64 % of clinker.

* Lime content (%): percentage of lime in a specimen

* Extra treatment (adding other binders like fibers)

* Curing duration (days): the time between the treatment and compaction of soil specimen, and the loads beginning.

* Curing temperature (°C): the temperature of the room where the sample is stored during the curing duration
■ Extra curing condition (adding another condition like immersion in water)

* Specimen characteristics

■ Volume (cm³)

■ Density (g/cm³)

■ Water content (%)

■ Compaction methods

* Test conditions

■ Type of test

■ Frequency (Hz)

■ SR (Stress Ratio) (-), The ratio between applied stress ($\sigma_{app}$) during the test and failure stress ($\sigma_{max}$) in a monotonic test

* Experimental results

■ Number of cycles to failure ($N_f$) (-)

Despite the variety of test methods, the dispersion of the experimental results is such that no consistent or significant trend can be attributed to any specific test configuration. In other words, the inherent variability in the data often exceeds the differences that might arise from the choice of fatigue test method. As a result, all test types are considered comparable in the context of this study, and their results are analyzed collectively within the database with statistical analysis. The use of statistical analysis is supported by the observation that fatigue performance metrics obtained from different methods generally fall within similar ranges when plotted against relevant parameters.

Not all data were directly available in the original publications, and some pretreatments were required to allow meaningful comparisons. For instance, USCS was adopted to harmonize soil types within the database because some soils were not described by this norm. In several cases, key values such as the number of cycles to failure and the stress ratio had to be extracted manually from graphs, introducing a margin of reading error. To minimize this uncertainty, a graph-reading application was used [48], allowing for more accurate axis definition and value extraction.

These error margins were quantified up to their maximum admissible values. For the stress ratio (SR), the error margin was defined as one-fourth of the smallest axis unit (0.1), resulting in an estimated error of ±0.025. For the number of cycles to failure, the error margin was calculated using a specific equation (Eq. (1)) that accounts for the logarithmic scale of the axis. This calculation yields a maximum possible error of 26 %. Although this value appears significant, it does not affect the general trends observed during the analysis. Furthermore, 26 % represents a worst-case scenario; the actual error is expected to be significantly lower but cannot be precisely determined.

$$ \frac{\Delta N_f}{N_f} = 10^{(\text{Smallest axis unit})} - 1 = 10^{0.1} - 1 = 0.26 \quad (1) $$

It should be noted that the database was systematically constructed. Therefore, several parameters cannot be applied to all soils and are considered as irrelevant as the "extra conditions" parameters, and the Plasticity Index parameter. The absence of this information does not reflect an oversight but rather the typical characteristics of the materials included in the studies. Regarding the Plasticity Index, a significant portion of the specimens tested are granular soils, such as sands and gravels, for which the Plasticity Index is not applicable due to their low or non-plastic behavior. This level of irrelevant data is expected, as detailed reporting of all testing conditions is not systematic across publications, especially for secondary parameters. These two parameters will not be included in the database analysis but remain valuable for providing a more comprehensive overview of the tests. Additionally, some parameters have <5 % irrelevant data; these are utilized in the analysis, but missing data are not considered when evaluating these specific parameters. In the database, irrelevant data are marked as 'Nan'.

## 2.2. Parameter analysis

In this section, a description of the parameter distributions is provided to assess the robustness and relevance of the analyses.

Fig. 1 illustrates the dispersion of categorical (non-numerical) parameters. Fig. 1.A shows the distribution of the type of cement used. The majority of specimens were treated with either CEM I (45 %) or CEM II (45 %), while CEM III was used in about 10 % of the cases.

Fig. 1.B show the classification of soils according to the USCS simplified system. The database covers a wide variety of soil types, ranging from clayey to gravelly soils. In total, 44 % of the samples are gravelly soils, 40 % sandy, 3 % silty, and 13 % clayey. Most sandy soils are classified as either silty sand or clayey sand, together making up two-thirds of the sandy samples. This underlines the diversity of soil types represented in the database. It is also worth noting that silty soils are underrepresented, suggesting a potential area for future studies on fatigue behavior. The diversity of the dataset is further reflected in the worldwide origins of the soils: the USA, Brazil, France, China, and India.

The compaction methods applied across the dataset, include static (one or two layers) and vibratory compaction, these being the most common methods.

The last non numerical parameter is the type of test use to evaluate the fatigue phenomenon. Fig. 1.C reveals that around 2/3 of the tests were two-point, three-point, or four-point bending tests, followed by splitting fatigue tests (15 %). This distribution can be compared with other material such as alkali active concrete [25], where approximately 80 % of the fatigue tests are bending test, while other configuration, including cyclic uniaxial compression can also be found. The cyclic uniaxial compression test has been directly compared for the same soil with bending tests in [3] providing an additional justification for including and comparing data obtained from different testing methods.

Fig. 2 presents histograms for the numerical parameters, Fig. 2.A shows that most soils were treated with 3–5 % cement, although some silty soils were treated with up to 9 %, reflecting different stabilization approaches. The 3–5 % range corresponds to typical field practice, being both technically effective and economically efficient. Using >5 % is generally not cost-effective but explains the data distribution. The few experiments with 9 % cement were included for exploratory purposes, providing the upper bounds of the dataset for this parameter. Some specimens were treated exclusively with lime, explaining the 0 % cement values. These trends are consistent with typical cement treatment practices [20].

Fig. 2.B shows curing duration in days. Three main clusters can be seen: the first around 0–28 days, the second around 90–120 days, and the last around 180 days. These clusters reflect varying experimental designs focusing on the effect of curing time.

Fig. 2.C shows a wide range of loading frequencies, from 1 Hz to 50 Hz. Higher frequencies, particularly 30 to 50 Hz, are mostly associated with two-point bending tests, aiming to simulate real fatigue loading conditions.

Fig. 2.D shows the number of cycles to failure. More than half of results lie between 10⁴ and 10⁶ cycles, though some values fall below 10³, fewer than 70 tests in total. This variation provides useful insight into fatigue behavior across different materials and conditions.

Fig. 2.E shows the Stress Ratio (SR), with >55 % of values concentrated between 0.5 and 0.7. However, some values exceed 1.0, reaching up to 1.6. This may result from sample heterogeneity, compaction variability, or the destructive testing process used to determine $\sigma_{max}$. Since fatigue alters the internal structure of treated soils, variations can emerge during testing.

We plotted the Casagrande diagram only for fine-grained soils, as PI, LL, and PL values are only available for materials with a significant fine fraction; consequently, gravelly soils are not represented. Fig. 3 shows that the Plasticity Index (PI) ranges from 6 % to 22 %, placing the samples within the CL-ML, CL, or ML classes, corresponding to low-plasticity silts and clays. These three parameters will not be used in
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Distribution of Non-Numerical Parameters, Fig. 1.A: Cement Classification, Fig. 1.B: Simplified classification of Soils, Fig. 1.C: Testing Methods Used.</figcaption></figure>

the subsequent analyses. Soil treatment is restricted to low-plasticity materials, as recommended by [1].

Regarding lime treatment, two distinct groups emerge: one at 1 % lime content, corresponding to around 50 specimens (mostly treated with both cement and lime), and another group treated with 5 % lime only. Over 90 % of soil are treated with cement.

Curing temperature is predominantly 20 °C, consistent with the standard [1], although a small number of specimens were cured at 50 °C to accelerate reactions.

Finally, the database includes five distinct specimen sizes, which align with the different types of tests performed. The specimen volumes correspond closely to the type of test performed, with five distinct specimen sizes identified in the database.

Table 1 summarized the objectives of each study [44,2,38], and [35] aimed to reach one million cycles. To achieve this, they carefully selected SR values, which generally fall between 0.5 and 0.7. In contrast, [39] explored different stress ratios for the same soil or different treatments under the same SR. These varying methodologies result in a broader dataset distribution, offering a more comprehensive understanding of fatigue behavior in treated soils.

Table 2 summarizes the numerical parameters, including the mean, standard deviation (std), first (Q1) and third (Q3) quartiles, median, minimum (min), and maximum (max) values. Regarding curing duration, three distinct groups can be identified: Q1 and the median both lie at 28 days, Q3 at 120 days, and the mean is 78 days, reflecting a broad distribution of experimental protocols. In addition, the table confirms that the database is primarily composed of sandy and gravelly soils, with median values of 41.5 % for sand content and 39 % for gravel content across all samples.

# 3. Results and discussion

## 3.1. Global analysis

### 3.1.1. Wöhler diagram

To analyze the database, a Wöhler diagram, also known as an S-N curve (Stress Number of cycles) [48], was used. This diagram shows the relationship between the stress ratio (SR) and the number of cycles to failure on a logarithmic scale. This type of representation is commonly

used in the literature to study material fatigue and to derive fatigue equations. Fig. 4 displays all the data from the database plotted on a Wöhler diagram, providing an overview of the results. Three distinct fatigue behavior regions are typically observed [49]: the first region is the low fatigue area, highlighted in red, which corresponds to high stress ratios and a low number of cycles to failure. The second region, shown in brown, covers stress ratios between 0.75 and 0.60 and features a slope where the number of cycles increases as the stress ratio decreases. The third region is the plateau area, marked in green, where the number of cycles exceeds one million and failure occurs randomly regardless of the stress ratio. In this study, the plateau begins at stress ratios below 0.60.

Fig. 4 shows some data points aligned at specific stress ratios, indicated by a black arrow. This is explained by the natural variability of fatigue behavior due to soil heterogeneity and the differences in tested parameters such as cement content, soil type, and curing duration for the same stress ratio.

### 3.1.2. Statistical matrix correlation

Correlations were performed to understand how the parameters interact, especially with the number of cycles to failure (Nf). Correlations can be positive or negative. A positive correlation means that as one variable increases, the second also increases, while a negative correlation means that when one variable increases, the second decreases. To select the appropriate test for assessing parameter correlations, the distribution of each parameter was evaluated. The Shapiro-Wilk test [50] was applied to all numerical parameters and revealed that none followed a normal (Gaussian) distribution. Therefore, to account for these distributions, the Spearman test [51] was chosen to analyze the correlations. Fig. 5 shows the correlation matrix obtained using the Spearman test for all data. The axes are composed by the parameters of the base and the correlation results is between -1 and 1. Some well-known relationships were confirmed, such as between the plasticity index (PI) and the percentage of silt and clay in the soils, the liquid limit (LL) and plastic limit (PL), density and water content, as well as water content and fine particle content. Some correlations were more surprising. Notably, there is a strong correlation of 0.74 between curing temperature (°C) and lime content.

Regarding correlations with the number of cycles to failure, the stress ratio (SR) has the strongest correlation at -0.72. The next most signif-
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Histograms of numerical parameters Fig. 2.A: histogram of cement content, Fig. 2.B: histogram of duration of cure, Fig. 2.C: histogram of frequency, Fig. 2.D: histogram of stress ratio (SR), Fig. 2.E: histogram of cycles to failure.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Casagrande diagram.</figcaption></figure>

icant correlations are with frequency (0.47) and density (−0.36). Other parameters show weaker correlations with $N_f$, including water content (0.34), cement content (0.32), curing duration (0.26), and fine particle content (0.31).

### 3.1.3. Principal component analysis (PCA)

PCA is a statistical method used to simplify complex datasets by transforming the original variables into a new set of uncorrelated variables called principal components. These components capture the maximum variance in the data while reducing its dimensionality [52]. The results are often visualized using a correlation circle, where two dimensions are shown on axes ranging from −1 to 1.

Fig. 6 presents PCA resulting from the whole database except PL,LL, PL, as PL, LL, PL are not measured on gravelly soils as explained previously. High values indicate that parameters contribute either positively or negatively to the axis dimension. The first two dimensions explain about 57 % of the variance in the dataset.

Fines particles content, water content, frequency, volume of specimen, cement content, gravel content, density, curing temperature and lime content are represented on Fig. 5. It can be seen that lime content and curing conditions are close to each other on the circle, which means these two parameters are correlated. Conversely, cement content is positioned opposite to them, indicating a negative correlation. Parameters that form a 90° angle have no correlation with each other. The other parameters affecting the first two dimensions correspond to well-known relationships such as liquid limit, water content, plasticity index, and fine particle content. Some expected relationships appear in Fig. 6: water content, and fine particles are positively correlated, while these
<table><caption>**Table 2**
Statistics of numerical parameters in the database.</caption><thead><tr><th></th><th>Mean</th><th>Std</th><th>Min</th><th>Q1</th><th>Median</th><th>Q3</th><th>Max</th></tr></thead><tbody><tr><td>Gravel content (%)</td><td>38.9</td><td>31.8</td><td>0.00</td><td>5.00</td><td>41.5</td><td>70.0</td><td>78.0</td></tr><tr><td>Sand content (%)</td><td>40.9</td><td>21.7</td><td>5.00</td><td>25.5</td><td>39.0</td><td>49.0</td><td>100</td></tr><tr><td>Fines particles content (%)</td><td>20.3</td><td>22.4</td><td>0.00</td><td>3.0</td><td>5.0</td><td>33.0</td><td>95.0</td></tr><tr><td>PI (%)</td><td>11.3</td><td>4.92</td><td>6.20</td><td>8.00</td><td>8.00</td><td>18.0</td><td>21.0</td></tr><tr><td>LL (%)</td><td>30.7</td><td>6.57</td><td>23.5</td><td>27.0</td><td>27.0</td><td>38.6</td><td>43.0</td></tr><tr><td>PL (%)</td><td>19.4</td><td>2.91</td><td>13.0</td><td>19.0</td><td>19.0</td><td>20.3</td><td>25.0</td></tr><tr><td>Cement content (%)</td><td>–</td><td>1.75</td><td>0.00</td><td>3.00</td><td>4.00</td><td>5.00</td><td>9.00</td></tr><tr><td>Lime content (%)</td><td>–</td><td>1.48</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>5.00</td></tr><tr><td>Duration of the cure (Days)</td><td>75.0</td><td>71.0</td><td>3.00</td><td>28.0</td><td>28.0</td><td>120</td><td>360.0</td></tr><tr><td>Curing conditions (°C)</td><td>23.9</td><td>9.07</td><td>20.0</td><td>20.0</td><td>20.0</td><td>20.0</td><td>50.0</td></tr><tr><td>Sample volume (cm³)</td><td>2.50</td><td>1.70</td><td>–</td><td>1.60</td><td>2.70</td><td>2.70</td><td>5.40</td></tr><tr><td>Density (g/cm³)</td><td>2.1</td><td>0.26</td><td>1.69</td><td>1.88</td><td>2.21</td><td>2.35</td><td>2.49</td></tr><tr><td>Stress ratio (-)</td><td>0.7</td><td>0.20</td><td>0.18</td><td>0.56</td><td>0.65</td><td>0.80</td><td>1.68</td></tr><tr><td>Frequency (Hz)</td><td>10.9</td><td>11.1</td><td>1.00</td><td>1.83</td><td>10.0</td><td>10.0</td><td>50.0</td></tr><tr><td>Number of cycles to failure (-)</td><td>1.57×10⁶</td><td>5.05×10⁶</td><td>4.00</td><td>6.34×10³</td><td>7.00×10⁴</td><td>4.94×10⁵</td><td>5.25×10⁷</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Wöhler diagram for all data.</figcaption></figure>

parameters are negatively correlated with sample density. Cement content shows a slight negative correlation with lime content, which makes sense since these binders are often used as alternatives rather than together in soil stabilization. Fig. 7 was plotted to better understand the unexpected correlation between lime content and curing temperature. It reveals that this correlation is due to test conditions. Fig. 7 shows curing temperature plotted against stress ratio, with lime content indicated by a color scale from yellow (0 %) to blue (5 %). Soils treated exclusively with a high lime content were cured at 50 °C, while the other tests were conducted between 20 °C and 30 °C. In some cases, 1 % lime was combined with cement. This explains why the Spearman test shows a strong correlation between high curing temperature and high lime content. However, this does not reflect the actual material behavior. Recognizing these types of correlations is important to avoid drawing incorrect conclusions.

## 3.2. Parameter-based analysis

In this section, the analysis focuses on parameters that have the most significant influence on fatigue resistance. Well-established factors are not detailed here, except for cement content as it is a key-parameter. The analysis of other parameters takes into account their potential interactions, though limitations remain due to variability in the experimental results. Only parameters for which sufficient data are available across the studied ranges (e.g., curing duration) are used to draw reliable conclusions. This ensures that the trends discussed reflect meaningful influences on fatigue behavior rather than effects of sparse data.

### 3.2.1. Curing duration parameter

The correlation between curing duration and $N_f$ is low (0.26), but the p-value is very small ($p = 2.52 \times 10^{-6}$), indicating that this correlation is statistically significant and not due to chance. Typically, a p-value greater than 0.05 is considered unacceptable.

To analyze more closely, curing duration was divided into three categories based on the distribution in the database (Table 2 and Fig. 2. B) and common practices reported in the literature [1]. These categories are: short term (0–28 days), long term (80–120 days), and very long term (>120 days). Spearman correlation tests were then performed separately for each category, yielding to:

* Short term (0–28 days): correlation of 0.077 with a p-value of $1.8 \times 10^{-3}$

* Long term (80–120 days): correlation of 0.506 with a p-value of $1.6 \times 10^{-9}$

* Very long term (>120 days): correlation of 0.543 with a p-value of $1.38 \times 10^{-7}$

These results reveal an interesting and original pattern. In the short term, there is almost no correlation between curing duration and $N_f$, meaning that curing time has little effect on fatigue failure during this period and other parameters appear to have a greater impact on fatigue resistance. In the long and very long-term, correlations are much stronger, making curing duration one of the most significant parameters related to $N_f$ after the stress ratio (SR). Moreover, the correlation increases with longer curing times. This can be explained by the fact that chemical reactions in treated soils are mostly completed around 90 days [1], which enhances the mechanical properties of the soil. It would be interesting to further investigate cement content, cement classification, and fine particle content using this short-, long-, and very long-term cure.

Fig. 8 presents Wöhler diagrams for each curing duration category. Fig. 8.A shows the short-term, Fig. 8.B the long-term, and Fig. 8.C the very long-term. Data covers almost all stress ratios between 0.8 and 0.5. 212 data represent short term, 114 data long term and 84, very long term. Moreover, the data are represented by square for CEM II and by round dot for CEM I and by color (dark violet to yellow) for the cement content. The short-term data (Fig. 8.A) displays a wide spread across three orders of magnitude for the same SR; the long-term (Fig. 8.B) narrows this spread to about two or three orders of magnitude; and the very long-term (Fig. 8.C) shows an even more linear trend in this semilogarithmic scale, with a distribution spanning only one order of magnitude.

Table 3 shows the average number of cycles to failure for different stress ratio categories and curing duration. To ensure reliability, the table provides an average only when >10 tests are available, avoiding calculations based on too few data points. This level of detail is rarely
<figure><img src="image_7.png" /><figcaption>Fig. 5. Correlation matrix of parameters by Spearman test.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. PCA correlation circle.</figcaption></figure>

found in the literature, especially for high stress ratios, where available experimental data remain scarce. It clearly shows that lowering the stress ratio leads to an increase in the average number of cycles to failure. Additionally, there is a significant increase, up to an order of magnitude, between the long and very long curing durations for the same stress ratio. This reflects a markedly different fatigue response even though the correlations are similar. For example, for 0.6 to 0.7 values of stress ratio, the average number of cycles is $2.3 \times 10^5$ for the long curing duration and rises to $1.1 \times 10^6$ for the very long curing

<figure><img src="image_9.png" /><figcaption>Fig. 7. Curing temperature display by SR sort by lime content.</figcaption></figure>

duration. Table 3 also shows that the results become more consistent as the curing duration increases. This trend is especially clear for the very long curing duration.

Curing duration has been investigated in previous studies [38], but with a limited number of distinct curing times and without detailed categorization. Analyses often relied on general distinctions between short term and long-term curing, based on common practice. In contrast, the present study introduces a more refined classification that reveals a clear difference in fatigue response when very long curing durations are considered separately. The results show that this third category is necessary to capture a shift in material behavior that is not visible when using only the standard two-level approach. Beyond a certain curing time, the material exhibits significantly enhanced and more consistent fatigue performance, underscoring the importance of rethinking how curing duration is incorporated into fatigue analyses. It is important to take this point into account for infrastructure applications
<figure><img src="image_10.png" /><figcaption>Fig. 8. Wohler diagrams by curing duration, sorted by the content of cement and by the classification of cement use, Fig. 8.A: short term (0–28 days), Fig. 8.B: long term (81–120 days), Fig. 8.C: very long term (>120 days).</figcaption></figure>

<table><caption>**Table 3**
Mean number of cycles to failure (-) as a function of curing duration categories (days) and stress ratio (-).</caption><thead><tr><th rowspan="2">Stress ratio (-)</th><th colspan="3">Curing duration (days)</th></tr><tr><th>0–28 days</th><th>81–120 days</th><th>>120 days</th></tr></thead><tbody><tr><td>0.4–0.5</td><td>2.23×10<sup>6</sup></td><td>-</td><td>-</td></tr><tr><td>0.5–0.6</td><td>7.96×10<sup>5</sup></td><td>4.49×10<sup>6</sup></td><td>5.38×10<sup>6</sup></td></tr><tr><td>0.6–0.7</td><td>1.84×10<sup>5</sup></td><td>2.28×10<sup>5</sup></td><td>1.13×10<sup>6</sup></td></tr><tr><td>0.7–0.8</td><td>2.75×10<sup>4</sup></td><td>-</td><td>2.05×10<sup>4</sup></td></tr><tr><td>0.8–0.9</td><td>3.10×10<sup>4</sup></td><td>7.70×10<sup>4</sup></td><td>2.16×10<sup>3</sup></td></tr><tr><td>0.9–1.0</td><td>3.46×10<sup>3</sup></td><td>5.41×10<sup>3</sup></td><td>-</td></tr></tbody></table>

### 3.2.2. Cement

#### 3.2.2.1. Cement content.
The second parameter with the strongest correlation to the number of cycles to failure is the cement content in the samples. The influence of cement content on fatigue resistance is already well established [34,52]. This relationship is illustrated in Fig. 8.A for the short curing duration, where a clear trend appears: for the same stress ratio, the number of cycles to failure increases with cement content between 1 and 6 percent. Beyond that range, analysis is limited due to insufficient data, as the database contains almost no samples with cement content of 7 percent or higher.

The variation in fatigue resistance related to cement content is particularly interesting when considering curing durations longer than 120 days (Fig. 8.C). In this case, the data points appear more mixed, without the clear trend observed in the short-term results. There are only
three exceptions to this pattern, silty soils with very high cement content (9 percent). A possible explanation is that extremely high cement content may actually reduce fatigue resistance, based on the limited observations available. Additional data would be necessary to confirm or refute this hypothesis.

These results are consistent with findings reported in [3,33], where increased cement content is generally associated with improved fatigue resistance.

### 3.2.2.2. Cement classification.
Cement classification is not usually considered as a key parameter when studying fatigue behavior in treated soils. This parameter has never been directly compared or systematically analyzed in relation to fatigue performance across different studies, making its role unexplored in the existing literature.

Fig. 8 presents data for three types of cement (CEM I, CEM II, and CEM III), but there is insufficient data for specimens treated with CEM III (<50) to draw any conclusions. Therefore, the analysis in this study will focus only on CEM I and CEM II. Initially, specimens treated with CEM II appear to show better fatigue resistance, especially for short-term curing. However, there is not enough long-term data across all stress ratios to make a precise analysis. The very long-term curing data, which is more consistent, further highlights the influence of curing duration. To confirm this observation, an analysis was performed using the median number of cycles to failure for each stress ratio in the short-term category (see Table 4). Table 4 shows that tests using CEM II exhibit about one order of magnitude higher number of cycles to failure than those using CEM I, for stress ratios between 0.5 and 0.8. These stress ratios were chosen because of the sufficient amount of data available to support the analysis. A second analysis used the Kruskal-Wallis test [53] to assess the relationship between cement classification (CEM I and CEM II) and the number of cycles to failure for stress ratios from 0.4 to 0.8. This non-parametric test was selected due to the non-normal distribution of the data. The test result exceeded 90, with a p-value of 1.8 × 10⁻²¹, indicating a very strong and statistically significant relationship between cement type and fatigue life in this range of stress ratios.

### 3.2.3. Frequency and fine particle content

Frequency and fine particle content are two parameters that have received limited attention in the context of fatigue behavior in treated soils. While they are sometimes mentioned in individual studies, their influence is rarely analyzed in a systematic way, and comparisons across different datasets remain scarce. In this section, we aim to explore the potential impact of these two parameters on fatigue performance, using the available data in the database to identify possible trends or correlations.

Figs. 5 and 6 suggested a strong correlation between the number of cycles to failure ($N_f$) and the loading frequency. However, the Wöhler diagrams in Fig. 9 do not clearly show any significant impact of frequency on $N_f$. The data are split into the short term, long term and the very long term. The tests were conducted using frequencies ranging from 0 to 50 Hz represented on the scale by colors from dark violet to yellow, with higher frequencies corresponding to yellow colors. Overall, the results do not reveal a consistent influence of frequency on fatigue life. This contrast between the correlation results and the Wöhler diagrams reflects a phenomenon already observed in other materials, such as those studied by Kovacs et al. [54].

<caption>**Table 4**</caption>

Median value of number of cycles to failure (-) per stress ratio and classification of cement, all data.

<table><thead><tr><th rowspan="2">Stress ratio (-)</th><th colspan="2">Classification of cement</th></tr><tr><th>CEM I</th><th>CEM II</th></tr></thead><tbody><tr><td>0.5-0.6</td><td>200 000</td><td>1 200 000</td></tr><tr><td>0.6-0.7</td><td>50 000</td><td>150 000</td></tr><tr><td>0.7-0.8</td><td>5 000</td><td>26 000</td></tr></tbody></table>

To better interpret this discrepancy, an additional graph was created, displaying frequencies alongside the corresponding stress ratios. Fig. 10 shows the relationship between loading frequency and the number of cycles to failure on a logarithmic scale for all data. Test frequencies range from 0 to 50 Hz, with most results concentrated around 10 Hz, 30 Hz, and 50 Hz. The color scale represents the stress ratio, with darker colors corresponding to higher values and lighter colors to lower values of SR. Fig. 10 reveals that the tests conducted at 50 Hz generally involve lower stress ratios compared to those conducted at frequencies between 1 and 10 Hz. This leads to longer fatigue lives being associated with higher frequencies and shorter lives with lower frequencies. As a result, it becomes difficult to establish a clear, isolated effect of frequency on $N_f$, and no conclusive statement can be made regarding the absence of a relationship.

Fig. 11 adds results by presenting Wöhler diagrams according to curing duration, with the results shown in three separate sub-graphs, with a color scale from dark violet to yellow based on fine particle content from 0 % to 100 % respectively. It appears that soils with 20 to 40 % fines exhibit slightly better fatigue resistance. This is particularly noticeable in Fig. 11. B at a stress ratio of 0.5, where a visible difference emerges between this range of fine content and the rest of the data.

Overall, this section highlights the need to interpret correlation results carefully. Correlations can suggest patterns, but they do not always reflect direct causation, especially when underlying conditions or interactions between variables are not fully understood. Further investigation into soil microstructure could provide valuable insights into these complex behaviors.

### 3.3. Impact of parameters and discussion

As previously discussed, the ranking of parameter influence is carried out by taking the curing duration into account. The analysis is divided into two main groups: one for short- and long-term curing, and another for very long-term curing. For the short- and long-term, the most influential parameter is SR, followed by cement content, cement type, and soil type. Beyond these, it becomes challenging to establish a clear and reliable ranking of the remaining parameters. In contrast, for the very long-term, the order of influence shifts. SR remains the most significant factor, but it is followed by curing duration, then cement content, and finally cement type.

To better understand the factors controlling fatigue performance in treated soils, the influence of key parameters was analyzed and compared across the available dataset.

As previously discussed, the influence of curing duration on fatigue life was analyzed by separating the data into short-, long-, and very long-term categories. It is important to note that [1] do not differentiate between long-term and very long-term curing, which can mask important trends and there is no difference in fatigue behavior between 180 and 360 days of cure in the reaction to fatigue behavior. Our analysis shows that curing duration has little effect for the fatigue resistance in the short term, whereas its influence becomes much stronger for long and very long-term curing. This highlights that fatigue resistance in treated soils increases significantly once hydration and pozzolanic reactions are more advanced. Consequently, studies that do not separate long and very long-term curing may underestimate the true impact of curing on fatigue performance.

Another parameter of interest is cement type. The analysis revealed clear differences between CEM I and CEM II, with CEM II showing more favorable behavior under cyclic loading. This observation can be linked to the chemical composition of CEM II, which contains supplementary cementitious materials. A parallel can be drawn with alkali-activated concretes, which share a similar composition profile. Several studies on alkali-activated concrete have also reported superior fatigue resistance compared to Portland cement-based concretes [25,54,55], reinforcing the idea that binders with alternative compositions can offer
<figure><img src="image_11.png" /><figcaption>**Fig. 9.** Wohler diagrams by curing duration, sorted by frequencies, Fig. 9.A: short term (0–28 days), Fig. 9.B: Long term (81–120 days), Fig. 9.C: very long term (>120 days).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Repartition of frequencies by the number of cycles to failure, sorted by SR.</figcaption></figure>

improved performance in cyclic loading conditions.

The role of loading frequency in fatigue behavior remains an open question for treated soils. When compared with concrete, which has been studied much more extensively, the literature also provides different conclusions: some studies report negligible influence of fre-

quency on fatigue performance, while others identify effects, particu- larly at higher frequencies [10,25]. In our dataset, no clear effect of frequency on N<sub>f</sub> was observed. However, this does not imply that fre- quency is irrelevant, as the distribution of stress ratios across test fre- quencies introduces a bias that prevents a mathematical isolation of the frequency effect. These discrepancies, observed in both concrete and treated soils, suggest that the influence of frequency may depend on the coupling with other factors, such as stress ratio or material composition.

For fine particles, one possible explanation for the observed results is the increased cohesion and enhanced mechanical behavior resulting from modifications in the soil microstructure following treatment. These changes may contribute to improved resistance under cyclic loading. However, this observation is limited by the available data, and further experiments or dedicated studies would be necessary to confirm and quantify this effect.

Finally, it should be acknowledged that the diversity of experimental conditions across the studies introduces limitations to the analysis. Not all factors could be explored in depth, either because of representa- tiveness issues or because certain conditions were not sufficiently documented. These aspects inevitably restrict the scope of the conclu- sions, and future work should aim to address them more thoroughly.
<figure><img src="image_13.png" /><figcaption>**Fig. 11.** Wohler diagrams by curing duration, sorted by fines particles percentage, Fig. 11.A: short term (0–28 days), Fig. 11.B: long term (81–120 days), Fig. 11.C: very long term (>120 days).</figcaption></figure>

## 4. Empirical model

In geotechnical engineering, understanding and predicting material fatigue are crucial for ensuring the durability and reliability of structures. This is particularly relevant with treated soils that are subjected to cyclic loading during their use as pavements, and embankments. Fatigue modeling helps engineers assess short-term, long-term, and very long-term performance and optimize design parameters to prevent failure. Table 5 summarizes the stress-based fatigue equations proposed for treated soils in the literature. It highlights a variety of approaches, including linear relationships in logarithmic space, logarithmic or power-law scaling. Each equation involves specific parameters, such as intercepts, slopes, regression coefficients, and reduction factors, which characterize the material response under cyclic loading. For instance, some models distinguish between base and adjusted slopes to account for reduction effects [39], while others relate fatigue life directly to stress levels [3,38,37,44,34] or normalized stress ratios. The diversity of these equations illustrates the different strategies employed to capture the fatigue behavior of treated soils and emphasizes the need for a model that can robustly predict fatigue across various conditions.

An empirical model was developed to describe the fatigue behavior based on the available database. Several existing models were designed to capture fatigue behavior in the high-cycle regime, where the number of cycles to failure is very large and the stress ratios are comparatively low giving linear relationships. Which is consistent with infrastructure applications such as roads and railways, where the design typically targets between 10⁵ and 10⁶ cycles [24]. While such formulations can provide acceptable fits in that domain, they tend to oversimplify the

<table><caption>**Table 5**
Stress-based equations of fatigue for treated soil in the literature.</caption><thead><tr><th>Articles</th><th>Equation of fatigue</th><th>Parameters of equations</th></tr></thead><tbody><tr><td>[39]</td><td>lg SR = a - b' lg Nf<br>lg SR = a - b lg Nf<br>b = (1 - R) b'</td><td>N: Equivalent fatigue life<br>a: Intercept<br>b': Base gradient (initial<br>the slope before adjustment)<br>b: Adjusted gradient (slope<br>after adjustment)<br>R: Reduction factor</td></tr><tr><td>[34]<br>[3]</td><td>lg Nf = c - d SR<br>lg (Nf) = aSR + b</td><td>c, d: Regression coefficients<br>a: Slope<br>b: Intercept<br>Nf: Fatigue life</td></tr><tr><td>[40]</td><td>Nf = a(1/t1)<sup>b</sup><br>Nf = (1/t2)<sup>b</sup></td><td>t1 = σ/1.3 MPa,<br>t2 = σ/Sf<sub>v</sub><br>Sf<sub>v</sub> is the flexural strength<br>a, b: regression coefficients</td></tr><tr><td>[35]<br>[36]<br>[37]</td><td>SR = 1 - a ln (Nf)<br>SR = a ln (Nf) + b<br>SR = A - B lg (Nf)</td><td>a: Slope of the S-N<sub>f</sub> curve<br>a, b: regression coefficients<br>A: Depends on compaction<br>B: Slope of the regression<br>curve</td></tr><tr><td>[38]</td><td>SR = A lg(Nf) + B</td><td>A: Slope<br>B: Intercept</td></tr></tbody></table>

behavior at the extremes and neglect the low-cycle region, where the material response is governed by different mechanisms. Previous studies [42,39] have proposed linear equations in a semi-logarithmic space, often passing through the point (SR = 1, Nf = 0), which represents the
theoretical maximum stress ratio causing failure in a single cycle. However, the shape formed by the data in Fig. 4 is closer to an S-curve, which is one of the three typical forms of fatigue curves identified in the literature [32,56]. Given this observation, an arctangent (arctan) function was selected to represent the relationship between stress ratio and number of cycles to failure. The arctan function presents several advantages for modeling fatigue behavior. It begins with an initial plateau at high stress ratios, capturing the relatively stable and low-cycle fatigue behavior. It then transitions smoothly through a steep decrease, representing the region where the number of cycles increases rapidly as the stress ratio decreases. Finally, it levels off into a second plateau at lower stress ratios, which reflects the high-cycle fatigue domain, where failures are more scattered and less directly governed by stress ratio alone. This form provides a more realistic and conservative representation of fatigue life over a broad range of stress ratios, particularly when compared to linear models that may oversimplify behavior at the extremes. Unlike straight-line approximations, which tend to fit better in the high-cycle domain but ignore the low-cycle fatigue behavior, the arctan formulation allows for a more rigorous representation across the entire range of stress ratios. This is made possible by the presence of sufficient data at both ends of the curve, particularly at high stress ratios (SR > 0.8), which are typically underrepresented in traditional fatigue datasets. The availability of these data points enables the identification of the plateau and inflection regions, justifying the use of a non-linear model.

To perform the regression and take in account the variability of the results in fatigue test, median values were calculated for each 0.1 increment in stress ratio, provided that there were at least 10 data points within the range. This method reduces the influence of outliers and local scatter, ensuring more robust trend detection. A theoretical point at (SR = 1, Nf = 0) was also added to the dataset, to anchor the upper limit of the curve and maintain consistency with physical understanding. The resulting model, built on an arctan function with four parameters, provides a continuous representation of fatigue behavior (Eq. (2)):

$$SR = \alpha \tan^{-1}(\beta \lg(N_f) + \gamma) + \delta \qquad (2)$$

α (-): Controls the vertical scale of the curve.

β (-): Controls the slope of the curve.

γ (-): Adjusts when the transition happens along the log (Nf) axis.

δ (-): Controls the overall vertical positioning of the curve.

Fig. 12 evaluates the effect of cement content on fatigue behavior using the Wöhler diagram. All available data were included in the analysis, but only two cement content ranges had sufficient data: 4 % (grey curve, square points) and 5 % (green curve, round point), which are the most commonly used contents in soil treatment. Two separate

models were developed, one for each cement content. The curves are nearly identical for stress ratios between 1 and 0.9. However, a divergence appears below this range, where samples with 5 % cement content show better fatigue resistance. Toward the end of the curve, this gap narrows. The quality of the fit is strong, with R² values of 0.932 for 4 % and 0.983 for 5 %.

Fig. 13 investigates the influence of cement classification on fatigue behavior with the Wöhler diagram. Three types were considered, but the analysis focused on CEM I (green curve, round point) and CEM II (green curve, round point) due to limited data for CEM III. At high stress ratios, there is no noticeable difference between CEM I and CEM II. However, a separation becomes evident around an SR of 0.85 and continues to increase as the number of cycles rises. In the high-cycle fatigue plateau, the stress ratio difference reaches approximately 0.1 between the two cement types. This is a substantial difference for the same number of cycles to failure and even exceeds the difference caused by varying cement contents. The models fit the data well, with R² values of 0.935 for CEM I and 0.992 for CEM II.

The final model examines the influence of curing duration. Due to data constraints at individual SR intervals, the model is based on the entire dataset. Fig. 14 presents Wöhler diagrams with arctangent regressions for three curing duration categories. A predicted point for one million cycles is highlighted for each regression to indicate the long-life fatigue threshold, which can be considered the start of the third fatigue region.

For the regression, each median point was assigned a weight based on the number of test results used to calculate it. The more data available for a given SR, the higher the weight attributed to its median, thereby giving greater influence to more reliable medians in the regression. Each point on the graph represents a test, color-coded by curing duration. Points circled in black are the predicted SR values for one million cycles. The lines represent the regression curves.

In Fig. 14 the grey curve corresponds to the -28 day category, which displays the most scattered data. Despite this, the model's performance is strong, with an R² of 0.978, calculated from the medians rather than individual tests. This distinction is important, as the variability in test results makes it impractical to assess the model using raw data alone. The violet curve, representing 81-120 days of curing, has an R² of 0.940. The green curve, for samples cured >120 days, shows an R² of 0.975. This last regression appears almost linear, likely due to a lack of data at high stress ratios (above 0.8), which prevents the upper curve region from being well-defined. This explains the more linear appearance of the curve.

The parameters of all regressions are summarized in Table 6. Parameter α controls how steeply the curve drops in the central

<figure><img src="image_14.png" /><figcaption>Fig. 12. Empirical model in Wöhler diagrams by cement content, an arctan empirical model is done for 4 and 5 % of cement content, SR is predicted for 10⁶ cycles.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Empirical model in Wöhler diagrams by cement classification, an arctan empirical model is done for CEM I and CEM II, SR is predicted for 10⁶ cycles.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 14. Empirical model in Wöhler diagrams by curing duration, an arctan empirical model is done for each curing category, SR is predicted for $10^6$ cycles.</figcaption></figure>

<table><caption>**Table 6**
Parameters of arctan empirical model for different parameters, SR prediction for $10^6$ cycles and their corresponding $R^2$ values associated to the median of different SR.</caption><thead><tr><th>Parameters</th><th></th><th>α (-)</th><th>β (-)</th><th>γ (-)</th><th>δ (-)</th><th>SR (-) at N<sub>f</sub> = 10<sup>6</sup></th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td rowspan="3">Curing duration (days)</td><td>0-28</td><td>-0.213</td><td>1.129</td><td>-4.966</td><td>0.710</td><td>0.48</td><td>0.978</td></tr><tr><td>81-120</td><td>-0.182</td><td>1.249</td><td>-5.346</td><td>0.747</td><td>0.55</td><td>0.940</td></tr><tr><td>&gt;120</td><td>-0.399</td><td>0.205</td><td>-0.912</td><td>0.705</td><td>0.58</td><td>0.975</td></tr><tr><td rowspan="2">Cement content (%)</td><td>4</td><td>-0.180</td><td>1.471</td><td>-6.114</td><td>0.746</td><td>0.52</td><td>0.932</td></tr><tr><td>5</td><td>-0.188</td><td>1.202</td><td>-5.194</td><td>0.746</td><td>0.55</td><td>0.983</td></tr><tr><td rowspan="2">Cement classification</td><td>CEM I</td><td>-0.211</td><td>1.306</td><td>-6.542</td><td>0.697</td><td>0.51</td><td>0.935</td></tr><tr><td>CEM II</td><td>-0.190</td><td>0.811</td><td>-1.112</td><td>0.745</td><td>0.63</td><td>0.992</td></tr></tbody></table>

transition zone, values stay in a narrow range between -0.180 and -0.213. The only clear exception is for the >120-day curing group, where a drop to -0.399, indicating a much sharper transition. This is in line with what was already observed in the previous figures: very long curing leads to a clearly different fatigue behavior.

Parameter $\beta$ adjusts the spread or width of the transition. The values also stay quite close overall again, except for two cases. Both the >120-day curing and the CEM II cement show lower $\beta$ values (0.205 and 0.811 respectively), suggesting a more compressed transition in those curves. That could reflect a stronger or more reactive structure in these specific conditions.

Parameter $\gamma$, which shifts the curve horizontally along the stress ratio axis, shows the biggest variation ranging from -6.542 to -0.912. $\gamma$ directly influences where the drop in cycles happens. Here again, the >120-day curing and CEM II cases stand out, with the least negative values (-0.912 and -1.112), meaning failure is delayed and overall fatigue resistance is better.

By contrast, $\delta$ the upper asymptote barely changes, with values always around 0.70 to 0.75. Tests were conducted using three different values (0.7, 0.725, 0.75) of $\delta$ to evaluate the sensitivity of the models presented in the article. Each value was applied across all models, and the resulting predictions were compared. The analysis showed that varying $\delta$ had minimal effect on the predicted outcomes, confirming that using a fixed value of $\delta$ = 0.725 represents a reasonable simplification without significantly compromising accuracy and performance.

Finally, looking at the predicted stress ratio for one million cycles, the same two categories emerge again. The >120-day curing reaches 0.58, and CEM II climbs even higher to 0.63. These are clear indicators that both very long curing and cement type play an important role in improving high-cycle fatigue resistance.

A similarity is observed between the fitting parameters for the 0-28 day and 81-120 day curing durations, while the parameters for curing durations beyond 120 days differ significantly. This reinforces the importance of treating the very long-term category separately, rather than merging it with the long-term group. Overall, the results of the empirical model appear to be quite satisfactory. The model provides a more reliable prediction at both high and low stress ratios and aligns well with the observed behavior in the database.

To assess the performance of the proposed arctangent model. Results were compared with those of the previously published equations, summarized in Table 5. They were compared with the same protocol as the arctan model, median values were used for the regression to reduce the influence of outliers and ensure robust trend detection, and a theoretical point at SR = 1, $N_f$= 0 was included to anchor the upper limit of the curve.

Table 7 presents the $R^2$ values obtained for different parameters, including curing duration, cement content, and cement classification, across all models. Most of the previously proposed models rely on two parameters, while a model incorporate four parameters [39], and another one is based on a single parameter [35]. The single-parameter model exhibits the lowest $R^2$ values, reflecting its limited capacity to capture the variability in fatigue behavior, whereas models with two or four parameters show similar performances, suggesting that adding a third and fourth parameter provides only marginal improvement. Overall, these models generally exhibit lower $R^2$ values compared to the arctan formulation with three parameters. In some cases, such as for curing durations longer than 120 days where the fatigue behavior is closer to linear, classical models approach the $R^2$ of the arctan model but never surpass it. In all other situations, the arctan model consistently achieves the highest $R^2$, confirming its superior ability to capture the
<table><caption>**Table 7**
Comparison of R² values for different models across various parameters.</caption><thead><tr><th>Parameters</th><th>Parameter values</th><th>[39] R<sup>2</sup></th><th>[34] R<sup>2</sup></th><th>[3] R<sup>2</sup></th><th>[35] R<sup>2</sup></th><th>[36] R<sup>2</sup></th><th>[37] R<sup>2</sup></th><th>[38] R<sup>2</sup></th><th>Arctan model R<sup>2</sup></th></tr></thead><tbody><tr><td rowspan="3">Curing duration (days)</td><td>0-28</td><td>0.762</td><td>0.814</td><td>0.814</td><td>0.790</td><td>0.814</td><td>0.814</td><td>0.814</td><td>0.976</td></tr><tr><td>81-120</td><td>0.843</td><td>0.892</td><td>0.892</td><td>0.868</td><td>0.982</td><td>0.982</td><td>0.982</td><td>0.938</td></tr><tr><td>&gt;120</td><td>0.935</td><td>0.964</td><td>0.964</td><td>0.960</td><td>0.964</td><td>0.964</td><td>0.964</td><td>0.974</td></tr><tr><td rowspan="2">Cement content (%)</td><td>4</td><td>0.765</td><td>0.818</td><td>0.818</td><td>0.794</td><td>0.818</td><td>0.818</td><td>0.818</td><td>0.931</td></tr><tr><td>5</td><td>0.828</td><td>0.880</td><td>0.880</td><td>0.856</td><td>0.880</td><td>0.880</td><td>0.880</td><td>0.982</td></tr><tr><td rowspan="2">Cement classification</td><td>CEM I</td><td>0.711</td><td>0.756</td><td>0.756</td><td>0.737</td><td>0.756</td><td>0.756</td><td>0.756</td><td>0.934</td></tr><tr><td>CEM II</td><td>0.826</td><td>0.876</td><td>0.876</td><td>0.852</td><td>0.876</td><td>0.876</td><td>0.876</td><td>0.991</td></tr></tbody></table>

overall trend across both low-cycle and high-cycle regimes.

In conclusion, the proposed arctan model provides a robust and continuous representation of fatigue behavior for treated soils across all conditions, taking in account variations in curing duration, cement content, and cement classification. The model successfully captures both the low-cycle and high-cycle regimes and reproduces the trends observed experimentally, confirming the validity of the relationships identified in the database.

Based on the analysis of all parameters, it was observed that the upper asymptote parameter $δ$ remains almost constant across all regressions, and can therefore be fixed at $δ=0.725$ without compromising accuracy. This leads to a simplified three-parameter formulation (Eq. (3)):

$$SR = \alpha \tan^{-1}(\beta \lg(N_f) + \gamma) + 0.725 \quad (3)$$

This simplification preserves the model's ability to describe the full S-shaped fatigue curve while making it more practical for engineering applications.

Furthermore, the results highlight a distinct behavior for curing durations exceeding 120 days, where the curve is more linear and do not follow the S shape. This confirms the importance of separating long-term and very long-term categories, as merging them would obscure significant trends and reduce the reliability of predictions. Overall, the arctan model provides a reliable and interpretable tool for assessing fatigue performance from short-term to very long-term loading conditions.

## 5. Conclusion

This study presents a comprehensive analysis of a database containing 410 laboratory tests on the fatigue resistance of treated soils. The tests, extracted from 12 published articles from 1984 to 2025, cover different soil types and experimental conditions. The main outcomes can be summarized as follows:

* A detailed description and statistical analysis of the database parameters was carried out, showing the influence of stress ratio, curing duration, cement content, and soil composition on fatigue resistance.

* The analysis confirmed the critical role of curing duration, with very long-term curing (>120 days) producing a distinct and more homogeneous fatigue response.

* Cement classification was shown to matter, with CEM II-treated specimens consistently performing better than CEM I.

* The presence of a moderate amount of fines (20-40 %) improves fatigue resistance, likely due to its effect on soil microstructure.

* The influence of frequency was investigated, and the observed correlations appear to be linked more to experimental constraints than to intrinsic soil behavior.

* An empirical fatigue model based on an arctan function was developed, effectively capturing the S-shaped trend of fatigue curves and providing consistent predictions across short, long, and very long-term curing durations.

* Comparison with existing formulations demonstrated that the proposed model achieves higher predictive accuracy, while also offering a robust framework.

This study demonstrates the complexity of fatigue behavior in treated soils and highlights the value of combining data analysis with empirical modeling. For future researches, several aspects should be investigated in greater details using a systematic protocol to isolate the effects of individual parameters:

* Cement type and composition: explore a broader range of cement formulations, including CEM III, to evaluate their influence on fatigue behavior.

* Loading frequency: conduct targeted studies where frequency is the sole varying parameter on a given type of treated soil to clearly assess its effect.

* Soil microstructure and composition: investigate the role of soil grading, fines content, and microstructural characteristics in fatigue resistance.

* Binder combinations: examine the combined effect of cement and lime, as well as the effect of cement alone, to understand their respective contributions.

* Curing temperature: evaluate the influence of curing temperature, including accelerated curing conditions, on fatigue performance.

In addition, future work should consider fatigue testing under triaxial conditions, which provides better control over stress paths and more representative insights into fatigue behavior, even though such tests are time-consuming and require a large number of cycles. These investigations will help improve understanding of the mechanisms controlling fatigue in treated soils.

## Data availability

Data are available at https://doi.org/10.57745/LKOCQ0.

## CRediT authorship contribution statement

Vincent Krzewinski: Writing – original draft, Validation, Software, Data curation, Conceptualization. Nathalie Dufour: Writing – review & editing, Validation, Supervision, Methodology, Funding acquisition, Data curation, Conceptualization. Christophe Chevalier: Writing – review & editing, Validation, Supervision, Methodology, Funding acquisition, Data curation, Conceptualization. Julien Borderon: Writing – review & editing, Methodology, Data curation. Muriel Gasc-Barbier: Writing – review & editing, Validation, Supervision, Methodology, Funding acquisition, Data curation, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
# Data availability

Data will be made available on request.

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