# Determination of Mohr–Coulomb Failure Criterion of Cement-Treated Materials Using Mixture Design Properties

Mario Castaneda-Lopez, Thomas Lenoir, Luc Thorel, Jean-Pierre Sanfratello

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Mario Castaneda-Lopez, Thomas Lenoir, Luc Thorel, Jean-Pierre Sanfratello. Determination of Mohr–Coulomb Failure Criterion of Cement-Treated Materials Using Mixture Design Properties. Infrastructures, 2025, 10 (10), pp.267. 10.3390/infrastructures10100267. hal-05306308

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Article

# Determination of Mohr–Coulomb Failure Criterion of Cement-Treated Materials Using Mixture Design Properties

Mario Castaneda-Lopez <fnref n="1" />,<fnref n="2" /> ID, Thomas Lenoir <fnref n="1" />,* ID, Luc Thorel <fnref n="2" /> ID and Jean-Pierre Sanfratello <fnref n="3" />

## Abstract

The compressive, tensile, and shear strength properties of two cement-stabilized soils (CSS) treated with 2% to 4% of cement are investigated for several different curing times at several densities. The measured Mohr–Coulomb (MC) shear strength features, cohesion (c), and friction angle (φ) are compared with values reported in the literature for similar materials and are subject to debate depending on the estimation methods used. In addition, an alternative geometric criterion based on indirect tensile strength (ITS) and unconfined compressive strength (UCS) is evaluated. The results show that the value of c determined using the alternative criterion is slightly higher than the value of c measured using the direct shear (DS) test. A relationship between mixture variables and c is established and validated by combining numerical and experimental approaches. The friction angle appears to be constant, independent of mixture parameters. This parameter is underestimated using the geometric approach.

**Keywords:** earthworks; cement-stabilized materials; ITS test; UCS test; cohesion; friction angle; correlation analysis

## 1. Introduction

Cement stabilization is a technique commonly used to improve the engineering and mechanical properties of in situ soils, according to the stress rates imposed by civil engineering projects [1]. Cement-stabilized soils (CSSs) are used in varying civil engineering applications, including embankments, slope protection, foundations, pavements, and railway construction [2,3]. Using in situ soils with hydraulic binders, by reducing reliance on quarry materials [4], minimizes some environmental impacts and reduces infrastructure costs [5].

It is well known that the cement content is a key factor in improving the mechanical properties of CSSs [6–8]. The amount of hydraulic binder present in CSSs typically ranges from 5% to 10% of the total dry material. However, the cement industry is a major contributor to anthropogenic greenhouse gas emissions, accounting for approximately 8% of global CO₂ emissions [9,10]. In road geotechnics, due to the large amount of material involved, reducing the cement content of CSSs while optimizing their mechanical properties can provide a real environmental benefit.

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updates

Academic Editor: Hugo Silva

Received: 11 July 2025

Revised: 28 September 2025

Accepted: 3 October 2025

Published: 9 October 2025

**Citation:** Castaneda-Lopez, M.; Lenoir, T.; Thorel, L.; Sanfratello, J.-P. Determination of Mohr–Coulomb Failure Criterion of Cement-Treated Materials Using Mixture Design Properties. *Infrastructures* **2025**, *10*, 267. https://doi.org/10.3390/infrastructures10100267

**Copyright:** © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

<footnote n="1">¹ Advanced Materials for Transportation Infrastructures Laboratory (MIT), Materials and Structures Department (MAST), Université Gustave Eiffel, F-44344 Bouguenais, France; mario-alexander.castaneda-lopez@univ-eiffel.fr</footnote>

<footnote n="2">² Centrifuges for Geotechnics Laboratory (CG), Geotechnical Engineering, Environment, Natural Hazards and Earth Sciences Department (GERS), Université Gustave Eiffel, F-44344 Bouguenais, France; luc.thorel@univ-eiffel.fr</footnote>

<footnote n="3">³ Independent Researcher, 78340 Les Clayes-sous-Bois, France; jpsanfratello@orange.fr</footnote>

<footnote>* Correspondence: thomas.lenoir@univ-eiffel.fr</footnote>
Improving CSSs relies on mechanical assessment, which is typically based on the unconfined compressive strength (UCS) and the indirect tensile strength (ITS) [2]. To optimize the amount of cement present in CSSs, the dosage methods must also take into account the effect of a number of mixture factors such as the curing time, dry density, moisture content, temperature, aggregate type, void volume, porosity-binder ratio, etc., and their potential variations due to the construction methods used [6,11–18]. Current research on strength models for CSSs typically uses two main types of approaches [8]: first, conventional approaches use initial relationship characterization parameters, such as the well-known soil water-to-cement ratio, and/or time-varying parameters. As a result, strength, mainly UCS, is expressed as a function of these variables [11–13,18]. The second approach is less commonly used and not standardized. Here, models are based on statistical correlations between parameters that influence strength, e.g., artificial neural networks, multilinear regression, machine learning, etc. [8,14,15,17].

CSSs constitute an intermediate class of geomaterials placed between classical soil mechanics and rock mechanics [19]. UCS and ITS are properties conventionally used in rock engineering [20]. However, they reproduce stress states that do not correspond to the stress tensor of structures due to service loads (i.e., service tensor) [21]. Appropriate failure criteria are thus needed for the comprehensive design of structures made with CSSs. There is a general understanding that the shear strength of CSSs can be represented using the Mohr–Coulomb (MC) failure criterion [19]. It is the simplest geotechnical model used in soil mechanics.

The MC shear strength parameters for CSSs, namely cohesion (c) and the friction angle (φ), are typically determined using triaxial tests [22–26]. However, most geotechnical applications of CSSs involve low confining stresses [27], and the critical stresses for these materials in structures are predominantly tensile [28]. Therefore, the mechanical range described by triaxial testing does not fully capture the service tensors of structures [21,29]. In addition, specimens used for triaxial testing are saturated, which does not accurately reflect the moisture conditions in the field [21]. As a result, the mechanical behavior determined via triaxial testing may not provide a reliable description of the performance of CSSs in the field.

To overcome the limitations imposed by triaxial testing, several alternative methods rely on the proportionality between the tensile and compressive strengths for both rocks [20,30,31] and CSSs [21,25,32–34]. Using the geometric method [32], the MC shear strength parameters of CSSs can be estimated using a geometric approach when UCS and ITS are known. It has two main advantages: (i) the segment of the failure envelope bounded by UCS and ITS is larger and encompasses tensile stress states beyond those captured by conventional confined triaxial tests, and (ii) it is less susceptible to measurement variability compared to triaxial-based estimations. Specifically, when using this method, obtaining MC shear parameters depends on the UCS (or ITS) and the ITS/UCS ratio, denoted as ζ.

This geometrical approach has been investigated using triaxial tests [24,25,34–36]. However, it must be validated using direct experimental measurements that correspond to the range of service tensors of geotechnical structures. A reliable and straightforward experimental alternative for determining MC shear strength parameters in this range of mechanical solicitations is the direct shear (DS) test. Nevertheless, only a limited number of studies in the literature have employed this method to evaluate the MC properties of treated materials [36–41]. Coupling the DS test with the geometric method circumvents the limitations associated with triaxial testing. Furthermore, the link between the mixture design parameters and the MC properties of CSSs has never been investigated. However, this question is fundamental in the field for estimating the safety factor of structures made with CSSs [21]. When combined with numerical and statistical correlations, coupling
the geometrical approach with experimental measurements can potentially incorporate variations on the mixture parameters inherent to field constraints when determining the *c-φ* properties of CSSs.

This study investigates the shear strength of two lightly cement-stabilized soils under varying field conditions at the laboratory scale. To this end, the compressive, tensile, and MC properties of two geomaterials treated with 3% cement were investigated using the UCS, ITS, and DS tests to ascertain a reference mixture preparation. To account for field conditions and generalize existing results [41], UCS and DS were experimentally measured by varying the mixture parameters (cement content, curing time, and degree of compaction) using Latin Hypercube Sampling (LHS). To compare the measured *c-φ* properties with the geometric approach [32], these mixture parameters were then incorporated as predictors in statistical models to generate numerical values of ITS [17]. Therefore, the numerical and geometric approaches were used to extend the link between mixture variables and tensile properties to incorporate the UCS and MC properties of the studied materials.

## 2. Materials and Methods

### 2.1. Materials

#### 2.1.1. Soils

Two soils, soil A and soil B, were studied. Both geomaterials were utilized and discussed in previous studies [17,41]. The results of characterization are shown in Figure 1.

Soil A was classified as A-6(11) based on the AASHTO classification [42], A2 based on the GTR classification [43], and CL based on the USCS classification [44]. Soil B was classified as A-2-6(0) based on the AASHTO classification [42], B6 based on the GTR classification [43], and SM-SC based on the USCS classification [44].

<figure><img src="image_3.png" /><figcaption>**Figure 1.** The particle-size distribution of the studied soils. F: fine; M: medium; C: coarse [45,46]. PL: plastic limit; LL: liquid limit; PI: plasticity index [47]; ρs specific mass [48] (from [17,41]).</figcaption></figure>

#### 2.1.2. Treatment and Sample Preparation

Both soils were treated with 3% cement (CEM II/A-LL 42.5 R) [49]. The cement content (CC) is defined as the weight of cement divided by the weight of dry material. Lime pre-treatment was not needed, as both mixtures exhibited adequate workability. For this CC, the compaction parameters, i.e., the optimum moisture content (OMC) and maximum dry density (MDD), were determined using the Proctor compaction test for standard compaction effort (600 kN × m/m³) [50]. For mixture A, OMC = 15.5% and MDD = 1.79 g/cm³. For mixture B, OMC = 10.0% and MDD = 1.98 g/cm³ [17].
For the reference mixture (V0), samples were prepared with CC = 3% at 96% MDD and at the OMC. Samples were tested at several curing times: 7, 28, 90, 180, and 360 days [17,41].

The mixture parameters were varied using Latin Hypercube Sampling (LHS). Samples were prepared at 10 varying conditions (V1 to V10) around V0. Varying factors were randomly varied within the same inference space of the reference variability study [17]: cement content, CC (from 2.0% up to 4.0%), and degree of compaction, DC (from 94.0% up to 98.5%). The curing time, CT, was delimited between 7 and 60 days. As water content does not significantly impact ITS for the range of possible variations on site (from 90% to 110% OMC), it was set to the OMC [17]. Replicates were not considered. Table 1 shows all the parameters assessed, their definition, and the levels of variation.

<table><caption>**Table 1.** Varying factors used for sample preparation for the reference (V0) and validation mixtures (V1 to V10).</caption><thead><tr><th>V</th><th>Curing Time, CT (Days)</th><th>Cement Content, CC (%) <sup>(a)</sup></th><th>Degree of Compaction, DC (%) <sup>(b)</sup></th></tr></thead><tbody><tr><td>V0 (ref.)</td><td>7, 28, 90, 180, and 360</td><td>3.00</td><td>96.0</td></tr><tr><td>V1</td><td>60</td><td>3.56</td><td>94.0</td></tr><tr><td>V2</td><td>42</td><td>4.00</td><td>95.5</td></tr><tr><td>V3</td><td>54</td><td>2.44</td><td>98.5</td></tr><tr><td>V4</td><td>36</td><td>2.00</td><td>97.5</td></tr><tr><td>V5</td><td>48</td><td>2.67</td><td>97.0</td></tr><tr><td>V6</td><td>30</td><td>3.78</td><td>94.5</td></tr><tr><td>V7</td><td>24</td><td>3.11</td><td>98.0</td></tr><tr><td>V8</td><td>18</td><td>3.33</td><td>95.0</td></tr><tr><td>V9</td><td>12</td><td>2.89</td><td>96.5</td></tr><tr><td>V10</td><td>7</td><td>2.22</td><td>96.0</td></tr></tbody></table>

For all experiments, samples were compacted using a double static mode in one layer into cylindrical molds to reach the target DC. After compaction, they were immediately extracted and placed in plastic bags to avoid significant variations in water content before testing. They were then cured in a controlled-temperature room at 20 °C. No saturation process was carried out before the experiments.

## 2.2. Experiments

As specimens were tested under unsaturated conditions (moisture content equal to OMC), the results are reported in terms of total stresses. UCS, DS, and ITS were measured for the reference V0 conditions [41]. UCS and DS were measured for the V1 to V10 varying conditions, whereas ITS values were estimated numerically.

For V0, UCS, and ITS tests were carried out on a Zwick/Roell Z150 press (Zwick/Roell, Ulm, Germany), and the applied load was measured with a force sensor TME F521TC (max. load = 25 kN for UCS and 10 kN for ITS). For V1 to V10, UCS samples were tested on an Instron 8872 press with a force sensor capacity of 25 kN. Direct shear (DS) tests were carried out using Shear test VJT, and the horizontal (shear) load was measured using a force sensor TME F501 TC (max. load = 5 kN).

### 2.2.1. Unconfined Compressive Strength Test

For the UCS test, cylindrical samples were subjected to an axisymmetric compressive stress up to breakage [51]. The crosshead speed was 0.1 mm/min. Vertical stress, $\sigma_{zz}$, measured in MPa, was calculated using Equation (1):

$$ \sigma_{zz} = 4F/\pi D^2 \quad (1) $$

<footnote>(a) CC as a percentage of the dry material weight; (b) DC as a percentage of MDD.</footnote>
with the applied force (F) measured in N and the diameter of the sample (D) measured in mm. The maximum value of σ<sub>zz</sub>, corresponding to the failure load, was equal to the UCS. As σ<sub>zz</sub> was a compressive load, its value was positive.

### 2.2.2. Indirect Tensile Strength Test

For the ITS test, cylindrical samples were placed horizontally, and a compressive load was applied along the cross-section diameter until failure [52]. The crosshead speed was 0.3 mm/min. Horizontal (tensile) σ<sub>xx</sub> and vertical σ<sub>zz</sub> (compressive) stresses, in MPa, were calculated using Equations (2) and (3), respectively,

$$ \sigma_{xx} = 2F/\pi HD \quad (2) $$

$$ \sigma_{zz} = -3\sigma_{xx} \quad (3) $$

with the applied force (F) measured in N and the diameter (D) and height (H) of the specimen measured in mm. The minimum value of σ<sub>xx</sub>, corresponding to the failure load, was equal to the ITS. As σ<sub>xx</sub> was a tensile (negative) load, σ<sub>zz</sub> was positive [53,54].

### 2.2.3. Direct Shear Test and Failure Criterion

In the DS test, cylindrical samples were placed in the shear box, and a vertical (normal) load was applied. Then, the shearing load was applied via the linear displacement of one half of the shear box relative to the other [55]. A shear rate of 0.12 mm/min was applied. Normal (σ) and shear (τ) stresses, measured in MPa, were calculated following Equations (4) and (5), respectively:

$$ \sigma = N/A \quad (4) $$

$$ \tau = T/A \quad (5) $$

with normal (N) and tangential (T) forces measured in N and the corrected shearing area A measured in mm² [56].

The results of DS testing were used to quantify the shear strength properties of both CSSs. Given the cohesive-frictional nature of CCS, the relationship between σ and the shear stress at failure (τ<sub>max</sub>) described by the Mohr-Coulomb failure criterion (Equation (6)) was employed in this study. Thus, the shear strength is expressed as a function of the internal friction angle (φ) and the cohesion intercept (c). Shear strength parameters (c and φ) were estimated for each tested mixture:

$$ \tau_{max} = c + \sigma \tan \varphi \quad (6) $$

Five levels of σ, ranging from 2 to 250 kPa, were used to determine c and φ. The experimental program is summarized in Table 2.

<table><caption>**Table 2.** The experimental program used for the reference mixture and validation analysis.</caption><thead><tr><th rowspan="2">Test</th><th rowspan="2">Reference</th><th rowspan="2">D (mm)</th><th rowspan="2">H (mm)</th><th colspan="2">Reference Mixture (V0)</th><th colspan="2">Validation Analysis (V1 to V10)</th></tr><tr><th>CT (Days)</th><th>Tested Samples/Mix.</th><th>CT (Days)</th><th>Tested Samples/Mix.</th></tr></thead><tbody><tr><td>UCS</td><td>[51]</td><td>100</td><td>200</td><td rowspan="3">7, 28, 90, 180, and 360</td><td>3</td><td rowspan="3">7 to 60</td><td>1</td></tr><tr><td>ITS</td><td>[52]</td><td>100</td><td>100</td><td>4</td><td>(a)</td></tr><tr><td>DS</td><td>[55]</td><td>60</td><td>35</td><td>15</td><td>5</td></tr></tbody></table>

<footnote>(a) Numerical values based on mixture parameters [17].</footnote>
## 2.3. Geometric Method to Estimate c-φ Properties

Compressive and tensile properties were used in the geometric method to determine the c-φ properties of both CSSs [32]. This method has previously been used for characterizing lime-treated soils [13,24], cement-treated mixtures [21], mixed treatments (lime, cement, and/or fly-ash) [25,32], and rocks [31].

This approach is represented in Figure 2, depicted in a {σ, τ} stress space when failure is reached. Here, the stress tensor for UCS (σxx = σ3 = 0 and σzz = σ1 = UCS) and ITS (σxx = σ3 = ITS and σzz = σ1 = -3 × ITS) are depicted as Mohr semi-circles [29,31,53]. The Mohr–Coulomb failure criterion corresponds to the tangent to UCS and ITS semi-circles. In terms of the ratio ζ = ITS/UCS, the friction angle (φζ) can be expressed using Equation (7), and the cohesion intercept (cζ) can be calculated using either the UCS (Equation (8)) or the ITS (Equation (9)) [32,34].

$$ \varphi_{\zeta} = \operatorname{asin}\left(\frac{1 - 4\zeta}{1 - 2\zeta}\right) \qquad (7) $$

$$ c_{\zeta,UCS} = UCS \times \frac{\left[1 - \left(\frac{1-4\zeta}{1-2\zeta}\right)\right]}{2\cos\left[\operatorname{asin}\left(\frac{1-4\zeta}{1-2\zeta}\right)\right]} \qquad (8) $$

$$ c_{\zeta,ITS} = ITS \times \frac{\left[2 - \left(\frac{1-4\zeta}{1-2\zeta}\right)\right]}{\cos\left[\operatorname{asin}\left(\frac{1-4\zeta}{1-2\zeta}\right)\right]} \qquad (9) $$

<figure><img src="image_4.png" /><figcaption>**Figure 2.** The estimation of c-φ properties based on ITS and UCS: the principle of the method, based on [32].</figcaption></figure>

## 2.4. Numerical Approach for ITS Estimation

As introduced in Table 2, the ITS values for the variability analysis (V1–V10) were based on a numerical approach. For the materials investigated in this study and for the same inference space, a prior statistical review performed using analysis of variance was conducted to identify the significant mixture variables and generate multilinear regression models for ITS [17]. These models are described by Equations (10) and (11) for mixtures A and B, respectively. The data used to generate these numerical models are available in a public repository [57]. This approach enabled the use of the mixture design parameters CT, CC, and DC as predictors to estimate numerical ITS values, denoted as ITSn:

$$ ITS_{n(A)} (\text{MPa}) = (-131.99 + 6.34\log_{10}\text{CT} + 6.90\text{CC} + 1.24\text{DC}) \times 10^{-2} \qquad (10) $$

$$ ITS_{n(B)} (\text{MPa}) = (-160.45 + 9.99\log_{10}\text{CT} + 4.79\text{CC} + 1.55\text{DC}) \times 10^{-2} \qquad (11) $$
For the sampling size used (n = 10), the authors estimated the mean error in ITS measurements to be 6.8% for mixture A and 4.8% for mixture B [17].

All calculations were conducted using Python version 4.1 [58]. The following scientific libraries were used: sciPy [59], numPy [60], pandas [61], statsmodels [62], matplotlib [63], and seaborn [64].

# 3. Results and Discussion

## 3.1. Reference Mixture (V0)

Hereafter, standard deviation (SD) is indicated in parentheses. For the reference mixture V0, merged results of UCS, ITS, and DS are detailed in Appendix A.1 [17,41]. Complete data are available in a public repository [57,65]. Numerical results for ITS (ITSn) are also included. For DS, the linear regression parameters c-φ are presented in the same table with the coefficient of determination, R².

### 3.1.1. UCS Tests

Mean UCS values varied between 1.53 and 2.41 MPa for mixture A and between 0.91 and 1.91 MPa for mixture B (Table A1). In general, the higher the curing time, the higher the UCS. SD-related uncertainties in measurements range between 0.01 and 0.15 MPa for mixture A and between 0.03 and 0.18 MPa for mixture B. The coefficient of variation (COV), defined as the ratio of the SD to the corresponding mean value, was under 9% for mixture A and under 12% for mixture B. Results from UCS experiments can be considered statistically relevant.

As UCS is the main indicator used for assessing CSS performance, a number of authors have addressed this feature. The measured values are, in general, higher than those reported in the literature. For instance, the reported values for well-graded sand treated with 2% and 4% CC were 0.18 MPa and 0.55 MPa after 14 days of curing [22]. The mean values for cement-treated silt (CC = 3%) were 0.50, 0.70, and 0.90 MPa after 7, 14, and 28 days of curing [2]. For poor-graded sand treated with 3% CC, UCS was 0.60 MPa for CT = 7 days [26]. After 7 days of curing with 3% CC and a high initial strength binder type, UCS ranged between 0.80 and 2.80 MPa for a well-graded silty sand when DC varied between 87% and 99% MDD [66]. For silty sand treated with 3% CC, UCS was 0.74 MPa after 7 days of curing [19]. For silt treated with 3% lime after 7 days of curing, UCS varied between 1.0 and 1.8 MPa [67].

The influence of saturation on CSSs is already visible for UCS, as it leads to a decrease in UCS values. For instance, for lateritic silt treated with 3% cement for 7 days, the UCS decreased from 1.34 MPa to 0.31 when soaked for 4 days prior to testing. For CC = 2.5% and CT = 7 days, UCS decreased from 0.39 MPa to 0 MPa for treated silt and from 0.37 to 0.23 MPa for treated silty sand when samples were soaked 24 h prior to testing [68]. The UCS for a silty sand treated with 3% CC was 1.80 MPa after 7 days of curing and 3.60 MPa after 28 days of curing. When samples were soaked 24 h before testing, UCS values, for the same curing period, decreased to 0.40 and 0.80 MPa, respectively [69]. The reported values for similarly treated mixtures in an unsoaked state were typically higher. For a clay-lime soil treated with three road hydraulic binders (5.5%), the UCS varied between 3.14 and 4.40 MPa after 90 days of curing in an unsoaked state [21]. With CC = 5% and CT = 90 days, the UCS has been reported to be 3.58 MPa for a clayey sand in an unsoaked state [70]. This discussion confirms the necessity of taking into account the real moisture conditions of materials in the field to determine mechanical properties.
### 3.1.2. Experimental ITS Tests

Mean ITS values varied between 0.14 and 0.23 MPa for mixture A and between 0.12 and 0.31 MPa for mixture B (Table A1). The higher the curing time, the higher the ITS. The SD was under 0.031 MPa for mixture A and under 0.014 MPa for mixture B. The COV for ITS was under 14% for mixture A and under 7% for mixture B. The results from UCS experiments can be considered statistically relevant.

Measured ITS values agree with the values reported in the literature. For instance, mean values for cement-treated silt (CC = 3%) were 0.05, 0.08, and 0.12 MPa after 7, 14, and 28 days of curing [2]. For sand [12], values ranged from 0.01 to 0.37 MPa when CC varied from 4 to 8% (7 < CT < 21 days), 3% to 9% (7 < CT < 28 days), and 1 to 12% (CT = 7 days), respectively. For cement-modified loess (CC = 5.5% and 7.0%), ITS varied between 0.33 MPa and 0.53 MPa for a CT of 90 days [21]. For cemented sands [71], the ITS ranged between 0.02 and 0.13 MPa (42 < CT < 90 days) when 2 < CC < 6%. When compared with similar materials, an average ITS of 0.50 MPa was obtained for recycled cement-treated mixtures, with 75% of the individual values being lower than 0.65 MPa [72]. For lime-treated soils, ITS values varied between 0.31 and 0.80 MPa (CT = 7 and 28 days) when lime content varied from 3 to 5% [32]. For mixed treatment, ITS values varied between 0.20 and 0.70 MPa when soils were treated with 5 to 6% hydraulic binder (7 < CT < 720 days) [70].

### 3.1.3. Numerical ITS Tests

Table A1 presents the numerical estimation of ITS based on mixture parameters (CC = 3%, CD = 96%) for each CT using Equations (10) and (11). The mean value of the ratio ITS<sub>n</sub>/ITS is 1.04 (0.09) for mixture A and 0.99 (0.06) for mixture B. These results corroborate the suitability of these regression models for estimating the tensile strength of the studied materials within the studied inference space.

### 3.1.4. DS Tests

Cohesion values ranged from 0.25 to 0.39 MPa for mixture A and from 0.18 to 0.42 MPa for mixture B, with a positive effect of CT (Table A1). The friction angle varied between 35.4° and 62.0° for mixture A and between 42.8° and 61.4° for mixture B, with an unclear effect of CT. R<sup>2</sup> values for the MC model (Equation (6)) exceeded 0.93 for three out of the five CT levels for mixture A and were greater than 0.82 for all CT levels for mixture B. These results show the suitability of the MC model for representing their failure characteristics through c-φ parameters from DS test measurements.

In general, the measured c values are higher than those reported in the literature for similar testing conditions. While the study of c-φ properties for CSSs is conventionally carried out using triaxial tests under consolidated isotropically drained (CID) conditions, reports based on DS tests conducted on non-saturated conditions are limited. As a result, the reported peak strength envelopes are reduced when compared to non-saturated conditions. A degree of c similar to that measured in this study is reported for higher amounts of binder measured under CID conditions. For instance, after 7 days of curing, c measured via CID triaxial tests was 0.28 MPa for silty sand when treated with 5% CC [19]. For cemented sand, c measured using DS increased from 0.10 to 0.25 MPa when CC was increased from 2.5% to 7.5% (CT = 28 days) [39]. For a cement-treated sandy loess, c measured using DS for a CT of 7 days increased from 0.03 MPa to 0.14 MPa when CC varied from 4% to 9% [73]. With 5% CC, values of c measured via the DS test for cemented sand increased from 0.06 MPa to 0.15 MPa after 7 and 28 days of curing, respectively, and the maximum value of c was 0.21 MPa for 10% CC after 28 days of curing [37]. For other treated materials, c values for two clayey soils treated with 20% pozzolana and 8% lime and tested with DS ranged from 0.32 to 0.38 MPa after 28 and 90 days of curing, respectively [74]. When treating clayey
sand with 7% lime and 25% fly-ash, *c* values from CID triaxial tests reached 0.40 MPa and 0.67 MPa at CTs of 28 and 90 days, respectively [25]. With values of *c* varying between 0.38 MPa for 7 days and 0.70 MPa for 90 days, CID triaxial results for lateritic silt treated with 3% CC are higher than the values measured in the present study [75].

The measured friction angle values for both mixtures are, in general, comparable to those reported in the literature. For mixture A, the friction angle value varied from 35.4° to 62.0°, and for mixture B it varied from 42.8° to 61.4°. The mean values of φ were 48.0° (11.8°) for mixture A and 51.3° (6.8°) for mixture B. The results are similar, with a higher statistical spread for mixture A, as observed in other tests. The results also suggest that there is a low variation associated with the curing period. Unlike cohesion, the absence of a defined pattern does not allow for establishing a clear correlation between mixture parameters and friction angle. Some references report an increase in this feature with CT, whereas others have observed relatively constant values. For instance, for a cemented sand at CT = 7 days, the average value of φ measured with triaxial CID was 41° [19], whereas values of φ for a lateritic silt treated with 3% CC and tested under CID conditions increased from 14° to 44° (7 < CT < 90 days) [75]. Measured values of φ using DS test increased with both CT and CC, varying from 37° to 50° (7 < CT < 28 days and 2.5% < CC < 7.5%) [39]. Similarly, for cement-treated sand, the values of φ obtained via the DS test increased with CT and CC, varying from 37° to 47° (7 < CT < 28 days and 2% < CC < 10%) [37]. Measured values of φ using the DS test for cement-treated sandy clay at CT = 7 days varied between 25° and 40°, with no effect of CC (4% < CC < 9%) [73]. For other treated materials, values of φ for two clayey soils treated with 20% pozzolana and 8% lime and tested with the DS test ranged between 30° and 35°, with no effect of CT (7 < CT < 90 days) [74]. When treating clayey sand with 7% lime and 25% fly-ash, φ values for triaxial tests conducted under CID conditions reached 33° and 45° at CTs of 28 and 90 days, respectively [25].

### 3.1.5. Correlation Analysis

The correlation matrices between the studied parameters are presented in Figure 3 [41]. Given that the UCS of the CSSs increases approximately linearly with CT on a logarithmic scale [6], this form is used in the correlation analysis. With higher Pearson coefficient values (R), mixture B showed higher correlations than mixture A for CT, UCS, ITS, and c. These relationships were all positive. For φ, however, in general, there was a lack of correlation when compared to the other variables (for CT, for example, R<sub>CT-φ</sub> was equal to 0.25 for both materials). Excluding φ, the relationships between CT and the studied features ranged from moderate to strong for mixture A (0.63 < R < 0.92), while for mixture B, they were consistently strong (R > 0.97). The correlation between c and UCS was moderate for mixture A (R<sub>c-UCS(A)</sub> = 0.55) and strong for mixture B (R<sub>c-UCS(B)</sub> = 0.97). The degree of correlation between ITS and c was also moderate for mixture A (R<sub>ITS-c(A)</sub> = 0.68) and strong for mixture B (R<sub>ITS-c(B)</sub> = 0.91). Similarly, the correlation between ITS and UCS ranged from moderate for mixture A (R<sub>ITS-UCS(A)</sub> = 0.39) to fairly strong for mixture B (R<sub>ITS-UCS(B)</sub> = 0.99). Considering the dispersion of the measurements within mixture A, these findings support the validity of applying linear relationships, i.e., using quotients between mechanical features, as this type of relationship can be considered reliable.

For coefficients linked to c, the c/ITS ratio was 1.90 (0.30) and 1.55 (0.19) for mixtures A and B, respectively; the c/UCS ratio was 0.18 (0.03) for mixture A and 0.23 (0.02) for mixture B. The low dispersion suggests that these ratios are not affected by CT. These values agree with the observations reported in the literature. The ratios of c/ITS and c/UCS for lightly cementitiously stabilized granular base materials were 1.88 and 0.20, respectively [32]. In the same study, the authors reviewed similar investigations, reporting ratios ranging between 1.76 and 1.83 for c/ITS and between 0.20 and 0.34 for c/UCS. For
cemented sand (CC = 2% and 4%), the ratio of c/ITS varied between 2.04 and 2.30, and the ratio of c/UCS varied between 0.23 and 0.30 depending on the binder content and relative density [22]. For similar materials, the ratio of c/UCS was 0.24 for sand-fly ash-lime [25] and 0.26 for lime-treated clayey sand after 90 and 360 days of curing [24]. For rocks, a computed ratio of c/ITS = 1.82 was successfully compared with experimental results for six types of rocks [31].

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Correlation matrices of the dataset for mixture A (**left**) and mixture B (**right**) (reference mixtures, V0) [41]. Colorscale varies from royal blue (−1) to deep red (+1).</figcaption></figure>

For the relationship between compressive and tensile strength, when $\xi$ = ITS/UCS was averaged, it varied in a narrow range of 0.10 (0.02) for mixture A and 0.15 (0.01) for mixture B. The linear relationship between compressive and tensile strength seems to be independent of CT [76,77]. These values agree with those reported in the literature for similar materials. For silty-cemented soil, the $\xi$ ratio ranged from 0.09 to 0.18 depending on the strength level [78]. This ratio was reported to range between 0.11 and 0.13 for cemented sand (CC = 2 and 4%) [22]. In cemented loess, this ratio was 0.11 [21]. The reported values for the $\xi$ ratio in cement-treated sand were 0.11 [79] and 0.15 [12]. The influence of CT was studied for cement-treated silt, with values of 0.15, 0.16, and 0.17 for 7, 14, and 28 days of curing, respectively [2]. For lime-treated clayey sand, the $\xi$ ratio was found to be equal to 0.15 for 90 and 360 days of curing [24]. With $\xi$ values varying from 0.12 to 0.13, the non-influence of gradation was evidenced in granular CSSs [80]. The $\xi$ ratio has been reported to be a scalar for other soils and cementing agents such as silt-lime mixtures, with $\xi$ = 0.16 [13]. The $\xi$ ratio was in the range of 0.08 to 0.28 for six types of rocks [31].

### 3.1.6. Geometric Method for Assessing $c-\varphi$ Properties

The mean results for mixture A are displayed in a {$\sigma - \tau$} space in Figure 4 for CT = 90 days and CT = 360 days. Here, the principal stresses in failure, i.e., the couple ($\sigma_3$, $\sigma_1$), are depicted, with the Mohr semi-circles representing UCS in light orange (0, UCS) and ITS in dashed light orange (ITS, -3ITS) [53]. The tangent to the Mohr semi-circles of UCS and ITS is depicted as a dashed black line. The experimental DS results are presented with red dots, which represent the measured couple ($\sigma$, $\tau_{max}$) at failure. The linear MC envelope, with c and $\varphi$ (Equation (6)), is presented with a red line.

Figure 4a shows that the MC failure envelope depicted is a very fair approximation of the tangent to the Mohr semi-circles of the ITS and UCS tests at failure. However, Figure 4b shows that this geometric approximation is not so straightforward. To address this issue, $c-\varphi$ properties for both materials are estimated for all studied CTs using this geometric method (Equations (7) and (8) (or Equation (9)) and the results are compared to those measured with the DS test. The application of this method involves making some assumptions regarding their $c-\varphi$ properties [24,25,27]: (i) the $\xi$ ratio is assumed to be a scalar
value for a given dosage, controlled by the frictional strength of the cement bonds, but varying within quite a narrow range. Consequently, (ii) the friction angle $\varphi_{\zeta}$, Equation (7) is treated as a constant. Finally, (iii) the cohesion $c_{\zeta}$ can be determined if either ITS or UCS and $\zeta$ are known, as they are considered interdependent.

<figure><img src="image_6.png" /><figcaption>**Figure 4.** The mean results for mixture A after (a) 90 days and (b) 360 days of curing. The UCS (light orange), ITS (light orange, dashed), tangent to UCS and ITS semi-circles (black line, dashed), DS (red, scatter plot), and MC envelope (red line) are depicted.</figcaption></figure>

The results are presented in Table A2. The mean ratio between the estimated and measured cohesion was 1.03 (0.13) for mixture A and 1.14 (0.13) for mixture B. For $\varphi$, this ratio was 1.07 (0.23) for mixture A and 0.71 (0.13) for mixture B. Due to the low variability in $\zeta$, and its independence from CT, the derived $\varphi_{\zeta}$ (Equation (7)) was nearly constant for both materials: 49.4° (5.5°) for mixture A and 36.0° (3.5°) for mixture B. These findings indicate that for either the measured or predicted MC parameters, the mechanical enhancement of the studied materials can be primarily quantified using $c$, while $\varphi$ (or $\zeta$) can be considered as scalar parameters.

### 3.1.7. Alternative Methods for Assessing $\zeta$

The correlation analysis for $c$, ITS, and UCS showed their interdependency, and the geometric method allowed us to quantify these relationships through the parameter $\zeta$. The conventional procedure for obtaining $\zeta$ is based on directly measuring UCS and ITS. This section discusses two alternative methods for obtaining $\zeta$: using either $c$ and UCS or $c$ and ITS.
- $\xi$, Based on c and UCS

For a given c/UCS ratio, Equation (8) allows us to back-calculate $\xi$. This value is defined as $\xi_{c,UCS}$. The mean values of $\xi_{c,UCS}$ were 0.09 (0.02) for mixture A and 0.13 (0.01) for mixture B. With a very similar dispersion to the reference method, $\xi_{c,UCS}$ also seems to be a constant value for both materials. When compared to the reference values of $\xi$, the mean ratio $\xi_{c,UCS}/\xi$ was 0.98 (0.19) for mixture A and 0.87 (0.12) for mixture B. This shows that this parameter is slightly underestimated when using c and UCS.

- $\xi$, Based on c and ITS

By analyzing the function defined in Equation (9), two main limitations related to its range and domain are identified. First, the function range has a limit inferior at c/ITS = 1.73 when the parameter $\xi$ (defined here as $\xi_{c,ITS}$) is equal to 0.17. In a {σ, τ} stress space (as shown in Figure 2), this value corresponds to the intersection of the Mohr semi-circle of ITS with the vertical axis. Since the geometric method is based on the principle that the MC envelope must be tangent to the Mohr semi-circle of ITS, the cohesion value cannot lie within the circle. Second, for a given c/ITS ratio, there are two valid solutions for Equation (9): one above and one below $\xi_{c,ITS} = 0.17$. Thus, it is necessary to delimit the domain of the function prior to analyzing it, which requires making a preliminary estimation of $\xi$.

To apply this method to the experimental results, the function domain is delimited for values of $\xi_{c,ITS} < 0.17$. For mixture A, 3 out of 5 measurements (CT = 7, 28, and 90 days) belonged to the range of the function (i.e., c/ITS > 1.73), and for mixture B, only 1 out of 5 belonged to the range of the function (CT = 90 days). The back-calculated values of $\xi_{c,ITS}$ were 0.09 (0.05) for mixture A and 0.11 for mixture B. These results highlight the limitations of the procedure, as most of the measurements lie outside the range of the function.

With a lower dispersion and no limitations in terms of range and the domain of the function, the method based on c and UCS could potentially be a good alternative for assessing $\xi$ when the tensile properties are not known.

## 3.2. *Variability Study (V1–V10)*

Merged results of UCS and DS are detailed in Appendix A.2. Complete data are available in a public repository [65]. Numerical results for ITS (ITSn) under these varying conditions are included.

### 3.2.1. Experimental and Numerical Results

The suitability of the MC criterion is confirmed under varying conditions (Table A3) with R² values between the model and experimental values of 0.97 (0.02) for both materials. The average values of $\phi$ were 52.9° (4.1°) for mixture A and 53.1° (1.3°) for mixture B. When normalized with regard to the mean from the reference V0 mixtures, the corresponding ratios were 1.10 for mixture A and 1.04 for mixture B. Thus, it corroborates that this parameter was nearly constant for both materials, even when the mixture composition varied. Experimental values of c and UCS varied as a function of the mixture variables, such as the numerical ITSn values (Equations (10) and (11)). For both mixtures, V2 (CT = 42 days, CC = 4% and DC = 95.5%) exhibited the highest values for these features, whereas V10 (CT = 7 days, CC = 2.22% and DC = 96.0%) exhibited the lowest values.

The interdependence between the involved variables is examined for the varied mixtures using a correlation analysis presented in Figure 5.

Correlations between mixture variables (log₁₀CT, CC, CD) were low. This confirms that the LHS is robust enough to generate near-random mixture parameters within the given inference space.
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Correlation matrices of the dataset for mixture A (**left**) and mixture B (**right**) (variability study, V1–V10). Colorscale varies from royal blue (−1) to deep red (+1).</figcaption></figure>

The correlations between the mixture parameters and the experimental measurements (UCS and c) were similar to the correlations between the mixture parameters and the numerical estimations (ITSn) for both materials. Therefore, UCS, c, and ITSn were highly correlated. This suggests that the numerical models (Equations (10) and (11)) used to generate ITSn are relevant for describing experimental UCS and c measurements. For material A, CC primarily impacts the experimental results of UCS and c, followed closely by CT. For material B, CT primarily impacted UCS and c, while the influence of CC was moderate. For both materials, DC was the least important parameter [17]. The V1–V10 experiments confirm the findings made for V0 regarding CT, UCS, ITSn, and c.

For mixture A, correlations were identified between φ and UCS and ITSn and c. However, the uncertainty of these correlations expressed as the square root of the sum of squared differences (e.g., 2.97 for UCS) was comparable to the experimental SD (4.09) of φ, and thus, these correlations could not be validated.

### 3.2.2. Geometric Method for Assessing the *c-φ* Properties for Variability Study

Based on UCS and ITSn, the geometric method was used to estimate the *c-φ* properties. As shown in Table A4, φ<sub>ξ</sub> remained almost constant, with mean values of 43.7° (5.9°) for mixture A and 38.6° (3.7°) for mixture B. The mean ratio between the estimated and measured values was 0.83 (0.09) for mixture A and 0.73 (0.07) for mixture B. As φ and φ<sub>ξ</sub> were constant, the ratio between them tended to also be constant. When compared to V0, the magnitude of the underestimation of φ<sub>ξ</sub> was similar for mixture B but more pronounced for mixture A For c, the mean ratio between the estimated and measured values was 1.34 (0.20) for mixture A and 1.32 (0.14) for mixture B. In contrast to φ, c was overestimated for both materials to a similar degree, differing from V0, where the estimated and measured values are in closer agreement.

The mean values of ξ̄ for the variability study agreed with the mean values for V0. The results for both mixtures determined using the conventional procedure (ξ̄ = f (ITS, UCS)) are depicted in Figure 6a for both materials. For the variability study (in white), the mean value of ξ̄ was 0.12 (0.02) for mixture A and 0.14 (0.01) for mixture B. The overall average of ξ̄ determined via this method for the reference and varied mixtures was 0.11 (0.02) for mixture A and 0.14 (0.01) for mixture B. These results suggest that for the whole investigated range of CT, CC, and DC, there is a straight proportionality between the tensile and compressive strengths. This proportionality can be treated as a scalar, independent of the composition of the mixture, as corroborated in the literature for other CSSs and cementing agents such as cemented-modified loess [21], cemented sands [12,27], lime-treated clayey sands [24], and fly ash-lime-treated sands [25]. In these investigations, values
of $\xi$ ranged between 0.11 and 0.18, which is also in agreement with the measured values for V0 and V1 to V10 for both materials.

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Measured values of $\xi$ for the reference (V0) and varied mixtures (V1–V10) using (a) the conventional procedure based on ITS and UCS and (b) the alternative method based on c and UCS.</figcaption></figure>

### 3.2.3. Estimation of $\xi$, Based on c and UCS

The alternative method based on c and UCS was employed to quantify $\xi$ for the variability study. The mean value of $\xi_{c,UCS}$ was 0.08 (0.02) for mixture A and 0.10 (0.01) for mixture B. As shown in Figure 6b for $\xi = f(c, UCS)$, these values, as displayed in white, were slightly lower than those obtained for V0, as shown in gray. For both materials, the dispersion of $\xi_{c,UCS}$ was very similar to that observed for V0. The overall average of $\xi_{c,UCS}$ for the reference and varied mixtures was 0.08 (0.02) for mixture A and 0.11 (0.02) for mixture B. These results suggest that this alternative method also captures the relatively constant nature of the parameter.

When comparing the combined results of all studied cases generated via the conventional method (Figure 6a) to those generated through the alternative method (Figure 6b), it is observed that this alternative method underestimates $\xi$. The overall mean value of the ratio $\xi_{c,UCS}/\xi$, considering both the reference and varied mixtures, was 0.78 (0.22) for mixture A and 0.76 (0.13) for mixture B. These findings suggest that the value of $\xi$, is more greatly influenced by the estimation method than by the mixture composition.

As suggested in Section 3.1.7, the results for $\xi_{c,ITS}$ are not considered. For varied mixtures, with mean values of c/ITSn of 1.40 (0.26) for mixture A and 1.35 (0.16) for mixture B, only one value of measured c/ITSn quotients among both mixtures belongs to the range of the function (Equation (9)).

### 3.3. Coupling Geometrical and Numerical Approaches to Assess Mechanical Properties Based on Mixture Variables

To generalize the results within the inference space defined by the reference and the varied mixtures, geometric and numerical approaches were combined. The numerical values of ITS (Equations (10) and (11)) and measured values of UCS were used to quantify $\xi_n = f(ITS_n, UCS)$. Since $\xi_n$ was treated as a constant value, the corresponding quotients c/UCS, c/ITSn, and $\varphi$ were also considered scalar values. For mixture A, the overall average value of $\xi_n = ITS_n/UCS$ was 0.11, resulting in $c_{\xi, n}/UCS = 0.21$ (Equation (8)), $c_{\xi, n}/ITS_n = 1.83$ (Equation (9)) and $\varphi_{\xi, n} = 45.4^{\circ}$ (Equation (7)). For mixture B, the average $\xi_n$ was 0.14, yielding $c_{\xi, n}/UCS = 0.24$, $c_{\xi, n}/ITS_n = 1.76$ and $\varphi_{\xi, n} = 37.9^{\circ}$.

For both materials, $\varphi$ was underestimated to a degree similar to that obtained for the reference and varied mixtures when $\xi$ was not averaged. For this parameter, the mean ratio
between the estimated and measured values was 0.91 (0.16) for mixture A and 0.73 (0.06) for mixture B.

Based on the c/UCS and c/ITS quantities, the geometric and numerical approaches were coupled to estimate ITS, UCS, and c using mixture variables as predictors.

### 3.3.1. Generalization of the Geometric Method: From UCS to Cohesion

Once $\xi_n$ was fixed, values of cohesion were calculated for all the studied cases using the measured values of UCS (Equation (8)). The mean ratio between the estimated and measured c values was 1.26 (0.17) for both materials. The overestimation of c was consistent with the results obtained when $\xi$ was not averaged for the varied mixtures. This demonstrates the advantage of considering the value of $\xi$ as a scalar.

### 3.3.2. Generalization of the Geometric Method: Coupling Mixture Variables to Assess UCS and Cohesion Values

Based on $IT_S_n$, numerical values of UCS ($UCS_{\xi,n}$) and cohesion ($c_{\xi,n}$) were derived using the fixed quotients $\xi_n$ and $c_{\xi,n}/IT_S_n$, respectively. These numerical values are detailed in Appendix B (Table A5) and compared in Figure 7 with measured values.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Measured values and predicted values of (a) UCS and (b) cohesion, based on mixture variables ($IT_S_n$) with a fixed $\xi$.</figcaption></figure>

The predicted and measured values of UCS (Figure 7a) and cohesion (Figure 7b) for both materials are depicted with the corresponding linear fits. For UCS, with a moderated correlation for mixture A ($R^2 = 0.73$), the magnitude of the error prediction seems to be associated with the magnitude of measurements. For mixture B, with a fairly strong correlation ($R^2 = 0.96$), a slope close to 1.0, and a negligible intercept value, the fit shows very good agreement between the measured and predicted (numerical) values. The mean ratio between numerical and measured values of UCS was 1.00 (0.19) for mixture A and 1.00 (0.09) for mixture B. For both materials, the predicted $UCS_n$ values were in close agreement with the measured ones. Regarding cohesion (Figure 7b), correlation was also stronger for mixture B ($R^2 = 0.86$) than for mixture A ($R^2 = 0.61$), with a general overestimation of c for both materials. Here, both slopes are also closer to 1.0, indicating that overestimation is not associated with the magnitude of observations. On average, the predicted values of c were 1.25 times higher than the measured values: this ratio was 1.25 (0.25) for mixture A and 1.25 (0.16) for mixture B. This overestimation was equal to the magnitude of the intercepts: 0.10 MPa for mixture A and 0.06 MPa for mixture B.
The overestimation of cohesion and underestimation of φ may be related to the
geometric approach employed. As tensile strength is calculated via an indirect method
(ITS), performing a direct measurement (direct tensile strength (DTS) test) could potentially
explain these variations [81].

DTS measurements on CSSs are limited due to experimental difficulties. However, it
is possible to compare the two geometrical approaches of the aforementioned study if it
is assumed that the indirect and direct measurements are equal for the studied materials
(i.e., ITS = DTS) [82]. Using the geometric method based on DTS [81] (with ξ = DTS/UCS)
reveals the following: (i) c calculated from ITS was overestimated compared with c from
DTS. The calculated ratio was 1.46 for mixture A and 1.55 for mixture B. (ii) φ from ITS was
underestimated compared with φ from DTS. The calculated ratio was 0.98 for mixture A
and 0.89 for mixture B. These coefficients are merely indicative. They confirm the influence
of the tensile strength measurement method on the results for the geometric approach.
Nevertheless, using these alternative geometric methods to assess c-φ properties based on
tensile properties is relevant.

3.3.3. From Mixture Parameters to Cohesion

In preliminary sections, the c/ITS quotients were multiplied by the numerical expres-
sions of ITS (Equations (10) and (11)) to estimate c values. These composed expressions,
which were not adjusted, are plotted in Figures 8 and 9 to represent the variation in the c
gain as a function of the mixture variables for mixtures A and B, respectively.

In Figures 8a and 9a, iso-contours of cohesion are plotted as functions of CT and
CC at DC = 96% MDD. Although the curves display a logarithmic shape, c increases
proportionally with CC variations. A 1% increase in CC (e.g., 2.5% to 3.5%) yielded gains
of ~0.13 MPa for mixture A and ~0.08 MPa for mixture B, independent of CT. This suggests
that CC governs the number of bonds between aggregates. The lower the CC, the lower
the number of bonds and, thus, the lower c.

The coarse material B (Figure 9a) exhibited higher gradients than the clayey mixture A (Figure 8a), reflecting different stabilization mechanisms. The higher occurrence of water in A enhanced early reactivity, but the occurrence of clay could affect the long-term quality of cement bonds between soil aggregates. Coarse-grained soils such as mixture B generally benefit more from cement stabilization [68], whereas fine-grained soil may require higher amounts of binder. For example, for 3% CC at a DC of 96% MDD, increasing CT from 7 to 360 days increased c by ~0.20 MPa for mixture A and ~0.30 MPa for mixture B.

Lower CC can be offset by a longer CT. For mixture A (Figure 9a), an ITS of 0.40 MPa could be achieved either after ~150 days of curing with 3.0% CC, or after ~50 days of curing with 3.5% CC. This is particularly relevant for short-term applications. Such relationships can guide binder optimization in practice and help to estimate performance variations due to dosage fluctuations.

Figures 8b and 9b show c as a function of CT and DC for 3% CC. Increasing DC from 94.0 to 98.5% MDD yields a gain of ~0.10 MPa in c for both mixtures, regardless of CT. Increasing DC also reduced the CT needed to reach a given c. For instance, for mixture B, c = 0.40 MPa can be reached after ~100 days of curing with DC = 96% MDD. This CT doubled when DC decreased to 94% MDD, and it halves when CT increased to 98.5% MDD. This reflects the role of DC in increasing contact points between soil aggregates in the matrix and, thus, in creating more cemented bonds. The stronger influence of DC agrees with previous observations [83]. This remark underlines the importance of compaction control, even in the long term.

Figures 8c and 9c show the combined effect of DC and CC for a CT of 90 days for
mixtures A and B, respectively. For mixture B, a reduction in CC from 3.0% to 2.0% can
be compensated by increasing the DC by 2.5% (e.g., from 94.5% to 97.0% MDD) to obtain $c = 0.35$ MPa. This shows the potential for achieving cement savings through improved compaction.

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Cohesion contour plots for mixture A: (a) CT vs. CC (DC = 96% MDD), (b) CT vs. DC (CC = 3%), and (c) DC vs. CC (CT = 90 days). Contour lines: c, measured in kPa.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Cohesion contour plots for mixture B: (a) CT vs. CC (DC = 96% MDD), (b) CT vs. DC (CC = 3%), and (c) DC vs. CC (CT = 90 days). Contour lines: cohesion, measured in kPa.</figcaption></figure>

## 4. Conclusions

We experimentally and numerically assessed the impact of varied dosages in two CSSs in the following inference space: cement content (from 2.0% to 4.0% of dry matter), the degree of compaction (from 94.0% to 98.5% of MDD), and curing time (from 7 to 360 days). A variability study for assessing ITS, UCS, and the $c-\varphi$ features was planned for both materials. A numerical approach was used to generate synthetic ITS values from statistical models using mixture variables as predictors. LHS was used as the space-filling technique. ITS predictions were coupled with the geometric method via the ITS/UCS ratio to assess the $c-\varphi$ properties of both materials. Measured and predicted values were compared. Based on the experimental results, geometric analysis, and numerical results, the following conclusions can be drawn:

* The overall performance of both sandy and clayey soils is significantly affected by the mixture variables. For the reference mixture (V0), the positive effect of CT was experimentally verified via ITS, UCS, and DS tests. For the varied mixtures (V1 to V10), the combined effect of mixture variables was examined using numerical values of ITS and experimental DS and UCS measurements. Correlation analysis suggests
that the numerical models (Equations (10) and (11)) used to generate ITS<sub>n</sub> are relevant for describing experimental UCS and *c* measurements. The appropriateness of the numerical approach used for ITS was assessed for V0.

* The DS test proved to be a reliable and straightforward method for assessing the shear strength of CSS. The shear strength of both materials was accurately represented by the Mohr–Coulomb failure criterion. For V0, the determination coefficient R² was higher than 0.80 for 3 out of 5 studied cases for mixture A and for all studied cases for mixture B. For the varied mixtures (V1–V10), R² was higher than 0.90 for all studied cases for both materials.

* For V0, *c* values ranged from 0.25 to 0.39 MPa for mixture A and from 0.18 to 0.42 MPa for mixture B. Lower values were measured after 7 days of curing, and higher values were measured after 180 days and 1 year of curing for mixture A and mixture B, respectively.

* The friction angle proved to be independent of the mixture variables and ranged between 48 and 53° for both materials.

* Findings relative to the *c*-φ properties indicate that for either the measured or predicted MC parameters, the mechanical enhancement of both CSSs can be quantified using *c*, while φ (or ξ) can be considered scalar parameters.

* The ratio ξ̄ = ITS/UCS seemed to be independent of mixture variables, and it varied within quite a narrow range. For V0, ξ̄<sub>A</sub> = 0.10 (0.02) and ξ̄<sub>B</sub> = 0.15 (0.01). For the varied mixtures (V1–V10), ξ̄<sub>A</sub> = 0.12 (0.02) and ξ̄<sub>B</sub> = 0.14 (0.01). These results suggest that for the studied inference space, there is a straight proportionality between the tensile and compressive strengths. This proportionality can be treated as a scalar.

* The parameter ξ̄ was incorporated into the geometric method, which was successfully validated via DS measurements. For the studied materials, this led to the overestimation of *c* from 60 to 100 kPa and the underestimation of φ from 5 to 15°. The overestimation of *c* and underestimation of φ are characteristic of the methodology employed, which involves performing indirect measurements of tensile properties. Nevertheless, using alternative geometric methods to assess *c*-φ properties based on tensile properties is relevant.

* The combined effects of mixture variables were assessed via contour plots for cohesion. These plots were generated using the c/ITS quotients and numerical expressions for ITS. These tools can be used for mixture optimization based on the technical specifications and operating conditions used.

* Long-term results can be achieved in several days by increasing CC. Reducing binder consumption can be achieved by increasing CT or DC. Specifically, the effects of compaction were shown to be relevant even in the long term for both materials.

**Author Contributions:** Conceptualization, T.L.; methodology, M.C.-L. and T.L.; validation, L.T.; formal analysis, M.C.-L.; investigation, M.C.-L. and T.L.; resources, T.L. and J.-P.S.; writing—original draft preparation, M.C.-L.; writing—review and editing, T.L. and L.T.; supervision, T.L. and L.T.; project administration, T.L. and J.-P.S.; funding acquisition, J.-P.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the French Ministry of Ecological Transition and Territorial Cohesion and the COLAS Company, grant number RP2-E20029.

**Data Availability Statement:** The original contributions presented in this study are included in the article and refs. [57,65]. Further inquiries can be directed to the corresponding author.

**Acknowledgments:** Authors thank the French Ministry of Ecological Transition and Territorial Cohesion and the COLAS company for their funding. The manuscript benefited from comments and suggestions by three anonymous reviewers and journal editors.
**Conflicts of Interest:** The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

**Abbreviations**

The following abbreviations are used in this manuscript:

<table><tr><td>AASHTO</td><td>American Association of State Highway and Transportation Officials</td></tr><tr><td>CC</td><td>Cement content</td></tr><tr><td>CID</td><td>Consolidated isotropically drained conditions</td></tr><tr><td>COV</td><td>Coefficient of variation</td></tr><tr><td>CSS</td><td>Cement-stabilized soils</td></tr><tr><td>CT</td><td>Curing time</td></tr><tr><td>DC</td><td>Degree of compaction</td></tr><tr><td>DS</td><td>Direct shear</td></tr><tr><td>DTS</td><td>Direct tensile strength</td></tr><tr><td>GTR</td><td>French Road Earthworks Manual</td></tr><tr><td>ITS</td><td>Indirect tensile strength</td></tr><tr><td>LHS</td><td>Latin Hypercube Sampling</td></tr><tr><td>LL</td><td>Liquid limit</td></tr><tr><td>MC</td><td>Mohr-Coulomb</td></tr><tr><td>MDD</td><td>Maximum dry density</td></tr><tr><td>OMC</td><td>Optimum moisture content</td></tr><tr><td>PI</td><td>Plasticity index</td></tr><tr><td>PL</td><td>Plastic limit</td></tr><tr><td>SD</td><td>Standard deviation</td></tr><tr><td>UCS</td><td>Unconfined compressive strength</td></tr><tr><td>USCS</td><td>Unified Soils classification system</td></tr></table>

**Appendix A. Experimental and Numerical Results**

*Appendix A.1. Reference Mixture (V0) [17,41,57,65]*

<table><caption>**Table A1.** Experimental results for reference mixture V0 and numerical estimation of ITS.</caption><thead><tr><th rowspan="2">Mix.</th><th rowspan="2">CT (Days)</th><th colspan="2">UCS (MPa)</th><th colspan="2">ITS (MPa)</th><th rowspan="2">ITS<sub>n</sub> (MPa)</th><th colspan="3">DS</th></tr><tr><th>Mean</th><th>SD</th><th>Mean</th><th>SD</th><th>c (MPa)</th><th>φ (°)</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td rowspan="5">A</td><td>7</td><td>1.53</td><td>0.05</td><td>0.14</td><td>0.02</td><td>0.13</td><td>0.25</td><td>50.2</td><td>0.94</td></tr><tr><td>28</td><td>1.65</td><td>0.15</td><td>0.15</td><td>0.01</td><td>0.17</td><td>0.35</td><td>36.3</td><td>0.57</td></tr><tr><td>90</td><td>2.41</td><td>0.01</td><td>0.18</td><td>0.01</td><td>0.20</td><td>0.37</td><td>55.9</td><td>0.93</td></tr><tr><td>180</td><td>1.77</td><td>0.11</td><td>0.23</td><td>0.01</td><td>0.22</td><td>0.39</td><td>35.4</td><td>0.74</td></tr><tr><td>360</td><td>2.11</td><td>0.02</td><td>0.22</td><td>0.03</td><td>0.24</td><td>0.34</td><td>62.0</td><td>0.94</td></tr><tr><td rowspan="5">B</td><td>7</td><td>0.91</td><td>0.03</td><td>0.12</td><td>0.01</td><td>0.12</td><td>0.18</td><td>49.6</td><td>0.98</td></tr><tr><td>28</td><td>1.20</td><td>0.03</td><td>0.17</td><td>0.01</td><td>0.18</td><td>0.27</td><td>53.1</td><td>0.94</td></tr><tr><td>90</td><td>1.53</td><td>0.18</td><td>0.21</td><td>0.01</td><td>0.23</td><td>0.39</td><td>49.5</td><td>0.93</td></tr><tr><td>180</td><td>1.72</td><td>0.07</td><td>0.26</td><td>0.01</td><td>0.26</td><td>0.39</td><td>42.8</td><td>0.82</td></tr><tr><td>360</td><td>1.91</td><td>0.04</td><td>0.31</td><td>0.01</td><td>0.29</td><td>0.42</td><td>61.4</td><td>0.86</td></tr></tbody></table>
<table><caption>**Table A2.** Measured $\xi$ and estimated ($c_{\xi}$, $\phi_{\xi}$) MC parameters for the V0 case.</caption><thead><tr><th rowspan="2">Mixture</th><th rowspan="2">CT (Days)</th><th rowspan="2">ξ</th><th colspan="2">Estimated</th><th rowspan="2">c<sub>ξ</sub>/c</th><th rowspan="2">φ<sub>ξ</sub>/φ</th></tr><tr><th>c<sub>ξ</sub> (MPa)</th><th>φ<sub>ξ</sub> (°)</th></tr></thead><tbody><tr><td rowspan="5">A</td><td>7</td><td>0.09</td><td>0.27</td><td>51.0</td><td>1.11</td><td>1.02</td></tr><tr><td>28</td><td>0.09</td><td>0.29</td><td>50.9</td><td>0.84</td><td>1.40</td></tr><tr><td>90</td><td>0.07</td><td>0.37</td><td>56.0</td><td>0.99</td><td>1.00</td></tr><tr><td>180</td><td>0.13</td><td>0.40</td><td>41.0</td><td>1.04</td><td>1.16</td></tr><tr><td>360</td><td>0.10</td><td>0.41</td><td>48.0</td><td>1.18</td><td>0.77</td></tr><tr><td rowspan="5">B</td><td>7</td><td>0.13</td><td>0.21</td><td>39.9</td><td>1.17</td><td>0.80</td></tr><tr><td>28</td><td>0.15</td><td>0.30</td><td>36.0</td><td>1.13</td><td>0.68</td></tr><tr><td>90</td><td>0.14</td><td>0.37</td><td>38.5</td><td>0.95</td><td>0.78</td></tr><tr><td>180</td><td>0.15</td><td>0.45</td><td>34.5</td><td>1.15</td><td>0.81</td></tr><tr><td>360</td><td>0.16</td><td>0.54</td><td>30.9</td><td>1.31</td><td>0.50</td></tr></tbody></table>

## Appendix A.2. Variability Study (V1–V10)

<table><caption>**Table A3.** Experimental and numerical results for the variability study (V1 to V10).</caption><thead><tr><th rowspan="3">V</th><th rowspan="3">CT (Days)</th><th rowspan="3">CC (%)</th><th rowspan="3">DC (%)</th><th rowspan="3">UCS (MPa)</th><th colspan="6">Experimental Approach</th><th colspan="2">Numerical Approach</th></tr><tr><th colspan="3">A</th><th colspan="3">B</th><th>A</th><th>B</th></tr><tr><th>c (MPa)</th><th>φ (°)</th><th>R<sup>2</sup></th><th>c (MPa)</th><th>φ (°)</th><th>R<sup>2</sup></th><th>ITS<sub>n</sub> (MPa)</th><th>ITS<sub>n</sub> (MPa)</th></tr></thead><tbody><tr><td>1</td><td>60</td><td>3.56</td><td>94.0</td><td>2.05</td><td>0.29</td><td>58.1</td><td>0.98</td><td>1.63</td><td>0.32</td><td>55.0</td><td>0.96</td><td>0.20</td><td>0.20</td></tr><tr><td>2</td><td>42</td><td>4.00</td><td>95.5</td><td>2.23</td><td>0.34</td><td>57.3</td><td>0.99</td><td>1.90</td><td>0.32</td><td>53.0</td><td>0.96</td><td>0.24</td><td>0.23</td></tr><tr><td>3</td><td>54</td><td>2.44</td><td>98.5</td><td>1.62</td><td>0.23</td><td>54.4</td><td>0.96</td><td>1.59</td><td>0.27</td><td>53.8</td><td>0.96</td><td>0.18</td><td>0.22</td></tr><tr><td>4</td><td>36</td><td>2.00</td><td>97.5</td><td>1.07</td><td>0.17</td><td>51.2</td><td>0.98</td><td>1.13</td><td>0.19</td><td>53.9</td><td>1.00</td><td>0.12</td><td>0.16</td></tr><tr><td>5</td><td>48</td><td>2.67</td><td>97.0</td><td>1.30</td><td>0.20</td><td>51.0</td><td>0.95</td><td>1.54</td><td>0.27</td><td>52.3</td><td>0.95</td><td>0.17</td><td>0.20</td></tr><tr><td>6</td><td>30</td><td>3.78</td><td>94.5</td><td>2.05</td><td>0.30</td><td>56.3</td><td>0.99</td><td>1.49</td><td>0.24</td><td>50.8</td><td>0.95</td><td>0.20</td><td>0.19</td></tr><tr><td>7</td><td>24</td><td>3.11</td><td>98.0</td><td>1.90</td><td>0.26</td><td>55.7</td><td>0.98</td><td>1.36</td><td>0.24</td><td>52.3</td><td>0.93</td><td>0.19</td><td>0.21</td></tr><tr><td>8</td><td>18</td><td>3.33</td><td>95.0</td><td>1.00</td><td>0.17</td><td>48.5</td><td>0.98</td><td>1.03</td><td>0.20</td><td>54.9</td><td>0.99</td><td>0.16</td><td>0.16</td></tr><tr><td>9</td><td>12</td><td>2.89</td><td>96.5</td><td>1.00</td><td>0.18</td><td>50.7</td><td>0.97</td><td>0.89</td><td>0.19</td><td>52.4</td><td>1.00</td><td>0.14</td><td>0.14</td></tr><tr><td>10</td><td>7</td><td>2.22</td><td>96.0</td><td>0.65</td><td>0.15</td><td>45.7</td><td>0.91</td><td>0.63</td><td>0.13</td><td>52.9</td><td>1.00</td><td>0.07</td><td>0.08</td></tr></tbody></table>

<table><caption>**Table A4.** Measured $\xi$ and estimated ($c_{\xi}$, $\phi_{\xi}$) MC parameters for variability study.</caption><thead><tr><th rowspan="3">V</th><th rowspan="3">ξ</th><th colspan="3">A</th><th colspan="3">B</th></tr><tr><th colspan="2">Estimated</th><th rowspan="2">c<sub>ξ</sub>/c</th><th rowspan="2">φ<sub>ξ</sub>/φ</th><th colspan="2">Estimated</th></tr><tr><th>c<sub>ξ</sub> (MPa)</th><th>φ<sub>ξ</sub> (°)</th><th>c<sub>ξ</sub> (MPa)</th><th>φ<sub>ξ</sub> (°)</th></tr></thead><tbody><tr><td>1</td><td>0.10</td><td>0.38</td><td>49.2</td><td>1.30</td><td>0.85</td><td>0.12</td><td>0.36</td><td>41.9</td><td>1.14</td><td>0.76</td></tr><tr><td>2</td><td>0.11</td><td>0.44</td><td>46.6</td><td>1.31</td><td>0.81</td><td>0.12</td><td>0.42</td><td>42.4</td><td>1.31</td><td>0.80</td></tr><tr><td>3</td><td>0.11</td><td>0.33</td><td>46.2</td><td>1.40</td><td>0.85</td><td>0.14</td><td>0.38</td><td>38.8</td><td>1.39</td><td>0.72</td></tr><tr><td>4</td><td>0.11</td><td>0.22</td><td>44.9</td><td>1.31</td><td>0.88</td><td>0.14</td><td>0.28</td><td>36.9</td><td>1.49</td><td>0.69</td></tr><tr><td>5</td><td>0.13</td><td>0.30</td><td>40.2</td><td>1.51</td><td>0.79</td><td>0.13</td><td>0.35</td><td>40.9</td><td>1.29</td><td>0.78</td></tr><tr><td>6</td><td>0.10</td><td>0.38</td><td>48.9</td><td>1.27</td><td>0.87</td><td>0.13</td><td>0.34</td><td>40.8</td><td>1.45</td><td>0.80</td></tr><tr><td>7</td><td>0.10</td><td>0.36</td><td>48.1</td><td>1.39</td><td>0.86</td><td>0.15</td><td>0.36</td><td>34.6</td><td>1.46</td><td>0.66</td></tr><tr><td>8</td><td>0.16</td><td>0.28</td><td>30.9</td><td>1.68</td><td>0.64</td><td>0.15</td><td>0.27</td><td>34.3</td><td>1.38</td><td>0.63</td></tr><tr><td>9</td><td>0.14</td><td>0.25</td><td>37.4</td><td>1.35</td><td>0.74</td><td>0.16</td><td>0.24</td><td>32.6</td><td>1.26</td><td>0.62</td></tr><tr><td>10</td><td>0.11</td><td>0.13</td><td>44.9</td><td>0.90</td><td>0.98</td><td>0.12</td><td>0.14</td><td>42.3</td><td>1.08</td><td>0.80</td></tr></tbody></table>
## Appendix B. Generalization of the Geometric Method

<table><caption>Table A5. Numerical values of UCS and c based on mixture variables and for all studied cases.</caption><thead><tr><th rowspan="3">V</th><th rowspan="3">CT<br>(Days)</th><th rowspan="3">CC<br>(%)</th><th rowspan="3">DC<br>(%)</th><th colspan="4">Mixture A, ξ = 0.11</th><th colspan="4">Mixture B, ξ = 0.14</th></tr><tr><th colspan="2">UCS (MPa)</th><th colspan="2">c (MPa)</th><th colspan="2">UCS (MPa)</th><th colspan="2">c (MPa)</th></tr><tr><th>UCS<sub>ξ,n</sub></th><th>UCS<sub>ξ,n</sub>/UCS</th><th>c<sub>ξ,n</sub></th><th>c<sub>ξ,n</sub>/c</th><th>UCS<sub>ξ,n</sub></th><th>UCS<sub>ξ,n</sub>/UCS</th><th>c<sub>ξ,n</sub></th><th>c<sub>ξ,n</sub>/c</th></tr></thead><tbody><tr><td>0</td><td>7</td><td>3.00</td><td>96.0</td><td>1.14</td><td>0.74</td><td>0.31</td><td>1.28</td><td>0.91</td><td>0.91</td><td>0.22</td><td>1.22</td></tr><tr><td>0</td><td>28</td><td>3.00</td><td>96.0</td><td>1.48</td><td>0.90</td><td>0.34</td><td>0.97</td><td>1.20</td><td>1.05</td><td>0.29</td><td>1.08</td></tr><tr><td>0</td><td>90</td><td>3.00</td><td>96.0</td><td>1.77</td><td>0.73</td><td>0.49</td><td>1.32</td><td>1.53</td><td>1.06</td><td>0.37</td><td>0.96</td></tr><tr><td>0</td><td>180</td><td>3.00</td><td>96.0</td><td>1.94</td><td>1.10</td><td>0.36</td><td>1.10</td><td>1.72</td><td>1.07</td><td>0.42</td><td>1.07</td></tr><tr><td>0</td><td>360</td><td>3.00</td><td>96.0</td><td>2.11</td><td>1.00</td><td>0.43</td><td>1.26</td><td>1.91</td><td>1.08</td><td>0.47</td><td>1.12</td></tr><tr><td>1</td><td>60</td><td>3.56</td><td>94.0</td><td>1.79</td><td>0.87</td><td>0.42</td><td>1.43</td><td>1.634</td><td>0.90</td><td>0.40</td><td>1.25</td></tr><tr><td>2</td><td>42</td><td>4.00</td><td>95.5</td><td>2.14</td><td>0.96</td><td>0.46</td><td>1.35</td><td>1.896</td><td>0.88</td><td>0.46</td><td>1.45</td></tr><tr><td>3</td><td>54</td><td>2.44</td><td>98.5</td><td>1.58</td><td>0.97</td><td>0.33</td><td>1.43</td><td>1.589</td><td>0.98</td><td>0.39</td><td>1.41</td></tr><tr><td>4</td><td>36</td><td>2.00</td><td>97.5</td><td>1.09</td><td>1.02</td><td>0.22</td><td>1.29</td><td>1.132</td><td>1.03</td><td>0.28</td><td>1.45</td></tr><tr><td>5</td><td>48</td><td>2.67</td><td>97.0</td><td>1.52</td><td>1.17</td><td>0.27</td><td>1.33</td><td>1.545</td><td>0.92</td><td>0.38</td><td>1.38</td></tr><tr><td>6</td><td>30</td><td>3.78</td><td>94.5</td><td>1.81</td><td>0.88</td><td>0.42</td><td>1.39</td><td>1.494</td><td>0.93</td><td>0.37</td><td>1.55</td></tr><tr><td>7</td><td>24</td><td>3.11</td><td>98.0</td><td>1.73</td><td>0.91</td><td>0.39</td><td>1.49</td><td>1.359</td><td>1.09</td><td>0.33</td><td>1.36</td></tr><tr><td>8</td><td>18</td><td>3.33</td><td>95.0</td><td>1.47</td><td>1.46</td><td>0.21</td><td>1.21</td><td>1.031</td><td>1.09</td><td>0.25</td><td>1.27</td></tr><tr><td>9</td><td>12</td><td>2.89</td><td>96.5</td><td>1.26</td><td>1.26</td><td>0.20</td><td>1.12</td><td>0.893</td><td>1.14</td><td>0.22</td><td>1.13</td></tr><tr><td>10</td><td>7</td><td>2.22</td><td>96.0</td><td>0.66</td><td>1.02</td><td>0.13</td><td>0.89</td><td>0.632</td><td>0.89</td><td>0.15</td><td>1.19</td></tr></tbody></table>

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