# Proposal of a homogenization model of the mechanical properties of a silt treated with lime and binder

Lucile Pigeot, Anthony Soive, Nathalie Dufour, Ahmed Loukili

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Lucile Pigeot, Anthony Soive, Nathalie Dufour, Ahmed Loukili. Proposal of a homogenization model of the mechanical properties of a silt treated with lime and binder. 2nd International Conference on Geomechanics and Geoenvironmental Engineering, Western Sydney University, Nov 2022, Sydney, Australia. pp.1-7. hal-05154242

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# Proposal of a homogenization model of the mechanical properties of a silt treated with lime and hydraulic binder

Lucile Pigeot¹, Anthony Soive², Nathalie Dufour², Jean-François Barthélémy³, Fabienne Dermenonville⁴, Ahmed Loukili⁵

¹PhD Student, GEM, Ecole Centrale de Nantes, Nantes, France

²Dr., Project-team GeoCoD, Cerema, Aix-en-Provence, France

³Dr., Cerema, UMR MCD, Cerema, Provins Cedex, France

⁴Head of Materials Engineering, Egis Structures et environnement, Seyssins, France

⁵Professor, GEM, Ecole Centrale de Nantes, Nantes, France

Corresponding author's E-mail: lucile.pigeot@ec-nantes.fr

## Abstract

Economic and environmental imperatives are prompting project managers to switch places to the use of in situ soils for the construction of structures. Generally, the poor geotechnical performances of these soils are improved using treatment and compaction. The material studied in the present article is a lime and hydraulic binder treated low clayey silt. This article focuses on the feasibility of a numerical model to predict the mechanical behaviour of this treated soil. Previous authors' SEM (scanning electron microscope) observations of treated silt enabled to propose a microstructure of the soil which would be composed of aggregates of natural soils surrounded by a shell of hydrates. A multi-scale numerical homogenization model is presented to validate these hypotheses. Homogenized mechanical properties obtained numerically are compared with the Young modulus obtained by measuring celerity of S-waves and P-waves in a soil specimen. The experimental results lie within the range of the numerically obtained results.

**Keywords:** Stabilized soil, Multi-scale homogenization, Binder hydration, Mechanical properties

## 1. INTRODUCTION

Soil improvement by treatment and compaction is increasingly used in large geotechnical projects. In order to optimise the structures, it is necessary to predict the behaviour of these materials. The timedependent mechanical gain from adding lime and cement to the soil can be measured by laboratory tests (Uniaxial Compression Test, triaxial test) as is done with concrete. In cementitious materials studies (Jacquemot, 2014), the compressive strength of mortars as a function of curing time is usually associated with the degree of advancement of hydration reactions, which can be estimated by measuring the evolution of the quantity of hydrates created (by X-ray diffraction (XRD) or thermogravimetric analysis (TGA)). In contrast, these experimental measurements do not provide results on treated soils. Indeed, the lime and cement dosage (<10%) produce a limited quantity of hydrates that does not enable them to be identified (Bouras, 2020; Maubec, 2010).

Then, to understand the origin of the mechanical gain, studies were carried out on the microstructure of soils before and after treatment. Authors have highlighted a change in terms of microstructure in treated soils: creation of bonds between soil grains and shells around the natural soil aggregates that are formed by the hydration products (Cabane, 2004; Lemaire, 2012).

A multi-scale homogenization model is presented to test the microstructural assumptions and pass from the hydrate scale to the soil behavior at the laboratory specimen scale. The objective of this
study is to test different structural schemes and estimate an amount of hydrate phases formed by using numerical homogenization methods. The model results are compared with measurements of the Young modulus obtained by measurements of P- and S-wave velocities (Vp and Vs) in soil samples.

# 2. MATERIAL CHARACTERIZATION

## 2.1. Compositions

The treated soil is a mixture of silt with 1% lime and 5% hydraulic binder. The soil used is a silt classified ML (silt low liquid limit) according to the Unified Soil Classification System (USCS). It was sieved to 0.2 mm and passed through a riffle box for dividing soil aggregates into representative sample increment for testing. The particle size identification performed by laser diffraction indicates a granular composition equivalent to 81.41% silt, 11.03% clay and 7.55% sand. The sieve size of 0.08 mm is higher than 80% and the maximum grain diameter Dmax equal to 0.1346 mm (ISO, 2020). The mineralogical composition was analyzed by X-ray diffraction (XRD): the silt is mostly composed of quartz, carbonate phases (Calcite and Dolomite), Silicates (Muscovite, Microcline and Clinochlore) Feldsphaths (Albite) clay minerals (Kaolinite) and Hematite. This composition is like several silty soil from northern France (Lemaire, 2012; Maubec, 2010). At a specimen scale, the natural soil is considered as a homogeneous entity in terms of mineralogical composition and water content.

Thermogravitational analysis (TGA) and X-ray diffraction (XRD) measurements are used to characterize the lime and binder employed. Lime contains most of quicklime (90% of CaO), low in hydrate (7% of Portlandite) and is slightly carbonated (0,5% of Calcite). The hydraulic binder used contains about 65% of slag and 35% of Portland Cement. From its composition, it can be compared to a CEM III/A (Stephant, 2015).

## 2.2. Experimental measurement

Experimental measurements of the Young modulus were performed on unsaturated soil specimens compacted in laboratory. The natural soil was brought to the optimal water content determined by Proctor tests, WOPN (OPN: Optimal Proctor Normal), then treated with 1% lime and 5% hydraulic binder. The mixture was statically compacted in a single layer to obtain a soil specimen of 75 mm in diameter and 15 cm high. The treated and the untreated specimen were characterized by the same parameters: a density of 19.9 KN/m³ and a degree of saturation Sr of 79%. The treated one was stored hermetically in a controlled atmosphere at 20 °C for 90 days. Indeed, according to the requirements of the GTS (Guide de Traitement des sols (LCPC/SETRA, 2000)), it is estimated that after 90 days of curing the soil has reached its final mechanical properties. Then, the evolution of the mechanical properties was estimated by measurements of the propagation of the Primary (Vp) and Secondary (Vs) waves velocities in the specimen. The tests were performed at atmospheric pressure, with piezoelectric transducers (Bender Element).

The corresponding Young's modulus (E) and Poisson's ratio (v) were calculated as follows:

$$E = 2\rho V_s + (1 + v) \quad \text{and} \quad v = \frac{Vp^2 - 2V_s^2}{2Vp^2 - V_s^2} \quad \text{with } \rho: \text{sample density (kg/m}^3) \quad (1)$$
<table><caption>Table 1: Young Modulus and Poisson ratio evolution from untreated to treated soil at 90 days of curing time.</caption><thead><tr><th></th><th>Vp (m/s)</th><th>Vs (m/s)</th><th>E (MPa)</th><th>v</th></tr></thead><tbody><tr><td>Untreated compacted soil</td><td>307</td><td>116</td><td>75</td><td>0,42</td></tr><tr><td>Treated compacted soil</td><td>498</td><td>250</td><td>436</td><td>0,38</td></tr></tbody></table>

Vp and Vs were calculated as an average of three stacks of 40 measurements each. Young's modulus of the treated soil sample is almost six times higher than that of the untreated sample (see Table 1).

## 3. NUMERICAL HOMOGENIZATION MODEL

To understand the origin of the experimentally measured mechanical gain and to represent this heterogeneous material, a homogenization model is proposed. The simulations were made possible by a python library developed at Cerema and allowing to implement mean-field estimates. The input data required for the model were the nature and volume fractions of the components (soil, pores, hydrates), their mechanical properties (Young modulus and Poisson ratio) and an hypothesis of their structural arrangement after treatment. Three different scales were used, based on the three pore size families identified by pore distribution analyses on similar treated silt: a nano-porosity, a micro-porosity and a macro-porosity (Nguyen, 2015).

### 3.1. Hydration

When added to humid soil, the anhydrous phases of lime and cement react with the water contained in the soil to form hydrated phases. To be more precise, the added lime reacts with the available water in the soil to form Portlandite (CH in cement notation) (Eq. 2). The hydraulic binder also hydrates and forms CSH (hydrated calcium silicates) and Portlandite (CH) (Eq. 4). The created Portlandite can then react with silicates and aluminates present in the soil (A-S) to form CASH (calcium aluminosilicate hydrates) (Eq. 3, 5). As explained earlier, experimental identification of hydrated phases in treated soils is difficult. To avoid this problem, it is possible to quantify analytically the phases formed. The equations proposed by Åhnberg were used to estimate the maximum amounts of CH, CSH and CASH formed by the addition of lime and binder in silt (Ahnberg, 2006). All the Portlandite (CH) created was supposed to react with aluminates and silicates (A-S) in the soil to form calcium aluminosilicate hydrates (CASH). For the lime:

$$100 \text{ kg CaO} + 32.1 \text{ kg H}_2\text{O} \rightarrow 132.1 \text{ kg CH} \qquad (2)$$

$$100 \text{ kg CH} + 109 \text{ kg AS} + 75 \text{ kg H}_2\text{O} \rightarrow 282 \text{ kg CASH} \qquad (3)$$

For the binder (65% slag and 35% Portland cement):

$$100 \text{ kg binder} + 25 \text{ kg H}_2\text{O} \rightarrow 113 \text{ kg CASH} + 11.17 \text{ kg CH} \qquad (4)$$

$$11.17 \text{ kg CH} + 12.18 \text{ kg AS} + 8.38 \text{ kg H}_2\text{O} \rightarrow 31.5 \text{ kg CASH} \qquad (5)$$

Other authors separate the action of the lime and the hydraulic binder. The lime, added first, only contributes to the flocculation of the soil into lumps. Thus, the Portlandite (CH) created by the hydration of CaO does not participate in the formation of the hydrate shell around the soil aggregates, which is only formed by the hydration of the hydraulic binder (Cabane, 2004). Both assumptions were tested in the model.
## 3.2. Microstructure

From Scanning Electron Microscopy (SEM) observations made on treated soils, (Lemaire, 2012) highlighted the presence of a hydrate layer surrounding lumps of natural soil (Figure 1 a)). Aggregates are formed because of the addition of lime which causes the soil to flocculate (Cabane, 2004). The added binder creates a cementitious paste around the aggregates, which during hydration will form a shell with bonds (Figure 1 b)). (Cabane, 2004) proposed a schematic representation of this stabilized silt, the aggregates are millimeter-sized and the hydrate shell is about 30 µm thick for 5% added binder.

<figure><img src="image_2.png" /><figcaption>Figure 1: From SEM observation (Lemaire 2012) a), and the representation proposed by (Cabane 2004) b), to scheme used in the model c)</figcaption></figure>

## 3.3. Modeling representation

In order to verify the microstructural hypotheses proposed by Cabane (Cabane 2004), a multi-scale homogenization approach is proposed to calculate the Young modulus of the treated soil. Therefore, parameters must be defined: the volume fractions of the elements (3.1), their mechanical properties, the arrangement of the particles (3.2) and the different scales.

First, the maximum amount of hydrates was calculated using Ahnberg's equations. Information on the mechanical properties of these hydrated elements has been obtained in the literature, from the results of nanoindentation measurements performed on cement pastes (Jennings et al., 2007). As the treated soil sample was unsaturated, the pores were estimated to be empty and thus with zero elastic moduli. The treated soil was represented as concentric spheres with the homogeneous natural soil surrounded by a homogeneous hydrate layer composed of CH, CSH, CASH and pores (Figure 2 c)). In the proposed model, there were three porosity scales: a nano-porosity at the hydrate shell scale, an intra-granular micro-porosity and an inter-granular macro-porosity. The nano-porosity which corresponds to hydrates was made up of more or less dense sheets (Stephant, 2015). It was estimated at between 10% (Tennis & Jennings, 2000) and 40% (Termkhajornkit et al., 2011) of the total volume of the hydrate matrix. The first step of the modelling consisted in finding the soil modulus without any porosity, using experimental measurement done previously on the porous untreated sample (Table 1). Soil aggregates were represented by a homogeneous soil matrix with a micro-porosity. To calculate its homogenized mechanical properties the Mori-Tanaka scheme (Mori & Tanaka, 1973), defined for matrices with inclusions, was used. Next, the self-consistent scheme (SC) (Hill, 1965) was employed to calculate the properties of the paste composed by the different hydrates identified and a nano-porosity.

In order to model the microstructure highlighted by previous authors, n-layer sphere geometry (Herve & Zaoui, 1993) applied with a self-consistent scheme (SC) was used with the soil layer surrounded by a layer of hydrate paste. The last scale corresponded to the homogeneous lumps within the
macroporosity that form the soil specimen. The homogenized properties at this scale were calculated by the self-consistent scheme (SC) (Hill, 1965).

# 4. RESULTS AND DISCUSSION

The first verification of the results of the models was done by comparing with the experimental results (see Table 1). The Young modulus measured experimentally on a specimen with 90 days of curing time reached 436 MPa. Assuming that the hydrates formed by the lime participate in the formation of the shell, the homogenized Young modulus obtained was 446 MPa. On the opposite, when testing the hypothesis that lime is only used to flocculate soils, the modulus obtained was 326 MPa. In both cases, the parameters set were as follows: the total porosity was 36%, the nanoporosity was 20%, the amount of hydrates corresponded to the maximum that can be formed with the added anhydrous phases and all the portlandite was transformed into CASH. Although these initial results validated the model, another structural arrangement was tested. If another microstructural hypothesis was adopted with a random distribution of natural soil, hydrates and pores and by using the self-consistent scheme (SC) the homogenized Young modulus obtained would be 200 MPa. Whereas the input volume fractions and mechanical properties were the same, this result shows the importance of the microstructure arrangement in the mechanical gain at the specimen scale.

Finally, a sensitivity analysis was done on the main parameters (see Figure 2) : the porosity of the hydrate shell, the presence or absence of lime in the hydrate shell, the amount of hydrate formed and the transformation of Portlandite with soil aluminates and silicates. To present this analysis, the Figure 2 shows the variation of the homogenised Young modulus (y axis, in MPa) as a function of the porosity within the hydrate shell (x axis, in %) for different cases.

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 2: Sensitivity of the model**</figcaption></figure>

First, the variation of the porosity of the hydrate shell is one of the most difficult parameters to estimate. Indeed, it is often measured by mercury intrusion porosimetry tests, but this method generally overestimates the volume of nano-pores. This is due to the low density of some of the hydrates formed such as CSH, especially in the case where the binder is slag (Stephant, 2015). As a result, the structures of these hydrates do not resist the mercury pressure required to measure them,
which can reach 150 MPa. So according to the bibliography the nanoporosity was ranged from 10% to 30%.

The solid lines represent the cases where 100% of the hydrates are formed from the anhydrous phases added to the soil and the dotted lines represent the case where only 80% of the hydrates are formed. Indeed, The Åhnberg equations only provide an indication of the maximum amount of hydrates that can be formed for a given amount of anhydrous phases. However, this suggests that free water was available for all phases to hydrate. So, to model an incomplete hydration or a temporal evolution, the quantity of hydrates created may be varied from 100% to 80%.

Finally, cases where the lime reaction products are included in the shell (blue and green curves) and where the lime is only used to flocculate the soil (orange curves) were also tested. Naturally, the moduli are higher when lime is involved in the formation of the hydrate shell. The difference between the green and blue curves is explained by the amount of Portlandite (CH) reacting with aluminates and silicates (A-S) to form CASH (calcium aluminosilicate hydrates) : in blue, all the CH has reacted and in green none. The moduli are quite different between the two cases because these hydrated phases do not have the same density and mechanical properties. In order to improve the model, a more accurate estimation of the hydrates formed with a consideration of the hydration kinetics and thus of the evolution of the mechanical properties as a function of time could be added. Similarly, a temporal evolution of the pore distribution could be proposed. Indeed, with the advancement of hydration the total porosity does not vary but the pore distribution evolves: the family of nano-pores corresponding to hydrates appears and the volume of micro-porosity decreases. The model could be also improved by taking into account saturation conditions closer to reality.

## 5. CONCLUSION

The main objective of this study was to verify by numerical homogenization techniques the microstructure of stabilized soils (grain and hydrate arrangement) by calculating the homogenized elastic properties of soil specimens and comparing them to Young Modulus measured in the laboratory. Numerous hypotheses were made based on references from the literature (nature, quantity of hydrates, structural arrangement). A sensitivity study of the model allowed to validate the proposed model. The numerical model of homogenization seems to validate the observations made by Cabane: aggregates of silts surrounded by a shell of hydrates. It also seems to validate the approximate quantity of hydrates formed from the Åhnberg equations.

## ACKNOWLEDGMENTS

The PhD project is financed by EGIS STRUCTURES ET ENVIRONNEMENT and Centrale Nantes.
We also wish to thank the Seine Nord Europe Canal Company.

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