Homogenised Elastic Properties of Lightweight Cemented Soils

Homogenised Elastic Properties of Lightweight Cemented Soils | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Homogenised Elastic Properties of Lightweight Cemented Soils Laura Perrotta, Enza Vitale, Giacomo Russo This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3409777/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Lightweight Cemented Soil (LWCS) is obtained by mixing soil, water and cement with air foam. This treatment allows a high workability of the mixture and good mechanical properties coupled with reduced unit volume weight, making the material suitable for many geotechnical applications. The complex and heterogeneous microstructure (a large pore system induced by foam in a cemented porous matrix) of the material requires a homogeneisation procedure for constitutive modelling of its hydromechanical behaviour. In this work, the stiffness modulus of LWCS was derived through the semi-analitycal Mean-Field Eshelby-based Homogenisation approaches, taking into account the chemo-physical evolution over curing time of the material. This procedure, never employed in the field of treated soils, requires several input parameters, as the artificial porosity induced by the addition of foam, the bulk and shear moduli and the Poisson’s ratio of LWCS samples. The artificial porosity was evaluated by means of X-Ray micro-CT scans, whereas the mechanical parameters were derived from experimental tests. The homogenised stiffness moduli, computed for different curing times, are in agreement with the available experimental results. The analytical homogenisation models allow predicting the elastic behaviour of the material and its evolution over time taking into account the role played by microstructure. soil treatment multi-scale analysis homogenisation Lightweight Cemented Soil (LWCS) Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 1 INTRODUCTION During the construction of major civil infrastructures, soils characterized by low mechanical properties usually need to be removed and replaced, leading to several economic and environmental drawbacks. Nowadays, this procedure is often avoided by using soil improvement techniques. In this way, the construction takes place in compliance with soil balance (i.e., reusing local materials considered unsuitable in their natural state), which is essential for the sustainability of civil works. For this reason, soil improvement techniques based on the use of binders are employed in several geotechnical engineering applications. A solution of increasing interest for reusing soils in geotechnical applications is Lightweight Cemented Soil (LWCS) [ 1 , 2 ]. LWCS is prepared by mixing soil, water and cement with air foam to have a low-density and self-levelling fresh paste which exibits relevant mechanical performances after setting of the binder. As shown in Fig. 1 , the LWCS sample is characterised by large voids, as footprints of foam-air bubbles, immersed in a cemented matrix of soil particles, cement-hydrated phases and smaller pores. Thanks to their reduced unit volume weight, LWCSs are advantageously used for different earthworks, such as cavities filling, trench backfilling [ 3 – 5 ] and also embankment construction [ 6 , 7 ] Experimental studies on the effects of cement, foam and initial water content on unit weight and shear strength of LWCS have been reported in literature, highlighting the evolution of physical and mechanical properties as a function of treatment parameters [ 8 – 10 ]. An insight into the role of foam content on chemo-physical evolution and microstructural features of the system on mechanical behaviour of LWCS have been recently provided by Vitale et al . (2020) [ 1 ]. The Authors showed that addition of foam does not alter chemo-physical evolution of the soil–cement–water system in terms of either cement hydration or pozzolanic reactions. For treated samples (with or without foam addition), precipitation of portlandite and progressive formation of cementitious compounds due to cement hydration process were generally observed since the very short term (i.e., within 24 h of curing). At increasing curing times, a gradual consumption of formed portlandite and progressive dissolution of clay minerals were found consistent with the development of pozzolanic reactions. The complex and heterogeneous microstructure of the material (a cemented porous matrix with a foam induced large pore system) requires a homogeneisation procedure for constitutive modelling of its hydromechanical behaviour, taking account of heterogeneities and different phases [ 11 , 12 ]. The advantage of this approach is the possibility of considering the role played by microstructure on the macroscopic behaviour of the material. The length scale used for the experimental observation determines the role of the inherent heterogeneity on material behaviour. Some constituents or phases in an heterogeneous material are identifiable only at or below a specific length scale. Each of the constituents can be regarded as homogeneous material, but may become heterogeneous when it is observed at a smaller length scale. Therefore, for LWCS a length scale of interest must be established, so that the microstructural features at smaller scale can be neglected [ 13 ]. Once the scale of interest has been defined, a homogenisation procedure can be introduced, starting from the physical reality of LWCS, in order to define the homogenised properties of the material, taking into account the heterogeneities of the sample that can be observed at such scale level. In this framework, compared to traditional approaches based on the continuum mechanics, homogenised material properties are determined by means of analytical, semi-analytical or numerical models that indeed account for the micromechanical features of the material through some parameters that depend on the approaches used. Homogenisation has been largely employed to predict the mechanical behaviour of heterogeneous materials. In the framework of solid mechanics, one of the most addressed problems was originally the determination of the mechanical properties of a polycrystal from those of a single crystal. In the last years, improved estimates have been pursued. The problem of a single ellipsoidal particle embedded in an unbounded domain of material under uniform exterior loading has been analytically solved by Eshelby (1957) [ 14 ]. His solution is relatively compact and easy to use, therefore has been a basis for the development of many approximated analytical methods, such as Mean Field Approaches (MFAs) [ 13 ]. These approaches are very convenient thanks to their low computational cost compared to numerical methods. MFAs have been employed in literature to estimate the elastic properties of concrete [ 11 , 12 ]; however, in the field of geotechnical engineering the homogenisation approaches, and in particular MFAs, are rarely implemented due to the difficulties in determining phase properties compared to other materials such as concrete and steel. MFAs have been employed to compute the effective elastic properties of porous materials [ 15 ], and gap-graded soils [ 16 ]. In the framework of treated soils, no previous studies based on this type of approaches were found in literature. In this study, MFAs have been implemented for the determination of elastic properties of LWCS, taking account of the chemo-physical evolution of the material over curing time. A multi-scale model of sample heterogeneities, as reference for the homogenisation process, is shown in Fig. 2 . In the first step of homogenisation, the four-phases system (solid particles, water, voids and cement-hydrated phases) is homogenised into a two-phase system (cementitious matrix, large voids induced by foam). Under the hypothesis of homogeneous and isotropic cementitious matrix, and assuming the large voids as spherical, the second homogenisation step allows to determine the elastic properties of the homogenised matrix. This second step of homogenisation process has been performed using simplified approaches based on Mean Field Homogenisation Methods. Elastic stiffness modulus of homogenised LWCS as function of curing time has been determined through analytical formulae. A comparison with the elastic stiffness of LWCS determined by available experimental tests has been performed. A three-phase system, made of matrix, voids and portlandite, has been also considered for further investigations on the suitability of the analytical methods. In the framework of the homogenisation methods based on Mean Field Approaches, the study highlights the valuable capability of Mori-Tanaka scheme of predicting the elastic properties of an extremely heterogeneous material like the LWCS over time, despite its simple analytical formulation. 2 HOMOGENISATION PROCESS The main interest of the homogenisation approach lies on the possibility of modeling mechanical behaviour of heterogeneous materials in the framework of continuum michromechanics [ 17 ] (Fig. 3 ). In the field of homogenisation, continuum micromechanics deals mainly with statistically homogeneous materials for which it is possible to define a Representative Volume Element (RVE) and an Equivalent Homogeneous Medium (EHM), which are equivalent to each other from a mechanical point of view. This statement implies that they should have the same overall responses to any kind of mechanical loading. In the EHM, the stress and strain fields \(\underset{\_}{\underset{\_}{\varvec{\sigma }}}\left(\underset{\_}{\varvec{x}}\right)\) and \(\underset{\_}{\underset{\_}{\varvec{\epsilon }}}\left(\underset{\_}{\varvec{x}}\right)\) must be the average values, over any RVE centred at \(\underset{\_}{\varvec{x}}\) (which is the macroscopic position vector in a global reference system), of the local stress and strain fields \({\underset{\_}{\underset{\_}{\varvec{\sigma }}}}_{\mu }\left(\underset{\_}{\varvec{y}}\right)\) and \({\underset{\_}{\underset{\_}{\varvec{\epsilon }}}}_{\mu }\left(\underset{\_}{\varvec{y}}\right)\) (with \(\underset{\_}{\varvec{y}}\) equal to the microscopic position vector, in a local reference system), [ 18 ]. \(\underset{\_}{\underset{\_}{\varvec{\sigma }}}\left(\underset{\_}{\varvec{x}}\right)\) and \(\underset{\_}{\underset{\_}{\varvec{\epsilon }}}\left(\underset{\_}{\varvec{x}}\right)\) are the stress and strain field derived at the macroscale by solving the boundary value problem of a homogeneous body constituted by this fictitious homogenous material. In many materials the microstructure is statistically homogeneous, i.e., the statistical descriptors of the geometrical arrangement do not depend on the position where are evaluated, being equal for any RVEs. In this case it is reasonable to define volume averaged properties, which are independent of the size and position of the volume considered, provided that it is sufficiently large. Therefore, while the problem concerned has a statistical nature (requiring the definition of probability density and n-point correlation functions), it is possible to remain within the framework of continuum micromechanics. This implies that the average properties of statistically identical specimens can be replaced by the volume average taken over a sufficiently large region of a single specimen, i.e., RVE ( ergodic hypothesis , [ 19 ]). In general, quasi-homogeneous sub-domains with known physical quantities (such as volume fraction, elastic or strength properties) represent the microstructure within each RVE. These sub-domains are referred to as material phases. Mechanical properties of the RVE can be estimated by means of continuum micromechanics from the aforementioned phase properties [ 18 ]. Therefore, the underlying idea of continuum micromechanics is that it is possible to separate a heterogeneous material into phases with average material properties. In this sense, the homogenisation is based on volume averaging over the RVE of the constitutive relations defined at the scale of the phases. It delivers the macroscopic properties of the RVE as a function of the microscopic phase properties, their volume fractions and their specific morphologies [ 11 ]. An insight into the homogenisation procedure is schematised in Fig. 4 . By choosing a representative domain Ω of a heterogeneous medium, it is possible to distinguish two types of phases, i.e., matrix and inclusions, with different elastic properties, respectively E (1) and E (2) . Thanks to the homogenisation procedure, an equivalent homogeneous medium is obtained, whose elastic properties are a combination IE* of the elastic properties of the single phases. For heterogeneous materials with a random microstructure (as in the case of LWCS) a stochastic approach is needed. The procedure requires the determination of the mechanical phases, whose geometric and mechanical characteristics need to be specified, and the statistical description of their spatial distribution. From a practical point of view, the the latter cannot be completely described, even in a statistical sense. Therefore, the constitutive behavior of the EHM can be only determined by making appropriate assumptions or approximations [ 18 ]. In order to overcome this limit, Mean Field Approaches can be used to describe the the microstructure of heterogeneuos materials, whose statistical information are limited. 3 MEAN FIELD APPROACHES Mean Field Homogenisation allows modelling composite materials characterised by one matrix phase and one or multiple inclusion phases with uniform properties. The more sophisticated mean-field homogenisation approaches, based on Eshelby's solution [ 11 – 13 ], need only few information on the microstructure to be employed, namely the volume fraction (i.e., the ratio between the volume of the considered phase and the total volume of the RVE), phase morphology (i.e., phase connectedness or disconnectedness), aspect ratio (i.e., the ratio between the major axis and the minor axis of a given phase) and the spatial orientation of the inclusions. Mean-Field Approaches can differ according to the selection of the concentration tensors, which link microscopic strain and stress fields with the corresponding macroscopic ones [ 13 ]. The most common approach for cement-based materials (in particular, concrete) is the Eshelbian-type ellipsoidal inclusion embedded in a reference medium, for which an estimate of the localization tensor is provided, depending on the Eshelby tensor [ 11 ]. Being characterised by a random microstructure, a reliable assumption for cement-based materials is to consider all phases as isotropic [ 11 ]. Moreover, in the current work, another reasonable approximation is describing the inclusions as spheres (Ferriero et al ., submitted). For spheroidal inclusions in an isotropic elastic matrix, Eshelby tensor can be estimated analytically and depends on the Poisson’s ratio of the homogeneous material (or, in the case of heterogeneous inclusions, on the Poisson’s ratio of the matrix) and on the aspect ratio of the inclusions. These assumptions yield explicit expressions for the homogenised bulk and shear moduli \({k}_{hom}^{est}\) and \({\mu }_{hom}^{est}\) [ 11 ]: \({\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{\varvec{h}\varvec{o}\varvec{m}}^{\varvec{e}\varvec{s}\varvec{t}}={3k}_{hom}^{est}\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{J}}}}}+{2\mu }_{hom}^{est}\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{K}}}}}\) 1 \({k}_{hom}^{est}=\sum _{p}^{}{c}_{p}{k}_{p}{\left(1+{\alpha }_{0}^{est}\left(\frac{{k}_{p}}{{k}_{0}}-1\right)\right)}^{-1}\times {\left[\sum _{p}^{}{c}_{p}{\left(1+{\alpha }_{0}^{est}\left(\frac{{k}_{p}}{{k}_{0}}-1\right)\right)}^{-1}\right]}^{-1}\) 2 \({\mu }_{hom}^{est}=\sum _{p}^{}{c}_{p}{\mu }_{p}{\left(1+{\beta }_{0}^{est}\left(\frac{{\mu }_{p}}{{\mu }_{0}}-1\right)\right)}^{-1}\times {\left[\sum _{p}^{}{c}_{p}{\left(1+{\beta }_{0}^{est}\left(\frac{{\mu }_{p}}{{\mu }_{0}}-1\right)\right)}^{-1}\right]}^{-1}\) 3 Where \({\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{\varvec{h}\varvec{o}\varvec{m}}^{\varvec{e}\varvec{s}\varvec{t}}\) is the estimate of the homogenised elasticity tensor, \({c}_{p}\) , \({k}_{p}\) and \({\mu }_{p}\) are respectively the volume fraction, the bulk modulus and the shear modulus of the phase p, \({k}_{0}\) and \({\mu }_{0}\) are the bulk and shear moduli of the reference medium and \({\alpha }_{0}^{est}\) and \({\beta }_{0}^{est}\) are homogenisation parameters of the reference medium. Mori-Tanaka Method In the Mori–Tanaka (MT) method [ 21 ], appropriate for materials that exhibit an evident matrix-inclusion morphology [ 11 ], the matrix phase is chosen as reference medium, i.e., \({\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{0}={\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{\varvec{m}}\) . In this case of study, the microstructure of LWCS can be simplified in matrix with spherical inclusions (i.e. voids induced by the foam), so Equations 2 and 3 can be specialised as follows: \(\frac{{k}_{hom}^{est}}{{k}_{m}}=1+{c}_{I}\frac{\raisebox{1ex}{${k}_{I}$}\!\left/ \!\raisebox{-1ex}{${k}_{m}$}\right.-1}{1+{\alpha }_{m}^{est}\left(1-{c}_{I}\right)\left(\raisebox{1ex}{${k}_{I}$}\!\left/ \!\raisebox{-1ex}{${k}_{m}$}\right.-1\right)}\) 4 \(\frac{{\mu }_{hom}^{est}}{{\mu }_{m}}=1+{c}_{I}\frac{\raisebox{1ex}{${\mu }_{I}$}\!\left/ \!\raisebox{-1ex}{${\mu }_{m}$}\right.-1}{1+{\beta }_{m}^{est}\left(1-{c}_{I}\right)\left(\raisebox{1ex}{${\mu }_{I}$}\!\left/ \!\raisebox{-1ex}{${\mu }_{m}$}\right.-1\right)}\) 5 \({\alpha }_{0}^{est}\equiv {\alpha }_{m}^{est}=\frac{{3k}_{m}}{{3k}_{m}+4{\mu }_{m}} and {\beta }_{0}^{est}\equiv {\beta }_{m}^{est}=\frac{{6(k}_{m}+2{\mu }_{m})}{5({3k}_{m}+4{\mu }_{m})}\) 6 where the subscripts I and m stand for Inclusions and matrix respectively. The general formulae 4 and 5 can then be rewritten taking into account porosity induced by foam, n foam : \(\frac{{k}_{hom}^{est}}{{k}_{m}}=1+{n}_{foam}\frac{\raisebox{1ex}{${k}_{foam}$}\!\left/ \!\raisebox{-1ex}{${k}_{m}$}\right.-1}{1+{\alpha }_{m}^{est}\left(1-{n}_{foam}\right)\left(\raisebox{1ex}{${k}_{foam}$}\!\left/ \!\raisebox{-1ex}{${k}_{m}$}\right.-1\right)}\) 7 \(\frac{{\mu }_{hom}^{est}}{{\mu }_{m}}=1+{n}_{foam}\frac{\raisebox{1ex}{${\mu }_{foam}$}\!\left/ \!\raisebox{-1ex}{${\mu }_{m}$}\right.-1}{1+{\beta }_{m}^{est}\left(1-{n}_{foam}\right)\left(\raisebox{1ex}{${\mu }_{foam}$}\!\left/ \!\raisebox{-1ex}{${\mu }_{m}$}\right.-1\right)}\) 8 Starting from Eq. 7 and Eq. 8, two assumptions have been considered in order to obtain simplified expression of Mori-Tanaka method for LWCS. Poisson’s ratio of the matrix was assumed equal to ν m = 0.2, being this value commonly assumed for cemented materials. Assuming bulk and shear moduli of voids equal to zero ( \({k}_{foam}={\mu }_{foam}=0\) ), the effective normalized bulk and shear modulus in Eq. 7 and Eq. 8 assume the same following hyperbolic form [ 15 ]: \({k}_{MT}^{h}\left({n}_{foam}\right)=\left(\frac{1-{n}_{foam}}{1+{n}_{foam}}\right){k}_{m}{\mu }_{MT}^{h}\left({n}_{foam}\right)=\left(\frac{1-{n}_{foam}}{1+{n}_{foam}}\right){\mu }_{m}\) 9 Self-Consistent Method In the Self-Consistent method (SC) [ 22 , 23 ] the reference medium is assumed coincident with the homogenised medium, \({\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{0}={\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{\varvec{h}\varvec{o}\varvec{m}}^{\varvec{e}\varvec{s}\varvec{t}}\) . Differently from the MT method, this method implies the solution of two nonlinear equations, in which \({k}_{0}\equiv {k}_{hom}^{est}\) and \({\mu }_{0}\equiv {\mu }_{hom}^{est}\) . In this study, the same assumptions on Poisson’s ratio and inclusions (voids) moduli (ν m = 0.2 and \({k}_{foam}={\mu }_{foam}=0\) ) yield to the same expressions of the homogenised values of \({k}_{MT}^{h}\) and \({\mu }_{MT}^{h}\) of the MT method (Eq. 9). Dilute Method The Dilute Method (DI) (Eshelby, 1957) [ 14 ] does not take into account the interaction among inclusions. In this study, with the same assumptions on Poisson’s ratio and inclusions (voids) moduli (ν m = 0.2 and \({k}_{foam}={\mu }_{foam}=0\) ), the effective normalized bulk and shear modulus are provided by the same expression and depend linearly on the artificial porosity n foam : \({k}_{DI}^{h}\left({n}_{foam}\right)=(1-2{n}_{foam}){k}_{m}{\mu }_{DI}^{h}\left({n}_{foam}\right)=\left(1-2{n}_{foam}\right){\mu }_{m}\) 10 Differential Method According to Bruggeman and Roscoe [ 24 , 25 ], the Differential method (DF) models the composite as a sequence of dilute suspensions [ 15 ]. In this study, under the same assumptions adopted for the previous approaches (ν m = 0.2 and \({k}_{foam}={\mu }_{foam}=0\) ), the effective normalized bulk and shear moduli of the inclusions are provided by: \({k}_{DF}^{h}\left({n}_{foam}\right)={\left(1-{n}_{foam}\right)}^{2}{k}_{m}{\mu }_{DF}^{h}\left({n}_{foam}\right)={\left(1-{n}_{foam}\right)}^{2}{\mu }_{m}\) 11 Homogenised elastic stiffness Starting from the input parameters of Mean Field Methods, the homogenised values of k h and µ h have been determined for the estimation of the homogenised elastic stiffness modulus of LWCS by means of Eq. 12 [ 11 ]. \({E}_{hom}^{est}=\frac{9{k}_{hom}^{est}{\mu }_{hom}^{est}}{3{k}_{hom}^{est}+{\mu }_{hom}^{est}}\) 12 The values of the homogenised stiffness modulus obtained have been then compared with experimental test results in order to validate the calculations. Homogenised stiffness modulus of LWCS was evaluated under the subsequent assumptions: LWCS is regarded as a two-phase material (i.e., matrix and voids) or as three-phase material (i.e., matrix, voids and portlandite); phases are considered as isotropic inclusions (i.e. voids induced by foam) are spherical; matrix is supposed to be elastic, isotropic and homogeneous; bulk moduli k I and shear moduli µ I of voids (i.e., inclusions) are equal to zero. 4 HOMOGENISATION INPUT PARAMETERS FOR LWCS The homogenisation process according to Mean Field Approaches with the introduced approximations requires the following input parameters: the artificial porosity n foam , the volume fraction of portlandite V P , the volumetric stiffness moduli k m , the shear moduli µ m , and the Poisson's coefficient ν m of the matrix. For this purpose, experimental results of microstructural (Ferriero et al., to be submitted ) and mechanical [ 1 , 2 ] tests on LWCS samples of cement and foam-treated kaolin were interpreted. Treated samples considered in the cited experimental studies were prepared by mixing Speswhite kaolin, Portland limestone cement, water and foam in different proportions. Main features of each constituent are reported in Table 1 . As reference material forming the matrix, cemented samples without foam were considered as well. Table 1 Main components of LWCS. Kaolin Cement Foam Speswhite kaolin supplied by Imerys Minerals, UK G s = 2.6 Specific surface area = 14 m 2 /g pH = 4.6 w L = 70%, w p = 32%, IP = 38% Portland limestone cement supplied by Buzzi, Italy CEM II/A-LL 42.5 R ISOCEM S/L supplied by Isoltech srl, Italy concentration 2.5% pH = 7.5-9 Cement treated samples were prepared by mixing speswhite kaolin with water to slurry at approximately twice the liquid limit of soil (i.e., 2w L = 140%) and dry cement with water (water/cement ratio = 0.5). The amount of cement by dry weight of soil considered for the treatment was 40% (i.e. KC40%). Lightweight cemented samples were prepared by adding 40% by volume of the mixture of preformed foam, (i.e. KCF40%). Foam was formed by mixing a surfactant solution and air with an industrial foam generator at density approximately equal to 75 g/L. Treated samples were poured into moulds and sealed in plastic bags for curing at room temperature. Curing time intervals ranging from 1 day to 28 days were considered for the experimental studies. Artificial porosity induced by foam and matrix porosity of LWCS samples have different characteristic dimensions (average diameter of 300 µm the former, 0.3 µm the latter), and can be considered as a two distinct systems of pores. For determining the artificial porosity n foam of LWCS samples as a part of total porosity, X-Ray computed microtomography scans were performed on one sample at increasing curing times, namely 24h, 7 and 28 days. This experimental technique allowed the detection of artificial porosity via image analysis, based on the characteristic dimension and the spherical shape of pores induced by foam. XRm-CT scans of the KCF40% sample over time are shown in Fig. 5 . From the results, it can be seen that although artificial porosity is not altered over time, the formation of shrinkage microfractures in the material results in increased porosity in the large pore size range. Moreover, the denser particles (white in the micrographs) in the system are attributed to portlandite precipitating over curing time as a consequence of the hydration of cement; the consumption of portlandite takes place in the long term, highlighted by a progressive diminution of the frequency of denser particles in the system. Image analysis was based on two different segmentation techniques (i.e., watershed segmentation and global thresholding segmentation) for the evaluation of artificial porosity over time (Fig. 6 ). Average values of the two methods for both artificial porosity and volume fraction of portlandite were considered as input in Mean Field Approaches, as reported in Table 2 . Table 2 Micro-CT artificial porosity. Curing Time [days] n foam [%] V p [%] 1 18.42 2.32 7 19.23 1.32 28 19.45 0.87 The elastic stiffness of cement treated and lightweight cemented samples (i.e., E KC40% and E KCF40% respectively) were evaluated from from direct shear tests, performed at 50kPa of vertical effective stress, for 24h cured samples (Fig. 7 ) and from unconfined compression tests performed after 7 and 28 days of curing from stress-strain curves at low strain level (Fig. 8 ). In order to determine Poisson’s ratios, one-dimensional compression curves of cement treated samples (KC40%) after 7 and 28 days of curing were considered (Fig. 9 ). Oedometer modulus (E oed ) of treated samples cured at 7 and 28 days has been determined in the reversible region for stress level of 50 kPa. From elastic stiffness (E KC40% ) of cement treated sample and oedometer modulus (E oed ), Poisson’s ratio of cemented samples was calculated from Eq. 13. With reference to 24h cured samples, Poisson’s ratio was assumed equal to 0.2 considering the early stage of the evolution of the system in terms of cement hydration process. \({E}_{oed}=\frac{{E}_{KC40\%}(1-{v}_{m})}{(1+{v}_{m})(1-2{v}_{m})}\) 13 Starting from the values of elastic stiffness and Poisson’s ratio, the bulk modulus k m and the shear modulus µ m of cement treated samples were calculated (Eqs. 14 and 15). The subscript m used for cemented samples stands for the values assigned to the matrix of the material. \({k}_{m}=\frac{{E}_{KC40\%}}{3*(1-2{v}_{m})}\) 14 \({\mu }_{m}=\frac{{k}_{m}}{2(1+{{v}_{m}}^{})}\) 15 A summary of physical and mechanical parameters of cement treated (KC40%) and lightweight cemented samples (KCF40%) over time is reported in Table 3 . Table 3 Physical and mechanical parameters of cement treated and lightweight cemented samples. Curing Time E KC40% E KCF40% E oed ν m k m µ m [days] [MPa] [MPa] [MPa] [-] [MPa] [MPa] 1 3.276 2.299 [-] 0.2 1.820 1.365 7 11.993 11.180 70.502 0.375 15.991 5.815 28 19.110 13.195 72.727 0.316 17.311 7.260 5 RESULTS AND DISCUSSION The homogenised values of the bulk modulus and the shear modulus of LWCS samples (i.e., \({k}_{hom}^{est}\) and \({\mu }_{hom}^{est}\) ) are reported in Table 4 . Table 4 Results of Mean Fields simplified formulae. Curing time \({k}_{MT}^{h}\) [MT] \({k}_{DI}^{h}\) [DI] \({k}_{DF}^{h}\) [DF] \({\mu }_{MT}^{h}\) [MT] \({\mu }_{DI}^{h}\) [DI] \({\mu }_{DF}^{h}\) [DF] \({E}_{\text{h}\text{o}\text{m}\_simplified}^{est}\) [MT] \({E}_{\text{h}\text{o}\text{m}\_simplified}^{est}\) [DI] \({E}_{\text{h}\text{o}\text{m}\_simplified}^{est}\) [DF] [days] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] 1 1.254 1.149 1.211 940 819 874 2.257 2.069 2.180 7 4.514 4.101 4.347 3.385 3.076 3.260 8.125 7.381 7.825 28 11.170 10.121 10.747 8.377 7.590 8.060 20.105 18.217 19.345 The stiffness moduli were calculated following the Mori-Tanaka method (MT), Dilute method (DI) and Differential method (DF) (Equations 9–11) with the assumption of Poisson’s ratio equal to \({v}_{m}\) = 0.2. The homogenised elastic stiffness ( \({E}_{\text{h}\text{o}\text{m}\_simplified}^{est}\) ) of LWCS was calculated following the Eq. 4 for each Mean Field Approach. The evolution over time of the \({E}_{\text{h}\text{o}\text{m}\_simplified}^{est}\) for the different approaches is compared with the experimental results in Fig. 10 . Assuming \({v}_{m}\) = 0.2, the evolution over time of artificial porosity n foam affects the homogenised stiffness and shear moduli \({k}_{hom}^{est}\) and \({\mu }_{hom}^{est}\) . Moreover, this assumption yields similar \({E}_{\text{h}\text{o}\text{m}\_simplified}^{est}\) values at different curing time for the three different Mean Field Approaches. After one day of curing, homogenised values and the experimental value of elastic moduli for lightweight cemented samples E KCF40% overlap. At increasing curing time (i.e., 7 days), the analytical methods tend to slightly underestimate the homogenised elastic stiffness, while slightly overestimate the experimental results after 28 days. Further evaluations of the homogenised elastic stiffness of LWCS have been obtained starting from Mori-Tanaka method (Eq. 6–8). In this case, the experimental values of Poisson’s ratio have been used for the calculation of thehomogenised bulk and shear moduli ( \({k}_{hom}^{est}\) ; \({\mu }_{hom}^{est}\) ), whereas the coefficients \({\alpha }_{m}^{est}\) and \({\beta }_{m}^{est}\) have been determined starting from the bulk and shear moduli of the matrix (k m KC40% and µ m KC40% ) corresponding to the experimental values of cement treated samples. On the basis of these hypothesis, the values of homogenised elastic stiffness ( \({E}_{\text{h}\text{o}\text{m}\_completed}^{est}\) ) have been calculated and reported in Table 5 . Table 5 Results of complete formula of Mori-Tanaka method. Curing time \({k}_{hom}^{est}\) \({\mu }_{hom}^{est}\) \({\alpha }_{m}^{est}\) \({\beta }_{m}^{est}\) \({E}_{\text{h}\text{o}\text{m}\_completed}^{est}\) [days] [MPa] [MPa] [-] [-] [MPa] 1 1.254 940 0.5 0.5 2.257 7 9.956 3.992 0.607 0.479 10.565 28 10.346 4.983 0.641 0.472 12.881 A comparison between the elastic stiffness determined with simplified and complete Mori-Tanaka methods and the experimental results is shown in Fig. 11 as function of curing time. Compared to simplified method, the calculation of elastic stiffness with the of whole formula of MT leads to values quite overlapping with the experimental results for all curing times. The formula of Mori-Tanaka has been also applied by considering a three phase system instead of two phase system. The portlandite (Ca(OH) 2 ), whose precipitation in the system is consequence of the hydration of cement, has been considered as one independent phase. The elastic properties of portlandite (i.e., Poisson’s ratio n p =0. 315, Young modulus E p =38823 MPa), reported in [ 11 ], have been considered in Equations 2 and 3 for the determination of the homogenised elastic stiffness \({E}_{\text{h}\text{o}\text{m}\_3phases}^{est}\) of the three phase system. Results of the homogeneisation process as function of curing time are reported in Table 6 . Table 6 Three-phases system results. Curing time \({k}_{hom}^{est}\) \({\mu }_{hom}^{est}\) \({E}_{\text{h}\text{o}\text{m}\_3phases}^{est}\) [days] [MPa] [MPa] [MPa] 1 1.563 1.172 2.814 7 13.468 4.959 12.951 28 14.502 6.180 15.703 The comparison of the elastic stiffness, determined with the two and three-phases MT methods, and the experimental results are shown in Fig. 12 as function of curing times. MT-method for the three phases system yelds to slightly overestimate the homogenised elastic stiffness \({E}_{\text{h}\text{o}\text{m}}^{est}\) . The Mori-Tanaka method in its complete formulation was the best analytical Mean Field method for evaluating the homogenized elastic stiffness of LWCS. The assumption of constant Poisson’s ratio adopted for the simplified MFA, equivalent to considering only the influence of artificial porosity (n foam ), strongly limits the capability of these approaches in predicting the homogenised stiffness moduli, in particular for materials as LWCS with evolving mechanical properties over time, as consequence of chemo-physical evolution of the system. Conversely, the good agreement between the experimental results and the homogenization analytical method derives from including the chemo-physical evolution of the cemented material by the variation of relevant parameters over time. In the case of the two-phase system, such prediction is very effective because the representative parameters of the two phases are well characterized as a function of curing time. In the case of the three-phase system, on the other hand, an overestimation of the value of the stiffness modulus with respect to the experimental values probably results from the assumption of a constant elastic modulus for portlandite phase in the system over time, without taking into account its variation over time as the pozzolanic reactions proceed. 6 CONCLUSIONS In the study the elastic stiffness moduli of LWCS have been derived by means of Mean-Field Eshelby-based Homogenisation Approaches, which have been proved an effective method for heterogeneous materials with physical and mechanical properties evolving with curing time. The relevant parameters for the implementation of homogenisation approaches have been derived by experimental results at different scale levels on LWCS samples. The following conclusions can be drawn: with the assumption of constant Poisson's ratio (i.e., ν m = 0.2), different simplified Mean Field Methods yield to similar homogenized elastic moduli \({E}_{\text{h}\text{o}\text{m}\_simplified}^{est}\) over curing time, underestimating the experimental values in the short term, and overestimating them in the long term. Only the influence of artificial porosity n foam and its variation over time is taken into account, limiting the efficiency of those methods in predicting the homogenized elastic stiffness the comparison with experimental data shows that the Mori-Tanaka method provides the most suitable homogenisation analytical method for LWCS complete formulation of the MT method for a two-phase system (i.e., matrix and voids) with input parameters evolving with curing time yields to very good predictions of the elastic stiffness and its evolution over time computation of homogenised elastic stiffness for three-phases system (i.e., matrix, portlandite and voids) with complete formulation of MT methods provides an overestimation of the stiffness values if relevant input parameters for the homogeneisation process, due to the chemo-physical evolution of the system, are not properly evaluated. List of Notation and Acronym The following notations are used in this paper : \(\underset{\_}{\varvec{x}}\) macroscopic position vector \(\underset{\_}{\varvec{y}}\) microscopic position vector \(\underset{\_}{\underset{\_}{\varvec{\sigma }}}\left(\underset{\_}{\varvec{x}}\right)\) macro-stress field (second order tensor) \(\underset{\_}{\underset{\_}{\varvec{\epsilon }}}\left(\underset{\_}{\varvec{x}}\right)\) macro-strain field (second order tensor) \({\underset{\_}{\underset{\_}{\varvec{\sigma }}}}_{\varvec{\mu }}\left(\underset{\_}{\varvec{y}}\right)\) micro-stress field (second order tensor) \({\underset{\_}{\underset{\_}{\varvec{\epsilon }}}}_{\varvec{\mu }}\left(\underset{\_}{\varvec{y}}\right)\) micro-strain field (second order tensor) \({c}_{p}\) volume fraction of the phase p \(\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{J}}}}}\) volumetric part of the identity tensor \(\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{K}}}}}\) deviatoric part of the identity tensor \({\alpha }_{0}^{est},{\beta }_{0}^{est}\) homogenization coefficients \({\mu }_{0}\) shear modulus of the reference medium \({k}_{0}\) bulk modulus of the reference medium \({\nu }_{0}\) Poisson’s ratio of the reference medium \({\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{\varvec{h}\varvec{o}\varvec{m}}^{\varvec{e}\varvec{s}\varvec{t}}\) estimate of homogenised elasticity tensor \({\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{0}\) fourth-order elasticity tensor of the reference medium \({\underset{\_}{\underset{\_}{\underset{\_}{\underset{\_}{\varvec{C}}}}}}_{\varvec{m}}\) fourth-order elasticity tensor of the matrix \({\mu }_{p}\) shear modulus of the phase p \({k}_{p}\) bulk modulus of the phase p \({k}_{hom}^{est}\) homogenised bulk modulus \({\mu }_{hom}^{est}\) homogenised shear modulus \({\mu }_{I}\) shear modulus of the inclusions \({k}_{I}\) bulk modulus of the inclusions \({\mu }_{m}\) shear modulus of the matrix \({k}_{m}\) bulk modulus of the matrix \({c}_{I}\) volume fraction of the inclusions \({\alpha }_{m}^{est},{\beta }_{m}^{est}\) matrix homogenization coefficients \({n}_{foam}\) artificial porosity \({\mu }_{foam}\) shear modulus of the voids \({k}_{foam}\) bulk modulus of the voids \({\nu }_{m}\) Poisson’s ratio of the matrix \({k}_{MT}^{h},{\mu }_{MT}^{h}\) Mori-Tanaka homogenised bulk and shear moduli \({k}_{DI}^{h},{\mu }_{DI}^{h}\) Dilute method homogenised bulk and shear moduli \({k}_{DF}^{h},{\mu }_{DF}^{h}\) Differential method homogenised bulk and shear moduli \({E}_{hom}^{est}\) homogenised Young modulus \({E}_{KC40\%}\) Young modulus of cement treated samples \({E}_{KCF40\%}\) Young modulus of LWCS samples \({E}_{oed}\) Oedometric modulus of the matrix \({E}_{\text{h}\text{o}\text{m}\_\text{s}\text{i}\text{m}\text{p}\text{l}\text{i}\text{f}\text{i}\text{e}\text{d}}^{est}\) Young modulus computed with simplified Mori-Tanaka method \({E}_{\text{h}\text{o}\text{m}\_completed}^{est}\) Young modulus computed with complete Mori-Tanaka method \({E}_{\text{h}\text{o}\text{m}\_3phases}^{est}\) Young modulus computed with complete Mori-Tanaka method considering three phases V p Volume fraction of portlandite ν p Poisson’s ratio of portlandite E p Young modulus of porlandite Declarations Data Availability The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request. Funding No funds, grants, or other support was received. Competing interests All authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. Author contributions All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by all authors. The first draft of the manuscript was written by Laura Perrotta and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript. References Vitale E, Deneele D, Russo G, De Sarno D, Nicotera M V, Pap R, & Urciuoli G (2020) Chemo-mechanical behaviour of lightweight cemented soils. Acta Geotechnica, 15: 933-945. De Sarno D, Vitale E, Nicotera M V, Papa R, Russo G, Urciuoli G (2020) Lightweight Cemented Soils: Mix Design, Production and Control. Lecture Notes in Civil Engineering. In: Geotechnical Research for Land Protection and Development: Proceedings of CNRIG 2019 7. Springer International Publishing, pp. 743-752. Satoh T, Tsuchida T, Mitsukuri K, Hong Z (2001) Field placingtest of lightweight treated soil under seawater in Kumamoto port. Soils and Foundations , 41.5: 145-154. Tsuchida T, & Egashira K (2004). The lightweight treated soil method: new geomaterials for soft ground engineering in coastal areas. CRC Press. Watabe Y, Itou Y, Kang MS, Tsuchida T (2004) One-dimensional compression of air-foam treated lightweight geo-material in microscopic point of view. Soils and Foundations, 44.6: 53-67. Jamnongpipatkul P, Dechasakulsom M, Sukolrat J (2009) Application of air foam stabilized soil for bridge-embankmenttransition zone in Thailand. In : Asphalt Material Characterization, Accelerated Testing, and Highway Management: Selected Papers from the 2009 GeoHunan International Conference, pp. 181-193. Miki H, Mori M, Chida S (2003) Trial embankment on softground using lightweight-foam-mixed in situ surface soil. In: Proc. XXIInd PIARC World Road Congress, Durban. Horpibulsuk S, Rachan R, Suddeepong A, Liu M D, Du Y J, (2013) Compressibility of lightweight cemented clays. Engineering Geology, 159: 59-66. Ma C, Cheng B (2015) Properties of a foamed concrete with soil as filler. Construction and Building Material, 76: 61-69. Zhang H, Liu M, Shuo Z, Zhao Z, Sun Y, Song X, Wang H, Zhang X, Wu J (2021) An experimental investigation of the triaxial shear behaviors of silt-based foamed concrete. Case Studies in Construction Materials15:e00713. Bernard O, Ulm F J, & Lemarchand E (2003) A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials. Cement and concrete research, 33.9: 1293-1309. Constantinides G, & Ulm F J (2004) The effect of two types of CSH on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling. Cement and concrete research, 34.1: 67-80. Ortolano González J M, Hernández Ortega J A, & Oliver Olivella X (2013) A comparative study on homogenization strategies for multi-scale analysis of materials. Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE). Eshelby J D (1957) The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the royal society of London. Series A. Mathematical and physical sciences, 241.1226: 376-396. Miled K, Sab K, & Le Roy R (2011) Effective elastic properties of porous materials: Homogenization schemes vs experimental data. Mechanics Research Communications, 38.2: 131-135. Shi X S, Zhao J, Yin J, & Yu Z (2019) An elastoplastic model for gap-graded soils based on homogenization theory. International Journal of Solids and Structures, 163: 1-14. Maghous S, Consoli N C, Fonini A, & Dutra V P (2014) A theoretical–experimental approach to elastic and strength properties of artificially cemented sand. Computers and Geotechnics, 62: 40-50. Zaoui A (2002) Continuum micromechanics: survey. Journal of Engineering Mechanics, 128.8: 808-816. Hashin Z (1983). Analysis of composite materials—a survey. Al Kassem G, & Weichert D (2009) Micromechanical material models for polymer composites through advanced numerical simulation techniques. In PAMM: Proceedings in Applied Mathematics and Mechanics (Vol. 9, No. 1, pp. 413-414). Berlin: WILEY‐VCH Verlag. Mori T, & Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta metallurgica, 21.5: 571-574. Budiansky B (1965) On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids, 13.4: 223-227. Hill R (1965) A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids, 13.4: 213-222. Bruggeman V D (1935) Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Annalen der physik, 416.7: 636-664.. Roscoe R (1952) The viscosity of suspensions of rigid spheres. British journal of applied physics, 3.8: 267. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3409777","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":238355298,"identity":"c45d38f0-aeee-4d22-99ed-93699c0074bf","order_by":0,"name":"Laura Perrotta","email":"","orcid":"","institution":"Scuola Superiore Meridionale","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Laura","middleName":"","lastName":"Perrotta","suffix":""},{"id":238355299,"identity":"2509560a-ee80-4789-9b5a-5e2e39510100","order_by":1,"name":"Enza 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17:06:00","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":33650,"visible":true,"origin":"","legend":"\u003cp\u003eMulti-scale model of LWCS samples heterogeneities\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/04057efdac9832dd965fc7a0.jpg"},{"id":44456749,"identity":"ced250fa-73ff-40e3-b387-1f4b3809ae16","added_by":"auto","created_at":"2023-10-11 17:22:00","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":46692,"visible":true,"origin":"","legend":"\u003cp\u003eHomogenization process (adapted from [17])\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/5d72160fa787f44acc697b84.jpg"},{"id":44452250,"identity":"3580ac6b-4a12-4cb2-980f-bb0a699888d3","added_by":"auto","created_at":"2023-10-11 16:58:00","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":35900,"visible":true,"origin":"","legend":"\u003cp\u003eContinuum micromechanics procedure of homogenization (adapted from [20])\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/02aa77213fa186c614c1e1e5.jpg"},{"id":44452256,"identity":"7a7f0a2c-e30b-452d-98ee-406793215244","added_by":"auto","created_at":"2023-10-11 16:58:00","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":40978,"visible":true,"origin":"","legend":"\u003cp\u003eMicro-CT scans for increasing curing time\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/8ea2c4399fa5fa4e99edde06.jpg"},{"id":44453841,"identity":"5e982c15-1e2a-40a5-9b23-9d85300e813b","added_by":"auto","created_at":"2023-10-11 17:06:00","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":53685,"visible":true,"origin":"","legend":"\u003cp\u003eXR micro-CT foam porosity as function of curing time\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/49cb75ee6963c51970a6551f.jpg"},{"id":44452252,"identity":"840170f8-e605-412d-aba9-42b102ca50e5","added_by":"auto","created_at":"2023-10-11 16:58:00","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":53373,"visible":true,"origin":"","legend":"\u003cp\u003eDirect shear tests\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/3fc8f3773db9808422f1866c.jpg"},{"id":44455269,"identity":"6e26128b-d3ac-4ec4-ba4a-dc418e95aab8","added_by":"auto","created_at":"2023-10-11 17:14:00","extension":"jpg","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":60073,"visible":true,"origin":"","legend":"\u003cp\u003eUnconfined compression tests.\u003c/p\u003e","description":"","filename":"8.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/cbe43b808eb3a02d2fbadb86.jpg"},{"id":44452261,"identity":"044cda5c-6001-40c5-985e-6ac616bbbb90","added_by":"auto","created_at":"2023-10-11 16:58:01","extension":"jpg","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":69801,"visible":true,"origin":"","legend":"\u003cp\u003eOne dimensional compression tests\u003c/p\u003e","description":"","filename":"9.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/c0800e3a32ed6ae664f8665f.jpg"},{"id":44452260,"identity":"e377922b-a5c2-411b-951c-dcea8948372f","added_by":"auto","created_at":"2023-10-11 16:58:01","extension":"jpg","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":52927,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of elastic stiffness modulus as function of curing time: Mean Field approaches vs experimental results\u003c/p\u003e","description":"","filename":"10.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/e1210c4db28adddbf90fbf84.jpg"},{"id":44452257,"identity":"325392ae-e697-4bac-9ac0-8961b12aefdb","added_by":"auto","created_at":"2023-10-11 16:58:00","extension":"jpg","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":50211,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of elastic stiffness modulus as function of curing time: MT-methods vs experimental results\u003c/p\u003e","description":"","filename":"11.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/54bfcf5da867e0e4af05e5b2.jpg"},{"id":44452259,"identity":"5e68282c-d8d8-4b82-b56d-2e8a0e5d43c4","added_by":"auto","created_at":"2023-10-11 16:58:01","extension":"jpg","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":48990,"visible":true,"origin":"","legend":"\u003cp\u003eEvolution of elastic stiffness modulus as function of curing time: MT-two-pahses vs MT-three-pahses\u003c/p\u003e","description":"","filename":"12.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/5b19343efeeba1f7f6b995f0.jpg"},{"id":46679217,"identity":"b80a2f82-c9b2-469b-bde7-7cdf7dac8616","added_by":"auto","created_at":"2023-11-17 20:22:31","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":877256,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3409777/v1/ba692f02-8a55-45f8-bb5d-7641de88ba21.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Homogenised Elastic Properties of Lightweight Cemented Soils","fulltext":[{"header":"1 INTRODUCTION","content":"\u003cp\u003eDuring the construction of major civil infrastructures, soils characterized by low mechanical properties usually need to be removed and replaced, leading to several economic and environmental drawbacks. Nowadays, this procedure is often avoided by using soil improvement techniques. In this way, the construction takes place in compliance with soil balance (i.e., reusing local materials considered unsuitable in their natural state), which is essential for the sustainability of civil works. For this reason, soil improvement techniques based on the use of binders are employed in several geotechnical engineering applications.\u003c/p\u003e \u003cp\u003eA solution of increasing interest for reusing soils in geotechnical applications is Lightweight Cemented Soil (LWCS) [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]. LWCS is prepared by mixing soil, water and cement with air foam to have a low-density and self-levelling fresh paste which exibits relevant mechanical performances after setting of the binder.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAs shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e, the LWCS sample is characterised by large voids, as footprints of foam-air bubbles, immersed in a cemented matrix of soil particles, cement-hydrated phases and smaller pores. Thanks to their reduced unit volume weight, LWCSs are advantageously used for different earthworks, such as cavities filling, trench backfilling [\u003cspan additionalcitationids=\"CR4\" citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] and also embankment construction [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]\u003c/p\u003e \u003cp\u003eExperimental studies on the effects of cement, foam and initial water content on unit weight and shear strength of LWCS have been reported in literature, highlighting the evolution of physical and mechanical properties as a function of treatment parameters [\u003cspan additionalcitationids=\"CR9\" citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. An insight into the role of foam content on chemo-physical evolution and microstructural features of the system on mechanical behaviour of LWCS have been recently provided by Vitale \u003cem\u003eet al\u003c/em\u003e. (2020) [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. The Authors showed that addition of foam does not alter chemo-physical evolution of the soil\u0026ndash;cement\u0026ndash;water system in terms of either cement hydration or pozzolanic reactions. For treated samples (with or without foam addition), precipitation of portlandite and progressive formation of cementitious compounds due to cement hydration process were generally observed since the very short term (i.e., within 24 h of curing). At increasing curing times, a gradual consumption of formed portlandite and progressive dissolution of clay minerals were found consistent with the development of pozzolanic reactions.\u003c/p\u003e \u003cp\u003eThe complex and heterogeneous microstructure of the material (a cemented porous matrix with a foam induced large pore system) requires a homogeneisation procedure for constitutive modelling of its hydromechanical behaviour, taking account of heterogeneities and different phases [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. The advantage of this approach is the possibility of considering the role played by microstructure on the macroscopic behaviour of the material.\u003c/p\u003e \u003cp\u003eThe length scale used for the experimental observation determines the role of the inherent heterogeneity on material behaviour. Some constituents or phases in an heterogeneous material are identifiable only at or below a specific length scale. Each of the constituents can be regarded as homogeneous material, but may become heterogeneous when it is observed at a smaller length scale. Therefore, for LWCS a length scale of interest must be established, so that the microstructural features at smaller scale can be neglected [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Once the scale of interest has been defined, a homogenisation procedure can be introduced, starting from the physical reality of LWCS, in order to define the homogenised properties of the material, taking into account the heterogeneities of the sample that can be observed at such scale level. In this framework, compared to traditional approaches based on the continuum mechanics, homogenised material properties are determined by means of analytical, semi-analytical or numerical models that indeed account for the micromechanical features of the material through some parameters that depend on the approaches used.\u003c/p\u003e \u003cp\u003eHomogenisation has been largely employed to predict the mechanical behaviour of heterogeneous materials. In the framework of solid mechanics, one of the most addressed problems was originally the determination of the mechanical properties of a polycrystal from those of a single crystal. In the last years, improved estimates have been pursued. The problem of a single ellipsoidal particle embedded in an unbounded domain of material under uniform exterior loading has been analytically solved by Eshelby (1957) [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. His solution is relatively compact and easy to use, therefore has been a basis for the development of many approximated analytical methods, such as \u003cem\u003eMean Field Approaches\u003c/em\u003e (MFAs) [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. These approaches are very convenient thanks to their low computational cost compared to numerical methods.\u003c/p\u003e \u003cp\u003eMFAs have been employed in literature to estimate the elastic properties of concrete [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]; however, in the field of geotechnical engineering the homogenisation approaches, and in particular MFAs, are rarely implemented due to the difficulties in determining phase properties compared to other materials such as concrete and steel. MFAs have been employed to compute the effective elastic properties of porous materials [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], and gap-graded soils [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. In the framework of treated soils, no previous studies based on this type of approaches were found in literature.\u003c/p\u003e \u003cp\u003eIn this study, MFAs have been implemented for the determination of elastic properties of LWCS, taking account of the chemo-physical evolution of the material over curing time. A multi-scale model of sample heterogeneities, as reference for the homogenisation process, is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn the first step of homogenisation, the four-phases system (solid particles, water, voids and cement-hydrated phases) is homogenised into a two-phase system (cementitious matrix, large voids induced by foam). Under the hypothesis of homogeneous and isotropic cementitious matrix, and assuming the large voids as spherical, the second homogenisation step allows to determine the elastic properties of the homogenised matrix. This second step of homogenisation process has been performed using simplified approaches based on Mean Field Homogenisation Methods. Elastic stiffness modulus of homogenised LWCS as function of curing time has been determined through analytical formulae. A comparison with the elastic stiffness of LWCS determined by available experimental tests has been performed. A three-phase system, made of matrix, voids and portlandite, has been also considered for further investigations on the suitability of the analytical methods.\u003c/p\u003e \u003cp\u003eIn the framework of the homogenisation methods based on Mean Field Approaches, the study highlights the valuable capability of Mori-Tanaka scheme of predicting the elastic properties of an extremely heterogeneous material like the LWCS over time, despite its simple analytical formulation.\u003c/p\u003e"},{"header":"2 HOMOGENISATION PROCESS","content":"\u003cp\u003eThe main interest of the homogenisation approach lies on the possibility of modeling mechanical behaviour of heterogeneous materials in the framework of continuum michromechanics [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn the field of homogenisation, continuum micromechanics deals mainly with statistically homogeneous materials for which it is possible to define a Representative Volume Element (RVE) and an Equivalent Homogeneous Medium (EHM), which are equivalent to each other from a mechanical point of view. This statement implies that they should have the same overall responses to any kind of mechanical loading. In the EHM, the stress and strain fields \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\underset{\\_}{\\varvec{\\sigma }}}\\left(\\underset{\\_}{\\varvec{x}}\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\underset{\\_}{\\varvec{\\epsilon }}}\\left(\\underset{\\_}{\\varvec{x}}\\right)\\)\u003c/span\u003e\u003c/span\u003e must be the average values, over any RVE centred at \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\varvec{x}}\\)\u003c/span\u003e\u003c/span\u003e (which is the macroscopic position vector in a global reference system), of the local stress and strain fields \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\varvec{\\sigma }}}}_{\\mu }\\left(\\underset{\\_}{\\varvec{y}}\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\varvec{\\epsilon }}}}_{\\mu }\\left(\\underset{\\_}{\\varvec{y}}\\right)\\)\u003c/span\u003e\u003c/span\u003e (with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\varvec{y}}\\)\u003c/span\u003e\u003c/span\u003e equal to the microscopic position vector, in a local reference system), [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\underset{\\_}{\\varvec{\\sigma }}}\\left(\\underset{\\_}{\\varvec{x}}\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\underset{\\_}{\\varvec{\\epsilon }}}\\left(\\underset{\\_}{\\varvec{x}}\\right)\\)\u003c/span\u003e\u003c/span\u003e are the stress and strain field derived at the macroscale by solving the boundary value problem of a homogeneous body constituted by this fictitious homogenous material. In many materials the microstructure is statistically homogeneous, i.e., the statistical descriptors of the geometrical arrangement do not depend on the position where are evaluated, being equal for any RVEs. In this case it is reasonable to define volume averaged properties, which are independent of the size and position of the volume considered, provided that it is sufficiently large. Therefore, while the problem concerned has a statistical nature (requiring the definition of probability density and n-point correlation functions), it is possible to remain within the framework of continuum micromechanics. This implies that the average properties of statistically identical specimens can be replaced by the volume average taken over a sufficiently large region of a single specimen, i.e., RVE (\u003cem\u003eergodic hypothesis\u003c/em\u003e, [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]).\u003c/p\u003e \u003cp\u003eIn general, quasi-homogeneous sub-domains with known physical quantities (such as volume fraction, elastic or strength properties) represent the microstructure within each RVE. These sub-domains are referred to as material phases. Mechanical properties of the RVE can be estimated by means of continuum micromechanics from the aforementioned phase properties [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. Therefore, the underlying idea of continuum micromechanics is that it is possible to separate a heterogeneous material into phases with average material properties. In this sense, the homogenisation is based on volume averaging over the RVE of the constitutive relations defined at the scale of the phases. It delivers the macroscopic properties of the RVE as a function of the microscopic phase properties, their volume fractions and their specific morphologies [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAn insight into the homogenisation procedure is schematised in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. By choosing a representative domain Ω of a heterogeneous medium, it is possible to distinguish two types of phases, i.e., matrix and inclusions, with different elastic properties, respectively E\u003csup\u003e(1)\u003c/sup\u003e and E\u003csup\u003e(2)\u003c/sup\u003e. Thanks to the homogenisation procedure, an equivalent homogeneous medium is obtained, whose elastic properties are a combination IE* of the elastic properties of the single phases.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFor heterogeneous materials with a random microstructure (as in the case of LWCS) a stochastic approach is needed. The procedure requires the determination of the mechanical phases, whose geometric and mechanical characteristics need to be specified, and the statistical description of their spatial distribution. From a practical point of view, the the latter cannot be completely described, even in a statistical sense. Therefore, the constitutive behavior of the EHM can be only determined by making appropriate assumptions or approximations [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eIn order to overcome this limit, Mean Field Approaches can be used to describe the the microstructure of heterogeneuos materials, whose statistical information are limited.\u003c/p\u003e"},{"header":"3 MEAN FIELD APPROACHES","content":"\u003cp\u003eMean Field Homogenisation allows modelling composite materials characterised by one matrix phase and one or multiple inclusion phases with uniform properties. The more sophisticated mean-field homogenisation approaches, based on Eshelby's solution [\u003cspan additionalcitationids=\"CR12\" citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e], need only few information on the microstructure to be employed, namely the volume fraction (i.e., the ratio between the volume of the considered phase and the total volume of the RVE), phase morphology (i.e., phase connectedness or disconnectedness), aspect ratio (i.e., the ratio between the major axis and the minor axis of a given phase) and the spatial orientation of the inclusions. Mean-Field Approaches can differ according to the selection of the concentration tensors, which link microscopic strain and stress fields with the corresponding macroscopic ones [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe most common approach for cement-based materials (in particular, concrete) is the Eshelbian-type ellipsoidal inclusion embedded in a reference medium, for which an estimate of the localization tensor is provided, depending on the Eshelby tensor [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. Being characterised by a random microstructure, a reliable assumption for cement-based materials is to consider all phases as isotropic [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. Moreover, in the current work, another reasonable approximation is describing the inclusions as spheres (Ferriero \u003cem\u003eet al\u003c/em\u003e., submitted). For spheroidal inclusions in an isotropic elastic matrix, Eshelby tensor can be estimated analytically and depends on the Poisson\u0026rsquo;s ratio of the homogeneous material (or, in the case of heterogeneous inclusions, on the Poisson\u0026rsquo;s ratio of the matrix) and on the aspect ratio of the inclusions. These assumptions yield explicit expressions for the homogenised bulk and shear moduli \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]:\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Taba\" border=\"1\"\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{\\varvec{h}\\varvec{o}\\varvec{m}}^{\\varvec{e}\\varvec{s}\\varvec{t}}={3k}_{hom}^{est}\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{J}}}}}+{2\\mu }_{hom}^{est}\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{K}}}}}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{hom}^{est}=\\sum _{p}^{}{c}_{p}{k}_{p}{\\left(1+{\\alpha }_{0}^{est}\\left(\\frac{{k}_{p}}{{k}_{0}}-1\\right)\\right)}^{-1}\\times {\\left[\\sum _{p}^{}{c}_{p}{\\left(1+{\\alpha }_{0}^{est}\\left(\\frac{{k}_{p}}{{k}_{0}}-1\\right)\\right)}^{-1}\\right]}^{-1}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{hom}^{est}=\\sum _{p}^{}{c}_{p}{\\mu }_{p}{\\left(1+{\\beta }_{0}^{est}\\left(\\frac{{\\mu }_{p}}{{\\mu }_{0}}-1\\right)\\right)}^{-1}\\times {\\left[\\sum _{p}^{}{c}_{p}{\\left(1+{\\beta }_{0}^{est}\\left(\\frac{{\\mu }_{p}}{{\\mu }_{0}}-1\\right)\\right)}^{-1}\\right]}^{-1}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eWhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{\\varvec{h}\\varvec{o}\\varvec{m}}^{\\varvec{e}\\varvec{s}\\varvec{t}}\\)\u003c/span\u003e\u003c/span\u003e is the estimate of the homogenised elasticity tensor, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({c}_{p}\\)\u003c/span\u003e\u003c/span\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{p}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{p}\\)\u003c/span\u003e\u003c/span\u003e are respectively the volume fraction, the bulk modulus and the shear modulus of the phase p, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{0}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{0}\\)\u003c/span\u003e\u003c/span\u003e are the bulk and shear moduli of the reference medium and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{0}^{est}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{0}^{est}\\)\u003c/span\u003e\u003c/span\u003e are homogenisation parameters of the reference medium.\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003e \u003cb\u003eMori-Tanaka Method\u003c/b\u003e \u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eIn the Mori\u0026ndash;Tanaka (MT) method [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e], appropriate for materials that exhibit an evident matrix-inclusion morphology [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], the matrix phase is chosen as reference medium, i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{0}={\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{\\varvec{m}}\\)\u003c/span\u003e\u003c/span\u003e. In this case of study, the microstructure of LWCS can be simplified in matrix with spherical inclusions (i.e. voids induced by the foam), so Equations 2 and 3 can be specialised as follows:\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"No\" id=\"Tabb\" border=\"1\"\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{k}_{hom}^{est}}{{k}_{m}}=1+{c}_{I}\\frac{\\raisebox{1ex}{${k}_{I}$}\\!\\left/ \\!\\raisebox{-1ex}{${k}_{m}$}\\right.-1}{1+{\\alpha }_{m}^{est}\\left(1-{c}_{I}\\right)\\left(\\raisebox{1ex}{${k}_{I}$}\\!\\left/ \\!\\raisebox{-1ex}{${k}_{m}$}\\right.-1\\right)}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{\\mu }_{hom}^{est}}{{\\mu }_{m}}=1+{c}_{I}\\frac{\\raisebox{1ex}{${\\mu }_{I}$}\\!\\left/ \\!\\raisebox{-1ex}{${\\mu }_{m}$}\\right.-1}{1+{\\beta }_{m}^{est}\\left(1-{c}_{I}\\right)\\left(\\raisebox{1ex}{${\\mu }_{I}$}\\!\\left/ \\!\\raisebox{-1ex}{${\\mu }_{m}$}\\right.-1\\right)}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{0}^{est}\\equiv {\\alpha }_{m}^{est}=\\frac{{3k}_{m}}{{3k}_{m}+4{\\mu }_{m}} and {\\beta }_{0}^{est}\\equiv {\\beta }_{m}^{est}=\\frac{{6(k}_{m}+2{\\mu }_{m})}{5({3k}_{m}+4{\\mu }_{m})}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e6\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003ewhere the subscripts \u003cem\u003eI\u003c/em\u003e and \u003cem\u003em\u003c/em\u003e stand for \u003cem\u003eInclusions\u003c/em\u003e and \u003cem\u003ematrix\u003c/em\u003e respectively.\u003c/p\u003e \u003cp\u003eThe general formulae 4 and 5 can then be rewritten taking into account porosity induced by foam, n\u003csub\u003efoam\u003c/sub\u003e:\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{k}_{hom}^{est}}{{k}_{m}}=1+{n}_{foam}\\frac{\\raisebox{1ex}{${k}_{foam}$}\\!\\left/ \\!\\raisebox{-1ex}{${k}_{m}$}\\right.-1}{1+{\\alpha }_{m}^{est}\\left(1-{n}_{foam}\\right)\\left(\\raisebox{1ex}{${k}_{foam}$}\\!\\left/ \\!\\raisebox{-1ex}{${k}_{m}$}\\right.-1\\right)}\\)\u003c/span\u003e\u003c/span\u003e7\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{\\mu }_{hom}^{est}}{{\\mu }_{m}}=1+{n}_{foam}\\frac{\\raisebox{1ex}{${\\mu }_{foam}$}\\!\\left/ \\!\\raisebox{-1ex}{${\\mu }_{m}$}\\right.-1}{1+{\\beta }_{m}^{est}\\left(1-{n}_{foam}\\right)\\left(\\raisebox{1ex}{${\\mu }_{foam}$}\\!\\left/ \\!\\raisebox{-1ex}{${\\mu }_{m}$}\\right.-1\\right)}\\)\u003c/span\u003e\u003c/span\u003e8\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eStarting from Eq.\u0026nbsp;7 and Eq.\u0026nbsp;8, two assumptions have been considered in order to obtain simplified expression of Mori-Tanaka method for LWCS. Poisson\u0026rsquo;s ratio of the matrix was assumed equal to ν\u003csub\u003em\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.2, being this value commonly assumed for cemented materials. Assuming bulk and shear moduli of voids equal to zero (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{foam}={\\mu }_{foam}=0\\)\u003c/span\u003e\u003c/span\u003e), the effective normalized bulk and shear modulus in Eq.\u0026nbsp;7 and Eq.\u0026nbsp;8 assume the same following hyperbolic form [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{MT}^{h}\\left({n}_{foam}\\right)=\\left(\\frac{1-{n}_{foam}}{1+{n}_{foam}}\\right){k}_{m}{\\mu }_{MT}^{h}\\left({n}_{foam}\\right)=\\left(\\frac{1-{n}_{foam}}{1+{n}_{foam}}\\right){\\mu }_{m}\\)\u003c/span\u003e \u003c/span\u003e9\u003c/p\u003e \u003cp\u003e \u003cb\u003eSelf-Consistent Method\u003c/b\u003e \u003c/p\u003e \u003cp\u003eIn the Self-Consistent method (SC) [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e] the reference medium is assumed coincident with the homogenised medium, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{0}={\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{\\varvec{h}\\varvec{o}\\varvec{m}}^{\\varvec{e}\\varvec{s}\\varvec{t}}\\)\u003c/span\u003e\u003c/span\u003e. Differently from the MT method, this method implies the solution of two nonlinear equations, in which \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{0}\\equiv {k}_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{0}\\equiv {\\mu }_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003eIn this study, the same assumptions on Poisson\u0026rsquo;s ratio and inclusions (voids) moduli (ν\u003csub\u003em\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.2 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{foam}={\\mu }_{foam}=0\\)\u003c/span\u003e\u003c/span\u003e) yield to the same expressions of the homogenised values of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{MT}^{h}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{MT}^{h}\\)\u003c/span\u003e\u003c/span\u003e of the MT method (Eq.\u0026nbsp;9).\u003c/p\u003e \u003cp\u003e \u003cb\u003eDilute Method\u003c/b\u003e \u003c/p\u003e\u003cp\u003eThe Dilute Method (DI) (Eshelby, 1957) [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] does not take into account the interaction among inclusions.\u003c/p\u003e \u003cp\u003eIn this study, with the same assumptions on Poisson\u0026rsquo;s ratio and inclusions (voids) moduli (ν\u003csub\u003em\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.2 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{foam}={\\mu }_{foam}=0\\)\u003c/span\u003e\u003c/span\u003e), the effective normalized bulk and shear modulus are provided by the same expression and depend linearly on the artificial porosity n\u003csub\u003efoam\u003c/sub\u003e:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{DI}^{h}\\left({n}_{foam}\\right)=(1-2{n}_{foam}){k}_{m}{\\mu }_{DI}^{h}\\left({n}_{foam}\\right)=\\left(1-2{n}_{foam}\\right){\\mu }_{m}\\)\u003c/span\u003e \u003c/span\u003e10\u003c/p\u003e \u003cp\u003e \u003cb\u003eDifferential Method\u003c/b\u003e \u003c/p\u003e \u003cp\u003eAccording to Bruggeman and Roscoe [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e], the Differential method (DF) models the composite as a sequence of dilute suspensions [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. In this study, under the same assumptions adopted for the previous approaches (ν\u003csub\u003em\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.2 and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{foam}={\\mu }_{foam}=0\\)\u003c/span\u003e\u003c/span\u003e), the effective normalized bulk and shear moduli of the inclusions are provided by:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{DF}^{h}\\left({n}_{foam}\\right)={\\left(1-{n}_{foam}\\right)}^{2}{k}_{m}{\\mu }_{DF}^{h}\\left({n}_{foam}\\right)={\\left(1-{n}_{foam}\\right)}^{2}{\\mu }_{m}\\)\u003c/span\u003e \u003c/span\u003e11\u003c/p\u003e \u003cp\u003e \u003cb\u003eHomogenised elastic stiffness\u003c/b\u003e \u003c/p\u003e\u003cp\u003eStarting from the input parameters of Mean Field Methods, the homogenised values of k\u003csup\u003eh\u003c/sup\u003e and \u0026micro;\u003csup\u003eh\u003c/sup\u003e have been determined for the estimation of the homogenised elastic stiffness modulus of LWCS by means of Eq.\u0026nbsp;12 [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e].\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{hom}^{est}=\\frac{9{k}_{hom}^{est}{\\mu }_{hom}^{est}}{3{k}_{hom}^{est}+{\\mu }_{hom}^{est}}\\)\u003c/span\u003e\u003c/span\u003e12\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe values of the homogenised stiffness modulus obtained have been then compared with experimental test results in order to validate the calculations.\u003c/p\u003e \u003cp\u003eHomogenised stiffness modulus of LWCS was evaluated under the subsequent assumptions:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eLWCS is regarded as a two-phase material (i.e., matrix and voids) or as three-phase material (i.e., matrix, voids and portlandite);\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ephases are considered as isotropic\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003einclusions (i.e. voids induced by foam) are spherical;\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ematrix is supposed to be elastic, isotropic and homogeneous;\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ebulk moduli k\u003csub\u003eI\u003c/sub\u003e and shear moduli \u0026micro;\u003csub\u003eI\u003c/sub\u003e of voids (i.e., inclusions) are equal to zero.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e"},{"header":"4 HOMOGENISATION INPUT PARAMETERS FOR LWCS","content":"\u003cp\u003eThe homogenisation process according to Mean Field Approaches with the introduced approximations requires the following input parameters: the artificial porosity n\u003csub\u003efoam\u003c/sub\u003e, the volume fraction of portlandite V\u003csub\u003eP\u003c/sub\u003e, the volumetric stiffness moduli k\u003csub\u003em\u003c/sub\u003e, the shear moduli \u0026micro;\u003csub\u003em\u003c/sub\u003e, and the Poisson's coefficient ν\u003csub\u003em\u003c/sub\u003e of the matrix. For this purpose, experimental results of microstructural (Ferriero et al., \u003cem\u003eto be submitted\u003c/em\u003e) and mechanical [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] tests on LWCS samples of cement and foam-treated kaolin were interpreted. Treated samples considered in the cited experimental studies were prepared by mixing Speswhite kaolin, Portland limestone cement, water and foam in different proportions. Main features of each constituent are reported in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. As reference material forming the matrix, cemented samples without foam were considered as well.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMain components of LWCS.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eKaolin\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCement\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eFoam\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eSpeswhite kaolin supplied by Imerys Minerals, UK\u003c/p\u003e \u003cp\u003eG\u003csub\u003es\u003c/sub\u003e = 2.6\u003c/p\u003e \u003cp\u003eSpecific surface area\u0026thinsp;=\u0026thinsp;14 m\u003csup\u003e2\u003c/sup\u003e/g\u003c/p\u003e \u003cp\u003epH\u0026thinsp;=\u0026thinsp;4.6\u003c/p\u003e \u003cp\u003ew\u003csub\u003eL\u003c/sub\u003e = 70%, w\u003csub\u003ep\u003c/sub\u003e = 32%, IP\u0026thinsp;=\u0026thinsp;38%\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePortland limestone cement supplied by Buzzi, Italy\u003c/p\u003e \u003cp\u003eCEM II/A-LL 42.5 R\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eISOCEM S/L supplied by Isoltech srl, Italy\u003c/p\u003e \u003cp\u003econcentration 2.5%\u003c/p\u003e \u003cp\u003epH\u0026thinsp;=\u0026thinsp;7.5-9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eCement treated samples were prepared by mixing speswhite kaolin with water to slurry at approximately twice the liquid limit of soil (i.e., 2w\u003csub\u003eL\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;140%) and dry cement with water (water/cement ratio\u0026thinsp;=\u0026thinsp;0.5). The amount of cement by dry weight of soil considered for the treatment was 40% (i.e. KC40%).\u003c/p\u003e \u003cp\u003eLightweight cemented samples were prepared by adding 40% by volume of the mixture of preformed foam, (i.e. KCF40%). Foam was formed by mixing a surfactant solution and air with an industrial foam generator at density approximately equal to 75 g/L. Treated samples were poured into moulds and sealed in plastic bags for curing at room temperature. Curing time intervals ranging from 1 day to 28 days were considered for the experimental studies.\u003c/p\u003e \u003cp\u003eArtificial porosity induced by foam and matrix porosity of LWCS samples have different characteristic dimensions (average diameter of 300 \u0026micro;m the former, 0.3 \u0026micro;m the latter), and can be considered as a two distinct systems of pores. For determining the artificial porosity n\u003csub\u003efoam\u003c/sub\u003e of LWCS samples as a part of total porosity, X-Ray computed microtomography scans were performed on one sample at increasing curing times, namely 24h, 7 and 28 days. This experimental technique allowed the detection of artificial porosity via image analysis, based on the characteristic dimension and the spherical shape of pores induced by foam. XRm-CT scans of the KCF40% sample over time are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFrom the results, it can be seen that although artificial porosity is not altered over time, the formation of shrinkage microfractures in the material results in increased porosity in the large pore size range. Moreover, the denser particles (white in the micrographs) in the system are attributed to portlandite precipitating over curing time as a consequence of the hydration of cement; the consumption of portlandite takes place in the long term, highlighted by a progressive diminution of the frequency of denser particles in the system. Image analysis was based on two different segmentation techniques (i.e., watershed segmentation and global thresholding segmentation) for the evaluation of artificial porosity over time (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAverage values of the two methods for both artificial porosity and volume fraction of portlandite were considered as input in Mean Field Approaches, as reported in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMicro-CT artificial porosity.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCuring Time\u003c/p\u003e \u003cp\u003e[days]\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003en\u003csub\u003efoam\u003c/sub\u003e\u003c/p\u003e \u003cp\u003e[%]\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eV\u003csub\u003ep\u003c/sub\u003e\u003c/p\u003e \u003cp\u003e[%]\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e18.42\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e2.32\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e19.23\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e1.32\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e19.45\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.87\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe elastic stiffness of cement treated and lightweight cemented samples (i.e., E\u003csub\u003eKC40%\u003c/sub\u003e and E\u003csub\u003eKCF40%\u003c/sub\u003e respectively) were evaluated from from direct shear tests, performed at 50kPa of vertical effective stress, for 24h cured samples (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e) and from unconfined compression tests performed after 7 and 28 days of curing from stress-strain curves at low strain level (Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e ).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn order to determine Poisson\u0026rsquo;s ratios, one-dimensional compression curves of cement treated samples (KC40%) after 7 and 28 days of curing were considered (Fig.\u0026nbsp;\u003cspan refid=\"Fig9\" class=\"InternalRef\"\u003e9\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eOedometer modulus (E\u003csub\u003eoed\u003c/sub\u003e) of treated samples cured at 7 and 28 days has been determined in the reversible region for stress level of 50 kPa. From elastic stiffness (E\u003csub\u003eKC40%\u003c/sub\u003e) of cement treated sample and oedometer modulus (E\u003csub\u003eoed\u003c/sub\u003e), Poisson\u0026rsquo;s ratio of cemented samples was calculated from Eq.\u0026nbsp;13. With reference to 24h cured samples, Poisson\u0026rsquo;s ratio was assumed equal to 0.2 considering the early stage of the evolution of the system in terms of cement hydration process.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({E}_{oed}=\\frac{{E}_{KC40\\%}(1-{v}_{m})}{(1+{v}_{m})(1-2{v}_{m})}\\)\u003c/span\u003e \u003c/span\u003e13\u003c/p\u003e \u003cp\u003eStarting from the values of elastic stiffness and Poisson\u0026rsquo;s ratio, the bulk modulus k\u003csub\u003em\u003c/sub\u003e and the shear modulus \u0026micro;\u003csub\u003em\u003c/sub\u003e of cement treated samples were calculated (Eqs.\u0026nbsp;14 and 15). The subscript \u003cem\u003em\u003c/em\u003e used for cemented samples stands for the values assigned to the matrix of the material.\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{m}=\\frac{{E}_{KC40\\%}}{3*(1-2{v}_{m})}\\)\u003c/span\u003e \u003c/span\u003e14\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\mu }_{m}=\\frac{{k}_{m}}{2(1+{{v}_{m}}^{})}\\)\u003c/span\u003e \u003c/span\u003e15\u003c/p\u003e \u003cp\u003eA summary of physical and mechanical parameters of cement treated (KC40%) and lightweight cemented samples (KCF40%) over time is reported in Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003ePhysical and mechanical parameters of cement treated and lightweight cemented samples.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCuring Time\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eE\u003csub\u003eKC40%\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eE \u003csub\u003eKCF40%\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eE\u003csub\u003eoed\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003eν\u003csub\u003em\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003ek\u003csub\u003em\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u0026micro;\u003csub\u003em\u003c/sub\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e[days]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e[-]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e3.276\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.299\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e[-]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e1.820\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e1.365\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e11.993\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e11.180\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e70.502\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.375\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e15.991\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e5.815\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e19.110\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e13.195\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e72.727\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.316\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e17.311\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e7.260\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e"},{"header":"5 RESULTS AND DISCUSSION","content":"\u003cp\u003eThe homogenised values of the bulk modulus and the shear modulus of LWCS samples (i.e., \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e) are reported in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eResults of Mean Fields simplified formulae.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"10\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c9\" colnum=\"9\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c10\" colnum=\"10\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCuring time\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{MT}^{h}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[MT]\u003c/p\u003e\u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{DI}^{h}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[DI]\u003c/p\u003e\u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{DF}^{h}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[DF]\u003c/p\u003e\u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{MT}^{h}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[MT]\u003c/p\u003e\u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{DI}^{h}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[DI]\u003c/p\u003e\u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{DF}^{h}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[DF]\u003c/p\u003e\u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_simplified}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[MT]\u003c/p\u003e\u003c/th\u003e \u003cth align=\"left\" colname=\"c9\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_simplified}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[DI]\u003c/p\u003e\u003c/th\u003e \u003cth align=\"left\" colname=\"c10\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_simplified}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003cp\u003e[DF]\u003c/p\u003e\u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e[days]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.254\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.149\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.211\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e940\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e819\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e874\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e2.257\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e2.069\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e2.180\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e4.514\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.101\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e4.347\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e3.385\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e3.076\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e3.260\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e8.125\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e7.381\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e7.825\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e11.170\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e10.121\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e10.747\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e8.377\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e7.590\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c7\"\u003e \u003cp\u003e8.060\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c8\"\u003e \u003cp\u003e20.105\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c9\"\u003e \u003cp\u003e18.217\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c10\"\u003e \u003cp\u003e19.345\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe stiffness moduli were calculated following the Mori-Tanaka method (MT), Dilute method (DI) and Differential method (DF) (Equations 9\u0026ndash;11) with the assumption of Poisson\u0026rsquo;s ratio equal to \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({v}_{m}\\)\u003c/span\u003e\u003c/span\u003e= 0.2. The homogenised elastic stiffness (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_simplified}^{est}\\)\u003c/span\u003e\u003c/span\u003e) of LWCS was calculated following the Eq.\u0026nbsp;4 for each Mean Field Approach. The evolution over time of the \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_simplified}^{est}\\)\u003c/span\u003e\u003c/span\u003e for the different approaches is compared with the experimental results in Fig.\u0026nbsp;\u003cspan refid=\"Fig10\" class=\"InternalRef\"\u003e10\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eAssuming \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({v}_{m}\\)\u003c/span\u003e\u003c/span\u003e= 0.2, the evolution over time of artificial porosity n\u003csub\u003efoam\u003c/sub\u003e affects the homogenised stiffness and shear moduli \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e. Moreover, this assumption yields similar \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_simplified}^{est}\\)\u003c/span\u003e\u003c/span\u003e values at different curing time for the three different Mean Field Approaches. After one day of curing, homogenised values and the experimental value of elastic moduli for lightweight cemented samples E\u003csub\u003eKCF40%\u003c/sub\u003e overlap. At increasing curing time (i.e., 7 days), the analytical methods tend to slightly underestimate the homogenised elastic stiffness, while slightly overestimate the experimental results after 28 days.\u003c/p\u003e \u003cp\u003eFurther evaluations of the homogenised elastic stiffness of LWCS have been obtained starting from Mori-Tanaka method (Eq.\u0026nbsp;6\u0026ndash;8). In this case, the experimental values of Poisson\u0026rsquo;s ratio have been used for the calculation of thehomogenised bulk and shear moduli (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003csub\u003e;\u003c/sub\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e), whereas the coefficients \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{m}^{est}\\)\u003c/span\u003e\u003c/span\u003e and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{m}^{est}\\)\u003c/span\u003e\u003c/span\u003e have been determined starting from the bulk and shear moduli of the matrix (k\u003csub\u003em KC40%\u003c/sub\u003e and \u0026micro;\u003csub\u003em KC40%\u003c/sub\u003e ) corresponding to the experimental values of cement treated samples. On the basis of these hypothesis, the values of homogenised elastic stiffness (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_completed}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003cb\u003e)\u003c/b\u003e have been calculated and reported in Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eResults of complete formula of Mori-Tanaka method.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"6\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCuring time\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{m}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\beta }_{m}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_completed}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e[days]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e[-]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e[-]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.254\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e940\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e2.257\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e9.956\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e3.992\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.607\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.479\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e10.565\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e10.346\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.983\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0.641\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c5\"\u003e \u003cp\u003e0.472\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c6\"\u003e \u003cp\u003e12.881\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eA comparison between the elastic stiffness determined with simplified and complete Mori-Tanaka methods and the experimental results is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig11\" class=\"InternalRef\"\u003e11\u003c/span\u003e as function of curing time. Compared to simplified method, the calculation of elastic stiffness with the of whole formula of MT leads to values quite overlapping with the experimental results for all curing times.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe formula of Mori-Tanaka has been also applied by considering a three phase system instead of two phase system. The portlandite (Ca(OH)\u003csub\u003e2\u003c/sub\u003e), whose precipitation in the system is consequence of the hydration of cement, has been considered as one independent phase. The elastic properties of portlandite (i.e., Poisson\u0026rsquo;s ratio n\u003csub\u003ep\u003c/sub\u003e=0. 315, Young modulus E\u003csub\u003ep\u003c/sub\u003e=38823 MPa), reported in [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], have been considered in Equations 2 and 3 for the determination of the homogenised elastic stiffness \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_3phases}^{est}\\)\u003c/span\u003e\u003c/span\u003e of the three phase system. Results of the homogeneisation process as function of curing time are reported in Table\u0026nbsp;\u003cspan refid=\"Tab6\" class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab6\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThree-phases system results.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCuring time\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({k}_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\mu }_{hom}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_3phases}^{est}\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e[days]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e[MPa]\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1.563\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.172\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e2.814\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e13.468\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e4.959\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e12.951\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e28\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e14.502\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.180\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e15.703\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe comparison of the elastic stiffness, determined with the two and three-phases MT methods, and the experimental results are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig12\" class=\"InternalRef\"\u003e12\u003c/span\u003e as function of curing times. MT-method for the three phases system yelds to slightly overestimate the homogenised elastic stiffness \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}}^{est}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe Mori-Tanaka method in its complete formulation was the best analytical Mean Field method for evaluating the homogenized elastic stiffness of LWCS. The assumption of constant Poisson\u0026rsquo;s ratio adopted for the simplified MFA, equivalent to considering only the influence of artificial porosity (n\u003csub\u003efoam\u003c/sub\u003e), strongly limits the capability of these approaches in predicting the homogenised stiffness moduli, in particular for materials as LWCS with evolving mechanical properties over time, as consequence of chemo-physical evolution of the system. Conversely, the good agreement between the experimental results and the homogenization analytical method derives from including the chemo-physical evolution of the cemented material by the variation of relevant parameters over time. In the case of the two-phase system, such prediction is very effective because the representative parameters of the two phases are well characterized as a function of curing time. In the case of the three-phase system, on the other hand, an overestimation of the value of the stiffness modulus with respect to the experimental values probably results from the assumption of a constant elastic modulus for portlandite phase in the system over time, without taking into account its variation over time as the pozzolanic reactions proceed.\u003c/p\u003e"},{"header":"6 CONCLUSIONS","content":"\u003cp\u003eIn the study the elastic stiffness moduli of LWCS have been derived by means of Mean-Field Eshelby-based Homogenisation Approaches, which have been proved an effective method for heterogeneous materials with physical and mechanical properties evolving with curing time. The relevant parameters for the implementation of homogenisation approaches have been derived by experimental results at different scale levels on LWCS samples. The following conclusions can be drawn:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003ewith the assumption of constant Poisson's ratio (i.e., ν\u003csub\u003em\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;0.2), different simplified Mean Field Methods yield to similar homogenized elastic moduli \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_simplified}^{est}\\)\u003c/span\u003e\u003c/span\u003e over curing time, underestimating the experimental values in the short term, and overestimating them in the long term. Only the influence of artificial porosity n\u003csub\u003efoam\u003c/sub\u003e and its variation over time is taken into account, limiting the efficiency of those methods in predicting the homogenized elastic stiffness\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ethe comparison with experimental data shows that the Mori-Tanaka method provides the most suitable homogenisation analytical method for LWCS\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ecomplete formulation of the MT method for a two-phase system (i.e., matrix and voids) with input parameters evolving with curing time yields to very good predictions of the elastic stiffness and its evolution over time\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003ecomputation of homogenised elastic stiffness for three-phases system (i.e., matrix, portlandite and voids) with complete formulation of MT methods provides an overestimation of the stiffness values if relevant input parameters for the homogeneisation process, due to the chemo-physical evolution of the system, are not properly evaluated.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e"},{"header":"List of Notation and Acronym","content":"\u003cp\u003e \u003cem\u003eThe following notations are used in this paper\u003c/em\u003e:\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\varvec{x}}\\)\u003c/span\u003e \u003c/span\u003e macroscopic position vector\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\varvec{y}}\\)\u003c/span\u003e \u003c/span\u003e microscopic position vector\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\underset{\\_}{\\varvec{\\sigma }}}\\left(\\underset{\\_}{\\varvec{x}}\\right)\\)\u003c/span\u003e \u003c/span\u003e macro-stress field (second order tensor)\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\underset{\\_}{\\varvec{\\epsilon }}}\\left(\\underset{\\_}{\\varvec{x}}\\right)\\)\u003c/span\u003e \u003c/span\u003e macro-strain field (second order tensor)\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\varvec{\\sigma }}}}_{\\varvec{\\mu }}\\left(\\underset{\\_}{\\varvec{y}}\\right)\\)\u003c/span\u003e \u003c/span\u003e micro-stress field (second order tensor)\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\varvec{\\epsilon }}}}_{\\varvec{\\mu }}\\left(\\underset{\\_}{\\varvec{y}}\\right)\\)\u003c/span\u003e \u003c/span\u003e micro-strain field (second order tensor)\u003c/p\u003e \u003cp\u003e \u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({c}_{p}\\)\u003c/span\u003e \u003c/span\u003e volume fraction of the phase p\u003c/p\u003e \u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{J}}}}}\\)\u003c/span\u003e \u003c/span\u003e volumetric part of the identity tensor\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{K}}}}}\\)\u003c/span\u003e \u003c/span\u003e deviatoric part of the identity tensor\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{0}^{est},{\\beta }_{0}^{est}\\)\u003c/span\u003e \u003c/span\u003e homogenization coefficients\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\mu }_{0}\\)\u003c/span\u003e \u003c/span\u003e shear modulus of the reference medium\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{0}\\)\u003c/span\u003e \u003c/span\u003e bulk modulus of the reference medium\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\nu }_{0}\\)\u003c/span\u003e \u003c/span\u003e Poisson\u0026rsquo;s ratio of the reference medium\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{\\varvec{h}\\varvec{o}\\varvec{m}}^{\\varvec{e}\\varvec{s}\\varvec{t}}\\)\u003c/span\u003e \u003c/span\u003e estimate of homogenised elasticity tensor\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{0}\\)\u003c/span\u003e \u003c/span\u003e fourth-order elasticity tensor of the reference medium\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\underset{\\_}{\\varvec{C}}}}}}_{\\varvec{m}}\\)\u003c/span\u003e \u003c/span\u003e fourth-order elasticity tensor of the matrix\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\mu }_{p}\\)\u003c/span\u003e \u003c/span\u003e shear modulus of the phase p\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{p}\\)\u003c/span\u003e \u003c/span\u003e bulk modulus of the phase p\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{hom}^{est}\\)\u003c/span\u003e \u003c/span\u003e homogenised bulk modulus\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\mu }_{hom}^{est}\\)\u003c/span\u003e \u003c/span\u003e homogenised shear modulus\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\mu }_{I}\\)\u003c/span\u003e \u003c/span\u003e shear modulus of the inclusions\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{I}\\)\u003c/span\u003e \u003c/span\u003e bulk modulus of the inclusions\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\mu }_{m}\\)\u003c/span\u003e \u003c/span\u003e shear modulus of the matrix\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{m}\\)\u003c/span\u003e \u003c/span\u003e bulk modulus of the matrix\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({c}_{I}\\)\u003c/span\u003e \u003c/span\u003e volume fraction of the inclusions\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\alpha }_{m}^{est},{\\beta }_{m}^{est}\\)\u003c/span\u003e \u003c/span\u003e matrix homogenization coefficients\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({n}_{foam}\\)\u003c/span\u003e \u003c/span\u003e artificial porosity\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\mu }_{foam}\\)\u003c/span\u003e \u003c/span\u003e shear modulus of the voids\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{foam}\\)\u003c/span\u003e \u003c/span\u003e bulk modulus of the voids\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\nu }_{m}\\)\u003c/span\u003e \u003c/span\u003e Poisson\u0026rsquo;s ratio of the matrix\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{MT}^{h},{\\mu }_{MT}^{h}\\)\u003c/span\u003e \u003c/span\u003e Mori-Tanaka homogenised bulk and shear moduli\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{DI}^{h},{\\mu }_{DI}^{h}\\)\u003c/span\u003e \u003c/span\u003e Dilute method homogenised bulk and shear moduli\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({k}_{DF}^{h},{\\mu }_{DF}^{h}\\)\u003c/span\u003e \u003c/span\u003e Differential method homogenised bulk and shear moduli\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({E}_{hom}^{est}\\)\u003c/span\u003e \u003c/span\u003e homogenised Young modulus\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({E}_{KC40\\%}\\)\u003c/span\u003e \u003c/span\u003e Young modulus of cement treated samples\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({E}_{KCF40\\%}\\)\u003c/span\u003e \u003c/span\u003e Young modulus of LWCS samples\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({E}_{oed}\\)\u003c/span\u003e \u003c/span\u003e Oedometric modulus of the matrix\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_\\text{s}\\text{i}\\text{m}\\text{p}\\text{l}\\text{i}\\text{f}\\text{i}\\text{e}\\text{d}}^{est}\\)\u003c/span\u003e \u003c/span\u003e Young modulus computed with simplified Mori-Tanaka method\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_completed}^{est}\\)\u003c/span\u003e \u003c/span\u003e Young modulus computed with complete Mori-Tanaka method\u003c/p\u003e\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({E}_{\\text{h}\\text{o}\\text{m}\\_3phases}^{est}\\)\u003c/span\u003e \u003c/span\u003e Young modulus computed with complete Mori-Tanaka method considering three phases\u003c/p\u003e\u003cp\u003eV\u003csub\u003ep\u003c/sub\u003e Volume fraction of portlandite\u003c/p\u003e\u003cp\u003eν\u003csub\u003ep\u003c/sub\u003e Poisson\u0026rsquo;s ratio of portlandite\u003c/p\u003e\u003cp\u003eE\u003csub\u003ep\u003c/sub\u003e Young modulus of porlandite\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNo funds, grants, or other support was received.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by all authors. The first draft of the manuscript was written by Laura Perrotta and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eVitale E, Deneele D, Russo G, De Sarno D, Nicotera M V, Pap R, \u0026amp; Urciuoli G (2020) Chemo-mechanical behaviour of lightweight cemented soils. Acta Geotechnica, 15: 933-945.\u003c/li\u003e\n \u003cli\u003eDe Sarno D, Vitale E, Nicotera M V, Papa R, Russo G, Urciuoli G (2020)\u0026nbsp;Lightweight Cemented Soils: Mix Design, Production and Control. Lecture Notes in Civil Engineering. In:\u0026nbsp;Geotechnical Research for Land Protection and Development: Proceedings of CNRIG 2019 7. Springer International Publishing, pp. 743-752.\u003c/li\u003e\n \u003cli\u003eSatoh T, Tsuchida T, Mitsukuri K, Hong Z (2001) \u0026nbsp;Field placingtest of lightweight treated soil under seawater in Kumamoto port. Soils and Foundations\u003cem\u003e,\u0026nbsp;\u003c/em\u003e41.5: 145-154.\u003c/li\u003e\n \u003cli\u003eTsuchida T, \u0026amp; Egashira K (2004). The lightweight treated soil method: new geomaterials for soft ground engineering in coastal areas.\u0026nbsp;CRC Press.\u003c/li\u003e\n \u003cli\u003eWatabe Y, Itou Y, Kang MS, Tsuchida T (2004) One-dimensional compression of air-foam treated lightweight geo-material in microscopic point of view. 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British journal of applied physics, 3.8: 267.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"soil treatment, multi-scale analysis, homogenisation, Lightweight Cemented Soil (LWCS)","lastPublishedDoi":"10.21203/rs.3.rs-3409777/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3409777/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eLightweight Cemented Soil (LWCS) is obtained by mixing soil, water and cement with air foam. This treatment allows a high workability of the mixture and good mechanical properties coupled with reduced unit volume weight, making the material suitable for many geotechnical applications. The complex and heterogeneous microstructure (a large pore system induced by foam in a cemented porous matrix) of the material requires a homogeneisation procedure for constitutive modelling of its hydromechanical behaviour. In this work, the stiffness modulus of LWCS was derived through the semi-analitycal Mean-Field Eshelby-based Homogenisation approaches, taking into account the chemo-physical evolution over curing time of the material. This procedure, never employed in the field of treated soils, requires several input parameters, as the artificial porosity induced by the addition of foam, the bulk and shear moduli and the Poisson\u0026rsquo;s ratio of LWCS samples. The artificial porosity was evaluated by means of X-Ray micro-CT scans, whereas the mechanical parameters were derived from experimental tests. The homogenised stiffness moduli, computed for different curing times, are in agreement with the available experimental results. The analytical homogenisation models allow predicting the elastic behaviour of the material and its evolution over time taking into account the role played by microstructure.\u003c/p\u003e","manuscriptTitle":"Homogenised Elastic Properties of Lightweight Cemented Soils","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2023-10-11 16:57:55","doi":"10.21203/rs.3.rs-3409777/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"52984c31-ed81-4948-81b5-7cf07f61199a","owner":[],"postedDate":"October 11th, 2023","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2023-11-17T20:14:21+00:00","versionOfRecord":[],"versionCreatedAt":"2023-10-11 16:57:55","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3409777","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3409777","identity":"rs-3409777","version":["v1"]},"buildId":"-HB7Z8yhvgn0wM9Nzuekk","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}