Mean Field Homogenisation allows modelling composite materials characterised by one matrix phase and one or multiple inclusion phases with uniform properties. The more sophisticated meanfield 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. MeanField 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 cementbased materials (in particular, concrete) is the Eshelbiantype 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 cementbased 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.
In the Mori–Tanaka (MT) method [21], appropriate for materials that exhibit an evident matrixinclusion 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, nfoam:
\(\frac{{k}_{hom}^{est}}{{k}_{m}}=1+{n}_{foam}\frac{\raisebox{1ex}{${k}_{foam}$}\!\left/ \!\raisebox{1ex}{${k}_{m}$}\right.1}{1+{\alpha }_{m}^{est}\left(1{n}_{foam}\right)\left(\raisebox{1ex}{${k}_{foam}$}\!\left/ \!\raisebox{1ex}{${k}_{m}$}\right.1\right)}\)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 MoriTanaka method for LWCS. Poisson’s ratio of the matrix was assumed equal to νm = 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
SelfConsistent Method
In the SelfConsistent 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 (νm = 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 (νm = 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 nfoam:
\({k}_{DI}^{h}\left({n}_{foam}\right)=(12{n}_{foam}){k}_{m}{\mu }_{DI}^{h}\left({n}_{foam}\right)=\left(12{n}_{foam}\right){\mu }_{m}\) 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 (νm = 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 kh and µh 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 twophase material (i.e., matrix and voids) or as threephase 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 kI and shear moduli µI of voids (i.e., inclusions) are equal to zero.