domain: phys.cond.cm-ms sde sdu spi

text:
Philippe Cosenza
email: philippe.cosenza@univ-poitiers.fr

Richard Giot

Stephen Hedan

Elastic moduli of clay minerals and their aggregates: A review

Keywords: clay minerals, elastic moduli, bulk modulus, shear modulus, transverse isotropy

The main objective of this review paper is to take a crossed look at all of the data of elastic moduli of clay minerals acquired from various experimental and theoretical methods. The comparison of these data of various origins, rarely made in practice, has been conducted qualitatively and quantitatively by calculating the Voigt-Reuss-Hill (VRH) averages of tensorial estimates. This comparison highlights four sets of results. First, the elastic constants of 2:1 nonswelling minerals can be reasonably estimated by molecular modeling of the density functional theory (DFT) type. Second, all the estimates on swelling minerals mainly obtained by molecular modeling confirm that their elastic properties are sensitive to the presence of interlayer water: they globally decrease with the number of water molecules existing in interlayer space. However, these theoretical estimates face a major difficulty: their experimental validation. The measurement on macroscopic analogs is of interest in this context. Third, most of the values provided by micromechanical inversion of geophysical or geomechanical data acquired on clay rocks or clayey composites and the unjacketed bulk modulus K s values classically measured in geomechanics are rather representative of clay mineral assemblages and not of a given single clay mineral. Considering these mineral assemblages, the mechanical interactions between minerals and the fundamental role of interlayer water, as shown in this review, should be considered. Fourth, the results of a micromechanical model, a Mori-Tanaka scheme, applied in particular to the isotropic VRH moduli suggest a solid nature of the elastic properties of interlayer water. The elastic stiffness of the latter would be rather representative of those of the mineral solid itself than those of ice, as the literature often suggests.

Introduction

Quantifying the elastic properties of clay minerals is of interest for several scientific issues and industrial applications. In geophysics, because clay minerals are a part of many sedimentary formations, geological reservoir characterization using seismic wave velocities requires the elastic parameters of mineral components to be known (e.g., [START_REF] Wang | Effective elastic properties of solid clays[END_REF][START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF][START_REF] Mondol | Elastic properties of clay minerals[END_REF]. In geomechanics, multiscale modeling of the mechanical behavior of clayey materials introduces the values of elastic constants of clay minerals at the lowest scale as input data (e.g., [START_REF] Giraud | Effective poroelastic properties of transversely isotropic rock-like composites with arbitrarily oriented ellipsoidal inclusions[END_REF]Ortega et al., 2007;[START_REF] Cariou | An original constitutive law for Callovo-Oxfordian argillite, a two-scale double-porosity material[END_REF]. In mineral physics, knowledge of the mineral elasticity of nanoscale clay minerals is of fundamental interest to understand the hydration and swelling processes of clay crystals or tactoids (e.g., [START_REF] Carrier | ESEM study of the humidity-induced swelling of clay film[END_REF]2014;[START_REF] Ebrahimi | Mesoscale properties of clay aggregates from potential of mean force representation of interactions between nanoplatelets[END_REF]. In polymer composite research and their industrial applications, smectite clays are used as fillers in polymer nanocomposites, and the reinforcement performance of the latter is controlled by the elastic properties of the present clay minerals (e.g., [START_REF] Chen | Elastic moduli of clay platelets[END_REF].

The elastic properties of clay minerals, especially those of swelling clays, are rarely found in handbooks (e.g., [START_REF] Schon | Physical properties of rocks: fundamentals and principles of petrophysics[END_REF][START_REF] Mavko | The rock physics handbook: Tools for seismic analysis of porous media[END_REF]. Review papers of these mechanical properties are also scarce [START_REF] Chen | Elastic moduli of clay platelets[END_REF][START_REF] Heinz | Clay minerals for nanocomposites and biotechnology: surface modification, dynamics and responses to stimuli[END_REF]. The review by [START_REF] Chen | Elastic moduli of clay platelets[END_REF], the most exhaustive to our knowledge, indicates elastic modulus values in the range of 178 to 265 GPa regardless of the hydration and swelling properties of clay minerals. [START_REF] Heinz | Clay minerals for nanocomposites and biotechnology: surface modification, dynamics and responses to stimuli[END_REF] reported in-plane elastic moduli of approximately 160 GPa lower than the range proposed by [START_REF] Chen | Elastic moduli of clay platelets[END_REF]. The scarcity and scattering of values stem from two facts: (a) clay minerals are often not sufficiently large for conventional experimental techniques, and (b) for swelling clay minerals, the hydric state, i.e., the amount of interlayer water molecules in crystals, must be correctly controlled. Despite these difficulties, attempts have been made to estimate the elastic properties of clay minerals using advanced experimental methodologies or indirect theoretical approaches. These attempts can be roughly divided into two groups.

The first group includes direct and indirect experimental methodologies. Direct experimental methodologies consist of directly measuring elastic parameters on single clay macrocristals [START_REF] Alexandrov | Elastic properties of rock-forming minerals. II: Layered silicates[END_REF][START_REF] Caslavsky | Examination of Imperfect Muscovite Crystals by X-Ray Diffraction Methods[END_REF][START_REF] Vaughan | Elasticity of muscovite and its relationship to crystal structure[END_REF][START_REF] Mcneil | Elastic moduli of muscovite mica[END_REF] or macroscopic analogs of swelling clay minerals [START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF][START_REF] Hedan | Measurement of the elastic properties of swelling clay minerals using the digital image correlation method on a single macroscopic crystal[END_REF]. Indirect methods aim at obtaining elastic properties from water-clay composites or mixtures using ultrasonic measurements (e.g., [START_REF] Wang | Effective elastic properties of solid clays[END_REF][START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF][START_REF] Mondol | Elastic properties of clay minerals[END_REF], atomic force acoustic microscopy [START_REF] Prasad | Measurement of Young's modulus of clay minerals using atomic force acoustic microscopy[END_REF], nanoindentation [START_REF] Bobko | The nano-mechanical morphology of shale[END_REF][START_REF] Du | Multiscale mechanical properties of shales: grid nanoindentation and statistical analytics[END_REF] or by coupling these techniques [START_REF] Deirieh | Nanochemomechanical assessment of shale: a coupled WDS-indentation analysis[END_REF]. These latter approaches can be considered indirect methods because they most often use upscaling models and a fitting procedure to back-calculate the elastic moduli of the clay minerals.

The second group of approaches addresses theoretical clay systems. Here, the elastic properties are primarily obtained from theoretical molecular techniques at the nanoscale: density functional theory (DFT) [START_REF] Santamarina | First-principles studies on the elastic constants of a 1: 1 layered kaolinite mineral[END_REF]Millitzer et al., 2011;[START_REF] Weck | Relationship between crystal structure and thermo-mechanical properties of kaolinite clay: beyond standard density functional theory[END_REF][START_REF] Zhao | Investigation on atomic structure and mechanical property of Na-and Mg-Montmorillonite under high pressure by firstprinciples calculations[END_REF] and molecular dynamics (MD) approaches [START_REF] Suter | Large-scale molecular dynamics study of montmorillonite clay: emergence of undulatory fluctuations and determination of material properties[END_REF]Mazo et al., 2008a,b;Ebrahimi et al., 2012;[START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF][START_REF] Honorio | Hydration phase diagram of clay particles from molecular simulations[END_REF][START_REF] Zhang | Nanoscale mechanical behavior of kaolinite under uniaxial strain conditions[END_REF][START_REF] Wang | Anisotropic elastic properties of montmorillonite with different layer charge densities and layer charge distributions through molecular dynamic dimulation[END_REF] are the most commonly used techniques. The results emphasized the crucial role of the interlayer water molecules on the mechanical response of the swelling clay crystal. These theoretical estimates of elastic properties are increasingly numerous but scarce compared to prior experimental data. This increase in studies is likely because theoretical approaches make it possible to provide estimates of a tensorial nature, which experimental approaches rarely can.

The objective of this paper is twofold. The first objective is to provide an updated review of quantitative data of elastic moduli of clay minerals by integrating the recent theoretical estimates obtained from molecular simulations. The second objective is to cross examine all of these data acquired from various experimental and theoretical methods. The comparison of these data of various origins is rarely made and would make it possible to validate both experimental and theoretical methodologies.

The direct and indirect experimental estimates used to determine the elastic moduli of clay minerals are presented in the first part of the paper. The second part addresses the theoretical approaches, mainly molecular-type simulations, which are able to achieve the full elastic tensor of clay minerals. A methodology to compare all of these estimates, especially to infer the elastic properties of interlayer water, is provided in the third section. The resulting discussion not only highlights the limitations of some approaches but also identifies the most robust estimates.

Experimental estimates

Methodological aspects

The experimental estimates of elastic constants of clay minerals can be roughly divided into three groups: (a) data acquired directly on natural clay macrocrystals, (b) those obtained on composites and clay powders mainly by the geophysical community and (c) estimates of Young's moduli in the plane parallel and perpendicular to the clay sheets or to the plane of isotropy in the case of mineral aggregates or analogs.

The data of natural macrocrystals were acquired using (a) elastic velocity measurements [START_REF] Alexandrov | Elastic properties of rock-forming minerals. II: Layered silicates[END_REF], (b) direct mechanical loading, i.e., by bending tests [START_REF] Caslavsky | Examination of Imperfect Muscovite Crystals by X-Ray Diffraction Methods[END_REF]1970) or uniaxial tests [START_REF] Hedan | Measurement of the elastic properties of swelling clay minerals using the digital image correlation method on a single macroscopic crystal[END_REF], which both provide a Young's modulus measurement, and (c) Brillouin spectrometry, which estimates elastic velocities [START_REF] Vaughan | Elasticity of muscovite and its relationship to crystal structure[END_REF][START_REF] Mcneil | Elastic moduli of muscovite mica[END_REF]. With the exception of the work of [START_REF] Hedan | Measurement of the elastic properties of swelling clay minerals using the digital image correlation method on a single macroscopic crystal[END_REF], which measured Young's modulus on vermiculite, most of the data have been obtained on nonswelling minerals, i.e., muscovite samples of millimeter size.

The geophysical communities in the 1990s and 2000s have developed more indirect approaches. The simplest indirect approach consisted of (i) measuring the moduli on composites or porous aggregates at different contents of the studied clay mineral and then in (ii) extrapolating mathematically these measurements to a value of 0% porosity (Hornby et al., 1994;[START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF]. Following the same approach, [START_REF] Wang | Effective elastic properties of solid clays[END_REF] measured the elastic wave velocities of composites made of clay particles embedded in an epoxy matrix of known elastic properties. An inverse analysis using a simple ad hoc upscaling approach or homogenization model made it possible to estimate the elastic moduli of clay particles considered to be isotropic. Therefore, all of these petrophysical works have provided estimates that are more representative of an aggregate of clay minerals than of a single clay mineral. In this aggregate of clay minerals, complex interactions between sheets (e.g. friction between sheets with or without possible buckling) may occur.

It is convenient to add a third group of experimental approaches, which is all the recent advanced techniques partly introduced by nanotechnology and digital image processing developments. Among these methods implemented at the micrometric and pluri-micrometric scales, nanoindentation approaches have been widely used on clay rocks [START_REF] Prasad | Measurement of Young's modulus of clay minerals using atomic force acoustic microscopy[END_REF][START_REF] Ulm | Experimental microporomechanics[END_REF]Ortega et al. 2007;Bobko andUlm, 2008 Zhang et al., 2009;Bobko et al., 2011;[START_REF] Magnenet | Determination of the matrix indentation modulus of Meuse/Haute-Marne argillite[END_REF][START_REF] Goodarzi | Predicting the elastic response of organic-rich shale using nanoscale measurements and homogenisation methods[END_REF][START_REF] Li | Shale anisotropy estimation from logs in vertical wells[END_REF]. To identify the mineralogical composition of the mineral or material in contact with the indenter, nanoindentation testing is often coupled with scanning electron microscopy (SEM) observation combined with wave dispersive spectroscopy (WDS) [START_REF] Deirieh | Nanochemomechanical assessment of shale: a coupled WDS-indentation analysis[END_REF][START_REF] Veytskin | Micromechanical characterization of shales through nanoindentation and energy dispersive x-ray spectrometry[END_REF]. These approaches use the same assumptions as the previous experimental approaches: the nanoindentation technique requires back-calculations and corrections to obtain the elastic constants. [START_REF] Zhang | Elastic modulus and hardness of muscovite and rectorite determined by nanoindentation[END_REF] postulated an a priori Poisson's ratio value of 0.25 for their crystals that were assumed to be isotropic, i.e., their actual layered structure of clay minerals was not considered. Similarly, [START_REF] Prasad | Measurement of Young's modulus of clay minerals using atomic force acoustic microscopy[END_REF] assumed a Poisson's ratio value of 0.3 and isotropic particles to measure the Young's modulus of kaolinite and dickite powders. In practice, measurements by nanoindentation combined with petrophysical and/or tomographic measurements are mainly used to calibrate or validate multiscale models derived from micromechanics (Ortega et al. 2007;[START_REF] Deirieh | Nanochemomechanical assessment of shale: a coupled WDS-indentation analysis[END_REF][START_REF] Abedi | Nanochemo-mechanical signature of organic-rich shales: a coupled indentation-EDX analysis[END_REF][START_REF] Mashhadian | Assessing mechanical properties of organic matter in shales: Results from coupled nanoindentation/SEM-EDX and micromechanical modeling[END_REF][START_REF] Li | Shale anisotropy estimation from logs in vertical wells[END_REF].

The nanoidentification methods mentioned previously require high-tech and costly devices. This is why authors have recently proposed circumventing this difficulty by developing lighter methodologies on macroscopic analogs of clay minerals. [START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF] used thin films obtained by evaporation of a clay suspension as analogs of swelling clay minerals.

These clay films of a few dozen micrometers were subjected to a tensile test at controlled humidity, during which the resulting deformations were measured by digital image correlation. The knowledge of the values of the tension applied and the measurements of the deformations made it possible to estimate the static Young's modulus of the films.

To end this brief panorama of experimental approaches, it is interesting to mention the unjacketed triaxial test used to determine the bulk modulus of the solid matrix (also called the unjacketed bulk modulus), denoted K s . The solid bulk modulus reflects the compressibility of the assembly of minerals that constitute the solid matrix of the rock. It is a fundamental parameter used in the formalism of poroelasticity (e.g., [START_REF] Coussy | Poromechanics[END_REF]. The unjacketed triaxial test, a macroscopic test carried out on a centimeter unjacketed sample, consists of measuring the volumetric deformation of the sample by varying the pore pressure, u, concomitantly with the confining pressure  3 (to keep constant the difference  3 -u). The determination of K s. has recently seen renewed interest since the formalism of poroelasticity and more generally that of poromechanics are applied for modeling the behavior of clayey rocks used for the underground storage of radioactive waste [START_REF] Ferrari | One-dimensional compression and consolidation of shales[END_REF][START_REF] Belmokhtar | Poroelasticity of the Callovo-Oxfordian claystone[END_REF][START_REF] Giot | Multi-stage water permeability measurements on claystone by steady and transient flow methods[END_REF][START_REF] Hu | Evaluation of anisotropic poroelastic properties and permeability of the Opalinus Clay using a single transient experiment[END_REF] or for the exploration of gas-and oil-producing, organic-rich, mudstone rocks [START_REF] Suarez-Rivera | Evaluating the poroelastic effect on anisotropic, organicrich, mudstone systems[END_REF].

Experimental estimates

The experimental estimates acquired on muscovite macrocrystals are presented in Table I, which shows two features. First, the two approaches, i.e., measurements by elastic wave velocities and those obtained by Brillouin scattering, provide the same estimates. As further shown, the robustness of these estimates will make it possible to validate the theoretical approaches used to determine the elastic constants of other clay minerals. Second, the elastic constants in the plane of the clay sheet, i.e., C 11 , C 22 , C 12 and C 66, are significantly higher than those normal to the basal plane, i.e., C 33 and C 44 , C 55 , C 13 and C 23 (see Supplementary Material for the mechanical significance of moduli C ij ). All other things being equal, this means that the muscovite sheet elastically deforms more easily in the perpendicular, out-ofplane direction than in the in-plane direction. It can also be noted that components C 15 , C 25 , C 35 and C 46 are rather low compared to the other components of the elasticity tensor. This justifies the hypothesis of orthotropic symmetry of clay minerals adopted by some authors (Mazo et al., 2008b[START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF].

The estimates established by the geophysical and petrophyiscal communities [START_REF] Wang | Effective elastic properties of solid clays[END_REF][START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF][START_REF] Mondol | Elastic properties of clay minerals[END_REF] are given in Tables 2, 3 and 4, which show the values of the elastic properties of nonswelling minerals (kaolinite, illite) and smectites, respectively. It should be recalled that these data were obtained by acoustic methods on clay composites or compacted clay powder.

The values given in Tables 2, 3 and 4 show three major features.

First, all of these estimates implicitly assume that the clay phase of rocks is isotropic. This assumption certainly has a practical origin. In fact, these values are generally intended as input data for commercial software used for the interpretation of logs and/or seismic inversion codes, which rarely consider the anisotropy of rocks. Thus, these values would rather be representative of aggregates of clay minerals that can often be considered isotropic (e.g., [START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF]. This last hypothesis is supported by the values of unjacketed bulk moduli, K s , given in Table 5, which are representative mostly of an assembly of clay minerals.

The values of K s are clearly of the same order of magnitude as the compressibility modulus values given in Tables 2, 3 and 4. The significant scattering of the latter estimates could also be explained in this way. They would account for elastic interactions between clay minerals in particles of extremely variable size in the clay composites or powders rather than the elastic properties of individual clay minerals themselves.

Tables 2, 3 and 4 also show a second important feature. They illustrate that the petrophysical approaches used in these studies discriminate 1:1 clay minerals (kaolinite) and 2:1 nonswelling clay minerals (illite and chlorite) (Figure 1). However, they do not distinguish between nonswelling clay minerals (kaolinite, illite, chlorite) and swelling smectite-type minerals (Figure 1).

On the one hand, data from [START_REF] Wang | Effective elastic properties of solid clays[END_REF] suggest that nonswelling clay minerals and especially 2:1 nonswelling clay minerals are elastically stiffer than swelling clay minerals.

However, this observation is not confirmed by the data of [START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF] and [START_REF] Mondol | Elastic properties of clay minerals[END_REF], from which a clear hierarchy between clay minerals cannot be identified (Figure 1). This inconsistency between these different datasets is partly explained by the scattering nature of these data but can also find an origin in a third feature: the uncontrolled hydration state of swelling clay minerals. Indeed, the latter can present in their interlayer space variable numbers of water layers up to three layers for Na-montmorillonite (e.g., [START_REF] Meunier | Clays[END_REF].

However, the control of this number of water layers, i.e., of this hydration state, requires a specific experimental procedure, i.e., a process of water adsorption, which was not considered in the indicated studies. The theoretical studies and the experimental data given below clearly show that the state of hydration strongly influences the elastic moduli of swelling clay minerals.

Indeed, the measurements of Young's moduli acquired in the in-plane and out-of-plane of the clay basal plane indicated in Table 6 show two interesting observations. On the one hand, the measurements of [START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF] clearly illustrate the influence of the hydration state of a clay crystal on its elastic response. The Young's modulus in the clay sheet, considered an isotropic basal plane, decreases sharply with water content. On the other hand, in the case of muscovite, these data confirm those of Table 1: the Young's modulus measured in the basal plane is clearly higher than that perpendicular to the clay sheet.

Theoretical approaches

Methodological aspects

The deployment of experimental methods on objects as small as clay minerals is delicate from a practical point of view and requires complex and costly technical developments. This is why many authors have sought to develop theoretical approaches by molecular modeling, requiring no experimental data to be acquired. Molecular modeling refers to a set of numerical approaches to simulate the evolution of a system of particles, generally molecules, over time.

In the case of clay minerals, two approaches have been preferred: density functional theory (DFT) [START_REF] Santamarina | First-principles studies on the elastic constants of a 1: 1 layered kaolinite mineral[END_REF]Millitzer et al., 2011;[START_REF] Weck | Relationship between crystal structure and thermo-mechanical properties of kaolinite clay: beyond standard density functional theory[END_REF][START_REF] Zhao | Investigation on atomic structure and mechanical property of Na-and Mg-Montmorillonite under high pressure by firstprinciples calculations[END_REF] and molecular dynamics (MD) (Suter et al., 2006;Mazo et al., 2008a,b;Ebrahimi et al., 2012;[START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF][START_REF] Honorio | Hydration phase diagram of clay particles from molecular simulations[END_REF][START_REF] Zhang | Nanoscale mechanical behavior of kaolinite under uniaxial strain conditions[END_REF][START_REF] Wang | Anisotropic elastic properties of montmorillonite with different layer charge densities and layer charge distributions through molecular dynamic dimulation[END_REF].

Without going into detail that would go beyond the scope of this paper, the molecular density functional theory explicitly considers the orbitals of the atoms involved, whereas molecular dynamics considers an atom as a point subjected to a force field, rather empirically defined.

The major interest of molecular dynamics lies in its ability to manage several million atoms.

However, these approaches require a posteriori experimental validation. Indeed, like all theoretical approaches, these are based on assumptions (i.e., mathematical formulation of the local force fields governing the interactions between molecules), which must be verified by experiment.

In parallel to the implementation of these numerical methods, some authors have preferred to employ lighter approaches based on micromechanical homogenization techniques, i.e., effective medium models or simple averaging (Ortega et al., 2007;[START_REF] Keller | Porosity anisotropy of Opalinus Clay: implications for the poroelastic behaviour[END_REF][START_REF] Louis | Computation of effective elastic properties of clay from X-ray texture goniometry data[END_REF]. For example, by using a micromechanical approach, Ortega et al. (2007) inverted acoustic and nanoindentation data obtained on several shale samples to estimate the elasticity tensor of the clay matrix considered as transverse isotropic. [START_REF] Louis | Computation of effective elastic properties of clay from X-ray texture goniometry data[END_REF] proposed considering a single clay elasticity tensor obtained by calculating averages of values of elastic constants estimated from published stiffness data and from a clay mineral orientation dataset obtained by X-ray texture goniometry.

Theoretical estimates

Tables 7, 8 and 9 show the theoretical estimates of the elastic constants C ij obtained by molecular modeling for 1:1 clay minerals, nonswelling 2:1 minerals (muscovite, pyrophyllite, illite) and montmorillonites, respectively. These tables highlight three important results. First, in the specific case of muscovite (Table 8), the estimates obtained by the DFT approach (Millitzer et al., 2011) are very similar to those provided by experimental approaches, i.e., by Brillouin scattering [START_REF] Vaughan | Elasticity of muscovite and its relationship to crystal structure[END_REF]McNeil and Grindtich, 1993) and in a lesser extent, by elastic wave velocity measurements [START_REF] Alexandrov | Elastic properties of rock-forming minerals. II: Layered silicates[END_REF] (Figure 2, compare also Table 1 andTable 8). This important result gives some confidence to the theoretical estimates provided by molecular modeling and especially the DFT approach.

Second, in the case of 1:1 minerals (Table 7), the different estimates of the C ij constants by DFT are consistent with each other (except for constant C 33 ). This is not the case for those obtained by MD (Figure 3). This inconsistency is probably due to the high sensitivity of the results with respect to the force field used in the MD calculations (e.g., [START_REF] Hantal | Atomic-scale modelling of elastic and failure properties of clays[END_REF]. It is important to emphasize here that the force field called Clayff commonly used in MD for clay minerals [START_REF] Cygan | Molecular models of hydroxide, oxyhydroxide, and clay phases and the development of a general force field[END_REF][START_REF] Hantal | Atomic-scale modelling of elastic and failure properties of clays[END_REF] significantly overestimates the modules in the plane of clay sheet [START_REF] Teich-Mcgoldrick | Molecular dynamics simulations of structural and mechanical properties of muscovite: pressure and temperature effects[END_REF]. The mathematical formulation of the optimal force field for clay minerals is still the subject of ongoing important research.

Third, the estimates in Table 9 confirm the experimental results in Table 6. The state of hydration, i.e., the number of water molecules in the interlayer space of montmorillonite-type minerals, has a major influence on the elastic moduli C ij . All these elastic constants significantly decrease with an increasing number of interlayer water layers. Despite the scattering between the different values in Table 9, this effect appears to be greater for elastic constants normal to the basal plane, i.e., C 33 , C 44 , C 55 and C 13, than those in the plan of the clay sheet (Figure 4). All these estimates are completed from molecular modeling with those obtained by approaches based on more or less sophisticated homogenization techniques (Ortega et al., 2007;[START_REF] Keller | Porosity anisotropy of Opalinus Clay: implications for the poroelastic behaviour[END_REF][START_REF] Louis | Computation of effective elastic properties of clay from X-ray texture goniometry data[END_REF] (Table 10). The estimates given in Table 10 are highly variable and depend fundamentally on the experimental input data chosen in these methodologies. In comparison with those of Tables 7, 8 and 9, the estimates of Ortega et al. (2007) and [START_REF] Keller | Porosity anisotropy of Opalinus Clay: implications for the poroelastic behaviour[END_REF] appear rather representative of aggregates of clay minerals and not of a given single mineral.

Discussion

Tools for the purpose of comparison

This review shows that the estimates obtained for the elastic moduli of clay minerals and, more generally, of clay solid aggregates are expressed in separate ways. Theoretical approaches usually determine the components of the elasticity tensor C ij according to different symmetries, e.g., orthotropic symmetry or transverse isotropic. The experimental approaches aim rather to measure Young's moduli along directions of symmetry or often by considering the clayey solid phase as isotropic. The comparison between all these estimates is therefore not straightforward.

To be able to carry out this comparison, a first approach would consist of "isotropizing" the tensors C ij to reduce all these estimates to a single dataset of moduli {K, G}, where parameters K and G are the bulk modulus and the shear modulus, respectively. In other words, according to [START_REF] Fedorov | Theory of Elastic Waves in Crystals[END_REF], the objective would be to average C ij over all directions to obtain a mean isotropic tensor <C ij > as follows:

(1) Following Voigt's notation, the moduli {K, G} associated with <C ij > can be written as (e.g., [START_REF] Bornert | Homogénéisation en mécanique de matériaux, Tome 1: Matériaux aléatoires élastiques et milieux périodiques[END_REF] 2) and (3) correspond to Voigt's approximations or Voigt moduli (e.g., [START_REF] Hill | The elastic behaviour of a crystalline aggregate[END_REF] and will be referred to hereafter K v and G v . Now, at this stage, all the theoretical estimates of Tables 7 to 10 can be reduced to a set of two equivalent "isotropized" moduli {K v , G v }, which allow us to compare these estimates with each other and obviously with the measured isotropic moduli of Tables 2 to 5. Following [START_REF] Fedorov | Theory of Elastic Waves in Crystals[END_REF], these expressions would provide the best effective bulk and shear moduli, which would minimize the mean square difference between the elastic constants of a given anisotropic material and an equivalent isotropic material characterized by {K v , G v }. However, this approach suffers from two drawbacks. First, the mean moduli {K v , G v } account for only 9 components of 21 independent elastic constants of the tensor C ij in the general case. Second, this formulation implicitly assumes that the strain is uniform throughout the considered representative elementary volume (e.g., [START_REF] Hill | The elastic behaviour of a crystalline aggregate[END_REF]. This is why two other options will be preferred further. The first option will not consider the average process defined by (1) but another averaging process named the Voigt-Reuss-Hill average or the VRH approximation:

(6) (7)
where K v and G v are the Voigt moduli given by Equations ( 4) and ( 5); K R and G R are the Reuss moduli written as follows (e.g., [START_REF] Hill | The elastic behaviour of a crystalline aggregate[END_REF]:

(8) (9)
where S ij are the components of the compliance tensor obtained by inversion of the elasticity stiffness tensor C ij (see Supplementary Material):

(10) Following their definition, the VRH moduli K VRH and G VRH a priori account for the 21 components of tensor C ij .

To compare theoretical estimates and experimental data often expressed in terms of in-plane or out-of-plane Young's modulus, a second option is to note that (see Supplementary Material):

(11) (12) (13)
where E 11 and E 22 are the in-plane Young's moduli (E 11 =E 22 for the transverse isotropic mineral) and E 33 is the out-of-plane modulus, i.e., perpendicular to the basal plane of the mineral. Components S 11 , S 22 and S 33 are determined by numerically inverting the stiffness tensor C ij (Equation (10))

Comparison between theoretical and experimental estimates using VRH moduli and inversion of elasticity tensor

G VRH .
In comparison with the estimates inferred from geophysical approaches displayed in Figure 1, the VRH values given in Figure 5a and Figure 5b show two distinctive features. First, the values of the VRH moduli appear to be less dispersed than those of Figure 1. As explained previously, the petrophysical estimates, in particular those of [START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF] and [START_REF] Mondol | Elastic properties of clay minerals[END_REF], are certainly associated with aggregates of minerals rather than with a single pure clay mineral. It is recalled here that these aggregates integrate complex mechanical interactions (e.g., friction between clay layers possibly associated with buckling processes, mechanical interactions between aggregates themselves) in relation to their spatial organization (spatial distribution of contacts between minerals) that are not taken into account, to our knowledge, in molecular simulations. Second, contrary to Figure 1 and2 are likely representative of aggregates of minerals rather than a pure and single clay mineral. This statement is also confirmed by noting that the bulk modulus K VRH is much higher than the unjacketed bulk modulus K s (Table 5) representative of clay aggregates.

Therefore, it is not justified to directly compare these petrophysical and geomechanical estimates to those obtained by molecular dynamics (e.g., [START_REF] Zhao | Investigation on atomic structure and mechanical property of Na-and Mg-Montmorillonite under high pressure by firstprinciples calculations[END_REF].

Table 12 shows the calculated values of the elastic moduli E 11 , E 22 and E 33 calculated from Equations ( 11), ( 12) and ( 13). E 11 and E 22 are the in-plane Young's moduli (E 11 =E 22 for the transverse isotropic mineral); E 33 is the out-of-plane modulus, i.e., perpendicular to the basal plane of the mineral. scattering, the perpendicular elastic modulus E 33 appears particularly sensitive to the increase in interlayer water layers between 2 water layers and 3 water layers (Figure 7), similar to the out-of-plane elastic constants C 33 , C 44 , C 55 and C 13 (Figure 4). The values of Table 12 displayed in Figure 7 also confirm that the values of the in-plane moduli E 11 and E 22 are close and thus justify the a priori choice of numerous authors to consider clay minerals as transverse isotropic materials (e.g., Alexandrov and Ryzhova, 1964;Ebrahimi et al., 2012;[START_REF] Hedan | Measurement of the elastic properties of swelling clay minerals using the digital image correlation method on a single macroscopic crystal[END_REF].

In a more precise way, the calculated theoretical values of Table 12 can also be compared to the experimental measurements given in Table 6. This comparison shows that the Young's moduli obtained by [START_REF] Prasad | Measurement of Young's modulus of clay minerals using atomic force acoustic microscopy[END_REF] and Carrier et al. (2016) that are much lower than those calculated in Table 12 are likely more representative of the moduli of clay particles or aggregates, i.e., at a larger scale than that of individual clay minerals. Moreover, the experimental estimates of the out-of-plane modulus obtained on macroscopic muscovite crystals [START_REF] Caslavsky | Examination of Imperfect Muscovite Crystals by X-Ray Diffraction Methods[END_REF] and macroscopic 2 water-layer vermiculite [START_REF] Hedan | Measurement of the elastic properties of swelling clay minerals using the digital image correlation method on a single macroscopic crystal[END_REF] are close to those theoretically calculated from the molecular simulations in Table 12. This agreement confirms once more the interest in working on macroscopic analogs of clay minerals. In this context, it should be noted that macroscopic vermiculite crystals considered as analogs of microscopic swelling clay minerals have been successfully used to investigate the diffusion properties of interlayer cations [START_REF] Tertre | Diffusion of water through the dual-porosity swelling clay mineral vermiculite[END_REF][START_REF] Asaad | Role of interlayer porosity and particle organization in the diffusion of water in swelling clays[END_REF].

Elastic properties of interlayer water

This review shows that the elastic properties of swelling clay minerals are highly sensitive to the presence of interlayer water existing in the interlayer space of smectite crystals. The elastic properties of montmorillonite generally decrease as the number of water molecules increases in interlayer space. At this stage, one may wonder about the elastic behavior of this interlayer water. Does this particular water show elastic properties, in particular a compressibility close to that of free water? Or alternatively, does it exhibit physical properties close to those of ice as suggested by many authors (e.g., [START_REF] Kurosaki | The dielectric behavior of sorbed water on silica gel[END_REF]Mc Cafferty et al., 1970;[START_REF] Saarenketo | Electrical properties of water in clay and silty soils[END_REF][START_REF] Cosenza | Electromagnetic determination of clay water content: role of the microporosity[END_REF][START_REF] Jacinto | Influence of water density on the waterretention curve of expansive clays[END_REF] Here, a simple way is proposed to address these issues. Following this simple approach, hydrated montmorillonite mineral is seen as a composite with randomly oriented platelets corresponding to interlayer water layers. This elastic composite is considered an isotropic elastic material whose elastic moduli, i.e., bulk modulus and shear modulus, are given by the VRH average (Table 11). As a result, its constituents, i.e., dry clay minerals and water interlayers, are isotropic elastic materials also allocated a bulk modulus and a shear modulus.

The corresponding VRH elastic moduli of dry minerals (0 water layer) and hydrated minerals (water layers 1, 2 and 3) are provided in Table 11. The elastic moduli of interlayer water, K iw and G iw , which are a priori unknown are determined by back-analyzing the VRH elastic moduli of hydrated minerals using (a) a Monte-Carlo-type procedure as explained further and (b) a simple homogenization scheme, i.e., a Mori-Tanaka scheme considering a composite with randomly oriented platelets [START_REF] Chen | Mori-Tanaka estimates of the overall elastic moduli of certain composite materials[END_REF]:

(14) (15)
where , , and are the bulk modulus of the hydrated tactoid, the shear modulus of the hydrated tactoid, the bulk modulus of the dry mineral, and the shear modulus of the dry mineral, respectively. In practice, the elastic moduli and depend on the considered hydrated state, i.e., the number of water layers, and their values are given in Table 11. The dry moduli and are also given in Table 11 and correspond to the and values at the 0 water layer. By assuming that water is adsorbed in the interlayer space as a uniform layer covering the entire basal space, the adsorbed water content can be approximated as follows:

(16)

where A s is the specific surface area of montmorillonite (expressed in m 2 .kg -1 ), h is the thickness of the interlayer water film in the interlayer space (m), and is the density of the hydrated tactoid (i.e., the mass of dry mineral, m s , divided by the total volume occupied by the dry mineral and the interlayer water, , expressed in kg.m -3 ):

(17)

where , , and are the volume of interlayer water, the mass of interlayer water, the volume of dry minerals, the dry density and the water density, respectively. In the calculations, the water density  w and solid density  s were taken to be equal to 1 g.cm -3 and 2.7 g.cm -3 , respectively. The mass of interlayer water can also be approximated by:

(18)

The combination of ( 17) and ( 18) gives:

(19)

From ( 16) and ( 19), the adsorbed water content can now be calculated as a function of the thickness of the interlayer water film, i.e., at different states of hydration:

(20)

The specific surface area of montmorillonite has been considered equal to 750 m 2 .g -1 [START_REF] Meunier | Clays[END_REF]. The thickness of the interlayer water film in the interlayer space, h, is also estimated from Meunier (2005) (see Table 13). In practice, only two hydrated states (one and two water layers) were considered. The hydrated state associated with three water layers induces a  value greater than 50% (i.e., greater than 48%, see Table 13), which can challenge the validity of a Mori-Tanaka-type scheme rarely used for such high concentrations of heterogeneities (e.g., [START_REF] Christensen | On the range of validity of the Mori-Tanaka method[END_REF].

The back-analysis used to estimate the values of the elastic moduli of interlayer water, K iw and G iw of each considered hydrated state (i.e., 1 or 2 water layers) is a Monte-Carlo-type procedure. This procedure allows the generation of a high number of values or realizations of K iw and G iw (typically one million) in a uniform random fashion between pairs of upper and lower bounds and thus a high number of values of moduli and by using ( 14) and ( 15) and considering one hydration state (i.e., 1 or 2 water layers). The upper and lower bounds for K iw are K w and K s , respectively (K w is the bulk modulus of free water taken equal to 2.5 GPa). The upper and lower bounds for G iw are 0 and G s , respectively. The values of K iw and G iw that were finally accepted corresponded to those that minimized the root-mean-square error (RMSE) calculated as follows:

(

) 21
where values of moduli and are the "observed" and values

given in Table 11 given for hydrated state j (one or two water layers); values of moduli and are the simulated values obtained by Equations ( 14) and ( 15) for the same hydrated state j. Note that the values of elastic moduli of interlayer water, K iw and G iw for each considered hydrated state have been inverted by considering separately the dataset of Ebrahimi et al. (2012) and that of [START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF]. Indeed, these two datasets are the only ones to provide values of elastic properties from the dry state (0 water layer) to the hydrated state, two water layers.

The results of this inversion procedure are displayed in Table 13, which shows very good agreement between the simulated values obtained from Equations ( 14) and ( 15) and the VRH values (i.e., RMS error values lower than 2 10 -2 %). Three comments can be made with respect to these results. First, considering the whole inverted values, it is observed that the bulk modulus values are substantially equal to 21 GPa, except for the significantly higher value (31 GPa) obtained for two water layers from the dataset of Ebrahimi et al. (2012).

Second, if the two datasets of Ebrahimi et al. (2012) and [START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF] are considered separately, it should be noted that the values of the inverted shear moduli vary little with the number of water layers in the interlayer space. In other words, for a solid, the shear behavior of adsorbed water varies little with the hydrated state. Third, regardless of the hydrated and considered datasets, the inverted values of K iw and G iw are significantly larger than those of ice. Indeed, the values of K iw and G iw for ice are in the range of [8][9][10][11] GPa [START_REF] Gow | Linear compressibility of ice[END_REF]Neumeier, 2018) and [3-4] GPa [START_REF] Sinha | Elasticity of natural types of polycrystalline ice[END_REF], respectively. The elastic properties of adsorbed water would therefore be more similar to those of the solid mineral itself than those of ice, as the literature often suggests.

Conclusion

This review confirms that there is a large diversity of experimental and theoretical estimates of the elastic moduli of clay mineral phases. These estimates are obtained experimentally on macrocrystals and macroscopic analogs or theoretically by molecular simulations and micromechanical inversions. These estimates are expressed as isotropic moduli or written following isotropic transverse symmetry.

Here, all the available data are compared using Voigt-Reuss-Hill averages, which account for all the components of the stiffness tensor and allow us to reduce most of these estimates to equivalent isotropic moduli. The qualitative and quantitative comparison of all these data highlights the following four sets of results.

First, the elastic constants of 2:1 nonswelling minerals can be reasonably estimated by molecular modeling of the DFT type. These experimentally validated numerical results can therefore be considered as first order estimates of the elastic coefficients of dry clay minerals.

The models obtained by MD provide much more dispersed results (Figure 3), which can be explained by the variety of local force fields used in the calculations. This review confirms that the mathematical formulation of the optimal force field for clay minerals is still the subject of ongoing important research.

Second, all the estimates obtained by molecular modeling on swelling minerals show that their elastic properties are sensitive to the presence of interlayer water: they globally decrease with the number of water molecules in the interlayer space. However, these theoretical estimates face a major difficulty: their experimental validation. The measurement on macroscopic analogs (e.g., macroscopic vermiculite crystals as analogs of microscopic swelling clay minerals) is again of interest in this context. It is recalled here that such analogues have been successfully used to study the diffusion properties of interlayer water.

Third, most of the values provided by micromechanical inversion of geophysical or geomechanical data acquired on clay rocks or clayey composites and the unjacketed bulk modulus K s values measured in geomechanics are rather representative of clay mineral assemblages and not of a given single clay mineral, as it is often awkwardly considered in literature. Considering these mineral assemblages, the mechanical interactions between minerals or clay aggregates (e.g., friction between clay minerals possibly associated with buckling processes, mechanical interactions between clay aggregates themselves) and the fundamental role of interlayer water, as shown in this review, should be considered. It should be noted that these complex mechanical interactions in relation to the spatial organization of clay minerals have not yet been fully integrated into molecular simulations to our knowledge.

Fourth, the use of a simple micromechanical model applied to the isotropic VRH moduli illustrates the solid nature of interlayer water. The elastic properties of adsorbed water would be more similar to those of the solid mineral itself than those of ice, as the literature often suggests. Once again, these last results should be validated by the experimental measurement of all the elastic constants of hydrated clay minerals, which still remains an experimental challenge.

Table captions

Table 1. Measured single-crystal elastic moduli for muscovite (*: given in [START_REF] Vaughan | Elasticity of muscovite and its relationship to crystal structure[END_REF].

Table 2. Values of elastic and acoustic properties of clay aggregates made from 1:1 clay minerals (kaolinite). ND: Not determined.

Table 3. Values of elastic and acoustic properties of clay aggregates made from nonswelling 2:1 clay minerals (illite and chlorite).

Table 4. Values of elastic and acoustic properties of clay aggregates made from swelling clay minerals (smectite/montmorillonite).

Table 5. Experimental values of the unjacketed bulk modulus measured on different clay rocks.

Table 6. Experimental values of Young's modulus of different clay minerals.

Table 7. Calculated values of elastic constants of 1:1 clay minerals (i.e., kaolinite and dickite).

Table 8. Calculated and measured values of elastic constants of nonswelling 2:1 clay minerals (muscovite, pyrophyllite, illite).

* given in [START_REF] Vaughan | Elasticity of muscovite and its relationship to crystal structure[END_REF].

-: value not indicated or not calculated/measured.

Table 9. Values of elastic constants of swelling 2:1 clay minerals (mainly montmorillonite) calculated by molecular dynamics (except Zhao's data calculated by DFT).

*value calculated by assuming transverse isotropic symmetry.

-: value not indicated or not calculated/measured.

Table 10. Calculated values of elastic constants for the clay matrix considered as transverse isotropic.

Table 11. Elastic moduli calculated following the Voigt-Reuss-Hill (VRH) average. Th: theoretical approaches; Exp: experimental approaches; DFT: density functional theory;

MD: molecular dynamics.

Table 12. The values of Young's moduli in the plane of clay minerals and perpendicular to clay minerals were both calculated from the values of the compliance tensor S ij obtained by experimental approaches and molecular simulations. Th: theoretical approaches; Exp: experimental approaches; DFT: Density functional theory; MD: Molecular.

Table 13. Inverted values of the elastic moduli of interlayer water, K iw and G iw , from two datasets (Ebrahimi et al., 2012 and[START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF].

(1) Input data taken from [START_REF] Meunier | Clays[END_REF] (2) Calculated from Equation ( 19).

(3) Calculated from Equation ( 20).

(4) Input data taken from Table 10.

Figure captions

Figure 1. Bulk modulus as a function of shear modulus experimentally determined for 1:1 and 2:1 clay minerals by the geophysical community (data taken from Tables 2, 3 and4).

Figure 2. Comparison of experimental elastic constants (Table 1) and numerical estimates from Millitzer et al. ( 2011) (Table 8) the whole being obtained on muscovite. 7) and on nonswelling 2:1 clay minerals (muscovite, pyrophyllite, illite seeTable 8). 10. Calculated values of elastic constants for the clay matrix considered as transverse isotropic.

2:1 clay minerals by the geophysical community (data taken from Tables 2, 3 and4). 1:1 clay minerals (kaolinite) [START_REF] Wang | Effective elastic properties of solid clays[END_REF][START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF][START_REF] Mondol | Elastic properties of clay minerals[END_REF] 2:1 nonswelling clay minerals (illite and Chlorite) [START_REF] Wang | Effective elastic properties of solid clays[END_REF] 2:1 swelling smectite-type clay minerals [START_REF] Wang | Effective elastic properties of solid clays[END_REF][START_REF] Vanorio | Elastic properties of dry clay mineral aggregates, suspensions and sandstones[END_REF][START_REF] Mondol | Elastic properties of clay minerals[END_REF] molecular dynamics simulations (MD) on kaolinite (Table 7) and on nonswelling 2:1 clay minerals (muscovite, pyrophyllite, illite seeTable 8). 11). 11) in a loglog scale. For sake of comparison, the values of bulk modulus and shear modulus determined by the geophysical community (Figure 1) are also given in grey and square-cross symbols. 11).

  , VRH estimates do not discriminate the moduli of 1:1 clay minerals, 2:1 nonswelling clay minerals and montmorillonite. The theoretical estimates of muscovite validated by experimental measurements (elastic velocity measurements and Brillouin scattering) are close to those inferred for 1:1 clay minerals (kaolinite and dickite) and dry montmorillonite.

Figure 6 ,

 6 Figure 6, which displays the VRH moduli as a function of the number of interlayer water

  ? For instance, considering water films on -Fe 2 O 3 , Mc Cafferty et al. (1970) suggested three stages at the solid-liquid interface: an immobile layer of adsorbed water linked by hydrogen bound to the hydroxyl layer, a second layer of water molecules that, on average, are singly hydrogen bonded and finally ordered ice-like layers.

Figure 3 .

 3 Figure 3. Comparison between the mean and standard deviation of elastic coefficients C ij obtained by density functional theory (DFT) and molecular dynamics simulations (MD) on kaolinite (Table7) and on nonswelling 2:1 clay minerals (muscovite,

Figure 4 .

 4 Figure 4. Mean values of elastic tensors C ij of swelling clay minerals as a function of the number of water layers in interlayer space. Mean values and corresponding extrema values were theoretically calculated from Mazo et al. (2008b), Ebrahimi et al. (2012) and Carrier et al. (2014). The filled and hollow symbols correspond to elastic constants in the plane of the clay sheet (i.e., C 11 , C 22 and C 12 ) and those normal to the basal plane (i.e., C 33 , C 44 , C 55 , and C 13 ), respectively.

Figure 5a .

 5a Figure 5a. VRH Bulk modulus as a function of VRH shear modulus calculated from experimental measurements and theoretical estimates (linear scale) (data taken from

  Figure 4. Mean values of elastic tensors C ij of swelling clay minerals as a function of the number of water layers in interlayer space. Mean values and corresponding extrema values were theoretically calculated from Mazo et al. (2008b), Ebrahimi et al. (2012) and Carrier et al. (2014). The filled and hollow symbols correspond to elastic constants in the plane of the clay sheet (i.e., C 11 , C 22 and C 12 ) and those normal to the basal plane (i.e., C 33 , C 44 , C 55 , and C 13 ), respectively.

Figure 5a .

 5a Figure 5a. VRH bulk modulus as a function of VRH shear modulus calculated from experimental measurements and theoretical estimates (linear scale) (data taken from Table11).

  Figure5b. VRH bulk modulus as a function of VRH shear modulus calculated from experimental measurements and theoretical estimates (taken from Table11) in a loglog scale. For sake of comparison, the values of bulk modulus and shear modulus determined by the geophysical community (Figure1) are also given in grey and square-cross symbols.

Figure 6 .

 6 Figure 6. VRH bulk modulus as a function of VRH shear modulus calculated from theoretical estimates and determined for different hydrated states of montmorillonite montmorillonite, i.e., for different water layers (see Table11).

Figure 7 .

 7 Figure 7. Mean values of the calculated Young's moduli of swelling clay minerals as a function of the number of water layers in their interlayer space. Mean values and corresponding extrema values were calculated from Mazo et al. (2008b), Ebrahimi et al. (2012) and Carrier et al. (2014). The filled and hollow symbols correspond to Young's moduli in the plane of the clay sheet (i.e., E 11 and E 22 ) and normal to the basal plane (i.e., E 33 ), respectively.

  Or, in the case of transverse isotropic symmetry (C 11 =C 22 , C 44 =C 55 , C 13 =C 23 and C 12 =C 11 -

	(5)
	Equations (
	:
	(2)
	(3)
	2C 66 , see Supplementary Material):
	(4)

Table 11 and

 11 Figures 5a, 5b and 6 show the calculated values of the VRH moduli, K VRH and

Table 11

 11 also shows the values of VRH Young's modulus, E VRH , and VRH Poisson's ratio,  VRH , both calculated from K VRH and G VRH. This finding highlights two results. First, the values of the VRH Poisson's ratio are included in a rather narrow range between 0.22 and 0.34. This could justify the a priori choice of a value of 0.25 by[START_REF] Zhang | Elastic modulus and hardness of muscovite and rectorite determined by nanoindentation[END_REF] and a value of 0.3 by[START_REF] Prasad | Measurement of Young's modulus of clay minerals using atomic force acoustic microscopy[END_REF]. Second, in comparison with Young's modulus estimates given in Tables 1, 2 and 3 (with the exception of chlorite estimates), E VRH values given in Table11are significantly higher. This again confirms that most of the petrophysical estimates of Tables1

Table 12

 12 confirms the two features observed in Table11. First, the calculated values of the elastic moduli E 11 , E 22 and E 33 do not discriminate the moduli of 1:1 clay minerals, 2:1 nonswelling clay minerals and montmorillonite. Second, similar to the VRH moduli, the elastic moduli E 11 , E 22 and E 33 are sensitive to changes in the interlayer water content. These moduli generally decrease with increasing interlayer water molecules. Despite the observed

Table 11

 11

	).
	Figure 5b. VRH Bulk modulus as a function of VRH shear modulus calculated from
	experimental measurements and theoretical estimates (data taken from Table 11) in a
	log-log scale to be compared to Figure 1.
	Figure 6. VRH Bulk modulus as a function of VRH shear modulus calculated from theoretical
	estimates and determined for different hydrated states of montmorillonite
	montmorillonite, i.e., for different water layers (see Table 11).
	Figure 7. Mean values of the calculated Young's moduli of swelling clay minerals as a
	function of the number of water layers in their interlayer space. Mean values and
	corresponding extrema values were calculated from Mazo et al. (2008b), Ebrahimi et
	al. (2012) and Carrier et al. (2014). The filled and hollow symbols correspond to
	Young's moduli in the plane of the clay sheet (i.e., E 11 and E 22 ) and normal to the
	basal plane (i.e., E 33 ), respectively.

Table 2 .

 2 Values of elastic and acoustic properties of clay aggregates made from 1:1 clay minerals (kaolinite). ND: Not determined.

	Young's Modulus E (GPa)	Poisson's ratio 	Bulk modulus K (GPa)	Shear Modulus G (GPa)	P-wave V p (m/s) velocity	S-wave V s (m/s) velocity	Comments Experimental conditions	Reference
	56.8 ± 10	0.28 ± 0.05	44.0 ±8	22.1 ± 2	5488	3010	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Kaolinite, poorly crystallized
	52.0 ± 9	0.32 ± 0.05	47.9 ± 8	19.7 ± 2	5509	2839	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Kaolinite well crystallized
	66.6 ± 5	0.31 ± 0.05	60.1 ± 5	25.3 ± 3	5889	3058	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Illite
	139.6 ± 4	0.36 ± 0.01	163.4 ± 5	51.4 ± 3	9056	4255	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Chlorite (ripidolite)
	202.2 ± 3	0.23± 0.005	127.0 ± 3	81.9 ± 2	9386	5527	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Chlorite, magnesium rich

Table 3 .

 3 Values of elastic and acoustic properties of clay aggregates made from nonswelling 2:1 clay minerals (illite and chlorite).

	Young's Modulus E (GPa)	Poisson's ratio 	Bulk modulus K (GPa)	Shear Modulus G (GPa)	P-wave V p (m/s) velocity	S-wave V s (m/s) velocity	Comments Experimental conditions	Reference
	51.0 ± 7	0.24 ± 0.03	34.7 ± 5	20.3 ± 4	4876	2795	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Na-Montmorillonite
	62.9 ± 3	0.29 ± 0.02	49.6 ± 3	24.4 ± 2	5927	3231	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Ca-Montmorillonite
	44.6 ± 3	0.30 ± 0.01	37.8 ± 3	17.1 ± 1	5169	2746	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Ca-Montmorillonite
	16.6 ± 1	0.20 ± 0.01	9.3 ± 1	6.9 ± 1	2780	1698	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							Smectite rich in Fe
	47.3 ± 2	0.29 ± 0.01	36.7 ± 2	18.4 ± 1	4904	2688	Clay particles diluted, at various	Wang et al. (2001)
							concentrations, in an epoxy matrix
							IIllite-Smectite mixed layer
							(60/40 ordered)
	10	0.23	6	4	2300	1300	Compacted powder of smectite	Vanorio et al.
								(2003)
	14	0.29	11	5.5	2750	1500	Compacted powder of Na-	Vanorio et al.
							Montmorillonite	(2003)
	32.9	ND	12.3	15.6	4640	3310	Dry compacted powder of smectite	Mondol et al.
								(2008)
	22.3	0.21	29.0	7.9	5060	2770	Brine saturated compacted powder of	Mondol et al.
							smectite	(2008)
	815

Table 4 .

 4 Values of elastic and acoustic properties of clay aggregates made from swelling 2:1 clay minerals (smectite/montmorillonite).

	816

Table 6 .

 6 Experimental values of Young's modulus of different clay minerals.

	821
	Elastic constants
	(GPa)

  Figure 2. Comparison of experimental elastic constants (Table 1) and numerical (DFT) estimates from Millitzer et al. (2011) (Table 8) the whole being obtained on muscovite. Elastic constants C 11 , C 22 , C 33 , C 44 , C 55 , C 66 , C 12 , C 13 and C 23 have been considered.

		Alexandrov and Ryzhova (1961)
		Vaughan and Guggenheim (1986)
		McNeil and Grinsditch (1993)
			)
		100
	modulus, K (GPa)	Experimental C ij (GPa)
	Bulk
		10
		10	100
		Numerical (DFT) C
		Shear modulus, G (GPa)

ij

(GPa)

Acknowledgments. The authors acknowledge financial support from the European Union (ERDF) and "Région Nouvelle Aquitaine".

 13. Inverted values of the elastic moduli of interlayer water, K iw and G iw , from two datasets (Ebrahimi et al., 2012 and[START_REF] Carrier | Elastic Properties of Swelling Clay Particles at Finite Temperature upon Hydration[END_REF].

(1) Input data taken from [START_REF] Meunier | Clays[END_REF] (2) Calculated from Equation ( 19).

(3) Calculated from Equation (20).

(4) Input data taken from