# A multi-scale modeling of grouted sand behavior

Pierre-Yves Hicher, Ching Chang, Christophe Dano

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Pierre-Yves Hicher, Ching Chang, Christophe Dano. A multi-scale modeling of grouted sand behavior. International Journal of Solids and Structures, 2008, 45 (16), pp.4362-4374. 10.1016/j.ijsolstr.2008.03.024. hal-01004899

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# Multi-scale modeling of grouted sand behavior

Pierre-Yves Hicher<sup>a</sup>, Ching S. Chang<sup>b</sup>, Christophe Dano<sup>a</sup>

<sup>a</sup>Research Institute in Civil and Mechanical Engineering, UMR CNRS, Ecole Centrale Nantes, Université de Nantes, Nantes, France

<sup>b</sup>Department of Civil Engineering, University of Massachusetts, Amherst, MA 01003, USA

The mechanical properties of sand: stiffness, cohesion and, to a less extent, friction angle can be increased through the process of grouting. A constitutive model adapted for cohe-sive-frictional materials from a homogenization technique which allowed us to integrate constitutive relations at the grain level has been developed to obtain constitutive equations for the equivalent continuous granular medium. A representative volume was obtained by mobilizing particle contacts in all orientations. Thus, the stress-strain relationship could be derived as an average of the behavior of these local contact planes. The local behavior was assumed to obey a stress-dependent elastic law and Mohr-Coulomb's plastic law. The influence of the cement grout was modeled by means of adhesive forces between grains in contact, which were added to the contact forces created by an external load. The intensity of these adhesive forces is a function of nature and amount of grout present inside the material and can be reduced due to a damage mechanism at the grain contact during loading. In this paper, we present several examples of simulation which show that the model can reproduce with sufficient accuracy the mechanical improvement induced by grouting as well as the damage of the grain cementation during loading.

Keywords:

Grouted sand

Microstructural model

Stress-strain relationship

Cementation damage

## 1. Introduction

Microstructural models for inelastic stress-strain behavior of granular material can be derived from properties of inter-particle contacts. If we take the mean behavior of all contacts in each orientation, the overall stress strain behavior can be obtained as an average of the contact behavior for all orientations. The basic idea is to view the packing as represented by a set of micro-systems. The inelastic behavior of each micro-system is characterized and the overall stress-strain relationship of the packing is obtained from an average of the behaviors of all the micro-systems. Micro-systems can be regarded as inter-particle planes (or mobilized planes) in the packing. Models based on inter-particle contact planes can be found in Jenkins and Strack (1993), Matsuoka and Takeda (1980), Chang et al. (1989), etc. An alternative way is to view the micro-systems as particle groups of different configurations. Models developed along this line can be found in Chang et al. (1992a,b), and more recently in Suiker and Chang (2004).

The microstructural plasticity model requires a mathematical or numerical description of the micro-systems, which is not readily available in the case of particle groups. But, in order to treat micro-systems as contact or mobilized planes, the approach of estimating the overall behavior by oriented planes can be linked to G.I. Taylor's (1948) concept, developed long ago in the slip theory of plasticity for polycrystalline materials by Batdorf and Budianski (1949). These ideas were applied by Pande and Sharma (1982) to rocks and soils in what they called the overlay model, and to concrete by Bazant et al. (1995) in the so-called microplane model.
Along these lines, we have developed a new stress-strain model which considers inter-particle forces and displacements (Chang and Hicher, 2005). A micro-macro link has been incorporated into the model between strain and inter-particle dis-placements for granular materials (Liao et al., 1997). By comparing predicted and measured triaxial loading results for sands of different void ratios and under different confining stresses, in both drained and undrained conditions, we demonstrated the ability of this model to reproduce the main features of fully saturated or dry sand behavior (Chang and Hicher, 2005).

The construction of underground structures on soft ground often requires the soil to be improved in order to ensure the safety and the stability of surrounding buildings. Grout permeation is an efficient technique for reducing permeability and for increasing stiffness and strength of coarse-to-medium grained soils with low initial mechanical properties. Since tradi-tional organic grouts, such as silicate gels, have proved to be hazardous for groundwater, they have been prohibited over the years. New grouts, therefore, such as very fine cement suspensions and mineral grouts, have been developed for about the past 20 years (Zebovitz et al., 1989; Benhamou, 1994). These new grouts present similar groutability performance as do pre-vious chemical solutions. Their considerable advantage is that they prove to be stable with time. We can therefore take into account their improved mechanical properties in design. Experimental studies have shown that cementation in sands, including grouted sands, influences their mechanical properties by adding cohesion and tensile strength, increasing stiffness, without changing significantly the friction angle (Airey, 1993; Clough and Sitar, 1981; Dano, 2001; Dano et al., 2004).

In order to be able to model the mechanical behavior of cohesive-frictional material, we have extended the capability of the microstructural plasticity model by including internal forces at the grain contacts in order to reproduce the effect of grain adhesion. In this paper, the adhesive forces simulate the effects of a cement grout incorporated inside the pores of the gran-ular material.

## 2. Microstructural model

In this model, we envision a granular material as a collection of contact particles. The deformation of a representative volume of the material is generated by the mobilization of contact particles in all orientations. Thus, the stress-strain relationship can be derived as an average of the mobilization behavior of local contact planes in all orientations. For contact planes in the $\alpha$th orientation, the local forces $f_j^x$ and the local movements $\delta_i^x$ can be denoted as follows: $f_j^x = \{f_n^x, f_s^x, f_t^x\}$ and $\delta_i^x = \{\delta_n^x, \delta_s^x, \delta_t^x\}$, where the subscripts n, s and t represent the components in the three directions of the local coordinate system as shown in Fig. 1. The direction normal to the plane is denoted as $\mathbf{n}$; the other two orthogonal directions, $\mathbf{s}$ and $\mathbf{t}$, are tangential to the plane.

### 2.1. Inter-particle behavior

#### 2.1.1. Internal adhesive forces

In order to take into account the effect of cement grout in the pores of the granular material, adhesive forces were added at each contact point to the local forces determined from the external stresses $\sigma$ applied on the granular assembly. The amplitude of those forces depends on the nature of the grout and on the concentration of the grout in cement particles. Extensive experimental work performed by Dano (2001) demonstrated the influence of these two parameters on the response of the grouted sand. Fig. 2 presents the evolution of the unconfined compressive strength $R_c$ of Fontainebleau sand at different density indexes grouted by a cement grout called Intra-J at different cement-to-water mass ratios C/W (see grain size distribution curve of the cement used in Intra-J grout in Fig. 3). One can see that the compressive strength of the grouted sand follows the same trend as the pure grout.

The amplitude of the adhesive force $f_{ad}$ has been correlated to the nature of the cement grout by taking a fraction of the unconfined strength of specimens of pure cement grout prepared at a given concentration C/W. SEM analysis of grouted sand specimens shows that the grout is not equally distributed inside the sand pores (Fig. 4). The adhesive force $f_{ad}$ can there-

<figure><img src="image_2.png" /><figcaption>Fig. 1. Local coordinate at inter-particle contact.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2. Evolution of the unconfined compressive strength $R_c$ with C/W.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. Grain size distribution curves of sand and grout.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. SEM photos of grouted Fontainebleau sand sample.</figcaption></figure>

fore represent only a statistical mean value of each adhesive force measured at each contact point. This mean value increases when the amount of cement increases inside the pores, due to either a higher cement concentration of the grout or the filtration of the cement particles during injection (see Section 5). Based on the results presented in Fig. 4, the compressive strength of the grout, $R_{c,PG}$, is related to the cement-to-water ratio, C/W, as follows:

$$R_{c,PG} = A_0 \times \left(\frac{C}{W}\right)^B \qquad (1)$$
The values of the two parameters A₀ and B are determined by matching the experimental data. In the case of the microfine cement grout Intra-J, A₀ was found equal to 45.3 MPa and B close to 2.2.

### 2.1.2. Elastic part

The contact stiffness of an orientation includes normal stiffness, $k_n^x$, and shear stiffness, $k_r^x$, of the contact plane. The elastic stiffness tensor is defined by

$$f_j^x = k_{ik}^{xe} \delta_k^{xe} \quad (2)$$

which can be related to the contact normal and shear stiffness

$$k_{ik}^{xe} = k_n^x n_i^x n_k^x + k_r^x (s_n^x s_k^x + t_i^x t_k^x) \quad (3)$$

The value of the stiffness for two elastic spheres can be estimated from Hertz-Mindlin's formulation (Mindlin, 1969). For sand grains, a revised form can be adopted, given by

$$k_n = k_{n0} \left( \frac{f_n}{G_g l^2} \right)^n; \quad k_t = k_{t0} \left( \frac{f_n}{G_g l^2} \right)^n \quad (4)$$

where $G_g$ is the elastic modulus for the grains, $f_n$ is the contact force in normal direction, $l$ is the branch length between the two particles, $k_{n0}$, $k_{t0}$ and $n$ are material constants.

The presence of adhesive forces on each contact plane affect the elastic properties given by Eq. (4). Effects of adhesive forces on the contact normal stiffness have been studied by Chang and Misra (1990). They showed that the normal stiffness of two bonded spheres could be expressed as a function of the normal force at contact and the adhesive force. Since, in our case, sand grains cannot be assimilated to spheres and coating is not regularly distributed within the grain assembly, we decided to adopt a simplified form of the stiffness considering adhesive force

$$k_n = k_{n0} \left( \frac{f_n^*}{G_g l^2} \right)^n \quad (5)$$

$$\text{with } f_n^* = f_n + \xi f_{ad} \quad (6)$$

where $\xi$ represents the relative influence of the adhesive force, function of the surface area of two neighbouring particles coated by the grout. $\xi$ increases with the degree of cementation. At low degrees of cementation, the bond area is relatively small as the cement coats the particles to form weak bonds between two neighbouring particles. With the addition of cement, the bond area increases and the bonds become stronger, contributing to a higher stiffness of the grouted sand.

Shear modulus of Fontainebleau sand grouted by Intra-J cement was measured by Dano (2001). For unconfined grouted samples, the elastic shear modulus $G_{max}$ was found to be a function of the cement-to-water ratio of the grout (Fig. 5).

$$G_{max}(\text{MPa}) = 26.8 \left(\frac{C}{W}\right)^{1.24} \quad (7)$$

For computing the stiffness of an unconfined grouted sand (i.e., null external loading), we set $f_n = 0$ in Eqs. (5) and (6). The initial stiffness is only due to the existence of adhesive forces from grouting, thus we obtain an elastic modulus proportional to $(f_{ad})^n$. Let us assume that the adhesive force $f_{ad}$ is proportional to the unconfined strength of the pure grout $R_{c,PG}$. Therefore, the elastic stiffness of a grouted sand is proportional to $(C/W)^{Bn}$. A value of B close to 2.2 was found for Intra-J grout (Eq. (1)).

<figure><img src="image_6.png" /><figcaption>Fig. 5. Elastic shear modulus of unconfined grouted Fontainebleau sand.</figcaption></figure>
Since $n$ was found equal to 0.5 for Fontainebleau sand, as for most of the sands, the unconfined elastic modulus of Fontainebleau grouted sand could be computed as proportional to $C/W$ at the power 1.1. This value can be compared to the measured value equal to 1.24. Considering the amount of dispersion in the measurement of $R_{c,PG}$, especially at high $C/W$ values, the proposed approach seems to give satisfactory results for determining elastic properties of cemented sands.

### 2.1.3. Plastic part

The elastic behavior does not have a coupling effect (i.e., there is no shear induced by normal movements). However, plastic sliding does not necessarily occur along the tangential direction of the contact plane. The sliding direction may be upward or downward, and the shear dilation/contraction takes place simultaneously. The dilatancy effect can be described by

$$ \frac{d\delta_n^p}{d\Delta^p} = \frac{T}{f_n} - \tan \phi_0 \qquad (8) $$

where $\phi_0$ is a material constant which, in most cases, can be considered equal to the internal friction angle $\phi_\mu$. This equation can be derived by assuming that the dissipation work for a contact plane due to both normal and shear plastic movements ($f_n\delta_n^p + T d\Delta^p$) is equal to the energy loss due to friction ($f_n \tan \phi_0 \Delta^p$) at the contact. Note that the generalized shear force $T$ and the rate of plastic sliding $d\Delta^p$ can be defined as

$$ T = \sqrt{f_s^2 + f_t^2} \quad \text{and} \quad d\Delta^p = \sqrt{(d\delta_s^p)^2 + (d\delta_t^p)^2} \qquad (9) $$

The yield function is assumed to be of Mohr-Coulomb type,

$$ F(f_j, \kappa) = T - f_n \kappa (\Delta^p) = 0 \qquad (10) $$

where $\kappa(\Delta^p)$ is an isotropic hardening parameter. The material behaves elastically when $F(f_j, \kappa) < 0$. Whereas $F(f_j, \kappa) > 0$ is inadmissible, Eq. (10) must be satisfied throughout the plastic response of the work hardening material. The work hardening function is defined by a hyperbolic curve in $\kappa - \Delta^p$ plane, which involves two material constants: $\phi_p$ and $k_{p0}$

$$ \kappa = \frac{k_{p0} \tan \phi_p \Delta^p}{|f_n| \tan \phi_p + k_{p0} \Delta^p} \qquad (11) $$

Eq. (10) means that when the shear contact force ratio $T/f_n$ increases, the plastic sliding $\Delta^p$ occurs. Thus, for an isotropically consolidated specimen, plastic sliding occurs immediately after the shear force is generated. Eq. (11) is a hyperbolic function asymptotic to the value of $\tan \phi_p$. It is noted that the hyperbolic function has been adopted to describe the shear behaviour of stress and strain at macro level (see, e.g., Duncan and Chang, 1970). It is assumed that the hyperbolic curve can also describe the shear behaviour at inter-particle contact level.

### 2.1.4. Interlocking influence

An important factor in granular modelling is the critical state concept. Under critical state, the granular material will remain at constant volume while it is subjected to a continuous distortion. The void ratio corresponding to this state is $e_c$.

The critical void ratio $e_c$ is a function of mean stress. The relationship has traditionally been written as follows:

$$ e_c = \Gamma - \lambda \log(p') \quad \text{or} \quad e_c = e_{\text{ref}} - \lambda \log \left( \frac{p'}{p_{\text{ref}}} \right) \qquad (12) $$

$\Gamma$ and $\lambda$ are two material constants, $p'$ is the mean stress of the packing, and $(e_{\text{ref}}, p_{\text{ref}})$ is a reference point on the critical state line.

The internal friction angle $\phi_\mu$ is a constant for the material. However, the peak friction angle, $\phi_p$, on a contact plane depends on the degree of interlocking by neighbouring particles, which can be related to the state of packing void ratio $e$ by

$$ \tan \phi_p = \left( \frac{e_c}{e} \right)^m \tan \phi_\mu \qquad (13) $$

where $m$ is a material constant (Biarez and Hicher, 1994).

For dense packing, the peak frictional angle $\phi_p$ is greater than $\phi_\mu$. When the packing structure dilates, the degree of interlocking and the peak frictional angle are reduced, which results in a strain-softening phenomenon.

### 2.1.5. Damage law for the bonds

The evolution of the shear modulus during a triaxial loading on a grouted specimen was measured using bender elements technique (Dano and Hicher, 2003). After an initial phase of constant stiffness, the results showed a continuous decrease of the shear modulus, which was analysed as a damage of the bonded contacts (Fig. 6). Therefore, a damage law was introduced in the expression of the adhesive force amplitude, expressed as follows:

$$ f_{\text{ad}} = f_{\text{ad}}^0 e^{-\eta_b(\rho - \delta_b)} \quad \text{for} \quad \rho > \delta_b \qquad (14) $$

$f_{\text{ad}}^0$ being the initial adhesive force, and $\eta_b$ the factor of damage.

$\rho$ is an equivalent local displacement expressed as
<figure><img src="image_7.png" /><figcaption>Fig. 6. Evolution of elastic shear modulus during triaxial loading.</figcaption></figure>

$$ \rho = \sqrt{(\beta \delta_n^t)^2 + \Delta^2} \qquad (15) $$

where $\delta_n^t$ correspond to the normal displacement in elongation. The value of $\delta_n^t$ is otherwise considered equal to zero. This hypothesis concerning the influence of the normal displacement in the damage law is linked to the fact that the tensile strength of the grout is only a small fraction of its compressive strength, which indicates that the rate of damage is much more important in traction than in compression, as in all cementitious materials (Dano et al., 2004).

The parameter $\beta$ represents the relative influence of the normal displacement versus the tangential displacement in the damage law. Usually, high values of $\beta$ (>10) are selected, due to a much greater influence of the tensile displacement compared to the tangential displacement on the grout damage.

$\delta_b$ corresponds to the limit value of $\rho$, under which no damage takes place (initial phase of constant stiffness).

Therefore, each contact produces a cohesive-frictional behavior. During loading, progressive damage of the cementitious bond leads to a decrease of adhesive forces, while the frictional shear resistance is progressively mobilized. Fig. 6 shows the model's response after the parameters introduced in the damage law, obtained by curve fitting, were determined (Table 2).

### 2.1.6. Elasto-plastic relationship

With the elements discussed above, a final incremental stress-strain relations of the material can be derived that includes both elastic and plastic behavior, given by

$$ \dot{f}_j^z = k_{ik}^{zp} \delta_k^z \qquad (16) $$

Detailed expression of the elasto-plastic stiffness tensor is given in Chang and Hicher (2005).

### 2.2. Stress-strain relationship

The stress-strain relationship for an assembly can be determined from integrating the behavior of inter-particle contacts in all orientations. In the integration process, a micro-macro relationship is required. Using the static hypotheses, we obtain the relation between the global strain and inter-particle displacement (we do not consider the finite strain condition)

$$ \dot{u}_{j,i} = A_{ik}^{-1} \sum_{z=1}^{N} \dot{\delta}_{j}^{z} l_{k}^{z} \qquad (17) $$

where the branch vector $l_k^z$ is defined as the vector joining the centres of two particles, and the fabric tensor is defined as

$$ A_{ik} = \sum_{z=1}^{N} l_{i}^{z} l_{k}^{z} \qquad (18) $$

The mean force and moment on the contact plane of each orientation are

$$ \dot{f}_j^z = \dot{\sigma}_{ij} A_{ik}^{-1} l_k^z V \qquad (19) $$

In Eq. (19), the stress increment can be obtained by the contact forces and branch vectors for all contacts (Christofferson et al., 1981; Rothenburg and Salvadurai, 1981), as follows:

$$ \dot{\sigma}_{ij} = \frac{1}{V} \sum_{z=1}^{N} \dot{f}_{j}^{z} l_{i}^{z} \qquad (20) $$
The problem is defined as follows:

Initially, we know the global variables ($\sigma_{ij}$ and $\epsilon_{ij}$) for the assembly and the local variables ($f_j^x$ and $\delta_j^x$) for each contact orientation. For any given loading increment, which can be stress control, strain control or mixed mode, 6 out of the 12 variables ($\Delta\sigma_{ij}$ and $\Delta\epsilon_{ij}$) are unknown. The objective is therefore to determine all global variables ($\sigma_{ij}$ and $\epsilon_{ij}$) and local variables ($f_j^x$ and $\delta_j^x$) at the end of each load increment. For a system with N inter-particle orientations, the number of unknown is $3N$ for $f_j^x$ and $3N$ for $\delta_j^x$. The total number of unknown is $3N + 3N + 6$.

The following equations must be satisfied:

(1) The local constitutive equation, i.e., Eq. (16): since there are three equations for each contact plane orientation, the total number of equations is $3N$: N being the total number of inter-particle orientations.

(2) Static hypothesis between global stress and local forces, i.e., Eq. (19): the number of equations is $3N$.

(3) Strain definition between global strain and local displacement, i.e., Eq. (17). The number of equations is 6 (strain is symmetric).

The total number of unknowns is the same as the total number of equations. Therefore, a solution can be determined. Using Eqs. (16), (17) and (19), the following relationship between stress increment and strain increment can be obtained

$$ \dot{u}_{i,j} = C_{ijmp} \dot{\sigma}_{mp}; \quad \text{where} \quad C_{ijmp} = A_{ik}^{-1} A_{mn}^{-1} V \sum_{\alpha=1}^{N} (k_{jp}^{ep})^{-1} l_k^{\alpha} l_n^{\alpha} \quad (21) $$

When the contact number N is sufficiently large in an isotropic packing, the summation of flexibility tensor in Eq. (21) and the summation of fabric tensor in Eq. (18) can be written in integral form, given by

$$ C_{ijmp} = A_{ik}^{-1} A_{mn}^{-1} \frac{NV}{2\pi} \int_{0}^{\pi/2} \int_{0}^{2\pi} k_{jp}^{ep}(\gamma, \beta)^{-1} l_k(\gamma, \beta) l_n(\gamma, \beta) \sin\gamma \, d\gamma \, d\beta; \quad \text{and} \quad (22) $$

$$ A_{ik} = \frac{N}{2\pi} \int_{0}^{\pi/2} \int_{0}^{2\pi} l_{i}(\gamma, \beta) l_{k}(\gamma, \beta) \sin\gamma \, d\gamma \, d\beta \quad (23) $$

The integration of Eqs. (22) and (23) in a spherical coordinate can be carried out numerically using Gauss integration points over the surface of the sphere. The locations of integration points in spherical coordinates represent discretized contact orientations; each orientation has a weighting factor. As schematically illustrated in Fig. 7, each integration point may represent an area of triangle on the surface of the sphere. We found that the results are more accurate by using a set of fully symmetrical integration points. From a study of the performance of different numbers of integration points, we found 37 points to be adequate.

### 3. Numerical simulations of triaxial tests on unbound sand

One can summarize the material parameters as:

- Normalized contact number per unit volume: $NI^3/V$;

- mean particle size, $d = 2R$;

- Inter-particle elastic constants: $k_{n0}$, $k_{r0}$ and $n$;

- Inter-particle friction angle: $\phi_{\mu}$ and $m$;

- Inter-particle hardening rule: $k_{p0}$ and $\phi_0$;

- Critical state for packing: $\lambda$ and $\Gamma$ or $e_{ref}$ and $p_{ref}$.

The contact number per unit volume changes during the deformation. According to the experimental data by Oda (1977) for three mixtures of spheres, the contact number per unit volume can be approximately related to the void ratio by

<figure><img src="image_8.png" /><figcaption>Fig. 7. Schematical illustration of discretization of the surface area for spherical integration.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 8. Numerical simulations of triaxial tests on Fontainebleau sand.</figcaption></figure>

<table><caption>**Table 1**

Material parameters for Fontainebleau sand</caption><thead><tr><th>e<sub>ref</sub></th><th>p<sub>ref</sub> (kPa)</th><th>λ</th><th>φ<sub>μ</sub> (°)</th><th>φ<sub>0</sub> (°)</th><th>m</th></tr></thead><tbody><tr><td>0.95</td><td>10</td><td>0.03</td><td>33</td><td>29</td><td>0.5</td></tr></tbody></table>

$$ \frac{N}{V} = \frac{3}{2\pi r^3 (1 + e)e} \qquad (24) $$

This equation is used to account for the evolution of contact number per unit volume.

Elastic parameters can be determined by using a standard procedure (Hicher and Chang, 2006). Plastic and frictional parameters were determined from experimental results on ungrouted sand. Other than critical state parameters, all the plastic parameters are related to inter-particle behavior. Standard values for $k_{p0}$ are the following: $k_{p0} = k_{n0}$. Therefore, for dry or saturated samples, only six parameters have to be determined from experimental results, which can all be determined from the stress-strain curves obtained from drained or undrained compression triaxial tests.

Drained triaxial tests have been performed by Dano (2001) on dense Fontainebleau sand (density index $I_D$ = 0.85). Fontainebleau sand is a uniform fine siliceous sand with sub-angular particles of mean size $d_{50}$ = 0.22 mm (see grain size distribution curve in Fig. 3). Fig. 8 presents the results of three triaxial tests at three different confining stresses. Numerical simulations were performed using the set of material parameters summarized in Table 1. The predictions show good comparison with experimental results up to 4% of strain. After 4% strain, the experimental data show a moderate softening behavior, which may be due to strain localization. The present constitutive model is for a material point, which is not capable to capture non-homogeneous strain within a specimen.

## 4. Numerical simulations of triaxial tests on grouted sands

We present in Figs. 9 and 10 two examples of Fontainebleau sand ($I_d$ = 0.95) samples reinforced by cement grout Intra-J at two different concentrations $C/W$ = 0.17 and 0.23. The influence of the cement content in the sand pores can be seen by the increase of the material strength with the increase of $C/W$ at every confining stress.

The macroscopic cohesion $c$ due to cemented intergranular bonds and the friction angle $φ$ due to inter-particle frictional contacts are determined by plotting the maximum strength envelope in the Mohr-Coulomb diagram (Fig. 11). For unconfined granular soils, one obtains a straight-line failure envelope with zero cohesion. For grouted sands, a straight-line failure envelope is obtained.
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Grouted Fontainebleau sand by Intra-J C/W = 0.175.</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Grouted Fontainebleau sand by Intra-J C/W = 0.23.</figcaption></figure>

ure envelope that is almost parallel to that of the uncemented granular soil is obtained. In other words, the value of the friction angle is approximately the same, probably since permeation grouting with low injection pressures does not cause any disturbance of the particle assembly. SEM photos (Fig. 4) showed that the grouting procedure, when realized at small or
<figure><img src="image_14.png" /><figcaption>Fig. 11. Mohr-Coulomb maximum strength envelopes for pure and grouted Fontainebleau sand.</figcaption></figure>

<table><caption>**Table 2**
Material parameters for Intra-J grout C/W = 0.175</caption><thead><tr><th>f<sub>ad</sub><sup>0</sup></th><th>ξ</th><th>η<sub>b</sub></th><th>δ<sub>b</sub></th><th>β</th></tr></thead><tbody><tr><td>0.01</td><td>20</td><td>0.003</td><td>0.0001</td><td>40</td></tr></tbody></table>

<table><caption>**Table 3**
Material parameters for Intra-J grout C/W = 0.23</caption><thead><tr><th>f<sub>ad</sub><sup>0</sup></th><th>ξ</th><th>η<sub>b</sub></th><th>δ<sub>b</sub></th><th>β</th></tr></thead><tbody><tr><td>0.015</td><td>25</td><td>0.003</td><td>0.0001</td><td>40</td></tr></tbody></table>

moderate values of the injection pressure, does not modify the initial position of the grains. The grout is located inside the pores, but does not penetrate the contact area between two grains. This can explain the minor influence of grouting on the frictional properties of grouted sands. Similar results have been found by Dupas and Pecker (1979) on mortars, and by Clough and Sitar (1981) on natural weakly cemented sands. The slight increase of the friction angle can be associated to the decrease of the void ratio with grout impregnation. The macroscopic cohesion c, corresponding to the intersection of the maximum strength envelope with the axis of the ordinates in the Mohr plane, increases with the cement-to-water ratio C/W of the grout.

Numerical simulations were performed with the set of elasto-plastic parameters determined in Table 1 for non-cemented Fontainebleau sand. The additional five parameters for the grout behavior are

f<sub>ad</sub><sup>0</sup>: initial adhesive force;

ξ: mobilised bonded surface;

η<sub>b</sub>: damage factor;

δ<sub>b</sub>: limit strain value of induced damage;

β: relative influence of elongated displacement versus sliding on grout damage.

They were determined from the stiffness decay curves (see example in Fig. 6). They are presented in Tables 2 and 3. A comparison with experimental results in Figs. 9 and 10 shows that the model is capable of reproducing with sufficient accuracy the mechanical behavior of grouted Fontainebleau sand. The difference in initial stiffness between experimental and numerical results is mainly due to the compliance of the loading frame which induced higher measured axial displacement. Its influence can also be seen on the beginning of the ε<sub>1</sub>-ε<sub>v</sub> curves.

## 5. Influence of cement content

Maalej (2007) studied the microstructure of the material using scanning electron microscopy and mercury porosimetry. He was able to quantify very precisely the amount of cement grout inside the sand pores after injection. For a given cement grout at a given concentration C/W, this amount can vary during the grouting process, due to the filtration of cement particles inside the sand pores. As a consequence, the concentration of the cement particles is higher near the point of injection and decreases when the distance towards the injection point increases.

Figs. 12 and 13 present experimental results and numerical simulations, respectively, for grouted Fontainebleau sand specimens with different cement contents under a confining pressure equal to 0.8 MPa. The volumetric cement concentration varied from 0.22 in specimen 1 to 0.075 in specimen 5. The adhesive forces could be correlated to the cement content using a single linear relation
<figure><img src="image_15.png" /><figcaption>Fig. 12. Grouted Fontainebleau sand. Experimental results for different grout concentrations.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. Grouted Fontainebleau sand. Numerical results for different grout concentrations.</figcaption></figure>

$$f_{ad}^{0} = 0.15\zeta (f \text{ in N}) \qquad (25)$$

$\zeta$ being the volumetric cement concentration (if $e_0$ is the initial sand void ratio and $e_f$ the void ratio after grouting, $\zeta = e_0 - e_f$).
<figure><img src="image_17.png" /><figcaption>Fig. 14. Grout concentration at a given time after the beginning of the injection process.</figcaption></figure>

This linear relation is in agreement with the results presented by Maalej (2007), which has showed that a linear relation
could be found between the macroscopic cohesion c and the cement content ζ. It is therefore possible to predict the mechan-
ical properties of an *in situ* grouted sand by taking into account the filtration process around the point of injection.

Chupin et al. (2003, in press) developed a model of transport with filtration in saturated porous media. The model is based on the advection-dispersion equation of miscible fluids and considers a phenomenological approach for the purpose of filtration. The permeability reduction of the porous medium, as a result of the filtration phenomenon, depends on the concentration of the filtrated component. The kinetic equation of filtration is assumed to be a function of the transported component. The resulting coupled system of equations was discretized by finite elements. Fig. 14 shows an example of a numerical simulation of grout propagation with filtration condition inside a sand volume. The cement grout was injected at a given point of the medium and the grout propagation was measured as a function of time. The figure shows that the grout concentration is higher near the point of injection due to the filtration of cement grains inside the sandy medium.

**6. Conclusion**

We have adapted a microstructural model developed for non-cohesive granular materials to the mechanical behavior of grouted sand. The model is based on the description of the inter particulate contact law which requires a simple relation between the vectors of forces and the relative displacements on a contact plane. Deformation of a representative volume is obtained by mobilizing particle contacts in all orientations. Thus, the stress-strain relationship can be derived as an average of the behavior of these local contact planes.

The introduction of grouting influence was made by introducing an adhesive force at each grain contact, function of the
nature and concentration of the cement grout. Experimental results demonstrated that the intensity of these adhesive forces
decreased during mechanical loading. Therefore, a damage mechanism was introduced at each grain contact. The intensity of
the damage was taken as a function of the relative displacement of two contacting grains.
Numerical simulations of triaxial tests demonstrated the ability of this new version of the model to reproduce the
mechanical properties of grouted Fontainebleau sand with various cement grout concentrations. An original aspect of this
multi-scale modeling is the model's capacity to take into account the influence of cement concentration in a simple manner.
If coupled to a numerical model of grout propagation with filtration, the model can provide the spatial evolution of the
mechanical properties of a sand layer impregnated by a cement grout, employing ordinary *in situ* injection techniques.

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