W Q Shen

Y J Cao

Z B Liu

J F Shao

A multiscale elastoplastic constitutive model for geomaterials with a porous matrix-inclusion microstructure

Keywords: Geomaterials, Pores and grains, Drucker-Prager, Macroscopic criterion, Multi-scale modeling

. This criterion is applied to describe the peak stresses of Callovo Oxfordian argillite obtained by the uniaxial and triaxial compression tests with different compositions and different confining pressures. Then, a complete constitutive model is established with a plastic hardening behavior and the non-associated plastic flow rule. The evolutions of the microstructure, such as the variations of the porosity and the volume fraction of inclusions with the loading process, can be fully considered by this micromechanics based model. By comparing the numerical results with the experimental data, the proposed model is able to capture the main features of the studied geomaterial with a porous matrix-inclusion microstructure.

Introduction

For heterogeneous geomaterials, porosity and volume fraction of mineral grains are two common characteristics which affect significantly their macroscopic mechanical behaviors.

According to many experimental and site observation, the composition of heterogeneous geomaterials, such as pore and mineral grains vary in space, for instance with the depth. In regard to the stability and safety of the engineering applications, it is particularly important and meaningful to study these influences by experimental methods or analytical and numerical approaches. During the last decades, many researchers have focused on the micromechanics based modellings to take into account the influence of material microstructure and also their evolutions on the macroscopic mechanics behaviors. This is the main advantage comparing with the phenomenological modellings. For example, to consider the effect of pore in porous materials, analytical yield criteria have been derived in [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: part1-yield criteria and flow rules for porous ductile media[END_REF] for a hollow sphere or cylinder with von Mises matrix in the framework of limit analysis. The effect of porosity was explicitly considered. Based on these famous works, many extensions have been done: a heuristic extension of the Gurson model [START_REF] Tvergaard | Influence of voids on shear bands instabilities under plane strain conditions[END_REF][START_REF] Tvergaard | Material failure by void growth and coalescence[END_REF] known as the GTN model which is widely used in structural computations; considering the porosity, pore shape and size effects: [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids-cas of axisymmetric prolate ellipsoidal cavities[END_REF][START_REF] Gologanu | Approximate models for ductile metals containing nonspherical voids-cas of axisymmetric oblate ellipsoidal cavities[END_REF][START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals[END_REF][START_REF] Pardoen | An extended model for void growth and coalescence[END_REF][START_REF] Monchiet | Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids[END_REF][START_REF] Monchiet | Macroscopic yield criteria for ductile materials containing spheroidal voids: An eshelby-like velocity fields approach[END_REF][START_REF] Keralavarma | A constitutive model for plastically anisotropic solids with non-spherical voids[END_REF][START_REF] Benzerga | Plastic potentials for anisotropic porous solids[END_REF][START_REF] Pardoen | Micromechanics-based model for trends in toughness of ductile metals[END_REF][START_REF] Monchiet | A Gurson-type model accounting for void size effects[END_REF][START_REF] Fritzen | Computational homogenization of elastoplastic porous metals[END_REF][START_REF] Brach | Nanoporous materials with a general isotropic plastic matrix: Exact limit state under isotropic loadings[END_REF]; taking into account the plastic anisotropy of matrix [START_REF] Monchiet | Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids[END_REF][START_REF] Keralavarma | A constitutive model for plastically anisotropic solids with non-spherical voids[END_REF][START_REF] Pan | Approximate yield criteria for anisotropic porous ductile sheet metals[END_REF][START_REF] Liao | Yield criteria for porous ductile sheet metals with planar anisotropy under plane stress conditions[END_REF] or tensioncompression asymmetry [START_REF] Oana | Analytic plastic potential for porous aggregates with matrix exhibiting tension-compression asymmetry[END_REF]; to consider the pressure-sensitive of geomaterials, some studies have been devoted to porous materials with a Drucker-Prager type compressible matrix [START_REF] Jeong | A new yield function and a hydrostatic stress-controlled model for porous solids with pressure-sensitive matrices[END_REF][START_REF] Guo | Continuum modeling of a porous solid with pressure sensitive dilatant matrix[END_REF][START_REF] Trillat | Limit analysis and Gurson's model[END_REF][START_REF] Maghous | Micromechanical approach to the strength properties of frictional geomaterials[END_REF][START_REF] Barthelemy | Détermination du critère de rupture macroscopique d'un milieu poreux par homogénéisation non linéaire[END_REF][START_REF] Shen | Evaluation and improvement of macroscopic yield criteria of porous media having a drucker-prager matrix[END_REF], a Green type matrix has been recently studied by [START_REF] Shen | Approximate criteria for ductile porous materials having a Green type matrix: Application to double porous media[END_REF][START_REF] Pastor | 3d-fem formulations of limit analysis methods for porous pressuresensitive materials[END_REF][START_REF] Fritzen | Computational homogenization of porous materials of Green type[END_REF][START_REF] Shen | Improved criteria for ductile porous materials having a green type matrix by using eshelby-like velocity fields[END_REF], or the Mises-Schleicher type [START_REF] Lee | Yield functions and flow rules for porous pressure-dependent strain-hardening polymeric materials[END_REF][START_REF] Durban | Plastic response ofporous solids with pressure sensitive matrix[END_REF][START_REF] Monchiet | Exact solution of a plastic hollow sphere with a mises-schleicher matrix[END_REF][START_REF] Shen | A new macroscopic criterion of porous materials with a mises-schleicher compressible matrix[END_REF][START_REF] Shen | An approximate strength criterion of porous materials with a pressure sensitive and tension-compression asymmetry matrix[END_REF]; some works focus on the double porous material with two populations of voids at different scales [START_REF] Shen | Approximate criteria for ductile porous materials having a Green type matrix: Application to double porous media[END_REF][START_REF] Vincent | Ductile damage of porous materials with two populations of voids[END_REF][START_REF] Vincent | Porous materials with two populations of voids under internal pressure: I. instantaneous constitutive relations[END_REF][START_REF] Shen | Effective strength of saturated double porous media with a drucker-prager solid phase[END_REF]. On the other hand for the influence of mineral inclusions on the overall mechanical behavior of composite, [START_REF] Maghous | Micromechanical approach to the strength properties of frictional geomaterials[END_REF][START_REF] Barthélémy | A micromechanical approach to the strength criterion of druckerprager materials reinforced by rigid inclusions[END_REF][START_REF] Dormieux | Résistance d'un composite à renforts rigides : le cas d'une matrice de drucker-prager avec règle d'écoulement plastique non associée[END_REF] obtained the macroscopic yield criteria for Drucker-Prager type matrix reinforced by rigid inclusions with associated or non-associated plastic flow rule by adopting the non-linear homogenization methods based on the modified secant method. For porous materials reinforced by inclusions whose size is much bigger than the one of pore, macroscopic yield criteria have been established with two steps of homogenization due to the well separation of pores and grains at two different scales, such as [START_REF] Garajeu | Effective properties of porous ideally plastic or viscoplastic materials containing rigid particles[END_REF] for von Mises type solid phase et [START_REF] Shen | A closed-form three scale model for ductile rocks with a plastically compressible porous matrix[END_REF][START_REF] Shen | A micro-macro model for clayey rocks with a plastic compressible porous matrix[END_REF][START_REF] He | Strength properties of a drucker-prager porous medium reinforced by rigid particles[END_REF] for Drucker-Prager type solid phase. The first upscaling is for the porous matrix with effect of pore at the microscopic scale, the second one is for the influence for grains at the mesoscopic scale. However, the exact solution can not be retrieved by these criteria for the special case of porous medium having a Drucker-Prager type matrix. This point will be improved in this work.

The purpose of this study is to propose a new macroscopic criterion for geomaterials with a porous matrix-inclusion microstructure. The size of pore is much smaller than the one of inclusion. The paper is organized as follows: based on the previous works of [START_REF] Shen | A closed-form three scale model for ductile rocks with a plastically compressible porous matrix[END_REF], a new explicit expression of the effective yield criterion is derived in section 2 to describe the elastoplastic mechanical behavior of the studied composite. It improves the predictions in the case of porous medium. In section 3, this yield criterion is applied to describe the peak stresses of Callovo-Oxfordian argillite with different compositions and different confining pressures. Then, a complete constitutive model is established in section 4 which takes into account explicitly the influence of the microstructure and their evolutions on the macroscopic mechanical behavior of the studied porous matrix-inclusion type geomaterials.

Macroscopic yield criterion for geomaterials with a porous matrix-inclusion microstructure

Problem statement

Pores and grains are two common properties of geomaterials. Normally, the mineral grains (such as quartz , calcite) increase the material strength, which is reduced quickly by the increase of volume fraction of pore. According to the experimental observations, the size of pore are usually much smaller than the one of grains, such as the geomaterial argillite.

Based on the type of microstructure, this type of material can be treated as "porous matrix -inclusion" composite as illustrated in Figure 1. In order to derive an analytical solution, the shapes of pore and inclusion are assumed to be spherical and randomly distributed. Ω denotes the total volume of the studied representative elementary volume; Ω s , Ω f and Ω i are the ones of solid phase, pore and inclusion, respectively. Based on these definitions, the volume fraction of pores at the micorscopic scale and the one of inclusions at the mesoscopic scale can be calculated respectively:

f = Ω f Ω f +Ω s , ρ = Ω i Ω = Ω i Ω f +Ω s +Ω i .

Big inclusions

Porous matrix

Small voids Solid phase Unlike the metal materials, the performance of geomaterial usually depends on the hydrostatic stress and it is pressure sensitive. There is a strength asymmetry between tension and compression. For example, the uniaxial compression strength is much higher than the uniaxial tensile strength. For this reason, the following Drucker-Prager yield criterion is adopted to describe the elastoplastic mechanical behavior of the matrix at the microscopic scale:

φ(σ) = σ eq + 3ασ m -σ 0 ≤ 0 (1)
in which σ m = tr(σ)/3 denotes the mean stress, σ eq = (3/2)σ : σ is the equivalent stress where σ the deviatoric part of the stress tensor, σ = σσ m I and I the second-order unit tensor. α and σ 0 are material parameters, the frictional coefficient and the strength in the case of purely shear loading, respectively.

Macroscopic criterion proposed by [1]

For the studied porous geomaterial with a microstructure "porous matrix -inclusion", a macroscopic yield criterion has been derived in [START_REF] Shen | A closed-form three scale model for ductile rocks with a plastically compressible porous matrix[END_REF] based on the works of [START_REF] Maghous | Micromechanical approach to the strength properties of frictional geomaterials[END_REF] with the modified secant method. This elastoplastic yield criterion (2) takes into account explicitly and simultaneously the influences of porosity f at the microscopic scale , the volume fraction of inclusion ρ at the mesoscopic scale and the pressure-sensitive of the solid phase by adopting the local yield criterion [START_REF] Shen | A closed-form three scale model for ductile rocks with a plastically compressible porous matrix[END_REF].

Φ = Θ Σ 2 eq σ 2 0 + 9 f 4 -9α 2 Σ 2 m σ 2 0 + 6(1 -f )α Σ m σ 0 - 3 + 2 f + 3 f ρ 3 + 2 f (1 -f ) 2 = 0 (2)
where Θ =

1+ 2 3 f +ρ( f -4α 2 ) 24α 2 -12 f -9 36α 2 -13 f -6 ρ+1
.

Based on this criterion for the composite with a "porous matrix -inclusion" microstructure, some special cases will be studied firstly:

-Special case 1: without the influence of inclusion (ρ = 0)

The macroscopic yield criterion (2) reduces to the one proposed by [START_REF] Maghous | Micromechanical approach to the strength properties of frictional geomaterials[END_REF][START_REF] Barthelemy | Détermination du critère de rupture macroscopique d'un milieu poreux par homogénéisation non linéaire[END_REF]:

1 + 2 3 f Σ 2 eq σ 2 0 + 9 f 4 -9α 2 Σ 2 m σ 2 0 + 6(1 -f )α Σ m σ 0 -(1 -f ) 2 = 0 (3)
where the square form of the local Drucker-Prager criterion (1) can be retrieved by taking f = 0.

-Special case 2: without the influence of pore ( f = 0)

One can simplify the yield criterion (2) as follows:

1 -4ρα 2 1 + 8α 2 -3 12α 2 -2 ρ Σ 2 eq σ 2 0 -9α 2 Σ 2 m σ 2 0 + 6α Σ m σ 0 -1 = 0 (4)
It is interesting to note that the volume fraction ρ has no influence on the value of Σ m σ 0 for a solid phase reinforced by the inclusions in the case of purely hydrostatic loading ( Σ eq σ 0 = 0). In the macroscopic criterion (2), the parameter ρ have an effect due to the presence of pore at the matrix (last item

3+2 f +3 f ρ 3+2 f (1 -f ) 2
). But this effect is very small and can almost be ignored. However, the inclusion content affects the deviatoric value Σ eq σ 0 .

-Special case 3: a pure porous material with a von Mises type matrix (α = 0, ρ = 0) This criterion (3) reduced to the one proposed by [START_REF] Castaneda | The effective mechanical properties of nonlinear isotropic composites[END_REF] without pressure sensitivity:

1 + 2 3 f Σ 2 eq σ 2 0 + 9 f 4 Σ 2 m σ 2 0 -(1 -f ) 2 = 0 (5)
As indicated in [START_REF] Maghous | Micromechanical approach to the strength properties of frictional geomaterials[END_REF][START_REF] Shen | Evaluation and improvement of macroscopic yield criteria of porous media having a drucker-prager matrix[END_REF][START_REF] Traxl | Multi-level homogenization of strength properties of hierarchicalorganized matrix-inclusion materials[END_REF] for a porous medium, the macroscopic yield criterion (3) or

(5) overestimates the material strength. With a Drucker-Prager type matrix, the exact value of Σ m σ 0 in the case of hydrostatic loading ( Σ eq σ 0 = 0) can be analytically calculated:

Σ m σ 0 =                    1 -f 2α (2α+1) 3α
, tensile hydrostatic loading

1 -f 2α (2α-1)

3α

, compressive hydrostatic loading

(6)
The corresponding one predicted by the criterion ( 3) is

Σ m σ 0 = 2 3 (2α ± f )(1 -f ) 4α 2 -f (7)
As illustrated in Figure 2 for the pure hydrostatic loading, the value [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals[END_REF] given by (3) (blue dashed line) is different from the exact one (6) (red solid line), especially in the compressive loading with high value of α and low and medium porosity f . In the tensile loading, the difference is diminished. Based on these observations, the prediction of the yield criterion (3) for the case of a porous material with a Drucker-Prager type matrix should be improved. It means that the porosity effect considered in the macroscopic criterion (2) for the porous geomaterial with a "porous matrix -inclusion" microstructure should be ameliorated, specially for the hydrostatic loading.

A heuristic macroscopic yield criterion based on criterion (2)

In order to improve the performance of the yield criterion (2) for geomaterials with a "porous matrix -inclusion" microstructure, some special conditions will be considered. As indicated in the above section, the influence of porosity f are not well predicted by (3) in the compressive loading. So, it is interesting to study firstly the case of porous material without grains (ρ = 0). Based on the exact solution of Σ m σ 0 given in ( 6) for a porous material with a Drucker-Prager type matrix in the hydrostatic loading, the performance of the macroscopic yield criterion (2) will be improved.

According to the expression of Σ m σ 0 given in ( 6), it can be rearranged as the following form:

f 2α (2α±1) = 1 -3α Σ m σ 0 (8)
which can be expressed as:

ln f 1 -3α Σ m σ 0 = ± 1 2α ln 1 -3α Σ m σ 0 ⇒ 1 -3α Σ m σ 0 f = e -± 1 2α ln 1-3α Σm σ 0 (9)
With the following mathematical property:

γ = e -ϑ ⇔ 2γ cosh(ϑ) -1 -γ 2 = 0 (10)
one can easily derive:

2 f 1 -3α Σ m σ 0 cosh 1 2α ln(1 -3α Σ m σ 0 ) A -f 2 -1 -3α Σ m σ 0 2 = 0 (11)
The Taylor's expression of the first term "A" in equation ( 11) can be calculated:

2 f 1 -3α Σ m σ 0 cosh 1 2α ln(1 -3α Σ m σ 0 ) A = 9 f 4 Σ 2 m σ 2 0 + 2 f 1 -3α Σ m σ 0 B +O Σ 4 m σ 4 0 (12)
Based on this observation, the yield criterion (3) can be arranged as follows with

Σ eq σ 0 = 0: 9 f 4 Σ 2 m σ 2 0 + 2 f 1 -3α Σ m σ 0 B -1 -3α Σ m σ 0 2 -f 2 = 0 (13)
The term "B" in equation ( 13) depends on the porosity f . Its expression is the same as the one in equation [START_REF] Benzerga | Plastic potentials for anisotropic porous solids[END_REF]. Based on this observation, one can find that the difference between the exact solution ( 6) and the one [START_REF] Gologanu | Recent extensions of Gurson's model for porous ductile metals[END_REF] proposed criterion (3) in the pure hydrostatic loading is due to the omission of the higher order term O(). In order to overcome this default, the omitted term should be taken into account.

Then, it is interesting to rearrange the macroscopic yield criterion (2) for geomaterials with a "porous matrix -inclusion" microstructure as follows:

Φ = Θ Σ 2 eq σ 2 0 + 9 f 4 Σ 2 m σ 2 0 + 2 f 1 -3α Σ m σ 0 B -1 -3α Σ m σ 0 2 -f 2 -(1 -f ) 2 f ρ 1 + 2 3 f = 0 (14)
The same term "B" as the one in equation ( 13) and ( 12) is found. It depends only on porosity f and the frictional parameter α and is independent of the volume fraction of inclusion ρ. Comparing with the expression of ( 11), ( 12) and ( 14), the term "B" in the criterion ( 14) will be replaced by the one "A" in equation [START_REF] Benzerga | Plastic potentials for anisotropic porous solids[END_REF] in order to take into account the omitted higher order term. Finally, the searched macroscopic yield criterion for the studied geomaterials with a "porous matrix -inclusion" microstructure can be expressed as follows:

Φ = Θ Σ 2 eq σ 2 0 +2 f 1 -3α Σ m σ 0 cosh 1 2α ln(1 -3α Σ m σ 0 ) -1 -3α Σ m σ 0 2 -f 2 -(1-f ) 2 f ρ 1 + 2 3 f = 0 (15)
It is worth to study some special cases of the obtained macroscopic yield criterion [START_REF] Fritzen | Computational homogenization of elastoplastic porous metals[END_REF].

When ρ = 0, this criterion reduces to the following one for elastoplastic behaviors of porous material with a Drucker-Prager type matrix:

1 + 2 3 f Σ 2 eq σ 2 0 + 2 f 1 -3α Σ m σ 0 cosh 1 2α ln(1 -3α Σ m σ 0 ) -1 -3α Σ m σ 0 2 -f 2 = 0 (16)
The exact solutions of tension and compression [START_REF] Gologanu | Approximate models for ductile metals containing nonspherical voids-cas of axisymmetric oblate ellipsoidal cavities[END_REF] in the pure hydrostatic loading is retrieved by the yield criterion [START_REF] Brach | Nanoporous materials with a general isotropic plastic matrix: Exact limit state under isotropic loadings[END_REF], which improves fundamentally the one (3). When α → 0 for porous materials with an incompressible von Mises type matrix, the yield criterion ( 16) can be simplified as:

1 + 2 3 f Σ 2 eq σ 2 0 + 2 f cosh 3 2 Σ m σ 0 -1 -f 2 = 0 (17)
2.4. Comparisons between the macroscopic criteria (2) and [START_REF] Fritzen | Computational homogenization of elastoplastic porous metals[END_REF] In this section, the mechanical performances predicted by the macroscopic yield criterion

(2) proposed in [START_REF] Shen | A closed-form three scale model for ductile rocks with a plastically compressible porous matrix[END_REF] and the new one [START_REF] Fritzen | Computational homogenization of elastoplastic porous metals[END_REF] derived in this work will be fully compared. As illustrated in Figure 3, the influences of the volume fraction of inclusions ρ at the mesoscopic scale, the matrix porosity f and the frictional parameter α of the solid phase at the microscopic scale on the macroscopic mechanical behavior are investigated. With the increase of porosity, one can see from Figure 3(a) that the resistance of the studied material decreases very quickly. In the case of tensile loading, the mechanical behavior predicted by the new criterion (15) (solid line) is very similar to the one given by criterion (2) (dashed line). But for the case of compressive loading, the strength predicted by [START_REF] Fritzen | Computational homogenization of elastoplastic porous metals[END_REF] are much smaller, specially for the case of pure hydrostatic loading. As shown in Figure 3

Comparisons between the macroscopic criterion (15) and experimental data

In the nuclear waste storage investigation, the layer of Callovo-Oxfordian argillite has been chosen as a possible geological disposal of radioactive wastes in France. Based on the experimental observations [START_REF] Chiarelli | Experimental Investigation and Constitutive Modeling of Coupled Elastoplastic Damage in Hard Claystones[END_REF][START_REF] Robinet | Mineralogie, porosite et diffusion des solutes dans l'argilite du Callovo-Oxfordien de Bure (Meuse/Haute-Marne, France) de l'echelle centimetrique a micrometrique[END_REF], the porosity of the argillite is mainly located in the clay matrix and varies from 11.04 to 13.84%. The typical mineralogical compositions are: 40 to 50% of clay matrix, 20 to 27% of calcite, 23 to 25% of quartz, 5 to 10% of minor minerals.

As the pores are totally embedded in the clay matrix, the porosity of the clay matrix is then equal to the overall porosity divided by the volume fraction of the clay matrix. The typical average value is taken here: f = 25%. The elastic properties of calcite and quartz grains are E 1 = 95GPa, v 1 = 0.27 and E 2 = 101GPa, v 2 = 0.06, respectively. Comparing with the ones of clay matrix E 0 = 3 GPa, v 0 = 0.3, the grains of calcite and quartz can be treated as rigid inclusions. The studied argillite is simplified as a composite having a "porous matrixinclusion" microstructure, as illustrated in Figure 1. The new macroscopic yield criterion [START_REF] Fritzen | Computational homogenization of elastoplastic porous metals[END_REF] will be adopted to describe the overall behavior of the argillite. The experimental data carried out in [START_REF] Chiarelli | Experimental Investigation and Constitutive Modeling of Coupled Elastoplastic Damage in Hard Claystones[END_REF] will be investigated. The mineralogical compositions vary with the depth. Three groups of samples have been prepared at different depths: 451.4 -466.8 m, 468.9 -469.1 m and 482.2 -482.4 m. Different uniaxial and triaxial compressive tests have Concerning the elastoplastic behavior of the solid phase, another expression of Drucker-Prager criterion is also widely used:

φ(σ) = σ d + T (σ m -c 0 ) ≤ 0 (18)
In this section, this form will be adopted for the complete constitutive model. Then, the corresponding macroscopic yield criterion (15) can be rewritten as:

Φ = Θ Σ 2 eq c 2 0 + 2 f 1 - Σ m c 0 cosh        3 2T 2 ln(1 - Σ m c 0 )        -1 - Σ m c 0 2 -f 2 -(1 -f ) 2 f ρ 1 + 2 3 f = 0 (19) in which Θ = 1+2 f /3 T 2 2 3 + 4 9 ρ 3 f 2T 2 -1 4T 2 -12 f -9 6T 2 -13 f -6 ρ + 1 .
The criterion [START_REF] Oana | Analytic plastic potential for porous aggregates with matrix exhibiting tension-compression asymmetry[END_REF] will be adopted here to describe these mechanical behaviors of the studied argillite. With the variation of depths, the composition of the argillite will change.

These variations can be taken into account explicitly by the yield criterion [START_REF] Oana | Analytic plastic potential for porous aggregates with matrix exhibiting tension-compression asymmetry[END_REF]. However, the same mechanical properties of the solid phase will be adopted for all tests: T = 0.88, c 0 = 14.5MPa, with f = 0.25. The comparisons between yield surfaces predicted by new criterions (19) (solid line) and experimental data (points) [START_REF] Chiarelli | Experimental Investigation and Constitutive Modeling of Coupled Elastoplastic Damage in Hard Claystones[END_REF][START_REF] Chiarelli | Modeling of elastoplastic damage behavior of a claystone[END_REF] are shown in Figure 4 with different compositions and different confining pressures: 0 MPa (Figure 4 and experimental data (points) with different confining pressures: 0, 5, 10, 20 MPa.

A complete constitutive model for geomaterials with a porous matrix-inclusion microstructure

Based on the new macroscopic yield criterion [START_REF] Oana | Analytic plastic potential for porous aggregates with matrix exhibiting tension-compression asymmetry[END_REF] which takes into account explicitly the influence of matrix porosity f at the microscopic scale and the effect of inclusion ρ at the mesoscopic scale, a complete constitutive modeling will be established in this section.

By considering the macroscopic plastic potential for plastic flow rule, plastic hardening and also the evolutions of volume fractions of pores and inclusions, it will be used to describe the macroscopic mechanical behavior of geomaterials with a porous matrix-inclusion microstructure.

Construction of a complete constitutive model

Elastic behavior of the studied composite

The microstructure and its evolution have influences on the elastic mechanical behavior of the composite. In order to take into account the effects of matrix porosity, solid phase dilatation and the volume fraction of inclusion, a two-step homogenization procedure will be adopted. Due to the studied microstructure, the first homogenization is for the influences of the porosity f and the elastic properties (κ s , µ s ) of the solid phase at the microscopic scale. The Mori-Tanaka homogenization scheme [START_REF] Mori | Average stress in a matrix and average elastic energy of materials with misfitting inclusions[END_REF] is used to determine the effective elastic properties (κ pm , µ pm ) of the porous matrix:

κ pm = 4(1 -f )κ s µ s 4µ s + 3 f κ s ; µ pm = (1 -f )µ s 1 + 6 f κ s + 2µ s 9κ s + 8µ s (20)
Based on [START_REF] Jeong | A new yield function and a hydrostatic stress-controlled model for porous solids with pressure-sensitive matrices[END_REF], the effect of big inclusion on the macroscopic elastic behavior (κ hom , µ hom ) is considered in the second upscaling:

κ hom = (1-ρ)κ pm 3κ pm +4µ pm + ρκ i 3κ i +4µ pm (1-ρ) 3κ pm +4µ pm + ρ 3κ i +4µ pm ; µ hom = (1-ρ)µ pm µ pm (9κ pm +8µ pm )+6µ pm (κ pm +2µ pm ) + ρµ i µ pm (9κ pm +8µ pm )+6µ i (κ pm +2µ pm ) (1-ρ)
µ pm (9κ pm +8µ pm )+6µ pm (κ pm +2µ pm ) + ρ µ pm (9κ pm +8µ pm )+6µ i (κ pm +2µ pm ) [START_REF] Guo | Continuum modeling of a porous solid with pressure sensitive dilatant matrix[END_REF] in which κ i , µ i are bulk and shear modulus of the inclusion.

Plastic behavior of the studied composite

The plastic behavior of the studied composite will be described by the macroscopic yield criterion [START_REF] Oana | Analytic plastic potential for porous aggregates with matrix exhibiting tension-compression asymmetry[END_REF] derived in this work. For most geomaterials, a plastic hardening phenomena can be observed in the experiment due to the plastic deformations generated during the loading. Based on the works of [START_REF] Shen | A closed-form three scale model for ductile rocks with a plastically compressible porous matrix[END_REF][START_REF] Shen | A micro-macro model for clayey rocks with a plastic compressible porous matrix[END_REF], the plastic hardening of the solid matrix at the microscopic scale is taken into account via the evolution of the frictional coefficient T as a function of the equivalent plastic strain ε p . Denoting T 0 the initial threshold and T m the asymptotic value of the frictional coefficient, the following exponential form is adopted for the hardening law:

T = T m -(T m -T 0 )e -b 1 ε p (22)
Then, the macroscopic yield criterion can be rewritten as:

Φ = ΘΣ 2 eq c 2 0 + 2 f 1 - Σ m c 0 cosh        3 2 T 2 ln(1 - Σ m c 0 )        -1 - Σ m c 0 2 -f 2 -(1 -f ) 2 f ρ 1 + 2 3 f = 0 (23) in which Θ = 1+2 f /3 T 2 2 3 + 4 9 ρ 3 f 2 T 2 -1 4 T 2 -12 f -9 6 T 2 -13 f -6 ρ + 1 .
The non associated flow rule is one of the main characteristic of geomaterials. Inspired by the work of [START_REF] Maghous | Micromechanical approach to the strength properties of frictional geomaterials[END_REF], a macroscopic plastic potential is proposed in the following form:

G = 1+2 f /3 T t 2 3 + 4 9 ρ 3 f 2 T t -1 4 T t-12 f -9 6 T t-13 f -6 ρ + 1 Σ 2 eq c 2 0 + 3 f 2 T t -1 Σ 2 m + 2(1 -f )c 0 Σ m (24)
in which t is the dilatancy coefficient which controls the transition between volumetric contractance and dilatancy under deviatoric loading. As the rate of volumetric dilatancy generally varies with plastic deformation history, it is assumed that t is a function of the plastic hardening variable:

t = t m -(t m -t 0 )e -b 2 ε p ( 25
)
where t 0 , t m and b 2 are used to describe the transition between volumetric contractance and dilatancy.

Generally, the non associated plastic flow rule is given by:

D p = λ ∂G ∂Σ (Σ, f, T , t) (26)
where D p denotes the macroscopic plastic strain rate and λ the plastic multiplier.

According to the energy condition initially introduced by [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: part1-yield criteria and flow rules for porous ductile media[END_REF]:

Σ : D p = Ω -1 Ω s σ : d dΩ = Ω -1 Ω s εp tc 0 dΩ (27)
the equivalent plastic strain rate of the solid matrix can be calculated:

εp = Σ : D p (1 -f )(1 -ρ) T c 0 + ( t -T ) Σ m 1-f (28)
The variations of the porosity and of the volume fraction of inclusion can be determined from the kinematical compatibility as follows:

ḟ = 1-f 1-ρ trD p -1 Ω s +Ω f Ω s trd p dΩ = 1-f 1-ρ trD p -(1 -f ) t εp ρ = -ρtrD p (29)

Application to the COx argillite

The complete constitutive model presented in the above section then is implemented in the finite element software Abaqus with a user subroutine and applied to describe the macroscopic mechanical behavior of COx argillite, considering the influence of the inclusions at the mesoscale scale, the matrix porosity and the dilatation of the solid phase at the microscopic scale and also their evolutions.

Identification of the model parameters

According to the experimental observations, the elastic properties of calcite and quartz grains are (Table 1). The obtained values are shown in Table 2:

E 1 = 95GPa, v 1 = 0.

Solid phase Inclusion

Elastic: E s = 5GPa, v s = 0.33

E i = 98GPa, v i = 0. 15
Plastic: T 0 = 0.1, T m = 0.88, b 1 = 200, t 0 = -1.1, t m = 0.35, b 2 = 150, c 0 = 14.5MPa
Table 2: Elastic and plastic model parameters for COx argillite.

Experimental validation

In order to fully assess the proposed constitutive model, a series of uniaxial and triaxial compression simulations with 0, 5, 10 and 20 MPa confining pressures are carried out for samples at different depth. As illustrated in Figure 7 to 9, the main features of the composite, such as the transition from contractancy to dilatancy, the effect of confining pressure and the influence of inclusion can be captured by the proposed model. Generally, a good agreement is found between the numerical simulations and experimental data. The evolutions of the porosity f and the volume fraction of inclusion ρ for the triaxial compression tests at depth 482.2 m are shown in Figure 10. According to the evolution law [START_REF] Shen | Improved criteria for ductile porous materials having a green type matrix by using eshelby-like velocity fields[END_REF], the variation of porosity ḟ depends on the macroscopic deformation of the composite and the one of the solid phase, while the variation of ρ depends only on the macroscopic deformation.

Taking the triaxial compressive test with 10 MPa as an example, the macroscopic volumetric deformation is in compressive state at the beginning of loading. The volume of inclusion Ω i is not changed at the mesoscopic scale. Then, the volume fraction of inclusion ρ increases and the porosity f decreases as the black lines shown in Figure 10(a) and 10(b). With the process of loading, there is a volumetric transition from contractancy to dilatancy of the composite. In this moment, the actual ρ will reduce due to the increasing volume of porous matrix at the mesoscopic scale. However, the porosity f still decreases because of the dila-tant of the solid phase at the microscopic scale. One finds that the total variations of f and ρ are small. With the same composition, the confining pressure increases the macroscopic volumetric contractancy and influences the microstructure evolution. These have been taken into account automatically by the proposed constitutive model.

Conclusions

In the present work, the effective mechanical behaviors of geomaterial with a pressuresensitive solid phase and pore at the microscopic scale and bigger inclusions at the mesoscopic scale is investigated. A new macroscopic yield criterion is derived for the studied porousmatrix-inclusion type composite. This criterion improves the existing ones, specially for the case of porous medium with a Drucker-Prager type matrix. By considering the influences of mineral inclusions and the matrix porosity, the proposed yield criterion is able to describe the peak stresses of the Callovo-Oxfordian argillite with different compositions and different loading pathes. Then, a complete constitutive model is established which takes into account the influence of microstructure and its evolution. By comparing with the experimental data, the main features of the studied porous matrix-inclusion type geomaterials can be captured by the proposed model, such as the transition from contractancy to dilatancy, the influence of porosity and volume fraction of inclusion, as well as the effect of confining pressure.

Figure 1 :

 1 Figure 1: Representative volume element of the studied geomaterial

( a )

 a Effects of porosity f , α = 0.2 (b) Effects of frictinal parameter α, f = 0.2

Figure 2 :

 2 Figure 2: Comparisons between the values of Σ m σ 0 predicted by (3) (blue dashed line) and the exact solution (6) (red solid line) as functions of f and α, respectively.

  Figure 3(c) by considering the frictional parameter α. With small value of α, the compressive mechanical behavior is similar to the tensile one, for example α = 0.05. The dissymmetry between tension and compression increases with the increasing of α. Figure 3(d) shows the values of Σ mσ 0 as functions of f and α in the case of pure hydrostatic compressive loading without the inclusion effects (ρ = 0). As indicated in the above section, the new criterion (15) (blue surface) retrieves the exact solution given by (6), which improves fundamentally the value given by (2) (red surface).

  (a) Effects of matrix porosity f (b) Effects of inclusion content ρ (c) Effects of frictional parameter α (d) Evolution of Σ m σ 0 with hydrostatic loading (ρ = 0): blue surface-(15), red one-(2)

Figure 3 :

 3 Figure 3: Comparisons between the yield surfaces predicted by the criteria (2) (dashed line) and (15) (solid line) with different f , ρ and α.

Figure 4 :

 4 Figure 4: Comparisons between yield surfaces predicted by new criterion (19) (solid line)

27 and E 2 =

 2 101GPa, v 2 = 0.06, respectively. Comparing with the ones of clay matrix E 0 = 3 GPa, v 0 = 0.3, the grains of calcite and quartz can be treated as rigid inclusions. Based on the average matrix porosity, the elastic properties of the solid phase at the microscopic scale can be calculated by adopting the inverse procedure of the first homogenization[START_REF] Jeong | A new yield function and a hydrostatic stress-controlled model for porous solids with pressure-sensitive matrices[END_REF]. One gets E s = 5GPa, v s = 0.33 for the solid phase. The plastic parameters are determined by the following procedure: the parameters are first identified by numerical fitting of stress-strain curves obtained from the hydrostatic compression test performed on a given mineralogical composition (see Figure5, ρ = 0.45, f = 0.25). The corresponding evolutions of the porosity and the volume fraction of inclusion are illustrated in Figure6. Then the obtained parameters are validated through the simulations of other tests performed on different mineralogical compositions, and different confining pressures

Figure 5 :

 5 Figure 5: Comparisons between numerical simulations (solid line) and experimental data (points) with the hydrostatic loading on a sample at depth 469.1 m, ρ = 0.45

Figure 6 :

 6 Figure 6: Evolutions of porosity f and volume fraction of inclusion ρ as a function of axial deformation with the hydrostatic loading on a sample at depth 469.1 m, ρ = 0.45.

  451.4m: ρ = 0.53

Figure 7 :

 7 Figure 7: Depth 1 : Comparisons between numerical simulations (solid line) and experimental data (points) with different confining pressures: 0, 5, 10, 20 MPa.

  468.9m: ρ = 0.66

Figure 8 :

 8 Figure 8: Depth 2 : Comparisons between numerical simulations (solid line) and experimental data (points) with different confining pressures: 0, 5, 10, 20 MPa.

20

 20

  482.3m: ρ = 0.55

Figure 9 :

 9 Figure 9: Depth 3 : Comparisons between numerical simulations (15) (solid line) and experimental data (points) with different confining pressures: 0, 5, 10, 20 MPa.

  Evolution of volume fraction of inclusion

Figure 10 :

 10 Figure 10: Evolutions of porosity f and volume fraction of inclusion ρ as a function of axial deformation with different confining pressures on samples at depth 482.2 m.

MPa √ √ √Table 1: Compressive tests at different depths with different confining pressures.

Acknowledgement This study was supported by the National Natural Science Foundation of China (Grant No. 11902069), in part by the Young Talent Program of Liaoning Province under Grant XLYC1807094 and the framework of Sino-Franco Joint Research Laboratory on Multiphysics and Multiscale Rock Mechanics.


I Bentegri

O Boukendakdji

E-H Kadri

T T Ngo

H Soualhi

Rheological and tribological behaviors of polypropylene fiber reinforced concrete

Keywords: Fiber reinforced concrete, Rheology, Tribology, Polypropylene fibers, Superplasticizer

Highlights

• Rheological and tribological properties of fiber reinforced concrete.

• Fibers shape, length and dosage influence on the fresh concrete

Introduction

According to ACI Committee report about the fibrous concrete worldwide application, fibers have been widely used in concrete all over the world. Indeed, large and interesting projects were established with the use of fiber-reinforced concrete (FRC), such as demountable panels for parking garages, runway slabs and nuclear waste container [START_REF] Mehta | CONCRETE Microstructure, properties and materials[END_REF], [START_REF] Othman | Conceptual design of ultra-high performance fiber reinforced concrete nuclear waste container[END_REF].

Massive quantities of several types of FRC are fabricated annually by dint of its low environmental impact and its enhancement properties in the hardened state, compared to conventional concrete [START_REF] Hassanpour | Lightweight aggregate concrete fiber reinforcement -A review[END_REF]. In fact, FRC has various technical advantages, including a low permeability [START_REF] Kakooei | The effects of polypropylene fibers on the properties of reinforced concrete structures[END_REF], high flexural strength [START_REF] Noushini | Mechanical and flexural performance of synthetic fibre reinforced geopolymer concrete[END_REF], good failure impact resistance [START_REF] Li | Investigation on mechanical properties and microstructure of high performance polypropylene fiber reinforced lightweight aggregate concrete[END_REF], ductility and cracks control [START_REF] Hassanpour | Lightweight aggregate concrete fiber reinforcement -A review[END_REF] besides the reduction of shrinkage and expansion rate of concrete [START_REF] Kakooei | The effects of polypropylene fibers on the properties of reinforced concrete structures[END_REF].

Nevertheless, it has been reported that the workability of fresh concrete was negatively affected by the introduction of fibers [START_REF] Güneyisi | Combined effect of steel fiber and metakaolin incorporation on mechanical properties of concrete[END_REF]- [START_REF] Mazaheripour | The effect of polypropylene fibers on the properties of fresh and hardened lightweight selfcompacting concrete[END_REF]. As a matter of fact, the influence of fibers on fresh concrete properties depends on their nature, length, shape and volume fraction [START_REF] Li | Combined effects of water film thickness and polypropylene fibre length on fresh properties of mortar[END_REF]. For instance, adding steel fibers reduces the concrete workability and requires more mixing time than ordinary concrete because of its high specific gravity [START_REF] Hassanpour | Lightweight aggregate concrete fiber reinforcement -A review[END_REF]. On the other hand, PPF reinforced concrete requires less mixing time than steel fiber reinforced one [START_REF] Yin | Comparative evaluation of virgin and recycled polypropylene fibre reinforced concrete[END_REF]. It is worth noting that, polypropylene fibers have a lot of advantages over other commercially available synthetic fibers because of their light weight, cost efficiency, low thermal conductivity and being resistant to attack by acids and alkalis. Besides, polypropylene fibers have hydrophobic levels, which protect them against wetting with cement paste and have no effect on the water dosage needed. Hence, the use of PP fibers in field concrete is gaining popularity nowadays.

In addition, it has been reported that adding polypropylene fibers have a low impact on air content and fresh density because of the low density of PPF unlike other types of fibers as steel fibers. Akça et al. [START_REF] Akça | Properties of polypropylene fiber reinforced concrete using recycled aggregates[END_REF] found that adding 1.5% of PPF (by volume) in concrete increases the air voids from 1 to 1.3%. In the same time Hassanpour et al. [START_REF] Hassanpour | Lightweight aggregate concrete fiber reinforcement -A review[END_REF] used PPF on lightweight aggregate concrete indicating that PPF have no effect on concrete fresh density.

In terms of PPF length effect on the workability, Hassanpour et al. [START_REF] Hassanpour | Lightweight aggregate concrete fiber reinforcement -A review[END_REF] reported that the workability of PPF reinforced lightweight concrete decreases with the increase of fibers length, and among the two previous fibers types, the lowest effect on the slump was attributed to PPF [START_REF] Hassanpour | Lightweight aggregate concrete fiber reinforcement -A review[END_REF]. This result was confirmed by Li et al. [START_REF] Li | Combined effects of water film thickness and polypropylene fibre length on fresh properties of mortar[END_REF] whose studied the combined effect of the water film thickness and PPF length reporting that increasing PPF length in mortar decreased significantly the slump flow. The workability loss of PPF reinforced concrete and mortars depends greatly on the friction between fibers and cement pastes as well as the use of admixtures. As the inclusion of PPF, causes an increase of the granular structure and a drop of the maximum compactness which increases the friction between fibers and cement particles and hinders the granulate movement during the test. This phenomenon results in a decrease of the slump value and an augmentation of the viscosity of mixture. Zhang et al. [START_REF] Zhang | How does adsorption behavior of polycarboxylate superplasticizer effect rheology and flowability of cement paste with polypropylene fiber?[END_REF] have investigated the flowability and rheology of polypropylene fiber reinforced cement paste, indicating that increasing PPF length and dosage increased evenly the apparent viscosity and the shear stress of the cement paste. In the same study, it has been proved that adding a superplasticizer in the cement paste showed three times higher flowability. Actually, the incorporation of a superplasticizer in FRC has also been suggested by other researchers as a technic to maintain concrete workability besides other solutions such as the augmentation of the paste amount in the mix or using pozzolanic additions [START_REF] Hassanpour | Lightweight aggregate concrete fiber reinforcement -A review[END_REF], [START_REF] Aydin | Self compactability of high volume hybrid fiber reinforced concrete[END_REF].

It is worth noting that slump test value is an important indicator of fresh properties, yet still not sufficient to characterize the fundamental rheological properties of concrete, namely plastic viscosity and yield stress [START_REF] Chidiac | Plastic viscosity of fresh concrete -A critical review of predictions methods[END_REF], [START_REF] Chidiac | Controlling the quality of fresh concrete-a new approach[END_REF], especially with the growing development of construction technics and complicated shapes of buildings which required a better understanding of the concrete fresh state. Despite this fact, only few investigations have been devoted to studying the rheological behavior of fiber reinforced ordinary concrete. Moreover, the tribological behavior of FRC has been ignored in the previous studies, even though concrete implementation by the traditional or pumping technics depends on the tribological properties of concrete such as the shear stress and the viscous constant [START_REF] Ngo | Measurement and modeling of fresh concrete viscous constant to predict pumping pressures[END_REF].

Therefore, the rheological and tribological behaviors of FRC were investigated in the present study. First of all, the control concrete (CC) formulation was selected with respect to the best rheological and tribological behaviors via the optimization of the paste volume. Thereafter, PPF was added with different lengths, shapes, and dosages in the first step. Afterward, a polycarboxylate superplasticizer (Sp) was introduced in the mixes with numerous percentages to maintain the concrete flowability, while the plastic viscosities and the viscous constants were determined and compared with those of the previous step. Finally, based on the practical use of fibers in construction sites, two fiber lengths were used in the same mix with different weight combinations and both rheological and tribological parameters were analyzed.

Experimental program

Materials

The cement used to formulate all concrete mixes is an ordinary Portland cement (CEM I 52.5N) manufactured by EQIOM cement plant in France. It is characterized by a density of 3.1 g/cm 3 and a specific surface area of 4300 cm 2 /g. Its chemical and mineral compositions are given in Table 1. In addition, Table 2 gives all characteristics of the coarse and the fine aggregates while Fig. 1 shows their grading curves. The 19 and the 54 mm fibrillated twist PPF are intended to be used for reinforcement and precast concrete due to their high tensile strength (689 MPa). However, 30 mm PPF have a wavy shape and lower tensile strength (486 MPa). At the same time, this type of fibers is more flexible than the others, thanks to its high elastic modulus (6.9 GPa), unlike 19 and 54 mm fibers (5.75 GPa). The properties of these fibers are depicted in Table 3. Using scanning electron microscope (SEM) analysis, the difference between fibers shape can be clearly seen in Fig. 3 and Fig. Lastly, the superplasticizer (Sp) used was polycarboxylate based type, with a solid mass content of 33.0 ±1.5 % and a specific gravity of 1.07 ± 0.02 .

Concrete mix design

First of all, knowing that the rheological and the tribological behaviors of concrete depend on mixture compositions and its workability, four concrete mixes (without fibers) namely CC1, CC2, CC3 and CC4 were produced in order to obtain the control formulation. All concrete mixes were formulated with the same optimal G/S ratio (1.44), determined by Dreux-Gorisse method [START_REF] Dreux | COMPOSITION DES BETONS : METHODE DREUX-GORISSE -BILAN DE CINQ ANNEES D'APPLICATION EN COTE D'IVOIRE[END_REF] using the grading curves of aggregates (Fig. 1). In addition, W/C ratio has been fixed at 0.5 unlike the cement paste volume which was different for each mixture, to obtain four various concrete mixes with several workabilities. The best formulation is chosen with respect to the best workability, rheology and tribology properties.

Once the optimum formulation was determined, PPF were added in mixes to study their effect on concrete rheological and tribological properties. Nine mixes were produced with three different fiber lengths (19, 30 and 54 mm) and three PPF percentages (0.12, 0.24, 0.36% by volume of concrete) namely FRC19, FRC30 and FRC54.

Noted that using fibers in concrete decreases its workability [START_REF] Zhang | How does adsorption behavior of polycarboxylate superplasticizer effect rheology and flowability of cement paste with polypropylene fiber?[END_REF], researchers proposed adding superplasticizer in FRC to improve its workability and fluidity. Therefore, the formulations made in the previous phase were reproduced by adding a different percentage of superplasticizer (0.2-0.4%) to obtain the optimum concrete workability without segregation.

In order to control micro and macrocracks in construction sites, FRC are often made with 2 different lengths in the same mix. Considering this fact, the last part aims to investigate this combined effect by studying three concrete mixes with different 19 mm/30 mm fibers ratios equal to 0/100, 30/70, 70/30 and 100/0 respectively which leads to obtain the best combination percentage. It is worth noting that the target strength class of all concrete formulations is C35/45. The detailed mix proportions of all phases are shown in Table 4.

Test protocol and measurements 168

The concrete mixes were prepared basing on the technical sheet of fibers and according to NF 169 EN 14845-1 standard, using a concrete mixer. Starting with mixing fine and coarse aggregates 170 for 1 minute, cement and fibers were then added and mixed for 2 minutes, and finally the 171 water containing the superplasticizer was added. 172

Testing program

After mixing, the fresh concrete was instantly subjected to the following measurements:

• Workability

In this study concrete workability was measured according to the standard NF-EN 12350-2 [START_REF] Afnor | NF EN 12350-2 Essais pour béton frais -Partie 2 : Essai d'affaissement[END_REF] using Abrams cone, in which fresh concrete was poured in three layers, each layer is tamped 25 times, and the cone is carefully lifted which makes concrete flowed. The slump was then measured as the drop-in height of the concrete mix.

• Rheological measurement

It has been reported by Wallevik et al. [START_REF] Wallevik | Relationship between the Bingham parameters and slump[END_REF] that two different concrete formulations with the same slump value may have different rheological properties (yield stress and plastic viscosity). In addition, previous studies proved that concrete flow depends on its rheological behavior during the implementation. As the slump test is related to the yield stress only which represents the flow initiating, however as soon as the flow begins it is the viscosity that influences the concrete flow. Therefore, several studies proved that the slump test is not efficient to characterize the fundamental rheological properties of concrete [START_REF] Wallevik | Rheology as a tool in concrete science: The use of rheographs and workability boxes[END_REF] [START_REF] Wallevik | Rheology as a tool in concrete science: The use of rheographs and workability boxes[END_REF].

A new concrete rheometer, developed by Soualhi et al. [START_REF] Soualhi | Design of portable rheometer with new vane geometry to estimate concrete rheological parameters[END_REF], was used to determine the rheological properties (Fig. 5). The apparatus is composed of an agitator with speed electronic control, a steel vane with a double U shape of 15 cm and 10.5 cm in height and diameter respectively and a cylindrical container (h=25 cm and d=30cm) with 18 square steel rods 9x9 mm 2 and 22 mm of length welded in the inner wall in order to optimize the adhesion of concrete to the container inner wall [START_REF] Soualhi | New model to estimate plastic viscosity of eco-friendly and conventional concrete[END_REF]. The rheological test consists to fill the metallic container with fresh concrete in two layers; each layer is tamped 25 times with a metal rod. Then the vane is placed in the center and plunged into the concrete mix. Thereafter, corresponding to the imposed rotation profile shown in Fig. 6, and for each speed level, the total torque is measured and recorded. The rotating vane creates two concrete layers during the rotation process. The first one is beyond the vane and moves with it while the second one still stagnant in the recipient borders. This phenomenon creates friction between the two concrete layers which allows to determine the concrete viscosity.

The plastic viscosity µ was then determined by assimilating concrete rheological behavior to a Bingham fluid to fit the shear stress and shear rate according to the following equation: . 0

τ τ µ γ = + (1)
It has been admitted by many authors in the literature that the concrete rheological behavior presents a laminar flow therefore the Bingham model is used for various types of concrete [START_REF] Wallevik | Relationship between the Bingham parameters and slump[END_REF], [START_REF] Burns | État des connaissances dans le béton projeté par voie humide[END_REF], [START_REF] Ferraris | Modified Slump Test to Measure Rheological Parameters of Fresh Concrete[END_REF], [START_REF] Ferraris | Modified Slump Test to Measure Rheological Parameters of Fresh Concrete[END_REF]- [START_REF] Soualhi | Optimisation de la viscosité des bétons à faible impact environnemental[END_REF]. It is worth noting that, the test protocol and the calculation method used are based on the Bingham model. Besides, the degree of accuracy is 5% and rage of confidence is 5.8% for the viscosity and 3.1% for the yield stress [START_REF] Soualhi | Design of portable rheometer with new vane geometry to estimate concrete rheological parameters[END_REF].

Furthermore, the Bingham model relates linearly the shear stress and the shear rate between adjacent layers of concrete and explains the load distribution of the shear stress which is divided in two parts:

-The yield stress τ0 that represents the minimal stress provided to overcome the aggregates frictions.

-When the yield stress excessed the shear stress the flow starts and the evolution of the shar stress is described by a linear equation Then the plastic viscosity represents the delay effect induced by the friction between the underlying layer during the flow as follows: 2)

0 0 2 2 1 0 1 2 1 ln 4 2 2 h M M h R M h R π τ τ π µ µ π τ   Ω = - -     (2)
Where Ω , M , h , 1 R , µ , and 0 τ present respectively the rotational speed (rad/s), torque (N.m), vane height (m); vane radius (m); plastic viscosity (Pa.s) and the yield stress (Pa).

Based on (Eq. 2), and using the assumed values of µ and 0 τ , the mean squared error is calculated as the difference between the calculated and the measured rational speed for each measured torque point (Eq. 3).

( )

calculated measured mse n Ω -Ω = (3)
Lastly, the plastic viscosity was obtained by minimizing the mse values [START_REF] Soualhi | New model to estimate plastic viscosity of eco-friendly and conventional concrete[END_REF].

• Tribological measurement

Given that pumpability of concrete depends on the steel-concrete friction occurred during the concrete flow through the pumping pipe, the purpose of this part is the determination of the viscous constant. Therefore, a new tribometer developed by Ngo et al. [START_REF] Ngo | Use of tribometer to estimate interface friction and concrete boundary layer composition during the fluid concrete pumping[END_REF] was used (Fig. 8).

The latter is composed of an agitator with electronic speed regulator and torque recorder, which is placed on the top of a smooth steel cylinder (h=10 cm, d=10.7 cm). The test consists in measuring the friction between the metal cylinder and concrete which allows to obtain the interfacial properties as the shear stress and the viscous constant of the boundary layer formed during the test as shown in Fig. 8.

Fig. 8. Equipment used for the determination of the tribological parameters [START_REF] Ngo | Use of tribometer to estimate interface friction and concrete boundary layer composition during the fluid concrete pumping[END_REF].

The measurement of the friction couple was performed with an imposed speed profile (Fig. 9), according to the following steps:

-As it is shown in Fig. 8(b), the first layer of fresh concrete was filled up to the half of the container and received 25 stocks by a metal rod. Then the steel cylinder was fixed to the agitator and centered in the middle of the container.

-After saving the data from the previous phase, the second layer of concrete was filled up to the top of the container Fig. 8(c). The same process of the previous step was followed and measurements were recorded [START_REF] Ngo | Measurement and modeling of fresh concrete viscous constant to predict pumping pressures[END_REF].

It is worth noting that, the first step consists on determining the friction between the bottom of the cylinder and concrete, while the second one permits to measure frictions between concrete and the cylinder (bottom plus lateral surface). As only the lateral frictions are required, the final results are obtained by subtracting the first step data from the second step ones. The viscous constant and ) was determined from the obtained measured torques using the empirical correlation (Eq. 4) and the interface law equation (Eq. 5) to fit the test results as follows:

0 T T kV = + (4) 0 t t τ τ ην = + (5)
Where T (N .m) and 0 T (N .m) are the torque imposed and the initial torque respectively, V (cycles/s) is the cylinder rotating speed and k (N.ms) is the linear coefficient. Besides, the interface parameters are the shear stress t τ (Pa), the viscous constant η (Pa s/m), the yield stress 0t τ (Pa) and the angular speed ν (m/s).

In addition, taking into account the cylinder shape and dimensions, the interfacial parameters were calculated using the following equations:

0 0 2 2 t t t T R h τ π = (6) ( ) 2 3 2 t t k R h η π = (7)
While t R (m) and t h (m) represent respectively the radius and the cylinder height.

Results and discussion

Validation of the control formulation (without fibers)

In this phase, the rheological and tribological behaviors of control concrete was evaluated in order to choose the optimal formulation with the best workability, rheological and tribological parameters. Thus 4 mixes namely CC1, CC2, CC3 and CC4 were performed with W/C and G/S of 0.5 and 1.44 respectively, while the cement paste volume was changed. Results of slump against cement paste volume are presented in Fig. 10. On one hand, it can be observed that the slump value increased by increasing the cement paste volume for the first three mixes.

However, it decreased beyond 0.380 m 3 of paste volume, and the mix obtained was segregated this is may be explained by the fact that the tested formulations have a W/C equal to 0.5 which results in a fluid matrix, and we found that the optimum paste volume is 0.380m 3 beyond that the aggregates are much distant that there is no relation between them. That's why the cement paste flows out of the granular skeleton and induced concrete segregation.

Consequently, this mix was eliminated from both rheological and tribological tests. On the other hand, as it can be seen in Fig. 11(a), plastic viscosity decreased progressively with increasing the cement paste volume. This is due to the lubricating effect of cement paste on the distance inter-aggregates, which creates more distances between aggregates and minimizes the friction. As a matter of fact, during the formulation of 1m 3 of concrete, the aggregates volume decreases in increasing the paste volume. Thus, for a high paste volume, the aggregate volume is low and the aggregates are more discarded and surrounded by the paste. This phenomenon enhances the aggregates movement in concrete during the flow and decreases the concrete rheological parameters. Indeed other studies proved that using a high volume of paste improves the rheological behavior of concrete [START_REF] Soualhi | New model to estimate plastic viscosity of eco-friendly and conventional concrete[END_REF]. Likewise, Fig. 11(b) shows a loss of viscous constant corresponding to the increase of cement paste volume. In fact, the paste cement in the mix reacts as a transporter of small particles (cement and sand) to produce the boundary layer in the mix. Consequently, using an important volume of paste reduced the interfacial friction, which facilitates the aggregates movement and enhances the tribological behavior of concrete. This result is consistent with the literature [START_REF] Chapdelaine | Etude fondamentale et pratique sur le pompage du béton[END_REF], [START_REF] Ngo | Relationships between concrete composition and boundary layer composition to optimise concrete pumpability[END_REF].

It is commonly known that the implementation of fresh concrete by traditional technique or by pumping technic depends on rheological or/and tribological parameters. Furthermore, as much as the rheological and tribological parameters are low as much the implementation is easier. According to this fact, and taking into account the segregation phenomenon, the optimal formulation was attributed to the third mix (CC3) which has the lowest plastic viscosity and viscous constant comparing to other mixtures (Table 5).

Effect of PPF dosage, length and shape on concrete

This part aims to evaluate fibers length and percentage effects on the rheological and the tribological concrete behavior. Hence, nine formulations have been tested using three different fiber lengths (19, 30 and 54 mm) with various percentages (0.12, 0.24 and 0.36 % from the total weight of concrete). Table 6 illustrates the obtained slump results as well as the rheological and the tribological parameters.

Workability results

According to Table 6, it appears that the slump value decreased for all mixes with the increase of the fibers length and dosage which explain that the concrete workability is negatively affected by the inclusion of PPF. For all cases, the mixture composed of 0.12% of PPF was classified as fluid concrete (slump class S4). However, increasing fibers percentage to 0.36% leads to obtain a plastic concrete (slump class S2), therefore the concrete behavior completely changed. These results are consistent with previous studies [START_REF] Zhang | How does adsorption behavior of polycarboxylate superplasticizer effect rheology and flowability of cement paste with polypropylene fiber?[END_REF], [START_REF] Mazaheripour | The effect of polypropylene fibers on the properties of fresh and hardened lightweight selfcompacting concrete[END_REF], [START_REF] Al-Tayyib | Effect of polypropylene fiber reinforcement on the properties of fresh and hardened concrete in the Arabian Gulf environment[END_REF].

Rheological and tribological results

Fig. 12 outlines the variation of the viscosity and the viscous constant as function of the percentage of fibers for different lengths. Results show that, for each given length, the tribological and rheological parameters increased with increasing PPF percentage.

Furthermore, the plastic viscosity results, presented in Fig. 12(a), indicates that the rheological behavior of mixes with 19 and 54 mm PPF was approximatively the same, while the results values are overlapped around the same trend line, concluding that fibers length has no effect on the rheological behavior of concrete. Moreover, FRC30 rheological behavior was different from the others (FRC19 and FRC 54) where the plastic viscosity trend is presented in two parts. In fact, from 0 to 0.12% of PPF the plastic viscosity remains constant. However, beyond this percentage, it increased significantly to 12 Pa.s and remains constant again. This difference of behavior between the fibrillated twist and wave fibers based concrete may be explained by the fact that PPF 30 (wave shape) are more flexible and they are characterized by a high capacity to adapt with granulates movement in the mix. Hence, they can easily fill into the voids between the fine aggregates particles and pass through them making a different distribution of that of 19 and 54 mm PPF [START_REF] Li | Combined effects of water film thickness and polypropylene fibre length on fresh properties of mortar[END_REF]. Furthermore, adding 0.36% for PPF30 have shown a slow increase of the plastic viscosity which is resulted by the fibers important dosage used in the mixtures. Results showed that as the fiber dosage is important as the friction intergranulate increased. These frictions are caused by the granulates rearrangement and the voids filling in mixtures. Besides, the 0.24% represents the optimum dosage of fiber in concrete.

For each given length, the plastic viscosity is more important when the fibers amount is high. This is explained by the fact that the plastic viscosity depends on the solid concentration in the mix [START_REF] Li | Combined effects of water film thickness and polypropylene fibre length on fresh properties of mortar[END_REF], [START_REF] Yin | Comparative evaluation of virgin and recycled polypropylene fibre reinforced concrete[END_REF] which is modified by the PPF inclusion, causing an increase of the granular structure and a drop of the maximum compactness. In addition, the inclusion of fibers hinders the granulate movement during the rheological test (the shearing). This phenomenon results in a decrease of the slump value and an augmentation of the viscosity.

Fig. 12(b)

represents the evolution of the viscous constant against PPF percentage. It can be established that for mixtures with only 0.12% of fibers the viscous constant increases when PPF were added. This is partly explained by the increase of both the concrete viscosity and the interfacial frictions between granulates and fibers. Furthermore, the viscous constant depends on the boundary layer formation which is hindered by the interfacial frictions rustling in an increase of the viscous constant.

Otherwise, FRC54 behavior is more disturbed than that of FRC19 and FRC30 and shows a blockage phenomenon with 0.36% of PPF. For this reason, FRC54 has been eliminated and both rheological and tribological tests have not been carried out for this mix. In fact, during the tests, a high risk of segregation was observed which has been confirmed in the obtained data.

( Even though 54 and 30 mm PPF have not the same shape, both FRC54 and FRC30 have approximatively the same viscous constant trend. This finding indicates that the fibers shape has no effect on the tribological behavior. At the same time, the viscous constant was not well defined by the test equipment because of the high dosage of fibers in mixtures which caused an inhomogeneous PPF distribution as it can be seen in Fig. 13.

Superplasticizer effect

Basing on the previous phase outcomes, it has been concluded that the concrete workability and both rheological and tribological properties shows a significant degradation with the augmentation of the PPF concentration. In order to fix this problem, many researchers in the literature proposed the addition of a superplasticizer in the mix [START_REF] Li | Investigation on mechanical properties and microstructure of high performance polypropylene fiber reinforced lightweight aggregate concrete[END_REF], [START_REF] Zhang | How does adsorption behavior of polycarboxylate superplasticizer effect rheology and flowability of cement paste with polypropylene fiber?[END_REF], [START_REF] Alberti | Fracture mechanics of polyolefin fibre reinforced concrete: Study of the influence of the concrete properties, casting procedures, the fibre length and specimen size[END_REF].

Through this part, the superplasticizer effect on rheological and tribological properties of concrete has been investigated. 8 mixes from the previous phase have been reformulated with the superplasticizer addition to highlight its effect on both rheological and tribological FRC behaviors. The Sp ratio was determined in order to maintain a constant slump (20± 2 cm) with a maximum ratio of 0.4% by the weight of cement. It should be noted that, beyond this limit, concrete segregation was observed. The Rheological and tribological results of FRCSp are resumed in Table 7.

Rheological and tribological behaviors of FRCSp

The evolution of the plastic viscosity and the viscous constant of FRC19 with and without superplasticizer are presented in Fig. 14. The results show a decrease of the two parameters when the superplasticizer is used. In fact, they are lower than those of the control concrete (CC).

Fig. 14(a)

indicates that the plastic viscosity of concrete without superplasticizer increased with increasing fibers dosage. However, the addition of the superplasticizer in the mix permits the regain of the concrete workability as well as its rheological and tribological properties despite the augmentation of fibers dosage. In addition, a stable behavior was observed reveling a good fibers distribution.

In regards to the viscous constant, presented in Fig. 14(b), it remains approximatively constant and equal to that of the control concrete (CC). This result confirms that the superplasticizer enhances the tribological behavior of FRC.

Rheological and tribological behaviors of FRC54Sp

The rheological and tribological behaviors of FRC54Sp are given in Fig. 15. It can be established that the plastic viscosity of FRC54Sp is nearly similar to that of CC3 and it remains constant by adding 0.2% of Sp. This explains that using the superplasticizer with PPF of 54 mm unheeded the negative effect of fibers in fresh concrete. However, adding a high percentage (0.24%) of 54 mm PPF increased significantly the plastic viscosity due to the important length and dosage.

At the same time, adding superplasticizer reduces slightly the viscous constant. As it can be seen in Fig. 15(b), FRC54Sp with 0.12 and 0.24 PPF percentages have almost the same viscous constant, similarly to that of FRC19Sp with 0.12 and 0.24 PPF percentages. It is worth remembering that19 and 54 mm PPF have the same shape which explains the similarity of

Rheological and tribological behaviors of FRC30Sp

Fig. 16 represents the rheological and tribological results of FRC30Sp. The rheological behavior shows a decrease of the plastic viscosity by adding 0.2% of Sp (Fig. 16(a)). Yet, increasing the quantity of PPF to 0.24%, while the Sp ratio remains constant, has always a bad effect on the plastic viscosity. This result permits to conclude that adding superplasticizer in FRC30Sp,with a high percentage of PPF, has no significant effect on its rheology.

From Fig. 16(b), it can be seen that the viscous constant of FRC30Sp decreased significantly with 0.2% of superplasticizer and 0.12% of PPF. Nevertheless, adding more quantity of PPF with the same ratio of Sp increased the viscous constant. It is worth noting that, adding more percentage of Sp with this high ratio of PPF (0.24%) induced a segregation phenomenon and that is why it was limited at 0.4%. Despite the fact that adding a Sp in concrete enhances the tribological parameters, an increase of the viscous constant was recorded with 0.36% of PPF due to the high percentage of both Sp and PPF which caused a concrete segregation problem. In fact, it has been reported that Sp can be absorbed by PPF using the total organic carbon analyses (TOC) manufactured by Shimadzu, TOC-LCPH which measures the amount of PCE absorbed by the cement particles in terms of TOC. In this study the TOC was measured for cement past without fibers and have been compared with TOC results of fibers cement pasts with different dosages and lengths of PPF, the results show that increasing fibers dosage and length increases the amount of TOC which prove the absorption of Sp [START_REF] Zhang | How does adsorption behavior of polycarboxylate superplasticizer effect rheology and flowability of cement paste with polypropylene fiber?[END_REF].

As the superplasticizer was absorbed by each fiber and cement particle, the rheological and tribological behaviors improvement is due to the electrostatic repulsion and steric hindrance effects of the superplasticizer in the mixes as they destroy the flocculation state of the cement particles and fibers. This process leads to decrease the interaction between fibers and cement particles. Consequently, the slump values increased and the rheological and tribological properties improved. These results are in accordance with Zhang et al. [START_REF] Zhang | How does adsorption behavior of polycarboxylate superplasticizer effect rheology and flowability of cement paste with polypropylene fiber?[END_REF] findings about the rheological behavior of fiber cement paste.

Effect of PPF mixed length

In terms of rheology, Fig. 17(a) shows the evolution of the plastic viscosity as a function of PPF19 dosages, from 0 to 100%. Up to 70% of PPF19 dosage, it appears that the plastic viscosity remains constant with the increase of PPF19 percentage at the expense of PPF30. In contrast, from 70 to 100% of PPF19, a significant increase was observed. It can be said that the combination of both shapes and lengths till 70/30 of 19 and 30 mm PPF respectively has eliminated the negative effect of PPF on the rheological behavior of concrete. This is due to the different shapes of the fibers used and the good fibers distribution in the mix thanks to the superplasticizer introduction.

At the same time, the viscous constant slightly increased (Fig. 17(b)) with increasing PPF19 percentage up to 70%, and decreased thereafter to become equal to the viscous constant of FRC19/30Sp with 30% of PPF19.

According to the data obtained, it can be concluded that the combination of two different lengths and shapes has a less significant effect on the rheological and tribological concrete behaviors than that of using only one length and shape. It is worth noting that the lowest viscosity was obtained by using 30/70% of 19 and 30mm PPF respectively.

Fig. 2 .

 2 Fig. 2. Polypropylene fibers (PPF).

Fig. 1 .Fig. 3 .

 13 Fig. 1. Grading curves of fine and coarse aggregates.

Fig. 4 .

 4 Fig. 4. SEM images of wave PPF.

Fig. 5 .

 5 Fig. 5. Equipment used for the determination of the rheological parameters [23].

Fig. 6 .

 6 Fig. 6. Imposed vane rotation speed profile [22].

Fig 7 .

 7 Fig 7. Contribution of solid phase and liquid phase to the shear strength of concrete[33].

Fig. 9 .

 9 Fig. 9. Imposed cylinder rotation speed profile [34].

Fig. 10 .

 10 Fig. 10. Slump test results of control concrete.

Fig. 11 .

 11 Fig. 11. Plastic viscosity of control concrete (a), viscous constant of control concrete (b).

Fig. 12 .

 12 Fig. 12. Evolution of plastic viscosity (a) and viscous constant (b) as function of fibers percentage.

Fig. 13 .

 13 Fig. 13. The 54 mm PPF concentration around the vane during the test.

Fig. 14 .

 14 Fig. 14. Evolution of plastic viscosity (a) and viscous constant (b) of FRC19Sp as function of fibers percentage.

FRC19SpFig. 15 .

 15 Fig. 15. Evolution of plastic viscosity (a) and viscous constant (b) of FRC54Sp as function of fibers percentage.

Fig. 16 .

 16 Fig. 16. Evolution of plastic viscosity (a) and viscous constant (b) of FRC30Sp as function of fibers percentage.

Fig. 17 .

 17 Fig. 17. Evolution of plastic viscosity (a) and viscous constant (b) of FRC19/30Sp as function of fibers percentage.

Table 1

 1 Chemical and mineral compositions of Portland cement (CEM I 52.5N).

	Chemical composition	Mineral composition
	Element	(%)	Element	(%)
	PAF	1.4	C3A	8.5
	INS	0.4	C3S	61.8
	SiO2	20	C4AF	10.7
	Al2O3	5.1	C2S	19
	Fe2O3	3.2	-	-
	CaO	64.5	-	-
	MgO	0.8	-	-
	SO3	3.2	-	-
	K20	0.82	-	-
	Na2O	0.23	-	-
	S 2-	0.01	-	-
	Cl -	0.06	-	-
	CO2	1	-	-
	CaO	2	-	-
	Na2Oeq	0.77	-	-

Table 2 Aggregates properties. Aggregates type Size (mm) Specific gravity (g/cm 3 ) Compactness Water absorption (%) Finesse modulus Gravel

 2

	(G)	6.3-20	2.42	0.61	2.3	-
	Sand (S)	0-4	2.79	0.64	0.9	2.32

Moreover, in the present investigation, PPF produced by SikaFibre ® were added to concrete mixes with 3 different lengths (19, 30 and 54 mm) as shown in Fig.

2

.

Table 3 PPF properties. 139 Length Diameter Shape Specific gravity Elastic modulus Tensile strength

 3

	(mm)	(mm)			(GPa)	(MPa)
	19	0.34	Fibrillated twist 0.92	5.75	689
	30	0.48	wave	0.91	6.90	486
	54	0.34	Fibrillated twist 0.92	5.75	689
	140

Table 4

 4 Mix proportions.

		166
	Phase Concrete	CP.V	W/C	G/S	L Fiber	Fiber	Cement	Water	Aggregate	Sand	Sp	Fresh	Air
			(m 3 )			(mm)	(%)	(Kg)	(Kg)	(Kg)	(Kg)	(%)	density	content
													(kg/m 3 )	(%)
	1	CC1	0.280	0.5	1.44 -	-	340	170	1071	744	-		2.2
		CC2	0.316					384	192	1032	717			2.1
		CC3	0.380					462	231	963	669			2.2
		CC4	0.395					480	240	947	658			2.3
	2	FRC19	0.380	0.5	1.44 19	0.120	462	231	963	669	-		2.0
							0.240							2.2
							0.360							2.1
		FRC30				30	0.120							2.1
							0.240							2.0
							0.360							2.1
		FRC54				54	0.120							2.1
							0.240							2.2
	3	FRC19Sp	0.380	0.5	1.44 19	0.120	462	231	963	669	0.2		2.1
							0.240					0.4		2.3
							0.360					0.4		2.3
		FRC30Sp				30	0.120					0.2		2.2
							0.240					0.2		2.3
							0.360					0.4		2.2
		FRC54Sp				54	0.120					0.2		2.2
							0.240					0.2		2.3
	4	FRC19/30Sp 0.380	0.5	1.44 19(50%) 0.060	462	231	963	669	0.2		2.3
						30(50%) 0.060
						19(70%) 0.084							2.2
						30(30%) 0.036
						19(30%) 0.036							2.2
						30(70%) 0.084
		167

Table 5

 5 Formulation of the control mix.

Table 6

 6 Rheological and tribological results of FRC.

	Phase Concrete	CP.V	W/C	G/S	LFiber	Fiber	Slump	µ	τ0	η	τ0t
		mixes	(m 3 )			(mm)	(%)	(cm)	(Pa.s)	(Pa)	(Pa.s/m)	(Pa)
	2	FRC19	0.380 0.5	1.44	19	0.12	19	9.9	153.0 214.7	40.9
							0.24	17	10.5	214.2 270.5	27.6
							0.36	11	14.8	182.5 315.7	32.4
		FRC30	0.380 0.5	1.44	30	0.12	20	7.1	114.0 122.9	70.4
							0.24	16	12.6	140.2 250.3	77.9
							0.36	10	11.7	159.4 141.8	71.2
		FRC54	0.380 0.5	1.44	54	0.12	16	9.3	250.0 256.5	75.5
							0.24	13	10.8	282.9 225.5	32.7

Table 7

 7 Rheological and tribological results of FRCSp.

	Phase Concrete	CP.V	W/C G/S LFiber	Fiber	Sp	Slump	µ	τ0	η	τ0t
	mixes	(m 3 )		(mm)	%	%	(cm)	(Pa.s)	(Pa)	(Pa.s/m)	(Pa)
	FRC19Sp 0.380 0.5	1.44 19	0.12	0.2 22	3.9	80.1	127.1	39.2
					0.24	0.4 22	4.1	39.8	151.3	27.6
					0.36	0.4 20	5.5	77.8	121.0	19.6
	FRC30Sp			30	0.12	0.2 22	2.6	69.5	64.9	29.6
					0.24	0.2 18	13.7	26.5	106.8	34.8
					0.36	0.4 20	13.1	45.2	228.7	25.0
	FRC54Sp			54	0.12	0.2 22	7.1	86.5	186.8	32.4
					0.24	0.4 19	12.1	169.2 177.8	45.8

Concluding remarks

This study investigates the PPF effect on both rheological and tribological behaviors of fresh concrete. Therefore 4 series of formulation have been done, where the concrete rheological and tribological behavior was evaluated without and with the addition of different lengths (19, 30, 54mm) and dosages (0.12; 0.24; 0.36%) of PPF. The inclusion of fibers in mixtures induced a workability loss besides a higher viscosity and viscous constant. Thus, a superplasticizer was added in order to enhance the fresh state concrete parameters.

Based on the obtained results, the main conclusions can be summarized as follows:

• Increasing the cement paste volume effectively enhances the rheological and tribological parameters of ordinary concrete.

• For the same PPF type, the slump decreases with increasing fibers length. However, this later does not affect the plastic viscosity.

• The augmentation of fibers percentage leads to increase the plastic viscosity of FRC. Furthermore, the use of fibrillated twist fibers permits to reduce the plastic viscosity more than wave fibers.

• In regards to the tribological test, the use of a superplasticizer is required in order to insure the smooth functioning of the tribometer.

• The addition of the superplasticizer have not affect the viscous constant. In contrast, adding 0.4% of Sp results in 3 times lower plastic viscosity comparing with FRC without superplasticizer.

• The use of different lengths and shapes of PPF in the same mixture improves the rheological and the tribological behaviors and enhances the fibers homogenization in the FRC.

• Among the various fibers investigated, 19 mm PPF with fibrillated shape seems to have the best effect in terms of rheological and tribological concrete behaviors.