Chedid Saade
email: chedid.saade@univ-eiffel.fr

Zheng Li
email: zheng.li@univ-eiffel.fr

Sandra Escoffier
email: sandra.escoffier@univ-eiffel.fr

Luc Thorel
email: luc.thorel@univ-eiffel.fr

Centrifuge and numerical modeling of the behavior of homogeneous embankment on liquefiable soil subjected to dynamic excitation

Keywords: Centrifuge modeling, Numerical modeling, Embankment, Ground liquefaction List of symbols CRR Cyclic resistance ratio CRR TX Cyclic resistance ratio of cyclic triaxial test

Earth dams, levees, berms, and embankments have played an important role in the development of infrastructure throughout history. Considered among the oldest forms of construction in civil engineering, these structures serve multiple purposes, including transportation, irrigation, flood control, and power generation. Their failure may cause a huge disaster and loss of lives and facilities [START_REF] Wu | 20 -embankments and dams[END_REF]. Under dynamic loading, dam failures like cracks, huge settlements or internal erosion can occur [START_REF] Veylon | Modélisation numérique du mécanisme de liquéfaction des sols : application aux ouvrages hydrauliques[END_REF] even in low to medium seismic zones [START_REF] Kawakami | Damage to the ground and earth structures by the niigata earthquake of june 16, 1964[END_REF][START_REF] Seed | Performance of earth dams during earthquakes[END_REF]. Historical cases, for example the Chilean earthquake in 1960 [START_REF] Duke | The chilean earthquakes of May 1960[END_REF], the San Fernando Earthquake in 1971 [START_REF] Seed | The slides in the san fernando dams during the earthquake of february 9, 1971[END_REF] and the Tohuko earthquake in 2011 [START_REF] Yamaguchi | Liquefaction in Tohoku district during the 2011 off the Pacific Coast of Tohoku Earthquake[END_REF][START_REF] Oka | Damage patterns of river embankments due to the 2011 off the pacific coast of tohoku earthquake and a numerical modeling of the deformation of river embankments with a clayey subsoil layer[END_REF], reveal that earthquake-induced liquefaction was the most significant cause of embankment failure.

Therefore, the study of the damage or even failure of embankment due to the earthquake-induced liquefaction is of considerable importance from both public safety and financial standpoints.

In recent decades, physical tests have been conducted to directly observe and study the behavior and failure mechanisms of embankments under dynamic loading. Shaking table tests have been utilized to investigate the embankment response to soil ground liquefaction. For instance, studies by [START_REF] Iwasaki | Soil liquefaction studies in japan: state-of-the-art[END_REF], [START_REF] Koga | Shaking table tests of embankments resting on liquefiable sandy ground[END_REF] and [START_REF] Park | Liquefaction of embankments on sandy soils and the optimum countermeasure against the liquefaction[END_REF] have reported significant deformations in the embankment, including crest settlement and lateral displacement of embankment sides. Besides the 1g shaking table tests, Ng centrifuge modeling could be regarded as a better option in the study of embankment failure due to liquefaction. Based on the theory of scaling laws and the principle of stress similitude [START_REF] Garnier | Catalogue of scaling laws and similitude questions in geotechnical centrifuge modelling[END_REF], centrifuge modeling allows to interpret the behavior of a large-scale prototype model with a reduced-size model that subjected to the artificial macro-gravity. Many centrifuge tests in the literature (e.g. [START_REF] Ng | Centrifuge modeling of loose fill embankment subjected to uni-axial and bi-axial earthquakes[END_REF], [START_REF] Okamura | Seismic stability of embankments subjected to pre-deformation due to foundation consolidation[END_REF], [START_REF] Higo | Study of dynamic stability of unsaturated embankments with different water contents by centrifugal model tests[END_REF] , [START_REF] Adapa | Seismic stability of embankments with different densities and upstream conditions related to the water level[END_REF] and [START_REF] Izawa | Seismic Design of Embankments in Consideration of Damage Process during Earthquakes[END_REF]) have focused on the responses of embankments resting on non-liquefiable ground under earthquakes actions. Centrifuge tests were conducted by [START_REF] Adalier | Embankment dam on liquefiable foundation dynamic behavior and densification remediation[END_REF], Pramaditya and Fathani (2021a) and [START_REF] Pourakbar | Dynamic centrifuge tests to evaluate the seismic performance of an embankment resting on liquefiable ground improved by unreinforced and reinforced soil-cement columns[END_REF] to investigate the response of the embankment to soil ground liquefaction. These studies found that liquefaction typically occurs beneath the ground surface near the toe of the embankment, while the soil beneath the embankment is less susceptible to liquefaction. Furthermore, [START_REF] Adalier | Foundation liquefction countermeaser for earthquake embankements[END_REF] investigated the effects of the thickness and position of the liquefiable layer on embankment behavior. Additionally, [START_REF] Park | Liquefaction of embankments on sandy soils and the optimum countermeasure against the liquefaction[END_REF], [START_REF] Okamura | Effects of remedial measures for mitigating embankment settlement due to foundation liquefaction[END_REF], Tiznado et al.

(2020), [START_REF] Li | Centrifuge study on the effect of the SCP improvement geometry on the mitigation of liquefaction-induced embankment settlement[END_REF], Pramaditya and Fathani (2021b) and [START_REF] Pourakbar | Dynamic centrifuge tests to evaluate the seismic performance of an embankment resting on liquefiable ground improved by unreinforced and reinforced soil-cement columns[END_REF] have highlighted the effects of liquefaction remediation on embankment response, taking into account different types of liquefaction reinforcement.

Numerical modeling or simulation is a versatile and important technical approach that has been utilized in the geotechnical earthquake engineering community to study soil liquefaction. The numerical simulations of earthquakeinduced liquefaction were mainly conducted on the continuum domain by exploiting either Finite Element Method (FEM) [START_REF] Finn | Evolution of dynamic analysis in geotechnical earthquake engineering[END_REF][START_REF] Elgamal | Numerical analysis of embankment foundation liquefaction countermeasures[END_REF][START_REF] Aydingun | Numerical analysis of seismically induced liquefaction in earth embankment foundations. Part I. Benchmark model[END_REF][START_REF] Okochi | Dynamic centrifuge model tests of embankment with a new liquefaction countermeasure by ground improvement considering constraint effect[END_REF][START_REF] Bhatnagar | Numerical analysis of earth embankment resting on liquefiable soil and remedial measures[END_REF][START_REF] Gobbi | Numerical analysis of soil liquefaction induced failure of embankments[END_REF][START_REF] Rapti | Liquefaction analysis and damage evaluation of embankment-type structures[END_REF][START_REF] Chakraborty | Earthquake response of embankment resting on liquefiable soil with different mitigation models[END_REF] or Finite Difference Method (FDM) [START_REF] Wu | Seismic response of an earth dam: finite element coupling analysis and validation from centrifuge tests[END_REF][START_REF] Bouraoui | Modélisation numérique de la déformation induite par la liquéfaction de barrages en terre soumis à des chargements sismiques[END_REF][START_REF] Ziotopoulou | Seismic response of liquefiable sloping ground: Class a and c numerical predictions of centrifuge model responses[END_REF]Boulanger et al., 2015;[START_REF] Boulanger | Nonlinear deformation analyses of an embankment dam on a spatially variable liquefiable deposit[END_REF][START_REF] Dinesh | Performance evaluation of PM4Sand model for simulation of the liquefaction remedial measures for embankment[END_REF]. The main objectives of these numerical simulations were to predict the behavior or the failure of the embankment due to earthquake-induced liquefaction.

In order to have a more accurate and realistic prediction of the complex liquefaction phenomena, in the literature, many constitutive models were proposed and then used in the simulation of liquefaction-related problems. Examples of the constitutive models for liquefaction analysis include the multi-surface plasticity model [START_REF] Elgamal | Modeling of cyclic mobility in saturated cohesionless soils[END_REF],

the bounding surface plasticity model -SANISAND [START_REF] Dafalias | Simple plasticity sand model accounting for fabric change effects[END_REF], the UBCSand model [START_REF] Beaty | UBCSAND Constitutive Model: Version 904aR[END_REF], the coupled effective stress WANG model [START_REF] Wang | Bounding surface plasticity model for liquefaction of sand with various densities and initial stress conditions[END_REF] the responses in terms of accelerations and excess pore pressures. In recent years, the PM4Sand constitutive model has proven to be successful in simulating soil liquefaction [START_REF] Rahmani | Investigation of the influence of permeability coefficient on the numerical modeling of the liquefaction phenomenon[END_REF]Chiaradonna et al., 2022) and in investigating its effect on the behavior of various geotechnical structures, such as dams (Boulanger and Ziotopoulou, 2015;[START_REF] Boulanger | Nonlinear deformation analyses of an embankment dam on a spatially variable liquefiable deposit[END_REF][START_REF] Lu | Seismic Performance Screening and Evaluation for Embankments on Liquefiable Foundation Soils[END_REF], traffic embankments [START_REF] Oblak | Fragility Assessment of Traffic Embankments Exposed to Earthquake-Induced Liquefaction[END_REF], structures with shallow foundation (Kassas et al., 2021b) and offshore pipelines [START_REF] Seth | Numerical modelling of the effects of liquefaction on the upheaval buckling of offshore pipelines using the PM4Sand model[END_REF]. PM4Sand is also one of the recommended models by United States Society on Dams (USSD) for the seismic analysis of embankments, dams, and levees (USSD, 2022).

This study investigates the behavior of a homogeneous embankment on liquefiable sand ground under dynamic loading, using a combined experimental and numerical approach. The liquefiable ground was constructed with the wet under-compaction method and the effect of the humid tamping on the liquefaction resistance of sand was carefully considered. Although a rigid (or strong) container without any additional absorbing materials was used in the centrifuge test, the impact of boundary effects was found to be limited. The outline of this paper is as follows. Firstly, the experimental set-up including the model geometry, model preparation and instrumentation is presented. Then, the numerical modeling of the centrifuge test is discussed in detail, including FEM mesh, boundary conditions, constitutive model calibration, and modeling execution procedure etc. Next, the experimental results in terms of the model responses and liquefaction pattern are presented and discussed. The performance of the numerical model is then verified by comparing the experimental and numerical results in terms of acceleration, excess pore pressure, model deformation pattern, settlement etc. Finally, the effects of the boundary conditions on the model responses are evaluated numerically with models with larger and different boundaries.

Experimental set-up of the dynamic centrifuge test

In the present study, the dynamic centrifuge test was conducted at the University Gustave Eiffel (Nantes Campus).

The centrifugal acceleration level was 60 g. To excite the model, a 1D shaking table embedded in the swinging basket was employed, as previously described by [START_REF] Chazelas | Original technologies for proven performances for the new LCPC earthquake simulator[END_REF]. This section provides a detailed description of the experimental set-up utilized in the dynamic centrifuge test.

Geometry and dimensions of the centrifuge model

The geometry of the centrifuge model is shown in Fig. 1. According to the scaling laws [START_REF] Garnier | Catalogue of scaling laws and similitude questions in geotechnical centrifuge modelling[END_REF], the reduced-size centrifuge model represents a prototype embankment of 6 m in height, which rests on a liquefiable layer 9 m thick. The slope of the embankment is determined to be 1/1.5, which is closed to the friction angle of the sand that was used in the embankment of the centrifuge model.

Materials of the centrifuge model

The sand used to reconstitute the ground and the embankment was Hostun HN31 sand. Hostun HN31 sand is a fine sub-angular to angular siliceous sand [START_REF] Benahmed | Comportement mécanique d'un sable sous cisaillement monotone et cyclique: Application aux phénomène de liquéfaction et de mobilité cyclique[END_REF][START_REF] Benahmed | Initial structure and static liquefaction properties of sand[END_REF] and its main properties are listed in Table 1. The ground layer is fully saturated with a fluid that has a viscosity 60 times (60 cSt) higher than pure water to satisfy the scaling laws in the centrifuge test at 60g [START_REF] Adamidis | Use of viscous pore fluids in dynamic centrifuge modelling[END_REF]. The viscous fluid was prepared by a mixture of water with hydroxypropyl methylcellulose (HPMC) [START_REF] Dewoolkar | A substitute pore fluid for seismic centrifuge modeling[END_REF][START_REF] Escoffier | Leap-ucd-2017 centrifuge test at ifsttar[END_REF].

Model preparation

The centrifuge model was prepared in two different steps. The ground layer and the embankment were prepared by different techniques proposed by [START_REF] Adalier | Embankment dam on liquefiable foundation dynamic behavior and densification remediation[END_REF]. In this section, a detailed procedure for preparing the soil ground and the embankment is presented.

Liquefiable ground layer

The liquefiable ground layer which had a total thickness of 150 mm was prepared by the under-compaction method proposed by [START_REF] Ladd | Specimen preparation and liquefaction of sands[END_REF]. The moist Hostun HN31 sand was used with a 5% content of viscous fluid for tamping. The ground soil was then compacted layer by layer using a rectangular tamper of 1.4 kg which was dropped from a fixed height of 70 mm. As the tamper completed one compact at each position, it moved on to the next, until the height of the soil layer reached the controlled under-compaction depth. The levelness of each soil layer after under-compaction was carefully checked. Totally, six layers of sand were prepared for the ground layer. The relative density I D was controlled at around 50.0%±2%.

Embankment

After completing the ground layer, a flat layer of dry dense Hostun HN31 sand with a thickness of 100 mm was then prepared using air pluviation. The pluviation was carried out with an automatic sand hopper [START_REF] Garnier | Physical models in geotechnics: state of the art and recent advances[END_REF] with a slot opening equal to 4 mm, a falling height of 750 mm and a constant horizontal moving speed of 100 mm/s. With these parameters, the relative density I D was estimated around 80.0%±1.5%. In order to achieve the desired shape without disturbing the dry sand of the embankment, the flat layer of dry sand was carefully and gradually trimmed using a vacuum cleaner (Fig. 3). Finally, the shape of the embankment was double-checked with a wooden mould. The centrifuge model was saturated with viscous fluid at 1g in a vacuum chamber following the saturation procedure proposed by [START_REF] Ueno | Methods for preparation of sand samples[END_REF] and [START_REF] Mulilis | The effects of method of sample preparation on the cyclic stress-strain behaviour of sands[END_REF]. The air in the model was removed by applying vacuum and then flushing CO 2 by two times. Then, viscous fluid was injected into the centrifuge model under an absolute pressure of 50 mbar from the bottom of the container. After the saturation process was completed and the water level reached 5 mm above ground surface, the degree of saturation of the ground layer was checked by using the "laser-floater" method proposed by [START_REF] Okamura | Preparation of fully saturated models for liquefaction study[END_REF]. The global degree of saturation was estimated to be approximately 99.0%.

Instrumentation

The type and location of the sensors are shown in Fig. 1 and the detailed information about the sensors is listed in Table 2. Totally, 20 accelerometers (A) and 11 pore pressures transducers (P) were installed to follow the soil motion and the pore pressure evolution during the dynamic event, respectively. For example, Fig. 4 illustrates the positioning of a 1D accelerometer at a certain depth in the sand. There were also three pairs of bender element located at three different depths. Three laser displacement sensors were installed to track the settlements of the crest and mid-slope of the embankment. Two potentiometer sensors were also placed at the free ground surface near the toe of the embankment.

Experimental program

The dynamic centrifuge test was performed in three stages: in-flight consolidation, bender element test and shaking/excitation. For the in-flight consolidation, the centrifuge model was gradually spun up 10 g by 10 g until up to 60 g. The model was then kept at 60 g for 10 mins to fully consolidate the model in K 0 condition and stabilize the initial geo-static stress and pore pressures. Fig. 5 shows the initial pore pressure profiles obtained from the theoretical calculation, FEM solution (its details were introduced in Section 3 hereafter.) and experimental measurement. The pore pressure profiles were also compared under the embankment and under the free ground. The experimental measurement agreed well with the numerical model and the theoretical calculation. The theoretical profile was calculated by assuming an absolute dry state of the embankment, with the water level exactly at the surface of the ground. However, during the centrifuge test, the embankment was in an unsaturated state (with an estimated fluid content of around 3.5%). At the interface between the embankment and ground, the actual water level in the center of the embankment was slightly higher than that of the ground surface. This effect was observed in the numerical model and experimental measurement after consolidation, as it is shown in Fig. 5. After in-flight consolidation and before the shaking, bender element tests were performed to measure the shear wave velocities at three different depths, see Fig. 1. Then, the model was subjected to a 1D horizontal sinusoidal base shaking. As it is shown in Fig. 6, the target input signal used in the test consisted of 20 full cycles with a dominant frequency of 1.5 Hz and an amplitude of 0.2 g in prototype scale. Due to the non-perfect response of the hydraulic shaker, the input signal was unavoidably noised by multiple high-frequency contents with small amplitudes. Fig. 6 also shows the actual input signal and its frequency contents measured at the base of the container (A1). As a result of the presence of these small amplitudes at higher frequencies, the peak acceleration of the actual input signal is therefore slightly higher than that of the target amplitude.

Numerical modeling

The numerical simulation of the centrifuge test was carried out on the OpenSees platform -"Open System for Earthquake Engineering Simulation", which is a well known finite element method based open source code [START_REF] Mckenna | OpenSees command language manual[END_REF]. In this section, a detailed description is provided regarding the FEM mesh, boundary conditions, constitutive laws, and calibration of the constitutive model parameters.

FEM mesh and boundary conditions

The numerical model was built at the prototype scale and the FEM mesh is shown in Fig. 7. The ground soil and embankment were modeled using 4-node quadrilateral SSPquadUP elements. The SSPquadUP element which has a mixed displacement-pressure up formulation enables the dynamic plane strain analysis of fluid saturated porous media [START_REF] Mcgann | Stabilized single-point 4-node quadrilateral element for dynamic analysis of fluid saturated porous media[END_REF]. There were 1728 elements and 756 elements for the soil ground and the embankment, respectively. According to the mesh sensitivity analyses, the results of the model with the current mesh density (mesh size ≈ 0.5 m) are accurate enough.

The boundary conditions in the numerical model were chosen to accurately reflect the actual conditions in the rigid container which was used in the dynamic centrifuge test. All the nodes at the base were fixed in both horizontal X and vertical Y directions. While the lateral sides of the model were fixed only in the horizontal X direction. Drained conditions were assigned to the surfaces of the ground and the embankment (blue dashed line); while undrained conditions were assigned to the bottom and the lateral sides.

Constitutive model and parameter calibration

In this study, the behavior of liquefiable sand was modeled with the PM4Sand constitutive law which was developed by Boulanger and Ziotopoulou (2015). This model describes the plasticity of sand controlled by the stress ratio related to the critical state and the bounding surface [START_REF] Dafalias | Simple plasticity sand model accounting for fabric change effects[END_REF]). The PM4Sand model has the capability of predicting the behavior of soil under cyclic loading for various relative densities. PM4Sand model requires 27 input parameters: 6 primary parameters and 21 secondary parameters (Boulanger and Ziotopoulou, 2015).

The primary parameters include the relative density I D , small-strain shear modulus coefficient G o , contraction rate parameter h po , the atmospheric pressure p A and two 'flag' parameters (FirstCall and PostShake). There are variety of methods to estimate the small-strain shear modulus either by field tests (through empirical correlations with SPT or CPT tests) or from laboratory element tests (for example bender element test). In this study, the small-strain shear modulus coefficient G o was directly estimated through the equations proposed by [START_REF] Azeiteiro | Critical state-Based interpretation of the monotonic behavior of Hostun Sand[END_REF] and Kassas et al. (2021b) for Hostun HN31 sand. The contraction rate parameter h po can be calibrated with the experimental results in terms of the cyclic resistance curve (CRR -N) of the sand. While the other secondary parameters can be either left with their default values or calibrated against the laboratory tests data (Boulanger and Ziotopoulou, 2015).

In this study, the variable h o that adjusts the ratio of plastic modulus to elastic modulus was taken as the minimum value 0.3, while the other secondary input parameters were left to their default values.

In the centrifuge model, although the same sand was used for the embankment and the liquefiable ground, according to the literature [START_REF] Mulilis | Effects of sample preparation on sand liquefaction[END_REF][START_REF] Tatsuoka | Cyclic undrained triaxial and torsional shear strength of sands for different sample preparation methods[END_REF][START_REF] Benahmed | Comportement mécanique d'un sable sous cisaillement monotone et cyclique: Application aux phénomène de liquéfaction et de mobilité cyclique[END_REF][START_REF] Sze | Failure Modes of Sand in Undrained Cyclic Loading : Impact of Sample Preparation[END_REF], the soil preparation method affects the liquefaction resistance. Therefore, when calibrating the PM4Sand model, it is essential to consider the effects of the preparation method i.e. the under-compaction method used for the liquefiable ground layer.

For the model parameters of PM4Sand which are not dependent on the sand preparation method i.e. G o , ϕ, Q, R, the values proposed by Kassas et al. (2021b) were adopted (see Table 3). On the other hand, the contraction rate parameter h po , which plays an important role in stimulating the liquefaction resistance of sand under cyclic loading, has to be calibrated carefully by taking into account the effect of the sand preparation method. The calibration of h po was carried out against the experimental data of CRR -N curve of Hostun HN31 sand prepared by moist tamping method. For the CRR -N curve, since the direct data of undrained cyclic simple shear test of Hostun HN31 sand with humid under-compaction method was rare, thus the experimental data has to be obtained indirectly from other type of cyclic test e.g. triaxial cyclic test. In this study, the isotropically consolidated undrained cyclic triaxial tests on Hostun HN31 sand of [START_REF] Gobbi | Characterization of liquefaction parameters for saturated soil under dynamic loading using laboratory tests and calibration of constitutive laws by numerical modelling[END_REF] and Gobbi et al. (2022a,b) were converted to undrained simple shear CRR -N form by Eq. ( 1) proposed by [START_REF] Idriss | Soil liquefaction during earthquakes[END_REF].

K 0 = 1 -sin ϕ 0.44 Poisson's ratio † [-] ν = K 0 1+K 0 0.3 Bolton's Constant † [-] Q 8.4 Bolton's Constant † [-] R 0.78 Permeability † [m/s] k = 2.92 × 10 -3 ×
CRR = 1 + 2K 0 3 CRR TX (1)
where CRR is the cyclic resistance ratio in simple shear form; CRR TX is the cyclic resistance ratio of cyclic triaxial test; K 0 is the lateral pressure coefficient for simple shear test.

An empirical equation Eq. ( 2) was used to fit the converted undrained simple shear CRR -N data. Eq. ( 2) is based on the equation originally proposed by Kassas et al. (2021b), with modifications on the parameters. The contraction rate parameter, h po , was then calibrated by conducting a series of numerical simple shear tests and matching the results with the CRR -N curve obtained from Eq. ( 2).

CRR = (-1.119 e + 1.14) N (0.24e-0.44) (2)
In this study, the simple shear test was carried out on a single element (SSPquadUP) under the initial vertical stress σ v = 100 kPa. The optimized value of h po was found to be 0.07 and the calibration results are shown in Fig. 8. It is shown that the calibration fits well with the curve globally. In case of higher CRR ratio (N ranges from 1 to 30) where the liquefaction is triggered after a few cycles, the fitting remains satisfactory. The main parameters of PM4Sand model used in this study are summarized in Table 3. For the embankment soil, the Drucker-Prager model was adopted. As it was mentioned in section 2.4.3, the centrifuge model was saturated at 1g with viscous fluid. Due to the capillary effect, when the soil ground was fully saturated at 1g condition, the 100 mm height embankment was also partially saturated with viscous fluid, of which the water content is around 25%. According to [START_REF] Caicedo | Centrifuge modelling of unsaturated soils[END_REF], when spin-up the centrifuge, due to the macro-gravity, the water content in the embankment can drop to around 2.0∼3.5%, which was also confirmed by measuring the water contents of the samples taken from the embankment after the test. As a result of the unsaturated embankment and the viscous fluid, the cohesion and strength of the embankment soil increased. Therefore, in this study the peak friction angle 38 • , the bulk modulus 173 kPa and the shear modulus 76 kPa were assumed as the parameters of the Drucker-Prager model for the unsaturated embankment soil according to [START_REF] Chen | Constitutive equations for engineering materials[END_REF].

The cohesion of soil was set at 65 kPa in order to fit the settlement of the embankment as presented in Fig. 18 (see Section 5.4).

Procedure of numerical simulation

The numerical simulation was conducted on the FEM platform OpenSees -Version 3.2.2 64-Bit (OpenSees, 2020) in three steps. Firstly, the consolidation of the model was performed. At this stage, all elements' material properties were switched to elastic. Then, the material properties were switched from elastic to plastic and the "Update FirstCall" is activated and a "plastic gravity analysis" step was performed to initialize the internal variables and to reach the initial stress state of the model (see Section 2.6). After the consolidation, the whole model was subjected to dynamic shaking by "UniformExcitation" [START_REF] Mazzoni | Opensees command language manual[END_REF]. The numerical model was excited by the actual input signal applied in the dynamic centrifuge test, see Fig. 6. During the dynamic excitation, a Rayleigh damping ratio of 3.0% was applied. The damping ratio was estimated through bender element test by the method proposed by [START_REF] Karl | Determination of the material damping ratio with the bender element test[END_REF].

The damping parameters were calculated with two modal frequencies 2.7 Hz and 6.7 Hz. These two frequencies were estimated by modal analysis with elastic material properties in Table 3.

In order to properly model the variation of the excess pore pressure, the hydraulic conductivity in the numerical model varied in two different stages: first, during the base shaking; and second, during the dissipation of excess pore pressure after shaking. During base shaking, the procedure proposed by [START_REF] Shahir | Employing a variable permeability model in numerical simulation of saturated sand behavior under earthquake loading[END_REF] was adopted. The permeability was updated at each step according to Eq. ( 3), in which the permeability varied in function of the excess pore pressure ratio [START_REF] Shahir | Employing a variable permeability model in numerical simulation of saturated sand behavior under earthquake loading[END_REF][START_REF] Dinesh | Performance evaluation of PM4Sand model for simulation of the liquefaction remedial measures for embankment[END_REF][START_REF] Basu | Observations and challenges in simulating post-liquefaction settlements from centrifuge and shake table tests[END_REF].

k =              k b = k i (1 + (α -1)r u β 1 ) (if r u < 0.95) k l = α k i (if r u ≥ 0.95) (3)
where k i is the initial permeability of the soil; k b is the permeability during the build up of pore pressure (r u < 0.95); k l is the permeability during liquefaction (r u ≥ 0.95). The two constants α and β 1 were taken as 1.2 and 1, respectively, as they were proposed by [START_REF] Basu | Observations and challenges in simulating post-liquefaction settlements from centrifuge and shake table tests[END_REF] for Hostun sand.

After the shaking, the "PostShake flag" in PM4Sand model was activated and the hydraulic conductivity was considered to decrease earlier before the excess pore pressure dissipation [START_REF] Shahir | Evaluation of variation of permeability in liquefiable soil under earthquake loading[END_REF]. It is recommended by [START_REF] Tobita | Variation of permeability of viscous fluid during liquefaction model testing[END_REF] that the permeability should decrease to a certain value lower than its initial value during the dissipation process. Therefore, Eq. ( 4) was used for the reduction of hydraulic conductivity, which starts at the middle of the last cycle of the input signal (t 2 = 18.1 s) and ends at the last cycle (t 3 = 18.7 s) as presented in Fig. 6(a). This equation is based on the one proposed by [START_REF] Shahir | Evaluation of variation of permeability in liquefiable soil under earthquake loading[END_REF], but takes into account the reduction of permeability by a constant factor of α 2 .

k d = k i α 2 (1 + (αα 2 -1) t -t 2 t 3 -t 2 β 2 ) (4)
where k d is the permeability in the decreasing stage; t 2 and t 3 are the start and end time of decreasing stage, respectively; and the constants α 2 and β 2 were taken as 40 and 1 in this study to have the best fit with the experimental results.

Experimental results analysis

This section presents and discusses the main results of the dynamic centrifuge test. Firstly, the bender element test results before the base shaking are presented. Then, during the base shaking, the results are presented in terms of acceleration responses, the excess pore pressure ratio, liquefaction paths, etc. Finally, the model deformation and embankment failure pattern are discussed. All the experimental results hereafter are presented at the prototype scale unless otherwise mentioned.

Bender elements results

The shear wave velocity V s is calculated from the distance and the travel time of the shear wave between the transmitter and receiver. Concerning the determination of the travel time, different definitions of first arrival point are proposed in literature [START_REF] Jovičić | Objective criteria for determining Gmax from bender element tests[END_REF][START_REF] Lee | Bender elements: Performance and signal interpretation[END_REF][START_REF] Murillo | Elastic parameters of intermediate soils based on bender-extender elements pulse tests[END_REF][START_REF] Kumar | Interpretation of bender element test results using sliding fourier transform method[END_REF]. The travel time considered in this study is the time between the zero point of the input signal and the zero point after the first bump as proposed by [START_REF] Lee | Bender elements: Performance and signal interpretation[END_REF] and adopted later by [START_REF] Kassas | Structure-Soil-Structure Interaction (SSSI) of adjacent buildings founded on liquefiable soil[END_REF]. After the calculation of the shear wave velocity (V s ), the small-strain shear modulus (G max ) was calculated using Eq. ( 5).

G max = ρ sat V 2 s ( 5
)
where ρ sat is the mass density involved in the wave propagation. The estimated small-strain shear modulus was then compared with the empirical equation (Eq. ( 6)) proposed by [START_REF] Azeiteiro | Critical state-Based interpretation of the monotonic behavior of Hostun Sand[END_REF].

Acceleration response

The time-histories of acceleration response at different positions are compared to the base shaking input signal (Fig. 6), and the results are shown in Fig. 10. In this study, the noises and the unexpected frequency components in the signals (Fig. 6) were filtered by a band pass filter with the cut-off frequencies from 0.33 Hz to 6.67 Hz (at the prototype scale).

Different patterns of response were observed with the acceleration profiles. The first pattern was shown at bottom of the model (A2, A9, A12, A15, and A18) where the acceleration responses were very close to the input signal. This pattern proves the good transmission of the shear motion from the base of the container to the sand. An acceleration amplification effect is observed for A6, A7 and A8 which were embedded in the embankment, and the amplification increases from the bottom to the top. At A8, the amplification ratio with respect to the input signal is 2.12, with a phase lag of about 0.5 s. For the acceleration profiles beneath the toe of the embankment (A11, A14, A13, and A16), asymmetrical acceleration responses are observed, which indicates a horizontal movement of the embankment.

These acceleration responses are in accordance with the work of [START_REF] Adalier | Embankment dam on liquefiable foundation dynamic behavior and densification remediation[END_REF]. These positive spikes were caused due to the irregular geometry of the model above these positions. The acceleration profiles in Fig. 10 also reveal that liquefaction has occurred in the free ground rather than in the soil beneath the embankment. It is clear that the accelerations at A17 and A20 show a significant attenuation after two cycles, where the liquefaction occurred and the shear waves cannot be transmitted. This behavior is in accordance with the study of [START_REF] Koga | Shaking table tests of embankments resting on liquefiable sandy ground[END_REF] for embankments resting on liquefiable sandy ground.

Excess pore pressure response

The variation of excess pore pressure ratio r u (ratio between excess pore pressure and initial vertical effective stress) at various locations in the model is shown in Fig. 11. Below the embankment (P1 to P8), the evolution of the pore pressure ratio follows the same tendency. In terms of r u , the excess pore pressure reaches approximately 50% of the initial vertical effective stress, which indicates that the soil beneath the embankment was not liquefied. While for P10 and P11 under the free ground, the r u increased rapidly after approximately two cycles and reached 1, which indicates the occurrence of liquefaction. The responses of excess pore pressure at P10 and P11 are in accordance with the acceleration responses presented in section 4.2 and confirm again the preference of liquefaction at the free ground [START_REF] Koga | Shaking table tests of embankments resting on liquefiable sandy ground[END_REF].

Excess pore pressure ratio at different positions

Stress path of liquefaction

Fig. 12 presents the stress paths of the central array under the embankment and the array under the free ground surface, respectively. The liquefaction stress paths were calculated from the acceleration and excess pore pressure records by the method proposed by [START_REF] Zeghal | Analysis of site liquefaction using earthquake records[END_REF]. Fig. 12(a) highlights that the stress paths below the embankment stabilized, and the vertical effective stress did not reach zero. On the other hand, for the stress paths under the free ground, the effective stresses show a significant decrease which tends to zero after two cycles of loading, see Fig. A great crest settlement was observed, which is noted by the dashed lines in Figs. 14 (a) and (b). The recorded crest settlement was recorded by the laser sensors D1 (Fig. 14 (c)). The crest settled around 0.7 m which was nearly 12.0% of the embankment height. This settlement was accompanied by a 0.2 m settlement of the middle slope of the embankment shown in Fig. 14 (d). In addition, a heaving of 0.04 m was observed at the free ground surface near the toes, see Fig. 14 (e). The heaving observed near the toes indicates a possible lateral spreading of the toes embankment (Fig. 14 (f)). The deformation pattern agrees well with the results reported by [START_REF] Doboku | Manual for Repair Methods of Civil Engineering Structures Damaged by Earthquakes[END_REF], [START_REF] Oka | Damage patterns of river embankments due to the 2011 off the pacific coast of tohoku earthquake and a numerical modeling of the deformation of river embankments with a clayey subsoil layer[END_REF], [START_REF] Adalier | Embankment dam on liquefiable foundation dynamic behavior and densification remediation[END_REF], [START_REF] Shahbodagh | Large deformation and failure analysis of river embankments subjected to seismic loading[END_REF] and [START_REF] Li | Centrifuge study on the effect of the SCP improvement geometry on the mitigation of liquefaction-induced embankment settlement[END_REF].

Performance verification of the numerical model -comparison of numerical and experimental results

The performance of the numerical model was verified by comparing the numerical results with the experimental results. The comparisons were conducted for the time-histories of acceleration, excess pore pressures and the deformation of the embankment, which are presented hereafter.

Acceleration response

Based on Anderson's Goodness of Fit (GoF) criteria [START_REF] Anderson | Quantitative measure of the goodness-of-fit of synthetic seismograms[END_REF], the acceleration times histories computed with numerical model were compared to the experimental data. Quantitative scores were estimated to classify the agreement into poor fit (score lower than 4), fair fit (score between 4 and 6), good fit (score between 6 to 8) and excellent fit (score higher than 8) [START_REF] Santisi D'avila | Nonlinear seismic response for the 2011 Tohoku earthquake: borehole records versus 1 Directional -3 Component propagation models[END_REF]. The score S related to the parameter p was estimated between numerical (p n ) and experimental results (p e ) using Eq. ( 7)

S {p n , p e } = 10 exp -p np e min(p n , p e ) spectral (Ssa, 5 % damping) and the Fourier spectra (Sfs). The comparison of numerical and experimental results was carried out for signals filtered with a band pass filter (0.33 Hz∼6.67 Hz). The total score for a specific position is considered as the mean value of all scores calculated at this position. Fig. 15 and Table 4 show the comparison of the acceleration responses between experimental and numerical results at different positions: at embankment crest (A8), under the embankment (A3), near embankment toe (A14) and under the free ground surface (A17). Overall, the computed acceleration time histories are in good agreement with the measured acceleration in the centrifuge test.

For A3 under the embankment, a symmetrical acceleration profile is observed in the centrifuge and in numerical model which is classified as an excellent fit. The numerical model accurately simulated the experimental results in the central part of the model beneath the embankment, where liquefaction did not occur. At the embankment crest, the acceleration (A8) computed with numerical model is in good accordance with the experimental results.

Overall, the scores are good, despite of the fair score of the acceleration spectral (Ssa). This low score can be explain by the underestimation of the acceleration amplitude after the third cycle which is observed in time histories acceleration. The observed difference at A8 could be attributed to the constitutive law utilized for the embankment.

As mentioned before, the embankment was in an actual unsaturated state, and the Drucker-Prager model employed can only approximately simulate the behavior of the unsaturated embankment as a simplified "c -ϕ" soil. Thus, the transmission of the shear wave in the unsaturated embankment cannot be effectively captured by the numerical model. In addition, the response under the toe of the embankment (A14) which is characterized by an asymmetrical acceleration is also captured in a good way by the numerical model. The numerical model successfully captured the asymmetrical response, but it underestimated the acceleration amplitude in the positive part. This discrepancy may be attributed to the same cause that affects the prediction of the acceleration at the embankment crest (A8).

The difference in amplitude after the first three cycles could be attributed to the limitations of the Drucker-Prager model when simulating the unsaturated soil. Under the free ground surface, the comparison between numerical and experimental results at position A17 shows some differences. These differences were classified as an overestimation of the acceleration amplitude after two cycles which was also reflected by the scores of the peak acceleration (Spga) and arias intensity (SIa). These differences of the amplitude after two cycles, when the initiation of the liquefaction occurred, could be attributed to the boundary effect of the numerical model. the embankment. In the case of pore pressure P5 under the embankment, the excess pore pressure ratio was well simulated for the first six cycles, but for the subsequent cycles, the model underestimated the excess pore pressure ratio. This may be attributed to the limitations of the current numerical model in accurately simulating the changes in pore pressure in the zone (near P5) where the soil transitioned from a fully saturated to a partially saturated state.

At the positions P10 and P11, the numerical model captured well the build-up and also the dissipation of excess pore pressure under the free ground surface.

Stress path of liquefaction

The stress paths calculated from the experimental data were compared to those extracted directly from the elements of the numerical model in Fig. 17. The numerical model captured a rapid decrease in the effective stress under the free ground surface as well as the mobilization and stabilization of the effective stress under the embankment. In terms of the number of cycles before liquefaction occurs, the numerical results indicate that liquefaction initiates after one cycle. This is earlier than what was observed in the centrifuge test, where liquefaction occurred after two cycles. It should be noticed that the liquefaction stress paths in the experiment were calculated from the 1D horizontal accelerations and the excess pore pressure nearby, which were not as precise as the stress paths obtained directly from the individual elements of the FEM model. computed by the numerical model

Settlement and deformation

Fig. 18 compares the numerical and experimental results for the settlements of the embankment crest and free ground surface, including a parametric study that investigates the influence of the cohesion of the embankment soil on embankment settlement. In this study, the optimum value was selected for a good fit with experimental results.

According to the parametric study shown in Fig. 18, it is found that the cohesion does not only influence the deformation of the embankment itself, but also the settlement of the free ground surface. The numerical model reflects the settlement and deformation trends that observed in the centrifuge test. Quantitatively, the settlement of the embankment crest measured in the centrifuge test is about 0.7 m, which is also captured by numerical model with 65 kPa of the cohesion. However, the heaving of the free ground surface from the numerical model using the selected cohesion (about 0.25 m) was larger than the experimental result (about 0.04 m). The significant difference in ground surface heaving may be due to the influence of the rigid numerical boundary conditions. Additionally, the difference may be attributed to the presence of the self-weight of the potentiometer which can increase the settlement and at the same time limit the heaving during the centrifuge test. For the model deformation pattern, the response of the numerical is presented in Fig. 19 at the end of shaking (t=18.26 s). As explained before, there was no important liquefaction of soil beneath the embankment as it is shown in the contour plot of the excess pore pressure distribution. For the soil under the free ground surface, the excess pore pressure increased largely and reached the maximum value, which indicates the liquefied zones (in yellow). The liquefied zones, which extend from the toes of the embankment to the free ground surface, result in lateral displacement of the soil beneath the embankment and significant embankment settlement. In addition, a lateral displacement of the toes of embankments was observed followed by the heaving of the free ground surface. The first model is similar to the original model presented in section 3, but the boundaries on the lateral side were changed from rigid to flexible conditions. The flexible free-field conditions (or periodic boundary conditions [START_REF] Basu | Observations and challenges in simulating post-liquefaction settlements from centrifuge and shake table tests[END_REF]) were realized by imposing the same displacement on the nodes at the same depth by command "EqualDOF" in OpenSees.

For the second model, the original model was extended by 150 m for each side of the ground soil as shown in Fig. 20. According to the analyses of [START_REF] Kassas | Structure-Soil-Structure Interaction (SSSI) of adjacent buildings founded on liquefiable soil[END_REF] and Kassas et al. (2021a) on the boundary effect of a rigid container, when the ratio of the boundary distance to the soil depth is larger than 8, the boundary effect of a rigid container is negligible. In this study, for the model with extended boundaries, the ratio between the distance of the embankment toe to the side and the ground soil depth is around 18.2. Therefore, it could be regarded that the boundaries do not affect the response of the embankment or the soil in its vicinity in the extended model. The new extended model keeps the same boundary conditions as the original model. The extended parts were modelled with the same type of element and material constitutive law. The same analysis procedure explained in section 3.3 was adopted.

In order to investigate the influence of the boundary condition on the response of the model, this section focuses on the comparison of experimental results with the three numerical models in terms of acceleration, excessive pore pressure, and settlements. The excess pore pressures obtained from the two new models were compared with experimental results in Fig. 23 and Fig. 24. Under the embankment (P8), the excess pore pressure ratios obtained by the extended model and model with free field conditions are similar to the original numerical model and the experimental results. It indicates the numerical boundary conditions or the original model and the rigid lateral sides of the container used in the centrifuge test did not affect significantly the pore pressure response under the embankment.

For the position near the boundary of the container (P10), the build-up and dissipation of excess pore pressure computed with the three numerical models are comparable to each other and are also comparable to the experimental results. Under the free ground surface, the pore pressure increased rapidly (with r u ≥ 0.9) after 2 cycles. The liquefaction initiation was captured by the numerical models which is in accordance with the experimental results. This indicates that the numerical boundary conditions of the original numerical model and the rigid container used in centrifuge test did not have significant impacts the variation of pore pressure for the locations under the free ground surface. The embankment deformation and ground surface displacement recorded during the experimental test were also compared to the results of the three numerical models (with free field condition, with and without extension). The comparison is shown in Fig. 25. The deformation pattern was similar in all models: the embankment crest settled, accompanied by the heaving of the ground surface at a distance from the embankment toe (also observed in Fig. 26).

At the embankment level, the settlement of the crest obtained by the model with free field condition (0.72 m) is similar to those of the model without extensions (0.70 m) (Fig. 25 (a)).

For the free ground surface near the toes of the embankment, the heaving simulated by the free field condition model (0.21 m) is close to the heaving in the original numerical model (0.23 m). For the model with extended boundaries, the crest settlement (0.82 m) is slightly larger than the others (experimental, original numerical model and free field condition model). The computed heave (0.14 m) is improved but still higher than the experimental results (0.04 m).

Therefore, for the two models without extensions, the numerical boundary conditions affect slightly the crest settlement and the heaving of free ground surface as shown in Furthermore, the deformation and the distribution of the excess pore pressure ratio of the three numerical models were presented in Fig. 26 after the shaking (t=18.26 s). A similar deformation pattern was observed in the three models. Under the embankment, the excess pore pressure distribution has the same pattern for the three models.

Under the free ground surface, the pore pressure distribution was similar in the three models up to a distance of around 5 m away from the lateral sides (marked in red in Fig. 26 (a)). In this zone, the pore pressure ratio obtained by the numerical model was less than those of the other two models. This difference allows to define the zone affected by the numerical boundaries [START_REF] Saade | Etude des effets de bord dans les essais centrifuges sous chargement sismique par modélisation numérique, in: 11èmes journées nationales de géotechnique et de géologie de l'ingénieur[END_REF]. For the positions close to the lateral sides of the container, due to the influence of the boundaries, the liquefaction potential decreases.

Conclusions

In this study, a dynamic centrifuge program was performed to have the direct observation and analysis of the behavior of a homogeneous embankment resting on liquefiable ground. Additionally, FEM numerical modeling was carried out using an advanced constitutive model -PM4Sand, and the performance of the numerical model was verified against the experimental results. With the verified numerical model, the boundary effects in the numerical simulation and dynamic centrifuge test were analysed and discussed. Based on the experimental and numerical results, the following conclusions can be drawn:

• In the centrifuge test, different patterns of time histories of the acceleration were observed. The signal was amplified with a ratio of approximately 2.12 from the bottom of the model to the crest of the embankment.

The positions under the toes of the embankment had an asymmetrical acceleration profile due to the irregular geometry of the model above those positions. For the positions under the free ground surface, the liquefaction initiation was observed after two cycles of loading.

• The soil liquefaction was confirmed by the excess pore pressures and the stress paths at various positions in the model. For the positions under the embankment with high confining pressure where liquefaction did not occur, a limited excess pore pressure ratio was observed (around 0.5) and the stress path stabilized before complete liquefaction occurred. For the position at a depth less than 6 m under the ground surface where the soil liquefied,

There was a significant increase in excess pore pressure after two cycles, accompanied by a notable decrease in vertical effective stress.

• Soil liquefaction resulted in a soft ground that allowed the embankment to slide approximately as a rigid block.

A settlement of 0.7 m was observed at the crest of the embankment, along with a 0.2 m settlement at the middle slope. Furthermore, a heaving of 0.04 m was observed at ground surface level near the toes, indicating a possible lateral spreading of the toes embankment.

• The performance of the numerical model was verified by comparing the acceleration, excess pore pressures and deformations against the experimental results. The numerical model provided reasonable simulation for liquefaction and deformation responses. The symmetrical responses under the embankment and the asymmetrical responses under the toes of the embankment were well captured. Furthermore, the numerical model accurately simulated the attenuation of acceleration near the ground surface after two cycles. Additionally, the build-up of the excess pore pressure was also well reproduced by the numerical model. The deformation trends of the embankment were reflected in the numerical model, which were characterized by settlement at the embankment level and heaving at the free ground surface.

• The boundary effects were analyzed using two new numerical models: an extended model (with over 150m on each side) and a model with free field conditions. Regarding the results, the results from the centrifuge test slightly differ from those from the extended model and the model with free-field conditions. This indicates that the boundaries of the rigid container do not have significant impacts on the responses of the centrifuge model.

Based on the numerical results, a zone can be defined in which the behavior of the soil is slightly affected by the numerical boundaries in the original model.

CRediT author statement

D

  50 Mean particle diameter e Void ratio e max Maximum void ratio e min Minimum Void ratio t Time t 2 Start time of decreasing stage of permeability t 3 End time of decreasing stage of permeability G max Maximum shear modulus G o Shear modulus coefficient G s Specific gravity g Earth gravity h po Contraction rate parameter h o Secondary input parameter that adjusts the ratio of plastic modulus to elastic modulus I D Relative density k Permeability k i Initial permeability k b Permeability in excess pore pressure build up stage k d Permeability in excess pore pressure decreasing stage k l Permeability in liquefaction stage K 0 Lateral earth pressure coefficient N Number of cycles required for liquefaction p A Atmospheric pressure p ′ Mean effective stress p e Experimental value p n Numerical value Q Bolton's constant in PM4Sand model R Bolton's constant in PM4Sand model r u Excess pore water pressure ratio S Anderson's score V s Shear wave velocity X Horizontal position Y Vertical position α Constant controls the variation of excess pore pressure α 2 Constant controls the variation of excess pore pressure β 1 Constant controls the variation of excess pore pressure β 2 Constant controls the variation of excess pore pressure ∆u Excess pore pressure ν Poisson's ratio ρ sat Saturated density of the soil ρ dmin Minimum dry density ρ dmax Maximum dry density σ v Initial vertical stress σ ′ v Initial vertical effective stress ϕ Critical state friction angle 1. Introduction

Figure 1 :

 1 Figure 1: Centrifuge model at model scale, N=1/60 (Dimensions in mm)

Figure 2 :

 2 Figure 2: Illustration of the rigid container

Figure 3 :

 3 Figure 3: Final shape of the instrumented centrifuge model

Figure 4 :

 4 Figure 4: Centrifuge model in preparation -installation of accelerometers

Figure 5 :

 5 Figure 5: Comparison of initial static pore pressures of the centrifuge and numerical model after consolidation with the theoretical hydrostatic pressures: (a) under the embankment and (b) under the free ground surface

Figure 6 :

 6 Figure 6: Time and frequency representations of the input signal (target and actual) in prototype scale (a) amplitude and (b) normalized frequency content

Figure 7 :

 7 Figure 7: FEM mesh of the numerical model (Dimensions in m)

Figure 8 :

 8 Figure 8: Calibration of PM4Sand constitutive model: Cyclic stress ratio vs. the number of cycles for liquefaction triggering

  in Fig.9that the experimental results and the empirical equation of G max were comparable, although dispersion was observed, which could probably be due to the precision of bender element measurement in the centrifuge test.

Figure 9 :

 9 Figure9: Comparison of the calculated small strain shear modulus with the empirical equation[START_REF] Azeiteiro | Critical state-Based interpretation of the monotonic behavior of Hostun Sand[END_REF]

Figure 10 :

 10 Figure 10: Acceleration responses at different positions in the model during dynamic excitation

Figure 11 :

 11 Figure 11: Excess pore pressure ratio at different positions in the model during dynamic excitation

Figure 13 :

 13 Figure 12: Stress paths at different positions (a) under embankment and (b) under the free ground surface. CSL and PT are Critical state Line and Phase Transformation Line for Hostun HN31 sand[START_REF] Benahmed | Comportement mécanique d'un sable sous cisaillement monotone et cyclique: Application aux phénomène de liquéfaction et de mobilité cyclique[END_REF], respectively.

Figure 14 :

 14 Figure 14: Deformation and settlement analyses: photos (a) before and (b) after the base shaking; (c) D1, Settlement of embankment crest; (d) D2, settlement of the middle slope of the embankment; (e) settlement at the free ground surface measured by potentiometer D3 and (f) embankment deformation pattern

Figure 16 :

 16 Figure 16: Comparison of numerical and experimental results in terms of excess pore pressure ratio under the embankment (a) P5 and (b) P8; under free ground surface (c) P11 and (d) P10

Figure 17 :

 17 Figure 17: Stress paths of liquefaction under the free ground surface and under the embankment (a) obtained from the centrifuge test and (b)

  Figure 18: Settlements from numerical simulation and centrifuge test at the: (a) embankment crest and (b) free ground surface

Figure 20 :

 20 Figure 20: FEM mesh of the new models: (a) with free field conditions and (b) with extended boundaries of the ground soil (unit in prototype scale)

Figure 21 :Figure 22 :

 2122 Figure 21: Comparison of experimental and numerical results (original model and model with extensions) in terms of amplitude and normalized frequency content of the acceleration at different positions: (a) A17 below the soil surface near embankment toe and (b) A20 near the container boundary

Figure 23 :Figure 24 :

 2324 Figure 23: Comparison of experimental and numerical results (original model and model with extensions) in terms of excess pore pressures at different positions: (a) P8 under embankment and (b) P10 below the ground surface

Figure 25 :

 25 Figure 25: Vertical displacement at: (a) embankment crest and (b) free ground surface near the toe of embankment

Figure 26 :

 26 Figure 26: Deformation and the distribution of excess pore pressure at the end of shaking of the 3 numerical models: a) original model b) model with extended boundaries and c) model with free-field conditions

Chedid SAADE :

 : Experiment execution, Numerical modeling execution, Data Analysis, Writing-Original draft preparation. Zheng LI: Conceptualization, Co-supervision, Investigation, Reviewing and Editing. Sandra ES-COFFIER: Conceptualization, Co-supervision, Investigation, Reviewing and Editing. Luc THOREL: Conceptualization, Supervision, Investigation, Reviewing and Editing.

Table 1 :

 1 Properties of Hostun sand HN31[START_REF] Benahmed | Comportement mécanique d'un sable sous cisaillement monotone et cyclique: Application aux phénomène de liquéfaction et de mobilité cyclique[END_REF][START_REF] Benahmed | Initial structure and static liquefaction properties of sand[END_REF]

	Sand	D 50 (mm)	e min	e max	G s	ρ dmin (g/cm 3 ) ρ dmax (g/cm 3 )
	Hostun HN31	0.35	0.656 1.049 2.65	1.33	1.6
	2.3. Soil container
	The model was built in a rigid container (see Fig 2) which has the inner dimensions of 800 mm (length) × 400
	mm (height) × 340 mm (width). The soil container has a transparent on one side that allows the direct observation by
	cameras during the test. Doubled-face tapes were attached at the bottom of the container to enhance interface friction
	and ensure good transmission of the shear waves in the sand.

Table 2 :

 2 Instrumentation of the centrifuge model

	Sensor type	Sensor notation Working range Manufacturing precision (%) Number
	Accelerometer	A	1 Hz-20 kHz	0.3 %	20
	Pore pressure transducer P	0-2 bars	0.2 %	11
	Laser displacement	D (in red)	16-120 mm	0.1 %	3
	Potentiometer	D (in green)	0-100 mm	0.1 %	2

Table 3 :

 3 Parameters used in the numerical model

	Parameter		Value
	Relative density	[%]	I D	50
	Void Ratio	[-]	e = e max -I D (e max -e min ) 0.83
	Shear modulus coefficient †	[-]	G o = 293 × (2.97-e) 2 (1+e)	712
	Critical state friction angle †	[ • ]	ϕ	33.8
	Atmospheric pressure	[kPa] p A	101.3
	Lateral earth pressure †	[-]

 (7) The items used for comparison of the time-histories acceleration at different positions are the peak acceleration (Spga), peak velocity (Spgv), peak displacement (Spgd), arias intensity (SIa), energy integral (SIv), acceleration

Data availability statement

The experimental data presented in the study are available upon direct request to the authors.

Acceleration (g) Time (s) Position SIa SIv Spga Spgv Spgd Ssa Sfs Total score

Excess pore pressure ratio

The numerical and experimental results of excess pore pressure ratio are compared in Fig. 16

Declaration of competing interest

The authors declare no conflict of interest.

Acknowledgment

The financial support of University Gustave Eiffel -Nantes Campus (ex. IFSTTAR) and of the Région Pays de la Loire is gratefully acknowledged as well as the technical supports from the French National Project -ISOLATE (Characterizing and Improving SOiLs AgainsT liquEfaction, Grant NO. ANR-17-CE22-0009). The authors would like also to thank the valuable support and help of the technical staff of the GERS-CG centrifuge lab. Many thanks to the kind help from Dr. Bogdan Muresan Paslaru and Mme. Laurence Lumiere at Laboratoire EASE, Département AME, Université Gustave Eiffel Campus de Nantes. The authors would also like to acknowledge all the valuable comments, suggestions and efforts devoted by the anonymous reviewers to improve the quality of this work.


Alexis Mifleur

Isabelle Suisse

André Mortreux

Mathieu Sauthier

Enantioselective nickel catalyzed butadiene hydroalkoxylation with ethanol: from experimental results to kinetics parameters

may come     Enantioselective nickel catalyzed butadiene hydroalkoxylation with ethanol: from experimental results to kinetics parameters

Introduction

Double bond hydrofunctionalization is one of the most fundamental reactions in organic synthesis leading to a large variety of compounds [1]. This reaction corresponds to the addition of hydrogen and another fragment across a carbon-carbon or carbon-heteroatom multiple bond and is generally promoted by a Brønsted acid or a transition metal catalyst [2][START_REF] Bezzenine-Lafollee | First-row late transition metals for catalytic alkene hydrofunctionalisation: recent advances in C-N, C-O and C-P bond formation[END_REF][START_REF] Greenhalgh | Iron-Catalysed hydrofunctionalisation of alkenes and alkynes[END_REF][START_REF] Yadav | Recent progress in transition metal catalysed hydrofunctionalisation of less activated olefins[END_REF][START_REF] Margrey | A general approach to catalytic alkene anti-Markovnikov hydrofunctionalization reactions via acridinium photoredox catalysis[END_REF][START_REF] Liu | Platinum-catalyzed hydrofunctionalization of unactivated alkenes with carbon, nitrogen and oxygen nucleophiles[END_REF][START_REF] Obligacion | Earth-abundant transition metal catalysts for alkene hydrosilylation and hydroboration[END_REF][START_REF] Chen | Asymmetric hydrofunctionalization of minimally functionalized alkenes via earth abundant transition metal catalysis[END_REF][START_REF] Ananikov | Topics in Organometallic Chemistry[END_REF]. From an atom-economy standpoint, this transformation is ideal with no production of wastes. With butadiene, the hydrofunctionalization reaction leads to the formation of various functionalized alkenes according to the nucleophile (Scheme 1) [START_REF] Hermann | Homogeneously catalyzed 1,3-diene functionalization -A success story from laboratory to miniplant scale[END_REF][START_REF] Hydrosilylation : Ibrahim | A highly chemoselective cobalt catalyst for the hydrosilylation of alkenes using tertiary silanes and hydrosiloxanes[END_REF][START_REF] Hydrocyanation : Bini | Ligand development in the Ni-catalyzed hydrocyanation of alkenes[END_REF][START_REF] Hydroalkoxylation ; Mifleur | Synthesis of C4 chain glyceryl ethers via nickel-catalyzed butadiene hydroalkoxylation reaction[END_REF][START_REF] Mifleur | Synthesis of short-chain alkenyl ethers from primary and bio-sourced alcohols via the nickel-catalyzed hydroalkoxylation reaction of butadiene and derivatives[END_REF][START_REF] Bigot | Nickelcatalysed hydroalkoxylation reaction of 1,3-butadiene: ligand controlled selectivity for the efficient and atom-economical synthesis of alkylbutenyl ethers[END_REF][START_REF] Hydrophosphinylation : Nie | Enantioselective coupling of dienes and phosphine oxides[END_REF][START_REF] Hydrophosphorylation ; Mirzaei | Palladium-catalyzed hydrophosphorylation of 1, 3-dienes leading to allylphosphonates[END_REF][START_REF] Hydroamination: A) Banerjee | Palladium-catalysed regioselective hydroamination of 1,3-dienes: synthesis of allylic amines[END_REF][START_REF] Hydroamidation: Banerjee | A general catalytic hydroamidation of 1,3-dienes: atom-efficient synthesis of N-allyl heterocycles, amides and sulfonamides[END_REF][START_REF] Hydroboration: Satoh | New convenient approach to the preparation of (Z)-allylic boronates via catalytic 1, 4-hydroboration of 1, 3-dienes with catecholborane[END_REF]. Butenyl products issued from the 1,2-or 1,4-addition are thus obtained as a mixture of several isomers: the two 1,2-addition products with the Markovnikov and anti-Markovnikov compounds and the two 1,4-aducts that differ from the (Z) and (E) configurations of the internal C=C double bond. Among these examples, the hydroalkoxylation is particularly attractive since this transformation using readily accessible alcohols as nucleophiles leads to the formation of allylethers via a catalytic and environmentally friendly process. This approach is on this point of view advantageous in comparison to the classical Williamson reaction (SN reaction on an allylic halide derivative) which produces high quantity of inorganic salts as co-products [START_REF] Trost | Atom economy-a challenge for organic synthesis: homogeneous catalysis leads the way[END_REF]. Nevertheless, this transformation has been relatively rarely described in the literature, unlike considering the telomerization reaction which corresponds to the dimerization of butadiene with addition of a nucleophile [START_REF] Fassbach | The telomerization of 1, 3-dienes-A reaction grows up[END_REF]. Various metals as rhodium [START_REF] Dewhirst | Reaction of rhodium trichloride with dienes[END_REF][START_REF] Kawazura | On the catalytic addition of alcohols to 1, 3-butadiene with rhodium trichloride hydrates[END_REF][START_REF] Kawazura | The regioselective addition of alcohols to isoprene with rhodium trichloride hydrates[END_REF], palladium [START_REF] Patrini | Selective synthesis of octadienyl and butenyl ethers via reaction of 1, 3-butadiene with alcohols catalyzed by homogeneous palladium complexes[END_REF][START_REF] Utsunomiya | Palladium-catalyzed equilibrium addition of acidic OH groups across dienes[END_REF], nickel [START_REF] Shields | Nickel-complex catalyzed telomerisation of butadiene with alcohols[END_REF][START_REF] Weigert | The Ni(0)-catalyzed addition of phenol to butadiene[END_REF][START_REF] Beger | Diene oligomerization. IX. Nickel complexcatalyzed dimerization and telomerization of butadiene in hydroxy group-containing media[END_REF][START_REF] Commereuc | Telomerization of butadiene with methanol using homogeneous catalysts of palladium and nickel[END_REF] as well as aluminum [START_REF] Boelens | Chemical structure and odor. VI. Geraniol derivatives[END_REF][START_REF] Behr | A sustainable route from the renewable myrcene to methyl ethers via direct hydroalkoxylation[END_REF][START_REF] Williams | Efficient hydroalkoxylation of alkenes to generate octane-boosting ethers using recyclable metal triflates and highly active metal triflate/Brønsted acid-assisted catalysts[END_REF] or gold [START_REF] Chandrasekhar | Gold-catalyzed intramolecular hydroalkoxylation/cyclization of conjugated dienyl alcohols[END_REF] have been used to design suitable catalysts. In our hands, the hydroalkoxylation of dienes with simple and bio-sourced alcohols has been performed in the presence of nickel precatalysts associated with chelating diphosphine ligands (Scheme 2) [START_REF] Hydroalkoxylation ; Mifleur | Synthesis of C4 chain glyceryl ethers via nickel-catalyzed butadiene hydroalkoxylation reaction[END_REF][START_REF] Mifleur | Synthesis of short-chain alkenyl ethers from primary and bio-sourced alcohols via the nickel-catalyzed hydroalkoxylation reaction of butadiene and derivatives[END_REF][START_REF] Bigot | Nickelcatalysed hydroalkoxylation reaction of 1,3-butadiene: ligand controlled selectivity for the efficient and atom-economical synthesis of alkylbutenyl ethers[END_REF]. With this catalytic system, high butadiene conversions and selectivities in alkylbutenyl ethers were obtained with low catalyst loadings. The major products are the branched butenylether (OC4 b) together with the linear isomer (OC4 l). The allylic structures of the branched or linear ethers formed and the absence of product issuing from the 1,2 anti-Markovnikov addition strongly support the involvement of π-allylnickel intermediates rather than a direct nucleophilic attack on the diene. [START_REF] Mifleur | Deciphering the mechanism of the nickel-catalyzed hydroalkoxylation reaction: a combined experimental and computational Study[END_REF]

+ ROH OR OR + Ni(cod)2 ligand toluene OC4 l OC4 b

Scheme 2. Nickel catalyzed butadiene hydroalkoxylation

As the branched ether bearing a stereogenic center is obtained as major isomer of the reaction, we were interested in the enantioselective version of the nickel catalyzed hydroalkoxylation reaction. This asymmetric version is relevant as it offers an access to enantiomerically enriched small building blocks.

Chiral allylethers are commonly found in bioactive molecules and industrially important pharmaceutical intermediates [START_REF] Trost | Asymmetric transition-metal-catalyzed allylic alkylations: applications in total synthesis[END_REF]. To our knowledge, the enantioselective hydrofunctionalization of dienes with alcohols is rare with only one recent example reported in the literature [START_REF] Tran | Ni-catalyzed regioselective hydroalkoxylation of branched 1,3-dienes[END_REF]. Herein we report our recent progress in the enantioselective butadiene hydroalkoxylation with ethanol as model substrate.

Experimental

Chemicals were purchased from Aldrich, Alfa Aesar, Acros, Linde Gas France (butadiene) and Strem. Ethanol was distilled over magnesium turnings. Toluene was obtained from a solvent purification system MBraun SPS-800. Conversions and selectivities were determined by gas chromatography on a Shimadzu 2010 apparatus equipped with an Equity-5 column (30 m, i.d. = 0.32 mm). Ee's were determined by chiral GC using a Chiraldex column with H2 as gas vector.

Catalytic test:

The catalytic solution is prepared in a glovebox by mixing Ni(cod)2 (3.8 mg, 0.014 mmol) and the ligand (0.021 mmol) in 3 mL toluene in a glass reactor closed by a Rotaflo® stopcock. Under nitrogen atmosphere, the tube was cooled at -15 °C and dry and degassed EtOH was added (10 mL). A precise amount of butadiene (1.5 mL, 17.2 mmol) was then condensed at low temperature and transferred in the reaction mixture via a cannula. The glass reactor was closed and heated to 80°C for 30 hours. After reaction, the mixture was cooled and vented before GC analyses with heptane as internal standard. Branched vs linear products could be differentiated.

Results and discussion

Our group has recently reported that the hydroalkoxylation reaction can be achieved with both high butadiene conversions and high selectivites when using a catalytic system featuring zerovalent nickel in the presence of particular chelating diphosphines with two phosphorous atoms bridged with four carbons [START_REF] Hydroalkoxylation ; Mifleur | Synthesis of C4 chain glyceryl ethers via nickel-catalyzed butadiene hydroalkoxylation reaction[END_REF][START_REF] Mifleur | Synthesis of short-chain alkenyl ethers from primary and bio-sourced alcohols via the nickel-catalyzed hydroalkoxylation reaction of butadiene and derivatives[END_REF][START_REF] Bigot | Nickelcatalysed hydroalkoxylation reaction of 1,3-butadiene: ligand controlled selectivity for the efficient and atom-economical synthesis of alkylbutenyl ethers[END_REF]. Among the commercially available and optically pure C4 diphosphines, BINAP is one of the most representative ligands with an axial chirality. BINAP is elsewhere used in asymmetric catalytic hydrogenations on an industrial scale, particularly for the synthesis of (-)-menthol [START_REF] Kumobayashi | Recent advances of BINAP chemistry in the industrial aspects[END_REF]. An explorative trial for butadiene hydroalkoxylation reaction with (R)-BINAP has been conducted and displayed a moderate conversion (44%). Nevertheless, the regioselectivity for the branched product was important (72%) and a promising enantiomeric excess has been obtained (42%) with a low catalyst loading (0.08 mol %) (Scheme 3).

+ EtOH

Ni(cod)2 (0.08 %) (R) -BINAP (0.12 %) toluene (3 mL) 3h, 80°C OEt * + OEt (excess) sel. : 72% ee : 42% Conv. : 44% Scheme 3. Nickel based butadiene asymmetric hydroalkoxylation with ethanol using (R)-BINAP ligand Thanks to this encouraging result, we screened a series of chiral ligands (Figure 1) for the butadiene hydroalkoxylation with ethanol using Ni(cod)2 as source of zerovalent nickel, according the standard conditions reported for the evaluation of this catalytic system [START_REF] Mifleur | Synthesis of short-chain alkenyl ethers from primary and bio-sourced alcohols via the nickel-catalyzed hydroalkoxylation reaction of butadiene and derivatives[END_REF]. The previous procedure used with BINAP has been reproduced: toluene was used as co-solvent to insure a good solubility of the catalyst in the reaction media and the reactions were conducted at 80 °C. A first series of catalytic results is reported in Table 1. We initially applied ligands with an atropoisomeric C2-symmetric biaryl backbone as BINAP: (S)- Butadiene dimers and telomers are occasionally observed as traces (up to 5%).The best combinations between yields and enantioselectivities were obtained with L1 and L4 (entries 1 and 5) with respectively 66% yield and 68% ee for L1 and 67% yield and 58% ee for L4. When the experiment was performed at lower temperature with L1 (entry 2), 60°C instead 80°C, the yield decreased to 31% while the selectivity into the branched isomer increased as well as the enantioselectivity (75% and 77% vs 71% and 68%, respectively). Among these atropisomeric ligands, L2 and L5 led to the highest OC4 yields, but provided low enantiomeric excesses with 15% and 27% ee respectively (Entries 3 and 6). The highest enantiomeric excess was obtained with the ligand L3 (69% ee) along with a very low yield (17%) (Entry 4).

O O O O PPh 2 PPh 2 (S) -SEGPHOS L1 PPh 2 PPh 2 MeO MeO (S)-MeO-BIPHEP L4 PPh 2 PPh 2 O O O O (R)-Synphos L2 PPh 2 PPh 2 OMe MeO MeO OMe (R)-Ph-Garphos L5 PPh 2 PPh 2 O O (S)-C3-TUNEPHOS L3 N N PPh 2 PPh 2 OMe MeO MeO OMe CTH-(R) -P-PHOS L7 PPh 2 PPh 2 O O (S)-Cyclohexyl Soniphos L6 O O N N PXyl 2 PXyl 2 OMe MeO MeO OMe CTH-(R)-Xylyl-P-PHOS L8

Segphos L1, (R)-Synphos L2, (S)-C3-Tunephos L3, (S)-MeO-Biphep L4, (R)-Ph-Garphos L5, (S)-Cyclohexyl

We next considered chiral ligands with ferrocenyl structures (L9-L12). A nickel catalyst bearing Bophoz L9 or Josiphos L10 provides high yields in butenyl ethers but with low enantiomeric excesses (94 % yield and 13 % ee) (Entries 10 and 11). A low conversion as well as a low ee are obtained with Taniaphos L11 (Entry 12). In this series of ligands, the most interesting enantiomeric excess obtained with ferrocenyl based ligand is with Walphos L12 with 53 % ee but with a rather low conversion (Entry 13).

As ligands L1 and L4 gave the best compromise in terms of butadiene conversions and enantiomeric excesses, variation of L1 and L4 structures either on their phosphorous atoms or on their backbones was considered (Figure 2). Results are reported in Table 2. The change of the phenyl substituents on used and aliquot samples of the reaction mixture were taken and analyzed by GC at regular time intervals (Figure 3). As expected, the global yield in ethers increases rapidly at the beginning of the reaction, then the reaction rate decreases, and the full conversion is reached after 28 hours. Besides, the butenylethers selectivities are moving towards a thermodynamic equilibrium via an isomerization along with an epimerization process. Firstly, the yield into the branched ether decreased regularly to the benefit of the linear one. This phenomenon of isomerization of the branched compounds (including both enantiomers) into the linear isomer follows a decreasing profile from 70 % to 54 %. Similarly, the major enantiomer within the branched ethers decreases from 77 % after 30 h. Noteworthy are the values observed at 46 h, which are respectively 53% of branched butenylethers and 54% of the major enantiomer. As expected, the enantiomeric excess is decreasing faster. This important evolution in the products ratio implies both isomerization and racemization processes with time. A cationic Ni-allyl complex has been proposed by our group to be the key intermediate during the formation of butenyl ethers [START_REF] Mifleur | Deciphering the mechanism of the nickel-catalyzed hydroalkoxylation reaction: a combined experimental and computational Study[END_REF]. The computed barriers, in the case of the non-chiral ligand Herein, we propose a qualitative kinetic model based on the reactions described in Scheme 4 in order to extract the kinetic parameters. Previous DFT calculations showed that the formation of this cationic π-allyl nickel is accessible by direct protonation by ethanol of the initially formed butadiene-Ni(0) complex leading to an ion-pair featuring a Ni-allyl cation along with free ethoxide as counteranion [START_REF] Mifleur | Deciphering the mechanism of the nickel-catalyzed hydroalkoxylation reaction: a combined experimental and computational Study[END_REF]. Based on these previous calculations, this step corresponds to the highest activation energy value of the overall reaction. Moreover, the difficulties encountered monitoring the concentration of nickel species lead up to simplify the kinetic model. As a consequence, the initial elementary steps: nickel-diene coordination and πallyl formation are considered as one simplified step with a rate corresponding to the RDS of this reaction, the π-allyl formation. Because the ethanol is both the nucleophile of this reaction and the solvent (so used in large excess), we applied an order degeneration simplification and thus the apparent rate constant kallyl

v 1 v -1 v 2 v -2 v 3 v -3 (2) (3)
𝑣𝑣 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝑘𝑘 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 [𝑏𝑏𝑏𝑏𝑏𝑏][𝑁𝑁𝑁𝑁 0 ] 𝑣𝑣 1 = 𝑘𝑘 1 [𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴] 𝑣𝑣 -1 = 𝑘𝑘 -1 [𝑏𝑏 1 * ][𝑁𝑁𝑁𝑁 0 ] 𝑣𝑣 2 = 𝑘𝑘 2 [𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴] 𝑣𝑣 -2 = 𝑘𝑘 -2 [𝑏𝑏 2 * ][𝑁𝑁𝑁𝑁 0 ] 𝑣𝑣 3 = 𝑘𝑘 3 [𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴] 𝑣𝑣 -3 = 𝑘𝑘 -3 [𝐴𝐴][𝑁𝑁𝑁𝑁 0 ] (4) 𝑑𝑑[𝐵𝐵𝑏𝑏𝑏𝑏] 𝑑𝑑𝑏𝑏 = -𝑣𝑣 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑑𝑑[𝑁𝑁𝑁𝑁 0 ] 𝑑𝑑𝑏𝑏 = -𝑣𝑣 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑣𝑣 1 -𝑣𝑣 -1 + 𝑣𝑣 2 -𝑣𝑣 -2 + 𝑣𝑣 3 -𝑣𝑣 -3 𝑑𝑑[𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴] 𝑑𝑑𝑏𝑏 = 𝑣𝑣 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 -𝑣𝑣 1 + 𝑣𝑣 -1 -𝑣𝑣 2 + 𝑣𝑣 -2 -𝑣𝑣 3 + 𝑣𝑣 -3 𝑑𝑑[𝑏𝑏 1 * ] 𝑑𝑑𝑏𝑏 = 𝑣𝑣 1 -𝑣𝑣 -1 𝑑𝑑[𝑏𝑏 2 * ] 𝑑𝑑𝑏𝑏 = 𝑣𝑣 2 -𝑣𝑣 -2 𝑑𝑑[𝐴𝐴] 𝑑𝑑𝑏𝑏 = 𝑣𝑣 3 -𝑣𝑣 -3
Next, the kinetic constants for the other elementary steps were estimated at 353 K by minimizing the least squares with the solver displays by Microsoft Excel® and the GRG (Generalized Reduced Gradient) non-linear solving method. A multistart with 100 samples sized population was preferred to limit local minima in our estimation.

Values of the kinetic parameters are reported in Table 3. Rate constants k1, k2 and k3 display values between 5.9 and 18.8 min -1 with k1 > k3 > k2. Contrariwise, the rate constants for the reverse ways are in different orders of magnitude, with values included between 10 -5 and 10 -1 L.mol -1 .min -1 following the same order than the direct way. The significant differences between the values obtained for the direct and indirect ways show that the formation of the butenylethers are fostered vs the formation of the allylic intermediate.

The model accuracy is very good as illustrated by the parity plot of Figure 4.

Table 3 Optimal parameters of the kinetic model a Analysis of the experimental data at the early stage, before that the isomerization and racemization processes get a significant impact on the rate law of the reaction, may help in validating this computational approach. As an apparent zero order applies until a c.a. 30 % conversion, an average rate corresponding to the initial rate determining step allyl complex formation is obtained. This experimental value v0 = kapp is found to be 3.3 L.mol -1 .min -1 with kapp including the ethanol concentration. Both values, obtained from computational approach and experimentally are concurring exactly. Experimental and simulated concentrations of products with time are reported in Figure 6. It appears that b1 * as kinetic product is preferentially formed at the beginning of the reaction while the rates of formation of b2 * and l are lower, but their selectivities increase regularly vs conversion. As the butadiene concentration decreases, the π-allyl complex intermediate resulting from the oxidative addition of the butenylethers on a low valent Ni(0) species (k-1, k-2, k-3) still remains the key species of the isomerization and epimerization processes. The affinity of the products with the chiral nickel complex illustrates the differences observed between the kinetic parameters. One can expect that the spatial configuration of the enantiomer b1* allows for its preferable interaction with the chiral catalyst according to a "lock and key" molecular recognition as opposed to the other enantiomer b2* and the linear ether l [START_REF] Trost | Asymmetric ligands for transition metal catalyzed reactions: 2-diphenylphosphinobenzoyl derivatives of C2 symmetrical diols and diamines[END_REF]. As a result, the rate of the reverse way for the elementary step (1) is much higher than the ones for (2) and (3). This way back of b1 * to produce the allylic nickel intermediate tends this major enantiomer to be gradually consumed while its opposite enantiomer b2 * and the linear isomer l accumulate in the reaction mixture. Equilibrium constants determined for the elementary steps of products formation from the allylic intermediate are superior to 1 and confirm the predominance of the direct ways against the reverse reactions. As expected from the higher decreasing rate of the ee vs the isomerization, K1 is the lowest equilibrium constant among these steps for the butenylethers formation and equilibration. K2 and K3 are respectively 4000 and 1100 times higher.

Conclusion

In conclusion, nickel catalyzed asymmetric hydroalkoxylation of butadiene with ethanol has been successfully performed in the presence of chiral diphosphine ligands. Enantiomeric excesses up to 77 % have been obtained with high yields. Nonetheless, the chemo-and enantioselectivity of the reaction are governed by the kinetic of the reaction: at the initial stage, the kinetic enantiomer is observed as the major isomer leading to high chemo-and enantioselectivities. Then, the enantiomeric excess and the selectivity into the branched isomer decrease with time due to racemization and isomerization reactions. Although higher enantiomeric excesses are obtained for short reaction time and are decreasing gradually, this catalytic system is very easy to carry out and can be considered as an interesting tool for the asymmetric synthesis of chiral allylic ethers. Further investigations are carried out in order to further increase the ee and to apply this new methodology to the synthesis of other chiral allylic ethers.

  Scheme 1. Hydrofunctionalization of butadiene

4 Figure 2 .

 42 Figure 2. Atropoisomeric diphosphine ligands

Figure 3 .

 3 Figure 3. Ethers yields, OC4 b selectivities, major enantiomer selectivities and OC4 b enantiomeric excesses as function of time (Conditions : Butadiene = 17.2 mmoL, Ni(cod)2/ligand/butadiene (0,08 mol % : 0,12 mol % : 1), EtOH = 10 mL, toluene = 3 mL, T = 80°C.)

  an initial kinetic preference for the formation of the branched butenyl ether product but the low calculated energy barrier for the C-O bond formation from the nickel π-allyl intermediate allows considering the reversibility of this step and the observed isomerization.

  corresponds to k[EtOH]. The next elementary steps leading to the formation of the products from this intermediate are supposed to be reversible (steps (1), (2) and (3)) as suggested by the experimental results (Figure 3, epimerization and isomerization of the butenyl ethers).

  Scheme 4 π-allyl nickel intermediate and products formed during hydroalkoxylation of butadiene

Figure 4 :

 4 Figure 4 : Proposed kinetic model

Figure 4

 4 Figure 4 Parity plot of model concentration vs experimental concentration

Figure 6

 6 Figure 6Experimental and simulated concentrations of products with time.

Table 1 .

 1 First series of chiral ligands used for the nickel catalyzed asymmetric hydroalkoxylation of butadiene with ethanol Variation of the chiral ligand for butadiene hydroalkoxylation a

			Me
	Ph 2 P	Fe	N	PPh 2
			Me

N Figure 1.

  Yields in butenyl ethers were measured in the 17-97 % range and selectivities for the branched chiral ether varied from 64 % to 75 %.

				Ni(cod)2 (0.08 %)
		+ EtOH (excess)	ligand (0.12 %) toluene (3 mL) 3 h, 80°C	* OEt OC 4 b	+	OC 4 l	OEt
	Entry		Ligand	Conversion b (%)	Selectivity OC4 b b (%)	Ee c (%)
	1	L1		(S)-Segphos	66	71	68
	2 d	L1		(S)-Segphos	31	75	77
	3	L2		(R)-Synphos	97	64	15
	4	L3	(S)-C3-TUNEPHOS	17	70	69
	5	L4	(S)-MeO-BIPHEP	67	72	58
	6	L5	(R)-Ph-Garphos	88	69	27
	7	L6	(S)-cyclohexyl Soniphos e	33	68	63
	8	L7	CTH-(R)-P-Phos	17	74	64
	9	L8	CTH-(R)-Xylyl-P-Phos e	87	68	45
	10	L9		BoPhoz	94	85	13
	11	L10		JOSIPHOS	94	87	13
	12	L11		TANIAPHOS	18	91	16
	13	L12		WALPHOS	32	92	53

a Butadiene : 17.2 mmoL, Ni(cod)2 /ligand /butadiene (0,08% : 0,12% : 1), EtOH : 10 mL, toluene : 3 mL, ; T : 80°C, t : 3 hours.

b

Calculated by GC analysis, from the amount of butenyl ethers, using n-heptane as internal standard. c Calculated by GC analysis using a Chiraldex column d Reaction conducted at 60°C e t : 4 hours Soniphos L6, CTH-(R)-P-Phos L7 and CTH-(R)-Xylyl-P-Phos L8 (entries 1-9).

Table 2 .

 2 Effect of Segphos and MeO-BIPHEP derivative ligands on the enantioselective butadiene

	hydroalkoxylation
		+ EtOH (excess)	Ni(cod)2 (0.08 %) ligand (0.12 %) toluene (3 mL) 3h, 80°C	* OEt OC 4 b	+	OC 4 l	OEt	6 7
	Entry		Ligand	Conversion b (%)	Selectivity OC4 b b (%)	8 Ee c (%)
	1	L1	(S)-Segphos	66		71	68
								9
	2	L1.1	(S)-DM-Segphos	89		74	64
	3	L1.2	(S)-difluorphos	7		66	48	10
	4	L4	(S)-MeO-Biphep	67		72	58
								11
	5	L4.1	(R)-Cl-MeO-Biphep	40		71	49
	6	L4.2	(S)-L4.2	42		64	50	12
	7	L4.3	(S)-L4.3	99		95	21
								13
	8	L4.4	(S)-L4.4	0		-	-
								14
								15
	Among the different catalysts used in this study, several of them displayed a high catalytic activity
	but led to a low enantioselectivity. This behavior suggested that undesired processes of isomerization /
	racemization of the different isomeric allylic ethers formed during the reaction could occur. We thus
	considered studying the evolution of the enantiomeric excess vs conversion on a selected example. The

combination of CTH-(R)-Xylyl-P-Phos L8 with Ni(cod)2 was chosen for this study as a high yield and a moderate enantiomeric excess were obtained with this catalytic system. Standard reaction conditions were a butadiene :

17.2 mmoL, Ni(cod)2/ligand/butadiene (0,08% : 0,12% : 1), EtOH : 10 mL, toluene : 3 mL, T : 80°C, t : 3 hours. b Calculated by GC analysis, from the amount of butenyl ethers determined using n-heptane as internal standard c Calculated by GC analysis using a Chiraldex column

Acknowledgments

We are grateful to the Region Nord-Pas de Calais for A.M.'s fellowship. We acknowledge the Ministry of Research and Technology, the Institut Universitaire de France, the ANR (project H2CAT: ANR-15-CE07-0018-01) and the CNRS for their financial support. We would like to extend our gratitude to Dr Léo Violet and Dr Carole Mutschler for their precious advices.


Juliette Faucher
email: juliette.caroline.faucher@vub.be.

Thomas Déhais

Béatrice Luais

Vinciane Debaille

Philippe Claeys

Steven Goderis

volatilization and hydrothermal alteration

come

The Rochechouart impact structure

Samples and methods

Drillings by Centre for International Research and Restitution on the Impact structure of Rochechouart [1] (CIRIR, 2017) recovered impactites from the deeply eroded Rochechouart structure (France, 206.92 ±0.32 Ma), formed on crystalline gneiss and granites from the Hercynian Massif Central [2]  50 samples from 6 drilled cores for petrography and bulk geochemistry  19 samples for isotopic analyses (Fe, Cu, Ge, Zn) Irregural transition between impactites and basement [3] + uneven distribution of impactite lithologies.

Main results

 Uneven distribution of the different impactites lithologies across the structure  Non-traditional stable isotopic ratios showing variations compared to the target lithologies ratios: → limited volatilization effects (heavy ratios) on all lithologies shown by elements with low volatility (Fe) and high volatility (Zn);

→ Ge seems to preserve volatilization effects in suevites, impactoclastites and yellow impact melt rocks but display post-impact mobilisation effects in red impact melt rocks.  The effect of the impactor contribution remains to be investigated.

  South and East → almost breccia-free (proximity collapsed central uplift ?) , Plagioclase, Musc ± augen K-Fds ±Gt Red Impact Melt Rocks Hypocrystalline brown-red groundmass, mineral and lithic clasts Groundmass = metasomatic K-fds, hydrothermal related Fe-Ti oxides and infills of voids/vesicules Yellow Impact Melt Rocks Hypocrystalline yellow groundmass, lithic and mineral clasts Groundmass = metasomatic K-fds (absence of Fe-Ti oxides) Limited extend in structure (Rochechouart SC15 and Recoudert SC17) Immiscibility with red impact melt rock Suevites Clastic groundmass bearing mineral + lithic target clasts + glass particules Only in Chassenon Butte SC2 (impact melt rock-free part of the structure) Polymict Impact Breccias Clastic brown-red to grey groundmass + mineral and lithic clasts Impactoclastites Ash-like stratified deposit of clastic material, similar to very fine suevite Only occurring on top of the Chassenon Butte sequence (SC1-3) Bearing impact spherules (composed of wavy to vermicular illite)

  Red impact melt rocks generally in isotopic ranges of target lithologies Suevites and impactoclastites : lighter signatures and higher Fe 2 O 3 contents than target lithologies → post-impact mobilization ?

	Zn	Ge	Cu	Fe
				T 50 (K)
	704	830	1034	1338
		Fe isotopic ratios

Acknowledgements

We thank the Barringer Family Fund for Meteorite Impact Research for funding this project. We appre-

Cu isotopic ratios

Large variability in isotopic ratios (-0.75 to +0.93 ‰) and Cu contents (2.52-115.5 ppm) Red impact melt rocks: negative correlation from gneiss contents to lighter isotopic signatures → mobilization by post-impact hydrothermalism ?

Ge isotopic ratios

Group 1 (impactoclastites, suevites, yellow impact melt rocks): heavier Ge isotopic signatures → volatilization ? Group 2 (impact melt rocks): lighter Ge isotopic signatures but higher Ge elementary content → post-impact mobilization ?

Zn isotopic ratios

Almost all lithologies display Zn isotopic ratios lighter than target lithologies → mobilization by postimpact hydrothermalism ? Impact cratering is a fundamental phenomenon of planetary surface evolution during which processes such as impact melting, mixing, volatilization, projectile contamination, and impact-induced hydrothermalism affect the target lithologies. Detailed petrography, along with the use of a set of 4 non-traditional stable isotopes covering a full range of volatilities aim to disentangle these processes.