Zheng Li
email: lizheng619@hotmail.com

Sandra Escoffier
email: sandra.escoffier@ifsttar.fr

Panagiotis Kotronis
email: panagiotis.kotronis@ec-nantes.fr

Centrifuge modeling of batter pile foundations under earthquake excitation

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Batter piles, also called inclined or raked piles, are widely used in civil engineering constructions where substantial lateral resistance is required. However, many building codes or standards like AFPS 1990 [START_REF] Afps | Recommandations AFPS 90[END_REF] and Eurocode-8 [START_REF]Eurocode8. Eurocode 8: design of structures for earthquake resistance-Part 5: foundations, retaining structures and geotechnical aspects[END_REF] do not recommend the use of batter piles in seismic regions. Although less conservative, codes like ACI318-05 and ACI318-11 recommend to pay attention to the potential damages at the junction of the batter piles and the superstructure. The main drawbacks of the batter piles often mentioned by engineers are the following: large forces induced onto the pile cap, reduction in bending capacity due to the axial forces, unfavorable rotation on the cap and residual bending moment due to the soil settlement before the earthquake [START_REF] Gerolymos | Evidence of beneficial role of inclined piles: observations and summary of numerical analyses[END_REF]. Several case histories, for example, the wharf failure in the port of Oakland in 1989 during the Loma Prieta earthquake and the port of Los Angeles in 1994 during the Northridge earthquake reveal the unsatisfactory performance of batter piles.

However, recent studies show that batter piles can have a positive performance. As reported by Gazetas and Mylonakis [START_REF] Gazetas | Seismic soil-structure interaction: new evidence and emerging issues[END_REF], batter piles, if properly designed, can play a beneficial role both for the structure they support and the piles themselves (reduction of the bending moments in the piles). Research studies from Pender [START_REF] Pender | Aseismic pile foundation design analysis[END_REF] and Berrill et al. [START_REF] Berrill | Case study of lateral spreading forces on a piled foundation[END_REF] also suggest important beneficial effects from batter piles. The argument about whether the use of batter piles in seismic regions is detrimental or beneficial is therefore still unsettled. In 2004, Harn [START_REF] Harn | Have batter piles gotten a bad rap in seismic zones (or everything you wanted to know about batter piles but were afraid to ask)[END_REF] pointed out that the poor performance of batter piles in past earthquakes may due to the lack of knowledge and analytical tools. Using displacement based design, advanced numerical tools and appropriately detailed batter piles can result in significant project savings.

Several experimental studies on the performance of batter piles in the static domain have already been performed. From 1972 to 1995 Meyerhof and his colleagues conducted experimental campaigns (reduced model scale at 1 g) to investigate the bearing capacity of batter piles in soils [START_REF] Meyerhof | The bearing capacity of rigid piles under inclined loads in sand. I: vertical piles[END_REF][START_REF] Meyerhof | The bearing capacity of rigid piles under inclined loads in sand. II: batter piles[END_REF][START_REF] Meyerhof | The bearing capacity of rigid piles under inclined loads in sand. III: pile groups[END_REF][START_REF] Meyerhof | Ultimate pile capacity for eccentric inclined load[END_REF][START_REF] Meyerhof | Bearing capacity of rigid piles under eccentric and inclined loads[END_REF][START_REF] Meyerhof | Behaviour of flexible batter piles under inclined loads in layered soil[END_REF][START_REF] Meyerhof | Bearing capacity of flexible batter piles under eccentric and inclined loads in layered soil[END_REF]. They proposed an empirical equation to predict the ultimate strength of batter piles under arbitrary load combinations (horizontal and vertical forces). With the development of the centrifuge modeling technique, static tests on batter piles have also been performed on reduced scale models [START_REF] Pinto | Centrifuge testing of plumb and battered pile groups in sand[END_REF][START_REF] Mcvay | Centrifuge modeling of laterally loaded battered pile groups in sand[END_REF][START_REF] Zhang | Centrifuge modelling of laterally loaded single battered piles in sands[END_REF][START_REF] Zhang | Effects of dead loads on the lateral response of battered pile groups[END_REF][START_REF] Escoffier | Centrifuge modelling of raked piles[END_REF]. It was observed that batter piles in general increase the horizontal resistance of the foundations.

Centrifuge dynamic tests on batter piles are relatively rare. Escoffier et al. [START_REF] Escoffier | Centrifuge modelling of raked piles[END_REF] performed centrifuge tests on pile foundations in dry sand by applying a horizontal impact loading on the pile cap with a magnetic hammer. Two pile groups were studied: a 1 Â 2 (two piles in one row) vertical pile group and a 1 Â 2 pile group with one batter pile. Two pile bearing conditions (floating and end bearing) were considered. The tests indicated a more complicated translation-rocking mode for the inclined pile group. Its stiffness was found higher and the resulting movement smaller than that of the vertical pile group. In both groups, the center of rotation of the cap was near the vertical face of the pile cap towards the front pile side. In terms of bending moment profiles, the presence of the batter pile resulted in a decrease of the maximum bending moment below the soil surface in both piles. In addition, the influence of the batter pile on the maximum bending moment at the pile cap interface seemed to be negligible in the front pile, whereas it resulted in an increase of the maximum bending moment in the rear pile. For the batter pile group, the compression load in the front pile and the tension load in the rear pile were increased by a factor of 1.7 and 2.0, respectively.

In 2012, Escoffier [START_REF] Escoffier | Experimental study of the effect of inclined pile on the seismic behavior of pile group[END_REF] performed dynamic centrifuge tests on batter piles. The dynamic loads were applied using an earthquake simulator [START_REF] Chazelas | Original technologies for proven performances for the new LCPC earthquake simulator[END_REF] at the bottom of the model. As in the previous study, two configurations (a 1 Â 2 vertical pile group and a 1 Â 2 pile group with one batter pile and end-bearing pile configuration) were studied using earthquake time histories. The results of the seismic tests showed that the batter pile resulted in an increase of the response frequencies corresponding the translation-rotation mode at the pile cap and reduced by two times the maximum horizontal acceleration of the cap. The analysis of the bending moments and axial loads in the piles indicated that with the batter pile, larger residual bending moments were developed and the direction of the axial loads in the piles did not change. Escoffier reported that for batter pile foundations the residual bending moments cannot be ignored in the evaluation of the performance of batter piles.

Okawa et al. [START_REF] Okawa | Dynamic behaviour of a group-pile foundation with inclined piles in loose sand[END_REF] performed centrifuge tests on pile groups embedded in loose sand. The first was a 2 Â 4 pile group composed of 8 vertical piles and the second a 2 Â 4 pile group with 4 batter piles with an inclination of °10 and 4 vertical piles. A short su- perstructure was placed on the footing to study both kinematic interaction and inertial effects. The presence of batter piles decreased the horizontal acceleration amplitude at the footing and the superstructure. Larger axial forces were observed in the batter piles.

Boland et al. [START_REF] Boland | Pile-supported Wharf centrifuge model[END_REF][START_REF] Boland | Pile-supported Wharf centrifuge model[END_REF] performed dynamic centrifuge tests with two models (named SMS02 and JCB01) constructed as a generalization of the pile supported wharf structures common at the Port of Oakland (POOAK), the Port of Los Angeles (POLA) and the Port of Long Beach (POLB) in California. In terms of maximum moments, results clearly showed an increase (from 1.4 to 3.9 times) in recorded pile moments when all vertical piles were used for nearly identical input base accelerations. The shear forces within the batter pile heads were greater (9-20 times) than those recorded in the vertical piles for the same dynamic event. Without batter piles, the shear forces in the vertical piles and the wharf displacements increased significantly [START_REF] Schlechter | Influence of batter piles on the dynamic behavior of pile-supported Wharf structures[END_REF].

Juran et al. [START_REF] Juran | Engineering analysis of dynamic behavior of micropile system[END_REF] conducted a series of centrifuge tests on vertical and batter micropile groups. The tests showed that with increasing inclination angle the natural frequency of the network system increased. Compared with vertical piles, a reduction of 40% of the response in terms of acceleration of the superstructure (pile cap) was observed when batter piles were used. In general, increasing the pile inclination resulted in smaller pile cap displacements, and larger axial forces and bending moments in the piles at the pile cap connections.

Giannakou et al. [START_REF] Giannakou | Seismic behavior of inclined piles[END_REF][START_REF] Giannakou | Seismic behavior of batter piles: elastic response[END_REF] studied numerically, in the time domain, the performance of batter piles. Both soil and batter pile groups were modeled using linear elastic constitutive models. Five inclinations were considered. The authors found that for seismic loadings and purely kinematic conditions, the negative reputation (larger bending moments and axial forces) of batter piles was more-or-less confirmed. However, when the total response was considered (kinematic and inertial response of the structural system), their influence can be beneficial. Among the different parameters studied, the performances of batter piles in terms of displacements, bending moments and axial forces depend on the ratio of the overturning moment versus the shear force transmitted to the piles from the superstructure.

Shahrour and Juran [START_REF] Shahrour | Seismic behaviour of micropile systems[END_REF] performed numerical analysis of the seismic behavior of a micropile system containing batter micropiles. It was found that the inclination of the micropiles allowed an effective mobilisation of their axial resistance, leading to an increase in the stiffness and a reduction in both shear forces and bending moments.

This paper presents a comprehensive experimental seismic centrifuge program on the performance of batter piles in order to better understand their behavior. total bending moment on pile P7 M P7 max,VS maximum total bending moment on pile P7 in ver- tical pile group with short superstructure N axial force in the pile N P7

axial force in pile P7 N P7 max,VS maximum Axial force in pile P7 in vertical pile group with short superstructure z depth below ground surface D external diameter of pile D pile center-to-center distance between piles (SSPSI), the influence of the height of the Gravity Center of the superstructure (C.G.) and of the base shaking signal characteristics.

All tests have been performed at 40 g centrifugal gravity level. Short (squat) and tall (slender) superstructures were used in order to study the influence of the inertial loading from different types of superstructure on the performance of batter piles. A symmetric configuration of batter piles with a °15 inclination is adopted and results are compared to those of a vertical pile group. For both configurations, rigid connections were imposed at the pile heads through a stiff pile cap. The different behaviors of the two configurations under dynamic excitations are analyzed and discussed.

Experimental set-up

Base shaking signals-selection of real earthquakes

Two of the main objectives of the experimental campaign are to determine the response frequencies of the soil-pile-superstructure system and to evaluate the performance of batter piles when subjected to multiple-frequencies of base shaking. With this in mind, three different earthquakes were selected. The first two earthquakes are the Northridge (Tarzana station 090) and the Martinique (Jara station EW) records. Both are broad-band frequency range, see Fig. 1(a) and(b). The use of these two earthquakes should enable the determination of the natural frequency of the soil-pile-superstructure system. The third signal is a recording of the Kobe earthquake (DAI8-G, N43W). Most of the energy of this signal comes from frequencies below 2.0 Hz, see Fig. 1 (c), lower than the prior response frequencies of both the soil column and the soil-pile-superstructure system. 1 In order to be in the capacity range of the shaker [START_REF] Chazelas | Original technologies for proven performances for the new LCPC earthquake simulator[END_REF], these three earthquake signals have been filtered to be within the frequency range of 0.5-8.75 Hz Hz at the model scale). According to the different purposes of the tests and the capacity of the shaker [START_REF] Chazelas | Original technologies for proven performances for the new LCPC earthquake simulator[END_REF], earthquake signals are often scaled down, which will be introduced in Section 2.3.

Build of the centrifuge models

The main elements of the physical centrifuge models are the soil deposit with controlled homogeneous density, the piles instrumented with strain gauges, the stiff pile caps and the superstructures. Accelerometers and laser sensors were also installed to record the behavior of the soil-pile-superstructure system.

Properties of the experimental set-up components: sand, piles, pile caps, superstructures and pile groups

The properties and design considerations of each component are introduced hereafter. Dense dry Fontainebleau sand (NE34) was used in all tests and put in place with an air pluviation technique (relative density of about 80%). The material properties of Fontainebleau sand are listed in Table 1. The model piles were made of aluminium alloy and had a hollow section. They were instrumented with strain gauges, which enabled the determination of the bending moment profile and the axial stress at different depths. The pile heads were rigidly connected together through stiff pile caps, which induced rotational restraint [START_REF] Mokwa | Rotational restraint of pile caps during lateral loading[END_REF]. The pile caps for both vertical and batter piles were made of two aluminium alloy pieces to ensure an easy assembly with bolts after the sand pluviation. The reason for doing this is to avoid the "shadowing effect" of the pile cap during the air pluviation process. Fig. 2 shows the assembly of the batter piles and the pile cap. It can be noticed that the piles were connected with a thin plate before the assembly. Its role was to provide a temporary support to the piles during the sand pluviation process. Table 2 gives the characteristics of the piles and the pile cap in model and prototype scales.

The lateral resistance of the pile cap can significantly increase due to the friction between the soil and the cap, especially when the pile cap is embedded beneath the ground surface, [START_REF] Beatty | Lateral test on pile groups[END_REF][START_REF] Kim | Pile cap soil interaction from full-scale lateral load tests[END_REF][START_REF] Liu | Cap-pile-soil interaction of bored pile groups[END_REF][START_REF] Zafir | Lateral response of large diameter drilled shafts: I-15/ US 95 load test program[END_REF]. In order to avoid soil-cap interaction, the piles were designed to stand at a certain distance above the ground surface, see Fig. 3. The gap between the pile cap and the ground surface is 28 mm at model scale (1.12 m at prototype scale). The spacing of the piles, i.e. the center-to-center distance between the piles, was 4 times the diameter of the piles (4D) for both pile group configurations. Floating pile group configurations were considered. In order to study the influence of the height of the Gravity Center of the superstructure (C.G.) on the responses of the pile foundations, two types of single degree of freedom superstructures were used, i.e. a short and a tall superstructure. Both superstructures were designed to have:

the same resonant frequency under fixed base conditions, the same top mass weight, the same total weight of the whole foundation-superstructure system.

Fig. 4 shows the two superstructures. They were made of three parts: the top mass, the supporting column and the base. The top mass was made from brass, the column and the base from aluminium alloy. The base was designed in a way to provide a fixed connection with the superstructure. After the superstructure installation, the base together with the pile cap constituted an enlarged new foundation, Fig. 5. The frequency of the fixed-based building was designed to be 2.0 Hz. Due to the 1.12 m distance between the pile cap and the soil surface, after the installation of superstructure, the height or the gravity center of the top mass was thus increased. The foundation superstructure system can be idealized using lumped masses, see Fig. 5. Since the column supporting the building cannot be regarded as weightless, according to Harris and Piersol [START_REF] Harris | Harris' shock and vibration handbook[END_REF], approximately 23% of the mass of the column should be included in the top mass while the rest 77% goes to the base. The gravity center of the pile foundation, H cap , is 1.92 m (including the gap between the pile cap and the soil surface); the gravity center H mass of the building was measured from the ground surface. The detailed characteristics of the pile foundations with short and tall superstructures are listed in Table 3. 2.2.2. Experimental procedure: preparing the sand-pile system Fontainebleau sand with 80% relative density was used for the soil pile system. The homogeneity of the sand deposit was controlled with the air pluviation technique. In centrifuge tests, the methods to install the piles are different from the engineering practice (where batter piles are installed into the soil by driving). Escoffier et al. [START_REF] Escoffier | Centrifuge modelling of raked piles[END_REF] discussed the influences of the different ways of pile installation on the final response.

Mass block

The following method was adopted in this experimental study. First, all piles were fixed and maintained to their precise positions in the Equivalent Shear Beam (ESB) container [START_REF] Zeng | Design and performance of an equivalent-shear-beam container for earthquake centrifuge modelling[END_REF][START_REF] Teymur | Experimental study of boundary effects in dynamic centrifuge modelling[END_REF] at 1 g. Vertical shear rods were attached to the bottom of the container to provide complimentary shear stresses, see Fig. 6. Then the sand was filled by air pluviation from the bottom up to the level of the ground surface. A temporary sustaining system was used to maintain the pile group during pluviation, composed of an aluminium plate, 2 two steel rods and thin plastic wires, 3 see Fig. 6. All pieces in the temporary supporting system had to be thin enough to minimize the "shadowing effect" during sand pluviation. 4 After pluviation, the pile cap was installed onto the pile head by tightening the bolts with a dynamic torque screwdriver. Finally, the superstructure was installed onto the top of the pile cap.

Sensors were installed to record the movement of the soil-pile-superstructure system. Their configuration (arrangement) in the centrifuge model is shown in Fig. 7. The acceleration of the soil column was tracked by a vertical array of accelerometers, CH03∼CH09, which located away from the piles. Comparison of the measured maximum accelerations and displacements with those from the free field tests 5 showed that these measured accelerations are representative of the free field condition and were not influenced by the pile group. The movements of the pile cap and of the superstructure were monitored by accelerometers CH10∼CH13. Furthermore, three high speed laser sensors were used on the superstructure. Several accelerometers were attached on one side of the container in order to capture the response of the container.

Experimental procedure: program and test repeatability

For each pile foundation superstructure configuration, 18 earthquake inputs were used, see Table 4. The Arias intensities [START_REF] Arias | A measure of earthquake intensity[END_REF] are shown in Fig. 8. The first two small earthquakes were used to estimate the frequency response of the soil-pile-superstructure system. The last two small earthquakes were used to study the dynamic responses of the different pile configurations after extreme earthquake events. In order to verify the repeatability of the experimental results, some tests were performed twice. Fig. 9 provides an example of the results for two identical tests performed on the batter pile group with tall superstructure. Results indicate a good repeatability. Fig. 6. Position of the piles in the ESB container. 2 The thin aluminium plate was used to maintain the inclination angle of °15 . 3 The steel rods and the plastic wire were used to minimize the impact on the piles. When the sand surface was about 5 cm higher than the tips of the piles, the wires were cut to free the pile tips. After the sand surface arrived at the prescribed height, the pluviation stopped and the steel rods were removed. 4 Shadowing effect: during sand pluviation the presence of items can influence the falling of sand into the container and thus affect the homogeneity of the sand deposit. 5 The free field tests were performed previously.

Definitions, assumptions

Several definitions and assumptions concerning the analysis and interpretation of the experimental results are introduced in this section.

Pile names definition

In the dynamic centrifuge tests, there is no front or rear pile as the sign of the loading changes. Piles are named hereafter P7 and P8 refering to their positions: the pile close to the "Porte", on the right-hand-side, is called P7 and the pile close to the "Pivot", P8 see Fig. 10.

Residual, dynamic and total bending moment

During an earthquake event, due to permanent soil deformations and the influence of the superstructure, embedded pile foundations may move from their initial position. Permanent lateral displacements may occur at the pile head and along the pile, see Fig. 11. The residual bending moment is defined as the difference between the bending moment at the beginning and at the end of the earthquake event, see Fig. 12. As shown in the same figure, the total bending moment can be decomposed into two parts: a dynamic and a residual part. 1. Martinique Jara earthquake is abbreviated to Mj; Northridge earthquake is abbreviated to Nr; Kobe earthquake is abbreviated to Kb. 2. Values of PGA (peak ground acceleration) and Arias Intensity are in prototype.

3. Attenuation refers to the reduction of the signal strength and is represented in decibels (dB)-calculated as 20 times the logarithm (base 10) of the output divided by the input.

Accumulation effect

It was observed during the tests that successive earthquake events influence the residual bending moment. However, if the same earthquake signal is applied 3 times, the evolution or the "accumulation" effect is found limited. Furthermore, the last small earthquakes do not induce new residual bending moments. Fig. 13 shows for example the evolution of the residual bending moment of the batter pile group with tall superstructure under different earthquake events.

Rotation (rocking) and horizontal displacement (translation) of the pile cap

In the experiments, the responses of the pile cap and the superstructure are monitored by sets of accelerometers as shown in Fig. 14. The vertical accelerations of the pile cap are recorded by the sensors CH12 and CH13. The horizontal accelerations of the pile cap and the superstructure are measured by the sensors CH10 and CH11, respectively. The sensor CH09 is far enough away from the pile foundation and it is used to capture the acceleration near the soil surface. The rotation of the pile cap can be derived from CH12 and CH13. The relative translation of the pile cap with respect to the soil surface is obtained by calculating the difference between the displacement of the pile cap and the displacement of the soil surface. The displacement time histories are obtained by double integration [START_REF] Escoffier | Seismic and sinusoidal tests on pile group[END_REF] of the measured acceleration time histories, the accuracy of the double integration is validated [START_REF] Li | Using centrifuge tests data to identify the dynamic soil properties: application to Fontainebleau sand[END_REF] by the laser sensors (L128, L134 and L135) 6 installed near the superstructure, see Fig. 7. Due to the frequency range of the accelerometers, the residual part of the rotations and the displacements cannot be monitored. Therefore, the displacements and rotations calculated from the measured accelerations represent only the dynamic components.

Overturning moment and total horizontal force (base shear)

The overturning moment (OM) and the base shear force (BS) are calculated from the inertia forces generated by the mass of the superstructure and the base using Eqs. ( 1 where M mass and M cap are the masses, H mass and H cap the C.G. and u " mass and u " cap the measured accelerations at the top mass and at the pile cap, respectively. The overturning moments and horizontal forces are calculated only from the measured horizontal accelerations, no other forces are considered.

Representative values

According to Table 4, each earthquake was applied 3 times. Nevertheless, it was observed that experimental data (such as acceleration, bending moment and axial force) recorded from the 3 successive identical inputs were similar, see Fig. 15 for example. This may due to the high density of the sand used in the experiments that limits the evolution of the behavior of the soil-pilefoundation system. The maximum values obtained during the three same earthquake events are chosen as the representative values (conservative approach).

From the experimental results it was also found that for both batter and vertical piles, bending moments (residual and total) and axial forces measured in pile P7 are higher than (or equal to) those measured in P8 (this is may due to the first loading direction). The maximum response in pile P7 is therefore considered as the representative value.

Data normalization-interpretation of the results

In order to evaluate the performance of the batter piles and the influence of the superstructures they support, cross comparisons were made between the different configurations. The percentage of increase ( > ) P 0% or decrease ( < ) P 0% of a measured quantity due to the use of batter piles is calculated using Eq. (3).

= -( ) ( )

P Q Q Q in % 3 max I max V max V , ,
, where P is the performance index of the batter piles, Q max I , and Q max V , the measured quantities on the batter and vertical pile configurations, respectively. In this way, results obtained on one configuration (batter pile group with tall or short building) are normalized with respect to the corresponding vertical pile group with the same superstructure. Coloured bars are used to represent the performance index P, see for example Fig. 16. A positive value corresponds to a negative behavior of the batter pile (symbol ⊖) compared to the vertical one; the symbol ⊕ indicates a beneficial behavior. In addition, maximum values obtained for the vertical pile groups are indicated on the top of the figure. In Fig. 16 for Fig. 12. Total, dynamic and residual bending moments. Fig. 13. Accumulation of residual bending moment-batter pile group with tall superstructure (results of P7, prototype scale): (a) residual bending moment under the Northridge À 9 dB earthquake, events 7, 8 and 9; and (b) evolution of the residual bending moment during all the 18 earthquake events. Fig. 14. Illustration of the response of the pile cap. 6 In the tests, laser sensors only measured the absolute displacements of the superstructure and it is difficult to determine the relative displacements of the superstructure with respect to ground surface. Therefore, the relative displacements of the superstructure with respect to ground surface are determined by pair of accelerometers. example, batter piles have a negative effect especially when they support a tall superstructure. Another different normalization rule is adopted for the stress profiles in the piles. Among all the pile configurations, the vertical piles with the short superstructure (VS) is selected as the reference configuration. The stress profiles are thus all normalized with respect to the maximum value recorded on the referenced configuration. The maximum value on the reference configuration is chosen regardless the location where it appears along the pile (either at pile head or other locations).

Zero initial stress state

In the tests, for both vertical and batter piles, the possible initial pile stresses (bending moments and axial forces) generated during the air pluviation, the assembling of the pile cap and the spinning up of the centrifuge were not measured. Therefore, the analysis of the performance of the batter piles is based on the assumption that the initial pile stresses are zero. Furthermore, since both the short and tall superstructures are designed to have approximately the same total weight, the initial states for all pile configurations are assumed identical. Although the inclination of piles may induce differences, this effect is not taken into account.

Performance of batter piles considering Seismic Soil-Pile-Superstructure Interaction (SSPSI)

In this part, the results are presented at the prototype scale unless otherwise mentioned. It has been checked that the performance of the container did not influence the frequency of the ground motions (almost the same frequency response of the container and the soil column).

Under seismic loading, piles that support superstructure are subjected not only to the kinematic interaction but also to inertia interaction. The inertial loads coming from the superstructure can significantly influence the behavior of the pile foundations [START_REF] Giannakou | Seismic behavior of batter piles: elastic response[END_REF]. In return, the behavior of the superstructure is influenced by the movement of the piles (Seismic Soil-Pile-Superstructure Interaction (SSPSI)). The following section provides the main results of the experimental campaign focused on the behavior of batter and vertical piles under SSPSI and put in evidence, among others, the significant influence of the superstructure and the characteristics of the base shaking signal on the performances of both batter and vertical piles.

Frequencies

The first two small earthquakes (6 base shaking events) were used to evaluate the frequency response of the different pile group configurations (IS: batter (inclined) -small building, VT: verticaltall building etc., see Table 5). For this, transfer functions are calculated from the accelerations measured on the top of the superstructure (CH11) or at the pile cap (CH10) and those measured near the soil surface (CH09). The estimated corresponding frequencies are listed in Table 5. For all cases, soil-structure interaction decreases the superstructure's natural frequency when comparing with fixed-based conditions (2.0 Hz). The percentage of decrease for configuration IS, VS, IT and VT are À3%, À 10% À 15% and À 19%, respectively.

The presence of batter piles reduces the effect of SSPSI on the frequencies, especially for the configuration with a short building. This is due to the fact that batter piles contribute to a higher horizontal stiffness. For the case of short and tall superstructures installed on batter piles, frequencies are 0.14 Hz and 0.08 Hz higher than for the vertical pile groups. An increase of the height of C.G. decreases the frequency response of the superstructure for both pile group configurations ( À 0.24 Hz and À 0.18 Hz for the batter and the vertical pile group, respectively). This effect is more pronounced for the batter pile configuration.

The presence of a superstructure modifies also the response of the pile cap. Previous tests without superstructures have shown that the frequencies of the batter and vertical pile groups are 10.6 Hz and 6.0 Hz, respectively. When introducing a short or a tall superstructure with similar rigid base frequency on batter piles, the frequency of the pile cap is found decreased (from 10.6 Hz to 6.28 Hz and from 10.6 Hz to 7.12 Hz, respectively).

Displacements and rotations of the pile cap

The maximum displacements and rotations of the pile cap are shown in Figs. 17 and 18, respectively. The pile cap movement is crucial as it is often used in design criteria. Looking at the pile cap maximum horizontal displacements, batter piles play a beneficial role both for short and tall superstructures. These results are in accordance with previous results [START_REF] Juran | Engineering analysis of dynamic behavior of micropile system[END_REF][START_REF] Giannakou | Seismic behavior of batter piles: elastic response[END_REF][START_REF] Giannakou | Seismic behavior of inclined piles[END_REF]. This performance is influenced by the characteristics of the base shaking signal and the height of the superstructure. However, these tests show that the beneficial influence is more pronounced for the short superstructure (30-50%) than for the tall one (8-34%), except for the Kb À4 dB earthquake (whose frequency content is below the frequency responses of the total system (soil, piles, superstructure)). Giannakou et al. [START_REF] Giannakou | Seismic behavior of inclined piles[END_REF][START_REF] Giannakou | Seismic behavior of batter piles: elastic response[END_REF] pointed out that the performance of batter pile depends on the ratio of the overturning moment versus the shear force transmitted to the piles from the superstructure. In terms of pile cap displacement, the beneficial role of batter piles is more highlighted for the case of short superstructure with small ratio of overturning moment (OM) to base shear (BS). The ratios of overturning moment to base shear for different configurations are summarized in Table 6. It can be concluded from the experimental results that short superstructure has small ratio of overturning moment to base shear which is shear force dominant; and tall superstructure has overturning moment dominant inertial force with larger ratio of overturning moment to base shear, see Table 6.

In terms of maximum rotations (Fig. 18), the performance of the batter piles is influenced by the base shaking characteristics (frequency content and amplitude) and the type of the superstructure. For small earthquakes, batter piles induce a higher rotation at the pile cap level. This effect is more pronounced for the configurations with a tall superstructure (except for the last small Northridge event). This result is in accordance with the conclusions of Giannakou [START_REF] Giannakou | Seismic behavior of inclined piles[END_REF] from linear elastic finite element simulations. This may due to the shear interface stiffness along batter piles is less than the vertical pile cap rotational stiffness. For stronger earthquakes however, batter piles have almost no effect for the case of a tall superstructure or only a slightly beneficial role when a short superstructure is supported. This effect may due to the more important nonlinear behavior of the soil developed during strong earthquakes.

Total base shear

The normalized maximum total base shear forces for all the different configurations and earthquake events are shown in Fig. 19. For a short superstructure, batter piles play a beneficial role as they reduce the maximum total base shear. Again, this beneficial role depends however on the frequency content and the amplitude of the base shaking signal. In particular for strong earthquakes, the reduction is less pronounced ( À 35% for the small Northridge earthquake against À 20% for the large Northridge earthquake). For a tall superstructure, the use of batter piles can be detrimental or beneficial depending on the frequency of the earthquake and its amplitude. Its beneficial influence is more important for strong earthquakes. In general, the positive role of batter piles is more pronounced for the case of a short superstructure with small ratio (about 3-5, see Table 6) of OM/BS (except for the 4th Kb earthquake).

Total overturning moment

Results are summarized in Fig. 20. Batter piles play a beneficial role as they reduce the maximum total overturning moments ( À11 to À 55%). As before, this beneficial role seems more pronounced for the case of a short superstructure with small ratio of OM/BS. The reduction is À14 to À 55% when short superstructure is supported against À 11 to À 25% for a tall superstructure (except for the 4th Kb earthquake).

Stresses in piles

In this section, the "generalized" stresses along the piles (bending moments and axial forces) resulting from the combined kinematic and inertial interactions are analyzed and discussed. The envelope curves of the maximum stresses and the normalized peak values are compared for the different pile groups and superstructure configurations.

Residual bending moments

Fig. 21 shows the accumulated residual bending moment profiles along the piles for successive earthquake events. The presence of batter piles influences the shape of the profiles inducing in general a "C" and not the "S" shape profile obtained for vertical pile groups. For both pile group configurations, the height of the superstructure affects the depth of the maximum residual bending moment. For a short superstructure, batter piles increase this depth from 4D to 6D during the first two small earthquakes. This depth however remains unchanged during and after the strong earthquakes.

Concerning the maximum values of the residual bending moment, as shown in Fig. 22, batter piles have a detrimental effect regardless the type of the supported superstructure. However, the height of the superstructure, the base shaking characteristics and the loading history influence the effect of the batter piles. Once the residual bending moment was developed during the strong earthquakes, the subsequent small earthquakes cannot induce additional residual bending moments. This detrimental behavior is more pronounced when a tall superstructure is supported , except for the strong 3rd Northridge earthquake.

Maximum bending moments

Fig. 23 shows the normalized maximum bending moment profiles. Because of the presence of the residual bending moments, the curves for subsequent earthquakes are dragged and distorted in an asymmetric way. As a consequence, a "C" shape is still noticeable for the batter pile group. However, it is difficult to precisely describe the shape of the bending moment envelope curves for the vertical pile group. In general, the maximum values of the total bending moments for the vertical pile group are equally important below and above the soil surface. It is also observed that batter piles result in a reduction of the bending moment at the pile head. This observation is not in accordance with the results obtained on micropiles by Juran et al. [START_REF] Juran | Engineering analysis of dynamic behavior of micropile system[END_REF] where higher bending moments were found at the pile head connections. This difference may due to the different superstructure configurations. In the tests performed by Juran et al. [START_REF] Juran | Engineering analysis of dynamic behavior of micropile system[END_REF], only a pile cap was considered as a superstructure. However, in this paper, two more slender (short and tall) superstructures are used. It may also due to a different ratio of pull out resistance over bending moment resistance of the pile (the micropiles used by Juran had lower bending moment resistance).

Fig. 24 presents the normalized maximum total bending moments. In general, batter piles reduce the maximum total bending moment. This effect is more pronounced for the case of a short superstructure with a small ratio of OM/BS. When a short superstructure is supported, the performance of batter piles seems to be not influenced by the frequency content of the base shaking signals. For the case of a tall superstructure, the performance of batter piles is influenced by the base shaking characteristics.

Axial forces

Since only 3 strain gauges were used to measure the axial force, it was not possible to identify the axial force profiles. The maximum total axial force is discussed hereafter (no important residual effects were observed). The distance between the bottom of the pile cap and the soil surface remained almost the same before and after the test, no obvious relative displacement was observed in the axial direction. According to Nazir and Nasr [START_REF] Nazir | Pullout capacity of batter pile in sand[END_REF], in dense sand, within a small range of axial displacement, piles did not exceed their axial capacities. Similar to the total bending moments, batter piles often reduce the maximum axial force (Fig. 25). This effect is however relatively small (most of the time less than 30%) and is influenced by the frequency content and the amplitude of the base signal. For the case of a tall superstructure, the beneficial role of batter piles is less influenced. This reduction effect of batter piles on the axial forces could due to the presence of the superstructures, since it has been shown in Sections 4.3 and 4.4 that, in general, the batter pile-superstructure systems attract less base shear forces and overturning moments.

Overall behavior

The overall experimental behavior of batter and vertical pile foundations with short and tall superstructures under successive earthquake excitations can be summarized as follows:

1. The presence of batter piles contributes to a higher horizontal stiffness (comparing to the vertical pile configuration) and thus causes an increase of the corresponding frequencies. 2. An increase of the height of the center of gravity (C.G.) of the superstructure amplifies the influence of the Seismic Soil-Pile-Superstructure Interaction (SSPSI) on the frequency response for both pile group configurations. The influence of the C.G. is more pronounced for batter piles. 3. Batter piles decrease the horizontal pile cap displacements. This effect is influenced by the characteristics of the base shaking signal and it is more pronounced for short superstructures. 4. Batter piles tend to induce higher pile cap rotations for small earthquakes, especially for tall superstructures. This may due to the shear interface stiffness along batter piles is less than the vertical pile cap rotational stiffness. However for strong earthquakes batter piles play a beneficial role in decreasing the pile cap rotations, especially for short superstructures. This effect may due to the more important nonlinear behavior of the soil developed during strong earthquakes. 5. Batter piles tend to decrease the total overturning moment and the total base shear, especially for short superstructures. 6. The use of batter piles induces higher residual bending moments. This detrimental performance is more pronounced for tall superstructures. Batter piles also influence the shape of the residual bending moment profiles. 7. Batter piles reduce the total bending moment. This beneficial performance is not so much influenced by the characteristics of the base shaking signal for the case of short superstructures. 8. Batter piles reduce the axial forces. This effect is probably due to the batter pile-superstructure systems that attract less base shear forces and overturning moments. This beneficial effect is less influenced by the characteristics of base shaking signal for the case of tall superstructures.

In general, batter piles supporting short superstructures i.e. where inertial shear forces dominate, seem to have a more significant beneficial effects.

Conclusions

In this paper, an experimental dynamic centrifuge modeling campaign is presented to study the behavior of pile foundations considering Seismic Soil-Pile-Superstructure Interaction (SSPSI). This study enriches the database of the effect of batter piles on the seismic response of pile group. The influence of the height of gravity center on the performances of batter piles is highlighted. Cross comparisons are carried out to investigate the performance of different pile foundations with vertical or batter piles, short or tall superstructures and various earthquake loadings. Although the use of batter piles seems to have in general a positive effect in terms of displacements and total forces, it is found that the performance can vary depending on the characteristics of the base shaking signal and of the height of the center of gravity of the supported superstructures.

This paper extends the work of Li et al [START_REF] Li | Centrifuge modeling of batter pile foundations under sinusoidal dynamic excitation[END_REF] to more general real earthquake loading and the results confirm the conclusions that batter piles are likely to have beneficial performance. In addition, these beneficial effects are more often observed for batter piles with a short superstructure. In order to be able to predict the beneficial or detrimental behavior of vertical and batter pile foundations, simplified numerical models based on the macroelement approach [START_REF] Li | Numerical study of the 3D failure envelope of a single pile in sand[END_REF][START_REF] Li | A hypoplastic macroelement for single vertical piles in sand subject to three-dimensional loading conditions[END_REF] have been recently developed to simulate the nonlinear behavior of pile foundations.

The initial stress state of the batter piles could also significantly influence their performance, which should attract attentions in future experimental and numerical campaigns.

Fig. 1 .

 1 Fig. 1. Frequency representations (prototype scale) of the base shaking signals: (a) Martinique Jara, (b) Northridge earthquake and (c) Kobe earthquake.

Fig. 2 .

 2 Fig. 2. Assembly of batter piles and pile cap pieces.

Fig. 4 .

 4 Fig. 4. Short and tall superstructures used in the experiments.

Fig. 5 .

 5 Fig. 5. Foundation-superstructure discretization.

Fig. 7 .

 7 Fig. 7. Configuration of the sensors in the centrifuge tests (at model scale in mm).

Fig. 8 .

 8 Fig. 8. Arias intensities.

Fig. 9 .

 9 Fig. 9. Results of the 12th Kobe À 4 dB strong earthquake, batter piles with tall superstructure (prototype scale): (a) responses of the top mass; (b) residual bending moment and (c) total bending moment.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. P7 and P8 piles in the batter and vertical pile groups.

Fig. 15 .

 15 Fig. 15. Results of the Nr À 4 dB strong earthquake, batter piles with tall superstructure, events 7, 8 and 9 (results of P7, prototype scale): (a) acceleration history at the top of superstructure (b) total bending moment and (c) axial force.

Fig. 16 .

 16 Fig. 16. Example of the normalized Residual Bending Moment (RBM) in prototype scale. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 18 .

 18 Fig. 18. Normalized maximum rotations of the pile cap.

Fig. 19 .

 19 Fig. 19. Normalized maximum total base shear forces (prototype scale).

Fig. 20 .

 20 Fig. 20. Normalized maximum total overturning moments (prototype scale).

Fig. 21 .

 21 Fig. 21. Normalized residual bending moment profiles (normalized by the vertical pile bending moment): (a) Mj À 1 dB (b) Nr À 20 dB (c) Nr À 9 dB (d) Kb À 4 dB (e) Mj À 1 dB (f) Nr À 20 dB. z is the depth below ground surface.

Fig. 22 .

 22 Fig. 22. Normalized maximum residual bending moments (prototype scale).

Fig. 23 .

 23 Fig. 23. Normalized maximum bending moment profiles (normalized by the vertical pile bending moment): (a) Mj À 1 dB (b) Nr À 20 dB (c) Nr À 9 dB (d) Kb À 4 dB (e) Mj À 1 dB (f) Nr À 20 dB. z is the depth below ground surface.

Fig. 24 .

 24 Fig. 24. Normalized maximum total bending moments (prototype scale).

Fig. 25 .

 25 Fig. 25. Normalized maximum axial forces (prototype scale).

  Several important factors are studied such as Seismic Soil-Pile-Superstructure Interaction

	Nomenclature	P7	one pile in the 1 Â 2 pile group, on the 'Porte' side
	Roman symbol	P8 BS	one pile in the 1 Â 2 pile group, on the 'Pivot' side base shear force acted on the foundation
	SSPSI Mj Nr Kb C.G. IS	Seismic Soil-Pile-Superstructure Interaction Martinique Jara earthquake (Jara station EW) Northridge earthquake (Tarzana station 090) Kobe earthquake (DAI8-G, N43W) center of gravity of superstructure inclined (batter) pile group with short superstructure	OM RBM RBM max maximum residual bending moment overturning moment acted on the foundation residual bending moment RBM P7 residual bending moment on pile P7 RBM P7 max,VS maximum residual bending moment on pile P7 in vertical pile group with short superstructure M total bending moment
			M P7
	VS	vertical pile group with short superstructure
	IT	inclined (batter) pile group with tall superstructure
	VT	vertical pile group with tall superstructure

Table 1

 1 Fontainebleau sand NE34: material properties.

	Soil	e min				e max	e	D r (%)
	Fontainebleau NE34	0.51				0.866	0.58	80%
	Note: e: void ratio; D r : relative density,	D	r	=	--e min e max e max e	.

Table 2

 2 Characteristics of the piles and of the pile cap.

	Item	Model scale 40 g	Prototype scale 1 g
	Pile-embedded depth	326.5 mm	13.06 m
	Pile-external diameter	18 mm	0.72 m
	Pile-bending stiffness	197 N m 2	505 MN m 2
	Pile-yield strength	245 MPa	245 MPa

Pile cap-dimensions 140 mm  40 mm  40 mm 5.6 m  1.6 m  1.6 m Cap-weight 5.5 N 331.24 kN Fig. 3. Geometrical characteristics of the batter pile and the vertical pile groups (prototype scale).

Table 3

 3 Main characteristics of the pile foundations with short and tall superstructures.

		Short		Tall
		Model scale	Prototype 1 g Model scale	Prototype 1 g
		40 g		40 g
	Top mass weight 0.85 kg	54.40 tonne	0.84 kg	53.67 tonne
	Foundation mass	0.20 kg	12.80 tonne	0.34 kg	21.85 tonne
	weight
	Total weight	1.05 kg	67.20 tonne	1.18 kg	75.52 tonne
	Height of C.G.	128.0 mm	5.12 m	211.0 mm	8.48 m
	H mass
	Height of C.G.	48.0 mm	1.92 m	48.0 mm	1.92 m
	H cap
	Frequency of	81.0 Hz	2.03 Hz	80.4 Hz	2.01 Hz
	building

Table 4

 4 Applied sequences of the real earthquake signals.

	No.	Excitations	Seismic signal	Attenuation (dB)	PGA (m/s 2 )	Arias intensity (m/s)
	Test 1	1, 2, 3	Martinique Jara	À 1	0.9	0.2
	Test 2	4, 5, 6	Northridge	À 20	1.35	0.2
	Test 3	7, 8, 9	Northridge	À 9	4.79	2.43
	Test 4	10, 11, 12	Kobe	À 4	3.83	3.51
	Test 5	13, 14, 15	Martinique Jara	À 1	0.9	0.2
	Test 6	16, 17 ,18	Northridge	À 20	1.35	0.2
	Note:

  ) and (2).

	OM	=	M mass mass mass u H "	+	M u H " cap cap cap	( ) 1
	BS	=	M mass mass u "	+	M u " cap cap	( ) 2

Table 5

 5 Frequencies of the different pile configurations.

	Configurations	Natural frequency of the super-	Natural frequency of the
		structure (Hz)			pile cap (Hz)
		(calculated by	CH CH 09 11	)	(calculated by	CH CH	10 09	)
	IS	1.94			6.28
	VS	1.80			4.26
	IT	1.70			7.12
	VT	1.62			5.30

Fig. 17. Normalized maximum displacements of the pile cap (prototype scale).

Table 6

 6 Ratios of overturning moment to base shear for the different configurations.

	Configurations	1. Mj	2. Nr	3. Nr	4. Kb	5. Mj	6. Nr
		À 1 dB	À 20 dB	À 9 dB	À 4 dB	À 1 dB	À 20 dB
	IS	4.13	3.70	4.67	4.22	4.04	3.64
	VS	3.90	3.44	4.35	4.60	3.81	3.36
	IT	7.06	6.46	6.57	6.86	5.89	6.21
	VT	8.33	9.19	6.95	6.91	8.59	7.48

From previous series of dynamic centrifuge tests, the natural frequencies of the soil column and the soil-pile-superstructure system are about 3.5 Hz and

2.0 Hz, respectively.

Acknowledgments

The financial support of IFSTTAR (Institut français des sciences et technologies des transports, de l′aménagement et des réseaux) and of the Région Pays de la Loire is gratefully acknowledged. The authors would like also to thank the valuable support and help from the technical staff of the IFSTTAR centrifuge team.


W Huang

L Schwan

V Romero-García

J.-M Génevaux

J.-P Groby

Digital sound absorbing metafluid inspired by cereal straws

Used as building biomaterials for centuries, cereal straws are known for their remarkable acoustic performances in sound absorption. Yet, their use as fibrous media disregards their internal structure made of nodes partitioning stems. Here, we show that such nodes can impart negative acoustic bulk modulus to straw balls when straws are cut on either side of a node. Such metafluid inspired by cereal straws combines visco-thermal diffusion with strong wave dispersion arising from quarter-wavelength resonances within straws. Large spectral bandgaps and slow sound regimes are theoretically predicted and experimental data from impedance tube measurements on an idealised 3D-printed sample layer are in good agreement with the theoretical model. Perfect absorption is achieved at wavelengths 13 times larger than the thickness of the metafluid layer, and slow sound entails an increased density of states causing a cascade of high absorption peaks. Such features could lead cereal straws to serve as cheap acoustic bio-metamaterials.

1

 From this perspective, biological systems can be a valuable source of inspiration: idealised synthetic replicas mimicking fauna and flora, 6 spiderweb structures, 7 or sperm-cell motion 8 have demonstrated astonishing performances in photonics, lattice mechanics, and micro-robotics.

Particularly interesting in acoustic and thermal insulation are cereal straws such as wheat, reed or rattan. They have been used as building materials for centuries due to their large availability throughout the world. Previous empirical, 9 phenomenological, 10 and theoretical 11 studies showed that viscous and thermal diffusion through the arrangement of stems play a dominant role in acoustic performances. Attention was mainly paid to the effects of the length, concentration, Node within stem

and relative orientations of the straws on sound absorption. While arrays of straws were usually idealised as classical visco-thermal fluids made of either solid or completely hollow straws, closer examination of cereal straws in Fig. 1 reveals that nodes within the stem partition the straw into tubular segments. Therefore, when cutting straws on either side of a node, the resulting pieces are neither solid obstacles in the path of the acoustic wave, nor open hollow straws channelling the flow, but rather double quarter-wavelength resonators (QWR) separated by the node.

Based on this observation, we designed a metafluid inspired by the cereal straws (MCS) made of the periodic repetition of double QWR, see Fig. 2. We theoretically show that their collective resonances entail an effective negative bulk modulus in the MCS around the QWR fundamental frequency resulting in both band gap opening and strong wave dispersion. In particular, the strong dispersion induced by the double QWR below its fundamental resonance generates a slow propagation of sound over a wide band in the low frequencies. Therefore, the Fabry-Perot resonances of a MCS layer appear in the deep subwavelength regime and its associated density of states is increased. Combining these effects with the viscous and thermal diffusion tailored by the concentration of double QWR, each low quality factor mode of the MCS layer can yield perfect absorption peaks due to a critical coupling between the inherent losses and the energy leakage of the MCS layer. [START_REF] Romero-García | Use of complex frequency plane to design broadband and sub-wavelength absorbers[END_REF] The theoretical predictions are supported experimentally by data from impedance tube measurements on prototypes fabricated by Fused Filament Fabrication. This manufacturing technique induces corrugation at the walls of the straws which has been advantageously used to enhance further attenuation in the manufactured sample. Perfect absorption of sound for wavelengths in air 13 times larger than the layer thickness has been achieved with large absorption value below the first resonance of the double QWR.

Idealisation of the cereal straw arrangement

The cereal straw arrangement is idealised by a three-dimensional Ω-periodic repetition of straight double QWR, as shown in Fig. 2. The unit cell Ω is a rectangular cube with the dimensions 1 × 2 × 3 in the directions of the Cartesian coordinate system (e 1 , e 2 , e 3 ) containing a double QWR. The axis of each double QWR is oriented along the direction e 1 and consecutive stacks of straws are separated by an air-gap to permit air to flow inside the straws. Each double QWR has the outer radius r o , the inner radius r i < r o , and the finite length h ≤ 1 . The volume of the double QWR is split by a rigid node into two cylindrical ducts Ω 1 and Ω 2 with the lengths h 1 and h 2 and the outside apertures Σ 1 and Σ 2 respectively. Denoting by Ω 0 the air domain of the unit cell outside the straw, the filling fractions

φ 0 = |Ω 0 |/|Ω|, φ 1 = |Ω 1 |/|Ω| and φ 2 = |Ω 2 |/|Ω|
are defined and the total porosity of the arrangement is φ = φ 0 + φ 1 + φ 2 . Besides, the outside walls of the straw are supposed to be corrugated periodically with the period c in the direction e 1 and the semi-elliptic profile of height h c in a longitudinal cross-section, see Fig. 2(d). Although, this corrugation is not fundamental to the singular behaviour of the metafluid, it is an inherent side-effect of the layer-bylayer 3D-printing process using Fused Filament Fabrication technology. It must be accounted for theoretically in the present case of tightly-packed straw arrangements, for which the corrugation height h c is of the same order as the spacing 2 -2r o between two neighbouring straws.

Sound propagation in the MCS

The propagation of air-borne acoustic waves in the MCS is studied under ambient conditions in the linear harmonic regime at frequencies ω with the implicit time factor e -iωt . Under the assumption that lattice sizes 1 , 2 , 3 are sufficiently small compared to the wavelength λ e = 2π/k e in air (typically below the Bragg limit 1 ≤ λ e /2), the idealised system is expected to behave as a metafluid, that is an effective homogeneous medium with strong wave dispersion induced by microstructural QWR resonances. The effective pressure P and particle velocity V in the MCS are governed by the following equations of mass conservation and generalized Darcy law, see Section Methods for details,

∇ • V = iωP B, and V = -K ∇P η, (1)
where η is the viscosity in air and where both the effective bulk modulus B and the Darcy permeability tensor K are complex-valued and frequency dependent. In particular, the Darcy tensor is diagonal in the coordinate system (e 1 , e 2 , e 3 ) due to the symmetries of the unit cell Ω,

K = K 1 e 1 ⊗ e 1 + K 2 e 2 ⊗ e 2 + K 3 e 3 ⊗ e 3 , (2)
where the principal permeabilities K n with n ∈ {1, 2, 3} are expressed by the Johnson formula, [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF] and the homogenisation theory [START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF][START_REF] Auriault | Homogenization of Coupled Phenomena in Heterogenous Media[END_REF] , see Section Methods. The effective bulk modulus B satisfies the following association rule when normalised by the bulk modulus B e of air,

B -1 = B -1 e (φ 0 β 0 + φ 1 β 1 + φ 2 β 2 ) , (3)
where β 0 is the compressibility factor considering the thermo-acoustic phenomena in the domain Ω 0 , and β 1 and β 2 are apparent compressibility factors induced by the resonances in the domains Ω 1

and Ω 2 . While β 0 is given by the Johnson-Lafarge model [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF][START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF] and the homogenisation theory, [START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF][START_REF] Auriault | Homogenization of Coupled Phenomena in Heterogenous Media[END_REF] the apparent compressibility factors β 1 and β 2 account for the fact that the QWRs act as secondary acoustic sources in the medium, see Section Methods. They can be expressed in the form,

β j = B j B e k j h j tan(k j h j ) -µk 2 e r i h j -1 , j ∈ {1, 2} , (4)
where k j = ω/ B j /ρ j is the wavenumber in the domain Ω j , with the effective density ρ j and bulk modulus B j in Ω j being derived from the Zwikker & Kosten model. [START_REF] Zwikker | Sound absorbing materials[END_REF] Moreover, the dimensionless factor µ = 1.2 in Eq. ( 4) corrects the radiation impedance at the aperture Σ j .

Hence, the quarter-wavelength resonance of the duct Ω j can cause β j and thus the bulk modulus B to become negative, [START_REF] Fang | Ultrasonic metamaterials with negative modulus[END_REF][START_REF] Boutin | Acoustics of porous media with inner resonators[END_REF]

Effective fluid parameters of the MCS

The MCS with the following geometry is considered, with n ∈ {1, 2} for the MCS, where P e is the atmospheric pressure and ρ e the density of air. For the sake of comparison, both quantities in the absence of the QWR, that is when r i = 0 and hence φ j = 0 for j ∈ {1, 2}, are also plotted. This latter system can be seen as a equivalent anisotropic fluid inspired by cereal straws (FCS). [START_REF] Torrent | Sound scattering by anisotropic metafluids based on twodimensional sonic crystals[END_REF] As shown in Fig. 4(a) and emphasised in the lower inset, the fundamental resonance of the double QWR induces negative bulk modulus B over the broad frequency range [2460, 3254] Hz. The low frequency limit of the bulk modulus is also magnified in the upper inset, which shows the transition between the isothermal and adiabatic regimes of the MCS around the characteristic thermal frequency ω θ /2π ≈ 8.24 Hz. [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF][START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF] Compared to the QWR resonance, this thermal transition occurs at much lower frequencies and with much smaller impact on the effective bulk modulus. At frequencies below the band gap, the wave speeds are much smaller than the wave speed in both the FSC and the air, resulting in a slow sound regime. In this regime, the wave speeds possess a plateau at low frequencies and slowly decrease to a value close to zero at the frequency of the lower bound of the band gap. It is worth noting that the group velocity, defined as v g = ∂ω/∂k M CS n is also much lower than the group velocity in both the FCS and in air and tend to zero at the frequency of the lower bound of the band gap. We notice also that the inherent losses of the MCS, translating in Im(k n ), are not negligible for the frequencies below the band gap. At frequencies above the band gap the wave speed is higher than that of both the FSC and the air, resulting in a supersonic regime. [START_REF] Boutin | Acoustics of porous media with inner resonators[END_REF][START_REF] Groby | The use of slow waves to design simple sound absorbing materials[END_REF][START_REF] Groby | Use of slow sound to design perfect and broadband passive sound absorbing materials[END_REF] Conversely, the wave dispersion and attenuation in the FCS are weak in every direction:

                       1 = 41 mm, 2 = 8.4 mm, 3 = 8.4 mm, h = 40 mm, h 1 = 30 mm, h 2 = 10 mm, r o = 4.0 mm, r i = 3.8 mm, c = 209 µm, h c = 77 µm. (5)
the sound speed along the principal directions are almost constant in the inertial-adiabatic regime above 100 Hz, with the sound speed being close to that of air in the direction e 1 but almost divided by 2 along the directions e 2 due to a tortuosity effect. For both the FCS and the MCS, the sound speed decreases down to zero as ω → 0 in the viscous regime below 100 Hz.

Critical coupling and perfect absorption

The slow sound effect and the inherent losses of the MCS discussed before can be used to design finite size perfect absorber samples. On the one hand, the frequencies of the Fabry-Perot resonances drastically decrease due to the slow sound regime of the MCS. As a consequence, these resonances happen for wavelengths much larger than the structure thickness, i.e., the MCS becomes deeply sub-wavelength. On the other hand, the MCS layer behaves as an open lossy and resonant system, characterised at the resonant frequencies, by both the leakage rate of energy (i.e., the coupling of the resonant elements with the propagating medium) and the inherent losses. In the reflection problem, the balance between the leakage and the losses activates the condition of critical coupling, enabling a perfect impedance matching to the background medium, and therefore generating a perfect energy absorption. [START_REF] Romero-García | Use of complex frequency plane to design broadband and sub-wavelength absorbers[END_REF][START_REF] Groby | Use of slow sound to design perfect and broadband passive sound absorbing materials[END_REF][START_REF] Jiménez | Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption[END_REF] The layer sample analysed here and depicted in Fig. 3 has been designed so that the layer of thickness L = 13 3 = 109.2 mm is critically coupled to air under plane wave excitation at normal incidence with λ e = 13L in the principal direction e 3 , see Section Methods.

In a reflection problem, the representation of the zeros and the poles of the reflection coefficient in the complex frequency plane is an efficient tool to interpret the critical coupling condition. [START_REF] Romero-García | Use of complex frequency plane to design broadband and sub-wavelength absorbers[END_REF] Introducing the complex frequency plane (complex frequency ω = ω + iω Im ), the reflection coefficient, R(ω), represents the scattering of the system. In the lossless case, the relation R * (ω) = 1/R(ω * ) is satisfied due to temporal invariance symmetry, where R * (ω) and ω * are the complex conjugate of R(ω) and ω respectively. Consequently, the reflection coefficient possesses pairs of poles and zeros at complex frequencies being complex conjugate with each other. More importantly, the imaginary part of the complex frequency of the pole represents the energy leakage of the system. In the lossy case, when inherent losses are introduced, the zeros and poles are not complex conjugates any more and in particular both are shifted downward in the complex frequency plane, if e -iωt is used. The zero of the reflection coefficient can thus intersect the real frequency axis, enabling perfect absorption when exactly located on this axis. Here, we pay attention to the frequencies below the band gap. The first absorption peak appears at 272 Hz. The ratio of the wave speed between the FCS and the MCS along the e 2 (=e 3 ) direction [in Fig. 5(c)] is in good agreement with the downshifting of this peak at low frequencies with respect to the first absorption peak of the FCS. More importantly, we observe perfect absorption for this peak (experimentally 99.6 % at 258 Hz), where the corresponding wavelength equals to 13 times the slab thickness. Thus, this system represents a sub-wavelength perfect absorber. A cascade of nearly perfect absorption peaks, due to the higher order Fabry-Perot resonances, is produced and perfectly grasper by the theory. Each of these nearly perfect absorption peak is associated with a low quality factor resonance arising from the coupling with viscous and thermal diffusion tailored by the concentration of double QWR and the corrugation. These low quality factors strongly contribute to enhance the absorption between two consecutive peaks. Nevertheless, a perfect absorption peak, theoretically predicted at the lower bound of the band gap, is in disagreement with the experiments. This is due to the fact that the theoretical model assumes an infinite number of the resonators along the structure depth, creating an accumulation point [START_REF] Jiménez | Quasiperfect absorption by subwavelength acoustic panels in transmission using accumulation of resonances due to slow sound[END_REF] , while this number is finite in practice. Effectively, the MCS sample comprises only 13 unit cells along the e 3 direction.

Conclusions

The metafluid inspired by cereal straws has been theoretically and experimentally reported for sub-wavelength perfect sound absorption. A straw bale constituted of straws cut on either side of a node is idealized by a three-dimensional periodic repetition of tightly packed double QWR. This idealised straw bale is homogenised, exhibiting slow sound region enabling to increase of the den-sity of state in the subwavelength regime together with viscous and thermal diffusion through the arrangement of stems. The MCS behaves like an open and lossy resonator, the absorption properties of which are tailored by critical coupling. Therefore, the designed MSC is a sub-wavelength perfect absorber. Moreover, the strong dispersion arising from the fundamental resonance of the double QWR induces a cascade of nearly perfect absorption peaks below the lower bound of the band gap. The absorption between consecutive peaks is strongly enhanced by the viscous and thermal diffusion through the anisotropic surrounding fluid. The designed MCS possess a 99.6% absorption peak at 258 Hz, which corresponds to a wavelength 13 times larger than of the slab thickness. The design of the present MSC offers both a comprehensive explanation of the acoustic properties of bio-sourced materials, such as wheat, reed, or rattan and large perspectives for the design of bio-inspired metafluid, notably through its anisotropy.

Methods

The Poro-acoustics with inner resonances. Using the theory of poro-acoustics with inner resonances [START_REF] Boutin | Acoustics of porous media with inner resonators[END_REF] , the effective pressure P and particle velocity V in the MCS are governed by the following equations of mass conservation and generalised Darcy law,

∇ • V = iω φ 0 β 0 B e P + Q 1 + Q 2 |Ω| , V = - K η ∇P, (6)
where Q j is the flux pulsed out from the aperture Σ j , and where the compressibility factor β 0 in Ω 0 is driven by the thermal permeability Θ according to,

β 0 = γ + iω (γ -1) Θ C p /(φ 0 κ). (7)
Equation ( 6) reveals that the QWRs act as secondary acoustic sources in the mass conservation: they radiate the flux Q 1 + Q 2 in the effective medium formed by the array of straws, as these latter modify the air flow path. Conversely to the case of FCS for which Q 1 = Q 2 = 0, the presence of the nodes within the straws makes each duct Ω j behave as a QWR emitting the flux,

Q j = |Σ j |P 1/Y j + iωρ e µr i with Y j = iω tan(k j h j ) k j B j , (8)
where |Σ j | = |Ω j |/h j is the surface area of the apertures, Y j is the surface admittance at the aperture Σ j . Substitution of Eq. ( 8) into (6) results in the equations of effective mass conservation and Darcy law in the MCS presented in Eq. ( 1). Furthermore, the effective density and bulk modulus of air in the ducts Ω j with j ∈ {1, 2} satisfy Zwikker & Kosten model: 17

ρ j ρ e = - J 0 (X v ) J 2 (X v ) , B e B j = γ + (γ -1) J 2 (X t ) J 0 (X t ) , (9)
where J 0 and J 2 are Bessel functions of the first kind and order 0 and 2, while X v = r i iωρ e /η and X t = r i iωC p /κ represent the duct radius normalized by the viscous and thermal skin-depths.

Thermal permeability Θ and principal Darcy permeabilities K n with n ∈ {1, 2, 3}, are given by Johnson formula, [START_REF] Johnson | Theory of dynamic permeability and tortuosity in fluid-saturated porous media[END_REF][START_REF] Lafarge | Dynamic compressibility of air in porous structures at audible frequencies[END_REF] K

n = K 0 n 1 -iω(2δ n /Λ n ) 2 /ω n -iω/ω n , (10)
Θ = Θ 0 1 -iω(2δ θ /Λ θ ) 2 /ω θ -iω/ω θ , (11)
where ω n and ω θ = O(ω n ) are characteristic viscous and thermal frequencies, δ n = η/(ρ e ω n )

and δ θ = κ/(C p ω θ ) are viscous and thermal skin depths at these characteristic frequencies, Λ n and Λ θ are characteristic viscous and thermal lengths and K 0 n and Θ 0 are the static values of K n and Θ as ω → 0. The characteristic frequencies ω n and ω θ are defined by:

ω n = φ 0 η (K 0 n ρ e τ ∞ n ), ω θ = φ 0 κ (Θ 0 C p ), (12)
where τ ∞ n is the high frequency tortuosity related to K n .

Cell problems. Following two-scale asymptotic homogenisation, [START_REF] Sanchez-Palencia | Non-Homogeneous Media and Vibration Theory[END_REF][START_REF] Auriault | Homogenization of Coupled Phenomena in Heterogenous Media[END_REF] the parameters Θ 0 and Λ θ characterising the thermal behaviour, and K 0 n , Λ n and τ ∞ n characterising the visco-inertial behaviour are given as volume and surface averages of Ω-periodic tests fields θ, u n and χ n defined on the air domain Ω 0 :

Θ 0 = 1 |Ω| Ω 0 θ dΩ, K 0 n = 1 |Ω| Ω 0 u n • e n dΩ, (13a)
Λ θ = 2 |Ω 0 | |Γ 0 | , Λ n = 2 Ω 0 (e n -∇χ n ) 2 dΩ Γ 0 (e n -∇χ n ) 2 dΓ , ( 13b
)
τ ∞ n = |Ω 0 | Ω 0 1 -∇χ n • e n dΩ, (13c)
Samples and experimental set-up. The reflection of a plane wave from a layer of material of thickness L arranged against a rigid backing is studied. The air/layer interface Γ 0 L is located at 3. Impedance tube measurements were performed in a tube with 4.2 cm × 4.2 cm square cross section. Assuming that plane waves propagate below the cut-off frequency of the tube (4200 Hz), the walls of the tube act as perfect mirrors to emulate a periodicity pattern in the directions e 1 and e 2 . The sample is placed at the end of the tube against a copper plug that closes the tube. A single microphone (1/4 inch pressure field B&K microphone type 4938) attached on a one-dimensional robotised arm is used. This technique provides the experimental absorption coefficient at normal incidence without cross-calibration of many microphones and with an enhanced accuracy at low frequencies. [START_REF] Groby | The use of slow waves to design simple sound absorbing materials[END_REF][START_REF] Groby | Use of slow sound to design perfect and broadband passive sound absorbing materials[END_REF] Input sine signals were generated and output signals from microphone were

Figure 1 :

 1 Figure 1: Photograph of a wheat straw and details of the inner node partitioning the stem

Figure 2 :

 2 Figure 2: Schematic of the idealised metafluid inspired by cereal straws. (a) Representation of

  which offers the possibility to open non-propagative bandgaps and generate slow sound regimes. Moreover, since the inner resonances only affect the mass conservation, non-propagative band-gaps are opened for any direction of propagation in the MCS, regardless of the anisotropy of the Darcy tensor.

Figure 3 :

 3 Figure 3: Cut view of the 3D-printed prototype manufactured by means of Fused Filament Fabrica-

Figures 4 (

 4 Figures 4(a) and 4(b) represent the effective normalised parameters φB/P e and -iωρ e K n /η

Figure 4 (Figure 4 : 2 1 Figure 5 :

 44215 Figure 4: Effective parameters. (a) Real (red line) and imaginary (blue line) parts of the normal-

Figure 6 :

 6 Figure 6: Complex frequency planes representing log(|R| 2 ) for the FCS (a) and for the MCS

Figure 6 (

 6 Figure 6(a) shows the complex frequency plane for the FCS sample. The pairs of pole/zero

Figure 6 (

 6 Figure 6(c) and 6(d) respectively show the complex frequency plane and the absorption coef-

  density ρ e = 1.213 kg/m 3 , adiabatic constant γ = 1.4, thermal conductivity κ = 2.5 × 10 -2 W/m/K, isobaric heat capacity C p = 1.219 × 10 3 J/K, and viscosity η = 1.839 × 10 -5 Pa.s were used for air at equilibrium, as well as the atmospheric pressure P e = 1.013 × 10 5 Pa, the bulk modulus B e = γP e , the sound speed c e = B e /ρ e and the air wavenumber k e = ω/c e .

x 3 = 0 ,

 30 and the rigid backing Γ b L at x 3 = -L. Both interfaces have the normal vector e 3 which is principal direction of the Darcy tensor. Moreover, normal incidence is considered, so that the pressure reflection coefficient and absorption coefficient read as R = (1 -Υ)/(1 + Υ) and A = 1 -|R| 2 where the normalised admittance Υ of the layer reads as follows,Υ = -i [B e c 3 /(Bc e )] tan(ωL/c 3 ). (17)The MCS geometrical parameters were tuned using optimisation Sequential Quadratic Programming. The minimised cost function is |R| 2 at the required frequency. The MCS and FCS samples were fabricated using a MakerBot R Replicator R 2X Experimental 3D Printer based on the Fused Filament Fabrication technology. Heated continuous filament of thermoplastic material is deposited layer-by-layer by the extruder head having a 400 µm diameter nozzle. The layer height setting is 200 µm, which corresponds to the period measured on the ESM cliché in Fig.

Acknowledgements This work was funded by the PAVNat project, RFI Le Mans Acoustique, Région Pays de la Loire. This article is based upon work from COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology).

where θ, u n and χ n satisfy the following cell problems, with χ n and ζ n having zero mean values over Ω 0 : ∆θ = -1, θ = 0 on Γ 0 , (14a)

In Eqs. ( 13) and ( 14), vector n is normal to the boundary Γ 0 = Σ 1 ∪ Σ 2 ∪ Σ lat where Σ lat is the lateral surface of the straws. The cell problems ( 14) are computed numerically by means of the Finite Element Method using commercial software COMSOL Multiphysics R . To account for the corrugation on the cell problems, similar cell problems are computed for the square array of infinite cylinders with corrugated lateral surface and values of the tests-fields hence obtained are substituted into Eq. ( 13) in the domain between the cylinders, while values of θ, u n and χ n in the air gap of thickness 1h in the direction e 1 is kept unchanged. A corrugation profile in the form of semi-ellipses in a radial cross-section is considered in the computations. It leads to:

for both MCS and FCS analysed in this work. To emphasize the effects of the corrugation, the parameters calculated for the smooth surface (h c = 0) were:

Author contributions J.P.G. conceived and designed the research. W.H. prepared the samples. W.H. and L.S. performed the analytical and numerical calculations. W.H., L.S. and J.P.G. performed the experiments.

All the authors contributed to the data analysis, discussion, and manuscript preparation.

Competing Interests The authors declare that they have no competing interests.

Correspondence Correspondence and requests for materials should be addressed to L.S. (email: logan.schwan@gmail.com).