Thibault P Gouache
email: thibault.gouache@isae.fr

Christopher Brunskill

Gregory P Scott

Yves Gourinat

Pierre Coste

Yang Gao

Regolith simulant preparation methods for hardware testing

Keywords: Regolith, Soil simulants, Density, Relative density, Preparation method

To qualify hardware for space flight, great care is taken to replicate the environment encountered in space. Emphasis is focused on presenting the hardware with the most extreme conditions it might encounter during its mission lifetime. The same care should be taken when regolith simulants are prepared to test space system performance. Indeed, the manner a granular material is prepared can have a very high influence on its mechanical properties and on the performance of the system interacting with it. Three regolith simulant preparation methods have been tested and are presented here (Rain, Pour, Vibrate). They should enable researchers and hardware developers to test their prototypes in controlled and repeatable conditions. The Pour and Vibrate techniques are robust but only allow reaching a given relative density. The Rain technique allows reaching a variety of relative densities but can be less robust if manually controlled.

Introduction

. Many studies focus on the influence of "environmental conditions" on space system performance. For instance the influence of ambient pressure and vacuum on drilling performance and power requirements were studied in [START_REF] Zacny | Laboratory drilling under martian conditions yields unexpected results[END_REF]. It was shown that lower atmospheric pressure could facilitate the evacuation of drillings. In other studies it was shown that different combinations of temperature and pressure could alter the friction between a drill bit and a drilled rock by modifying the stability of the layer of adsorbed water and the layer of oxides [START_REF] Zacny | Friction of drill bits under Martian pressure[END_REF]). The influence of gravity has regularly been studied. Its effect on bearing capacity was evaluated empirically and numerically in [START_REF] Bui | Numerical and experimental studies of gravity effect on the mechanism of lunar excavations[END_REF]. The recently developed Landing & Mobility Test Facility at DLR, German Aerospace Center, is equipped with a robotic arm to compensate the excess of gravity on Earth when compared to Martian or Lunar conditions [START_REF] Witte | A vehicle dynamics test facility for planetary surface mobility[END_REF]). The influence of temperature and vacuum on space system performance are not only studied, but also taken into account in hardware qualification (for instance, American Military Standard MIL-STD-810 or European Cooperation for Space Standardization ECSS-E-10-03A). In contrast very little emphasis is put on the influence of regolith simulant relative density.

. The surface of the Moon and Mars are covered by a layer of granular material (GM) called regolith. A GM can be compacted to different levels. It can be in a very aerated configuration with a high volume of voids compared to the volume of grains. It can be very compacted with a very low volume of voids compared to the volume of grains. Or it can be in any intermediate state. The degree of compaction of a GM is given by its relative density. It is considered 0 % if its density is at a minimum and 100 % if its density is at the maximum (Eq. 1). Rarely is a GM found at either of these extremes in nature. Some publications refer to regolith with the term soil. [START_REF] Seiferlin | Simulating Martian regolith in the laboratory[END_REF] have pointed out that this is not entirely correct since the term soil supposes important transformations due to biological activity but also uses soil to refer to regolith. In this article, soil is also used to designate regolith but this does not imply biological activity.

relative density = density -minimum density maximum density -minimum density

(1)

. The influence of relative density on a granular material's mechanical properties has clearly been established. The effect of relative density on the results of sand compression tests is reported in [START_REF] Mahmood | Effect of specimen preparation method on grain arrangement and compressibility in sand[END_REF]. The variation of internal angle of friction with relative density of five Martian soil simulants is illustrated in Perko et al. (2006). The mechanical properties of SSC-1 and SSC-2 (Surrey Space Centre Mars Simulant-1 and -2, the regolith simulants used to test the preparation techniques in this paper, see 4.1) were investigated experimentally in Scott and Saaj (2009b). It was shown that relative density has an impact on their mechanical properties. In [START_REF] Mahmood | Effect of specimen preparation method on grain arrangement and compressibility in sand[END_REF], it was also observed that for a given relative density achieved, the manner it is achieved affects the physical properties of the prepared GM. Indeed, vibration and pluviation techniques were used to obtain the same relative density of a well-sorted, medium grain size quartz and feldspar beach sand. The preparation manner was shown to affect the fabric (grain orientation) and influence the mechanical properties of the sand.

. The successful exploration of Mars is believed to depend on our ability to predict performance of systems interacting with regoliths (Perko et al. (2006)). Recent publications have illustrated the impact of GM relative density on space system performance. The Optical Probe for Regolith Analysis was inserted in JSC Mars-1 regolith simulant that had been prepared at two levels of density. After a few centimetres of penetration (10 cm) the force required to penetrate the simulant was orders of magnitude higher for the compacted JSC Mars-1 [START_REF] El Shafie | Penetration Forces for Subsurface Regolith Probes[END_REF]). Static penetration tests in UK4 sand, another Martian regolith simulant, showed the influence of compaction on the penetration forces [START_REF] Seiferlin | Simulating Martian regolith in the laboratory[END_REF]). The mobility performance of rovers will also be effected by different relative densities of the regolith encountered [START_REF] Brunskill | The effect of soil density on microrover trafficability under low ground pressure conditions[END_REF]). It was shown empirically that the traction forces a legged-robot can generate depend on the relative density of the granular medium (Scott and Saaj (2009a); [START_REF] Scott | Steps towards characterising legged microrover performance on compressible planetary soils[END_REF]). Experimental works have showed that the performance of the bioinspired dual-reciprocating planetary drill is highly influenced by the relative density of the excavated regolith simulant [START_REF] Gouache | Experimental Parametric Evaluation of Dual-Reciprocating-Drilling Mechanism Performance[END_REF]).

. To enable hardware developers and researchers to take into account this effect, robust sample preparation techniques must be developed. This article will expose three different soil preparation techniques, their advantages and drawbacks.

Challenges and advantages of repeatable granular soil preparation techniques

. The difficulty of preparing Granular Materials (GM) in a repeatable and robust manner has been acknowledged. In [START_REF] Okamoto | An evaluation of the dry pluviation preparation technique applied to silica sand samples[END_REF], it was shown that low relative density samples are hard to create in a repeatable manner. Moreover the compaction process of GM is not precisely understood and modelled because of complex relaxation processes [START_REF] Knight | Density relaxation in a vibrated granular material[END_REF].

Convection and segregation of particles can also be observed when a GM is compacted through vibration [START_REF] Knight | Vibration-induced size separation in granular media: The convection connection[END_REF]; [START_REF] Ehrichs | Convection in vertically vibrated granular materials[END_REF].

Once compacted, a GM's density can oscillate around a dense state when subject to continued vibrations [START_REF] Nowak | Density fluctuations in vibrated granular materials[END_REF]).

. Despite these difficulties, it is worthwhile developing methods allowing soil preparation with sufficient consistency. Indeed such methods would allow different hardware solutions to be tested in similar conditions. This would facilitate system performance comparison. In [START_REF] Mahmood | Effect of specimen preparation method on grain arrangement and compressibility in sand[END_REF], authors conclude on the need to specify relative density and preparation method of granular materials in standardized tests. Additionally, tests conducted with carefully prepared regolith simulants can be used to calibrate numerical simulations or develop analytical models.

. Moreover being able to control soil properties through relative density and testing hardware in precisely controlled conditions allows the calibration of the system. This transforms a "purely functional space sub-system" (a drill or a locomotion system for instance) into a "scientific sub-system" (a soil mechanics instrument). This generally increases the scientific return of the mission. This was done with the Rock Abrasion Tool on the NASA Mars Exploration Rovers. By comparing the energy necessary to grind Martian rocks and the energy needed during benchmark tests done on Earth, it was determined that Martian rocks are analogue to a range of Earth rocks from low strength basalts to gypsum (for grindability) in [START_REF] Myrick | The RAT as a rock physical properties tool[END_REF]. The Viking Lander 2 robotic arm was used to displace rocks in order to gain access to the regolith under them. By analyzing the forces developed by the robotic arm during these operations, the mass of each displaced rock was evaluated [START_REF] Thomson | Extracting scientific results from robotic arm support operations: A technique for estimating the density and composition of rocks on Mars[END_REF]).

. Finally, having different reference regolith simulant preparation techniques allows a more robust qualification process. It will be possible to test and qualify planetary systems interacting with regolith in a larger number of conditions. Or it will be possible to identify one soil condition as "the most challenging scenario" and apply the corresponding preparation method for the qualification test.

Proposed regolith preparation techniques

. The three techniques investigated in this publication are presented. Figure 1 is a schematic of these three techniques.

Pour > 50 cm Controlled height

Vibration

Controlled speed

Vibrate Rain

Figure 1: The three Granular Material (GM) preparation techniques explored in this publication: Pour (from a height above 50 cm), Vibrate (while pouring the GM in container) and Rain (a curtain of GM is deposited by layers from a controlled height and at a controlled speed).

Rain

. A variety of raining or pluviation techniques have been investigated in other publications [START_REF] Bica | A Study of Free Embedded Cantilever Walls in Granular Soil[END_REF]). Their common point is that the GM is deposited into the substrate container in the same manner rain would fill a bucket. In the tests presented here, we have used the curtain raining technique. In this method, a hopper with a slit in its bottom is filled with sand and a "curtain" of falling GM is created. The hopper is moved back and forth across the substrate container. Each pass of the hopper deposits a layer of GM in the substrate container. The size of the slit in the hopper, the height of the hopper and the speed of the hopper must be controlled. The flow of GM in the hopper used in this publication was generated only by gravity and not by more complex systems (like an air activated system in [START_REF] Bica | A Study of Free Embedded Cantilever Walls in Granular Soil[END_REF]).

. A fixed slit width was used for all the tests reported here (5 mm). The flow rate of the two soils tested in this paper (SSC-1 and SSC-2, see 4.1) out of the hopper with a 5 mm slit was evaluated. The hopper was filled and placed above a scale. An automatic image acquisition system (1 Hz frequency) monitored the mass accumulation on the scale. Each soil simulant was rained three times on the scale. Figure 2 presents the accumulated mass versus time for all the performed runs and Table 1 presents the flow rates obtained.

SSC-1 SSC-2

Mean flow rate (g/s) 55.12 40.54 Standard deviation (g/s) 1.79 0.69

Table 1: SSC-1 and SSC-2 flow rate out of the hopper.

Pour

. The GM is poured directly into the substrate container from a bucket. When the sand is poured into its final container, it is first accelerated during its free-fall. As it accelerates, turbulence appears, mixing sand and air. To enable the sand to accelerate sufficiently (approaching its terminal velocity) and to allow the turbulence to fully develop a minimum pouring height is required. In [START_REF] Okamoto | An evaluation of the dry pluviation preparation technique applied to silica sand samples[END_REF], it is reported that sand fall heights above 50 cm have no influence on the density of the prepared sample. First tests confirmed this and a pouring height of 50 cm was chosen.

Vibrate

. The GM is poured into the substrate container that is being vibrated. The vibrations compact the sand as it is poured into the container. No specific constraints were put on the manner the sand is poured and the duration of the vibration. The only constraint in this method is to pour the sand into a vibrating container. Here the vibrations were stopped immediately after pouring. The containers were vibrated by an Endecotts sieve shaker (rough order of magnitude of amplitude and frequency of vibrations are 1 cm and 5 Hz).

Test procedures

4.1. Soils tested . The three different methods presented here were used to prepare two Martian regolith simulants available at the Surrey Space Centre (SSC): SSC Mars Simulant-1 (SSC-1) and SSC Mars Simulant-2 (SSC-2). SSC-1 is a medium grained quartz sand (particle size from 1.3 mm to 63 µm and some silt) and SSC-2 is a fine grained garnet sand (particle size from 90 µm to less than 45 µm) (Scott and Saaj (2009b)).

Minimum and maximum density determination

. To establish the relative density of a granular material, the minimum and maximum density must be known. The minimum density of SSC-1 and SSC-2 was determined by following the ASTM D4254-00 standards in ASTM (2000) (obtained by turning up and down a contianer half full of the granular material). The maximum density of SSC-1 and SSC-2 is given by estimation in Scott and Saaj (2009b)

Volumetric density and volumetric relative density evaluation .

To determine the density obtained by a given preparation method, the two regolith simulants were prepared in substrate containers of known mass and volume. After completely filling the container and clearing the excess material, the mass of the container filled with regolith simulant was evaluated. The mass measurements of the empty and full container and the measurement of the container volume allowed to calculate the density of the prepared SSC-1 or SSC-2. Densities obtained in such a manner will be referred to as Volumetric Density (VD) and Volumetric Relative Density (VRD) throughout this paper. Three cylindrical Containers (C) were used for this evaluation: a metal bin of 14.5 L (height of 29 cm and diameter of 25 cm) named C-1, a 13.5 L plastic bin (height of 27 cm and diameter of 25 cm) named C-2 and a transparent 245 mL glass (height of 155 mm and diameter of 45 mm) named C-3.

4.4. In-situ density and relative density evaluation . The main difficulty in GM local density evaluation is sample disruption. Because GMs do not have a fixed structure, most sampling techniques will change the arrangements of the grains and modify the density. However, previous testing of one method showed it had negligible effect on the grain arrangement in a sample -the resin impregnation technique [START_REF] Bica | A Study of Free Embedded Cantilever Walls in Granular Soil[END_REF]).

. In the resin impregnation technique an epoxy resin was injected through a 2.1 mm diameter needle (1.6 mm inner diameter) into a prepared GM sample. As can be seen in Fig. 3, the needle and syringe are placed in the prepared GM. The tip of the needle is filled with metal wool to prevent grains from penetrating the needle and blocking the resin flow. The resin is then poured in the syringe and allowed to flow unaided down the needle and into the GM. Such a resin was chosen because it has very little volume changes during the curing process and thus causes minimal disruption to the prepared GM [START_REF] Bica | A Study of Free Embedded Cantilever Walls in Granular Soil[END_REF]; [START_REF] Clayton | A resin impregnation technique for the determination of the density variations in completed specimens of dry cohesionless soil[END_REF]). Once cured, the resin is hard and it can be taken out of the GM sample. Resin spheres are thus obtained.

The mass of sand in the resin sphere is obtained by subtracting the mass of resin impregnated. The volume of the sphere is obtained by measuring the Archimedes force on the resin sphere when it is fully submerged in water.

The in-situ density is then measured by dividing sand mass by sphere volume. By using formula 1, the in-situ relative density is obtained.

Results

Volumetric relative density

. The Pour and Vibrate techniques were used to prepare samples of SSC-1 and of SSC-2 in the 14.5 L metal cylindrical container (C-1) by a first Operator (Op) named Op-1. The mean and standard deviation of the VRD data set obtained for each soil simulant and each technique are presented in the second column of Table 3 (Op-1 C-1).

. The Vibrate and Pour procedures are simple and can be done manually.

In order to assess the robustness of these protocols to different operators and interpretations, new samples of SSC-1 and SSC-2 were prepared by a different operator (Op-2) using the vibrate and pour protocols. The data collected is presented in the third column of Table 3 (Op-2 C-1). This operator was also asked to prepare vibrated samples in the 13.5 L plastic container (C-2) to assess the influence of the container nature on the vibrate technique The data is presented in the fifth column of Table 3 (Op-2 C-2). The differences in mean VRD obtained after changing operator and changing container nature are also reported in Table 3 (columns ∆ Op and ∆ C). Two different Operators (Op-1 and Op-2) and two different containers (C-1 and C-2) were used. C-1 is the 14.5 L metal container and C-2 is the 13.5 L plastic one. Each data series is named after the operator and the container used. ∆ Op is the difference between data series Op-1 C-1 and Op-2 C-1 (only the operator has changed from one to another). ∆ C is the difference between Op-2 C-1 and Op-2 C-2 (only the nature of the container has changed). x represents experiments that were not conducted.

Mean VRD (Standard deviation) in %

Op-1 C-1 Op-2 C-1 ∆ Op Op-2 C-2 ∆ C SSC-
first operator. The mean and standard deviation of the VRD data set for each soil simulant and each preparation technique are presented in the second column of Table 4 under Op-1 C-3. To illustrate the influence of using a much smaller container, the difference between this data series and Op-1 C-1 is presented in the third column (∆) of Table 4.

Mean VRD (Standard deviation) in % Op-1 C-3 ∆ SSC-1 Pour -0.3 (0.9) 7.7 SSC-2 Pour -1.7 (0.9) 1.3 SSC-1 Vibrate 64 (1.3) 19 SSC-2 Vibrate 60 (3.5) 17

Table 4: Mean value and standard deviation of Volumetric Relative Density (VRD) data set measured after using Pour or Vibrate preparation techniques on SSC-1 and SSC-2 to fill a 245 mL cylindrical container (C-3) by first operator (Op-1). The difference with data series Op-1 C-1, Table 3, is reported in column ∆.

. Previous works have shown the influence of lateral speed of hopper and raining height on the final relative density achieved [START_REF] Okamoto | An evaluation of the dry pluviation preparation technique applied to silica sand samples[END_REF]; [START_REF] Scott | Steps towards characterising legged microrover performance on compressible planetary soils[END_REF]). A series of tests were thus lead to systematically assess the VRD achieved with the Rain method depending on hopper speed and raining height. The Rain technique was used to fill the 245 mL transparent cylindrical container (C-3). Two heights, low and high (respectively 3 cm above container top and 23 cm above container height), and two hopper speeds, slow and fast (24 and 72 sweeps per minute) were chosen. Three to four runs for each soil simulant in each raining configuration were conducted.

The mean value of the VRD data sets obtained are presented Fig. 4 with its standard deviation as error bars. . Negative relative densities have been reported here. This has also been observed by other authors (Perko et al. (2006)). It is clearly indicated in the ASTM D 4254-00 minimum density test description that the ASTM minimum density test does not insure an absolute minimum density (ASTM (2000)). Thus a lower density may be reached, which then corresponds to the negative relative density found in the results above.

In-situ density

. A series of SSC-1 samples prepared in C-2 with the Vibrate and Pour techniques were impregnated with resin. The mean and standard deviation of the VRD and in-situ relative density obtained are presented in Table 5 . The in-situ relative density is higher than the VRD obtained. The difference is higher in the poured case (+23%) than in the vibrated case (+14%). The higher value of the in-situ relative density could in part be explained by compaction under self weight in the sample. This would also explain why the difference is higher in the low density sample than in the high density sample. Indeed a low density sample will present a more important density gradient with depth than a compacted sample. However it is likely that a part of the difference is due to a bias between the two evaluation methods.

The standard deviation of the in-situ relative density is also much higher than the one for the VRD. Indeed, the in-situ density evaluation requires many more measurements than the VD. Measurement errors on the amount of resin impregnated may in part explain this higher dispersion of densities obtained. The determination of resin sphere volumes must be done with great care and can be influenced by the resin sphere absorption of water (though the sphere is coated with resin to limit this effect). Moreover some in-situ density evaluations allowed only a few grams of resin to be impregnated in the sand (due to potential clogging of needle by grains of material).

The smaller the resin sphere, the more sensitive the in-situ density value obtained is to the errors described above. For the data presented in Table 5, all tests injecting less then 10 g of resin were discarded. If this threshold is set to 20 g of impregnated resin, the mean in-situ relative density obtained for vibrated SSC-1 is closer to the VRD value in Table 5. (76% compared to 71%) and the standard deviation is much smaller (3.7% compared to 14%).

Evaluation of preparation techniques

6.1. Different densities achieved . Samples prepared with the Vibrate technique and the Pour technique have two very distinct relative densities: very low values for pouring (around 10%) and very high values for vibrating (around 80%). These two techniques allow soils to be prepared at two very different and extreme levels of relative density and are thus a very good manner of testing hardware in two different conditions for the same GM. The Rain technique allows to reach a wider variety of densities than the Pour and Vibrate techniques by varying usercontrolled parameters (raining height, hopper speed). To the knowledge of the authors, no theoretical models allow to choose flow rate, hopper speed and hopper height for a given target density, as these models would certainly vary with each GM. These parameters would thus be determined experimentally for each GM and each desired density.

. The densities and relative densities achieved with a given preparation technique depend on the material prepared. This was also found with the sample preparation methods used in Perko et al. (2006). For instance, though the difference between the VRD of vibrated SSC-1 and vibrated SSC-2 are small, they are significant enough to be differentiated. Thus if new soil simulants are prepared using the techniques proposed in this article, it would be essential to experimentally assess the acheived relative density of the new simulant.

6.2. Robustness . The differences in rained densities obtained by modifying raining height and hopper speed illustrate the need to precisely control hopper speed and height. This is not possible without some level of hardware investment.

. Varying the operator and the final container for the Pour and Vibrate technique have allowed to identify sources of variations in the final density achieved. The differences observed between the densities achieved by two different operators with the Pour method in the 14.5 L metal container (C-1) are expected to be related to the variation in pouring speed. This is a clear limitation of the simple manual method proposed here. It illustrates the need to calibrate the method for each operator or to invest in hardware to make this method operator independent. The differences observed between the poured densities in (C-1) and the 245 mL container (C-3) are expected to be due to self compaction of the GM under its own weight. (C-1) being much taller than (C-3), the soil on the bottom of the taller container is subject to much more over-head weight and is thus compacted. The differences observed between the vibrated densities achieved in (C-1), the plastic 13.5 L container (C-2) and (C-3) are expected to be due to a different transmission of the vibrations from the shaker to the container. At the beginning of the vibration preparation, the 245 mL container had a tendency to bounce under vibration whereas the larger ones (C-1 and C-2) did not. This modified the vibrations applied to the GM and thus the final density achieved. Similarly, due to the different properties of metal and plastic, the vibrations are transfered to the prepared sample differently. This explains the differences observed experimentally between C-1 and C-2. This illustrates the need to calibrate each vibration setup.

. Once prepared, gravity alone has no effect on granular material over-time. Indeed an arrangment of particles in a gravity field is said to be in a metastable state. However if an exterior perturbation (shock or vibration) injects enough energy in the arrangment, it can leave its meta-stable state and evolve towards another meta-stable state. It will evolve towards a lower energy level in the given gravitational field (which corresponds to a higher density state).

The lower the initial density the more sensitive the granular material will be to perturbations. Since it is almost impossible to ensure that the prepared material will receive no shocks or vibrations it should be expected that the density of the poured samples will progressively increase over time. The vibrated samples being in a very high state of density will be much less sensitive to such perturbations. It is thus recommended to use the poured samples directly after preparation and minimize the vibrations and shocks it could be subject to.

. However, for a given granular material and preparation technique done by a given operator with a given setup, the standard deviations observed are very low for all the data sets presented here. The soil simulants can thus be prepared in a repeatable and controlled manner.

6.3. Soil sample fabric . Preparing SSC-1 and SSC-2 in the 245 mL transparent container (C-3) has allowed direct observations of layering and granular material fabric. Figure 5 presents four pictures of rained SSC-1 from high and low heights and at slow and fast hopper speeds. Layers and particle segregation in each layer are clearly visible. The size of the layer is very logically linked to hopper speed. Indeed, the slower the hopper translation speed, the more time it takes to complete a pass over the container. Since the soil flow rate out of the hopper is constant, more soil is deposited in a single pass, thus creating thicker layers.

. Figure 6 is a picture and a schematic of SSC-1 prepared with the Rain method at high height and low speed. The particle segregation or layer structure can clearly be seen in this figure. The bigger particles are in the lower portion of each layer and the finest particles are on the top portion of the layer. The aerodynamic forces have more influence on the finer particles than on the large particles. Thus the finer particles settle after the largest ones. It is interesting to note that, even though raining from the high level at a slow hopper speed and raining from the low level at a fast hopper speed gave very close densities, the internal structures obtained are very different.

. Similar images were taken for the rained SSC-2 samples to examine the particle layering, however, it was more difficult to observe layering in the SSC-2 samples. Since the particle size distribution of SSC-2 is much tighter than SSC-1 particle size distribution, much less segregation will take place. Without significative segregation it is difficult to detect layers visually.

. The poured and vibrated SSC-1 and SSC-2 samples were also photographed. Figure 7 presents these pictures. The SSC-2 samples are very homogenous (because of the very tight particle distribution as explained above). The SSC-1 samples present some local fabric variations but do not have a very organised and layered structure like the rained samples. Generally, it can be said that the Poured and Vibrate techniques prepare much more homogenous samples than the Rain technique. The slight particle segregation that was observed in the poured and vibrated SSC-1 samples is expected to be due to variations in pouring rate during the sample preparation.

. The resin spheres also allow the observations of granular material fabric. Figure 8. right presents a resin sphere obtained after impregnating into poured SSC-1. The resin spheres obtained after pouring and vibrating are generally round. The resin sphere in Figure 8. left was obtained by impregnating resin into SSC-1 after it was rained into a large soil bin for leg soil interaction experiments [START_REF] Scott | Steps towards characterising legged microrover performance on compressible planetary soils[END_REF]). The layers created by the Rain technique are clearly visible in the resin "sphere". The peaks present on the surface of the rained resin sphere are made of the finer particles of SSC-1. This is explained by capillary action. Indeed the capillary forces in particulate materials are inversely proportional to the distance between particles. Finer particles have smaller distances between them. Thus the resin, pulled out by stronger capillary forces, spreads more in the fine particulate.

6.4. Summary . Table 6 is a summary of the main characteristics of the three proposed granular material preparation methods studied in this paper. The relative densities achieved, the fabric of the granular material, the hard-ware requirements and the critical points of each technique are presented.

Conclusions

. In this paper, three granular material preparation techniques have been presented (Rain, Pour and Vibrate)and tested with two soil simulants (SSC-1 and SSC-2). The densities achieved were measured by volumetric methods and by resin impregnation. All three techniques allow precise and repeatable results. However, variations in hardware used and in operators can cause variations in the final density achieved. Additionally the relative density obtained by a given preparation technique depends on the granular material used. It is thus recommended to assess the density of the prepared granular material each time the setup or the granular material is changed. If high precision on the relative density achieved is needed over a long period of time or with different operators, it is recommended to invest in hardware to execute these preparation techniques. Furthermore, this paper has illustrated the impact of preparation techniques on soil fabric (with resin impregnation or transparent containers). . Apart from testing techniques and identifying sources of dispersion, this paper shows the necessity to systematically assess the relative density achieved and to describe how this relative density was achieved for a given experimental setup. Indeed numerous publications describing tests done on space hardware interacting with regolith simulants do not include such details. It is thus very difficult to repeat or to compare results. A precise description of regolith stimulant preparation technique, of relative density achieved and an indication of dispersion observed should be common practice in publications.

Poured

. It is not possible to recommend one preparation technique over another for creating regolith samples for space hardware testing. Indeed depending on the application, a layered granular material preparation or a homogenous sample may be preferred. But above all, the most challenging scenario will be different for two different applications. For a locomotion system, it seems loosely compacted materials are very difficult to traverse, as the recent trapping of NASA Martian rover Spirit has shown (NASA ( 2009)). For a drilling or penetration system, the high density case is the most demanding in terms of over-head force requirements [START_REF] El Shafie | Penetration Forces for Subsurface Regolith Probes[END_REF]). Thus to ensure space system performance, if the most stringent environment is not clearly identified, it seams prudent to test systems interacting with granular materials at high and low relative densities. The techniques presented here can be used to do so.

Acknowledgments

.

Figure 2 :

 2 Figure 2: Graph of cumulative mass obtained when raining SSC-1 and SSC-2 regolith simulants from the hopper.

.Figure 3 :

 3 Figure 3: Schematic of resin impregnation set-up with needle placed in the granular material and filled with resin.

Figure 4 :

 4 Figure 4: Mean value of volumetric relative density data set measured after using Rain preparation techniques on SSC-1 and SSC-2 to fill a 245 mL cylindrical container. The speed of hopper and height of hopper were varied.

Figure 5 :Figure 6 :

 56 Figure 5: Pictures of SSC-1 prepared in transparent container with Rain method. The layering and segregation of particles in each layer are visible in each picture. The differences in soil fabric caused by the slow or fast speed of the hopper are clearly visible. A fast hopper speed creates thin layers and a slow hopper speeds creates thick layers. Hopper height has no visible influence on layering.

Figure 7 :

 7 Figure 7: Pictures of poured and vibrated SSC-1 and SSC-2 samples. The SSC-2 samples are very homogenous. The SSC-1 samples present some slight fabric variations.

  (5 % above the maximum observed density after compaction by pressure on top of the sample). The values obtained are reported in Table2.

	Simulant Maximum density (kg/m 3 ) Minimum density (kg/m 3 )
	SSC-1	1794	1383
	SSC-2	2503	1948

Table 2 :

 2 Minimum and maximum density of SSC-1 and SSC-2.

Table 3 :

 3 Mean value and standard deviation of Volumetric Relative Density (VRD) data sets measured after using Pour or Vibrate preparation techniques on SSC-1 and SSC-2.

	1 Pour	7.4 (4.4)	15 (2.1)	7.6	x	x
	SSC-2 Pour	-0.4 (1.1) 7.7 (1.3)	8.1	x	x
	SSC-1 Vibrate 83 (4.6)	87 (1.6)	4.2	71 (2.9) 15.8
	SSC-2 Vibrate 87 (1.8)	77 (1.0)	9.5	69 (3.4)	7.8

Table 5 :

 5 .

	SSC-1	Mean VRD (%)	Standard deviation (%)	Mean ISRD (%)	Standard deviation (%)
	Pour	15	2.1	38	11
	Vibrate	71	2.4	82	15

Mean value and standard deviation of Volumetric Relative Density (VRD) and Insitu Relative Density (ISRD) measured after using Pour or Vibrate preparation techniques with SSC-1 in the 13.5 L container (C-3).

  The authors thank the European Space Agency for their support, particularly for fostering research through the Networking and Partnership Initiative. The authors thank the French Ministry of Research for funding PhD grants. ASTM, 2000. 4254-00 Standard Test Methods for Minimum Index Density and Unit Weight of Soils and Calculation of Relative Density, in: American Society for Testing and Materials.


Amir Shahdin

Joseph Morlier
email: joseph.morlier@isae.fr

Guilhem Michon

Laurent Mezeix

Christophe Bouvet

Yves Gourinat

Application of Modal Analysis for Evaluation of the Impact Resistance of Aerospace Sandwich Materials

Keywords: Structural Integrity Assessment, Composite Sandwich beams, Vibration Testing, Impact Resistance

Impact resistance of different types of composite sandwich beams is evaluated by studying vibration response changes (natural frequency and damping ratio). This experimental works will help aerospace structural engineer in assess structural integrity using classification of impact resistance of various composite sandwich beams (entangled carbon and glass fibers, honeycomb and foam cores). Low velocity impacts are done below the BVID limit in order to detect damage by vibration testing that is hardly visible on the surface. Experimental tests are done using both burst random and sine dwell testing in order to have a better confidence level on the extracted modal parameters. Results show that the entangled sandwich beams have a better resistance against impact as compared to classical core materials.

Introduction

The aim of composite sandwich structures is to increase the stiffness and specific strength and to reduce the weight so it is advantageous to employ them in aerospace applications where the challenge is to produce light structures. However damage due to impact in these structures may negate many of the benefits of sandwich construction. The facesheets can be damaged through delamination and fibre breakage; the facesheet and core interface region can be debonded and the core can be damaged through crushing and shear failure mechanisms. Safe and functional effectiveness of stressed sandwich structures can often depend on the retention of integrity of each of the different materials used in its manufacture. For aeronautical structures, a field where this problem has been quite studied, the components have to undergo low energy impacts caused by dropped tools, mishandling during assembly and maintenance, and in-service impacts by foreign objects such as stones or birds. In these low energy impacts normally, a small indentation is seen on the impact surface. This level of damage is often referred to as Barely Visible Impact Damage (BVID).

Although not visually apparent, low energy impact damage is found to be quite detrimental to the load bearing capacities of sandwich structures, underscoring the need for reliable damage detection techniques for composite sandwich structures. In recent years, vibration based damage detection has been rapidly expanding and has shown to be a feasible approach for detecting and locating damage [START_REF] Doebling | A summary review of vibration-based damage identification methods[END_REF][START_REF] Yan | Development in vibration-based structural damage detection technique[END_REF][START_REF] Carden | Vibration based condition monitoring: A review[END_REF]. Any structure can be considered as a dynamic system with stiffness, mass and damping. Once some damages emerge in the structures, the structural parameters will change, and the frequency response functions and modal parameters of the structural system will also change. This change of modal parameters can be taken as the signal of early damage occurrence in the structural system. Shift in natural frequency is the most common parameter used in the identification of damage [START_REF] Khoo | Structural damage assessment using vibration modal analysis[END_REF][START_REF] Gadelrab | The effect of delamination on the natural frequencies of a laminated composite beam[END_REF][START_REF] Yam | Damage detection of composite structures using dynamic analysis[END_REF]. However, in structures made of composite materials there seems to be a tendency to use damping as a damage indicator tool, as it tends to be more sensitive to damage than the stiffness variations, mainly when delamination is concerned [START_REF] Shahdin | Correlating low energy impact damage with changes in modal parameters: A preliminary study on composite beams[END_REF][START_REF] Adams | Damping in composites[END_REF][START_REF] Gibson | Modal vibration response measurements for characterization of composite materials and structures[END_REF][START_REF] Saravanos | Effects of delaminations on the damped dynamic characteristics of composites[END_REF]. A comprehensive review of low-velocity impact responses of composite materials is presented by Richardson and Wisheart [START_REF] Richardson | Review of low-velocity impact properties of composite materials[END_REF]. Dear et al. [START_REF] Dear | Impact damage processes in composite sheet and sandwich honeycomb materials[END_REF] studied the impact toughness of different lightweight sandwich panels and composite sheet materials. They emphasized on the degree of damage inflicted on the contact surface, throughthickness and rear surface of the materials when subjected to different impacts. Vaidya et al [START_REF] Vaidya | Impact and post-impact vibration response of protective metal foam composite sandwich plates[END_REF] studied composite sandwich structures composed of aluminum foam core and found it optimal for resisting low-velocity impacts. Similarly several novel sandwiches have been developed with a view to enhance the impact toughness [START_REF] Hosur | Manufacturing and low-velocity impact characterization of foam filled 3-D integrated core sandwich composites with hybrid face sheets[END_REF][START_REF] Sutherland | Impact characterisation of low fiber-volume glass reinforced polyester circular laminated plates[END_REF][START_REF] Abdullah | The impact resistance of polyporopylene based fiber-metal laminates[END_REF].

In this paper we have made Low Velocity Impact (LVI) and post impact vibration tests to measure the impact resistance. More precisely we evaluated the resistance against impact of entangled sandwich materials in comparison with standard sandwiches with honeycomb and foam cores, uniquely based on vibration test results (i.e., decrease in natural frequency which signifies loss of rigidity, and increase in damping which corresponds to friction in damaged zones). Vibration tests verify the presence of high damping in the entangled sandwich specimens making them suitable for specific applications like the inner paneling of a helicopter cabin, even if the structural strength of this material is on the lower side.

Material and Specimen

Six sandwich beam specimens are studied in this article, Fig. 1. The main emphasis of this article is on the two entangled sandwich beams with carbon and glass fibers as core materials. The honeycomb and foam sandwich beams are only presented for comparison purposes. Therefore comparison with standard sandwich beams is essential in order to evaluate the performance of entangled sandwich materials.

Fig. 1 The six sandwich beams tested in this article with (a) carbon prepregs and (b) glass woven fabric as skin materials Sandwich beams have entangled carbon fiber, honeycomb and foam as core materials. For the skin, three of them have unidirectional carbon-fiber/epoxy prepregs of T700/M21 (four plies each with a stacking sequence of [0/90/90/0]); the other three are made of glass woven fabric, impregnated with epoxy resin (two plies each with a total thickness of 0.5 mm containing 50 % of resin). The thickness of the skins in case of glass woven fabric is kept similar to carbon fiber skins. The honeycomb sandwich beams are made of Nomex-aramid honeycomb core (nominal cell size of 6.5 mm and a core thickness of 10 mm). In case of the foam sandwich beams, the foam core has also a thickness of 10 mm. The core, in case of the carbon entangled sandwich beam, consists of carbon fibers that are made of a yarn of standard carbon filaments having a diameter of 7 μm. The length of the carbon fibers is 10 mm and their elastic modulus is 240 GPa. In case of the glass entangled sandwich beam, the core consists of glass fibers made of glass filaments having a diameter of 14 μm. The length of the glass fibers is 10 mm with an elastic modulus of 73 GPa. For the cross-linking of carbon and glass fibers, epoxy resin and injection hardener are used. A better vaporization is achieved if the resin is heated up to 35°C before being sprayed on the carbon and glass fibers. This allows the mixture of resin and hardener to become less viscous. The fabrication of these entangled sandwich beams is a relatively complex process. The carbon and glass fibers are cut with the help of a fiber-cutting machine. A blow of compressed air then separates the fibers. The fibers vaporized by the resin are then placed in the mold between the two skins. A fiber core density of approximately 200 kg/m 3 is chosen for the entangled sandwich core in case of both carbon and glass fibers.

Experimental Methods

High quality vibration tests

The experimental equipment used for vibration testing is shown in Fig. 2. The experimental set-up is that of a free-free beam excited at its center, based on Oberst method [START_REF] Wojtowicki | New approach for the measurements of damping properties of materials using oberst beam[END_REF]. The test specimen is placed at its center on a force sensor which is then assembled on a 100N electrodynamics shaker. The response displacements are measured with a non-contact and high precision laser vibrometer. The shaker, force sensor and the laser vibrometer are manipulated with a data acquisition system.

Fig. 2. Diagram of the experimental set-up

Each specimen is tested with two types of excitations: burst random and sine dwell. Response is measured at 27 points that are symmetrically spaced the length of the beam to have reliable identifiable mode shapes. Burst random excitation permits to have rapidly the overall dynamic response of the structure (natural frequencies and mode shapes). It is a broadband type excitation signal. 50% burst percentage is used for burst random excitation in the frequency range of 0-2650Hz. However if we need precise damping measurements then sine-dwell testing becomes inevitable. Its advantage is its capability of detecting non-linear structural dynamic behavior. As this method requires larger acquisition times, acquisition is carried out only around the first four bending modes previously identified by burst random testing.

The modal parameters are extracted for burst random and sine-dwell testing respectively. The estimation method is a new non-iterative frequency domain parameter estimation method based on weighted least squares approach. One of the specific advantage of this technique lies in the very stable identification of the system poles and participation factors as a function of the specified system order, leading to easy-to-interpret stabilization diagrams.

Low Velocity Impact tests

The sandwich beams tested in this article are damaged by drop weight impacts below the Barely Visible Impact Damage limit (BVID), in order to simulate damage by foreign impact objects. The impact tests are carried out by a drop weight system as shown in Fig. 3-a, and a detailed cut away of the drop assembly is shown in Fig. 3-b. The impactor tip has a hemispherical head with a diameter of 12.7 mm. A force sensor is placed between the impactor tip and the free falling mass of 2 kg. The velocity before the impact is measured with an optical velocity sensor from which the energy of impact can be verified.

Fig. 3. Arrangement of the test equipment for the impact test (a) and detailed cutaway of the drop assembly (b)

The impact energy is chosen in such a way that each sandwich beam has approximately the same level of damage (i.e., below the BVID limit which is nearly invisible on the surface). The impact parameters and the indentation depths measured for the six sandwich beams are listed in Table 1. The data obtained during the drop weigh impact tests carried out on the beams are presented in Fig. 4. In Fig. 4-a and 4-b, the impact forces are drawn as a function of time for the six sandwich beams. These curves are globally smooth and almost sinusoidal at low impact energy. It can also be seen, from the force-displacement plots (Fig. 4-c and4-d), that after the first damage in the classical sandwiches with honeycomb and foam cores, there is a decrease in the force signal followed by oscillations which signifies damage and loss of rigidity in the material. But in case of both glass and carbon fiber entangled sandwich beams (Fig. 4-c and4-d), after the appearance of first damage the material continues to rigidify which is shown by a progressive increase of force signal. It can be observed from the force displacement curves of the honeycomb and foam core sandwiches (Fig. 4-c and4-d) that the energy dissipation seems mostly due to the rupture mechanism. However in case of entangled sandwiches the behavior is different, it is possible that the energy dissipation might be predominantly due to damping as no oscillations or force signal loss is observed (i.e., no apparent damage signs).

Shift in modal parameters due to damage

The tested sandwich beams have two states. First one is the undamaged state (UD) and the second is the damage state due to two impacts (D1). Vibration tests are carried out on the six sandwich beams after each of these two states. The effect of impact damage on the modal parameters of the three types of sandwich beams is studied in the following sections with the help of frequency and damping changes between the undamaged (UD) and the damaged case (D1) defined by Eq. 1 and 2:

Change in frequency between UD and D1, (Δf

) = UD D UD f (k) f (k) f (k) -1 (1)
Change in damping between UD and D1,

(Δ ζ ) = D UD UD (k) (k) (k) ζ -ζ ζ 1 (2)
where f UD (k) is the damped natural frequency for the undamaged specimen for the k th mode and f D1 (k) is the damped natural frequency for the specimen damaged at two impact points (D1) for the k th mode. The effect of physical properties on poles in the complex s-plane is illustrated in Fig. 6. It can be observed that a change in stiffness affects only the frequency, while changes in mass and structural damping affect both modal damped frequency (ω d ) and modal damping ( σ ). For this study, the primary interest is to study the decrease in the modal damped frequency (ω d ) and the increase in modal damping ( σ ) due to damage in the sandwich specimens [17].

Results and discussions

Effect of impact damage on modal parameters

Frequency and damping ratios are the global parameters of the specimen, and are extracted from measurements carried out on the 27 measurement points. The modal parameters (natural frequency and damping) help in monitoring globally the health of a specimen. As a result of impact damage, there is a decrease in the natural frequencies for the six sandwich beams as discussed before in section 3.3. It can be noticed that this decrease is less prominent in case of both the carbon and glass entangled sandwich beams as compared to the honeycomb and foam sandwich beams. But the interesting fact is that for all the sandwich beams, even as the impact damage does not produce a visible damage on the surface, the change in frequency between the undamaged and the damaged cases is quite noticeable. This proves that there is a notable loss of rigidity without any signs of damage on the surface. The fact that the change in natural frequency between the damaged and the undamaged case is small in the entangled beams as compared to the foam and honeycomb beams can be seen in Fig. 7, which presents a comparison between the frequency response functions of the undamaged and the damaged cases for the six sandwich beams for the 1st bending mode. The frequency response functions presented in Fig. 10 are obtained with the sine-dwell testing. For the six sandwich beams, the shift in natural frequencies is slight in case of entangled sandwich beams proving that they have a loss of rigidity that is less pronounced as compared to the honeycomb and foam sandwich beams. Furthermore in case of the three carbon sandwich beams (Fig. 7-a), the frequency response functions of the carbon entangled sandwich beam are more acute (smaller in width) as compared to the honeycomb and foam sandwich beams. This phenomenon is less evident in case of the glass entangled sandwich beams (Fig. 7-b). Therefore it can be concluded that the entangled sandwich beams show a better resistance to impact as compared to the honeycomb and foam sandwich beams, whereas all the beams have more or less the same level damage. The shift in modal parameters is less in case of the entangled sandwich beams, which signifies that they possess better impact toughness as compared to the standard sandwich beams with honeycomb and foam as core materials. Furthermore, the results underline the fact that the damping change ratios are more prominent than the frequency change ratios. The maximum damping change ratio is 310 % whereas the maximum frequency change ratio is 25 %. It can be concluded from the above results that damping seems more sensitive to damage than the natural frequency variations in case of honeycomb sandwich beams. So it is reasonable to assume that damping may be used instead of natural frequency as a damage indicator tool for structural health monitoring purposes. However, the fact that damping is a parameter that is relatively difficult to estimate as compared to natural frequency has to be taken into account.

Conclusion

The aim of the article is to evaluate the resistance against impact of entangled sandwich beams by the pole shift which signifies loss of rigidity (decrease in natural frequency) and increase in friction in damaged zone (increase in damping ratio). A simple case of symmetrical impacts is studied and Low Velocity Impacts are done below the BVID limit in order to detect damage by vibration testing that is hardly visible on the surface. From the results it can be concluded that both carbon and glass fiber entangled sandwich beams show a better resistance to impact (based only on decrease in natural frequency) as compared to the honeycomb and foam sandwich beams, whereas all the beams have more or less the same level damage.

Fig. 4 .

 4 Fig. 4. Impact test data (a,b) force-time (c,d) force-displacement for the carbon and glass sandwich beams

Fig. 5 .

 5 Fig.5. Sandwich test beams with location of the two impact points, 27 measurement points and one excitation point (Point 14)

Fig. 6 .

 6 Fig. 6. Movement of pole due to mass stiffness and damping effect

Fig. 7

 7 Fig. 7 Comparison of the frequency response functions estimated by sine-dwell testing for the undamaged case (UD) and damaged at 2 points (D1) for the 1 st bending mode for (a) Point 11 for the three carbon sandwich beams and for (b) Point 21 for the three glass sandwich beams

Table 1 . Impact test parameters of the sandwich beams Indentation just after impact (mm)

 1

	Type of Specimen	Energy of			Velocity of impact
		Impact			(m/s)
		(J)	Point 1	Point 2
	Entangled Carbon	4.9	0.1	0.15	2.21
	Honeycomb Carbon	3.9	0.1	0.5	1.98
	Foam Carbon	3.8	0.1	0.2	1.98
	Entangled Glass	6.2	0.2	0.15	2.49
	Honeycomb Glass	3.8	0.2	0.4	1.98
	Foam Glass	3.8	0.15	0.25	1.98


Pierre Jehel
email: pierre.jehel@ecp.fr

Hugues Vincent

Thomas Rodet

DAMPING MODEL-FREE ANALYSIS OF A STOCHASTIC MULTI-SCALE REINFORCED CONCRETE FRAME ELEMENT FOR EARTHQUAKE ENGINEERING APPLICATIONS

Keywords: Earthquake engineering, damping, stochastic multi-scale material model, frame structure, discrepancy forces, Bayesian inference

The viscous damping forces commonly added in earthquake engineering applications lack a sound physical justification. Consequently, whether or not they accurately represent what they are added for is uncertain. In this paper, this latter source of uncertainty is removed, appealing to the concept of discrepancy forces, in the analysis of the capabilities of a nonlinear reinforced concrete frame elements to represent actual seismic behavior. Frame elements are modeled as fiber elements accommodating uniaxial concrete and steel behavior laws; two different nonlinear cyclic concrete behavior laws are considered: a stochastic multi-scale concrete model along with a more classical one. Concrete parameters are identified from experimental data using Bayesian inference technique. Finally, the capabilities of the two models to represent the actual response of a reinforced concrete frame element tested on a shaking table are compared in an objective way, that is without any side effect resulting from the action of uncertain damping forces.

Introduction

For running accurate predictive computational analyses of nonlinear structures in seismic loading, practitioners have to cope with the dual issue of i) explicitly modeling the main seismic energy absorption mechanisms in the structural model, and ii) adding damping forces that represent all the energy dissipative mechanisms that are not otherwise accounted for in the inelastic structural model. There is no structural model which is always capable of perfectly grasping altogether the numerous energy dissipative mechanisms activated in inelastic structures in seismic motion. Consequently, the addition of damping forces necessarily lacks a sound physical justification and whether or not those forces accurately represent what they are added is uncertain. More specifically, denoting u the displacement field and using an upper dot to represent time derivation, basic equilibrium equation for solving earthquake engineering problems read:

f ine (ü; t) + f dam ( u; t) + f hys (u; t) = f ext (t) (1)
In addition to the inertia (f ine ), resisting (f hys ) and seismic along with static external (f ext ) forces, it is common practice to add viscous damping forces (f dam ) "to simulate the portion of energy dissipation arising from both structural and nonstructural components (e.g., cladding, partitions) that is not otherwise incorporated in the model" (FEMA P695, 2009, §6.4.4.). In inelastic time history analyses, such damping models are not built on a sound physical basis but rather following an ad hoc method 1 i i "2014-10-02˙CSM7˙PJ-HV-TR" -2014/10/2 -18:34 -page 2 -#2 i i i i i i

Damping model-free analysis of a stochastic multi-scale frame element for earthquake engineering P. Jehel, H. Vincent, and T. Rodet allowing for adding dissipation in the system while remaining easy to be implemented (see the studies on Rayleigh damping in [START_REF] Jehel | Initial versus tangent stiffness-based Rayleigh damping in inelastic time history seismic analyses[END_REF]; [START_REF] Charney | Unintended consequences of modeling damping in structures[END_REF]; [START_REF] Hall | Problems encountered from the use (or misuse) of Rayleigh damping[END_REF]).

In the case of reinforced concrete structures, there is abundant literature on concrete models that have been developed and implemented in structural elements in the purpose of grasping the salient features of concrete response in seismic loading as a step toward, ultimately, computing accurate forces f hys (t). This is the topic of section 2 in this paper where two types of uniaxial concrete models to be implemented in fiber elements will be briefly introduced: a stochastic multi-scale model along with a more classical phenomenological model with internal variables.

When running numerical inelastic time history analyses and then comparing model outputs with experimental measures to assess the validity of a simulation, one actually checks the validity of the combined action of both the structural (f hys ) and the damping (f dam ) forces altogether. Determining which of the inelastic structural model or the damping model is accurate, or which is not, is a kind of chickenand-egg problem. Indeed f hys (u) and f dam ( u) are interdependent due to the fact that the displacement field is affected by both the structural and the damping model, making it practically impossible to determine whether a poor overall model response is due to a poor structural model or to a poor damping model. As a consequence, in case the accuracy of f hys (u) only has to be investigated, it is important to carry out a damping model-free analysis. This is desirable for instance to compare the respective capabilities of two different concrete behavior laws to represent actual seismic nonlinear response of a frame element. To that purpose, the concept of discrepancy forces will be introduced in the third section of this paper.

Before closing the paper with some conclusions, numerical applications will be presented. The parameters of the two concrete behavior laws introduced in section 2 will be identified with Bayesian inference technique and then, the discrepancy forces pertaining to one column of a reinforced concrete frame tested on a shaking table will be computed. This provides a damping model-free comparison of the respective performances of both concrete models in structural inelastic time history seismic analysis, that is a comparison that is not possibly biased by the effects of uncertain damping forces.

Uniaxial concrete behavior laws

Uniaxial concrete response in compressive cyclic loading

In the earthquake engineering community, there is an abundant literature on the formulation and numerical implementation of material models developed for representing the main features of concrete response in cyclic loading. Typical axial stress with respect to axial strain response of a concrete specimen in cyclic compressive loading is depicted in figure 1. We provide here a few references to illustrate four particular categories of concrete models:

(i) Analytical stress-strain relations (power laws, exponential laws,...) identified from experimental observations (e.g. [START_REF] Wong | VecTor2 & Formworks User's Manuals[END_REF]Computers & Structures Inc. (2007)).

(ii) Stress-strain phenomenological laws derived from thermodynamical principles with internal variables (see e.g. [START_REF] Richard | Continuum damage mechanics based model for quasi brittle materials subjected to cyclic loadings: Formulation, numerical implementation and applications[END_REF]; [START_REF] Jehel | Towards robust viscoelasticplastic-damage material model with different hardenings / softenings capable of representing salient phenomena in seismic loading applications[END_REF]). This can be interpreted as a multi-scale approach, as by [START_REF] Maugin | The Thermodynamics of Nonlinear Irreversible Behaviors -An Introduction[END_REF], in the sense that the internal variables, which are the history of the concrete response, convey at macro-scale the effects of mechanisms occurring at lower scales.

(iii) Probabilistic elasto-plasticity model based on the computation of the time evolution of the probability density function of the stress (e.g. [START_REF] Jeremić | Probabilistic elasto-plasticity: formulation in 1D[END_REF]).

(iv) Concrete meso-scale (aggregates, cement paste and so-called interfacial transition zone around aggregate grains) is explicitly represented as a spatial truss with enhanced kinematics within the finite element method, for instance in the work of [START_REF] Benkemoun | Failure of heterogeneous materials: 3D meso-scale FE models with embedded discontinuities[END_REF]. Due to the computation costs, the approach is not tractable for structural analyses but it is interesting to observe that characteristic features of the uniaxial concrete response at macro-scale can be explained by this representation of concrete meso-scale.

Figure 1: Strain-stress concrete experimental response in pseudo-static cyclic compressive loading (adapted from a figure by [START_REF] Ramtani | Contribution to the modeling of the multi-axial behavior of damaged concrete with description of the unilateral characteristics[END_REF]). Σ is the (homogeneous) compression stress in a concrete test specimen, that is the load in the hydraulic cylinder of the testing machine divided by the area of the specimen cross section; E is the homogeneous deformation field, that is the displacement of the cylinder divided by the specimen height.

At structural level, those concrete laws can be integrated in 2D/3D finite elements. For earthquake engineering applications, the fiber element approach, as described e.g. by [START_REF] Davenne | Efficient approaches to finite element analysis in earthquake engineering[END_REF], is particularly well adapted because it provides a good compromise between computational efficiency and refined description of the physics. Implemented with displacement-based formulation and Euler-Bernoulli kinematics assumption, fiber elements call for 1D material behavior law at each numerical integration point in the control sections of the beam (for another beam element formulation, see e.g. the work of [START_REF] Taylor | A mixed finite element method for beam and frame problems[END_REF]).

Virtual internal potential increment for such a beam element of length L and section S is numerically approximated as:

δW int := L S δE(x) Σ(x) dSdx 1 ≈ 2 l=1 N F F =1 A F δE F l Σ F l W l (2)
where A F is the cross section area, E F l and Σ F l are the axial strain and stress in fiber F in control section l, and W l denotes quadrature weight and length. Σ F = Σ(E F ) is the uniaxial behavior law of fiber F material, which, in reinforced concrete structures, is either concrete or steel. In common practice, concrete is considered as homogeneous over the beam so that Σ(x) is the same concrete law at any material point x in the beam. The number of fibers N F has to be sufficiently large for the structural response of the model to be independent of it.

A uniaxial stochastic multi-scale nonlinear cyclic concrete law

Thereafter, we will consider another type of concrete material model, which we refer to as "multi-scale stochastic" because it is based on the homogenization of a randomly generated heterogenous structure at a meso-scale. A detailed description of the model can be found in [START_REF] Jehel | On damping created by heterogeneous yielding in the numerical analysis of nonlinear reinforced concrete frame elements[END_REF]. The principal motivation for developing such an approach comes from the fact that concrete exhibits a "continuous range of micro-structural dimensions", as stated by [START_REF] Stroeven | Stochastic heterogeneity as fundamental basis for the design and evaluation of experiments[END_REF], when observed with progressive magnification. At first glance, with human eyes, the bi-phasic nature of concrete is obvious (mixture of aggregates and cement paste). Then, zooming in the cement paste would reveal other heterogeneities so that several other phases would appear at microscopic scale (water, pores, air voids, C-S-H, CH, ettringite,...).

Seeking meaningful information in these heterogeneities to explain characteristic features of the concrete response at macro-scale looks compelling. However, there is obviously both too little knowledge, from a physical point of view, and too much data to process, from a computational viewpoint, to realistically think (i) Concrete uniaxial homogeneous response at macro-scale Σ = Σ(E) results from uniaxial heterogeneous behavior at mesoscale σ = σ(ǫ(x)), such that:

Σ F l := Σ(E F l ) = 1 |R| R σ(ǫ F l ) dR ≈ 1 N 2 f N 2 f f =1 σ ǫ(x F,f l ) (3)
where |R| is the size of a representative square area R divided into N f × N f equal square parts, and

ǫ(x F,f l ) = E(x F l ) ∀x F,f l ∈ R (4)
(ii) Concrete meso-scale response is represented by an elasto-plastic law with linear kinematic hardening. At any material point, yielding is parameterized by the yield stress σ y . This is achieved using classical computational model with internal variables [START_REF] Simo | Computational Inelasticity[END_REF]).

(iii) The heterogeneity at meso-scale is conveyed by the fluctuations of the yield stress σ y (x), as depicted in figure 2. These fluctuations are represented by a 2D homogeneous log-normal stochastic field over R, built as the translation of a Gaussian field generated with the Spectral Representation Method as presented by [START_REF] Shinozuka | Simulation of multi-dimensional Gaussian stochastic fields by spectral representation[END_REF], and with triangular power spectral density function.

(iv) Concrete has no strength in tension.

Figure 3 shows macroscopic response at each numerical integration point x F l in the beam. This has been computed with N f = 64, |R| = d × d, correlation length in both orthogonal directions ℓ c = d/10. We refer the reader to [START_REF] Jehel | On damping created by heterogeneous yielding in the numerical analysis of nonlinear reinforced concrete frame elements[END_REF] for further discussion on the choice of these parameters. This set of parameters will remain unchanged for any result shown throughout this paper. It is obvious in figure 3 that the homogenized response at macro-scale exhibits much richer behavior than the local response at meso-scale (see figure 2). An interesting feature is in particular the hysteresis during unloading-reloading cycles. Besides, figure 3 shows that R is a representative area in the sense that there is practically independence between the random realization of the meso-structure and the response at macro-scale.

As already appealed for by [START_REF] Charmpis | The need for linking micromechanics of materials with stochastic finite elements: A challenge for materials science[END_REF], albeit in a different context, there is a need here for interaction between structural and material scientists so that the assumptions made to account for the heterogeneous nature of concrete at meso-scale be supported by experimental observations.

A uniaxial concrete law with internal variables

A second kind of concrete model will also be used in the numerical application that follows. This model is based on the work presented in [START_REF] Jehel | Towards robust viscoelasticplastic-damage material model with different hardenings / softenings capable of representing salient phenomena in seismic loading applications[END_REF] where a uniaxial material model is developed in the framework of thermodynamics with internal variables, along with enhanced kinematics accommodating discontinuities in the displacement field. The model is capable of representing salient phenomena that can be experimentally observed in the stress-strain response of a concrete representative elementary volume in cyclic uni- minus standard deviation (--) response at macro-scale obtained from a sample of 2,000 meso-structures. [bottom] One realization of concrete meso-structure. Targeted mean and standard deviation of the marginal log-normal law are µ = 35.0 MPa and s = 51.9 MPa. This figure shows in particular that, for the parameterization of the meso-scale chosen, concrete response at macro-scale almost is independent of the realization of the meso-structure.

axial loading (see figure 4): brittle behavior in tension, quasi-brittle behavior in compression with strain hardening, appearance of residual deformation, stiffness and strength degradations, hysteresis in unloading-reloading cycles. Besides, the model is designed for accommodating loading-rate dependent effects. To control these mechanisms altogether, 10 internal variables and a set of 13 parameters are required. Not all the capabilities of the model will be exploited thereafter in order to reduce the number of parameters to handle.

Damping model-free analysis of structural seismic response

Now, we want to compare this two concrete models regarding their capability of grasping the physical inelastic mechanisms occurring in structural time history inelastic seismic simulations. The question which is at stake here is about defining a strategy to have an objective comparison of both concrete material models. As will be illustrated in this section, in earthquake engineering problems where experimental data only is available from seismic tests, the answer to this issue is not that obvious because of the common addition of Rayleigh damping forces in the numerical model "to simulate the portion of energy dissipation arising from both structural and nonstructural components (e.g., cladding, partitions) that is not otherwise incorporated in the model" (FEMA P695, 2009, §6.4.4.). This latter "model" being here the inelastic structural elements that incorporate either of the concrete models presented in previous section.

Classical formulation of a nonlinear structural earthquake engineering problem

The classical formulation for time-history analysis of nonlinear structure in seismic motion reads, in its digitized form: At each time step t n , find the relative displacement, velocity with

r := f ext (t n ) (6) -Mü(t n ) + C(t n ) u(t n ) + f hys (u; t n )
where M and C are the mass and damping matrices while f hys and f ext are the inelastic structural response forces and the external (static and seismic) loading vectors.

The assumption of Rayleigh damping is generally made to calculate the viscous forces:

C(t n ) = α M + β K(t n ) (7)
where K = ∂f hys /∂u is the structural tangent stiffness matrix. Coefficients α and β most commonly are set once for all at the beginning of the simulation. Most commonly too, they are identified so that the same modal damping ratio ξ is generated at structural vibration modes I and J, with I = 1 and J defined so that 90% or 95% of the structural mass is activated by the first J modes. It has been acknowledged many times that the actual amount of seismic energy dissipated by Rayleigh damping throughout inelastic time-history analyses is difficult to control (see e.g. [START_REF] Jehel | Initial versus tangent stiffness-based Rayleigh damping in inelastic time history seismic analyses[END_REF]; [START_REF] Charney | Unintended consequences of modeling damping in structures[END_REF]; [START_REF] Hall | Problems encountered from the use (or misuse) of Rayleigh damping[END_REF]). Practically, this means that at a time t n during the analysis, there is some uncertainty in the value of the actual damping ratio: it can be twice or three times higher or lower than the targeted damping ratio initially set by the user.

The issue here is not to discuss the pros and cons of using Rayleigh damping, but to avoid the bias such a damping model can introduce when comparing two concrete models. Indeed, two different concrete models implemented in the same structural model yield two different resisting forces vectors and, consequently, two different tangent stiffnesses. Then, using two different concrete models -as well as two different meso-structures in the case of the multiscale stochastic concrete model -not only yields two different structural forces vectors (f hys ) but also two different damping forces vectors f dam = C u or, in other words, two different actual modal damping ratios ξ. Comparing the outputs of both simulations thus means comparing two different pairs of inelastic structural model plus damping model, not only two different structural models. Hence a comparison of the concrete laws that turns out to be biased by the uncertainty on the actual modal damping ratio.

Sensitivity of analysis outputs to viscous damping ratio

Then, a natural question is whether or not the outputs of interest in a structural seismic analysis are sensitive to the modal damping ratio generated by Rayleigh damping model. [START_REF] Lee | Seismic demand sensitivity of reinforced concrete shear-wall building using FOSM method[END_REF] run inelastic time history analyses of a reinforced concrete structure with beams, columns and walls modeled by fiber elements. A bilinear stress-strain (1D) relationship is used for the reinforcement steel and a modified Kent-Park stress-strain (1D) relationship with zero tensile strength for concrete fibers. Uncertainties are introduced in the seismic ground motion, in the strength and stiffness of concrete and steel, as well as in the modal damping ratio used to identify the Rayleigh damping coefficients α and β (ξ has truncated Gaussian distribution with a mean of 5% and a COV of 40%). The study reports that for some engineering demand parameters of interest for performance-based design, viscous damping ratio is the second source of uncertainty right after ground motion. A two-story two-bay reinforced concrete frame structure is modeled in [START_REF] Jehel | A look into uncertainty in structural seismic performance arising from additional Rayleigh damping in inelastic time history analysis[END_REF] with fiber elements accommodating bilinear stress-strain (1D) relationship for the steel rebars and the model introduced above in section 2.3 for concrete fibers. Viscous damping is added in the inelastic time history analyses with Rayleigh model and a random damping ratio uniformly distributed between 1% and 10%. The analysis shows in particular that, for some of the ground motions used, the maximum inter-story drift ratio can be significantly sensitive to the Rayleigh damping ratio.

Ideally, one wants to be able to compare the capabilities of two different concrete material models to grasp experimentally observed structural behaviors in a way that is independent of any side effect coming from uncertain viscous damping ratio. However, in practice, for earthquake engineering applications, one has generally only access to experimental data from shaking table tests, that is data that one attempts to predict numerically with the addition of uncertain damping forces. The concept of discrepancy forces is introduced right below as a way to circumvent this issue.

The discrepancy forces as an objective approach for structural model comparison in seismic loading

The concept of discrepancy forces in computational dynamics has been introduced in Jehel (2014) and is briefly presented here. Suppose one has a N -degree-of-freedom (DOF) FE model of a structure that is tested on a shaking table. Then, suppose both the relative displacement and relative acceleration time histories are recorded during the experimental test at N e ≤ N DOFs of the FE mesh of the structure and gathered in vectors y e (t) and ÿe (t) of size N e each. Then, the discrepancy forces are defined as the forces needed in the system to satisfy the dynamic balance equation without the uncertain damping forces vector. Sorting the DOFs so that the N e free DOFs that are monitored during the experimental test as well as the N b DOFs that are controlled by imposed boundary conditions are gathered, it comes (see equation ( 6)):

  f dis,e (t) f dis,f (t) f dis,b (t)   :=   f ext,e (t) f ext,f (t) f ext,b (t)   -M   ÿe (t) ÿf (t) ÿb (t)   -   f hys,e (y e , y f , y b ; t) f hys,f (y e , y f , y b ; t) f hys,b (y e , y f , y b ; t)   (8)
In an ideal situation, the displacement and acceleration time histories would have been recorded for all the DOFs of the FE mesh. Then, f hys (y e ; t) could be computed in a quasi-static nonlinear structural analysis, and then, because M and f ext are known a priori, the discrepancy forces could all be calculated according to equation ( 8). However, in practice, there is usually not such an amount of data recorded and, to compute f dis , some other assumptions have to be made to cope with this lack of knowledge on the experimental response of the structure. A practical case where the dynamic effects for the N f DOFs not monitored during shaking table test are neglected is presented in [START_REF] Jehel | A critical look into rayleigh damping forces for seismic performance assessment of inelastic structures[END_REF] as well as, in a simpler case though, in the next section.

Once the discrepancy forces are computed, they provide information on the capability of the structural model to represent the experimental response of the structure or, in other words, the forces that should be added in the system to account for the mechanisms either not or poorly represented by the inelastic structural response f hys . Discrepancy forces are independent of the damping model and consequently do not carry the uncertainty this latter brings in the analysis.

Numerical application

Shaking table test of a frame structure

Experimental data recorded during the shaking table test of a half-scale ductile momentresisting reinforced concrete frame is used (see figure 5). Four concrete blocks were used to simulate concentrated gravity loads in every beam span. The N04W component of the ground motion recorded in Olympia, Washington on April 13, 1949 was selected for the test program. Tests have been carried out for two analogous structures designed with two different assumptions: either a shear force reduction factor R = 2 or R = 4. This latter case is considered in this paper. The detailed presentation of the test and design assumptions can be found in [START_REF] Filiatrault | Seismic performance of ductile and nominally ductile reinforced concrete moment resisting frames. I. Experimental study[END_REF].

Recorded data during shaking table test are the horizontal displacement and acceleration time histories at both floors of the frame, along with the acceleration of the shaking table (seismic ground motion).

Numerical modeling assumptions

Finite element structural model

Fiber element is used to model one column of the 2D frame structure (see mesh in figure 6). Column is modeled by two identical elements.

Element sections are divided into 8 layers (8×1 fibers). Column is fixed at its base; at its top, 1st-floor horizontal displacement time history recorded during shaking table test is applied while vertical DOF is left free and rotation is assumed to be null. This latter assumption is expected to introduce some error in the results because, during shaking table test, the structural joint rotated. Rotation has not been measured though, hence this hypothesis.

Reinforcement steel behavior model

Identification of the concrete models

Uniaxial compression tests have been carried out on specimens of the concrete used to build the frame. According to these data, we use here Young modulus C c = 27.5 GPa and ultimate compression stress Σ ′ c = 27.75 MPa. In the absence of information on the response of confined concrete, we model all the concrete fibers as unconfined, although enhanced ductility properties generally are associated to the fibers inside the stirrups. Besides, in the absence of experimental data in cyclic loading, the identification is performed so that only the backbone curve of the numerical response fits the experimental data.

(i) For the multi-scale concrete model, referred to as M 1 in the following, kinematic hardening coefficient H p at mesoscale is uniform and taken as null. Indeed, this parameter controls the tangent modulus as deformation E becomes large and, concrete tangent modulus can take zero value before softening (softening phase is not modeled here). Then, the remaining parameters to be identified are the mean µ (ii) Concerning the concrete model presented in section 2.3 above, hereafter referred to as M 2 , a simplified version of it is used here so that only two parameters have to be identified too. As illustrated in figure 9 [bottom], concrete compressive response is assumed to be first elastic (slope C c ); then, as the stress reaches Σ f , continuum damage is activated and its evolution is controlled by parameter K d ; finally, once Σ ′ c has been passed, plasticity is activated and the stress remains equal to Σ ′ c . Accordingly, (Σ f , K d ) is the set of parameters to be identified.

Identification

is performed numerically using Monte Carlo simulations for Bayesian inference based on a random walk Metropolis-Hastings sampling algorithm [START_REF] Robert | Monte Carlo Statistical Methods[END_REF]. Note that M 2 is not a trilinear model because of the continuum damage model (see e.g. [START_REF] Ibrahimbegovic | Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods[END_REF] for details on the numerical implementation of continuum damage models). The following parameters have been identified as the expected a posteriori (expectation of the posterior parameter distribution): for M 1 , µ = 30.5 MPa and s = 28.8 MPa, and for model M 2 , Σ f = 16.32 MPa and K d = 14.4 GPa. How the numerical models match the experimental data is shown in figure 9. From these pictures, one can draw the observations gathered in table 1. We recall that a simplified version of model M 2 is used here and that enhanced performance can be expected from the full version of M 2 .

Discrepancy forces

Two models for the analyzed column (see figure 6) are built with same steel behavior law and both concrete laws M 1 and M 2 introduced in section 2 with the coefficients identified in section 4.2.3. The recorded time histories during shaking table test consist in the horizontal relative displacement (y e ) and relative acceleration (ÿ e ) at the top of the analyzed column, along with the shaking table acceleration (ü g ). Resisting forces f hys in the column are computed solving the following quasi-static nonlinear problem:

  0 0 0   :=   f sta,e + F f sta,f f ext,b   -   f hys,e (y e , y f , 0) f hys,f (y e , y f , 0) f hys,b (y e , y f , 0)   (9)
where the displacement y e is applied to the structure, F is the reaction to this imposed displacement and f sta is the static part of the external loading. Static loading is applied before and kept constant during seismic motion. Due to a lack of recorded data, we assume that dynamic effects for the vertical and rotational DOFs at the top of the column can be neglected, that is ÿf = 0. Finally, using the experimentally recorded accelerations ÿe and üg , the discrepancy forces f dis are computed: introducing equations (9) in equations ( 8) along with the assumption ÿf = 0, it comes:

  f dis,e f dis,f f dis,b   :=   f sta,e -M üg f sta,f f ext,b   -   M ÿe 0 0   -   f sta,e + F f sta,f f ext,b   (10) Accordingly, f dis,f = f dis,b = 0 and f dis,e = -M (ÿ e + üg ) -F (11)
f dis,e provides a damping model-free quantity that can be used as a measure of the capacity of the column model to represent the behavior that has been experimentally observed during shaking were capable of perfectly representing the experimental response of the analyzed column, discrepancy forces would be null. The lack of experimental information about the vertical and rotational DOFs introduces some uncertainty in the computation of f dis,e , which unlike the uncertainty brought by the damping model in a direct time history seismic analysis could be tackled by using complementary experimental data. Discrepancy force f dis,e is computed for the column modeled either with concrete model M 1 or M 2 . Figure 10 shows F (t) and f dis,e (t) for both models. One can observe that, in the structure with model M 1 , both the reaction and the discrepancy forces are lower than in the structure with model M 2 . Theoretically, the lower the discrepancy forces, the more accurate the structural model, which leads to conclude that model M 1 better grasps the actual physical mechanisms in concrete than model M 2 . Nevertheless, this conclusion is counterbalanced by the other fact that these forces have been computed according to some hypotheses that are likely to introduce errors in the results, such as zero-rotation at the top end of the column. With additional data collected during the shaking table test, these errors could however be removed. Note finally that, very roughly, the computation of the reactions F (t) with model M 2 is a matter of less than a minute on standard laptop with 2.53 GHz processor whereas it takes several but less than ten minutes with model M 1 .

Conclusions

In this paper, the capability of a fiber element with an inelastic stochastic multi-scale nonlin- ear concrete material model to numerically represent the seismic response of a reinforced concrete column observed during shaking table test is investigated. For the sake of comparison, same analysis is carried out using a more classical inelastic concrete model with internal variables. The results of such an investigation can be biased by the uncertainty brought by the damping forces added in inelastic time history seismic analyses. The concept of discrepancy forces for computational mechanics problems allows for removing this uncertainty. Nevertheless, due to a lack of recorded experimental data, some assumptions have to be made before calculating the discrepancy forces, which also introduces errors in the analysis. However, these latter errors, unlike those coming from uncertain damping forces, could be removed with complementary experimental measures.

The stochastic multi-scale nonlinear concrete material model used in this work simulates a uniaxial behavior at macro-scale that results from the homogenization of a spatially variable response at meso-scale. This spatial variability is conveyed by the fluctuations of a 2D homogeneous non-Gaussian random field. Some parameters of the digitized random field are set so that response at macro-scale is almost independent of the realization of the mesostructure. The mean and standard deviation of the marginal log-normal distribution selected for generating the random fields are identified with Bayesian inference along with MCMC simulations to fit experimental concrete uniaxial compressive response.

The numerical simulations presented in this paper lie in-between the two following bounds, both of which are related to the need for further experimental investigation:

(i) At small scales, interaction with material scientists is necessary to collect experimental evidence that would support the choices made to account for the heterogeneous nature of concrete (correlation function, inelastic response at mesoscale, marginal distribution, random parameter,...).

(ii) At large scales, interaction between experts in computational mechanics and experts in the design of experimental seismic tests is necessary for experimental protocol and finite element model to be jointly designed at an early stage in the analysis process, so that there is sufficient experimental data recorded to compute the discrepancy forces at a maximum of the DOFs of the finite element model. First author is supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme (proposal No. 275928). Second author is supported by the French Ministry of Higher Education and Research as a pupil of ENS Cachan. Both supports are gratefully acknowledged. First two authors also thank Prof. Deodatis at Columbia University for hosting them during the preparation of this work.

  -Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June[15][16][17][18] 2014

  Damping model-free analysis of a stochastic multi-scale frame element for earthquake engineering P. Jehel, H. Vincent, and T. Rodet about a computer program that would explicitly account for the full range of concrete microstructures in structural engineering applications. As a trade-off, we propose an approach based on the following set of assumptions (we refer the reader to[START_REF] Jehel | On damping created by heterogeneous yielding in the numerical analysis of nonlinear reinforced concrete frame elements[END_REF] for further details):

Figure 2 :

 2 Figure 2: Compressive cyclic behavior at meso-scale. The yield stress σy(x) fluctuates over the concrete representative section R: two local responses at two distinct material points x1 [left] and x2 [right] are represented in the figure.

-

  10-02˙CSM7˙PJ-HV-TR" -2014/10/2 -18:34 -page 5 -Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June 15-18, 2014

Figure 3 :

 3 Figure 3: [top] Mean (-) along with mean plus andminus standard deviation (--) response at macro-scale obtained from a sample of 2,000 meso-structures. [bottom] One realization of concrete meso-structure. Targeted mean and standard deviation of the marginal log-normal law are µ = 35.0 MPa and s = 51.9 MPa. This figure shows in particular that, for the parameterization of the meso-scale chosen, concrete response at macro-scale almost is independent of the realization of the meso-structure.

Figure 4 :

 4 Figure 4: Stress (Σ)-strain (E) concrete response at macro-scale obtained using the model developed in Jehel et al. (2010) in the framework of thermodynamics with internal variables.

  Damping model-free analysis of a stochastic multi-scale frame element for earthquake engineering P. Jehel, H. Vincent, and T. Rodet and acceleration fields (u, u, ü) such that r (u, u, ü; t n ) = 0(5)

  -Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June[15][16][17][18] 2014

  Figure 5: RC frame structure tested on the shaking table at École Polytechnique Montreal. The frame is 5-meter wide (2 × 2.5 m) and 3-meter high (2 × 1.5 m). Each beam supports additional masses to account for service static loads.

Figure 6 :

 6 Figure 6: In black: analyzed column of the frame. M concentrates a quarter of the total mass of the frame structure.

  Young modulus C s = 224.6 GPa, yield stress Σ y = 438 MPa and ultimate stress Σ u = 601 MPa have been experimentally measured during uniaxial tests on longitudinal steel rebars. An elasto-plastic model with kinematic hardening is used to represent steel response in cyclic loading. The model implemented according to these latter parameters is shown in figure 7.

Figure 7 :

 7 Figure 7: Steel longitudinal rebar constitutive law.

  -Proc. of the 7th International Conference (CSM-7) G. Deodatis and P.D. Spanos (eds.) Santorini, Greece, June 15-18, 2014 and standard deviation s of the marginal log-normal law used to generate the random meso-structures. The effects of those two parameters on the response at macroscale are illustrated in figure8.

Figure 8 :

 8 Figure 8: Macro-scale concrete response for different targeted values for the mean µ and standard deviation s of the random field: [top] µ = 24.5 (--), 35.0 (-), 45.5 (• • • ) MPa and s = 51.9 MPa; [bottom] µ = 35 MPa and s = 35.0 (--), 51.9 (-) and 78.3 (• • • ) MPa.

  Damping model-free analysis of a stochastic multi-scale frame element for earthquake engineering P.Jehel, H. Vincent, and T. Rodet

Figure 9 :

 9 Figure 9: Dashed line represents the experimental backbone curve; there is no experimental data available for the unloading-reloading cycles. For same loading path: [top] Fitted model M1 (mean of 2,000 meso-structures); [bottom] Fitted model M2.

Figure 10 :

 10 Figure 10: Black is for the column with concrete model M1, gray with model M2. Horizontal reactions with imposed relative displacement y e [top] and discrepancy forces pertaining to the horizontal DOF [bottom] at the top of the column.

Table 1 :

 1 table test. If the structural model Comparative analysis of the capability of models M1 and M2 to reproduce salient features of the compressive cyclic response of a concrete specimen. "Local hysteresis" is used for "hysteresis in unloading-reloading cycles".

	Feature	M 1	M 2
	Backbone curve accurate fair
	Local hysteresis	yes	no
	Loss of stiffness	no	yes
	Residual def.	yes	yes

Acknowledgements


Emmanuel Bertrand

Xuan Mi Meyer

Elizabeth Machado-Maturana

Barbara Rega

Anne-Sophie S Guillard

Alain Kondjoyan

J L Berdague

Jean-Louis Berdagué

Multi-response Modelling of the Maillard reaction in a model cheese

Keywords:

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

MULTI-RESPONSE MODELING OF THE MAILLARD REACTION IN A MODEL CHEESE

E. BERTRAND 1, 2 , X.M. MEYER 3,4 , E. MACHADO-MATURANA 1 , B. REGA 5,6 , A.S. GUILLARD 2 , A.KONDJOYAN 1 , J.L. BERDAGUE Processed cheese derives from a secondary milk processing step that involves mixing and heating dairy (cheese, butter and milk powders) and non-dairy products (emulsifiers). This processing yields a homogeneous product, usually spreadable, with a shelf-life often longer than 6 months. During cheese processing and storage, lipid oxidation, caramelization and Maillard reactions occur and produce odour-active compounds. In this study, a methodological approach was used in order to (i) identify odorants responsible for flavor attributes or compouds involved in the reaction chain, (ii) monitor the evolution of these markers during the heat treatment applied to the matrix, (iii) establish an observable reaction scheme and (iv) model and predict the evolution of these compounds during thermal operations. In this aim, a model cheese and a cooking cell were elaborated. Various couplings of gas chromatography with olfactometry were used to identify odorous compounds. Twodimensional comprehensive chromatography allowed a semi-quantitation of trace and ultratrace compounds, while precursors were quantitated by high performance liquid chromatography. An observable reaction scheme of the Maillard reaction was extracted from these data and makes the multi-response modeling step possible despite a partial quantitation of the volatile compounds. Finally, we obtained a formal model combining 19 components (including four odorants) connected by 14 stoechiometric balanced reactions. This model makes it possible to predict the evolution of these components depending on the initial content of lactose, galactose and according to the heat treatment applied to the cheese matrix.

This work was carried out with the financial support of the ANR-Agence Nationale de la Recherche -The French National Research Agency under the Programme National de Recherche en Alimentation et nutrition humaine , project ANR-06-PNRA-023REACTIAL "Prediction and control of the appearance or disappearance of reactional markers during food process and conservation " . ; observable reaction scheme; multi-response modeling;