C Brugger

T Palin-Luc

P Osmond

M Blanc

A new ultrasonic fatigue testing device for biaxial bending in the gigacycle regime

Keywords:

A new fatigue testing device has been developed to test specimens under biaxial loading at 20 kHz. The specimen is a flat smooth disc. It is placed on a torus frame and cyclically loaded in compression at the center of its upper face. Consequently disc bending generates a biaxial proportional stress state at the center of the specimen lower face. Any positive loading ratio can be applied. This device has been tested and is well functioning on specimens made of a cast aluminum alloy used to produce cylinder heads. Preliminary results in VHCF regime are compared with literature results obtained under similar stress state but in HCF regime and at 20 Hz only.

Introduction

Many components in several industries are loaded in very high cycle fatigue (VHCF) regime, either at high frequency (wheels of high speed trains, blades in aircraft turbojet engines, etc.) [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF] or at low frequency during decades (artificial heart, mooring chains for off-shore petroleum platforms, etc.) [START_REF] Palin-Luc | Fatigue crack initiation and growth on a steel in the very high cycle regime with sea water corrosion[END_REF][START_REF] Perez-Mora | Very high cycle fatigue of a high strength steel under sea water corrosion: a strong corrosion and mechanical damage coupling[END_REF]. Testing specimens up to 10 9 or 10 10 cycles in a realistic time requires to use a very high loading frequency (20 or 30 kHz). Several devices using the ultrasonic testing technique have been developed all over the world since Mason's work in the 1950s [START_REF] Mason | Piezoelectric crystals and their application in ultrasonics[END_REF]. A significant effort in this field has been done since the end of the last century [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF][START_REF] Bathias | Piezoelectric fatigue testing machines and devices[END_REF]. Specimens can be tested under tension (R = À1 or R > 0), torsion (R = À1 or R > 0) or bending (R > 0), either at room, low or high temperature, in air or in liquid environment [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF][START_REF] Palin-Luc | Fatigue crack initiation and growth on a steel in the very high cycle regime with sea water corrosion[END_REF][START_REF] Perez-Mora | Very high cycle fatigue of a high strength steel under sea water corrosion: a strong corrosion and mechanical damage coupling[END_REF][START_REF] Mason | Piezoelectric crystals and their application in ultrasonics[END_REF][START_REF] Bathias | Piezoelectric fatigue testing machines and devices[END_REF], but up to very recently there was no machine for testing specimens under multiaxial loading. However, it is known that during their life many industrial components are submitted to multiaxial loadings that may lead to a number of cycles close to one billion or more. Fatigue cracks may initiate in areas experiencing multiaxial stress states. That is the reason why a new fatigue testing device has been developed to test specimens under biaxial loading at 20 kHz.

After presenting a brief state of the art on ultrasonic fatigue testing machines, the principle of a new biaxial bending device and the associated stress state are presented hereafter. The preliminary results obtained on a cast Al-Si alloy in VHCF regime are then presented and compared with those obtained in literature under similar stress state but in HCF regime and under lower loading frequency.

State of the art: ultrasonic fatigue testing machines

According to the literature, the first ultrasonic fatigue testing device was developed by Mason in the 1950s under fully reversed tension [START_REF] Mason | Piezoelectric crystals and their application in ultrasonics[END_REF][START_REF] Mason | Ultrasonic fatigue[END_REF]. An interesting review of ultrasonic fatigue testing machines is given by Bathias in [START_REF] Bathias | Piezoelectric fatigue testing machines and devices[END_REF]. The basic principle of an ultrasonic fatigue testing machine is to apply to a specimen an axial sinusoidal displacement at an ultrasonic frequency (typically 20 kHz). The specimen is designed so that it has a natural frequency (or mode) at this frequency. An ultrasonic fatigue testing machine is made with: (i) a generator applying an electric sinusoidal signal (at 20 kHz) to (ii) a piezoelectric converter that converts this electric signal in a longitudinal vibration at the same frequency, and (iii) a horn for amplifying the vibration finally applied at one end of the specimen. The generator is controlled by a computer so that the resonance of the whole system (piezo-electric converter, horn and specimen) is kept all the test long together with the demanded displacement amplitude.

This principle has been used by several authors to design special apparatuses for testing specimens under fully reversed tension. Coupled with electromechanical or servo-hydraulic testing machines, such equipment can be used for gigacycle fatigue tests under tension with several positive R ratios. A machine has been developed for three point bending test with R > 0 too [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF]. All these machines allow tests on smooth or notched specimens under uniaxial stress state. Some authors [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF][START_REF] Stanzl-Tschegg | High frequency method for torsion fatigue testing[END_REF][START_REF] Mayer | Ultrasonic torsion and tension-compression fatigue testing: measuring principles and investigations on 2024-T351 aluminium alloy[END_REF] have also developed torsion testing machines working like uniaxial ones, by pulse and pause [START_REF] Stanzl-Tschegg | High frequency method for torsion fatigue testing[END_REF][START_REF] Mayer | Ultrasonic torsion and tension-compression fatigue testing: measuring principles and investigations on 2024-T351 aluminium alloy[END_REF] or continuously [START_REF] Nikitin | A new piezoelectric fatigue testing machine in pure torsion for ultrasonic gigacycle fatigue tests: application to forged and extruded titanium alloys[END_REF]. Furthermore, the ultrasonic testing technique can be used for testing specimens at room temperature with air cooling if needed, or at high temperature [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF][START_REF] Wagner | Ultrasonic fatigue tests at high temperature on an austenitic steel[END_REF], in cryogenic environment [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF] or in corrosive liquid environment [START_REF] Palin-Luc | Fatigue crack initiation and growth on a steel in the very high cycle regime with sea water corrosion[END_REF][START_REF] Perez-Mora | Very high cycle fatigue of a high strength steel under sea water corrosion: a strong corrosion and mechanical damage coupling[END_REF]. But these testing machines apply a uniaxial loading on the specimen.

Vieira, De Freitas et al. [START_REF] Vieira | Development of a very high cycle fatigue (VHCF) multiaxial testing device[END_REF] recently presented a horn and specimen design allowing for fully reversed proportional tensiontorsion ultrasonic testing. For a given load ratio, diagonal hollows are machined in the horn, so that it turns a longitudinal vibration into a coupled longitudinal and torsional one. In this case, the specimen has a triple hourglass geometry, so that the first natural frequency for tension matches the third one for torsion. Only the central neck of the specimen is submitted to both tension and torsion.

New ultrasonic biaxial bending device

Principle

The new fatigue testing device presented hereafter is designed for testing in bending under ultrasonic loading frequency a flat smooth specimen with a disc geometry [START_REF] Blanc | French patent N FR1357198[END_REF]. Its principle is similar to the testing apparatus proposed by Koutiri et al. [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF][START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF] but this last one was mounted on a servo-hydraulic testing machine working around 20 Hz only. The specimen is placed on a frame with a torus ring, so that the contact zone between the lower face of the disc and the frame is a circle. A compressive loading is applied at the center of the upper face using a hemispherical indenter (Fig. 1a). Like in a three point bending test, this leads to the bending of the disc.

Using an electromechanical testing machine and an ultrasonic loading device, both a static load and a sinusoidal displacement (at 20 kHz) are applied at the center of the specimen. Under the common assumption that the material remains macroscopically elastic in gigacycle fatigue regime, any positive loading ratio can be applied. In practice, to assure uninterrupted contact between specimen and indenter, loading ratios very close to zero are avoided; R > 0.05 is recommended.

The ultrasonic loading device, partly illustrated in Fig. 1b, is classic [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF]. It consists of a 20 kHz electric generator, a piezoelectric converter, a booster and a horn to amplify the sinusoidal axial displacement like in 3 point ultrasonic bending [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF][START_REF] Bathias | Piezoelectric fatigue testing machines and devices[END_REF]. In order to apply a non-zero mean load, this device is attached to an electromechanical testing machine using bars and hollowed discs fastened to the center of the booster, which is a vibration node. Finally, a servocontrol system adjusts in real time both the amplitude of displacement and the loading frequency to match the natural frequency of the whole device. For that reason, each part (booster, horn and specimen) must be carefully designed, so that their natural frequency for axial displacement matches 20 kHz. The next section details, for some components, how their geometry was determined by modal analysis with finite element analysis software.

Geometry of the specimen and device

First, in order to perform a modal analysis, both the density and the dynamic modulus (at 20 kHz) of the tested material are experimentally measured by using a cylindrical bar as explained in [START_REF] Bathias | Gigacycle fatigue in mechanical practice[END_REF] for designing tension-compression specimens. The specimen geometry (Fig. 2a) is described by only two parameters: diameter and thickness of the disc. These parameters are determined iteratively using a free-free modal analysis computed with a commercial FEA software. The ideal geometry corresponds to a first natural frequency -associated with biaxial bending (Fig. 2b)equal to 20 kHz. Additionally, other natural modes have to be far enough from 20 kHz (and multiples) to avoid the risk of parasite vibrations. In practice we recommend at least ±2 kHz from 20 kHz. For a given stress state at the center of the lower face, contact forces rapidly increase with thickness. On the other hand, since the hemispherical indenter is located at the center of the upper face, the stress state at the center of the lower face (where cracks will initiate) might be disturbed by the loading if the disc is too thin. For the application described in the next section, a compromise has been found by fixing the thickness equal to 6 mm. The radius of the circular ring in Fig. 1a is given by the location of the vibration nodes on the specimen (Fig. 2b). This choice minimizes the relative displacement between the specimen and the frame, then the frictional-induced heating. Please note that dimensions in Fig. 2a were determined for the cast Al-Si alloy tested in the next section. These dimensions remain valid for other Al-Si alloys with slightly different density and dynamic modulus, but we recommend to perform the modal analysis again for every new material.

Associated stress state

Experimental analysis of the stress state

Theoretically, disc bending generates an equi-biaxial proportional stress state at the center of the specimen's lower face, and the stress level is proportional to the center's displacement [START_REF] Koutiri | Effect of high hydrostatic stress on the fatigue behavior of metallic materials[END_REF]. Three calibration specimens were instrumented with strain gauge rosettes glued in the center of the lower face. Tests were performed for different amplitudes of the displacement, with a static load assuring a positive load ratio. Three centering pins are used to keep the specimen centered on the frame ring while the static load is applied. In order to avoid friction during cyclic loading, they are removed before applying the sinusoidal displacement. After measuring strain amplitudes using both a wide band conditioning device (Vishay 2210) and a high speed data recorder, stresses amplitudes were computed assuming an isotropic linear elastic behavior of the material (because of testing conditions in the VHCF regime). Since the results are almost proportional to the displacement, Table 1 summarizes the results on 3 specimens for a given 10 lm amplitude of displacement at the center of the specimen.

Considering the uncertainties related to the experimental mea-surements (location of the strain gauges, gauge factors, etc.), stress state can be considered equi-biaxial.

Reproducibility of the stress state

During a 10 9 cycles run-out test, the static force applied by the electro-mechanical testing machine is almost constant (±0.2%; Fig. 3a). The resonance frequency of the whole device slightly decreases then stabilizes (À0.2%; Fig. 3b), possibly due to the specimen's heating or to an evolution of the boundary conditions (penetration of the hemispherical indenter; compression of the PTFE layer between the specimen and the circular ring, see Section 4.1). The displacement amplitude can be monitored but not recorded. However, the accuracy on displacement amplitude should be better than the accuracy on strain amplitudes measured using strain gauges under ultrasonic loading, which is lower than 2% (10 lm/ m). We can conclude that the loading applied to the specimen is stable, with a good reproducibility and a good accuracy.

Application to a cast Al-Si alloy

This ultrasonic biaxial fatigue testing device has been tested and validated on a cast aluminum alloy used to produce cylinder heads and previously investigated by Koutiri et al. [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF][START_REF] Koutiri | Effect of high hydrostatic stress on the fatigue behavior of metallic materials[END_REF]. Since cast materials may contain casting defects (porosities, shrinkages, etc.), and because cylinder heads are submitted to high hydrostatic pressure loadings during a very high number of loading cycles, a safe fatigue design requires to determine the fatigue strength of this material under a similar stress state in the gigacycle regime.

Table 1

Stresses at the center of the lower face of the specimen for an amplitude of displacement of 10 lm. Nevertheless, the objective here is only to validate the new testing device.

Material and specimen

The material is the cast AlSi7Cu05Mg03 T7. Its conventional yield stress is 250 MPa [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF][START_REF] Koutiri | Effect of high hydrostatic stress on the fatigue behavior of metallic materials[END_REF]. The specimen's geometry and microstructure are illustrated in Figs. 2a and4a respectively. Specimens were machined from cast cylinder heads. In order to get enough material volume, cores were diminished prior to casting (Fig. 4b). This allows for microstructure parameters similar to real components (in terms of dendrite arm spacing or DAS, porosities, and hardness), even if Koutiri [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF] showed that diminishing casting cores leads to larger porosities. The circular ring has a 34 mm diameter. In order to reduce friction, a 0.4 mm thick PTFE layer is fixed on the ring using polyimide tape.

Testing conditions and fatigue test results

Fatigue tests were performed in air, at room temperature, with R = 0.1 load ratio. Cyclic loading is stopped after 10 9 cycles, or when the loading frequency drops down to 19,500 Hz due to fatigue crack initiation and propagation. Both static load and sinusoidal displacement levels were determined using calibration specimens following the procedure previously described in Section 3.3.1.

Since ultrasonic loading may generate self-heating, specimen cooling is necessary. To cool the specimen, dry compressed air flow was orientated with an air gun towards the specimen upper face (Figs. 1b and5a). In order to quantify self-heating, surface temperature has been measured using an infrared camera (FLIR SC 7000) during some tests for which the specimen was previously mat black painted for ensuring a high emissivity (Fig. 5a). It was not possible to place the camera perpendicular to a specimen surface, because both upper and lower faces of the specimen are obstructed (by the ultrasonic loading device and by the frame, Fig. 1b). However, for measuring relative temperature variation, DT, compared with the initial room temperature, such configuration can be used. It was not possible either to observe the lower face of the specimen, where self-heating associated with cyclic loading should be the larger. But according to the high thermal conductivity, temperature is sufficiently homogeneous. Indeed, although the compressed air flow is directed towards the specimen upper face, the area labelled 1 in Fig. 5a was only 2-3°warmer than the one labelled 2.

Temperature was averaged on the areas labelled 1 and 2 in Fig. 5a. After 2.5 Â 10 6 cycles, the mean temperature increasecompared to room temperature -almost stabilizes at 42-52-57 °C for maximal stresses equal to 130-140-150 MPa respectively. Self-heating is more pronounced than during tensioncompression ultrasonic testing due to the much larger specimen volume. However, the precipitation strengthened Al-Si alloy tested in this study is in T7 condition, which involves a final heat treat- ment at 200 °C during 5 h. For a maximum stress equal to 140 MPa (leading to the gigacycle regime, see Fig. 6), the specimen temperature reaches 75 °C during 14 h, the duration of a test up to 1E9 cycles at 20 kHz. Such temperature should be low enough to be neglected compared to the final heat treatment.

The first fatigue test results are illustrated in Fig. 6. Additional results obtained in HCF regime for the same material and similar stress state but at 20 Hz [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF][START_REF] Koutiri | Effect of high hydrostatic stress on the fatigue behavior of metallic materials[END_REF] are also presented for comparison purpose.

Discussion

For maximum stress levels equal or lower than 140 MPa, the first results obtained with the new ultrasonic biaxial testing device show a longer fatigue life than the results by Koutiri et al. [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF][START_REF] Koutiri | Effect of high hydrostatic stress on the fatigue behavior of metallic materials[END_REF]. The number of tested specimens at 20 Hz and 20 kHz is too low for being statistically representative. Nevertheless, in addition to the temperature effect discussed in the previous section, and to a possible frequency effect, various sources of discrepancy were identified on these preliminary results. It has to be pointed out that we used a hemispherical indenter whereas Koutiri et al. used a ring indenter generating a constant stress state in a 10 mm diameter disc of the lower face. Consequently the highly stressed volume is larger on the tests at 20 Hz by Koutiri et al. than in our case at 20 kHz. The fatigue life is thus a little bit shorter at 20 Hz because the probability to find a critical defect is higher.

In order to estimate the highly stressed volume of material (submitted to 90% of the maximum normal stress), disc bending was simulated using a commercial FEA software, assuming an isotropic and homogeneous defect-free material with a linear elastic behavior and no sliding between the frame ring and the specimen lower face. Fig. 7 shows the radial (r rr ) and circumferential (r hh ) stresses in the median section of the disc. Zones with a uniform grey color are under negative stress. Color bands show positive stress values by 5% intervals. The maximum values are identical and located at the center of the lower face (where the stress state is then equi-biaxial), but the two profiles differ. The radial (r rr ) and circumferential (r hh ) stresses are 10% lower on a 3 mm and 4 mm diameter disc, respectively. Along the specimen thickness, the two stresses decrease by 10% after 0.25 mm. This value is related to the disc thickness (6 mm) and should then be slightly larger in this study than in the one by Koutiri et al., who used 5 mm thick discs. Even so, the highly stressed volume is smaller in our case. This highly stressed volume difference seems to counterbalance the dif- ference in fatigue life compared to the Koutiri's results, leading to similar median fatigue strengths.

In term of stress amplitude, the median fatigue strength at 10 9 cycles is close to 63 MPa (corresponding to a maximal stress equal to 140 MPa). Fig. 8a illustrates, in a Dang-Van diagram [START_REF] Dang-Van | Biaxial and multiaxial fatigue[END_REF], both the experimental median fatigue strengths at 2 Â 10 6 cycles obtained by Koutiri [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF] on smooth specimens made in the same cast Al alloy and the threshold line at 2 Â 10 6 cycles identified from torsion (R = À1) and tension (R = 0.1) fatigue test data on smooth specimens tested at classic low frequency. Furthermore, the loading paths corresponding to existing ultrasonic fatigue testing machines are shown: torsion (R = À1), tension (R = À1), tension or 3 points bending (R > 0). The loading path corresponding to the specimens tested at the stress level corresponding to 10 9 cycles with the new ultrasonic biaxial testing device presented here is illustrated. Even if the results of this study are preliminary data only, it is clear that these results allow us to question the assessment of the fatigue strength given by the Dang-Van criterion for high hydrostatic stress states even at 2 Â 10 6 cycles. Indeed, our experimental fatigue strength at 10 9 cycles under biaxial stress state is above this threshold line (Fig. 8a). The same conclusion is valid for the Crossland criterion too [START_REF] Crossland | Int. Conf. on Fat. of Metals[END_REF], as shown in Fig. 8b. Furthermore one has to point out that additional studies have to be carried out to propose an efficient criterion for multiaxial fatigue strength assessment in VHCF regime.

For maximum stress levels equal or greater than 150 MPa (Fig. 6), another difference between experimental procedures must be accounted for: the stop criterion. Indeed, the test results presented in this paper were stopped when the resonance frequency decreased from about 19,900 Hz to 19,500 Hz. Such loading frequency drop is due to a rigidity loss associated to a large macroscopic fatigue crack propagation. Indeed, macroscopic cracks are either unique or branched but always extended almost to the frame ring when the test stops (Fig. 9). Their length at the specimen surface is thus around 17 mm. Additional investigations are needed to quantify the number of cycles associated with this propagation, but first observations indicate it might exceed 10 7 cycles. However, one can note that 10 7 cycles represent 1% of 10 9 cycles only. The stop criterion used by Koutiri et al. leads to 6 mm long macroscopic fatigue cracks, and then to a shorter fatigue life.

Fractographic analyses

After ultrasonic fatigue testing, the cracked specimens are not broken in two parts (Fig. 9a). For an easy observation of the fatigue crack path, each specimen was fractured under quasi-static monotonic loading. To do that, the disc was placed between a circular ring and a hemispherical indenter (like on the ultrasonic testing machine, but with a larger diameter) and loaded in compression under displacement control using an electromechanical classic testing machine. When macroscopic fatigue crack is unique (Fig. 9a), quasi-static loading generates two new cracks, and specimen is finally broken in four parts (Fig. 9b).

Koutiri [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF] observed multiple initiation sites on the fracture surfaces of disc specimens tested in bending in HCF regime, unlike specimens tested under uniaxial loading. We made the same observation in VHCF regime: at least 6 initiation sites are visible on the fracture surfaces of the specimen presented in Fig. 10. However, we could not determine in which order crack initiations occurred. As it can be seen in Fig. 9, the surface quality of the specimens used for these preliminary tests was poor. Additional tests on polished specimens are required to clarify whether multiple crack initiation is due to the surface quality or to the multiaxial stress state.

Koutiri et al. [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF][START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF] also showed that the tested cast aluminum alloy contains relatively small ($100 lm) casting defects (shrinkages, pores), and that most of the fatigue cracks initiate on these casting defects. Indeed, such defects are responsible for fatigue crack initiation both in HCF and in VHCF regime. This can be seen in Fig. 11 where two fatigue crack initiations are very close to shrinkages with sizes close to 100 lm.

Finally, Koutiri et al. [START_REF] Koutiri | Effet des fortes contraintes hydrostatiques sur la tenue en fatigue des matériaux métalliques[END_REF][START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF] and Le et al. [START_REF] Le | Multiaxial high cycle fatigue damage mechanisms associated with the different microstructural heterogeneities of cast aluminium alloys[END_REF] also observed some crack initiations on eutectic silicon particles, oxide films and iron based intermetallic particles. Fig. 12 illustrates a crack initiation that occurred without any casting defect, and authors suspect crack initiation on Si or iron based intermetallic particles, nevertheless exact localization of the crack initiation can't be precisely achieved.

Conclusion and prospects

A new ultrasonic fatigue testing device generating a biaxial proportional stress state with a positive loading ratio has been designed and successfully tested. For validating this equipment, VHCF tests were performed on a cast aluminum alloy already tested in the literature in HCF regime under a similar stress state. The preliminary results are consistent with data from the literature but some sources of discrepancy were identified and should be addressed to correctly determine the fatigue strength of materials under high hydrostatic stress in the gigacycle regime. Self-heating is more pronounced than during tension-compression ultrasonic testing. In future work the stop criterion should be improved to detect smaller cracks, as well as the surface quality of the specimens. Preliminary results show that both Dang-Van and Crossland criteria do not give good assessment of the VHCF strength under high hydrostatic stress. Fracture mechanisms are also consistent with the literature for the tested cast aluminum alloy: multiple crack initiations occurred, either on casting defects (pores or shrinkages), or without any casting defect. Additional work has to be done on specimens with a better surface quality to identify the crack initiation area, as two competing parameters play a role: the biaxial stress state which is maximal on the surface, and the presence of subsurface defects in cast materials.

Fig. 1 .

 1 Fig. 1. (a) Principle and (b) picture of the ultrasonic biaxial bending device.

Fig. 2 .

 2 Fig. 2. (a) Specimen geometry for a first natural mode at 20 kHz and (b) cut view of the associated deformed shape at 5 given moments and displacement by 10% intervals.

Fig. 3 .

 3 Fig. 3. (a) Static force and (b) frequency evolution during a 10 9 cycles run-out test.

Fig. 4 .

 4 Fig. 4. (a) Microstructure of the AlSi7Cu05Mg03 T7 according to Koutiri [13] and (b) cylinder heads without or with diminished casting cores.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Temperature measurement by IR camera during ultrasonic fatigue test: (a) areas (1 and 2) used for temperature averaging and (b) averaged temperature increase.

Fig. 7 .

 7 Fig. 7. Radial (r rr ) and circumferential (r hh ) stresses in the median section of a disc.

Fig. 8 .

 8 Fig. 8. (a) Dang Van and (b) Crossland diagrams for the cast AlSi alloy: loading paths associated with existing ultrasonic fatigue machines and new ultrasonic biaxial bending device; and comparison of experimental result at 10 9 cycles under biaxial bending with threshold at 2 Â 10 6 cycles.

Fig. 9 .

 9 Fig. 9. Macroscopic fatigue crack on the lower face of the specimen (a) after testing and (b) after breaking it under monotonic quasi-static loading.

Fig. 10 .

 10 Fig. 10. Macroscopic fatigue fracture surface (rmax = 140 MPa; R = 0.1; N = 3.46 Â 10 8 cycles): upper part and lower part of the fractured specimen shown in Fig. 9b.

Fig. 11 .

 11 Fig. 11. Two fatigue crack initiations close to shrinkages (rmax = 160 MPa; R = 0.1; N = 2.7 Â 10 8 cycles).

Fig. 12 .

 12 Fig. 12. Fatigue crack initiation without any casting defect (rmax = 140 MPa; R = 0.1; N = 1.3 Â 10 8 cycles).

Please cite this article in press as: Brugger C et al. A new ultrasonic fatigue testing device for biaxial bending in the gigacycle regime. Int J Fatigue (2017), http://dx.doi.org/10.1016/j.ijfatigue.2016.12.039

C. Brugger et al. / International Journal of Fatigue xxx (2017) xxx-xxx Please cite this article in press as: Brugger C et al. A new ultrasonic fatigue testing device for biaxial bending in the gigacycle regime. Int J Fatigue (2017), http://dx.doi.org/10.1016/j.ijfatigue.2016.12.039 Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org)


Emmanuel Renaux
email: renaux@telecom-lille1.eu

Gilles Vanwormhoudt
email: vanwormhoudt@telecom-lille1.eu

Christophe Tombelle
email: tombelle@telecom-lille1.eu

A Multiview Framework Driven by Use Cases to Support the Design of Service Components

Keywords: service component, service design, use case, view, UML, traceability, engineering process

In the field of software technologies, the adoption of Service Oriented Architecture for IS has been accompanied by the introduction of new service component models which are well-suited to realize services. While these service components offer new capabilities to simplify the construction and maintenance of IS, the issue of their design needs to be further examined as current existing works do not sufficiently bridge the gap between requirements and service component models. In this paper, we present a multi-view framework to support the design of services along the development process. The framework is grounded on the notion of Logical Service Block which represents a service from Use case to assembly model and acts as a pivot between views ensuring the consistency and the traceability of services at different stages and levels of detail. The proposed framework has been implemented to provide a UML2 tool that helps designers in elaborating views and projecting services towards a specific platform like SCA.

I. INTRODUCTION

Information systems (IS) have been the battlefield of successive campaigns of re-engineering for new features and new technologies. Recently, the technology change was the Service Oriented Architecture (SOA) [START_REF] Bell | Service-Oriented Modeling: Service Analysis, Design, and Architecture[END_REF]. This architecture framework has been adopted to partition IS in coarse grained, loosely coupled and autonomous services which encompass one or more business or technical discrete functions. SOA adoption has led to propose new approaches and new software artifacts for their design and implementation. The introduction of several service component models [START_REF] Karmarkar | Assembly of business systems using service component architecture[END_REF], [START_REF] Kramer | Component meets service: what does the mongrel look like?[END_REF], like SCA, OSGI and Fractal is one of these novelties. The services are then provided by service components which may in turn use services provided by other components and be assembled into composites to provide specific business capabilities. While service component models promise several benefits in terms of evolution and maintenance, the design of complex IS based on such components remains an important issue. In particular, questions about how to identify such service components, how to elaborate their external and internal structure and how to put them in a consistent assembly still need to be investigated. Existing works only offer partial responses. On one side, some works provide top-down solutions to identify services from the requirements by using an approach driven either by business process either by use cases [START_REF] Arsanjani | SOMA: A method for developing serviceoriented solutions[END_REF], [START_REF] Bocchi | A use-case driven approach to formal service-oriented modelling[END_REF]. It appears however that most of these works do not consider the rest of the development process. The SOAML specification recently adopted by OMG [START_REF] Elvesoeter | Model-driven service engineering with SOAML[END_REF] fills the gap but its notion of service is transverse to components and therefore is not well align with current service component models. On the other side, some works enable the modeling of service components during the design phase, mainly in the scope of the UML notation. According to these works, this modeling is achieved either by reusing existing constructs and diagrams [START_REF] Weiss | Modeling of Services and Service Collaboration in UML 2.0[END_REF] either by extending UML with appropriate profiles [START_REF] Ermagan | A UML2 Profile for Service Modeling[END_REF]. Unfortunately, we observe that the modeling of service components promoted by these works is generally not related with the requirements about services. In addition, service components are often considered as technological concerns. In this paper, we propose a multi-view framework driven by use cases to support the design of service components from requirements to implementation. The novelty of this approach compared to existing works is explained in Section II. In the framework, each view addresses a specific concern of the design process and models parts of the services or their assembly (see Section III). The framework is based on the notion of Logical Service Block which represents a service from requirements to design and acts as a pivot between views ensuring their consistency and enabling round-trip derivations. This pivot supports the traceability of services at different levels of detail and consequently improves the quality and readability of system models. The framework has been implemented to provide a UML2 tool (see Section IV) that helps designers during the elaboration of views and the projection of services toward a specific platform like SCA.

II. USE CASE AND SERVICES

Despite intensive SOA adoption in recent years, no unified methodological approach to identify and design services from business requirements [START_REF] Bell | Service-Oriented Modeling: Service Analysis, Design, and Architecture[END_REF] has been reached yet. Instead, top-down and bottom-up approaches have been proposed. In this paper, we are interested in top-down ones, that can be further divided into those driven by the business process and those driven by use cases. A lot of top-down approaches [START_REF] Arsanjani | SOMA: A method for developing serviceoriented solutions[END_REF], [START_REF] Elvesoeter | Model-driven service engineering with SOAML[END_REF], [START_REF] Azevedo | A Method for Service Identification from Business Process Models in a SOA Approach[END_REF] follow the process-driven method either by formulating guidelines to identify services or by deducting services from process models in a systematic manner. However, process-driven approaches also commonly bear some difficulties: need of a complete knowledge of the enterprise processes, too much process-oriented and too large scope for small and medium-sized systems.

Compared to process-driven approaches, the main arguments in favor of use-case driven ones are: first, use cases offer a systematic and intuitive means to capture functional requirements with a focus on the value added for the user, secondly, they can drive the whole development process. Regarding the relationship between use cases and services, we have found works with different focus. A first group has explored use cases as a way to characterize and integrate existing software services in relation with business requirements [START_REF] Zhao | From Use Case Model to Service Model: An Environment Ontology Based Approach[END_REF], [START_REF] Alkkiomaki | Integration Use Cases -An Applied UML Technique for Modeling Functional Requirements in SOA[END_REF]. A second group of works, which is more specifically related to our proposal, addresses the issue of constructing new service-based systems from use cases capturing requirements. We review these works hereafter.

Authors of [START_REF] Huayou | A Service-Oriented Analysis and Modeling Using Use Case Approach[END_REF] propose a description of services based on several related use cases derived from parts of the Use Case diagram: scripts like participants, system or extensions paths. To complete this description, a process and guidelines to direct the analysis are given. The work described in [START_REF] Bocchi | A use-case driven approach to formal service-oriented modelling[END_REF] also proposes to derive the structure of services from the Use Case diagram. For that purpose, each use case is assumed to map a service-oriented artifact and associations between primary actors and use-cases serve to identify interfaces of such services. In [START_REF] Pz | Modeling and design of SOA[END_REF], the mapping of use cases to services is also considered. Business and technical criteria to decide how to group use cases relatively to a particular business service component are given to drive this mapping process. [START_REF] Knauss | Using the Friction between Business Processes and Use Cases in SOA requirements[END_REF] also discusses use case as a requirements acquisition method which is applied to SOA requirements. The authors mainly explain how to exploit use case as a supplement to the model of business processes but they do not provide a guideline to identify the relevant services. More recently, [START_REF] Lago | A Pragmatic Approach for Analysis and Design of Service Inventories[END_REF] makes an extensive usage of use cases to produce service inventories from business requirements. In this work, each business service is itself modeled as a collection of candidate services represented by use cases marked with stereotype for services types and aggregated with inclusion relationship. In addition, service repositories are related to participants in a clustered Use Case diagram.

To sum up the reviewed works, few ones make the association between use cases and services by just providing guidelines while others aim to offer an explicit representation of services within the Use Case diagram. For the latter category, we have found works that just rely on a particular arrangement or interpretation of existing use case constructs. Some other works extend use case with new concepts and relationships using the profile mechanism. Although these works provide a useful basis to relate use cases and services, they mainly focus on the Use Case diagram but not on all other steps of the development process. Exceptions are [START_REF] Bocchi | A use-case driven approach to formal service-oriented modelling[END_REF], [START_REF] Pz | Modeling and design of SOA[END_REF] but in these works, the issue of designing service components is only treated superficially and no support is provided to maintain the consistency of designed services from requirements to implementation. In the next section, we propose a multi-view framework in response to this problem.

III. OUR MULTI-VIEW FRAMEWORK FOR MODELING SERVICE-BASED IS

A. Overview

In our framework, the notion of Logical Service Block (LSB) is the main piece of design process of a system. This unit of modeling represents the service from requirements to design and groups elements related to a same service. Thanks to this notion, we facilitate the traceability of services at each stage and support a decomposition of the system that enables to partition and reduces the design effort. In addition to the notion of LSB, our framework provides several views to describe a system. Each view addresses a specific concern related to a particular LSB or an assembly of LSBs. The notion of LSB is omnipresent but each view has a unique way to represent it with different constituents depending on the view. As the semantics is the same in each view, an LSB acts as a pivot that links the content of each views. This pivot is a support for traceability of services at different levels of detail and consequently improves the quality and readability of system models.

Figure 1 gives an overview of our framework and its underlying iterative process. In this process, each step corresponds to a view. A specific view can be elaborated iteratively, while remaining consistent with other views. For instance, a designer can draft the Use Case view, then proceed with the Collaboration Analysis view to refine the LSBs. By getting them into details, new requirements may be identified that cause the designer to revise the contents of the corresponding LSB within the Use Case view.

In the following, we explain each view in more detail and illustrate them with a management system for hotel rooms.

B. Detail of views and illustrations 1) Use Case view:

The Use Case view (UCv) concerns the requirements and is founded on the UML use case diagram with adding of the LSB notion and externalUse relationship. A UCv allows to decompose a system in LSBs, by grouping use cases related to the same service. For our example (see figure 2), we obtain two LSBs, one for reservation and occupation of the room and one for accountancy concerns. As we can see, each LSB contains the subset of the use cases related with the service it contributes to. These use cases will be realized by elements enclosed within the LSB in other views of the system. Standard relationships between use cases, extend and include, are kept but their use is limited to the inside of an LSB. To cross LSB boundaries, we add the non-standard externalUse relationship which explicitly shows the functional dependencies between service use cases. Relations that cross boundaries, either between actors and services, either between services, will be captured into interfaces between services in other views. In Figure 2, the externalUse relation between the two LSBs, expresses the interactions between the use cases of these services. Designers are free to choose the best decomposition, the one that reduces interactions and dependencies through boundaries, or simply the one that takes into account IS constraints. UML activity diagrams, modeling business process flows, can help in grouping use cases. As we will see after, this decomposition is maintained in subsequent views to make the corresponding model simpler to understand, change, and maintain.

2) Collaboration Analysis view:

The Collaboration Analysis view (CAv) describes the realization of identified use cases in terms of collaborations and objects. Like in UML, a collaboration defines a set of roles played by objects. This concept precises how a use case is realized. To enhance the traceability, collaborations are grouped like use cases following the LSB they describe.

In figure 3, we can see that each use case contained in an LSB is transformed into a set of one or several collaborations which define the roles played by the objects to realize the use case. These roles are attached to participating objects through collaboration bindings. This way, each participation of an object in a collaboration is represented by a link characterizing its role in this interaction. For instance, a customer object is enrolled in the CheckOutGuest collaboration and has a guest role within it. Roles played by an object will determine its interfaces in the design view. Ports appear in this view to hold required and provided interfaces to implement dependencies detected in the UCv. Here, the Booking provided interface realizes the relationship between Guest actor and Book a room use case. These interfaces allow actors to request services and services to be connected. An object can play a role in several collaborations specified for an LSB but it does not inevitably play the same role. However, as one goal is to minimize dependencies between services, we do not authorize the enrollment of an object in collaborations within another LSB. If needed, it is possible to add required interfaces to define connections between services as actors interactions. For instance, Payment required interface comes from externalUse relation of the UCv.

3) Interaction view (Iv): The Interaction view (Iv) partially models the collaborations identified for an LSB in the CAv. This view represents message exchanges between the roles of a collaboration during the execution of a use case scenario. In addition, to give more details about a particular collaboration, this view is also intended to prepare the design of a service within the Service Design view. Furthermore, this view can help to discover LSBs and to establish their boundaries iteratively with the other views.

We use UML-like sequence diagrams to represent the interactions of a collaboration in terms of roles but we introduce some new specific role types to better characterize the kind of behavior attached to a role and render the interactions of an LSB with external services and actors. The Control role manages all the steps described in the scenario. The Entity role is a kind of business object easily identified in specifications. It contains the information of the system as well as the business processes. Boundary roles are ones played by the service to respond to actors. Finally, the External Control role represents required functions provided by another system or service. Boundary and External Control roles can be considered respective representations of the provided and required ports. 4) Service design view: The Service Design view (SDv) describes the software elements, their structure and the relationships supporting the behavior of an LSB described in its Interaction views. In this view, an LSB provides interfaces that expose its business methods and organize them according to the points of view of the client service.

The control and the entity parts contain respectively the business processing and information management. The control parts encapsulate processing due to use case scenarios and manage all its steps. It plays the intermediate role between ports and entity parts. The entity parts manage the business objects, containing facility to access data as well as the business processes. In the example, the DeskManagement LSB service has provided interfaces corresponding to control roles described in the Iv (e.g. CheckoutGuest, figure 5). Provided interfaces are implemented by one or several parts. Entity roles are good candidates to become Entity Parts of a service as the HotelRoomManager. Some parts, such as ComputeBill, need data or processing from another service. To do that, they delegate to required ports like Payment.

Service Design View

5) Assembly View (Av): As shown by Figure 6, the Assembly view (Av) has the same scope as the UCv, i.e. the representation of the whole system in terms of LSBs, but gives a black box view of services and focuses on their connections. Each logical service is only represented with its ports and these ports are connected to ports of other services when it needs their operations. A connector always links a port specifying a required interface to a port specifying a provided interface of distinct LSBs. The connections between LSBs are the realization in the design view of their related externalUse relations in the UCv. Given this last view and all the previous ones, we reach a complete model of the system with consolidated LSBs. This

View Stereotype Target metaclasses System «System» Package UCv «UseCaseView» Package UCv «LogicalServiceBlock» Package UCv «externalUse» Relationship CAv «CollaborationView» Package CAv «LogicalServiceBlock» Component CAv «Required» Port CAv «Provided» Port CAv «LogicalServicePart» CollaborationUse Iv «ControlRole» LifeLine Iv «EntityRole» LifeLine Iv «BoundaryRole» LifeLine Iv «ExternalRole» LifeLine SDv «LogicalServiceBlock» Component SDv «ControlPart» Property SDv «EntityPart» Property SDv «delegate» Relationship Av «AssemblyView» Package Av «LogicalServiceBlock» Component Table I STEREOTYPES DEFINED FOR THE PROFILE
modeling is completely independent of any technological aspects and follows the classical Model Driven Architecture approach, it is then possible to translate these services into artifacts corresponding to a technological platform like SCA as explained in the following section.

IV. TOOLING

To experiment and validate our framework, we have developed a modeling tool integrated into Eclipse environment. This tool provides a set of functions to help the designer with the creation of views, the maintenance of the consistency of the complete model and its projection to specific platforms.

A. Metamodeling

Most of the concepts in our framework are close to UML2 ones. As a result, we have implemented them in the form of a profile for the UML modeling tools included with the Eclipse environment. Table I gives the set of stereotypes and metaclasses targeted, used to represent these concepts and each separate view. The capacity of a stereotype to be applied to distinct UML concepts is exploited to render the same concept in distinct views (see Component and Package for example). Inversely, we also use the capacity to apply distinct stereotypes to a same concept in order to render views of a system with separate packages.

B. View Consistency

We have defined OCL constraints to guarantee the consistency of views. These constraints are classified in two categories: 1) Intra-view constraints check the correct application of the proposed stereotypes on UML2 elements and check properties on the structure of stereotyped elements within a view; 2) Inter-views constraints concern with the consistency between views. They check properties between elements of a view with those included in other views. Hereafter, we give some OCL constraints defined for our profile in each category. Inter-views constraints are expressed in the context of the «System» root package to simplify the comparison of elements from multiple views.

/ / Query t o t e s t i f an e l e m e n t h a s a s t e r e o t y p e c o n t e x t E l e m e n t d e f : h a s S t e r e o t y p e ( s t e r e o t y p e : S t r i n g ) : Boolean = s e l f . g e t A p p l i e d S t e r e o t y p e s ( )->one ( s : S t e r e o t y p e | s . name = s t e r e o t y p e ) ) / / I n t r a-v i e w C o n s t r a i n t : A l l << e x t e r n a l U s e >> r e l a t i o n s h i p s m u s t l i n k UseCases All the OCL constraints have been implemented using the OCL engine included in the Eclipse environment. They can be checked at any time during the design to test the model consistency and its conformity with our abstract model. As explained in section III-A, the dependencies between the contents of some views and the contents of other ones ensure traceability along the design. To facilitate the creation of a view from another one, we provide a set of model transformations (see deductive derivations in Figure 1). Each transformation takes a profiled model containing the input view and completes the model by adding the target view. The input view must include all the elements required to produce the output view. The result is not intended to be complete and must be considered as a first sketch to be refined and improved. In Figure 7, we can observe that the derived view lacks some elements included in the CAv of figure 3 like role bindings or parts for instance. These elements cannot be deduced from the input view. The designer will generally add them afterwards. All the provided transformations have been written using the QVT language.

C. View derivation

D. Projection to the Service Component Architecture

The objective of our modeling framework is to obtain a system specification which is independent of technological concerns. In order to target a specific platform of services, a transformation of models. In our modeling tool, we propose a QVT transformation that enables the construction of a Service Component Architecture (SCA) model from a model based upon our profile. The provided transformation uses the profiled UML2 as the source metamodel and the SCA metamodel as target one. For this transformation, only the SDv of each LSBs and the Av of a profiled UML2 model are considered to produce the resulting SCA model. In the resulting model, nor bindings of services and references nor SCA components implementations are specified in order to let the designers choose the best technical platform to implement their services (JAVA, EJB, WSDL, ...).

V. CONCLUSION

In this paper, we propose a multi-view framework to deal with service boundaries very early in the development process and to better ensure their traceability. The proposed framework has been experimented in actual projects from large business domains, i.e. insurance, banking, health care. Two case studies experimenting the framework are described in [START_REF] Renaux | Component based approach to design Information System[END_REF], [START_REF] Renaux | Learning Management System component-based design: a model driven approach[END_REF], one to design an E-Learning system and one to refund parts of IS of French waterways. All along these experiments, it has been felt that applying services division early in the process and that maintaining consistency with a tool is more efficient. Furthermore, the framework has also enabled and accelerated human interactions, making all stakeholders participating in the service identification of their IS. Finally, traceability has been maintained more easily and the impact of an evolution is more quickly detected.

A first perspective aims to synchronize views in our modeling tools. Currently, views are not updated to reflect the changes in others. To avoid consistency problems, we want to propose a solution that automatically propagates the changes in dependent views or, if not possible, provides the designer with suggestions to make the update manually. In the scope of this paper, we have mainly focused on functional requirements which are generally the first and main focus of interest during services design. We acknowledge that non-functional requirements are also an issue of SOA [START_REF] Becha | Non-functional Properties in SOA a Consumer's Perspective[END_REF] and we are currently studying policies (security, transaction) of service component model to address this dimension in the scope of our approach. Finally, the last perspective concerns the reusability of services modeled with our approach, particularly with regard to their viewbased structuring. This perspective will be investigated with the goal to make reusable LSBs a key accelerator for the design of service-oriented systems.

Figure 1 .

 1 Figure 1. Overview of the iterative process and its 3 scopes

Figure 2 .

 2 Figure 2. Use case view of the Hotel Management System

Figure 3 .

 3 Figure 3. Collaboration Analysis View of the DeskManagement Service

Figure 4 .

 4 Figure 4. An interaction View of the Checkout Guest Collaboration

Figure 5 .

 5 Figure 5. Service Design View of the DeskManagement service

Figure 6 .

 6 Figure 6. Assembly view of the Hotel Room Management system

  f r o m << L o g i c a l S e r v i c e B l o c k >>. c o n t e x t Dependancy i n v : s e l f . h a s S t e r e o t y p e ( ' e x t e r n a l U s e ' ) i m p l i e s ( s e l f . s o u r c e . owner . h a s S t e r e o t y p e ( ' L o g i c a l S e r v i c e B l o c k ' ) and( s e l f . t a r g e t . owner . h a s S t e r e o t y p e ( ' L o g i c a l S e r v i c e B l o c k ' ) ) / / I n t r a-v i e w C o n s t r a i n t : S o u r c e and t a r g e t UseCase o f a l l << e x t e r n a l U s e >> r e l a t i o n s h i p s m u s t n o t b e l o n g t o t h e same LSB . c o n t e x t Dependancy i n v : s e l f . h a s S t e r e o t y p e ( ' e x t e r n a l U s e ' ) i m p l i e s ( s e l f . s o u r c e . owner <> s e l f . t a r g e t . owner ) / / I n t e r-v i e w s C o n s t r a i n t : R e q u i r e d P o r t o f a LSB i n t h e CAv m u s t match << e x t e r n a l U s e >> r e l a t i o n s h i p i n t h e u s e c a s e UCVview c o n t e x t P a c k a g e i n v : s e l f . h a s S t e r e o t y p e ( ' System ' ) i m p l i e s l e t u s e c a s e v i e w = s e l f . n e s t e d P a c k a g e ->s e l e c t ( p : P a c k a g e | p . h a s S t e r e o t y p e ( ' UseCaseView ' ) ) i n l e t u s e c a s e L S B = u s e c a s e v i e w . a l l O w n e d E l e m e n t s ( )-> s e l e c t ( e : E l e m e n t | e . h a s S t e r e o t y p e ( ' L o g i c a l S e r v i c e B l o c k ' ) ) i n l e t c o l l a b v i e w = s e l f . n e s t e d P a c k a g e ->s e l e c t ( p : P a c k a g e | p . h a s S t e r e o t y p e ( ' C o l l a b o r a t i o n V i e w ' ) i n l e t c o l l a b L S B = d e s i g n v i e w . a l l O w n e d E l e m e n t s ( )->s e l e c t ( e : E l e m e n t | e . h a s S t e r e o t y p e ( ' L o g i c a l S e r v i c e B l o c k ' ) ) i n c o l l a b L S B->f o r A l l ( c : Component | usecaseLSB->e x i s t s ( p : P a c k a g e | c . name = p . name and p->s e l e c t ( e : E l e m e n t | e . h a s S t e r e o t y p e ( ' e x t e r n a l U s e ' ) ) . s i z e ( ) = c . ownedPort ->( p o r t : P o r t | p o r t . h a s S t e r e o t y p e ( ' R e q u i r e d ' ) ) . s i z e ( ) )

Figure 7 .

 7 Figure 7. Example of view derivation


Stéphane Girardon

Louis Denaud

Guillaume Pot

Istie Rahayu

on the effective modulus of elasticity of poplar laminated veneer lumber

Keywords: Populus, Juvenile wood, Mature wood, Laminated Veneer Lumber

Modelling the effects of wood cambial age on the effective modulus of elasticity of poplar laminated veneer lumber

Introduction

Laminated veneer lumber (LVL) is an engineering wood product made from veneer sheets glued together layer by layer to form panels or beams. Veneers are mainly obtained from a rotary peeling process which consists in peeling a log from the outside to the core. Peeling lines are computer-controlled, providing access to different types of data from the logs [START_REF] Thibaut | Wood machining with a focus on French research in the last 50 years[END_REF]. Significant progresses in wood mechanical behaviour knowledge are needed to manage these log properties into a LVL model.

Furthermore, mechanical properties of LVL are strongly dependent on the mechanical properties of each layer. Juvenile wood is the name given to wood created at the beginning of the radial growth, hence in a zone close to the pith. Juvenile wood usually has a lower density and less mechanical properties than mature wood.

Stéphane Girardon; Louis Denaud; Guillaume Pot; Istie Rayahu

The rotary peeling process can easily separate mature wood from juvenile wood by sorting the veneers by their radial position. However, the impact of juvenile wood on LVL is difficult to apprehend experimentally [START_REF] Burdurlu | The effects of ply organization and loading direction on bending strength and modulus of elasticity in laminated veneer lumber (LVL) obtained from beech (Fagus orientalis L.) and lombardy poplar (Populus nigra L.)[END_REF]), due to the many factors which have an effect on mechanical properties during timber growth cycles. In this work, a model is proposed which considers the impact of juvenile and mature wood, i.e. cambial age. The study focuses on poplar wood species which are typically used in peeling and present fast growth, thus a high ratio of juvenile wood. This study is based on poplar wood properties in general, but principally on the measurements of [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF], who studied several poplar cultivars: A4A, Alcinde, Brenta, Dvina, I-214, Koster, Lambro, Lena, Mella, Polargo, Soligo, Taro, Trichobel, Triplo. The model developed is also based on numerous measured properties from the literature [START_REF] Bao | Differences in wood properties between juvenile wood and mature wood in 10 species grown in China[END_REF][START_REF] Fang | Clonal and within-tree variation in microfibril angle in poplar clones[END_REF][START_REF] Bjurhager | Mechanical characterization of juvenile European aspen (Populus tremula) and hybrid aspen (Populus tremula Populus tremuloides) using full-field strain measurements[END_REF][START_REF] Bremaud | Changes in viscoelastic vibrational properties between compression and normal wood: roles of microfibril angle and of lignin[END_REF][START_REF] Hein | Correlations among microfibril angle, density, modulus of elasticity, modulus of rupture and shrinkage in 6-year-old Eucalyptus urophylla E. grandis[END_REF].

The aim is to obtain the behaviour of LVL structural elements (estimation of the modulus of elasticity) from the pith to the bark of an average tree considering different peeling scenarios. In order to achieve this, a representative log is virtually peeled and veneer sheets are sorted according to their category: juvenile or mature wood. Finally, juvenile and mature wood test specimens are reconstituted from each group and simulation results are compared to experimental values from [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF].

Assessment of mechanical properties variations with cambial age

The longitudinal elastic modulus of wood is influenced by many parameters from the micro to the macro scale, such as Micro Fibril Angle (MFA), density and humidity [START_REF] Cousins | Elastic modulus of lignin as related to moisture content[END_REF][START_REF] Cave | Stiffness of wood in fast-grown plantation softwoods: the influence of microfibril angle[END_REF][START_REF] Evans | Rapid estimation of solid wood stiffness using SilviScan[END_REF][START_REF] Evans | Rapid prediction of wood stiffness from microfibril angle and density[END_REF]. The material's humidity is not studied here because it is managed during manufacturing due to adhesive process requirements.

As a first step, this study focuses on three main parameters, which are MFA, annual growth ring width and specific gravity. Their variations according to cambial age are taken into account in the model by interpolating experimental data of the literature. As described below, when authors didn't fit their measurements of the properties, a sigmoid function has been fitted on their values. Sigmoid functions allow both to smooth experimental data and provide monotonous curves which have an asymptotic behaviour at the boundary of the measurements.

Density

Wood density is an important quality attribute, related to many properties like stiffness and strength. Wood density is highly variable due to genetic, environmental, or silvicultural management. As a result, the specific gravity varies throughout the life of the tree.

The following specific gravity data (Fig. 1) come from Paillassa et al. (2013) where 14 cultivars were analysed. Each cultivar was characterized using 3 logs of 7 meters long from 3 different trees. Cultivars had an average density of 330kg.m -3 . Wood density at 12% moisture content was calculated for each wood sample per cambial age. Fig. 1 shows the average specific gravity based on cambial age.

For modelling purposes, a sum of sigmoid function (Eq. ( 1)) was fitted by using Mathematica fitting method to the experimental values: 𝜌(𝐶 𝑎 ) = 319.99 + 37.46 1 + ⅇ -1.848(𝐶 𝑎 -14.11) + 40.31 1 + ⅇ 0.8703(𝐶 𝑎 -2.574)

(1) where specific gravity (𝜌 in kg.m -3 ) is predicted from cambial age (𝐶 𝑎 in years). The fitted density variation profile observed in Fig. 1 is comparable to the results proposed by [START_REF] Senft | Juvenile wood: processing and structural products considerations[END_REF] for poplar species (Popuplus Deltoid). The specific gravity decreases during the very first years of the tree's life before stabilization. Then it starts to increase from 10 to 16 years old. The density then seems to reach a new threshold, but the data available in the literature are limited for old poplars, entailing lower accuracy in this range. As observed for Larch, Douglas-fir and Scots Pine in literature [START_REF] Karlman | Wood Density, Annual Ring Width and Latewood Content in Larch and Scots Pine[END_REF][START_REF] Filipescu | Modeling regional and climatic variation of wood density and ring width in intensively managed Douglas-fir[END_REF], the specific gravity reaches a horizontal asymptote after a few decades (17 years on average for the studied poplar cultivars), which is obtained thanks to sigmoid functions of Eq. (1).

Micro fibril angle

MFA is an inherent property at the wood cell level, where wood compounds have a privileged direction due to growing conditions. MFA variation can depend on cambial age [START_REF] Cave | Stiffness of wood in fast-grown plantation softwoods: the influence of microfibril angle[END_REF]. Fig. 2 shows the average MFA with respect to cambial age for seven poplar clones, obtained from an experimental study [START_REF] Fang | Clonal and within-tree variation in microfibril angle in poplar clones[END_REF]. A sum of sigmoid function (Eq. ( 2)) was fitted to the experimental values (extract from their article), for modelling purposes: 𝑀𝐹𝐴(𝐶 𝑎 ) = 25.82 -40.30 1 + ⅇ -0.3947(𝐶 𝑎 -2.187) + 25.81 1 + ⅇ -1.202 (𝐶 𝑎 -1.542) (2) where the MFA (in degrees) is predicted from cambial age (𝐶 𝑎 in years). After 12 years, the MFA fitted by the sigmoid function reaches a constant value.

Specific Modulus of Elasticity

According to [START_REF] Bremaud | Changes in viscoelastic vibrational properties between compression and normal wood: roles of microfibril angle and of lignin[END_REF], there exists a strong relationship between the dynamic specific modulus of elasticity and the MFA. Here we use this relationship, which has been successfully tested for several softwood species [START_REF] Bendtsen | Mechanical and anatomical properties in individual growth rings of plantation-grown eastern cotton-wood and loblolly pine[END_REF][START_REF] Reiterer | Experimental evidence for a mechanical function of the cellulose microfibril angle in wood cell walls[END_REF][START_REF] Sedighi-Gilani | Microfibril angle nonuniformities within normal and compression wood tracheids[END_REF]. Indeed, poplar is usually classified with softwoods in terms of its mechanical properties. A trigonometric function (Eq. ( 3)) was fitted by [START_REF] Bremaud | Changes in viscoelastic vibrational properties between compression and normal wood: roles of microfibril angle and of lignin[END_REF] to their experimental values:

𝐸 𝜌 (𝑀𝐹𝐴) = 1 0.03426Cos[𝑀𝐹𝐴] 4 + 0.2053Cos[𝑀𝐹𝐴] 2 Sin[𝑀𝐹𝐴] 2 + 0.6797Sin[𝑀𝐹𝐴] 4 (3)
where the specific modulus (𝐸/𝜌 inMPa.m 2 .kg -1 ) is predicted from the MFA (in degrees).

The above relationships will enable us to obtain the modulus of elasticity according to the radial position in the log.

Annual growth ring width

As well as specific gravity, environmental factors affect the width of each annual growth ring. The widths of the growth rings obtained in the study of Paillassa et al. (2013) are presented in Fig. 4 with respect to cambial age. A sigmoid function (Eq. ( 4)) was fitted to the experimental values for smoothing purposes:

𝑤 𝑟𝑖𝑛𝑔 (𝐶 𝑎 ) = -1.026 - 18.77 1 + ⅇ -0.3475(𝐶 𝑎 -6.727) + 25.90 1 + ⅇ -1.696(𝐶 𝑎 -1.285) (4)
where the average annual growth ring width (𝑤 𝑟𝑖𝑛𝑔 in mm/year) is predicted from the cambial age (𝐶 𝑎 in integer years). The average annual growth ring width is supposed to tend to a constant value after 16 years. This hypothesis for normal wood can be observed for other species in literature [START_REF] Adamopoulos | Ring width, latewood proportion and density relationships in black locust wood of different origins and clones[END_REF][START_REF] Guller | Variations in the radial growth and wood density components in relation to cambial age in 30-year-old Pinus brutia Ten. at two test sites[END_REF][START_REF] Campelo | Which matters most for the formation of intra-annual density fluctuations in Pinus pinaster: age or size?[END_REF]. Notice that for this study, the model will not be used for trees older than 16 years old.

The radius based on cambial age is then deduced by integrating the annual growth ring width for each year of growth (Eq. ( 4)), with the assumption of a constant growth rate for a given year (cf. Fig. 4). This enables us to link the equations based on cambial age to equations based on the log radius:

𝑟(C a ) = ∫ 𝑤 𝑟𝑖𝑛𝑔 ̅̅̅̅̅̅̅(y)𝑑y C a 0 with 𝑤 𝑟𝑖𝑛𝑔 ̅̅̅̅̅̅̅(𝑦) = 𝑤 𝑟𝑖𝑛𝑔 (𝐶 𝑎 ) for 𝑦 ∈ ]𝐶 𝑎 -1, 𝐶 𝑎 ]
(5) where the radius (𝑟 in mm) of the log is determined from the ring width (𝑤 𝑟𝑖𝑛𝑔 ̅̅̅̅̅̅̅ in mm.year -1 ).

Model building

The model was developed using Wolfram Mathematica Software (2015). The model consists of four different phases, detailed in Fig. 5. First step is to determine the modulus of elasticity along the generatrix curve of the peeling process using the equations above (Eq. ( 1) to ( 5)). The second step is to divide the virtual peeling into sheets and sort them into two groups, "juvenile" and "mature". The chosen width of the sheets is 500mm, based on the experimental work of [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF]. The juvenile and mature groups were defined by their average age, calculated from the annual growth ring width which they originated from. The age threshold between mature and juvenile wood was chosen as 10.6 years, which corresponds to the measured average experimental value in the study of [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF]. Each sheet is virtually subdivided into 𝑛 𝑠 subsamples of 20 mm width (the final width of the test specimens). In the third step, for each group the sheets are assembled into 𝑛 𝑙 layers to represent a LVL panel of 21 mm thickness. The number of constituent panel layers depends on the thickness of the sheets. Sheet thicknesses of 3 mm and 5.25 mm were used, leading to seven layers for the 3 mm sheets and four layers for the 5.25 mm sheets. In this work, it is assumed that sheets can be randomly assembled upside down or not, and the layering is also randomised into a group. Finally, in the last step of the model, the panels are cut into test specimens of 20 mm in width. Then the effective modulus of elasticity of each test specimen is calculated from each layer's properties as described below.

Virtual peeling

The first step is to determine the maximum ribbon length depending on the log diameter. In cross-section, the peeling process corresponds to following a spiral curve (Fig. 5). The equation of a spiral in polar coordinates is given by Eq. ( 6), where 𝑟 𝑠 is the spiral radius (mm) with respect to the angle θ (in rad), 𝑟 𝑖 is the initial log radius (in mm) and 𝑡 is the peeling thickness (in mm).

𝑟 𝑠 (𝜃, 𝑡) = 𝑟 𝑖 -𝑡 • 𝜃 2 π (6)
The length of this curve can be calculated by the integration of Eq. ( 6), which corresponds to the ribbon length. Second-order length is neglected due to linear variations in radius with respect to the angle. Thus the ribbon length from the core is determined by the following equation ( 7):

𝑙(𝑟 𝑖 , 𝑟 𝑓 , 𝑡) = ∫ √ 𝜕𝑟 𝑠 (𝜃,𝑡) 𝜕𝜃 2 + 𝑟 𝑠 (𝜃, 𝑡) 2 𝑑𝜃 2𝜋• r i 𝑡 2𝜋• r f 𝑡 (7)
Where 𝑙 (in mm) is the ribbon length from the core and r f (in mm) the final log radius.

The integration domain depends on both the final and the initial log radius. Indeed, logs are not peeled up to the log centre; there remains a peeler core of r f radius. Initial and final log radii were chosen to be respectively 200 and 35 mm for comparison purposes with the study of [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF].

3.2

Mechanical properties computation

Using the above equations (Eq. ( 1) to ( 7)) taken from the literature, it is then possible to determine the wood's modulus of elasticity along the peeling sheet by composing them (Eq. ( 8)) to obtain parametric coordinates on cambial age basis:

{ 𝑙 = 𝑙(𝑟 𝑖 (𝐶 𝑎 , 𝑡), 𝑟 𝑓 , 𝑡) 𝐸 = 𝐸 𝜌 (MFA(𝐶 𝑎 )). 𝜌(𝐶 𝑎 ) (8)
Where 𝑙 (in mm) is the ribbon length from the core and 𝐸 (in MPa) is the modulus of elasticity. These values are parametrized by the cambial age (𝐶𝑎).

In Fig. 6, solid lines represent some examples for different peeling thicknesses showing the modulus of elasticity with respect to ribbon length for an initial log radius of 200 mm and a kernel log radius of 35 mm. Because of the 35 mm kernel radius, the model does not no use any extrapolated value towards the centre of the logs. Indeed, 35 mm radius corresponds to a cambial age higher than 3 years old, and all data in use starts to 1 year old.

Values beyond 200 mm radius are shown in dashed line in Fig. 6. These values are almost constant due to the asymptotic behaviour of sigmoid curves, but they are not used in the model. However, if the model were used beyond 200 mm, the extrapolation would not produce outliners due to sigmoid function property.

Calculation of Flatwise and Edgewise modulus of elasticity

Since the modulus of elasticity varies into each layer of each panel, it is necessary to compute an effective modulus of elasticity for each test specimen. The effective modulus of elasticity corresponds to the apparent modulus of elasticity for an equal and homogenous section. Moreover, this modulus of elasticity depends on the loading direction, because the impact of each layer on the mechanical behaviour of the LVL is different in flatwise or edgewise configuration [START_REF] Burdurlu | The effects of ply organization and loading direction on bending strength and modulus of elasticity in laminated veneer lumber (LVL) obtained from beech (Fagus orientalis L.) and lombardy poplar (Populus nigra L.)[END_REF]). This will be discussed in the next paragraph. In the model, the interface connection stiffness is considered higher than the material's raw stiffness. This is a prerogative in European standard (EN 14374) concerning LVL panels.

Edgewise

In an edgewise bending test, each layer can be considered as independent. Thus, the effective modulus of elasticity is determined by calculating the sum of the stiffness of each layer, as in the following Eq. ( 9), according to Voigt principle:

(𝐸) 𝑒 = ∑ 𝐸 𝑖 𝐼 𝑖 𝑛 𝑙 𝑖=1 𝐼 𝑡 (9)
With:

{ 𝐸 𝑖 Modulus of elasticity of the i th layer MPa 𝐼 𝑖 Local inertia of the i th layer mm 4 𝐼 𝑡 Inertia of the homogeneous section mm 4 𝑛 𝑙 number of layer -

Flatwise

In a flatwise bending test, the location of layers with different mechanical properties has more influence on global properties than in an edgewise bending test, due to a different stress rate between border and central layers. To calculate the effective modulus of elasticity in the flatwise direction, the position of neutral section axis has first to be determined thanks to the following equation (10) (cf. Fig. 7):

𝑧 0 = ∑ 𝑧 0,𝑖 𝐸 𝑖 𝑆 𝑖 𝑛 𝑙 𝑖=1 ∑ 𝐸 𝑖 𝑆 𝑖 𝑛 𝑙 𝑖=1 (10)
With, in accordance with Fig. 7:

{ 𝑧 0 Distance from the neutral axis to an arbitrary reference mm 𝑧 0,𝑖 Distance from the i th layer neutral axis to an arbitrary reference mm 𝑧 𝑖 Distance from the i th layer neutral axis to the global neutral axis mm Thus, the distance between each layer's neutral axis and the global neutral axis, 𝑧 𝑖 is deduced by equation ( 11):

𝑧 𝑖 = 𝑧 0,𝑖 -𝑧 0 (11)
Then, the effective modulus of elasticity can be determined by the following Eq. ( 12), according to Steiner's principle:

(𝐸) 𝑓 = ∑ 𝐸 𝑖 𝐼 𝑖 +𝑆 𝑖 𝑧 𝑖 2 𝑛 𝑙 𝑖=1 𝐼 𝑡 (12)
With:

{ 𝐸 𝑖 Modulus of elasticity of the i th layer

MPa 𝐼 𝑖 Inertia of the i th layer mm 4 𝐼 𝑡 Inertia of the whole section mm 4 𝑆 𝑖 Section of the i th layer mm 2 𝑧 𝑖 Distance from the i th layer neutral axis to the neutral axis mm

Stochastic approach

As each virtual peeling process is based on a randomised assembly process, the process is repeated a thousand times to identify and enhance a tendency. Indeed, the peeling process always gives the same veneer for a given thickness value, but there are several combinations of sheet arrangements after the primary cutting. These combinations are also increased by including the possibility to turn each sheet upside-down. For a given set of veneer sheets, the sheet positioning influences the flatwise modulus of elasticity, whereas it does not impact the edgewise modulus of elasticity. Consequently, a great number of processes have to be performed to obtain results close to experimental conditions. The number of possible combinations for a single 𝑛 𝑙 layer LVL is given by Eq. ( 13) below:

𝑛 𝑐𝑜𝑚𝑏 = 𝑛 𝑙 !•2 𝑛 𝑙 2 (13)
Where 𝑛 𝑐𝑜𝑚𝑏 is the number of possible combinations and 𝑛 𝑙 is the number of layers. Symmetric cases are considered in the number of combinations; consequently the number is halved.

For the log size used, there are respectively 175 and 250 test specimens per log for respectively juvenile and mature groups. Note that the number of specimens does not depend on veneer thickness, but only on the volume of the test specimens. The number of possible combinations is determined with binomial coefficients, which gives the following results (Eq. ( 14)) for the studied cases (4 layers of 5.25 mm thickness and 7 layers of 3 mm thickness over a total number of test specimens of 175 or 250): Due to the numerous ways to choose 𝑛 𝑙 sheets from a set of 175 or 250 elements, a stochastic approach is chosen in order to obtain results within a reasonable computing time.

{ ( 175

Results

The model can be used to predict the effective elastic modulus of a parameterized LVL sample. It could be used with different scenarios but requires validation through a comparison with experimental data, which is done in the following.

4.1

Experimental results

A study of [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF] was carried out on ten different poplar species, where juvenile and mature wood LVL had their moduli of elasticity measured. A summary of the results is shown in Table 1. The dynamic MOE of LVL composed of juvenile wood appears to be significantly lower than that of mature wood, with a ratio of 0.858. The flatwise MOE is also slightly lower than the edgewise MOE, while there is no significant effect of veneer thickness. According to the authors, the differences in specific gravities are low and cannot alone explain these observations.

Model results

Our model was computed with the following parameters to fit the experimental set up presented by [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF]: initial log radius: 200 mm; kernel log radius: 35 mm; sheet thicknesses: 3 and 5.25 mm; sheet length: 500 mm; test sample width: 20 mm; and juvenile cambial age limit: 10.6 years.

Fig. 8 and Fig. 9 show the distribution of test samples with respect to the modulus of elasticity. Common statistical values are given in Table 2 andTable 3. As expected in the light of the input parameters, mature groups have a higher average modulus of elasticity value ( 8225 ± 220 MPa) than juvenile groups ( 6461 ± 449 MPa) (cf. Table 2), which leads to a modulus of elasticity ratio of 0.79 between juvenile and mature groups. Each juvenile group has a higher dispersion than mature groups, which is shown by coefficients of variation from 5.03 up to 8.86% for juvenile groups and only from 1.94 to 3.35% for mature groups.

The dispersion of modulus of elasticity is higher in a flatwise configuration than in an edgewise configuration, due to the greater influence of border layers in flatwise cases. Dispersion also increases between 3 mm and 5.25 mm LVL thicknesses because of the number of layers used. Indeed, the fewer layers there are, the greater influence they have.

A comparison between experimental and model MOE values can be made thanks to Table 1 andTable 2, where the measurements are classified into groups according to whether the wood is juvenile or mature, and sheet thicknesses (3 and 5.25 mm). The comparison shows that modelled MOE are systematically lower than experimental ones, and that the standard deviation is higher for the experimental data.

Juvenile LVL shows a significantly lower MOE in both the model and experimental results. No clear effect of the veneer thickness or the loading direction appears in the model, while in the experimental results there is no significant effect of veneer thickness, but a statistically significant effect of loading direction on the dynamic MOE exists. Nevertheless, this effect appears non-existent when looking at the static MOE, showing that this result has to be discussed (contrary to the effect of juvenility).

Discussion

The model is based on the assumption that all poplar trees and cultivars behave in the same way in terms of growth, MFA variation and juvenilemature transition, and thus the coefficient of variation (6.95 and 2.67%) is obviously lower than in the experimental results from (around 15% regardless of the group).

The experimental ratio between the dynamic MOE of juvenile and mature wood is 0.876, which is higher than the model prediction (0.785). However, the results of [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF] present a higher coefficient of variation (Table 1). The study's ratio is included between 0.69 and 1.06 by considering both standard deviations of juvenile and mature modulus of elasticity. Indeed, uncertainty of a ratio can be calculated from the standard deviation of each part of this ratio. Therefore the results obtained with the model are close or similar to those from [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF]. However, several reasons could have induced different ratios. From the experimental point of view, there exist difficulties in distinguishing juvenile from mature wood in [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF]. Indeed, groups were made by visual observation of false heartwood but false heartwood does not necessarily correspond exactly to juvenile wood. Furthermore, a part of the studied logs was quite young, with a first quartile equal to 13.25 years, which leads to a minor number of mature annual growth rings to obtain a reliable long term evolution of mature wood properties. From the modelling point of view, there is a lack of experimental input data for mature wood in the literature [START_REF] Thibaut | Wood machining with a focus on French research in the last 50 years[END_REF], entailing an assumption of steady parameters for extra dataset domain for raw material properties. By doing so, the MOE of mature wood is bounded in the model, and the dispersion of LVL composed of mature wood is very low in comparison with experimental measurements. Thus the MOE can be underestimated, especially for mature wood, leading to the higher ratio between the juvenile and mature wood MOE observed in the model.

Despite the observed differences between experimental and modelled MOE values, this study takes the stand of not modifying experimental values from the literature to obtain a first model without entering modifications. In this way, it is easier to understand the influence of each property's behaviour without any behaviour correction. The difference can rely on the relationship between the specific modulus and MFA (Fig. 3) based on [START_REF] Bremaud | Changes in viscoelastic vibrational properties between compression and normal wood: roles of microfibril angle and of lignin[END_REF]. These experimental results show a significant dispersion in the ten to twenty degrees range. Furthermore, they were performed on a coniferous tree, which shows similar mechanical properties to populus, but poplar has a different internal structural anatomy, like a broad-leaved tree. In [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF], the effect of thickness and bending direction cannot be concluded, since dynamic and static MOE give contrary results. Indeed, dynamic MOE results show an influence of bending direction but not of thickness, whereas static MOE results give the opposite conclusion. Modelling shows no influence from either thickness or bending direction on the average MOE. However, compared to edgewise bending, the model shows an increase in standard deviation for flatwise bending, as well as for 5-layer LVL compared to 3-layer LVL.

The results obtained by [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF] indicate that maturity is a source of variance of the effective modulus of elasticity after performing an ANOVA test on the dynamic and static test results; this variance also occurs with the model.

Conclusion

The paper proposes a modelling method to predict LVL elastic behaviour. This model is based on experimental measurements to create an average poplar log, which is used in a simulated peeling process. This study shows a correct mechanical behaviour prediction by the developed model compared to experimentally determined behaviour from [START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF], in particular with the decrease in MOE in LVL made of juvenile wood. The model, featuring a stochastic approach, is also able to predict LVL behaviour according to veneer thickness and loading direction. Indeed, model results show no influence of these parameters on the average MOE. However, the model shows that dispersion is higher for flatwise than for edgewise bending. Greater veneer thickness also increases the dispersion. Model accuracy could be improved by leading a measurement campaign to determine a reliable specific modulus of elasticity for poplar cultivar. Indeed, this study relies on softwood specific modulus of elasticity and could underestimate the specific MOE value on poplar with respect to specific gravity.

From a practical perspective, this model could be used to highlight the necessity or not of sorting veneer sheets in an industrial process by showing the impact of juvenile wood in LVL products. It could also help to estimate the impact of a peeling strategy to build LVL billets as regards veneer thickness.

7

Tables

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 9 .

 12345679 Fig. 1. Variation in specific gravity with cambial age for poplar cultivar (measurements from (Paillassa et al. (2013))

Table 1 .

 1 : The effects of veneer thickness, cultivar, maturity and sample position on static and dynamic MOE and density (measurements from[START_REF] Rahayu | Ten new poplar cultivars provide laminated veneer lumber for structural application[END_REF]

	Group	n	Dynamic MOE (MPa)	Static MOE (MPa)	Density (kg.m -3 )
	3 mm	1203	8707.18±1334.7	8201.54±1333.6	414.58±35.3
	5.25 mm	604	8774.34±1551.6	8415.64±1516.1	394.91±39.7
	Mature	905	9298.49±1373.7	8879.49±1336.6	408.25±37.3
	Juvenile	902	8157.86±1204.8	7664.24±1185.6	401.26±38.4
	Flatwise	949	8654.64±1401.8	8267.40±1419.1	407.91±37.8
	Edgewise	858	8811.50±1418.1	8278.70±1381.4	408.11±38.2

Table 2 .

 2 Predicted MOE (MPa) of each group of LVL

	Group	n	Min	Max	Average	STD	COV (%)
	3mm	238,000	5171	8793	7271	931	12.80
	5.25mm	238,000	4837	8819	7254	970	13.38
	Juvenile	285,600	4837	7511	6461	449	6.96
	Mature	190,400	7649	8819	8225	220	2.67
	Edgewise 238,000	5075	8805	7269	927	12.75
	Flatwise	238,000	4837	8819	7256	974	13.42

Table 3 .

 3 Predicted MOE (MPa) of juvenile and mature groups of LVL

	Group	Juvenile		Mature
	Thick-
	ness	3mm	5.25mm	3mm	5.25mm

Acknowledgment

This research was carried out at the Laboratoire Bourgogne des Matériaux et des Procédés (LaBoMaP), Ecole National Superieure d'Arts et Métiers (ENSAM), Cluny, Bourgogne, France. We wish to thank our partners FCBA (especially Alain Berthelot who perform physical analysis of poplar samples) and CNPF IDF for their active collaboration on this large research program. We also wish to thank our partners and funders of the Xylomat Technical Platform from the Xylomat Scientific Network funded by ANR-10-EQPX-16 XYLOFOREST.