Romain Viala
email: romain.viala@univ-fcomte.fr

Vincent Placet

Scott Cogan

Simultaneous non-destructive identification of multiple elastic and damping properties of spruce tonewood to improve grading

Keywords: Tonewood grading, Non-destructive testing, Identification, Wood variability, Finite element model updating, Modal analysis, Violin and guitar making

The tonewoods used in musical instruments are finely selected and are subject to current and upcoming availability issues. Wood grading assesses the quality of the tonewood and depends on both subjective and objective criteria. Traditionally, grading criteria based on mechanical properties consider mainly the longitudinal direction. This paper investigates the elastic and damping properties of spruce wood used in the making of acoustical guitars and violins. In this context, the mechanical grading properties will be studied in multiple directions. A non-destructive vibrational characterization method is used to simultaneously identify at least six mechanical properties of wood and leads to the determination of the elastic and damping parameters of wood along the different material directions. The main objective is to quantify the variability of the mechanical properties in different material directions as a function of the grading attributed by the wood seller. The results show that material anisotropy decreases as the grade decreases. This is attributed to strong increase of the specific elastic modulus in the radial direction and the specific shear modulus in longitudinal and radial plane. Moreover, grading seems to be assigned as a function of specific elasticity and loss factor in the longitudinal direction, which is tied in with the preferences of instrument makers. Following the assumption that high specific modulus is desirable, the classification based on instrument makers choices can be improved for new grading criteria that consider directions than longitudinal. Moreover, the substitution of wood species and the proposal of new materials exhibiting equivalent elastic and damping properties can be proposed.

Highlights

• Simultaneous non-destructive determination of at least six elastic and damping properties of tonewood.

• Description of elastic and damping properties as a function of tonewood grades.

• Decrease of specific longitudinal modulus and increase of longitudinal loss factor for decreasing grades.

• Increase of specific elastic modulus in radial direction and specific shear modulus in longitudinal and radial plane for decreasing grades.

wood that is used to construct musical instruments plays a dominant role in the sound quality and timbre of the instrument. The availability of woods considered to be the most effective for guitars, cellos and double bass making is becoming a concern, and a better understanding of the wood selection criteria and wood seller grading becomes necessary. Generally, instrument makers attribute a pri-mary importance to the longitudinal direction of tonewood. Nevertheless, recent studies [START_REF] Viala | Model-based effects screening of stringed instruments[END_REF], [START_REF] Viala | Towards a model-based decision support tool for stringed musical instrument making[END_REF] based on simulations of the structural dynamics of musical instruments showed that other directions also influence the dynamic behaviour of musical instruments, in particular the radial elastic modulus and the longitudinal-radial shear modulus. However, data is less abundant along the non longitudinal directions and thus are rarely used for tonewood grading. Meanwhile, previous work suggests that the current selection criteria might be improved to account for a wider range of material properties. By considering the other material directions, the choice of wood species used can be broadened and new materials can be proposed. The objective of this study is to simultaneously evaluate multiple elastic and damping parameters of spruce tonewood samples in a non-destructive way at the scale of the full sample and to discuss the link between tonewood grading and elastic and damping material parameters.

Nomenclature

• L : longitudinal direction.

• R : radial direction.

• T : tangential direction.

• ρ : density. : specific shear modulus in LR plane.

• η L : loss factor in L direction.

• η R : loss factor in R direction.

• η LR : loss factor in LR plane.

• T : temperature

• RH : relative humidity

• MC : moisture content

• MFA : micro-fibril angle.

Introduction

Chordophones such as acoustic guitars and quartet instruments are mainly constructed of resonant boxes that amplify and transmit string vibrations to the air. The resonant boxes are primarily made of wood parts that are assembled with glue. Instrument makers sculpt and assemble the wood parts according to traditional and empirical methods. In the early stages of making an instrument, instrument makers choose the wood species and samples that will be assembled according to their own techniques and tastes that may be different from those of tonewood sellers.

Instrument maker's point of view

The term "tonewood" or "resonance wood" refers to wood used for acoustic guitars and instruments from the violin family [START_REF] Bucur | Handbook of Materials for String Musical Instruments[END_REF]. Instrument makers carefully select the wood used for musical instrument making based on both objective or subjective criteria, such as physical, mechanical, aesthetic, workability, and availability as seen in figure 1. It has to be pointed out that some idealizations and folklore may have skewed assertions about tonewood quality [START_REF] Allen | Fatto di Fiemme' : Stradivari's Violins and the Musical Trees of the Paneveggio[END_REF]. The importance given each of these classification criteria vary according to each instrument maker [START_REF] Carlier | Tonewood Selection : Physical Properties and Perception as Viewed by Violin Makers[END_REF] as do the different types of evaluations considered (auditive, tactile, visual). There is no clear trend for a preferred evaluation method, and a majority of instrument makers use mainly subjective selection criteria, and moderately scientific approaches, despite their interest for studies on tonewoods [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF]. According to the studies [START_REF] Carlier | Tonewood Selection : Physical Properties and Perception as Viewed by Violin Makers[END_REF] and [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF], instrument makers consider the wood to be one of the most determining factors for sound quality, especially the wood used for back and top plates, prior to the geometrical making choices. Instrument makers also insist that the prior acoustical properties of wood evolve differently if the instrument is played or not.

Scope

From the pragmatic viewpoint of the instrument maker, a straight fibre and grain is preferred due to wood workability issues. Indeed, the lack of defects such as nodes is directly related to structural integrity. Aesthetic considerations are involved for the final appearance of the instrument, based on the tastes of musicians and instrument makers. Special patterns like flames for maple or indented rings for spruce are selected for aesthetic as well as mechanical considerations. Indeed, a recent study pointed out that flames in maple changed the mechanical properties of the wood [START_REF] Alkadri | Relationships between anatomical and vibrational properties of wavy sycamore maple[END_REF]. Moreover, the indented rings of spruce, also known as bear claw, reduce anisotropy in the wood [START_REF] Buksnowitz | Indented rings (hazel growth) of Norway spruce reduce anisotropy of mechanical properties[END_REF]. As with most biological materials, wood shows a high variability in mechanical properties between the different species, subspecies, individuals and the same individual. Tonewood is previously selected by the specialised vendor through a dedicated selection process different than for wood used in other applications. Instrument makers mainly buy tonewood from these sellers [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF], and their selection is mostly based on their own defined quality and the prices given by the seller. For musical instruments, the average price of spruce tonewood, as purchased by the wood seller, is close to 500 e per m 3 . Thanks to the selection process, the variability is reduced. Most of the time samples showing nodes and twisted grain are rejected. At the end of the sorting process, tonewood is graded according to a nomenclature that depends on the logger, who adapts the selection as a function of market choices and evolution.

Tonewood seller criteria

Following a discussion with a local speciality wood seller 1 , tonewood grading follows the set of guidelines summarized in table 1. It should be pointed out that the guidelines and classification criteria are not systematic, and the attribution of a grade may vary. The main criteria are the cutting orientation (with medullary rays clearly visible due to quarter sawing), the density, the percentage of latewood, the uniformity of the annual growth rings and their width. The seller pointed out that the selection process differs between plates (for guitar making) and quarters (for violin making). For plates, it is very important that the grain be not twisted at the full scale of the sample. Moreover he adapts the grades as a function of request, market and availability. During the sorting process, he also favours the homogeneity of the annual rings to avoid strong expansion, and a gradual narrowing from the center of the tree to the bark. The seller also remarked that since the 1980's, instrument makers are more demanding, especially about the longitudinal stiffness and density for the soundboard, and the prices can now range between 10 € and 300 € per pair [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF]. At the same time, studies have shown that an important quality for resonance wood plates was a high radiation ratio associated with a low density [START_REF] Ono | Study on Young's modulus and internal friction of wood in relation to the evaluation of wood for musical instruments[END_REF]. The best grades generally exhibit a low density, the absence of defects and compression wood, and homogeneous annual ring growth. The lower grades tend to exhibit one of the previous defects or inhomogeneous annual rings, and the different grades in lower qualities exhibit these features with different amplitudes and frequency of occurrence. If violin quarters are light in weight, exhibit a straight grain in longitudinal direction, show narrow and constant annual ring width and no compression wood, they are likely to be sorted to the best quality category. Generally, 2nd choice and 1.B may also show some blue stained fungal attacks or compression wood. The density is an important factor in the selection and grading process. The type of tonewood may differ depending on the growth location, and the seller reported that the spruce wood sold by Italian sellers was generally lighter. The position along the height of the tree may vary and he indicated that tonewood is generally selected from the first five meters of the log. Meanwhile, some loggers select tonewood from up to eight to twelve meters. Hence individual loggers may have preferences for selecting wood, above and below the center of gravity of the tree, close to eight meters for a thirty meters height tree. In the case of plates, the sound velocity in the radial direction is one of the criteria for the selection, where a high speed, and so a high specific modulus, is favoured in the radial direction. This last feature suggests that the mechanical properties are also taken into account for plates.

Material properties

In terms of materials properties, the "quality" of tonewood is associated first and foremost with physical (density) and mechanical properties (elasticity and damping), but also to sound velocity, radiation ratio and loudness, which are described in [START_REF] Spycher | Assessment of resonance wood quality by comparing its physical and histological properties[END_REF]. Among the different ways to evaluate the choices of instrument makers when selecting tonewood, the mechanical behavior can potentially provide objective and quantifiable criteria. The different parameters used to quantify the material properties are the elastic moduli (both specific and non specific), the loss factors and the density. Generally, mechanical studies of tonewood consider the density, the longitudinal specific modulus E L ρ and the corresponding loss factor η L . The work presented here focuses on the species used for the making of guitar and violin soundboards, namely European spruce, Picea abies. It is usually considered that spruce tonewood exhibits specific properties, such as high specific modulus in longitudinal direction. Generally, lower densities are observed for the best ranked woods [START_REF] Buksnowitz | Resonance wood [Picea abies] -evaluation and prediction of violin makers' quality-grading[END_REF]. In this study the material properties of the wood are compared as a function of the grading and hence price. The specific studies, dedicated mostly to spruce, are based on a wide variety of experimental methods [START_REF] Brémaud | Anisotropy of wood vibrational properties : Dependence on grain angle and review of literature data[END_REF]. These studies have highlighted an intra and inter individual variability. The different sources of variability in spruce wood are generally well-known and correspond to both macroscopic and mesoscopic structure [START_REF] Dünisch | Relationship between anatomy and vibration behaviour of softwoods and hardwoods[END_REF]. Moreover microscopic features are also at the source of the wood variability. The orientation of the cellulose microfibrils which constitute the S 2 layer of the cell wall of the tracheid, named microfibril angle (MFA) is a soruce of variability. Moreover the degree of cristallinity of the cellulose is also a source of variability. It has been higlighted in [START_REF] Fabisiak | Variation in the Microfibril Angles in Resonance and Non-Resonance Spruce Wood ( Picea abies [L.] Karst.)[END_REF] that spruce tonewood exhibits a microfibril angle (MFA) that is two times lower than regular spruce. This study also showed that the MFA is smaller in latewood than in earlywood. The different studies have led to the mean and standard deviation of spruce properties in all directions as reported in the table 2. Spruce tonewood, in comparison to other softwoods, exhibits higher E L , lower η L , and higher anisotropy ratio [START_REF] Brémaud | Characterisation and categorisation of the diversity in viscoelastic vibrational properties between 98 wood types[END_REF]. In [START_REF] Haines | On Musical Instrument Wood[END_REF], it was observed that the stiffness of tonewoods is ge-nerally higher than the values reported for general woods, which instrument makers assume indicated good quality. A higher specific modulus has been associated with a higher loudness of musical instruments, based on a higher acoustic conversion efficiency value [START_REF] Obataya | Vibrational properties of wood along the grain[END_REF]. It is generally alleged that these characteristics are favourable for instrument dynamics and acoustics, but no objective studies have clearly associated the quality of an instrument with its material properties. Moreover, from a structural dynamics point of view, it has been emphasized that the longitudinal stiffness of the wood used for soundboards and violin backs in the longitudinal direction was not the most influential parameter in comparison to the radial elasticity and longitudinalradial shear stiffness [START_REF] Viala | Model-based effects screening of stringed instruments[END_REF]. Meanwhile, data for the properties of tonewood along the non longitudinal directions is lacking. At this point, two main propositions are raised. It is generally supposed that high specific modulus is better for instruments dynamics, hence E R ρ and G LR ρ should be higher for high grades. In the other cases, high E R ρ and G LR ρ are not associated with good quality woods. So, the tonewood mechanical properties may not be considered as the most influential properties on the dynamics or acoustics of instruments. The last proposition is not in adequation with violin maker criteria for wood choice [START_REF] Carlier | Violin making " tonewood " : comparing makers ' empirical expertise with wood structural / visual and acoustical properties[END_REF].

Objectives

In this paper, a methodology is proposed to simultaneously identify several mechanical tonewood properties, in particular E R and G LR . The objective is to provide a means for monitoring the mechanical behaviour of the material from the raw material to the instrument where traditional destructive methods are not applicable. In this work, the mechanical properties of tonewood will be studied using a non-destructive method enabling the measurement of several elastic and damping parameters in different directions simulaneously in contrast to other techniques which evaluate one direction at a time and require calibrated specimens ( [START_REF] Brémaud | Anisotropy of wood vibrational properties : Dependence on grain angle and review of literature data[END_REF], [START_REF] Bucur | [END_REF], [START_REF] Sotomayor Castellanos | Dynamic characteristics of 22 woods determined by the transverse vibration method[END_REF]). In the present study, we propose to use a nondestructive inverse method based on finite element model updating (FEMU) and 3D vibratory field measurements (3DVF). This approach is motivated by the fact that wood samples used for instrument making are generally plate or quarter shaped. Traditional mechanical tests usually need to cut specimens in raw material which may preclude the future utilisation for instrument making. Moreover, the rarity and the cost of the sample may also be arguments for the use of non-destructive testing. The value of musical instruments, both old and new, is also an obstacle for performing mechanical studies that is alleviated by a non-destructive technique. The proposed approach is non-invasive, fast, easy to set-up, and requires reduced preparation time for samples. In addition, dynamic responses such as the vibratory modes are more representative of the global behaviour of the material in contrast to static properties that are generally driven by local phenomena. These benefits have led to the development of numerically based identification methods over the last thirty years ( [START_REF] Friswell | Finite element model updating in structural dynamics[END_REF], [START_REF] Sol | Identification of anisotropic plate rigidities using free vibration data[END_REF]) to estimate both anisotropic loss factors and stiffnesses of composite plates [START_REF] Mcintyre | On measuring the elastic and damping constants of orthotropic sheet materials[END_REF] or arbitrarily shaped parts [START_REF] Battaglia | Vibration-based identification of mechanical properties of orthotropic arbitrarily shaped plates : Numerical and experimental assessment[END_REF]. It has been noted that the measured elastic modulus E dynamical is generally close to 1.1 times E static in the longitudinal direction [START_REF] Haines | Determination of Young's modulus for spruce, fir and isotropic materials by the resonance flexure method with comparisons to static flexure and other dynamic methods[END_REF] and 1.2 times larger in the radial direction, thus leading to discrepancies in the trends obtained which each method. In 2016, a review paper [START_REF] Tam | Identification of material properties of composite materials using nondestructive vibrational evaluation ap-proaches : A review[END_REF] made a list of the works relating to the identification of material properties using vibrational approaches. This paper highlights the fact that, despite its potential, this method has mostly been applied to flat plate specimens and not more complex shape parts, which is covered in this study by the application to violin quarters. The proposed method is described in detail in [START_REF] Viala | Identification of the anisotropic elastic and damping properties of complex shape composite parts using an inverse method based on finite element model updating and 3D velocity fields measurements (FEMU-3DVF) : Application to bio-based composite violin sou[END_REF] and applied to a violin soundboard made of biobased composite. The FEMU-3DVF method deals with a finite element model and and experimental set-up comprising an acoustic exciter (speaker) and a Doppler effect laser 3D vibrometer. This allows the determination, through the measurement of velocities at the surface of the specimen in the three material directions, of the experimental modal basis. The identification process consists in the minimisation of the discrepancies between the frequencies of the matched test-analysis modes. In what follows, specimens will be tested using the FEMU-3DVF method and the mechanical parameters are reported as a function of the grading of each sample in the results section. Finally, conclusions are drawn based on the results and perspective studies are proposed.

Material and methods

Materials

The tonewood species studied is European spruce, Picea abies. The samples have been provided by a local sawmill specialised in tonewood, Le bois de lutherie, Fertans, France in the form of plates and quarters. The plates are used for guitar making and are flat. The quarters are bulk pieces with trapezoidal cross sections and are used for the back and the top of the violin where they are carved to create an arch. The dimensions of the quarters vary, and figure 2 (a) reports the average values. For all the quarters, the dimensions are measured with a rule and a calliper. The precision of the length measured with the rule is ± 0.5 mm and the precision of the trapezoidal shape, measured with a calliper is ± 0.1 mm. The samples are weighed with a KERN ® weighing machine whose precision is ± 0.1 g. The average relative error on the evaluation of the mass is equal to ± 0.02 %. The volume is evaluated from the dimensions of the parts. The relative error on the estimation of the volume is estimated to ± 1.4 %. The density is evaluated as the ratio between the mass and the volume. The relative error in the evaluation of the density of the quarters is close to 1.5 %. The impact of measurement errors on the identification of material parameters is detailed in the section 6 of the data in brief document. Ten spruce quarters have been studied and are labelled i with i varying from 1 to 10. Table 4 in the results section indicates for each quarter the label, references, and harvest year given by the first two digits. Their density and their "grading rank" as sold by the sawmill are also given. Note that some spruce quarters exhibit indented rings patters, known as bear-claw wood, described in [START_REF] Račko | The onset of hazel wood formation in Norway spruce ( Picea abies [ L .] Karst .) stems[END_REF].

The wood used for guitar soundboards is sold in the form of plates. Figure 2 (b) shows a sample plate and its dimensions. The dimensions of the plates vary, and the figure only indicates an average value. The dimensions of the plates are measured with the same apparatus as the quarters while a thickness calliper is used to evaluate their thickness with Considering the conditioning values, and using sorption isotherms [START_REF] Simpson | Specific Gravity , Moisture Content , and Density Relationship for Wood[END_REF], the moisture content MC is comprised between 9 and 11 %.

Mixed numerical-experimental method description

The numerical-experimental method used is fully described in [START_REF] Viala | Identification of the anisotropic elastic and damping properties of complex shape composite parts using an inverse method based on finite element model updating and 3D velocity fields measurements (FEMU-3DVF) : Application to bio-based composite violin sou[END_REF]. The measurement of the velocity field is made with a Polytec PSV-500 vibrometer as shown in the figure 2 (c). The excitation is produced by a loudspeaker, between 10 and 2000 Hz for the plates and 100 to 5000 Hz for the quarters. The acoustic excitation on the sample leads to strain levels close to 150.10 -9 (flexural strain) and 200.10 -9 (torsional strain). A modal analysis using the Polymax method is performed on the nonnormalized frequency response functions using an in-house numerical tool called MODAN developed using MATLAB ® . The measured and synthesized frequency response functions are provided in the data in brief document. The modal dampings of each modes are evaluated with the modal analysis. The modal damping of the purely torsion and flexural modes are used for the loss factors estimation. The loss factors η L , η R and η LR are estimated as the double of the modal damping ξ of the modes 2, 4 and 1 (shown in the figure 4 and 6 of the data in brief document), respectively. In the identification process, the eigenmodes computed using a finite element model of the structures are compared to the experimentally evaluated modes. The identifiability of the elastic and damping parameters is based on the sensitivity analysis of each mode. The error in the matched frequencies [START_REF] Allemang | A correlation coefficient for modal vector analysis[END_REF] is reduced using a first order sensitivity algorithm. At the end of the process, the discrepancies between experimental modal basis and computed ones are reduced (as shown in the data in brief document) and the elastic parameters are identified. The error in the estimation is based on algorithm convergence and the evaluation of the geometry and density of the samples. The FEMU-3DVF method has been applied to thirty spruce samples, divided in twenty plates and ten quarters. Preliminary results like sensitivity analyses and identification er-rors due to measurement and method errors are given in the data in brief document.

Results

Density evaluation

The values of the densities of the samples (quarters and plates) are given as box and whiskers plots in the figure 3. The mean value (0.45 g.cm -3 ) is in accordance with the usual values for spruce (0.43 g.cm -3 ) from [START_REF] Brémaud | Anisotropy of wood vibrational properties : Dependence on grain angle and review of literature data[END_REF]. The density of all the samples are given as a function of the grading in the figure 3. It is shown that high grade spruce is lighter than the other grades, as observed in [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF], and its dispersion is at least 5 times smaller, as expected [START_REF] Buksnowitz | Resonance wood [Picea abies] -evaluation and prediction of violin makers' quality-grading[END_REF], but it should be pointed out that the number of high grade samples is lower and the samples are only plates. Nevertheless, the number of samples is very small for the first grading and this needs to be taken into account when interpreting the results.

Identified material properties for spruce quarters and plates

The matched eigenfrequency errors are given before and after finite element updating process in the data in brief document for each specimen. The average absolute error of matched eigenfrequencies is very low and implies a good correlation between the experimental and numerical modal bases thus validating the underlying assumptions for the material's mechanical behaviour, and especially the identifiability of the elastic parameters. Once the finite element model updating is performed, the values that minimize the error of each elastic parameters are used in the following paragraph that lists the material properties. The identification errors attributed to geometrical measures and method algorithm are given in the section 6 of the data in brief document. Depending on the sample type studied, the sum of each type of error is close to 4 and 5 % for plates and quarters, respectively. Initially, the average values of the identified properties on the spruce wood are given in the table 2. The results for each sample and specimen type are also provided. The global results, given in the table 2 are in adequation with the standard values of spruce available in literature [START_REF] Brémaud | Anisotropy of wood vibrational properties : Dependence on grain angle and review of literature data[END_REF]. The standard deviation of each parameter is high, even for a highly selected material such as spruce tonewood. Generally, spruce tonewood is more rigid in the longitudinal direction and more anisotropic than standard softwoods. The shear modulus, G RT is much lower than the one predicted by [START_REF] Guitard | Modèles prévisionnels de comportement élastique tridimensionnel pour les bois feuillus et les bois résineux[END_REF], and in accordance with the one usually measured for spruce woods. Figure 4 gives the dispersion in the elastic and damping values for all the gradings and specimen type mixed in the results cases "all". It should be pointed out that the variability covers a wide span of values depending on the parameter. In the longitudinal direction, the covariance of E L ρ and η L are equal to 10 and 13 %, respectively. In the radial direction, the covariance is equal to 20 % for E R ρ and 10 % for η R . In the LR plane, the specific shear modulus covariance is equal to 14 %, and, for the corresponding loss factor η L R, equal to 16 %. Finally, the covariance of the density is smaller and equal to 6 %.

Plate specimens

Based on the results of the table 3, the longitudinal modulus of spruce plate specimens ranges from 8.7 to 17.6 GPa, for densities ranging between 0.4 and 0.5 g.cm -3 which is in accordance with what was expected based on Guitard's models [START_REF] Guitard | Modèles prévisionnels de comportement élastique tridimensionnel pour les bois feuillus et les bois résineux[END_REF]. Nevertheless, these relations are not systematically verified. The loss factor in the L direction ranges from 0.6 to 1.0 %, and in R direction from 1.4 to 2.2 %. It has to be highlighted that a high η L is often correlated with low η R , which can be explained by the MF orientation. It is shown that the specific rigidity in the L direction covers a broad range, from 20 to 35 GP a.g -1 .cm 3 . In the radial direction, the values of the specific elastic modulus range from 1.46 to 2.78 GP a.g -1 .cm 3 between the minimal and maximal values. In the longitudinal radial plane, the values of the specific shear modulus range from 1.48 2.44 GP a.g -1 .cm 3 .

Quarter specimens

The properties identified on the quarters are reported in the table 4. The mean density is equal to 0.46 g.cm -3 . The average values of the Young's moduli in L and R directions are equal to 13.8 and 1.2 GPa, respectively. The loss factors in the same directions are equal to 0.75 and 1.75 %, respectively. In the LR plane the average value of the shear modulus is equal to 0.8 GPa, and the loss factor in this plane is equal to 1.3 %. The averaged values of the shear moduli in the RT and TL planes are equal to 47 MPa and 0.85 GPa, respectively. As a reminder, the usual values are given in the table 2. The averaged values of the specific Young's moduli in L and R directions are equal to 29.4 and 2.8 GPa, respectively. In the LR plane the specific shear modulus averaged value is equal to 1.8 GPa. The usual values of the specific parameters are also given in the table 2.

Comparison of the mechanical parameters as a function of grading

This subsection compares the mechanical properties with respect to the grading levels. In order to evaluate the correlation between instrument maker choices (and/or price grading) and mechanical properties, the specific elastic and damping properties are regrouped in box and whiskers plots, shown in the figure 4. It is observed that the loss factor in the L direction tends to increase when the grading decreases, as observed in [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF], which is not observed in the other material directions. Moreover, in the radial direction, the loss factor tends to decrease when the grading decreases. In the LR plane and R direction, the loss factors increase when the grading decreases, but surprisingly are also low for the "lowest" grade. It is shown that, depending on the grading, the evolution of the rigidities and loss factors in the L and R directions and the LR plane vary. Considering the specific mechanical parameters, trends for specific moduli in R direction and LR plane are observed. The specific longitudinal modulus mean value decreases when the grading decreases but the variaiblity remains very high to exhibit clear trends, it is nevertheless lower for lower quality, as observed in [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF]. The contrary is observed for E R and G LR . These observations tend to contradict the commonly suggested criteria guiding instrument makers choices, namely that choosing the best wood for instrument making implies a lower specific rigidity in R and LR directions. Nevertheless, it has been shown in [START_REF] Viala | Model-based effects screening of stringed instruments[END_REF] that these moduli are particularly influential on the dynamical behaviour of the musical instruments. The clear dispersion of the material properties indicate that it is possible to find in lower qualities woods that exhibits mechnaical properties close the the best one, which was also pointed out in [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF].

Sample Grading Reference Density

E L ρ (GP a.g -1 .cm 3 ) E R ρ (GP a.g -1 .cm 3 ) G LR ρ (GP a.g -1 .cm 3 ) η L (%) η R (%) η LR (%) I 1.A 13C100

Sample Grade Ref. Density

E L ρ (GP a.g -1 .cm 3 ) E R ρ (GP a.g -1 .cm 3 ) G LR ρ (GP a.g -1 .cm 3 ) η L (%) η R (%) η LR (%) G RT (MPa) G T L (GPa) 1a 1.A 10Z120

Conclusions

The main objective of this work was to simultaneously identify the material properties of spruce tonewood for different gradings at the scale of the raw material. For this purpose, plate and quarter specimens were tested using the FEMU-3DVF method. The results show that at least three rigidities, E L , E R and G LR as well as the damping properties in these corresponding directions can be determined successfully at once. The identified parameters using this type of metohds were comparable to the properties determined on the same specie by numerous previous studies, using dynamic and static direct methods. Moreover, the method used provides a global evaluation, that prevents from intra specimen high variability pointed out in [START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF], which is equivalent to inter specimen variability It has been shown that the assumption that high specific modulus and low damping is a key choice for instrument acoustics is verified in the longitudinal direction. However, this hypothesis is shown to be invalid in the radial and longitudinal directions which can even exhibit inverse trends. As these directions are also influential on the dynamics of the violin and guitars, the usual selection criteria are controversial and a fine description of the impact of these directions on the dynamics and acoustics of the full instrument is required to provide an objective assessment of the impact of wood choices, which is still considered to play a prime role in the perceived overall "quality" of a musical instrument. This study highlights the fact that a previously selected biological material may exhibit high variability and the identified parameters exhibited variations depending on the directions, ranging from 10 % (longitudinal direction) and 20 % (radial direction). The covariance of the density for the considered samples is equal to 6 %. Hence, a wood selection based uniquely on a single objective property like density is not sufficient to ensure a small variability of elastic and damping parameters. The high variability in material properties can be considered as an opportunity for substitution materials, either biological or manufactured, that exhibit elastic and damping parameters comprised within the intrinsic variability of tonewood. In conclusion, this study provides a starting point and incentive to update the classification and grading procedures of spruce tonewood. Towards this end, two main steps are still required. Firstly, the impact of each parameter on the vibro-acoustics of each instruments must be quantified. Secondly, the association between tonewood mechanical properties and musical instrument sound preferences has to be studied.

  modulus in L direction.• E R ρ : specific modulus in R direction. • G LR ρ

Figure 1 :

 1 Figure 1: Summary of the criteria for tonewood grading.

Figure 2 :

 2 Figure 2: Dimensions of the quarters (a) and plates (b), (c), experimental setup of the FEMU-3DVF method applied to wooden plates.

Figure 3 :

 3 Figure 3: Box and whiskers plots of the density as a function of gradings. The horizontal line inside the boxes corresponds to the median. The lower and upper boundaries of the boxes correspond to the lower and upper quartiles respectively. The lower and upper limits of the dashed lines correspond to the 9 and 91 % percentiles respectively.

Table 1 :

 1 Spruce tonewood grading key and price as a function of the gradings for both plates (guitars) and quarters (violins), according to a personal communication with tonewood seller.

	Grade	Anatomic features	Aesthetic features	Density	Defaults	Price (€)	Mechanical parameters
			Plates (600 × 200 × 5 mm, pair)
	1.S	Low MFA, homogeneous	Medullary rays	Lightest	No	65	High radial modulus
	1.A	Homogeneous	Medullary rays	-	No	45	-
	1.B	Irregular growth rings	Compression wood	-	No	30	-
	2nd	Irregular growth rings	Small blue stain fungi/compression wood	Possibly heavy	Resin pockets accepted	18	-
			Quarters (450 × 150 × 28 mm, pair)
	1.S	Low MFA, homogeneous	Medullary rays	-	No	43	High longitudinal modulus
	1.A	Homogeneous	Medullary rays	-	No	30	-
	1.B	Irregular growth rings	Some compression wood	-	No	20	-
	2nd	Irregular growth rings	Small blue stain fungi/compression wood	Possibly heavy	Resin pockets accepted	12	-

Table 2 :

 2 Summary of the mechanical parameters of spruce, and comparison with bibliography values[START_REF] Brémaud | Anisotropy of wood vibrational properties : Dependence on grain angle and review of literature data[END_REF],[START_REF] Carlier | Revisiting the notion of " resonance wood " choice : a de-compartementalised approach from violin makers ' opinion and perception to characterization of material properties[END_REF] (labelled as Ref.). For the presented results, the value of the MC is comprised between 9 and 11 %.

		Parameter	Mean	RSD (%)	Range	Ref. mean	Ref. RSD (%)	Ref. range
	Density ρ (g.cm -3 )	0.45	7	0.39-0.51	0.42	13	0.32-0.55
		E L (GP a)	12.8	12	8.7-17.6	13.5	21	9-16.2
		E R (GP a)	1.01	22	0.6-1.93	0.92	21	0.54-1.12
		G LR (GP a)	0.81	15	0.57-1.2	0.93	17	0.7-1.2
	E L ρ (GP a.g -1 .cm 3 )	29.0	10	21-35	29.7	11	18-36
	E R ρ (GP a.g -1 .cm 3 )	2.28	20	1.46-3.82	2.2	18	1.7-3.6
	G LR ρ	(GP a.g -1 .cm 3 )	1.85	14	1.3-2.4	2.2	-	-
		η L (%)	0.73	12	0.55-1	0.68	11	0.6-1.0
		η R (%)	1.7	10	1.35-2.2	2.0	19	1.55-2.7
		η LR (%)	1.2	17	0.7-1.6	1.8	6	1.6-2.0
		G RT (GP a)	0.046	22	0.031-0.073	0.035 [32]	-	0.03-0.045
		G T L (GP a)	0.79	21	0.5-1.2	0.8	-	0.48-1.12

Table 3 :

 3 Identifed mechanical properties for spruce plate samples.

Table 4 :

 4 Identified material properties for spruce quarter samples.

				0.415	27.2	1.53	1.96	0.65 1.9 1.05
	II	1.A	13I81	0.44	27.3	1.46	1.82	0.65 1.9 1.25
	III	1.A	13F286	0.43	20.7	1.87	2.4	0.85 1.80 1.25
	IV	1.B	9J75	0.425	29.4	1.56	1.78	0.60 2.2 1.40
	V	1.B	13F247	0.46	28.0	2.29	2.06	0.7	2	1.55
	VI	1.B	9H356	0.46	29.2	2.0	1.73	0.6 1.72 1.28
	VII	2nd	10J256	0.48	23.1	2.3	2.09	0.85 1.4	1.1
	VIII	2nd	10H322	0.4	21.7	2.62	2.44	1	1.40 0.95
	IX	2nd	10N447	0.41	24.5	2.57	2.23	0.65 1.45 0.8
	X	1.S	14A257	0.41	30.5	1.75	1.67	0.6 1.75 1.05
	XI	1.S	14A155	0.40	29	1.98	1.48	0.6 1.65 0.85
	XII	1.S	14A87	0.41	30.35	1.59	1.58	0.65 1.80 1.05
	XIII	1.A	14C69	0.43	30.7	1.74	1.94	0.65 1.65 1.15
	XIV	1.B	12F23	0.455	31	2.13	1.97	0.6	1.4	1.3
	XV	1.B	15E238	0.50	35.4	2.35	1.74	0.6	1.8 1.45
	XVI	1.B	13J27	0.405	23.5	2.78	2.44	0.85 1.45 1.45
	XVII	2nd	08	0.47	31	2.52	1.30	0.7	2	-
	XVIII	2nd		0.45	29.7	2.41	1.64	0.65 1.65	-
	XIX	2nd	08	0.43	27.7	2.54	2.32	0.9 1.40 0.65
	XX	1.S	12	0.41	32.7	2.59	1.81	0.9 1.75 1.45
	Average	-	-	0.435	28.1	2.1	1.9	0.7	1.7 1.2

  Figure 4: Average rigidities and loss factors as a function of the grading. The horizontal line inside the boxes corresponds to the median. The lower and upper boundaries of the boxes correspond to the lower and upper quartiles respectively. The lower and upper limits of the dashed lines correspond to the 9 and 91 % percentiles respectively. The out of bounds crosses correspond to extreme values out of 9 % percentiles.

		x 10 4
		3.6				1
		3.4
	-3 )					0.9
	E L /ρ (MPa.g.cm	2.4 2.6 2.8 3			η L (%)	0.6 0.7 0.8
		2.2				0.5
		1.S 1.A 1.B 2nd All		1.S 1.A 1.B 2nd All
			Grading		Grading
		3500
		3000
		2500
		2000
	E
		1500
		1.S 1.A 1.B 2nd All
			Grading
					0.51	32.0	3.06	1.49	0.9	1.90 1.55	49	0.83
		2a	1.A	11K5	0.505	26.9	3.82	2.37	0.75	1.5	1.4	73	1.12
		3a	1.A	110131	0.43	28.2	2.45	1.34	0.75 1.75 1.45	30	0.63
		4a	1.A	13A360	0.46	34.5	2.94	1.92	0.7	2.2	1.35	36	0.68
		5a	1.B	10Q89	0.43	31.0	2.24	1.40	0.75	1.8	1.4	42	0.575
		6a	1.B	13C58	0.48	33.2	2.58	1.71	0.7	2.0	1.35	44	0.76
		7a	1.B	13C38	0.49	29.5	2.33	2.7	0.7	1.65 1.30	47	0.87
		8a	2nd	96WG46	0.43	20.7	3.09	2.10	0.75	1.4	1.1	65	1.2
		9a	2nd	98WI40	0.45	31.4	1.84	1.60	0.70 1.75 1.25	54	0.93
		10a	2nd	96WG67	0.44	26.9	3.5	1.99	0.75	1.4	1.25	48	0.85
	Average	-	0.46	-	29.4	2.8	1.8	0.75 1.75 1.35	47	0.85

R

/ρ (MPa.g.cm

Acknowledgements

Funding : This work has been performed in the Framework of EUR EIPHI (ANR-17-EURE-0002). The authors acknowledge Bernard Michaud, from the sawmill "Le bois de lutherie", Fertans, FRANCE.


Livia Avila De Oliveira

Gabriela Luiza

Corat Coura

Maikson Luiz

Passaia Tonatto

Hallak Tulio

Vincent Panzera
email: panzera@ufsj.edu.br

Fabrizio Placet

Scarpa

Corat Coura

Maikson Luiz Passaia

Lívia Ávila De Oliveira

Cota Coura

Maikson Luiz Passaiatonatto

Túlio Hallak Panzera

Vincent Placet

Fabrizio Scarpa

A novel sandwich panel made of prepreg flax skins and bamboo core

Keywords: sandwich panel, flax fibre, bamboo, thermosetting, flexural and shear properties

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

Introduction

Sandwich panels are lightweight structures employed in many engineering applications, such as aerospace and automotive components, military devices and civil infrastructure [START_REF] Cabrera | Design and manufacture of all-PP sandwich panels based on co-extruded polypropylene tapes[END_REF][START_REF] Hu | Dynamic crushing of the circular-celled honeycombs under out-of-plane impact[END_REF][START_REF] Gholami | Optimal design of a honeycomb core composite sandwich panel using evolutionary optimization algorithms[END_REF].

The advantage of the use of sandwich panels lies in their high energy absorption and outstanding mechanical performance [START_REF] Wang | Mechanical behaviors of inclined cell honeycomb structure subjected to compression[END_REF][START_REF] Wang | On the influence of structural defects for honeycomb structure[END_REF][START_REF] Wang | Experimental investigation on bending behavior of honeycomb sandwich panel with ceramic tile face-sheet[END_REF]. A sandwich panel is composed by two thin, stiff and strong sheets/skins commonly made of metals (steel, aluminium) or polymers (polycarbonate, polypropylene). The skins are separated by a thick core made of thin-walled hexagonal, circular or random shaped cells and low-density material [START_REF] Hg | Analysis and design of structural sandwich panels[END_REF][START_REF] D'mello | Synergistic energy absorption in the axial crush response of filled circular cell honeycombs[END_REF].

In recent years, the awareness of global environment challenges has led researchers to study alternative materials for the development of sustainable sandwich panels [START_REF] Oliveira | Sustainable sandwich composite structures made from aluminium sheets and disposed bottle caps[END_REF][START_REF] Dutra | Investigations on sustainable honeycomb sandwich panels containing eucalyptus sawdust. Piassava and cement particles[END_REF][START_REF] Smardzewski | Wooden sandwich panels with prismatic core -Energy absorbing capabilities[END_REF]. The use of fibrous resources available in agriculture is the most widely explored way of developing those sustainable materials because of their availability and accessibility, biodegradable properties, renewability, low cost, low density and specific mechanical properties for secondary structural applications [START_REF] Ramesh | Plant fibre based bio-composites: Sustainable and renewable green materials[END_REF][START_REF] Ku | A review on the tensile properties of natural fiber reinforced polymer composites[END_REF][START_REF] Pickering | A review of recent developments in natural fibre composites and their mechanical performance[END_REF][START_REF] Regazzi | Coupled hydro-mechanical aging of short flax fiber reinforced composites[END_REF][START_REF] Fiore | Aging resistance of bio-epoxy jute-basalt hybrid composites as novel multilayer structures for cladding[END_REF].

The use of flax fibres dates back to 5000 BC as the first known fibre extracted from plants to reinforce matrices [START_REF] Fiore | Effects of aging in salt spray conditions on flax and flax/basalt reinforced composites: Wettability and dynamic mechanical properties[END_REF][START_REF] Ramesh | Flax (Linum usitatissimum L.) fire reinforced polymer composite materials: A review on preparation, properties and prospects[END_REF]. Currently, flax is the most widely used bio-fibre due to its good mechanical properties and versatility. Flax is also widely available [START_REF] Zivkovic | Influence of moisture absorption on the impact properties of flax, basalt and hybrid flax/basalt fiber reinforced green composites[END_REF]; it can be found in simple household textiles, furnishing fabrics and decoration accessories [START_REF] Yan | Flax fibre and its composites -A review[END_REF]. Flax has also been developed into prepreg textiles for more technological applications, such as transport, wind energy, sports and leisure [21].

Another natural material that has gained popularity within the scientific community is bamboo. Bamboo has been widely used to build permanent and temporary structures [START_REF] Li | Axial load behavior of structural bamboo filled with concrete and cement mortar[END_REF]. In addition to all the advantages of bio-based plants, bamboo features a fast growth rate, a high strength-to-weight-ratio, good flexibility and lightweight characteristics that are compatible with engineering requirements for use in civil constructions [START_REF] P.-H. Lee | Development of a hollow cylinder test for the elastic modulus distribution and the ultimate strength of bamboo[END_REF][START_REF] Li | Mechanical performance of laminated bamboo column under axial compression[END_REF][START_REF] Krause | On the influence of Dendrocalamus giganteus bamboo microstructure on its mechanical behavior[END_REF][START_REF] Jakovljevic | The influence of humidity on mechanical properties of bamboo for bicycles[END_REF]. In addition, the circular geometry of the bamboo stem offers the potential to be used in developing sustainable circular core materials for sandwich structures.

Besides the correct selection of the materials for skins and cores, skin-core bonding is another critical factor that directly affects the mechanical performance of sandwich panels because the adhesion to the single sections of the core are the structurally weakest parts of the sandwich [START_REF] Cen | Mechanical behavior of novel GFRP foam sandwich adhesive joints[END_REF]. Two types of bonding are commonly used in composite structures: mechanical and adhesive [START_REF] Cen | Mechanical behavior of novel GFRP foam sandwich adhesive joints[END_REF]. The adhesive bonding is the most widely used because of the intrinsic low stress concentrations, effective weight reduction, uniform stress and load distribution [START_REF] Calik | Effect of adherend shape on stress concentration reduction of adhesively bonded single lap joint[END_REF]. The strength of bonded joints depends on several factors, such as the type of the specific joint, its geometric parameters, the bonding materials and the charcateristics of the adhesive [START_REF] Domingues | Experimental and numerical failure analysis of aluminium/composite single-L joints[END_REF]. Epoxy adhesives represent the most common type of structural adhesive due to their relatively high modulus and strength [START_REF] Kinloch | Toughening structural adhesives via nano and micro-phase inclusions[END_REF].

The engineering use of natural or bio/eco-based materials is promising, but unfortunately leads to a variability of the mechanical and physical properties of the structures made of those solids. The scarcity of large datasets of data with statistichal robustness also does not help to completely understand the physics and the mechanics underpinning the mechanical behaviour of bio-based structures. A numerical analyses was performed to establish the elastic constants of bamboo structures along their transverse axes [START_REF] Garcia | Experiments with rings to determine the anisotropic elastic constants of bamboo[END_REF]. Some studies have used a finite element method based on graded finite elements to obtain orthotropic constitutive properties through the bamboo wall [START_REF] Candelaria | Determination of the properties of bambusa blumeana using full-culm compression tests and layered tensile tests for finite element model simulation using orthotropic material modeling[END_REF][START_REF] Silva | Modeling bamboo as a functionally graded material: lessons for the analysis of affordable materials[END_REF]. Some experimental and numerical studies related to the flexural properties of sustainable sandwich composites made from a circular bamboo core and biodegradable skins have been investigated recently. Darzi et al. [START_REF] Darzi | Numerical study on the flexural capacity of ultra-light composite timber sandwich panels[END_REF] carried out a numerical study on the flexural capacity of ultralight sandwich panels made of plywood faces and bamboo core, using a proposed Ritz method and a validated Finite Element Analysis (FEA). Hartoni et al. [START_REF] Hartoni | Effect of core and skin thicknesses of bamboo sandwich composite on bending strength[END_REF] investigated, through experimental analysis, the flexural strength of sandwich composites made with bamboo core and multiplex skins, varying the thickness of the core and skin. Loth and Forster [START_REF] Loth | Comboo -a novel core material for composite sandwich structures from renewable sources[END_REF] developed and characterised a sandwich panel made of bamboo core and glass/flax fibre skins comparing flexural strength with other commercial core materials. In this context, there is much to understand using the bamboo ring as the core material of sandwich panels, e.g. the effects of bamboo species and geometric characteristics, core packing, type of adhesives and skins on different loads and respective failure modes, in addition, besides relating the experimental and numerical analysis. Furthermore, circular cores of bamboo rings lead to a substantial stress concentration in the skins and adhesives under bending loads because of the high strength and stiffness of those rings. The high stress concentrations decrease the structural efficiency of the panels. In order to better understand such effects, two types of adhesives and cell dimensions are also evaluated in this work. The sandwich panels with unidirectional prepreg flax skins and the bamboo core are assessed here by performing a full factorial design of experiment (DoE) and a finite element analysis to better understand the behaviour of the panels under three-point bending tests.

Materials and Methods

Materials

The skins of the sandwich panel are made of unidirectional flax fibre reinforced composite (Figure 1a). The architecture of the uni-directional skin laminate consists in flax fibres pre-impregnated with fire-retardant epoxy polymer XB 3515 GB (Huntsman). The epoxy is also combined with Aradur 1571 BD and Accelerator 1573 BD. The prepreg flaxtape of 220 g/m 2 grammage (fibre+resin) is supplied by EchoTechnilin (Lineo, France), considering an impregnation weight ratio of 50/50 fibre/matrix. A fibre/matrix volume fraction of 56/43% is estimated by analyzing the surface of the image obtained by Tescan Mira3 scanning electron microscope operating at 20 kV. The Tescan Essence software is used to recognise and determine the surface area of each constituent.

The core of the sandwich panels is composed of treated bamboo rings (Figure 1b)

Bamboo treatment

Bamboo culms are harvested and left upright for two weeks to drain and ferment the starch and sugar present in the culms [START_REF] Garcia | Experiments with rings to determine the anisotropic elastic constants of bamboo[END_REF]. Subsequently, the bamboo rings are cut by a bandsaw, with the dimensions shown in Figure 3a. The rings are immersed in a 3% (m/v) boric acid solution and 1% (m/v) copper sulfate for seven days to prevent biological degradation (Figure 3b). Finally, the bamboo rings are oven dried at 50 o C for three days before being used in the manufacture of the sandwich panels (Figure 3c).

Sandwich panels

Prior to the sandwich manufacturing, the polymer adhesives are prepared by hand mixing the components for five minutes, i.e., AW 139 + hardener XB 5323, from Huntsman and SR1124 resin + SD4775-1 hardener, from Sicomin. The adhesives are spread separately on the flax skins using a 1.5 mm deep zigzag spatula for the Huntsman adhesive, while a brush is used for for the Sicomin polymer (Figure 4a). The bamboo rings are then bonded with the adhesive to the flax composite skins in a hexagonal packing (Figure 4b). No adhesive is used between the adjacent bamboo rings. The opposite skin is filled with adhesive and then placed over the bamboo core. Cold bonding is performed by applying a uniform pressure of 3.

Characterisation

Characterisation of the single material components

The polymeric adhesives.

Characterisation of the sandwich panels

Sandwich panels are characterised by the three-point bending tests, as shown in Figure 6.

The tests are carried out using a Roell Amsler with a 25 kN load cell at 6 mm/min, with spans of 130 and 190 mm for the sandwich panels made with the Ø20 and Ø30 mm bamboo rings, respectively, according to ASTM C393 [START_REF]Standard Test Method for Core Shear Properties of Sandwich Constructions by Beam Flexure[END_REF]. The mechanical responses measured in the test are flexural strength and modulus [START_REF]Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials[END_REF], considering an equivalent homogeneous material; skin stress [START_REF]Standard Test Method for Core Shear Properties of Sandwich Constructions by Beam Flexure[END_REF][START_REF] Bitzer | Honeycomb Technology: Materials, Design, Manufacturing, Applications and Testing[END_REF]; core shear stress and core shear modulus [START_REF] Hg | Analysis and design of structural sandwich panels[END_REF][START_REF] Astm D | /D7250M -16[END_REF]. The equivalent density of the sandwich panels is also assessed by measuring the dimensions and mass of the panels using a caliper and a precision scale.

Figure 6. Characterisation of the sandwich panel via three-point bending test.

Finite element (FE) model

The Finite element (FE) model of sandwich panel under the three-point bending is develop to compare and benchmark against the experimental results. A non-linear FE model is created using the Abaqus TM /Standard software version 6.14 in order to obtain the cohesive failure. Figure 7 shows the 3D representative model of the panel, with the flax skins and bamboo core, which are modelled as orthotropic materials. The engineering constants of the flax reinforced composites are calculated using simplified micromechanics equations reported by Chamis [START_REF] Chamis | Simplified Composite Micromechanics For Predicting Microstresses[END_REF] and calibrated based on the elastic modulus (ET) obtained during the tensile tests.

The bamboo engineering constants are obtained based on their microstructural characteristics.

Bamboo is known for its sclerenchyma (fibrous phase) and parenchyma constituents [START_REF] Palombini | Bionics and design: 3D microstructural characterization and numerical analysis of bamboo based on X-ray microtomography[END_REF]. Its cross section shows a fibre bundle gradient that increases the fibre volume fraction from the inner side to outer side. This behaviour changes the mechanical properties along the thickness.

The bamboo cross section is therefore divided into three layers (inner, middle and outer) as shown in Figure 7 (detail A). The inner, middle and outer layers correspond to 60%, 30% and 10% of the total thickness. Different elastic constants calculated according to the fibre volume fraction are assigned to each layer (15 wt% for the inner, 35 wt% for the middle and 50 wt% for the outer layer [START_REF] Dixon | The structure and mechanics of Moso bamboo material[END_REF]). The elastics constants of the bamboo layers and flax reinforced composites used in the FE model and their explanation are shown in Table 1. Skins and bamboo are not modelled using failure criteria or degradation law, given that the main failure is debonding between the skins and the bamboo. Flax composite skins are modelled using S4 linear shell elements and three integration points each layer. The bamboo core is meshed using SC8R continuum shell elements. Each layer of bamboo is modelled using one elements through the thickness, resulting in three elements. The mesh convergence study is carried out with three different mesh refinements.

The nomenclature used corresponds to the following parameters: ls/ws /lc/dc, where ls is the number of elements along the skin length, ws is the number of the elements along the skin width, lc is the number of elements along the core length and dc is the number of elements along the core perimeter. The following meshes are used: MESH 1 totalling 1012 elements; MESH 2 with a total of 3000 elements and MESH 3 -with 6520 elements. After performing the simulations the convergence study shows that the results presented using MESH 3 do not differ significantly in relation to the ones from MESH 2. In addition, MESH 3 has a computational time 150% higher when compared to MESH 2. MESH 2 has been, therefore, chosen to perform all analyses.

The main objective of the model is to simulate the bending behaviour (force vs.

displacement curve) and to understand the failure mode, and therefore develop a model that can simulate the structural behavior of this type of sandwich. These results are presented and detailed in section 3.2.2. Generally, the mechanical properties vary close to the height of the culm. However, the chosen range (Ø20 -30 mm) is small to observe major material variations and the results therefore are

Results

Individual material components of the sandwich panels

Sandwich panel

Table 5 presents the mean (average) values and standard deviations of the responses related to replicates one and two for the sandwich panels, which are statistically interpreted in section 3.2.1. This research involving sustainable sandwich panels made of eco-friendly composite skins and bamboo core is very innovative in the literature. The only similarity found in the literature was reported by Hartoni et al. [START_REF] Hartoni | Effect of core and skin thicknesses of bamboo sandwich composite on bending strength[END_REF], in which sandwich panels made of plywood skins and bamboo rings core were tested under three-point bending. The only variable assessed was flexural strength, which ranged from 6 to 10 MPa, being very close to the results obtained in the present study, as shown in Table 5. Table 6 shows some additional characteristics of the sandwich panels. The percentage of the non-adhesive area is determined by considering the total panel area minus the average surface contact area of the bamboo rings, the whole then divided by the total area of the panel.

Twenty-seven [START_REF] Cen | Mechanical behavior of novel GFRP foam sandwich adhesive joints[END_REF] bamboo rings of Ø20 mm compose the smaller sandwich panels, resulting in a cross-sectional contact area of 3711 mm 2 and a non-adhesive area of 65.6%. On the other hand, twenty-four (24) bamboo rings of Ø30 mm are used for the larger sandwich panels, resulting in a cross-sectional contact area of 5184 mm 2 and a non-adhesive area of 76%.

Statistical Analysis

Table 7 presents the DoE/ANOVA analysis of the responses. Significant effects with Pvalues less than 0.05 and those in bold indicate higher-order effects that will be interpreted by the effect plots. The main effect of a factor should only be interpreted individually if there is no other evidence of significant interactions among factors. The R 2 -adjusted varies from 90.3% to increases respectively, when compared to sandwich panels made with the other manufacturing parameters. Although the compressive strength of the two bamboo rings is similar (Table 4, group A), the flexural strength is mainly related to the core-face adhesion. The thickness of the adhesive can affect the quality of the bonding of the panels, which leads to a lower variation when considering a core with fewer bamboo rings (24 rings when designed with Ø30 mm). The adhesive S presents superior results in terms of flexural strength due to its lower viscosity (Table 4); the S adhesive therefore better penetrates in the rough surface of the flax skins and pores of the bamboo rings. The flexural modulus ranges from 2.2 GPa (Ø20 mm, adhesive H) to 4.2 GPa (Ø30 mm, adhesive S). Figure 10 shows a behaviour similar to the one of the flexural strength, showing increases of 54% and 25% for bamboo rings Ø30 mm and adhesive S, respectively. This increase in modulus for larger bamboo rings is attributed to the higher surface contact area with the polymer (5184 mm 2 , Table5), and this offers higher stiffness and lower deflection during the elastic deformation. Although the H adhesive shows a higher indentation modulus (Table 4), very rigid adhesives can however negatively affect the mechanical performance of the panels, since a brittle adhesive does not follow the deformation of the faces, causing premature cracking and phase debonding.

Figure 10. Main effect plot for the mean (average) flexural modulus.

Skin Stress

Skin stress ranges from 26.9 MPa (Ø20 mm, adhesive H) to 48.9 MPa (Ø30 mm, adhesive S). Figure 11 shows the main effect plots for the average skin stress provided by the bamboo diameters and the type of adhesive used. The graphs show trends similar to the ones related to the bending properties, since this response is calculated assuming that the faces take up all the bending load by compressive and tensile forces [START_REF] Bitzer | Honeycomb Technology: Materials, Design, Manufacturing, Applications and Testing[END_REF]. Skin stress is directly related to the ability of the skin to withstand the load applied to the panel due to its bonding to the core. A 40% increase of skin stress is observed for Ø30 mm bamboo rings (Figure 11a) due to the larger surface contact area with the polymer, while an increase of 23% is noted for adhesive S case (Figure 11b) because of its lower viscosity and higher ductility that improve the bonding of the sandwich panel elements and follows the deformation of the skins.

Figure 11. Main effect plot for the mean (average) skin stress.

Shear Properties

The core shear stress ranges from 0.64 MPa to 0.89 MPa. Figure 12 shows the main effect plot of the adhesive type factor for the mean core shear stress. The core shear stress is closely related to the mechanical properties of the core constituents and their bonding to the faces. Since the two bamboo rings present similar compressive stress and modulus, only the specific type of adhesive used significantly affects this response, with a 30% increase when adhesive S is considered. This fact can be attributed to the lower viscosity and rigidity of this type of adhesive (see Table 4), which improves the core-face adhesion and longer creep-like deformations; the elements therefore tend to bond for longer and avoid the onset of premature cracks in the sandwich beam.

Figure 12. Main effect plot of adhesive type factor for the mean core shear stress.

The core shear modulus ranges from 32.7 MPa (Ø30 mm, adhesive H) to 48.1 MPa (Ø20 mm, adhesive S). An opposite behavior to bending is observed for this response in terms of the diameter effect (Figure 13). Smaller bamboo diameters offer higher shear rigidity due to the larger number of bamboo rings per area and higher adhesive area, which increases the shear constraints, especially when the S adhesive is used. This result also implies that the higher ductility of the S-adhesive and the larger number of Ø20 mm bamboo rings lead to a positive interaction effect that increases the core shear modulus of the panels. Panels made with Hadhesive or Ø30 mm rings with S-adhesive reach equivalent means evidenced by Group B (Tukey test).

Figure 13. Second order interaction effect plot for the mean core shear modulus. Figure 15a shows the state of the samples after failure in the experimental tests and Figure 15b shows the shape deformation in the numerical model. Mirroring is applied to the model symmetry planes for an improved visualization. Similar failure in all conditions are found, where the sudden drop in the force-displacement curves is caused by debonding between the face and the adhesive. This behaviour shows that the failure criteria and the degradation law in the cohesive elements are activated. In general, the onset of debonding mainly occurs on the bottom skin, but in some cases it also occurs on the top surfaces due to the buckling effect of the skins, leading to interfacial pull-out stresses. Some panels also reveal a loss of the bamboo ring, as shown in Figure 15a for the Ø20 mm -S panels. This failure mode also shows a better bamboo-polymer bonding rather than skin-polymer one, which is explained by the presence of porosity in the bamboo cross section that favours an improved interlocking effect. The numerical results also show a similar failure behaviour when compared to the experimental ones. Debonding is mostly found between the bottom skin and the bamboo.

Experimental and numerical comparisons

Conclusions

Sustainable sandwich panels made of unidirectional prepreg flax skins and bamboo rings as circular core material are here developed, characterised and investigated through a statistical experimental design and a finite element model. The effects of adhesive type and bamboo diameter on the equivalent density, flexural and shear properties of the panels are identified.

The main conclusions of this work are the following:

i. The equivalent density of the sandwich panels is affected not only by the density of the adhesive, but mainly by the percentage of non-adhesive area in the core, resulting in a decrease when larger bamboo rings (Ø30 mm) are used;

ii. The sandwich panels with Ø30 mm bamboo rings have a higher superficial contact area with the adhesive, resulting in better core-face bonding with higher flexural properties and skin stress. In contrast, the sandwich panels made with the Ø20 mm bamboo rings reveal an increase in the core shear modulus due to the larger number of bamboo rings per area and higher adhesive area, increasing shear constraints;

iii. The type S adhesive has a lower viscosity and indentation modulus, which positively affects all responses, increasing skin stress, bending and shear properties; iv. All experimental conditions exhibit debonding between the bottom skin and the adhesive, indicating that the polymeric adhesive plays an important role in the mechanical performance of the sandwich panels.

v. The finite element model predicts a good match in terms of load versus displacement curves. The modelling using three-layer bamboo cross sections (inner, middle and outer) and the use of ortotropic properties provides a good agreement with the stiffness of the structure. In addition, the cohesive surface elements provide a substantially correct prediction of the failure and type of damages. The model corroborates the experimental data, which show that the main failure mode of the panels is the debonding between the skin and the bamboo rings.

Finally, the sandwich panels reveal a promising application for secondary structural components, where low cost, moderate stresses and biodegradable characteristics are required, expanding the horizons of the future of sustainable engineering structures.

approximately 3 -Figure 1 .

 31 Figure 1. (a) Skins and (b) core of the sandwich panels.

2. 2 Figure 2 .

 22 Figure 2. Manufacturing tehcniques of the skins prior to autoclave: (a) hand lay-up and (b) vacuum compaction.

Figure 3 .

 3 Figure 3. Bamboo treatment process: (a) dimensions of the cell cores, (b) immersion in solution and (c) drying.

Figure 4 .

 4 Figure 4. Sandwich manufacturing process: (a) polymer spread (Left -Huntsman, Right -Sicomin), (b) bonding of bamboo rings (left -Ø20 mm, right -Ø30 mm), (c) sandwich panels.

Figure 5 .

 5 Figure 5. Characterisation of the single material components: (a) bamboo ring and (b)

A

  quarter model of the panel is modelled using symmetric boundary conditions to represent the three-point bending. The support and the indenter are modelled as analytical rigid surface (3D rigid surface defined in Abaqus) and fixed to a reference point (RP). Boundary conditions (BCs) are applied to RPs. All displacements and rotations are not allowed in the support RP. All displacements and rotations are also not allowed in the indenter RP except for a maximum imposed displacement of 3 mm along the Y-axys (δy). The results of the reaction forces are extracted for the indenter RP and the displacements are monitored in the same point.Cohesive surfaces are used to model the contact between the skins and the bamboo core.Quadratic traction is used to calculate the failure initiation and the degradation of the cohesive surface. Table2shows the cohesive zone parameters used in the model. The damage initiation stress parameters are used to establish the initial failure stress along the normal direction (tnn) and the transverse one (tss = ttt). The fracture energy parameters are used to identify the fracture energy along the normal direction (G1c) and the transverse energy (G11c=G111c); penalty stiffness parameters are used to establish the stiffness of the cohesive layer along the normal direction (Knn) and in the transverse one (Kss=Ktt). It is well known that these interface properties are totally dependent on the type of adhesive used[START_REF] Katsivalis | Development of Cohesive Zone Models for the Prediction of Damage and Failure of Glass/Steel Adhesive Joints[END_REF]. Furthermore, it is very difficult to obtain these values in open literature or through mechanical tests for the specific application considered in this work. In view of this, convergence studies are carried out to identify the cohesive zone surfaces from the experiments (Mode I, Mode II and Mixed Mode), based on datasheet of the Huntsman [37] and Sicomin[38] adhesives. The contact between elements of the bamboo core core is modelled as normal (frictionless hard and tangential behavior). In addition, the contacts between the skins and the support and skin/indenter are modelled using normal behaviour, both hard and tangential without friction. Some input parameters, especially cohesive zone are tuned to minimize the gap between experimental and numerical results. The average of the maximum loads obtained during the three-point bending tests are used to calibrate the largest load drop value. Adjustment is performed using two different samples made with the (i) bamboo Ø20 mm/Huntsman adhesive and the (ii) bamboo Ø20 mm/Sicomin adhesive. These two configurations are used to minimize the difference between the experiment and the numerical analysis for the two different types of adhesives used, and then calibrate the overall finite element model.

Figure 7 .

 7 Figure 7. The 3D FE model used to simulate the three-point bending test: mesh representation, boundary conditions and dimensions.

  within the same average, as shown by group A. The compressive modulus is slightly larger for the Ø30 mm bamboo rings. According to open literature[START_REF] Gottron | Creep behaviour of bamboo[END_REF][START_REF] Xu | Mechanical properties of structural bamboo following immersion in water[END_REF] the fibre volume increases with the culm height, thus, larger diameter bamboo rings (Ø30 mm) have a larger fraction of lignocellusosic matrix and this plays a significant role on matrix-dominated compression properties. The ultra-micro dynamic hardness test shows a superior indentation modulus for the Huntsman (H) adhesive. On the other hand, the density and viscosity of Sycomin (S) adhesive are lower.

Figure 9 .

 9 Figure 9. Main effect plot for the mean (average) flexural strength.

Figure 14

 14 Figure14shows the load versus displacement curves obtained by the three-point bending tests (average of experimental curves) and the finite element model for all experimental conditions. A similar trend is noted, which is described by a long elastic deformation with adhesive crack initiation, followed by a sudden drop. The highest flexural strength occurs in the S -30 mm sample (Condition 4), followed by the S -20 mm (Condition 2), H -30 mm (Condition 3) and H -20 mm (Condition 1) samples. A good match between the numerical and experimental model is evident in the initial and linear part of the curves. The results also show a good level of adjustment of the mechanical properties and constituents imparted on the model, mainly in relation to the division of the bamboo cross section into three layers. The model also predicts the drop load of the curves after the maximum load, with the presence of a subsequent. residual strength in the numerical and experimental results. The damage is quantified in terms of maximum load, dividing the average residual load by the maximum one. As expected, the structure damage increases when the maximum load increases. In addition, although the model considers a local condition, due to the method used in the implementation, such as symmetry conditions and simplified mesh elements, it uses a full-scale model for sandwich applications in structural conditions.

Figure 14 .

 14 Figure 14. Load versus displacement curves in three point bending test for experimental conditions: (a) H -20 mm, (b) S -20 mm, (c) H -30 mm and (d) S -30 mm.

Figure 15 .

 15 Figure 15. The failure mode of the sandwich panels: (a) experimental and (b) shape deformation in the numerical model.

Table 1 .

 1 Engineering constants considered in FE model for the phases.

	Parameter	Definition	Bamboo Inner Middle Outer	Flax composite
	E !! (GPa)	Longitudinal elastic modulus 6.51	14.98	21.32 34.89
	E "" = E ## (GPa)	Transverse elastic modulus	0.273 0.406	0.560 4.350
	ν !" = ν !# = ν "#	Poisson's ratio in plane 1-2/1-3/2-3	0.37	0.34	0.31	0.33
	G !" = G !# = G "# (GPa)	Shear modulus in plane 1-2/1-3/2-3	0.096 0.141	0.192 4.290

Table 2 .

 2 Cohesive zone parameters used in the model.

	Adhesive	Direction Normal	Damage initiation stress (N/mm 2 ) tnn	Fracture energy (N/mm) G1c	Penalty stiffness (N/mm 3 ) Knn
		Transverse	tss = ttt	G11c, G111c	Kss, Ktt
	Huntsman		0.8	0.1	10
	Sicomin		1.3	0.1	10
	2.5 Statistical experimental design

A full factorial design 2 2 is established to investigate the effect of the bamboo diameter (20 mm / 30 mm) and adhesive type (Huntsman [H] / Sicomin [S]) on the equivalent density, flexural and shear properties of sandwich panels, resulting in four experimental conditions (Table

3

). Six specimens (two replicates) are fabricated for each experimental condition, with a total of 24 specimens. The Minitab software v. 18 is used to perform the Design of Experiment (DoE) and Analysis of Variance (ANOVA).

Table 3 .

 3 Full Factorial Design (2 2 ).

	Experimental	Bamboo	Adhesive
	Condition	Diameter (mm)	Type
	1	20	H
	2	20	S
	3	30	H
	4	30	S

Table 4

 4 and modulus of 313 MPa and 35.8 GPa, respectively. The maximum compressive strength and density are equivalent for the two bamboo diameters (group A).

shows the mechanical and physical properties of the constituents of the sandwich panels. The letters represent the Tukey's comparison test, in which statistically equivalent means (averages) show a similar group of letter. Flax reinforced composites feature a maximum tensile strength

Table 4 .

 4 Mechanical and physical results of the individual material phases of the sandwich panels.

		Tensile	Compressive		Density		Ultra Micro Hardness	Viscosity
		σmax	ET	σmax		EC		ρ		EIT
		[MPa]	[GPa]	[MPa]		[GPa]		[g/cm 3 ]		[GPa]		[mPa.s]
	Flax reinforced composite	313.01 (±17.88)	35.79 (±2.26)	-		-		-		-		-
	Bamboo ring Ø20 mm	-	-	172.11 (±11.12)	A	11.93 (±0.95)	B	0.904 (±0.029)	A	-		-
	Bamboo ring Ø30 mm	-	-	173.95 (±12.03)	A	13.85 (±1.19)	A	0.925 (±0.048)	A	-		-
	Adhesive H	-	-	-		-		1.60*		4.721 (±0.675)	A	90000* (25 o C)
	Adhesive S	-	-	-		-		1.17*		4.213 (±0.393)	B	1620* (20 o C)
	*Obtained by the manufacturers [37,38].

Table 5 .

 5 Mean and standard deviation values for the DoE responses.

			Equivalent	Flexural	Flexural	Ultimate skin	Ultimate core	Core Shear
		E.C.	Density	Strength	Modulus	Stress	Shear Stress	Modulus
			(g/cm 3 )	(MPa)	(GPa)	(MPa)	(MPa)	(MPa)
		1	0.491 (±0.019)	6.33 (±0.88)	2.16 (±0.58)	28.28 (±1.73)	0.69 (±0.04)	34.45 (±9.69)
	R1	2 3	0.470 (±0.008) 0.404 (±0.004)	8.37 (±0.85) 9.47 (±0.84)	2.85 (±0.27) 3.80 (±1.06)	36.98 (±2.19) 42.40 (±3.24)	0.89 (±0.05) 0.71 (±0.05)	43.93 (±9.31) 34.07 (±2.43)
		4	0.390 (±0.009) 11.55 (±1.72)	4.23 (±0.30)	48.95 (±7.96)	0.89 (±0.03)	34.94 (±3.32)
		1	0.489 (±0.001)	6.48 (±0.47)	2.16 (±0.64)	26.89 (±4.55)	0.64 (±0.10)	36.48 (±12.24)
	R2	2 3	0.471 (±0.007) 0.402 (±0.005)	8.98 (±0.15) 9.49 (±1.89)	3.07 (±0.72) 3.43 (±0.21)	35.57 (±3.21) 39.67 (±7.93)	0.89 (±0.07) 0.69 (±0.12)	48.13 (±12.29) 32.73 (±3.25)
		4	0.391 (±0.015) 11.32 (±1.68)	4.24 (±0.41)	47.92 (±7.51)	0.87 (±0.03)	36.60 (±0.62)

Table 6 .

 6 Characteristics of manufactured sandwich panels.

	Type	Bamboo rings per panel	Surface contact area (mm 2 )	Non-adhesive area (%)
	Sandwich panel Ø20 mm	27	3711.01	65.64
	Sandwich panel Ø30 mm	24	5183.63	76.00

Acknowledgements

LAdO and THP are grateful to the Bristol Composites Institute (ACCIS) and the University of Bristol for the logistic support provided during the manufacturing and testing of the panels. The authors would like to thank SSUCHY for the flaxtape supply, the Brazilian Research Agencies, CAPES (PhD scholarship) and CNPq (PQ 309885/2019-1), for the financial support provided.

99.9%, indicating models of high predictability. ANOVA is validated by the Anderson-Darling normality test, where P-values ≥ 0.05 (0.367 -0.989) imply data of normal distribution.

Equivalent Density

The equivalent density of the sandwich panels ranges from 0.390 g/cm 3 (Ø30 mm, adhesive S) to 0.491 g/cm 3 (Ø20 mm, adhesive H). An approximate 4% decrease is observed when the sandwich panel adhesive changes from H to S (Figure 8); this is attributed to the lower density of the type S adhesive (Table 4). There is also a reduction of 17% when the bamboo diameter changes from Ø20 mm to Ø30 mm. Although the density of the bamboo rings is similar (Table 4, group A), his decrease is due to the smaller number of Ø30 mm bamboo rings per sandwich panel [START_REF] Li | Mechanical performance of laminated bamboo column under axial compression[END_REF] and the greater non-adhesive area (76%) in comparison to the Ø20 mm sandwich panels (Table 6).

Figure 8. Second order interaction effect plot for the mean (average) equivalent density.

Bending Properties

The flexural strength of the sandwich panels ranges from 6.3 MPa (Ø20 mm, adhesive H) to 11.5 MPa (Ø30 mm, adhesive S). The highest flexural strength is reached for sandwich panels with Ø30 mm bamboo rings and S adhesive, with 39% (Figure 9a) and 27% (Figure 9b)

Declaration of interests

The authors declare that there is no conflict of interest.