Héloïse Vigié
email: heloise.vigie@ensiacet.fr

Thalita De Paula
email: btdepaula@outlook.fr

Martin Surand
email: cm.surand@gmail.com

Bernard Viguier
email: bernard.viguier@ensiacet.fr

Low Temperature Strain Rate Sensitivity of Titanium Alloys

Keywords: Ti-β21S, Ti-6Al-4V, tensile testing, strain rate sensitivity

Titanium alloys are widely used in many industrial applications such as in aeronautics due to their combination of good mechanical properties, excellent corrosion resistance and low density. The mechanical behaviour of titanium alloys is known to exhibit a peculiar dependence on both deformation temperature and strain rate. Titanium alloys show significant room temperature creep and they are very sensitive to dwell fatigue and sustained load cracking. This behaviour is related to the viscosity of plastic deformation in titanium alloys, which can be represented by a strain rate sensitivity (SRS) parameter. The present study aims to compare the tensile behavior of two different titanium alloys, Ti-6Al-4V and β21S, which exhibit dissimilar microstructures. Results of tensile tests, performed under constant strain rate and including strain rate changes, are reported in terms of flow stress, ductility and SRS over a wide range of temperatures.

Introduction

Titanium alloys are currently used in the aerospace industry due to their good mechanical properties, excellent corrosion resistance and low density. Ti-6Al-4V is the titanium alloy most commonly used in the structural parts of airplanes and low temperature parts of turbine aircraft engines [1]. It is known to be a α + β alloy and has various microstructures depending on the thermo-mechanical treatment applied. Ti-β21S is a more recent alloy, developed by Timet USA in 1989 [START_REF] Bania | Oxidation resistant titanium-base alloy[END_REF] to be used as a matrix in composite materials. It has a better creep resistance than Ti-6Al-4V [START_REF] Weiss | [END_REF] and it is a β metastable titanium alloy. The mechanical behaviour of Ti-6Al-4V has been investigated by many authors [4]. The flow stress of this alloy is sensitive to the imposed strain rate [5] and as noticed for zirconium alloys [6,7] this sensitivity depends on the temperature. The mechanical properties of the Ti-β21S versus temperature and strain rate has been less studied [START_REF] Fanning | The Minerals[END_REF][START_REF] Rosen | [END_REF]. The aim of this study is to compare the mechanical behaviour of these two alloys with different microstructures against temperature and strain rate. The mechanical properties are characterized starting from room temperature up to 700°C by means of imposed strain rate tensile testing. The strain rate sensitivity is studied by strain rate jumps during the imposed strain rate tensile tests.

Material and experimental procedure

The Ti-6Al-4V alloy is part of a hot-forged billet, with the chemical composition as given in Table 1, obtained through a three-step thermo-mechanical treatment. The first step consisted of die forging in the α + β temperature domain, just below the β-transus temperature. The last two steps consisted of heat treatments in the α + β domain. These two steps lead to the α precipitates coarsening and to the releasing the internal stresses. The final microstructure obtained is a duplex microstructure with primary α grains and α/β lamellar grains, see Fig. 1a. Both types of grains show an equiaxed shape and are roughly the same size, about 25 µm. In the lamellar grains, 2 µm wide α lamellae are separated by very thin β layers. The composition of the Ti-β21S alloy, prepared in 1.0 mm thick rolled sheets, is also given in Table 1. The sheets were provided in a solution-state and underwent a precipitation heat treatment at 538°C for 8h under vacuum. The final microstructure shows α precipitates at the grain boundaries and within β grains as presented in Fig. 1b. The size of the α precipitates is heterogeneous, exhibiting bands of coarse precipitates along the grain boundaries and a fine and homogenous α precipitation in the centre of the grains. An MTS 810 servo-hydraulic machine, equipped with a radiation furnace and a 100kN load cell was used for tensile testing at the imposed displacement rate. Two kinds of specimens were used for the tensile tests: cylindrical specimens with threaded heads, a gage diameter of 5 mm and a gage length of 32 mm were used for the Ti-6Al-4V alloy, while flat specimens with a gage length of 30 mm and a cross section of (7 x 1) mm 2 were used for the Ti-β21S alloy. In both cases, three thermocouples were used to control the specimen temperature. They were spread over the gage length of the specimen and the temperature difference was always found to be lower than 5°C between the three thermocouples. Tensile tests were performed from 20°C to 700°C and an imposed strain rate of 1.6x10 -4 s -1 for Ti-6Al-4V and 4.4x10 -4 s -1 for Ti-β21S was applied. The evolution of mechanical behaviour of these alloys versus the temperature was recorded in terms of yield stress and ultimate tensile stress, expressed as true stress and the ductility of the alloy was followed by the evolution of the amount of plastic strain to rupture. In order to study the SRS of the flow stress for both alloys, another series of tensile tests were performed which included strain rate jumps during plastic deformation. The strain rate was changed from the monotonous strain rate previously mentioned to a strain rate ten times as high (approximately from about 10 -4 s -1 up to 10 -3 s -1 ). The SRS parameter was calculated for each strain rate change with the equation: ln /ln , where the subscript 1 (resp. 2) refers to stress and strain rate values before (resp. after) the strain rate jump.

(a) (b)

Results and discussion

The evolution of the tensile properties for the two alloys, the 0.2% offset plastic strain flow stress σ 0.2 , the ultimate tensile stress σ m and the plastic strain up to rupture are plotted against the deformation temperature in Fig. 2. The mechanical properties of the two alloys are clearly different at temperatures below 600°C. On the one hand, the strength of Ti-β21S is much higher than that of Ti-6Al-4V, e.g. at room temperature the ultimate tensile strength of the Ti-β21S alloy is close to 1600 MPa while that of Ti-6Al-4V is about 1150 MPa. On the other hand Ti-6Al-4V exhibits a plastic elongation that is at least twice that of the Ti-β21S alloy up to 400°C. The difference between the properties of the two alloys vanishes when the temperature increases; at 600°C the mechanical properties are quite similar for both alloys. It is also important to note that the tensile properties evolve as a function of the increasing temperature in a similar and non-monotonous manner for both alloys. In both cases three temperature domains are defined as indicated in the graphs of Fig. 2 and previously described in e.g. [4]. The SRS of the flow stress was measured for both alloys by means of strain rate jumps. The SRS parameter is plotted against temperature (T) in Fig. 3. As noted for the temperature dependence of the flow stress, the SRS evolution seems to be quite similar for both alloys. The variation of the m parameter versus temperature of the Ti-6Al-4V alloy confirms the existence of three temperature domains. This is less clear for the Ti-β21S alloy. However, this apparent difference must be related to the way the SRS parameter was measured. Indeed for both alloys and for T < 450°C the strain rate jumps result in a transient behaviour followed by a stabilized curve, forming a yield in stressstrain curve. For the Ti-6Al-4V the complete curve was recorded and m values were calculated from the σ 2 stabilized level, leading to zero or even negative m values for 250 < T < 450°C. By contrast, due to their limited ductility, the flat Ti-β21S specimens broke very rapidly upon strain rate changes and only the transient stress could be recorded. The recorded values of m thus remain positive, while the stabilized values would be close to zero, similar to what is measured for Ti-6Al-4V specimens. By comparing the tensile properties of Ti-6Al-4V and Ti-β21S alloys, very different values for flow stress and ductility can be noticed. However, common trends for the evolution of flow stress versus strain rate and temperature are also present. For both alloys, it seems that three temperature domains can be defined corresponding to low (T < 250°C), intermediate (250 < T < 450°C) and high (T > 450°C) temperatures, as delineated by vertical dash lines in Figs. 2 and3. The existence of such temperature domains has already been reported for the Ti-6Al-4V alloy [START_REF] Rosen | [END_REF]10] as well as for other alloys with hcp structure such as zirconium alloys [6,7,11]. These domains together with the values of SRS parameters indicate that deformation mechanisms involve dynamic strain aging (DSA) phenomenon, resulting from the interaction of the hexagonal structure dislocations with interstitial atoms such as oxygen [6,11]. The present results thus indicate that the deformation mechanisms in the hexagonal α phase have a predominant influence on the mechanical properties of both Ti-6Al-4V and Ti-β21S alloys. Consistently, the higher strength of Ti-β21S than that of Ti-6Al-4V may be related to a much smaller size of α phase precipitates in the Ti-β21S compared to the relatively wide α laths in Ti-6Al-4V.

Figure 3: Evolution of the strain rate sensitivity parameter versus temperature of the Ti-6Al-4V and the Ti-β21S alloys.

Summary

The evolution of the tensile properties of the Ti-6Al-4V and the Ti-β21S alloys was investigated across a wide range of temperatures. While the tensile properties, such as flow stress and ductility, are quite different for both alloys, the effect of temperature on the evolution of these properties presents common features. The evolution of tensile properties and SRS in relation to temperature leads to the definition of three temperature domains for the plastic deformation in both alloys, which are related to DSA phenomenon. These observations suggest that microscopic mechanisms controlling the plastic deformation of Ti-6Al-4V and Ti-β21S alloys present strong similarities despite their different microstructures.

Figure 1 :

 1 Figure 1: General microstructure of (a) Ti-6Al-4V and (b) Ti-β21S alloys.

Figure 2 :

 2 Evolution of the 0.2% offset plastic strain flow stress σ 0.2 , the ultimate tensile stress σ m and the rupture strain versus the temperature of the Ti-6Al-4V (a) and the Ti-β21S (b).

Table 1 :

 1 Chemical composition of the Ti-6Al-4V and Ti-β21S alloys (in wt %).

	Element	Mo	Al	Nb	V	O	Fe	C	N	Others	Ti
	Ti-6Al-4V	-	6.44	-	3.87	0.187	0.109	0.003	0.004	< 0.4	Balance
	Ti-β21S	15	3	2.7	-	0.1	0.4	0.05	0.05	-	Balance


Mohamed Benchikhp
email: benchikhi-mohamed@yahoo.fr

Rachida El Ouatib

Lahcen Er-Rakho

Sophie Guillemet-Fritsch

Jean Jacques Demai

Bernard Durand

Low-temperature molten salt synthesis and characterization of Cu 2 ZnSnS 4 ultrafine powders

Keywords:

Nanoparticles Molten sait

Nano-crystalline particles of Cu2ZnSnS 4 have been successfully synthesized by a relatively low-temperature molten sait technique. The reaction of metal chlorides with KSCN at 400 °C for 24 h led to the Cu2ZnSnS4 phase. The phase purity of Cu2ZnSnS4 nanocrystals has been confirmed by X-ray diffraction and Raman spectroscopy. The morphology and the crystallite size were evaluated by scanning and transmission electron microscopies. The Cu2ZnSnS 4 nanocrystals possess an optic.al band gap of 1.5 eV, very close to the theoretical optimum for solar energy conversion.

Introduction

Quaternary chalcogenides 1 2 -II-IV-Vl 4 compounds have attracted much attention in the past three decades thanks to their interesting optical, electrical, and electronic propreties (1)(2)(3)(4). Among them, the quaternary Cu 2 ZnSnS 4 is a good candidate for low-cost terrestrial photovoltaic applications because of its high absorption coefficient of 104 cm -1 in visible spectrum range. The band gap energy of this semiconductor is in the range 1.45 -1.55 eV which is very close to the theoretical optimum for solar energy conversion. ln addition. CZTS is consisting of the earth-abundant and non-toxic elements (2)(3)(4). Solar cells based on Cu 2 ZnSnS 4 have reached conversion efficiency as high as 8.4% (1 ]. However, it is so far below the physical limit, known as the Shockley-Queisser (SQ) li mit, of about 31% efficiency under terrestrial conditions (5 ].

Numerous methods have been developed to prepare this complex chalcogenide, such as laser removal (6), sputtering (4), evaporation (7), electrodeposition (3), nanoparticle or suspension printing (8) pyrolysis of molecular precursors (9) and solvothermal method (10). However, most of these methods usually require sophisticated instrumentation and some of them use environment-unfriendly reagents such as H 2 S, organometallic precursors or toxic solvents. ln the molten state, the minerai salts are ionic liquids where chemical reaction can be carried out as in common solvents. Thus. they enable chemical reactions leading to highly pure inorganic powders. lt's known that for the synthesis of mixed oxides from mixtures of parent oxides. the addition of a molten sait decreases the reaction temperature and increases the reaction rate (11 ]. Thus, various oxide powders, characterized by high specific surface areas with excellent chemical homo geneity, were prepared by reaction of transition metals with oxo-salts of alkali-metals (nitrites, nitrates. carbonates. sulfates, etc.) [11,12]. Recently, high-pure binary and ternary sulfides have been successfully obtained by molten salts synthesis at a temperature lower than 500 • C [13,14].

In this work, we report a simple route for the synthesis of Cu 2 ZnSnS 4 nanoparticles by a molten salt method at a relatively low temperature. KSCN was used as the solvent for the reaction. The structural, morphological, and optical properties of the powder obtained are given.

Experimental details

Copper (II) chloride dihydrate (CuCl 2 , 2H 2 O, sigma-Aldrich), tin (IV) chloride pentahydrate (SnCl 4 , 5H 2 O, Ficher Scientific), and zinc (II) chloride anhydrous (ZnCl 2 , sigma-Aldrich) were used as the metallic sources. Potassium thiocyanate (KSCN, Prolabo) was used both as the solvent and sulfurizating agent.

Copper, zinc and tin sources in appropriate proportions and an excess of KSCN were intimately mixed. Then, the mixture was heated under nitrogen flow in a vertical furnace at 350 and 400 • C for 24 h. The heating and the cooling rates were stated at 2 • C/min. After cooling and solidification of the molten medium, the black precipitate was extracted from the excess of salt by subsequent washings with distilled water and ethanol. The precipitate was dried at 60 • C for 12 h.

The crystal structure and the nature of the phase present in the sample were investigated by X-ray diffraction analysis using a Bruker D4 Endeavor X-ray diffractometer (CuK␣ = 0.154056 nm and CuK␣ = 0.15404 nm, operating voltage 40 kV and current 40 mA). Raman spectroscopy was performed using a dispersive laser spectrophotometer (Jobin-Yvon Labram HR 800) with an excitation wavelength of 633 nm. The morphology and the particle size were investigated by scanning (Jeol-JSM6400) and transmission (Jeol 2010) electron microscopies.

Optical absorption spectra were recorded on a perkin Elmer Lambda 19 spectrophotometer. Cuvettes used for the experiments are made of Suprasil quartz from Hellma Analytics. In order to avoid artifacts, the cuvettes contribution to the absorption has been systematically subtracted from the raw data.

Results and discussion

Fig. 1 shows XRD patterns of the sulfide formed by reaction of metal chlorides in molten KSCN with the molar ratio KSCN/Cu = 15 at 350 and 400 • C for 24 h. At 350 • C, kesterite phase Cu 2 ZnSnS 4 (JCPDS 26-0575) was identified as the main phase besides minor phases which could be binary sulfides. This indicates that the formation of quaternary chalcogenide phase has not completed at 350 • C. On increasing the temperature to 400 • C, the peaks of the minor phases are no longer present in the XRD patterns. This temperature is very close to the MSCN (M = K, Cu, Ag) decomposition temperature reported in the literature [15]. These results demonstrated that the quaternary phase CZTS can be obtained at 400 • C by the present molten salt method. This temperature is lower than that require by other methods [16,17]. The effect of the molar ratio KSCN/CIS on the formation quaternary phase was investigated. The XRD patterns of samples obtained at 400 • C/24 h with various KSCN/Cu ratios are given in Fig. 2. The XRD pattern of powder prepared with SCN/Cu = 5 identifies a main phase CZTS besides minor intermediates phases, indicating incomplete transformation of the metal chlorides. When the molar ratio was 10, the CZTS peaks height increased but significant amounts of intermediates phases still remained. On increasing the ratio to 15, the XRD pattern identifies only CZTS phase, indicating that the formation of CZTS phase had been completed. We note that the full width at half maximum of XRD peaks decrease with the KSCN/Cu molar ratio which signifies that the salt content improves the crystalline quality of CZTS. The lattice parameters calculated according to equation 1/d 2 = (h 2 + k 2 )/a 2 + l 2 /c 2 are a = 5.463 and c = 10.943 Å. These values are close to the reported data (JCPDS 26-0575, a = 5.434 Å and c = 10.848 Å,). According to the literature, ZnS (JCPDS 5-0566) and Cu 2 SnS 3 (JCPDS 89-4714) have a very similar diffraction pattern with Cu 2 ZnSnS 4 , consequently, it is not possible to exclude the presence of these phases in our samples. Hence, Raman investigation was performed to fully characterize the sample. The Raman spectrum (Fig. 3) shows a strong peak at 338 cm -1 and two secondary peaks at 287 and 367 cm -1 . Similar Raman spectrum has been described by Altosaar et al. for CZTS [16]. These signals are consistent with the vibration involving the motion of sulfur atoms of quaternary chalcogenides and are attributed to Cu 2 ZnSnS 4 structure [2,7,16]. On the other hand, the Raman spectrum does not show any peaks coinciding with the frequency of vibrational modes reported for Cu 2 SnS 3 , and ZnS compounds [18,19]. This confirms the phase purity of the Cu 2 ZnSnS 4 nanoparticles.

In these experiments, the thiocyanate plays four important roles: (1) solvent, (2) reducing agent, (3) chelating agent, and (4) sulfur source. When KSCN was added to the mixture of metal salts, the color turns rapidly. This indicates the formation of metal-thiocyanate (Cu(SCN) n 2-n , Zn(SCN) m 2-m , Sn(SCN) p 4-p ) complexes. The pre-formed complexes decompose under temperature effect and release the metal cations and SCN -in the reaction medium. The thiocyanate ions reduce Cu 2+ into Cu + at a temperature above 173 • C [20]. According to the literature [20], the thermal decomposition of KSCN starts at 275 • C with S 2-and (CN) 2 formation. Finally, Cu + , Zn 2+ and Sn 4+ react with S 2-to form Cu 2 ZnSnS 4 . The various reactions involved in this process are given in Eqs. ( 1) -( 4):

Cu 2+ + 2SCN -↔ [Cu(SCN) 2 ] * ↔ Cu + + ½(CN) 2 + SCN -+ S (1)
SCN -→ CN -+ S (2) 2CN -+ S → S 2-+ (CN) 2 (3)
2Cu + + Zn 2+ + Sn 4+ + 4S 2-→ Cu 2 ZnSnS 4 (4)
SEM observation of the as-synthesized samples (Fig. 4a) shows largely agglomerated particles with a poly-disperse size distribution. The particle sizes varied in the range of 2-3 m. TEM image (Fig. 4b) shows that these particles are composed of elongated or more or less spherical elementary grains with an average size in the range 40-70 nm. A crystallite size estimate of 50-60 nm was obtained using the Scherrer equation on the X-ray diffraction pattern of Cu 2 ZnSnS 4 , demonstrating that our nanoparticles are single crystals. The formation of lower grain size suggests a higher nucleation rate due probably to the high concentration of metal cations in the molten medium. UV-vis absorption spectroscopy was used to evaluate the optical properties of the Cu 2 ZnSnS 4 nanocrystals (Fig. 5). It is well known that Cu 2 ZnSnS 4 is a direct gap semiconductor, so the absorption coefficient in the region of strong absorption obeying the equation [21]:

␣h = A(h -E g ) 1/2 ( 5
)
Where ␣ is the absorption coefficient (cm -1 ), h the energy of the incident photons (eV) and A a constant. The optical band gap energy of the Cu 2 ZnSnS 4 nanoparticles can be estimated from the (␣h ) 2 versus h graph by extrapolating the linear absorption edge part of the curve. As shown in Fig. 5 (inset), the value is around 1.48 eV, which is very close to the theoretical optimum for solar energy conversion in a single-band-gap device. This value is in agreement with the ones of the literature, 1.48 eV for samples prepared by solvothermal method [2] and 1.5 eV for Cu 2 ZnSnS 4 films deposited by electrodeposition followed by a sulfuration step [3] or 1.51 eV for samples prepared by RF sputtering from binary compounds (Cu 2 S, ZnS and SnS 2 ) [4]. The band gap of Cu 2 SnS 3 and ZnS are 0.96 and 3.41 eV respectively, providing further evidence that these phases are not present in our sample [18,19].

Conclusion

In summary, Cu 2 ZnSnS 4 nanoparticles have been successfully synthesized by reaction of CuCl 2 , ZnCl 2 and SnCl 4 with molten KSCN at 400 • C with the molar proportion KSCN/Cu = 15. The phase purity of Cu 2 ZnSnS 4 nanoparticles was confirmed using XRD and Raman spectroscopy. The grain size was in the range 40-70 nm. The optical absorption spectrum shows that the Cu 2 ZnSnS 4 nanocrystals exhibit a band gap of 1.5 eV suitable for application in the photovoltaic conversion of solar energy. This method is a facile and low-cost technique for preparing high-quality chalcogenide nanocrystals.

Fig. 1 .

 1 Fig. 1. XRD patterns of samples resultant from 24 h firing at (a) 350 and (b) 400 • C with molar ratio KSCN/Cu = 15. The standard pattern of Cu2ZnSnS4 (JCPDS file no. 26-0575) is provided at the bottom of this figure.

Fig. 2 .

 2 Fig. 2. XRD patterns of samples resultant from 24 h firing at 400 • C with various KSCN/Cu molar ratios: (a) 5, (b) 10, (c)15. The standard pattern of Cu2ZnSnS4 (JCPDS file no. 26-0575) is provided at the bottom of this figure.

Fig. 3 .

 3 Fig. 3. Raman spectrum of Cu2ZnSnS4 nanoparticles.

Fig. 4 .

 4 Fig. 4. (a) SEM and (b) TEM micrographs of Cu2ZnSnS4 nanoparticles.

Fig. 5 .

 5 Fig. 5. The UV-vis absorption spectrum of Cu2ZnSnS4 nanoparticles. Inset shows the plot of (␣h ) 2 versus photon energy. The band gap of Cu2ZnSnS4 was estimated to be1.48 eV. The powder is dispersed in ethanol for recording the spectrum.

Acknowledgments

This work was supported by two French-Moroccan projects: Volubilis Partenariat Hubert Curien (PHC n o : MA 09 205) and Projet de Recherches Convention Internationale du CNRS (CNRS-CNRST n o : w22572).


F Errico

M Ichchou

S De Rosa

F Franco

O Bareille

Investigations about periodic design for broadband increased sound transmission loss of sandwich panels using 3D-printed models

Keywords: Sound Transmission, Periodic Structure Design, Metamaterials, Sandwich Structures

Two types of sandwich panels are designed by using the periodic structure theory. A double-wall panel with mechanical links and a sandwich panel with rectangular core are studied. An oriented optimization of the elastic bending waves' propagation versus the acoustic wavenumbers is achieved by using shifted core walls and by keeping the mass and stiffness of the system constant. Standard and optimized configurations are 3D-printed and sound transmission measurements are carried out by using a facility with an uncoupled reverberant-anechoic configuration. The experimental evidences of enlarged bending band-gaps and deformation mechanisms are proved using a reverse approach based on the acoustic radiation of the panels.

Introduction

In transport industry, the requirements for light and stiff structures often lead to sandwich structural solutions. The design of the core of sandwich panels can induce different anisotropies keeping a high stiffness-to-mass ratio. On the other hand, while a reduced mass is an advantage for fixed structural resistance, it generally induces lower vibroacoustic performance. Therefore, the design of sandwich panels for reduced acoustic radiation, limiting the mass of the system, has received more and more attention in the last decades [START_REF] Lang | Optimal acoustic design of sandwich panels[END_REF][START_REF] Moore | Sound transmission loss characteristics of sandwich panel constructions[END_REF][START_REF] Dym | Transmission of sound through sandwich panels: A reconsideration[END_REF][START_REF] Dym | Transmission of sound through sandwich panels[END_REF].

One of the main reasons for a larger acoustic radiation of sandwich panels is the shear core effect in the mid-frequencies [START_REF] Lang | Optimal acoustic design of sandwich panels[END_REF][START_REF] Moore | Sound transmission loss characteristics of sandwich panel constructions[END_REF][START_REF] Dym | Transmission of sound through sandwich panels: A reconsideration[END_REF][START_REF] Dym | Transmission of sound through sandwich panels[END_REF][START_REF] Kurtze | New wall design for high transmission loss or high damping[END_REF][START_REF] Franco | Structural-acoustic optimization of sandwich panels[END_REF]; the transition from global bending to core shear is fundamental [START_REF] Fahy | Sound and structural vibration: radiation, transmission and response[END_REF]. The effects of this wavemode transition has been investigated by describing panel using an equivalent shear core [START_REF] Clarkson | Modal density of honeycomb plates[END_REF][START_REF] Han | Modal density of sandwich panels based on an improved ordinary sandwich panel theory[END_REF][START_REF] Renji | Modal density of composite honeycomb sandwich panels[END_REF]. Some authors tried to design the core geometry in order to optimize the sound transmission.

Palumbo et al. [START_REF] Palumbo | Development of quiet honeycomb panels, Virginia[END_REF] and Grosveld et al. [START_REF] Grosveld | Finite element development of honeycomb panel configurations with improved transmission loss[END_REF] proved experimentally the increase of vibroacoustic performance of honeycomb-cored sandwich plates when periodical voids and recesses are included in the original geometry. While this approach creates regions of reduced bending stiffness, a strong benefit is observed in the sound transmission loss for large frequency bands. A brilliant work is also proposed by Hambric et al. in [START_REF] Hambric | Acoustically tailored composite rotorcraft fuselage panels, Virginia[END_REF] where a complex structural honeycomb panel is optimised by altering the original design, targeting a different the bending wave speed in the media versus the acoustic one.

On the other hand, with the rise of modern numerical techniques as the Wave Finite Element Method (WFEM), detailed investigations on shear-core transitions and the acoustic radiation of sandwich structures, have been conducted. The WFEM, applied also within this work, is used for the dynamic analysis of periodic structures by exploiting the periodic links among neighbouring nodes and substructures [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF][START_REF] Mead | Wave propagation in continuous periodic structures: research contributions from southampton[END_REF][START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF]. The single elementary cells are modelled instead of the whole panel, without requiring any homogenisation or analytical description of the core.

Baho et al [START_REF] Baho | On global bendingshear core transition effects for the vibroacoustic of sandwich structures: Analytical and numerical investigations[END_REF], for instance, used a wavemode energy method, coupled with a wave assurance criterion, to predict the transition frequency of honeycomb panels modelled using a single periodic cells. In addition, the core geometry effects on the shear-core transition and sound transmission loss of honeycomb sandwich plates, are investigated in [START_REF] Zergoune | Assessments of shear core effects on sound transmission loss through sandwich panels using a two-scale approach[END_REF]. Zergoune et al. [START_REF] Zergoune | Assessments of shear core effects on sound transmission loss through sandwich panels using a two-scale approach[END_REF] show the transmission loss sensitivity to most of the classic topological parameters of honeycomb cored sandwich panels, such as angles between walls, thickness and deformation of hexagonal core cells.

Similarly, sandwich panels with arbitrary shaped core are studied in [START_REF] Droz | Vibro-acoustic optimisation of sandwich panels using the wave/finite element method[END_REF]. Again, a comparative study of periodic cells that share the same mass-to-stiffness ratio, is performed to achieve a significant variation of the transition frequency and modal density, compared to classic honeycomb.

Alternative approaches, quite efficient when a narrow-band frequency issue has to be solved, are connected to the use of resonant periodically distributed elements [START_REF] Claeys | On the acoustic radiation efficiency of local resonance based stop band materials[END_REF][START_REF] Claeys | On the potential of tuned resonators to obtain lowfrequency vibrational stop bands in periodic panels[END_REF][START_REF] Liu | Broadband locally resonant metamaterial sandwich plate for improved noise insulation in the coincidence region[END_REF][START_REF] Liu | Investigation of the sound transmission through a locally resonant metamaterial cylindrical shell in the ring frequency region[END_REF]. In this case, the drawback is connected to the addition of mass to the structure and to the difficulty in targeting broadband enhancements of the acoustic performance of the panels.

Within the present paper, two common configurations of sandwich panels are modified using the periodic structure theory. The aim is to induce alterations of the bending wave propagation of the panels versus frequency, in order to increase the vibroacoustic performance, keeping the same mass and, optionally, the same bending stiffness as the original designs. A double-wall panel with mechanical links and a sandwich panel with rectangular core are optimised using shifted core walls. In the first case, the periodicity effects are evident and an enlarged band-gap for bending waves is created. In the second case, the main effects are observed on the shear-core transition and core energy absorption through an increased deformation mechanism. All considerations and enhancements are experimentally proved.

It is worth highlight that the optimization proposed here is not connected to any minimization of a specific cost function or parameter; the main idea of the paper is to prove the performance of optimised designs guided by the physical analysis of waves propagation characteristics. A proper topological optimization, exploring a large number of configurations, might lead to different variants of the proposed design, with higher performance.

The paper is organised as follows: Section 2 describes the numerical procedure used for predicting the dispersion curves versus frequency; Section 3 describes the proposed designs; Section 3D-printed panels; Section 5 presents the numerical and experimental results.

A Numerical Approach for Elastic Waves in a Periodic Media

A free wave propagating along a structure is assumed to take the form of a Bloch wave [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF].

Any wavefield in the periodic media is linked with a complex propagation constant, function of the distance among particles. This is physically representative of a magnitude attenuation and phase shift within the periodic structure. This reflects in periodic systems exhibiting passbands and stopbands; i.e. each disturbance can propagate freely only in specific frequency ranges, otherwise they decay with distance [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF].

The numerical approach used here to investigate the waves' dispersion in the periodic media is the wave finite element method. It is a FE-based method whose advantage stands in requiring the modelling of a single unit periodic cell, instead of a whole structure (Fig. 1). In fact, by exploiting the links among nodes using complex propagation constants, the dynamics of the whole periodic structural system can be analysed [START_REF] Brillouin | Wave propagation in periodic structures: Electric filters and crystal lattices[END_REF][START_REF] Mead | Wave propagation in continuous periodic structures: research contributions from southampton[END_REF][START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF].

With reference to Fig. 1, the dynamic stiffness equation of the unitary segment can be written as:

Dq = f + e, (1)
where q, f and e are the vectors of nodal degrees of freedom (DoFs), internal and external forces, respectively; D is the dynamic stiffness matrix. To analyse the dynamics of the whole periodic system, displacements (q) and forces (f and e) at any point of the cell can be liked to a reduced subset of degrees of freedom (q r ) by exploiting periodic links, as follows (see Fig. 1 for nodes subsets ordering):

q A = Iλ Y q F ; q R = Iλ X q L ; q 2 = Iλ X q 1 ; q 3 = Iλ Y q 1 ; q 4 = Iλ X λ Y q 1 ; (2)
being the propagation constants:

λ X = e -ik X L X , λ Y = e -ik Y L Y , (3)
where k X and k Y are wavenumbers of the waves in the plane x -y; L X and L Y represent the cell lengths along the same directions, respectively, and I is the identity matrix.

For complex structures requiring a fine mesh, a modal reduction is highly suggested to reduce the order of the model. The displacement vector inside a single unit cell can be substituted by a subset of stationary constrained internal modes, achieving a significant reduction of the number of inner DOFs. The displacements vector can be expressed as in Eq. 4, using a projection matrix (G) that includes static boundary modes (Ψ B ) and component modes (Ψ C ):

  q nI q I   = G   q nI P mod   ; G =   I 0 Ψ B Ψ C   (4)
where P mod represents modal participation factors vector for the set of retained modes; q nI is the whole set of degrees of freedom excluded the ones belonging to the internal nodes. The static boundary modes Ψ B and component modes Ψ C can be derived as in a Craig-Bampton (CB) framework [START_REF] Zhou | Wave finite element method based on reduced model for one-dimensional periodic structures[END_REF][START_REF] Zhou | Multi-scale modelling for two-dimensional periodic structures using a combined mode/wave based approach[END_REF]. Finally the stiffness and mass matrices of the condensed system can be calculated by pre and post multiplying for the projection matrix G, defined in Eq. 4:

M C = G T MG; K C = G T KG, (5)
where the subscript C refers to the mass and stiffness matrices of the cell being condensed. Then, at each frequency step, the modal participation factors can be statically condensed and the condensed dynamic stiffness equation D C obtained [START_REF] Zhou | Wave finite element method based on reduced model for one-dimensional periodic structures[END_REF][START_REF] Zhou | Multi-scale modelling for two-dimensional periodic structures using a combined mode/wave based approach[END_REF]. Assembling the links in Eq. 2 in a global periodicity matrix Λ and pre-multiplying Eq.1 by the Hermitian of this matrix, a reduced dynamic stiffness equation can be written:

D r q r = Λ H [D C ]Λq r = Λ H Λf r + Λ H Λe r . (6)
where D r is the dynamic stiffness matrix of the reduced model; q r , f r and e r are the displacements, internal forces and external forces of the reduced model, respectively. Given the equilibrium of the internal forces between consecutive cells, only potential external forces are considered [START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF], thus

Λ H Λf r = 0.
The problem in Eq. 6 is a three-parameters eigenproblem in ω, λ X and λ Y , whose eigenvectors and eigenvalues are the propagating wavemodes and the corresponding constants of propagation. It is here solved by imposing the frequency and one wavenumber to zero, in order to study singularly the wave propagation in x and y directions.

Shifted-Walls Design

A core design with shifted walls is here proposed for two common types of sandwich panels. First a double-wall panel with rectilinear mechanical links (Fig. 2a) is analysed for a trapezoidal shape of the stiffener (Fig. 2b). A similar idea has been applied to a sandwich panel with rectangular core (Fig. 5a), where the core walls have been folded in one direction (Fig. 5b).

The frequency band targeted for the optimization is 250 Hz to 10 kHz, corresponding to the measurements frequency range available for the facility used within this work (see Sec. 4).

Double-Wall Panel with Mechanical Links

The initial double-wall panel (Fig. 2a) is 1.06 cm thick (total thickness) and has a 1.0 cm spacing between consecutive stiffeners, in the periodicity direction (axis x in Fig. 2). The thickness of the skins and core walls is 0.6 mm. A unit cell, as the one illustrated in Fig. 2a, is modelled with finite elements (ANSYS Shell181) and the eigenvalue problem developed in Section 2 solved to get the wave dispersion in the media.

Full geometrical details for the designs analysed are given: the double wall panel with mechanical link is illustrated in Fig. 3, while the sandwich panel with rectangular core is illustrated in Fig. 4.

The modified design, proposed here for increasing the flexural band-gap region versus frequency, in the periodicity direction, is characterised by a trapezoidal (top-view) shape of the stiffer (Fig. 2b). The same global thickness and stiffeners' spacing as the classic double-wall panel is used; the skins are 0.6 mm-thick and the core walls are 0.45 mm-thick, in order to keep the same mass of the system.

The idea of using such a geometrical shape comes from the usually low bending and shear stiffness of this kind of panels in the direction normal to the stiffener envelope (axis x). The addition of oblique elements, periodically repeated in the x-y plane due to the periodicity of the system, has a main function: the core shear stiffness increases with respect to the classic design where the bending of the mechanical link was proving the whole core-shear stiffness in the x direction.

Rectangular-Core Panel

For the rectangular-cored panel, the initial design (Fig. 5a) has a total thickness of 1.06 cm and a 1.0 cm spacing between consecutive stiffeners, in both directions (Fig. 5a). As for the double-wall panel, both skins and core-walls are 0.6 mm-thick. Similarly to the previous case, the modified design proposed here is characterised by a trapezoidal (top-view) shape of the stiffener in one direction (axis y in Fig. 5b), while the rectilinear geometry is kept in the other direction.

To keep the same mass, global thickness and stiffeners spacing of the original system, the skins are 0.6 mm-thick and the core walls are 0.36 mm-thick. Hereby, the target is to tailor the elastic waves dispersion in the panel, against the acoustic wavenumbers. Differently from the previous case, the presence of additional components of the stiffener in the x direction (Fig. 5b) does not have the function of increasing the core-shear stiffness. The idea is to induce a larger deformation mechanism for the folded core walls both for bending in y and core-shear in x.

Experimental Set-Up

The work flow followed for the analysis and testing of the presented design configurations is illustrated in Fig. 6. Using the method described in Section 2, the numerical dispersion curves and transmission loss are computed and compared. Then, a CAD model is developed and transformed into a CAM input file for a Stratasys F ortus 450mc industrial 3D-printer with a maximum printable volume of 40 x 35 x 40 cm 3 (Fig. 6). The material used for the modelling and 3D-printing of the panels is the ABS-M30 (Acrylonitrile butadiene styrene) [START_REF] Croccolo | Experimental characterization and analytical modelling of the mechanical behaviour of fused deposition processed parts made of abs-m30[END_REF]. The material has been experimentally characterised and has the following properties: E = 1.8 GPa; ρ = 988 kg/m 2 ; ν = 0.32.

Both the 3D-printed panels with standard and optimised designs are shown in Fig. 8.

The sound transmission measurements were performed in an un-coupled reverberant-anechoic room as in Fig. 7. The reverberant room has volume of 2.50 x 1.40 x 1.75 m 3 and reflective surfaces are installed inside to increase the diffusiveness of the reverberant room; the calculated Schroeder frequency is ≈ 600 kHz [START_REF] Schroeder | The Schroeder frequency revisited[END_REF]. The acoustic excitation is generated using four speakers installed at the four top corners of the reverberant room, with an uncorrelated white noise input from 50 to 10 kHz.

The transmitted sound power is measured using a Bruel & Kjaer sound intensity probe with two half-inch microphones and a 6.0 mm spacer (see Fig. 7). The incident sound power is obtained by the sound pressure level measurements in the cabin room, averaged among four half-inch microphones disposed as in Fig. 7. The anechoic conditions of the receiving room are simulated by covering the room walls with absorbing layers, whose distance from the tested panel is larger than 2 m. The sound transmission loss (TL) is finally calculated, assuming that the excited and radiating surfaces are the same, as:

TL = L p -L i -6.18, (7)
where L p is the average pressure level measured in the reverberant room, L i the average sound intensity level over the surface of the test-panels in the semi-anechoic room, while the -6.18 factor arises from reference values in dB conversion [START_REF] London | Transmission of Reverberant Sound Through Single Walls[END_REF].

Results

Waves Tailoring versus Acoustic Wavenumbers

The approach presented in Section 2 is used for studying the waves' propagation in the periodic structures proposed in Section 3. The main idea is to evidence two phenomena: an increase of the bending band-gap in the periodicity direction, for the double-wall panel with mechanical links; an increase of bending wavenumbers versus the acoustic ones, for the sandwich panel with rectangular core.

In Fig. 9, the dispersion curves for the two designs illustrated in Fig. 2 are compared in the direction orthogonal to the stiffening elements. The band-gap present in the standard design extends from 3.3 kHz to 5.4 kHz and an acoustic coincidence is observed around 8.4 kHz. Differently, for the optimized design proposed in Fig. 2, the band-gap is strongly enlarged in frequency from 3.6 kHz to 9.1 kHz, while a coincidence is observed only at the end of the frequency band of interest (≈ 10 kHz). A double advantage is thus observed both in structural waves' filtering (band-gap) and coincidence shift, keeping the same mass of the system.

In Fig. 10 the eigenvectors of Eq. 6, which represent wavemodes of the wave-type propagating in the media with the wavenumbers of Fig. 9, are showed for two frequencies. In particular, the wavemodes with the highest wavenumbers of the set of waves propagating at 4.5 kHz and 6.5 kHz are shown for the two designs in Fig. 2. It is interesting to observe how, in Fig. 10, the cell modal wave deformations are completely different. The increase of core-shear stiffness in the periodicity direction, for the optimized design, induces almost null modal out-of-plane displacements of the skin surfaces. In opposition, the wavemodes of the classic design still present mode-shapes efficient for sound transmission (non-null out-of-plane displacements).

Differently, for the classic sandwich panel with rectangular core, band-gaps are not present in the frequency range of interest (see Fig. 11). Here, a coincidence effect is not observed, but the bending waves follow closely and almost parallel the acoustic wavenumbers versus frequency. This means that the sound transmission loss is expected to be reduced in a large frequency bandwidth.

The target of the optimised design, in this case, has been to enlarge the distance between the structural and acoustic wavenumbers for a large frequency band, by inducing a larger deformation mechanism of the folded core walls. Coherently, in Fig. 11, the waves' dispersion, both in x and y directions, is characterised by higher wavenumbers: the larger distance from the acoustic ones is expected to induce an enhanced sound transmission loss. In addition, a band-gap formation in x, at the end of the frequency band, is also observed for the optimised design. The wave shapes in x (wavemodes) at 4 and 6 kHz are presented in Fig. 12 for bending waves. As said before, the folded walls design offers a larger core deformation mechanism and this is clearly observable in Fig. 12 for both frequencies: a part of the vibrational energy is absorbed through the core deformation.

In this way, the shear core waves reduce their speed and the corresponding wavenumbers increase, as targeted.

Sound Transmission Loss

The sound transmission loss of the two couples of designs is numerically simulated for equivalent infinite panels.

The effects of the design optimisation, observed in terms of waves in Figs. 9 and 11, are specular in the transmission loss curves presented in Fig. 13. In Fig. 13a, the double-wall panel with mechanical links is studied and the effect of the enlarged band-gap effect is visible in shifting the coincidence frequency: starting from 1.5kHz, the sound transmission loss of the optimised panel is constantly larger than the one of the standard plate design with the same mass.

On the other hand, in Fig. 13b, as observed from the dispersion curves, while a real coincidence region is not observed for the standard sandwich plate design, a drop of the transmission loss is still evident due to the proximity of the structural and acoustic wavenumbers/wavelengths. Differently, for the optimised design, the induced distance of structural and acoustic wavenumber results in a constant increase of the sound transmission loss versus frequency, in absence of evident drops: from 1.5 kHz to 10 kHz, the new design assures a larger sound transmission loss with the same global mass of the panel.

To experimentally prove the increase of sound transmission loss for the proposed designs, the 3D-printed panels are installed and clamped in the TL facility, as in Fig. 7. The expected and real weights of the printed panels are reported in Table 1. It is observed in Table 1 that, when the geometrical complexity of the single cell design increases (optimised designs), the discrepancies between the expected and real weight of the 3D-printed panels increases too. To compare mass-normalised results, the approach proposed by De Rosa in [START_REF] Rosa | Models and comparisons for the evaluation of the sound transmission loss of panels[END_REF] is used to scale the measured transmission loss curves of the optimised panels versus the ones of the lighter (standard; see Table 1) panels.

In Fig. 14, the measured sound transmission losses (mass-normalised) of the two design couples proposed in Section 3 are compared. While the total surface of the plates is 0.40 x 0.35 m 2 , the exposed panel surface is 0.34 x 0.29 m 2 . In Fig. 14a, the measured sound transmission loss trends are in good agreement to the simulated ones for an equivalent infinite plate in Fig. 13a: as expected, the increase of sound transmission loss starts from 1.5 kHz and continues up to the end of the frequency band of interest. It must be highlighted that the damping induced by the 3D-printing, the installation and the clamping in the TL cabin might be high. It is also observed by the measured transmission loss curve around the coincidence region (≈ 8.4 kHz) for the standard DWP in Fig. 14a, with respect to the simulated drop in Fig. 13a.

Similarly, in Fig. 14b, the same trends of the simulation Fig. 13b, for the sandwich plate with rectangular core, are experimentally observed. Again, the reduction of transmission loss expected for the standard design is highly damped with respect to the simulation. However, the agreements is satisfying and validates the expected trends. The increase of sound transmission loss, as in the previous case, appears in a very large frequency band that goes from 1 kHz to 10 kHz, at least, for a fixed mass of the panel.

Although there is a general agreement between expected results and experimental evidence, some discrepancies are present. For example, in both cases the sound transmission loss increase starts in frequencies a bit lower than the expected ones. This should be addressed to two factor that act simultaneously. First, as discussed before, the Schroeder frequency of the TL cabin is ≈ 600 kHz and this can have an influence on the reliability of the measurements in the low frequency bands. Furthermore, the uncertainties connected to the 3D-printing and the effective differences among the ideal models and the real structures are not perfectly controlled and might, obviously, induce discrepancies by the predictions and the measurements.

Concluding Remarks

Two types of sandwich panels are analysed in terms of waves' propagation and sound transmission loss by using periodic structure theory. A double-wall panel with mechanical links and a sandwich panel with rectangular core are considered. A core optimization using shifted core walls is proposed targeting the structural waves' propagation versus the acoustic wavenumbers, forcing the mass to be fixed to the value of the original designs.For the double-wall panel with mechanical links an enlarged bending band-gap is achieved in a large frequency band, shifting the acoustic coincidence almost out of the frequency band of interest. For the sandwich panel with rectangular core, an increased core deformation mechanisms is achieved distancing the structural(bending) and acoustic wavenumbers versus frequency.

Standard and optimized configurations are 3D-printed and sound transmission loss measurements are carried out using a small facility with uncoupled reverberant and semi-anechoic configuration.

Numerical simulations and experimental tests evidence an increased vibro-acoustic performance of the new designs. The transmission loss measurements showed that, even keeping the same total mass of the panels, the sound transmission loss is increased in a very large frequency band that goes from ≈ 1.5 kHz to 10 kHz.

Figure 1 :

 1 Figure 1: Cell model extracted from a periodic structural system with periodicity along the x-y plane. a) Illustration of a cell within a full FEM model. b) Node ordering following the schemese in Section 2.

Figure 2 :

 2 Figure 2: Cell models extracted from a Double-Wall Panel with Mechanical links. a) Standard design. b) Proposed Optimal Design.

Figure 3 :

 3 Figure 3: Cell model extracted from the optimized double-wall panel with mechanical links. Distances in millimetres.

Figure 4 :

 4 Figure 4: Cell model extracted from the optimized sandwich panel with rectangular core. Distances in millimetres.

Figure 5 :

 5 Figure 5: Cell models extracted from a Sandwich Panel with Rectangular Core. a) Standard design. b) Proposed Optimal Design with shifted walls.

Figure 6 :

 6 Figure 6: Work-flow scheme for panel design optimisation based on numerical and experimental data.

Figure 7 :

 7 Figure 7: Illustration of the test facility with un-coupled reverberant-anechoic rooms for transmission loss measurements.

Figure 8 :Figure 9 :

 89 Figure 8: The 3D-printed panels side-view following the standard (up) and optimised(down) designs in: a) Fig. 2 -Double-Wall panel with mechanical links; b) Fig. 5 -Sandwich panel with rectangular core.

Figure 10 :

 10 Figure 10: Wavemodes of the Double-Wall Panel designs for waves propagating in x direction; eigenvector solutions of the eigenvalue problem in Eq. 6. a) f = 4.5 kHz; b) f = 6.5 kHz.

Figure 11 :

 11 Figure 11: Dispersion Curves of the Sandwich Panel with Rectangular core designs; eigenvalue solutions of the eigenvalue problem in Eq. 6.

Figure 12 :

 12 Figure 12: Wavemodes of the Sandwich Panel with Rectangular core designs for waves propagating in x direction; eigenvector solutions of the eigenvalue problem in Eq. 6. a) f = 4.0 kHz; b) f = 6.0 kHz

Figure 13 :

 13 Figure 13: Calculated Transmission Loss for infinite panels the proposed designs. a) Double-Wall panels designs; b) Rectangular-Cored Sandwich panel designs.

Table 1 :Figure 14 :

 114 Figure 14: Measured Transmission Loss (mass-normalised) for the 3D-printed panels. Total panel surface: 0.40x0.35 m 2 . Exposed panel surface: 0.34x0.29 m 2 . a) Double-Wall panels designs; b) Rectangular-Cored Sandwich panel designs.

Acknowledgments

This project has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 675441. Nassardin Guenfoud, Benoit Minard and Stephane Lemahieu, from Ecole Centrale de Lyon, are gratefully acknowledged for the help in 3D-printing and building the adaptive clamping system of the cabin.