G Tufano

C Droz

O Bareille

A.-M Zine

B Pluymers

W Desmet

M Ichchou

Wavenumber identification technique for axial-symmetric structures

This paper investigates the broadband vibro-acoustic and dynamic behavior of different structures, such as flat and curved panels and cylindrical shells. The aim of this work is to investigate propagation and dispersion features through the analysis of the wavenumber domain. The technique used to investigate the vibro-acoustic behavior is a wave propagation based technique, implemented to obtain the propagative and evanescent waves using a harmonic force excitation. The employed techniques are used to obtain the harmonic displacement field, given as primary input for the wavenumber domain analysis. To deeply investigate the dispersion features, a Discrete Fast Fourier Transform (DFFT) technique is used. The whole displacement field could be obtained by experimental tests or by numerical simulations; in our case, it is calculated using a classical Finite Element Analysis (FEA), by a commercial Finite Element package, or by an in-home algorithm, developed in a commercial numerical computing software.

Introduction

Flat and curved panels are widely used in the field of transportation engineering. In the aerospace engineering, the shape of the most employed structures for the aircrafts and for the space launchers can be considered to be axial-symmetric (the fuselage of the aircrafts, the first, the upper stages and the fairings of the launchers, for example). The dynamic behavior and the vibro-acoustic response of these structures are a fundamental challenge for the industry. Indeed, the most important aspect is the understanding of the waves dispersion characteristics under different loading conditions. The identification of the propagation parameters is a fundamental issue in vibro-acoustics. In the last years, some analytical formulations were developed for flat and/or homogeneous structures and they can be found in [START_REF] Fahy | Wave propagation in damped, stiffened structures characteristic of ship construction[END_REF], [START_REF] Leissa | Vibration of Shells[END_REF] and [START_REF] Qatu | Vibration of Laminated Shells and Plates[END_REF]. These analytical formulations are limited to the low-frequency domain, in which the mode are well defined; in the mid-high-frequency range classic analytical approach do not give a good estimation of the waves dispersion characteristics. In this bandwidth some statistical and energetic approaches were developed in [START_REF] Lyon | Statistical Energy Analysis of Dynamical Systems: Theory and Applications[END_REF], [START_REF] Cotoni | A statistical energy analysis subsystem formulation using finite element and periodic structure theory[END_REF] and [START_REF] Chronopoulos | Predicting the broadband response of a layered cone-cylinder-cone shell[END_REF]. The identification of the wave propagation parameters is one of the most important challenges in vibroacoustics. In literature several technique are available, especially for one-dimensional structures, mainly based on the loss factor identification, by the research of the complex wavenumber [START_REF] Mcdaniel | A wave approach to estimating frequency-dependent damping under transient loading[END_REF]. In their work, McDaniel et al. developed a strategy to estimate the frequency-dependent loss-factor in one-dimensional structures, identifying the complex wavenumber of flexural and evanescent waves at each frequency. In the last decades, some applications to two-dimensional structures have been proposed in literature. For example, Ferguson et al. [START_REF] Ferguson | The estimation of wavenumbers in two dimensional structures[END_REF] proposed a technique to identify the dominant wavenumber in a considered area, using a windowed field of the displacement of the structure, computing the correlation between the normal velocity measurement and an harmonic wave-field. Another method for the identification of the wavenumber was developed by Grosh et al. [START_REF] Grosh | Complex wave-number decomposition of structural vibrations[END_REF], based on Prony series. An interesting algorithm for the identification of the complex wavenumber and of the damping loss factor was developed by Berthaut et al. [START_REF] Berthaut | K-space identification of apparent structural behavior[END_REF]; the proposed method is called Inhomogeneous Wave Correlation (IWC) method and it allows to identify the complex flexural wavenumber in the broadband frequency range. An application of this method to one-and two-dimensional sandwich structures can be found in [START_REF] Ichchou | Multi-mode wave propagation in ribbed plates: Part I, wavenumber-space characteristics[END_REF] and an application on ribbed plates was proposed by Ichchou et al. [START_REF] Ichchou | Multi-mode wave propagation in ribbed plates. Part II: predictions and comparisons[END_REF] and [START_REF] Ichchou | Identification of effective sandwich structural properties via an inverse wave approach[END_REF]; Cherif et al. [START_REF] Cherif | Damping loss factor estimation of two-dimensional orthotropic structures from a displacement field measurement[END_REF] proposed the damping loss factor identification for flat plates; the identification of the material characteristics of the sandwich panels was proposed by Droz et al. [START_REF] Droz | A new procedure for the determination of structural characteristics of sandwich plates in medium frequencies[END_REF]; Van Belle et al. [START_REF] Van Belle | On the impact of damping on the dispersion curves of a locally resonant metamaterial: Modelling and experimental[END_REF] proposed the estimation of the flexural dispersion curve for a plate with local resonators; Van Damme et al. [START_REF] Van Damme | Measuring dispersion curves for bending waves in beams: A comparison of Spatial Fourier Transform and Inhomogeneous Wave Correlation[END_REF] applied this method to identify the mechanical properties of a classical timber plate. This paper shows the comparison between the classical Discrete Fast Fourier Transform (DFFT) approach and the Inhomogeneous Wave Correlation (IWC) method in the k-space analysis. Both methods are applied on different structures: flat and curved panels and cylindrical structures. The wave field is obtained using either a full Finite Element Analysis (FEA) or a Wave Finite Element Method (WFEM) (this method is widely used in waves propagation investigations and it is well-described in [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], [START_REF] Renno | Calculating the forced response of cylinders using the wave and finite element method[END_REF], [START_REF] Droz | A reduced formulation for the free-wave propagation analysis in composite structures[END_REF] and [START_REF] Errico | The modelling of the flow-induced vibrations of periodic flat and axial-symmetric structures with a wave-based method[END_REF]).

Methodology

In this section, two different wavenumber identification techniques are presented. The classical approach in the wavenumber domain analysis is the Continuous Fourier Transform (CFT) and its equivalent in the discrete domain, the Discrete Fourier Transform (DFT). The second method adopted in this work is the Inhomogeneous Wave Correlation method (IWC). Both methods use the harmonic displacement field as primary input, usually defined in the (x, y) plane (see Eq. ( 1), in which the symbol ŵ indicates that the displacement field is frequency-dependent), either from a harmonic excitation or from a temporal Fourier transform.

w(x, y, t) = +∞ 0 ŵ(x, y)e iωt dω (1)
The two methods are explained in the next sections; DFT and CFT are briefly described for the sake of clarity.

Spatial Fourier Transform

The DFT assumes two hypotheses:

1. the displacement field ŵ is given in an uniform grid

(x i = idx, y j = jdy) (1≤i≤N 1 -1,1≤j≤N 2 -1)
, in which N 1 and N 2 are the number of the acquisition points along the axes x and y, respectively, dx and dy are the space increments along the axes x and y, respectively;

2. the field is assumed to be 2D-periodic, i.e.:

∀i, j ∈ N 2 , ŵ(idx, jdy) = ŵ((i/N 1 )dx, (j/N 2 )dy) (2)
The spatial mesh has to be small enough to satisfy the Nyquist criterion, at the least half of the smallest wavelength of interest; on the other hand, the span of the measurement has to be large enough to have an acceptable resolution in the wavenumber domain.

In the discrete domain, we can assume ∆k x = 2π N 1 dx and ∆k y = 2π N 2 dy as a basis for the complex functions space; the family of exponential functions with discrete wavenumbers can be written:

(k xp = p∆k x , k yq = q∆k y ) (1≤p≤N 1 -1,1≤q≤N 2 -1)
. Consequently, the displacement field can be written as:

ŵ(x i , y j ) = N 1 -1 p=0 N 2 -1 q=0 ŵ(k xp , k yq )e i(kxpx i +kyqy j ) (3)
The DFT ŵ -→ ŵ assumes the following expression:

ŵ(k xp , k yq ) = 1 N 1 N 2 N 1 -1 i=0 N 2 -1 j=0 ŵ(x i , y j )e -i(kxpx i +kyqy j ) (4)
The periodicity condition, expressed in Eq. ( 2), imposes that also the DFT is periodic:

ŵ(k x , k y ) = ŵ k x + 2π dx , k y = ŵ k x , k y + 2π dy (5)
In this work, the DFT is performed in MATLAB and no additional windowing or signal processing was applied.

Inhomogeneous Wave Correlation method

Using the same displacement field as for the DFT, the dispersion relation can be found using a different approach: the Inhomogeneous Wave Correlation method. For a fixed frequency f 0 , the spatial response is correlated with an inhomogeneous running wave (Eq. ( 6)), in which the terms θ, γ, 2π k are the heading angle, the attenuation factor and the apparent wavelength, respectively. The wave attenuation is strictly connected to the damping by this relationship: γ = ηcϕ 2cg , with η, c ϕ , c g being the damping loss factor, the phase velocity and the group velocity, respectively. This wave is defined as follows: σk,γ,θ (x, y) = e -ik(θ)(1+iγ(θ))(x cos θ+y sin θ)

The unknown wavenumber k = k + ik can be found by the correlation between the inhomogeneous wave and the complete wave field, as the location of the maximum of the normalized correlation function:

IW C(k, γ, θ) = | S ŵ • σ * k,γ,θ dx dy| S | ŵ| 2 dx dy • S |σ k,γ,θ | 2 dx dy (7)
where * denotes the complex conjugate. The identification of a complex wavenumber for a given direction θ leads to the maximization of the function (k, γ) -→ IW C(k, γ, θ); typically, the IWC has a welldefined maximum, denoting the point where the measured signal correlates best with the inhomogeneous wave σk,γ,θ (x, y).

In practical applications, the wave field is measurable in discrete points, so the integration over the entire surface S in Eq. ( 7) is replaced by a finite weighted sum:

S dx dy =⇒ N i=1 ρ i S i (8)
where ρ i is the coherence of the measured data at each point (ρ i = 1 if the coherence is not available) and S i is an estimation of the surface around the point i.

IW C(k, γ, θ) = | N i=1 ŵ(x i , y i ) • σ * k,γ,θ (x i , y i )S i | N i=1 | ŵ(x i , y i )| 2 S i • N i=1 |σ k,γ,θ (x i , y i )| 2 S i (9)
Considering the well-known relationships between arcs and angles, and the transformation from the Cartesian coordinates reference system to the cylindrical one, the IWC method can be applied also on curved structure. Moving from the couple of coordinates (x, y) to the couple of coordinates (x, ϕ) and taking in account the circumferential wavenumber (Eq. ( 10)) instead of the one along the y axis, the new formulation of the IWC method can be easily developed.

k θ = k x R = k x ∆x ∆θ ( 10
)
where ∆θ is the angular distance.

In this work, the IWC method is implemented in MATLAB, for both reference systems.

Numerical applications and results

In this section, the methods previously explained are applied to a flat (Fig. 1) and a curved (Fig. 2) panels and to a cylindrical shell (Fig. 3). The numerical models are built using a commercial Finite Element (FE) package. The numerical simulations of the two panels are obtained by a full FE analysis while the cylindrical shell is analyzed with a WFEM approach. The harmonic displacement field is obtained by numerical simulations using an in-home algorithm developed in MATLAB. The displacement field is calculated for different locations of the harmonic punctual excitation; in the specific case hereby shown, the force is located in the middle of the plate, on the bottom surface. The normal displacements are then extracted and used as primary input for the FFT and IWC calculations. All the simulations were performed considering with free-free boundary conditions. A constant structural damping is assumed and fixed at 1%; the dynamic matrix has the following expression: These displacements are used both in the FFT and IWC method, in order to obtain the wavenumber domain; the k-space plots at these two different frequencies are shown in Fig. 4c and4d, respectively. In these two figures, a perfect agreement in the peaks estimation is obtained between the two methods. At each heading angle of the inhomogeneous wave, the IWC method estimated the same value of (k x , k y ) obtained with the FFT, predicting the classical ring shape expected for the isotropic panel.

D = K(1 + iη) -ω 2 M . The

Curved panel

In this section, the previous methods are applied on a curved panel. The employed material is a common alloy (E = 70.0 × 10 9 Pa, ν = 0.33 and ρ = 2700.0 kg /m 3 ) and the geometric properties are: L x = 1.0 m, R = 0.5 m and t = 0.002 m. The FE model of the curved panel is shown in Fig. 2. The FE model of the structure is assembled in a commercial FE software and the dynamic analysis is computed using an in-home algorithm developed in MATLAB. The excitation is punctual and located in the middle of the structure, normal to the inner surface; this location has been selected in order to excite many natural modes as possible. The forced responses and the wavenumber domain at two different frequencies are shown in Fig. 6. The transversal displacement field is obtained by projecting the displacement at each point from the global reference system of the structure into the local one; the out-of-plane displacements are shown in Fig. 6a and 6b, for the two considered frequencies.

Conclusions

This paper compares two wavenumber identification techniques manly devoted to the the measurement of the dispersion of the bending waves in different structures. The IWC method shows a good agreement with the classical FFT approach and with the Kircchoff-Love theory in the estimation of the k-space and of the flexural dispersion curve. The IWC method is also applied to curved and axial-symmetric structures, showing a good agreement with the classical FFT approach, predicting the complex wavenumber at a fixed frequency and identifying the complete θ-dependent dispersion curve; the classical 8-shape is also well identified, with a good estimation of the maxima in the wavenumber domain at different heading angles of the inhomogeneous wave. In the IWC algorithm, the introduction of the attenuation coefficient γ (with the associated damping loss factor coefficient) offers the possibility to separate near-field from far-field, reducing the effect of the finite size of the structures; the algorithm allows to identify and remove the wavenumbers with the imaginary part greater than the real part, which are associated to the near-field propagation and exhibiting a high apparent loss factor.

Figure 1 : 3 . 1

 131 Figure 1: Flat panel: Finite Element Model

Figure 2 :Figure 3 :

 23 Figure 2: Curved panel: Finite Element Model

  3D displacement field at 1100.0 Hz. (c) Comparison FFT-IWC at 500.0 Hz. (d) Comparison FFT-IWC at 1100.0 Hz.

Figure 4 :Figure 5 :

 45 Figure 4: Flat panel: displacement field and wavenumber estimation at two different frequencies

Figure 6 :

 6 Figure 6: Curved panel: displacement field and wavenumber estimation at two different frequencies

  3D displacement field at 4100.0 Hz. (c) Comparison FFT-IWC at 735.0 Hz. (d) Comparison FFT-IWC at 4100.0 Hz.

Figure 7 :

 7 Figure 7: Cylinder: displacement field and wavenumber estimation at two different frequencies

Acknowledgments

This project has received funding from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 675441.


Nataliya Perevoshchikova

Jordan Rigaud

Yu Sha

Martin Heilmaier

Barrie Finnin

Xinhua Elena Labelle

Xinhua Wu

Optimisation of selective laser melting parameters for the Ni-based superalloy IN-738 LC using Doehlert's design

  L'archive ouverte pluridisciplinaire

Introduction

Selective laser melting (SLM) is of huge interest for production of complex shaped metal parts for gas turbine industry and some nuclear applications [START_REF] Muktinutalapati | Materials for Gas Turbines -An overview[END_REF][START_REF] Kelbassa | Manufacture and repair of Aero engine components using laser technology[END_REF]. The SLM process is a rapid additive manufacturing technology suited to the net-shape fabrication of high value metallic components with complex shape. This method uses a laser, which scans over and selectively melts metal powder in a powder bed to build 3D components layer by layer to the design in a Computer-Aided Design (CAD) file.

SLM provides a potential alternative to the casting process, facilitating more optimal designs in terms of component mass and performance.

The microstructure and grain size resulting from the SLM process is controlled through a large number of key parameters [START_REF] Muktinutalapati | Materials for Gas Turbines -An overview[END_REF][START_REF] Kelbassa | Manufacture and repair of Aero engine components using laser technology[END_REF] potentially enhancing mechanical properties compared to conventionally cast structures by reducing or eliminating microstructural elements that impair mechanical properties [START_REF] Linn | Evaluation of property scatter of Ni-base alloy in 738 LC[END_REF][START_REF] Mackay | The development of directional coarsening of the gamma-prime precipitate in superalloy single crystals[END_REF]. Despite these advantages, an issue that needs to be addressed using SLM to manufacture metal components for aerospace industry is structural integrity, which is impaired by induced thermal residual stress and micro-cracking as well as micro-pore formation.

Thermal residual stress

Thermal residual stresses are formed during casting, welding as well as during SLM processes [START_REF] Dantzig | Solidification[END_REF][START_REF] Mercelis | Residual stresses in selective laser sintering and selective laser melting[END_REF][START_REF] Harrison | Reduction of micro-cracking in nickel superalloys processed by Selective Laser Melting: A fundamental alloy design approach[END_REF]. During SLM, thermal residual stresses are experienced during heating and cooling of molten top layers [START_REF] Mercelis | Residual stresses in selective laser sintering and selective laser melting[END_REF][START_REF] Harrison | Reduction of micro-cracking in nickel superalloys processed by Selective Laser Melting: A fundamental alloy design approach[END_REF]. The rapid heating of the top layer combined with the relatively low heat conductivity of the material creates a steep temperature gradient. The top layer tends to rapidly expand due to rapid heating putting the cooler lower layer in tension inducing elastic tensile strain. Once the top layer melts, it is stress relieved and the lower layer goes back to being flat. During cooling, the top layer tends to shrink due to thermal contraction, while the underlying layer tends to expand due to the heat transfer from the melt pool. This causes tensile stresses in the top layer and compressive stresses in the cooler underlying layer creating conditions conducive to hot cracking. A further feature of the SLM process is temperature cycling below the solidus temperature, which additionally contributes to the evolution of thermal residual stresses. If these thermal residual stresses are of sufficient magnitude, they trigger crack formation and propagation in the manufactured parts.

Micro-cracking in IN 738

IN-738 LC is a low carbon nickel based superalloy which consists of high volume fraction of γ' (Ni 3 (Al, Ti)) intermetallic strengthening phase in the face centred cubic solid solution matrix, γ. Cracking in alloys of type IN-738 typically results from three mechanisms: liquation cracking, strain-age cracking or solidification cracking [START_REF] Ojo | Liquation microfissuring in the weld heat-affected zone of an overaged precipitation-hardened nickel-base superalloy[END_REF][START_REF] Ojo | On the role of liquated γ` precipitates in weld heat affected zone microfissuring of a nickel-based superalloy[END_REF][START_REF] Ojo | Study of the fusion zone and heat-affected zone microstructures in tungsten inert gas-welded INCONEL 738LC superalloy[END_REF].

Liquation cracking is a type of hot cracking with a liquid phase involved that takes place during the final stages of solidification. Here, liquation involves the formation of a γ+γ' eutectic below the melting point of the alloy. Liquation can be either constitutional liquation or liquation by segregation.

When temperature increases, γ' particles dissolve progressively. If the temperature reaches the melting point of the γ' phase before it is completely dissolved, the γ' particle melts (constitutional liquation). If the particle dissolves completely before temperature reaches its melting point, no γ' liquation takes place, however, γ' dissolution leads to a local enrichment of the γ matrix in Al and Ti decreasing the melting point at that location [START_REF] Tancret | Thermocalc and Dictra simulation of constitutional liquation of gamma prime during welding of Ni based superalloys[END_REF].

Liquation by segregation is a consequence of elemental segregation during cooling of the alloy. During crystallization the liquid concentration tends to reach the eutectic phase composition decreasing the melting point at that location.

During cooling the γ' particle precipitation and growth at the grain boundaries leads to grain boundary deformation inducing strain and causes strain-age cracking. Solidification cracking does not occur during casting and SLM and only develops during welding in the final stages of solidification caused by residual stresses (contraction) generated as the weld pool solidifies.

The SLM process uses welding techniques at much higher rates. Therefore, cracking during SLM process is similar to that in welding: liquation cracking and strain-age cracking [START_REF] Zhong | Boundary liquation and interface cracking characterization in laser deposition of Inconel 738 on directionally solidified Ni-based superalloy[END_REF]. In addition, cold cracking occurs below the solidus temperature during cooling when the material is subjected to thermal cycling.

Thermal stresses, micro-cracking and micro-pores can be reduced through control of SLM parameters including heated platforms [START_REF] Mercelis | Residual stresses in selective laser sintering and selective laser melting[END_REF][START_REF] Harrison | Reduction of micro-cracking in nickel superalloys processed by Selective Laser Melting: A fundamental alloy design approach[END_REF][START_REF] Zhong | Boundary liquation and interface cracking characterization in laser deposition of Inconel 738 on directionally solidified Ni-based superalloy[END_REF][START_REF] Carter | The influence of the laser scan strategy on grain structure and cracking behaviour in SLM powder-bed fabricated nickel superalloy[END_REF][START_REF] Cloots | Investigations on the microstructure and crack formation of IN738LC samples processed by selective laser melting using Gaussian and doughnut profiles[END_REF]. Generally, increasing the temperature of the platform can reduce thermal stresses [START_REF] Mercelis | Residual stresses in selective laser sintering and selective laser melting[END_REF][START_REF] Harrison | Reduction of micro-cracking in nickel superalloys processed by Selective Laser Melting: A fundamental alloy design approach[END_REF]. Cracking sensitivity can be reduced during SLM through the optimisation of the alloy. In order to address cracking sensitivity, it was shown that Si and Mn contents for Hastelloy-X [START_REF] Tomus | Controlling the microstructure of Hastelloy-X components manufactured by Selective Laser Melting[END_REF] and Si for IN-738 [START_REF] Engeli | Gamma prime precipitation strengthened nickel-based superalloy for use in powder based additive manufacturing process[END_REF][START_REF] Engeli | Effect of Si on the SLM processability of IN738LC[END_REF] need to be kept as low as possible, but the opposite is the case for increasing mechanical properties [START_REF] Zhu | A new way to improve the superalloys[END_REF][START_REF] Holt | Impurities and trace elements in nickelbased superalloys[END_REF]19]. Hot Isostatic Pressing (HIP) can be employed to close pores and cracks and then the microstructure can be improved after HIP using heat treatment [START_REF] Kunze | Texture, anisotropy in microstructure and mechanical properties of IN738LC alloy processed by selective laser melting (SLM)[END_REF][START_REF] Rickenbacher | High temperature material properties of IN738LC processed by selective laser melting (SLM) technology[END_REF], however, surface cracks and open porosity may still remain [START_REF] Rickenbacher | High temperature material properties of IN738LC processed by selective laser melting (SLM) technology[END_REF].

The objectives of the present work are twofold: first, optimal SLM parameters (power, speed and hatching distance) for IN-738 LC alloy powder, yielding a microstructure with a porosity content of <0.5% and without micro cracks should be found. Second, a successful experimental methodology should be developed that minimises the number of experiments and the time to achieve these optimal SLM parameters. In order to reduce the number of experiments, a Doehlert design approach will be used [START_REF] Doehlert | Uniform shell designs[END_REF]. This method is widely used in chemistry [START_REF] Taragano | Application of Doehlert designs for water activity, pH and fermentation time optimization for Aspergillus niger pectinolytic activities production in solid-state and submerged fermentation[END_REF] but application to the optimisation of SLM parameters has not been demonstrated in literature so far. The experimental approach will be exemplified with the Doehlert matrix response variable, relative density, by comparing Archimedes method with microstructural assessments of pores and cracks from image analysis. In addition, utilizing tensile testing of vertically built samples at room temperature the mechanical properties of SLM IN-738 LC followed by standard heat treatment (HT) [START_REF]Alloy IN-738 Technical Data[END_REF] and by HIP + HT, respectively, will be compared to those of conventionally manufactured cast material [START_REF]Alloy IN-738 Technical Data[END_REF]. HIP was applied to close possible pores and cracks in SLMed material, particularly cracks, which are strictly prohibited in rotating parts in aeroengines. Slow cooling rate during HIP can cause grain and precipitate growth, HT following the HIP treatment has, thus, been applied to establish the desired γ/γ' microstructure as expected from the standard heat treatment. As we deliberately restrict ourselves on vertically built samples, this work will not address the impact of anisotropy on mechanical properties.

Design of experiments 2.1 Selection of parameters

SLM parameters, such as building method, layer thickness, laser offset, laser type and scanning strategy influence the as-built microstructure [START_REF] Kruth | Part and material properties in selective laser melting of metals[END_REF]. The effects of power (W), scanning speed (mm/s) and hatching distance (mm) on the final microstructure of asbuilt part are studied in this paper. The hatching distance is the distance between two parallel melt pools scanned closely together as schematically depicted in Fig. 1. These three control parameters are key parameters and interact with each other to influence a key factor in the metal SLM process referred to as the laser energy density, E (J/mm 3 ) [START_REF] Gu | Influences of energy density on porosity and microstructure of selective laser melted 17-4PH stainless steel[END_REF]. The laser energy density is given by

E = P v⋅ h⋅ t (1)
where, P is laser power (W), v is scanning speed (mm/s), h is hatching distance (mm) and t is layer thickness (mm).

Figure 1

Even though the energy density is a crucial indicator of energy input, it may not be a good indicator to predict the crack and porosity level of SLM processed components [START_REF] Gu | Influences of energy density on porosity and microstructure of selective laser melted 17-4PH stainless steel[END_REF]. In [START_REF] Zhong | Boundary liquation and interface cracking characterization in laser deposition of Inconel 738 on directionally solidified Ni-based superalloy[END_REF], an obvious decrease in crack size and numbers of SLM IN-738 LC was achieved with low heat input. In [START_REF] Tomus | Controlling the microstructure of Hastelloy-X components manufactured by Selective Laser Melting[END_REF], microstructure characterization revealed that the pore formation during SLM of Hastelloy-X was triggered by very high scanning speed.

Statistical design

Doehlert matrices present the advantage of being easily expanded in both the variables space and the experimental space [START_REF] Quignon | Use of Doehlert matrices for study of poliovirus-1 adsorption[END_REF]. The combined effects were represented by means of a polynomial model [START_REF] Taragano | Application of Doehlert designs for water activity, pH and fermentation time optimization for Aspergillus niger pectinolytic activities production in solid-state and submerged fermentation[END_REF] for each parameter. The surface methodology was then applied to represent these effects.

The experimental approach uses Doehlert design [START_REF] Doehlert | Uniform shell designs[END_REF][START_REF] Karima | [END_REF] to develop a series of experiments to investigate the influence of the three processing parameters (scanning speed , power and hatching distance) and arrive at a set of optimal parameters to meet the component integrity objectives pertaining to porosity and cracking.

An experimental field was defined as P = 270 -370 W, v = 850 -2250 mm/s and h = 0.09 -0.13 mm. Each experiment was performed twice in order to validate the Fisher test and to provide a stronger statistical basis for analysis.

Most physical and chemical phenomena in nature can be represented -to a certain extent -by a segment of a parabola. If this is the case, the result of Doehlert design with three parameters (power, scanning speed and hatching distance) will be a 2 nd order equation with 1 constant term, 3 linear terms, 3 interaction terms and 3 square terms:

Y = a 0 + a 1 X 1 + a 2 X 2 + a 3 X 3 + a 11 X 1 2 + a 22 X 2 2 + a 33 X 3 2 + a 12 X 1 X 2 + a 13 X 1 X 3 + a 23 X 2 X 3 (2)
where X 1 , X 2 , and X 3 are coded factors, and Y is estimated response. In this study, the relative density of sample (ρ, %) is the evaluation indicator of response Y, taking into account the influence of three variables, power (X 1 ), scanning speed (X 2 ) and hatching distance (X 3 ). Each experiment is located by its three coded values.

The coded values and the corresponding experimental values are used to set up the experiments and the model respectively. The experimental values (Z i ) were calculated from the coded values (X i ) using

ܼ = ܼ + ܺ ∆ܼ (3)
where Z i is the experimental value, Z 0i is the central value, X i the coded value and Z i the range.

Coefficients a 0 …a 23 in Eq. 2 were calculated from the vector of experimental response (Y), which is relative density measured by Archimedes method and matrix of coded factors X 1 , X 2 , X 3 using

∆ ܽ = (ܺ ௧ ܺ) ିଵ ܺ ௧ ܻ (4)
where X t is the transpose of the X matrix.

Typically, the Doehlert statistical design allows the description of a region around an optimal response and contains k 2 +k+1 points for k variables [START_REF] Karima | [END_REF]. For three variables, thus, a set of 13 experiments is required and the uniform distribution of the experiments can be mapped in a three-dimensional space, Fig. 2. The 13 experiments are equidistant from a central experiment having the coded values (0,0,0) and are distributed on a sphere with a radius of 1. The coded (X i ) and experimental values (Z i ) of these three variables are listed in Table 1. Locations of 13 experiments in the threedimensional space, Fig. 2. The coded values and locations of 13 experiments were chosen according to experimental matrix in [START_REF] Lundstedt | Experimental design and optimization[END_REF]. The origin of the 14 th experiment listed in Table 1 (14 th experiment) and plotted in Fig. 2 (dot labelled with number 14)

is coming from the difference between experimental design and developing method based on a global maximum search which is the search for highest relative density in the present case. To find the global maximum, firstly, all critical points need to be established, i.e all points where the first derivative of a function Y (X 1 , X 2 , X 3 ) in Eq.

2 equals zero. Secondly, the function Y at all critical points and all endpoints of the experimental field has to be evaluated to find the global maximum. One set of coded values (14 th experiment) listed in Table 1 and plotted in Fig. 2 corresponds to global maximum of function Y.

The significance of the coefficients in Eq. 2 was evaluated by multiple regression analysis based upon the F-test.

Materials and experimental procedures

Gas atomized IN-738 LC powder having a size range of 20-70 µm supplied by Carpenter (USA) with chemical composition as detailed in Table 2 was used in the current work. An example of as-received particle morphology is shown in Fig. 3. The powder contains a small number of fine particles.

Table 2 Figure 3 The liquidus temperature of IN-738 LC with the chemical composition listed in Table 2 is 1343 °C and the solidus temperature is 1291 °C (estimated using Thermo-Calc 2015a with the TCNI8-database [START_REF] Andersson | Computational tools for materials science[END_REF]). SLM processing was conducted under the following conditions: platform temperature was 100 °C, atmosphere was argon gas at 5 bar pressure with an oxygen level maintained at < 0.1 %, build layer thickness was 40 µm and laser spot beam was 100 µm. The chess scanning strategy was used to build all test coupons and is described schematically in Fig. 4. The hatching pattern rotates in each layer by 67 ° for the chess strategy.

Figure 4

Fourteen test coupons with cubical dimensions of 10*10*10 mm were manufactured according to the various combinations of processing SLM parameters shown in Table 1 using the EOSINT M280 machine. Each experiment was then repeated. After building up the coupons by SLM, they were cut off from the substrate plate by electrodischarge (wire cut) machining.

The Archimedes method was employed to measure the relative density of test coupon according to ASTM standard [29]. For this method, the balance Jewel Lab FA2204B with capacity of 220 g and accuracy of ± 0.1 mg was used. All test coupons with as-SLM surfaces were submerged in ethanol in an ultrasonic bath for 10 minutes to clean the surfaces. The mass of each coupon was measured three times in both, air and ethanol. The relative density (%) of coupon at 20 °C and at 1 atm was calculated as follows

ߩ = ఘ ఘ ೌ × 100 (5)
where

ߩ = (ߩ ௧ -ߩ ) × ݉ ݉ -݉ ௧ + ߩ
where ρ theoretical for IN-738 LC is 8177.1 kg/m 3 [START_REF] Limkitjaroenporn | Determination of mass attenuation coefficients and effective atomic numbers for Inconel 738 alloy for different energies obtained from Compton scattering[END_REF]. The density of air (ρ air ) at 20 °C and at 1 atm is 1.2754 kg/m 3 . The ethanol density (ρ ethanol ) at 20 °C is 789 kg/m 3 . ρ p is the coupon density; m a is the mass of the coupon in air at RT; m f is the mass of the coupon in ethanol at RT. Error bars are reported based on repeat experimental measurement.

During density measurement using the Archimedes method the formation of bubbles (which is an indication of pores open to the surface of the sample) was not observed.

For this reason, the part surface was not sealed with lacquer as recommended in the ASTM standard [29].

All coupons were cut in the middle of the cube and one of the cutting surfaces was prepared for metallographic analysis, as shown in Fig. 5. For each set of parameters in Table 1, two samples were taken. Metallographic samples were prepared using standard metallographic approaches. Samples were examined with a Nikon Eclipse Light Microscope. Optical micrographs were taken gradually across the sample's surface perpendicular to build direction. The relative density of pores and cracks from all surfaces was evaluated using ImageJ software [START_REF] Schneider | NIH Image to ImageJ: 25 years of image analysis[END_REF]. The final relative density of samples excluding defects was analysed. The error bars reported in the following sections are based on repeat image analysis measurements.

Figure 5

Some of the samples were electrolytically etched in a solution of 10 vol. % oxalic acid with 6Volt and 0.4 Ampere for a duration of 26 seconds and examined with Nikon Eclipse Light Microscope and Backscatter electron detector (BSE) using JEOL 7001F field emission gun scanning electron microscope (FEG SEM).

Tensile samples were built in vertical direction using optimal SLM processing parameters. The samples were built as cylinders of 70 mm height and 9 mm in diameter. After building up the samples by SLM, they were cut off from the substrate plate by electrodischarge (wire cut) machining. The samples were machined to their end contour (M8) after standard solid solution heat treatment in air at 1120 °C for 2 hours with air cool, followed by precipitation hardening heat treatment at 850 °C for 24 hours with air cool. This treatment will be abbreviated HT in what follows.

Comparatively, HIP at 1120 °C for 2 hours under 150 MPa in argon with 3°C/min heating and cooling rates was carried out followed by standard heat treatment. A commercial-scale HIP vessel (EPSI, Belgium), using argon as the pressurising gas, was used for HIP. Consequently, these samples are designated HIP+HT.

Tensile testing was performed at room temperature (RT) and a strain rate of 4 x 10 -3 s - 1 using an Instron 100 kN Model 5982 with an extensometer attached to the gage section for continuously monitoring the strain during the test. Three repeat experiments were performed for vertical samples and the error bars reported in the following sections are based on these repeat experimental measurements.

Results and discussion

Relative density

The relative density with error bars of test coupons built using the SLM processing parameters from Table 1 was evaluated using both, Archimedes method and microstructure image analysis and plotted as a function of energy density (Fig. 6).

Additionally, values of relative density measured by Archimedes method and microstructure image analysis are listed in Table 1. The relative density predicted by Eq. 2 for SLM processing parameters from Table 1 is plotted as well in Figure 6. predicted by the three approaches is in the range of 1 %. From these results it appears that simply maximizing the energy density is not a suitable approach for reducing the propensity for cracking and pores in SLMed IN-738 LC. Instead, the parameter combinations that yield to energy density around 70 J/mm 3 appears to correspond to the optimal SLM processing window because they provide the highest relative density and lowest level of pore and crack densities.

Results

Figure 6

The complex effect of the three processing parameters on relative density measured by Archimedes method, image analysis and calculated by Eq. 2 is presented in Fig. 7.

In order to evaluate the power effect on relative density in isolation, data for a given fixed scanning speed and hatching distance were plotted in Figs. 7e-g -g agrees well with all experimentally measured data points, we believe that the trend of reducing the relative density with increasing power for 0.09 mm hatching distance would likely be confirmed by experimental methods.

Figure 7

A relative density of as-built SLM parts of 99.5 % and above is promising and the relative density measured by image analysis lies in this range. However, since image analysis is always performed on 2-dimensional sections of the 3-dimensional microstructure, it may not give an accurate representation of the actual pore and crack distribution in the bulk of the material. Thus, at least the associated scatter during the density measurement is expected to be larger than for the Archimedes method as the latter always takes the whole sample volume into account. The accuracy of relative density measured by Archimedes method can be somewhat compromised by the roughness of the net-finished (SLM) surface, however, the variance in measured density for this method is lower, see Figs. 6 and7.

Iso-relative density lines calculated with Eq. 2 for hatching distances of 0.09 mm, 0.11 mm and 0.13 mm, respectively, as a function of power and scanning speed are shown in Fig. 8. Only relative densities above 98 % were considered. For 0.09 mm hatching distance, the maximum relative density ~99.4 % determined using the Archimedes method for laser powers in the range from 270 W to 300 W and for scanning speeds from 845 mm/s up to 1225 mm/s, is consistent with the predicted response by Eq. 2, Fig. 8a. This window covers the set of optimal processing parameters predicted by Archimedes method and image analysis. For 0.11 mm hatching distance, at least 99.3 % relative density has been calculated for powers from 270 W to 282 W and for scanning speeds from 845 mm/s up to 900 mm/s, Fig. 8b.

Finally, for 0.13 mm hatching distance, at least 98.9 % relative density is predicted for a very narrow processing window, i.e. ranging for the laser power from 271 W to 272 W and for the scanning speed from 850 mm/s to 855 mm/s, see inset in Fig. 8c.

These results suggest that the highest relative density as evaluated by Archimedes method and microstructure image analysis can be obtained in a wide operating window of suitable processing parameters (scanning speed and power, see Fig. 8a) provided the hatching distance is fixed at 0.09 mm. Further, the results also reveal the hatching distance to be a crucial parameter in the set of SLM parameters as the relative density reduces dramatically with an increase in hatching distance. The same tendency is tracked for power and scanning speed, particularly for scanning speed, whose operating windows are both narrowed with hatching distance increase suggesting that there is the synergy between hatching distance, scanning speed and power.

Figure 9

Electrolytic etching of samples as displayed in Fig. 9 reveals some common microstructural features, Fig. 10. Columnar grains are visible and defined by dendrite orientation. The size of melt pool is found to be approximately 100 µm corresponding well to the size of the laser spot beam. The growth direction of grains appears to be along the build direction and along the thermal gradient. Micro-cracks such as the one shown in Fig. 10b where frequently located in the melt pool, whereas micro-pores, From these results it appears that the micro-pores are observed in the samples built with high scanning speed (above 1316 mm/s) and large hatching distance. As the pores locate between the melt pools, Fig. 10c, this suggests that the scanning speed is too fast resulting in poor fusion of the melt pool to the previous layer and that a large hatching distance triggers this effect. Lower scanning speeds and a small hatching distance need to be used in optimal parameters providing good overlap between layers.

Microcracks occurred due to the material's hot cracking sensitivity where cracks are in the form expected from hot cracking: they are primarily intergranular, see Fig. 11, as this is the path of least resistance. Micro-cracks are observed in the samples built with high power (above 300 W) and high hatching distance (above 0.09 mm) and scanning speed (below 1000 mm/s). The optimal processing parameters provide the reduction of crack formation indicating that these processing parameters reduce the crack sensitivity of material.

The industrial standard heat treatment [START_REF]Alloy IN-738 Technical Data[END_REF] has been applied to our SLM IN-738 LC sample produced using optimal processing parameters. The microstructure of the SLM sample subjected to the standard heat treatment is shown in Fig. 12. Obviously, the heat treatment strongly influences microstructural features. While the as-SLM material shows melt-pools as exemplified with Figs. 10 and 11, heat treatment erases these melt-pools and the microstructure consists of large γ-matrix columnar grains, Fig. 12a. Inside these γ-grains a bi-modal distribution of fine secondary γ' particles situated between coarser primary γ' precipitates are visible, (see the higher magnification micrograph Fig. 12b). This bimodal particle microstructure compares well with the one observed for as-cast material [START_REF] Stevens | The dependence of creep rate on microstructure in a γ' strengthened superalloy[END_REF].

Figure 12

Mechanical properties

Tensile tests were performed on the SLM HT and HIP+HT samples, respectively.

Engineering stress-strain curves are plotted in Fig. 13. For each set of experiments, at least 3 tests were performed and these data are plotted in Fig. 13. In Table 3 the tensile test results are summarized and compared with fully heat-treated cast material [START_REF]Alloy IN-738 Technical Data[END_REF]. Clearly, the properties from standard cast material were exceeded by the SLM material in terms of both, yield strength (being ~4%-9% higher) and UTS (~27%-33%). More importantly, the average elongation at fracture is doubled as compared with the cast material. Assessing heat-treated SLM samples versus HIP+HT ones yields very similar values for all measured properties, i.e. yield strength, UTS and elongation at fracture. In addition, the consistency of the tensile test data is excellent providing confidence in the uniformity of the heat and HIP treatments. From the tensile properties observed, the HIP treatment has not improved the results significantly.

The relative density of SLMed followed by HT and SLMed followed HIP+HT material measured by Archimedes method is ~99.4 % and 99.8 %, respectively.

Walston et al. [START_REF] Walston | The role of the γ/γ ' eutectic and porosity on the tensile behavior of a single-crystal nickel-base superalloy[END_REF] studied the effect of porosity level in the range from 0.035 % after (HIP+HT) to 1 % (HT) on room-temperature tensile properties of a cast single-crystal

Ni-based superalloy. Their results suggest that porosity levels in this range have no effect on tensile properties. The level of porosity in the current material investigation after HT and HIP+HT is the same and is within the porosity range reported by Walston in [START_REF] Walston | The role of the γ/γ ' eutectic and porosity on the tensile behavior of a single-crystal nickel-base superalloy[END_REF]. Thus, it would be expected also to have no influence on mechanical properties.

It may be concluded that by applying our optimised processing parameters SLM of IN 738 LC the level of pores and cracking can be reduced to the extent that HIP is not necessary in terms of tensile properties sensitivity. Two methods, Archimedes method and image analysis were used in this study to assess relative density of SLM produced samples with sets of processing parameters as listed in Table 1 show coherency in prediction with predicted response (Y) by Eq.

2.

By using optimised process parameters (low heat input, medium scanning speed and small hatching distance) which provides medium energy density, samples of IN-738 LC with a macroscopic porosity < 0.5 % and free of micro-cracks can be manufactured by SLM. The synergy between hatching distance, scanning speed and power appears to be significant for micro-pore and micro-crack formation. An obvious decrease in crack and pore numbers of SLM IN-738 LC powder alloy was achieved with reduced hatching distance and reduced scanning speed, and reduced power.

The mechanical properties of SLM IN-738 LC produced using optimal processing parameters followed by standard heat treatment [START_REF]Alloy IN-738 Technical Data[END_REF] demonstrated superior strength and ductility properties as compared to those of conventionally manufactured castmaterial [START_REF] Mackay | The development of directional coarsening of the gamma-prime precipitate in superalloy single crystals[END_REF]. Additional HIP of SLM material did not yield a noticeable effect on tensile properties indicating that the level of both, porosity and crack density might be acceptable without HIP.

Optimisation of Selective Laser Melting parameters for IN-738 using Doehlert's design

Figure 1 : Schematic describing the hatching distance between two parallel melt pools.

Figure 2: Three-dimensional geometry of experimental space with 13 experimental points for three different coded variables (e.g. X 1 , X 2 and X 3 ). The bottom figure

shows the projection of X 3 axis on X 1 and X 2 , 2-dimension space.

Figure 8 4. 2

 82 Figure 8

Fig. 10c ,

 10c Fig. 10c, are usually observed between adjacent melt pools. Fig. 11 details location and propagation of the micro-crack shown in the overview, Fig. 10b, indicating that crack propagation is clearly intergranular.

Figure 10 Figure 11

 1011 Figure 10

Figure 13 5

 13 Figure[START_REF] Carter | The influence of the laser scan strategy on grain structure and cracking behaviour in SLM powder-bed fabricated nickel superalloy[END_REF]

Figure 3 :

 3 Figure 3: Morphology of gas atomised powder particles.

Figure 4 :

 4 Figure 4: Schematic diagram describing chess scanning strategy.

Figure 5 :

 5 Figure 5: Schematic of SLM built cubes. The hatched area was prepared for metallographic analysis.

Figure 6 :

 6 Figure 6: Relative density of samples built using SLM processing parameters from

Figure 8 :Figure 9 :Figure 10 :Figure 11 :Figure 12 :

 89101112 Figure 8: Iso-relative density lines calculated with Eq. 2 for (a) 0.09 mm, (b) 0.11 mm and (c) 0.13 mm hatching distances as a function of power and scanning speed.

Figure 13 :

 13 Figure 13: Engineering stress-strain curve for the SLM processed IN738-LC built in vertical direction followed by heat treatment and SLM followed by HIP and heat treatment.

Table 1

 1

	versus energy density calculated according to Eq. 1. Circles represent data
	from Archimedes method, triangles refer to image analysis using ImageJ software and
	squares represent response predicted (Y) by Eq. 2. Black full symbols correspond to
	the lowest relative density; red full symbols to the highest relative density; blue full
	symbols represent the highest energy density.

Table 1

 1

measured (circles) using Archimedes method and (triangles) image analysis using ImageJ software, and calculated with response predicted by Eq. 2 (squares) as a function of (a-b) scanning speed and (c-d) hatching distance and (e-g) power. Data of relative density versus power is presented with fixed scanning speed and hatching distance.

Table 1 :

 1 Experimental matrix created by the Doehlert design method for free variables, experimental variables and relative density measured by Archimedes method and microstructure image analysis.

		Coded values		Experimental values	Relative density, %
	Experiment X 1	X 2	X 3	Power,	Speed,	Hatching	Archimedes	Image
					W	mm/s	distance,	method	analysis
							mm
	1	0	0	0	320	1550	0.11	98.84	99.32
	2	1	0	0	370	1550	0.11	98.86	98.99
	3	0.5	0.866 0	345	2250	0.11	97.24	97.13
	4	-0.5	0.866 0	295	2250	0.11	95.3	95.71
	5	-1	0	0	270	1550	0.11	98.32	99.19
	6	-0.5	-0.866 0	295	850	0.11	99.17	99.67
	7	0.5	-0.866 0	345	850	0.11	99.16	99.62
	8	0.5	0.289 0.816	345	1784	0.13	98.13	98.22
	9	-0.5	0.289 0.816	295	1784	0.13	96.65	97.08
	10	0	-0.577 0.816	320	1084	0.13	98.67	99.45
	11	0.5	-0.289 -0.816 345	1316	0.09	99.02	98.22
	12	-0.5	-0.289 -0.816 295	1316	0.09	99.18	99.89
	13	0	0.577 -0.816 320	2016	0.09	98.66	99.81
	14	-1	-0.577 -0.816 270	1084	0.09	99.35	99.99

Table 2 :

 2 Chemical composition of IN738LC used in the current work.

	Ni	Cr	Co	Ti	Al	C	W	Ta	Mo	Nb
	wt.% Bal.	16	8.64	3.46	3.4	0.11	2.58	1.81	1.82	0.99
	Fe	Mn	Si	B	S	Cu	Zr	O	N	H
	wt.% 0.08	<0.01 0.04	0.01	0.002 0.01	0.03	0.0169 0.0034 >0.005

Table 3 :

 3 Engineering tensile properties of cast and SLM processed IN738-LC followed by standard heat treatment at room temperature.

	Description	σ 0.2 , MPa	σ UTS , MPa	ε uniform , %
	Cast reference	896	1034	7
	SLM-HT: vertical	981±12	1450±16	14±1.1
	SLM-HIP-HT:vertical	932±4	1350±22	14±1.3


G Tufano

C Droz

M Ichchou

O Bareille

A.-M Zine

W Desmet

B Pluymers

ON THE STRUCTURAL CHARACTERIZATION THROUGH K-SPACE METHODS: ASSESSMENTS AND VALIDATIONS

Keywords: Periodic structure, wavenumber, damping loss factor, dispersion curves

The physical characteristics of periodic structures are employed to analyze the vibroacoustic response of several complex structures in the wavenumber domain. This work is focused on the analysis of elastic periodic structures, designed in order to obtain a wave attenuation in certain frequency bands, generating the so-called band-gaps (mainly related to the Bragg's effect), and on the identification of the material properties of complex structures. In this context, an inverse wavenumber correlation method is developed to obtain the dispersion characteristics and the damping information of complex periodic structures. The wavenumbers and the damping loss factor are identified exciting the structure by a unit harmonic force and using the complete vibrational field as primary input.

INTRODUCTION

In literature, the complex wavenumber recovery of a vibrating structure is an open issue. Several methods are available to correctly extract the real part of the wavenumber, but the estimation of the related damping information is still an open challenge. In the domain of punctual harmonic excited structures, McDaniel et al. [START_REF] Mcdaniel | Estimation of structural wave numbers from spatially sparse response measurements[END_REF][START_REF] Mcdaniel | A wave approach to estimating frequency-dependent damping under transient loading[END_REF] developed a semi-analytical approach based on damped plane wave propagation (1D guided waves), using an expression of the plane wave of type e ±ikx , where i is the imaginary unit, k is the complex wavenumber and x is the spatial coordinate. Different methods based on Prony series have been developed over the past years; Grosh et al. [START_REF] Grosh | Complex wave-number decomposition of structural vibrations[END_REF] applied their method on cylindrical shells subjected to point harmonic excitation, obtaining a good estimation of the dispersion relation. Using a similar approach presented by Grosh et al., [START_REF] Grosh | Complex wave-number decomposition of structural vibrations[END_REF] another method, called Inhomogeneous Wave Correlation (IWC) method has been developed by Berthaut et al.; [START_REF] Berthaut | K-space identification of apparent structural behavior[END_REF] the proposed approach estimates the flexural wavenumber and the dispersion relation of a vibrating plate under punctual harmonic excitation, correlating the complete displacement field and a damped traveling plane wave, of the type e ±ik(x cos θ+y sin θ) , where θ is the propagation angle of the inhomogeneous wave. An interesting application of the IWC method is presented in Ichchou et al., [START_REF] Ichchou | Multi-mode wave propagation in ribbed plates: Part I, wavenumber-space characteristics[END_REF] obtaining the complete θ-dependent dispersion relation of a ribbed plate. The presented approaches based on plane waves propagation suffered of a disadvantage related to plane wave hypothesis: the vibrational field should be acquired in a steady state condition, trying to avoid the excitation region where the plane wave assumption is not valid. One of the main advantage of the IWC method is the estimation of the complex wavenumber, with the related damping information; Cherif et al. [START_REF] Cherif | Damping loss factor estimation of two-dimensional orthotropic structures from a displacement field measurement[END_REF] applied the IWC method to an aluminum panel and to a two different laminated composite panels with honeycomb core, correlating the measured vibrational field and the damped inhomogeneous wave; the flexural wavenumber and the damping loss factor are both successful estimated. Van Damme et al. [START_REF] Van Damme | Measuring Dispersion Curve for Bending Waves in Beams: A Comparison of Spatial Fourier Transform and Inhomogeneous Wave Correlation[END_REF][START_REF] Van Damme | Energy Distribution and Exchange Between Spatial Harmonics in Bending Wave Phononic Crystals[END_REF] obtained the dispersion relation and the complex wavenumber for complex 1D structures, showing a good description of the Bragg's band gaps, adopting a 1D formulation of the IWC method. An enriched formulation of the IWC method is shown in Van Belle et al., [START_REF] Van Belle | On the impact of damping on the dispersion curves of a locally resonant metamaterial: Modeling and experimental validation[END_REF] where they obtained the dispersion relation of a metamaterial plate with periodic resonators; this formulation takes in account the location of the force to have a better estimation of the wave attenuation. An IWC formulation for curved and axial-symmetric structures is shown in Tufano et al., [START_REF] Tufano | Wavenumber identification technique for axial-symmetric structures[END_REF] obtaining a good description of the k-space domain.

To describe the vibrational field of a structure subjected to punctual harmonic excitation, the Green's functions and the related image source method are widely used in literature. An application of the method of images is shown in Gunda et al. 11 ; the authors used this approach to describe the harmonic response of beams and rectangular plates. Based on the method of the images, Cuenca et al. [START_REF] Cuenca | The image source method for calculating the vibrations of simply supported convex polygonal plates[END_REF] described the vibrational field of a finite plate with simply supported boundary conditions by the reconstruction of the Green's function of a point excited plate. The same method has been applied by Cuenca et al. [START_REF] Cuenca | Measurement of the complex bending stiffness of a flat panel covered with a viscoelastic layer using the image source method[END_REF] to estimate the equivalent material properties of a plate covered by a viscoelastic layer. Recently, Roozen et al. [START_REF] Roozen | Estimation of plate material properties by means of a complex wavenumber fit using Hankel's functions and the image source method[END_REF] used the Green's function of a point excited plate, by using a set of Hankel's functions, into a complex wavenumber fit procedure, making a comparison between the acquired vibrational field and the Green's function; the procedure here described allows to retrieve the complex wavenumber and the equivalent material properties. All the methods based on plane wave assumption are affected by some limitations due to the nature of the plane wave itself. The vibrational field should be acquired in a steady-state condition, sufficiently far from the excitation location (this is a singularity point where the plane wave hypothesis is not valid) and trying to distinguish between the direct, reflected and evanescent fields. In this paper, a method is proposed to combine advantages of the IWC method and of the image source method, which uses a set of Hankel's functions to obtain the vibrational field of a point excited structure. The main advantage of this approach is the possibility to be close the excitation region, avoiding the problems related to the plane wave assumption. The proposed approach estimates the complex wavenumber and the dispersion relation of several complex structures, showing a good description of the periodicity effect (Bragg's band gaps); the damping loss factor is also well estimated.

METHODOLOGY

The complex dispersion relation of different structures is here estimated by a wave correlation method. The presented approach estimates, at each frequency f 0 , the complex wavenumber as the point of best agreement between the acquired vibrational field and the Green's function of the point excited structure. The Green's function is derived by the classical Kirchhoff's thin plate theory [START_REF] Leissa | Vibration of plates[END_REF] ; according to this theory, for an isotropic, homogeneous plate and of a constant thickness h, the flexural vibration, in terms of transverse displacement w, is expressed as follows

D∇ 4 w( x,t) + ρh ∂ 2 w( x,t) ∂t 2 = P( x 0 ,t) (1)
where

D = Eh 3 12(1-ν 2 )
is the bending stiffness, E is the Young's modulus, ν is the Poisson's coefficient, ρ is the mass density, ∇ 4 is the biharmonic operator, t is the time variable, P is the force per unit area and x, x 0 are the position vectors of the acquisition and excitation points, respectively. Assuming an harmonic point excitation, the expression of the force becomes P( x 0 ,t) = F( x 0 )e iωt , being ω the angular frequency; consequently, the steady state response of the transversal vibrational field is w( x,t) = u( x)e iwt . Substituting these two expressions in Eq. 1, the governing equation assumes the following form

D∇ 4 u( x) -k 4 u( x) = F( x 0 ) (2)
where k 4 = ρhω 2 /D is the flexural wavenumber. For a plate of infinite lateral dimensions, the solution of Eq. 2 is given by

G ∞ (x, y) = 1 8k 2 D H 1 0 (kr) -H 1 0 (ikr) (3)
with G ∞ the Green's function of the infinite plate, H 1 0 the Hankel's function of first kind and order 0 and r = xx 0 , yy 0 is the distance between the acquisition (x, y) and excitation (x 0 , y 0 ) points.

At each frequency f 0 , the complex flexural wavenumber is found as the point of best agreement between the measured vibrational field and the Green's function expressed in the Eq. 3.

Considering an acquisition region of area S in the plane (x, y), indicating with w and w the measured vibrational field and the one described by Eq. 3, respectively, the normalized correlation function has the following expression

F (k R , k I ) = S w • w(k R , k I ) * dx dy S | w| 2 dx dy • S |w(k R , k I ) 2 dx dy| (4)
where * denotes the complex conjugate. For discrete points measurement, the integrals in Eq. 4 are replaced by summations over the entire domain. The complex wavenumber is identified as the location of the maximum of the correlation function. From the estimated wavenumber amplitude, an estimation of the damping loss factor can be obtained, at each frequency, by the following relationship 2

η = ℑ(k 4 ) ℜ(k 4 ) (5)
By choosing a proper direction of propagation, Eq. 3 can be applied to a narrow beam subjected to punctual harmonic excitation. Keeping the same notation and indicating with L the length of the acquisition region, the correlation function showed in Eq. 4 becomes

F (k R , k I ) = L w • w(k R , k I ) * dx L | w| 2 dx • L |w(k R , k I ) 2 dx| (6)
The dispersion curves of the different structures have been validated using the Wave Finite Element Method (WFEM) [START_REF] Brillouin | Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices[END_REF][START_REF] Mace | Finite element prediction of wave motion in structural waveguides[END_REF][START_REF] Manconi | Modelling wave propagation in two dimensional structures using finite element analysis[END_REF] ; this method is not here described.

NUMERICAL RESULTS AND VALIDATIONS

The proposed approach is applied on different structures. Firstly, two different beams are taken in account: isotropic and periodic (with varying material and cross-section along the length). Then, application cases on two different plates are shown, taking in account an isotropic and homogeneous plate and a composite one, made of three different layers in the thickness direction.

Isotropic and periodic narrow plates

The first application of the proposed approach has been conducted on a simple isotropic beam. The total length of the beam is 1.0 m and the cross-section has dimensions 10 x 1 mm 2 ; the finite element model of the beam is shown in Fig. 1a. The employed material is a general thermoplastic polymer (ABS) with Young's modulus E = 1.0 GPa, density ρ = 980.0 kg m -3 and Poisson's coefficient ν = 0.35; two constant values of structural damping η are assumed: 2% and 4%. An analytical expression of the dispersion relation can be derived using Timoshenko's model for vibrating beams, resulting in

EIk 4 -ρAω 2 -ρI + EIρ KG k 2 ω 2 + ρ 2 I KG ω 4 = 0 ( 7
)
where G is the shear modulus, I = bh 3 /12 is the second moment of area, A is the beam's cross-section area and K = 5/6 is a constant to take in account the shear force variation.

A periodic beam is also investigated, to prove the feasibility of the proposed approach in describing the band-gaps due to the geometric periodicity of the structure. The finite element model of the unit cell of the considered structure is shown in Fig. 1b. The unit cell has total length of 0.04 m and it's characterized by a double periodicity: material and cross-section. The The aluminum part has a cross-section of dimensions 10 x 2 mm 2 ; the ABS part has dimensions 20 x 2 mm 2 . The propagative part of the dispersion curve for the isotropic beam is shown in Fig. 2a; the proposed IWC approach exhibits a very good agreement with the analytical model (cfr. Eq. 7) and the WFE method. In Fig. 2b is shown the damping loss factor estimation; in both cases, the identified complex wavenumbers give a reasonable estimation of the structural damping previously introduced.

The dispersion curve of the periodic beam are shown in Fig. 3; the dispersion curve calculated with the WFE method are limited to the first Brillouin zone, where the wavelengths are longer than one period of the structure (Fig. 3a). The real part of the wavenumber has periodicity π/∆, with ∆ the length of the unit cell, showing multiples band-gaps, but always with

k ℜ ∈ [0, π/∆].
The dispersion curve estimation with the proposed approach is shown in Fig. 3b, being in good agreement with the one calculated with the WFE method, identifying the same wide band-gaps zone. The proposed IWC approach takes in account the whole structure, avoiding the periodicity issue previously described in the wavenumbers estimation.

Isotropic and laminated plates

The proposed approach has been extended to 2D structures, an isotropic plate, made of ABS, and a composite one, constituted of three layers: aluminum-ABS-aluminum, of thickness 1 -8 -1 mm, respectively; both plates have geometrical dimensions 1.0 x 0.6 x 0.01 m 3 . The finite element models of the two unit cells are shown in Fig. 4; both unit cells are used in the WFE method to validate the results obtained with the proposed IWC approach. For both test cases, two different constant values of structural damping η are assumed: 2% and 5%. For what concern the isotropic plate, an analytical expression of the dispersion relation can be derived by Kirchhoff's thin plate theory, obtaining the following expression for the flexural wavenumber

k = √ ω ρh D 1/4 (8)
The propagative part of the dispersion curve is shown in Fig. 5a; the proposed approach is in good agreement with the analytical formulation of Eq. 8 and the WFE method, showing a low overestimation in the high-frequency region. In Fig. 5b, the damping loss factor estimations are shown; the relation of Eq. 5 is used, based on the estimated complex wavenumbers.

The propagative part of the dispersion curve for the composite plate is shown in Fig. 6a; a good agreement between the proposed approach and the WFE method is obtained in the whole frequency band of investigation. A very good identification of the damping loss factor is performed, and shown in Fig. 6b. Using an inverse approach based on the estimated wavenumber, the equivalent bending stiffness and the equivalent Young's modulus of the structure have been calculated by the following formulas, directly derived from Eq. 2

D eq = ω 2 ρh k 4 and E eq = D eq 12(1 -ν 2 ) h 3 (9)

CONCLUDING REMARKS

In this work, different ideas are successfully implemented in this extended IWC approach, showing a very good prediction of the complex dispersion relation of periodic narrow plates and laminated structures. An estimation of the damping loss factor is obtained; the calculated values of η fast converge to the structural damping introduced. The feasibility of this method in estimating the wave attenuation in a structure is successfully demonstrated. The proposed approach allows to identify the resonance zones (stop-bands) due to the geometrical periodicity of the structure, showing a good agreement with the WFE method, based on the unit cell theory, avoiding some limitations related to the latter method. The proposed approach can be used in an inverse way to estimate the frequency dependent equivalent material properties of complex structures, when analytical models are not available or difficult to obtain.

( a )

 a Isotropic beam (part). (b) Periodic beam (unit cell).

Figure 1 :

 1 Figure 1: Finite element models of the isotropic and periodic beams

Figure 2 :

 2 Figure 2: Dispersion curve and Damping Loss Factor estimation for the isotropic beam

  Dispersion curves (IWC).

Figure 3 :

 3 Figure 3: Real and imaginary dispersion curves of the periodic beam

  (a) Isotropic plate (unit cell). (b) Composite plate (unit cell).

Figure 4 :

 4 Figure 4: Finite element models of the isotropic and composite plates

Figure 5 :

 5 Figure 5: Dispersion curve and Damping Loss Factor estimation for the isotropic plate

Figure 6 :

 6 Figure 6: Dispersion curve and Damping Loss Factor estimation for the composite plate

Figure 7 :

 7 Figure 7: Dispersion curve and Damping Loss Factor estimation for the composite plate

ACKNOWLEDGMENT

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 675441.