Elia Zgheib
email: eliazgheib@utt.fr

Akram Alhussein

Mohamed Fares

Khaled Khalil

Manuel François

Multilayered models for determining the Young's modulus of thin films by means of Impulse Excitation Technique

Keywords: elasticity constants, multilayers, dynamical resonant method, coatings, physical vapor deposition, uncertainty

This work presents a methodology to determine the Young's modulus of an individual coating in a multilayered system by means of the Impulse Excitation Technique (IET). In this technique, the composite beam is excited by an impulse and the frequencies of the first four bending modes are extracted. They are used in a one-dimensional model to obtain the Young's modulus of the coating. Based on two different theories: the Flexural Rigidity of a Composite Beam (FRCB) and the Classical Laminated Beam Theory (CLBT), different models proposed in the literature for bilayer beams have been extended to describe a beam composed of N dissimilar layers. Moreover, an enhanced model was developed based on the laminated theory. It takes into account the shift of the neutral axis after each deposited film, which makes it applicable for any film thickness. The reliability of the proposed model is investigated by comparing its predicted frequency to those of existing models for bilayers. It is also compared to finite element analysis of beams composed of two and three dissimilar layers. A metrological study was performed to quantify the most influencing factors on the global uncertainty. The methodology was applied to beams composed of three layers (N=3) with titanium and niobium thin films deposited by DC magnetron sputtering. The most accurate models are applied to obtain the Young's modulus of Ti and Nb films in the Ti/Nb/(AISI 316 or Glass) multilayer beam. The films microstructure and morphology were analyzed by X-ray diffraction and scanning electron microscopy.

Introduction

Thin film deposition is a common technology used to enhance surface properties. The films are elaborated in various ways to protect materials from different types of damage [START_REF] Makhlouf | Protective coatings for automotive, aerospace and military applications: current prospects and future trends[END_REF][START_REF] Li | Atom doping in α-Fe2O3 thin films to prevent hydrogen permeation[END_REF] and to obtain functional surfaces. Numerous coating types have proven to be useful in a wide range of applications [START_REF] Sertel | Development of MgO: TiO2 thin films for gas sensor applications[END_REF][START_REF] Lee | Characterization of mesoporous silica thin films for application to thermal isolation layer[END_REF][START_REF] Tiron | Beryllium thin films deposited by thermionic vacuum arc for nuclear applications[END_REF][START_REF] Achache | Sputtered gum metal thin films showing bacterial inactivation and biocompatibility[END_REF]. These various types of coatings can be used individually or in combination (multiple layers) depending on the requirements of the operating environment [START_REF] Deng | Mechanical behavior of multilayered nanoscale metal-ceramic composites[END_REF][START_REF] Williams | Magnetic thin films and multilayers in data storage applications[END_REF][START_REF] Romero | Nanometric chromium/chromium carbide multilayers for tribological applications[END_REF][START_REF] Ziegele | DLC and metallic nanometer multilayers deposited by laser arc discharge[END_REF]. Multilayer structures containing metals, polymers and ceramics are widely used for modern technologies. Over the last twenty years, the investigation into the mechanical behavior of multilayer metallic thin films has attracted the attention of many researchers [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF][START_REF] Nallamshetty | Young's modulus of Cu/Cr multilayer films[END_REF]. For reliable and safe operation, knowledge of the elastic behavior of each individual layer and of the multilayers as a whole is of particular importance. The rigidity of these coatings has an important role to prevent cracks and structural failure. For example, if the rigidity of the coating does not correspond to the flexural stiffness of the base material, shear cracking can be produced [START_REF] Kubisztal | Elastic properties of epoxy system coatings on aluminum substrate[END_REF].

Elasticity constants are the main parameters that affect the response of a film to mechanical stresses. Young's modulus is one of these important parameters describing the elastic deformation of materials. Even if this property is particularly difficult to determine for thin films, the Young's modulus of a coating is crucial in terms of load carrying capacity [START_REF] Papachristos | Young's modulus, hardness and scratch adhesion of Ni-P-W multilayered alloy coatings produced by pulse plating[END_REF]. Many static and dynamic techniques were developed to determine the elastic constants of bulk and coated materials [START_REF] Bellan | Evaluation of Young modulus of CVD coatings by different techniques[END_REF][START_REF] Radovic | Comparison of different experimental techniques for determination of elastic properties of solids[END_REF][START_REF] Tan | Anisotropic elastic properties of thermal spray coatings determined via resonant ultrasound spectroscopy[END_REF][START_REF] Thomasová | Young's moduli of sputter-deposited NiTi films determined by resonant ultrasound spectroscopy: Austenite, R-phase, and martensite[END_REF]. In the following, we will focus on the determination of the coating Young's modulus using a vibrational approach which is an accurate method to determine the mechanical behavior of materials. The Euler-Bernoulli bending beam theory was initially used to develop mathematical relations without taking into account the effects of shear and rotation. Subsequently, Timoshenko's equation was simplified to consider both effects by adding some correction factors [START_REF] Pickett | Equations for computing elastic constants from flexural and torsional resonant frequencies of vibration of prisms and cylinders[END_REF]. These factors are determined as a function of the sample geometry and different vibration modes. After 1999, these improved equations have led to the formulation of the standardized ASTM-E1876 test method. The Impulse Excitation Technique (IET) is one of the vibrational techniques used to determine elastic modulus [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF][START_REF] Bellan | Evaluation of Young modulus of CVD coatings by different techniques[END_REF][START_REF] Gadaud | Application of the dynamical flexural resonance technique to industrial materials characterization[END_REF][START_REF] Gadaud | Bulk and coated materials shear modulus determination by means of torsional resonant method[END_REF][START_REF] Mazot | Détermination du module d'Young de dépôts par flexion dynamique: Application aux systèmes bicouche et tricouche[END_REF]. It is nondestructive and one of the less intrusive techniques, used with high precision to measure the elasticity constants of a wide range of materials such as metals, composites, ceramics, etc. [START_REF] Radovic | Comparison of different experimental techniques for determination of elastic properties of solids[END_REF][START_REF] Pickett | Equations for computing elastic constants from flexural and torsional resonant frequencies of vibration of prisms and cylinders[END_REF][START_REF] Whiting | Evaluation of elastic moduli in thin-film substrate systems by the two-layer vibrating reed method[END_REF] and especially coatings [START_REF] Gadaud | Bulk and coated materials shear modulus determination by means of torsional resonant method[END_REF][START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF][START_REF] Hoy Benítez | Vibration modeling and testing of bilayer beams for determination of film elastic modulus[END_REF]. The principle of this vibrational test is to correlate the resonant frequencies of the tested specimen with its mechanical properties through a suitable model.

Using vibrational beam theory, several authors have developed a composite model (substrate + coating) for measuring the elastic modulus of a thin film. Berry et al. [START_REF] Berry | Vibrating reed internal friction apparatus for films and foils[END_REF] were the first who used the IET to measure the Young's modulus of a composite system (substrate + film). The test relies on a two-step fundamental resonance frequency measurement of the sample: one measurement before deposition for the substrate and another one after deposition for the whole beam. After Berry et al., many researchers have used IET to test samples of various films with different thicknesses and elaboration techniques. For instance, Kim et al. [START_REF] Kim | Elastic modulus measurement of thin film using a dynamic method[END_REF] have used IET to test titanium (Ti)-coated silicon (Si) wafer specimens with different geometries, produced by radio frequency (RF) magnetron sputtering. They have shown that the IET gives accurate results compared to a static method based on beam bending under a dead weight. Peraud et al. [START_REF] Peraud | Determination of Young's modulus by a resonant technique applied to two dynamically ion mixed thin films[END_REF] have measured the Young's modulus of 2.2 µm thin silicon carbide (SiC) and of 2.5 µm thin nickel-titanium alloy (NiTi) manufactured by the Dynamic Ion Mixing (DIM) method. They have shown that IET allows us to determine the Young's modulus of very thin coatings. Cho et al. [START_REF] Cho | Elastic modulus measurement of multilayer metallic thin films[END_REF] have evaluated the Young's modulus of different metallic films: Ni, Co, Cr and Ti thin films deposited by RF magnetron sputtering with different thicknesses on Si wafers. In terms of film thickness variation, they found that the Young's modulus of Ni films decreased as the thickness of the film increased. For Cr and Co films, they did not find any significant difference. Etienne et al. [START_REF] Etienne | Elastic modulus determination of a thin layer[END_REF] have measured the Young's modulus of titanium nitride (TiN) of 0.2 µm of thickness elaborated by ion implantation. Bellan et al. [START_REF] Bellan | Evaluation of Young modulus of CVD coatings by different techniques[END_REF] have evaluated the elasticity constants of 35 µm thick SiC and 95 µm thick Pyrolytic Carbon (PyC) films elaborated by Chemical Vapor Deposition (CVD). They have shown that IET gives good and accurate results. The crystalline thin films are also used in the nanometric range such as in nano and microelectronic devices [START_REF] Deng | Mechanical behavior of multilayered nanoscale metal-ceramic composites[END_REF][START_REF] Williams | Magnetic thin films and multilayers in data storage applications[END_REF][START_REF] Romero | Nanometric chromium/chromium carbide multilayers for tribological applications[END_REF][START_REF] Ziegele | DLC and metallic nanometer multilayers deposited by laser arc discharge[END_REF]. Hoy-Benítez et al. [START_REF] Hoy Benítez | Vibration modeling and testing of bilayer beams for determination of film elastic modulus[END_REF] have tested bilayer beams in a cantilever configuration and they measured the Young's modulus of gold (Au) films of 100 nm thickness deposited on polysulfone (PSF) substrates of 130 µm thickness. They found similar results to those reported in the literature.

For multilayer coatings, Cho et al. [START_REF] Cho | Elastic modulus measurement of multilayer metallic thin films[END_REF] have developed a model, based on the Flexural Rigidity of a Composite Beam (FRCB), to determine the film Young's modulus of a bi-coated substrate. They applied this model to Cr/Ti bilayer thin films manufactured by RF magnetron with different sputtering times deposited on Si wafers. They suggested that the Young's moduli of multilayer films might be obtained by repeatedly using a two-layer composite model. Recently, López-Puerto et al. [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF] have developed a model based on the classical laminated beam theory (CLBT) and they applied it to obtain the elastic modulus of 200 nm aluminum (Al) and 250 nm thick Au films in an Al/Au/Kapton (125 µm) multilayer. After analyzing their model, they found that it provides great accuracy for the prediction of natural frequency in multilayered systems composed of thin films of less than 250 nm deposited on a thicker substrate.

The problem of determining the natural frequency of a multilayer as a function of the elasticity constants of thin films becomes complex for more than two layers but few analytical solutions exist for this situation requiring the use of advanced laminated theories [START_REF] Ying | The mechanical spectra of deposited materials by a composite reed vibration method[END_REF][START_REF] Hanada | Elastic constants of amorphous thin films in the systems SiO2-Al2O3 and AlPO4-Al2O3[END_REF]. Many studies have been made based on this laminated theory [START_REF] Pagano | Exact solutions for rectangular bidirectional composites and sandwich plates[END_REF][START_REF] Foraboschi | Three-layered plate: elasticity solution[END_REF][START_REF] Pagani | Refined dynamic stiffness elements applied to free vibration analysis of generally laminated composite beams with arbitrary boundary conditions[END_REF][START_REF] Dozio | Exact vibration solutions for cross-ply laminated plates with two opposite edges simply supported using refined theories of variable order[END_REF][START_REF] Rao | Vibration of Continuous Systems[END_REF] however; their solutions present a lack of a closed-form analytical expression due to mathematical complexity. Therefore, a one-dimensional solution seems to be inexistent to predict the resonant frequencies of multilayers as a function of elastic modulus for any case of coating-to-substrate thickness ratio at different flexural modes. Also, concerning the vibrational mode, a lack of information concerning the determination of Young's modulus using different modes measured by IET provides the motivation to carry out a thorough study and to develop new models which allow the determination of elastic modulus values.

The aim of this work is to develop new models able to determine the elasticity modulus of "N" arbitrary films in a multi-coated structure. It presents a vibrational approach to determine the i th (i > 0) harmonic resonant frequency as a function of the elasticity constants. The developed models take into account the shift of the neutral axis after deposition, which is essential in the case of layers with different elastic properties. They also take into account the effect of layering, the sources of error and the uncertainty of measurement. An investigation of the measurement accuracy will enable us to extract the true values of frequencies leading to the Young's modulus of coatings.

Firstly, a review of all the models developed in the literature to determine the Young's modulus of coatings and the theoretical background of the IET is presented. Secondly, an enhanced analytical formulation based on the classical laminated beam theory is developed to determine the macroscopic Young's modulus of "N" films (N > 0). It takes into account the shift of the neutral axis after deposition, which is an important factor affecting the solution, as was recently shown [START_REF] Slim | An enhanced formulation to determine the Young's and shear moduli of thin films by means of Impulse Excitation Technique[END_REF][START_REF] Slim | Elasticity Constants of a Two-Phase Tungsten Thin Film[END_REF]. An extension to multilayer structure is developed using the analytical models. The results were compared with the Finite Element Model (FEM) in order to validate the developed models. Finally, the models were validated experimentally and were applied to Ti and Nb films deposited by pulsed-DC magnetron sputtering on different substrates.

Vibrational modeling

Homogeneous beam

In the analysis of the flexural vibrations of a uniform isotropic beam, the Euler-Bernoulli equation is considered, neglecting the effects of shear and rotational inertia. The motion of a beam with length L, width b and thickness h, cross-sectional area S, mass density ρ, elastic modulus E, and second moment of area Iy subjected to flexural (z) vibrations can be described by the following differential equation, where w is the displacement along z [START_REF] Rao | Vibration of Continuous Systems[END_REF]:

( , ) + ( , ) = 0 (1)
This differential equation can be solved analytically to determine the modal shapes and the natural frequencies of the vibration modes. In the present study, a free-free (FF) beam with a rectangular cross-section is used and the characteristic equation corresponding to these boundary conditions is:

ℎ -1 = 0 (2)
By solving Eq. ( 2), the non-dimensional Eigen frequencies Xn for the first four mode shapes were determined. Then, the resonant frequencies can be determined from the following equation [START_REF] Rao | Vibration of Continuous Systems[END_REF]:

( ) = 2 (3)
Where the subscript "0" corresponds to the substrate. By inverting Eq. ( 3), the corresponding Young's modulus can be determined from the following expression:

= 2 " ( ) # (4)
The IET can be used to determine the Young's modulus, in the case of FF beam with a rectangular cross-section, according to the following equation [START_REF] Astm E | Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson[END_REF]:

= $ % & ℎ ' ( " (%) # (5)
Where k1 ≈ 0.9465. Tf is a geometrical correction factor introduced to take into account the shear and rotary inertia effects [START_REF] Astm E | Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson[END_REF]:

' ( =
Where ν0 is the substrate Poisson's ratio. For the case of hs / L ≤ 0.05, Eq. ( 6) can be simplified into the following expression:

' ( = 1 + $ 0 ℎ 1 (7)
Where k2 ≈ 6.585

Composite beam structure

The composite beam considered herein is a FF beam of length L, width b with rectangular cross-sectional area and total thickness ht constituted of a substrate and N dissimilar coatings (Fig. 1a); each one indicated by the index k (k=1,…, N) as shown in Fig. 1b. The vibration measurements of the resonance frequencies are conducted sequentially, first for the substrate, then after each deposited film. This order of measurements requires the determination of the elastic modulus of the k th film (k > 0) before depositing the following one (k+1) th . Using these frequencies, the Young's modulus of each individual film can be determined through analytical models that are proposed in the literature based on two different theories: the flexural rigidity of a composite beam (FRCB) and the classical laminated beam theory (CLBT).

Models based on FRCB

The FRCB theory assumes that the coatings are perfectly homogeneous and adherent to the substrate and the expression of the "n" resonant frequencies of the composite beam can be determined, using the Euler-Bernoulli solution (Eq. 3) as follows:

; ( ) = 2 < < < < (8)
Where EtIt is the flexural rigidity of the entire composite beam expressed as the sum of the flexural rigidity of each layer (Eq. 9). Moreover, the product ρtSt is expressed as the summation of the corresponding product of each layer (Eq. 10) [START_REF] Gere | Mechanics of Materials[END_REF].

< < = = > > ; >? (9)
< < = = > > ; >? [START_REF] Ziegele | DLC and metallic nanometer multilayers deposited by laser arc discharge[END_REF] The subscripts "0", "k", "t" and "N" correspond to substrate, k th film, whole beam and the total number of coatings, respectively.

Many models were developed [START_REF] Gadaud | Application of the dynamical flexural resonance technique to industrial materials characterization[END_REF][START_REF] Mazot | Détermination du module d'Young de dépôts par flexion dynamique: Application aux systèmes bicouche et tricouche[END_REF][START_REF] Whiting | Evaluation of elastic moduli in thin-film substrate systems by the two-layer vibrating reed method[END_REF][START_REF] Berry | Vibrating reed internal friction apparatus for films and foils[END_REF] with analytical expressions for a FF composite beam (substrate + coating) with a rectangular cross-sectional area to determine the Young's modulus of a single layer coating (N=1). Some of these models [START_REF] Mazot | Détermination du module d'Young de dépôts par flexion dynamique: Application aux systèmes bicouche et tricouche[END_REF][START_REF] Berry | Vibrating reed internal friction apparatus for films and foils[END_REF] assumed that the neutral axis remains fixed after deposition (it always corresponds to the mid-plane of the substrate cross-section). With this assumption, no shifting of the neutral surface is considered and a systematic error increases with the film thickness. Contrariwise, Pautrot's model [START_REF] Gadaud | Application of the dynamical flexural resonance technique to industrial materials characterization[END_REF][START_REF] Mazot | Détermination du module d'Young de dépôts par flexion dynamique: Application aux systèmes bicouche et tricouche[END_REF][START_REF] Whiting | Evaluation of elastic moduli in thin-film substrate systems by the two-layer vibrating reed method[END_REF] was developed in order to take into account the shift of the neutral axis after deposition (Fig. 2a). This shift can be determined using the equilibrium of the axial forces [START_REF] Gere | Mechanics of Materials[END_REF] generated during pure bending (Fig. 2b). Developing Eq. ( 8) by taking into consideration the shift in the expressions of second moments of area of both substrate and coating of Eq. ( 9), we obtain the analytical expression developed by Pautrot et al., expressing the ratio of the frequencies of the substrate and the coated substrate [START_REF] Gadaud | Application of the dynamical flexural resonance technique to industrial materials characterization[END_REF][START_REF] Mazot | Détermination du module d'Young de dépôts par flexion dynamique: Application aux systèmes bicouche et tricouche[END_REF][START_REF] Whiting | Evaluation of elastic moduli in thin-film substrate systems by the two-layer vibrating reed method[END_REF]:

@A (% B = (A C% A D% E + 1) (A C% A D% + 1) + 3A C% A D% (A D% + 1) @A F% A D% + 1B(A C% A D% + 1) (11)
Where:

A D% = ℎ % ℎ (12)
A F% = % (13)

A C% = % (14)
A (% = % (15)
Then, by inverting Eq. ( 11), the Young's modulus of the coating can be calculated using the following expression:

% = 2A D% & G@A D% + A F% A D% B@A (% B -4A D% E -6A D% -4A D% + 04A D% & H@1 + A D% A F% B@A (% B -1I
+ H4A D% E + 6A D% + 4A D% -@A D% + A F% A D% B@A (% B I 1

.J K (16)
Whiting et al. [START_REF] Whiting | Evaluation of elastic moduli in thin-film substrate systems by the two-layer vibrating reed method[END_REF] have provided a simplified expression for a more direct calculation of the film elastic modulus with further simplification related to the thin film geometry as:

% = 3 L 2A (% + A F% A D% -2 A D% M (17)
Where Rh1 << 1 and the terms containing powers of Rh1 greater than one may be neglected. The more general equation, Eq. ( 16), will be employed in this work to increase the accuracy. Note that all the models discussed above, for the resonant frequency of a bilayer beam (substrate + coating), are based on FRCB of isotropic materials and they neglect material anisotropy and damping contributions such as material viscoelasticity or other hysteretic effects [START_REF] Gamboa | A vibrating reed apparatus to measure the natural frequency of multilayered thin films[END_REF].

2.2.2.

Model based on CLBT In this section, a simplified one-dimensional (1D) model based on CLBT is used to determine the Young's modulus of an individual thin film in a multilayer beam structure. It fulfills Kirchhoff's law [START_REF] Hyer | Stress Analysis of Fiber-Reinforced Composite Materials[END_REF] and it is applicable for one, two and multiple layers [START_REF] Ying | The mechanical spectra of deposited materials by a composite reed vibration method[END_REF][START_REF] Hanada | Elastic constants of amorphous thin films in the systems SiO2-Al2O3 and AlPO4-Al2O3[END_REF]. This theory assumes that the beam is symmetrical about the mid-surface of the composite beam with respect to the z-axis (Fig. 3) and neglects the effect of shear deformations with no coupling between bending and extension [START_REF] Hoy Benítez | Vibration modeling and testing of bilayer beams for determination of film elastic modulus[END_REF][START_REF] Ying | The mechanical spectra of deposited materials by a composite reed vibration method[END_REF]. These two assumptions are not respected in most cases where the film is deposited on one side of the substrate. However, it can be used as an approximation in the case of an asymmetric beam of several thin films deposited on a thicker substrate. Obviously, the higher the ratio of Young's moduli and the thicknesses ratio, the higher the error associated with this model is [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF][START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF]. In the case of a laminated beam, the bending stiffness matrix Dij (3x3) can be expressed as follows [START_REF] Hyer | Stress Analysis of Fiber-Reinforced Composite Materials[END_REF]:

N OP = Q R S OP (>) T UT = V W V WXY 1 3 = R S OP (>) (T > E -T >Z% E ) ;[% >?% \ ], ^= 1, 2 _ 6 $ = 1, … , a + 1 ( 18
)
Where zk is the coordinate of the k th layer and R S OP (>) are the transformed reduced stiffness elements, the function of the reduced stiffness including information on elastic modulus, Poisson's ratio, and the orientation (ϴ) of the k th layer [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF][START_REF] Hyer | Stress Analysis of Fiber-Reinforced Composite Materials[END_REF].

Working with the x-y axes as principal axes for both substrate and film leads to obtain a zero-layer orientation (ϴ = 0) [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF]. For isotropic substrate and film materials, the stiffness elements become:

R S %% (>) = R S (>) = R %% (>) = R (>) = > 1 -(. > ) (19) R S % (>) = R % (>) = . > > 1 -(. > ) (20) R S %b (>) = R S b (>) = 0 (21) R S bb (>) = R bb (>) = > 2 (1 + . > ) (22)
Where:

Ek: Elastic modulus of the k th layer. ʋk: Poisson's ratio of the k th layer.

The flexural moment per unit width My, producing transverse deflections w can be expressed in terms of the curvature as [START_REF] Hyer | Stress Analysis of Fiber-Reinforced Composite Materials[END_REF]:

c = - 1 U %% ; ( 23
)
Where d11 is the (1,1) element of the bending matrix

[d]=[D] -1 .
For a symmetrical laminated beam, d11 can be expressed as:

U %% (;) = N Ud [N] (24)
Ud [N] = N %% N -N % (25)
Where det [D] is the determinant of the matrix Dij.

The vibrating laminated beam subjected to a flexural moment My with transverse displacements w can be expressed, according to CLBT, by the following equation [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF]:

c = g(( (;) (26)
Where ρeff is the weighted area density (in kg/m 2 ) expressed as:

g(( (;) = Q > UT = = > ℎ > ; >? @∑ D W i Wjk B/ Z@∑ D W i Wjk B/ (27)
By substituting Eq. ( 23) into Eq. ( 26), the following differential equation of the transverse vibration of a symmetric laminated beam is obtained:

1 U %% (;) g(( (;) & & + = 0 (28)
Where the term d11 contains the properties of each layer (Young's modulus, Poisson's ratio and thickness). Notice that Eq. ( 28) has the same form as Eq. ( 1) for a homogeneous isotropic beam when carrying out the following transformations:

→ 12 ℎ E U %% (;) (29) → g(( (;) ℎ (30)
By substituting Eqs. ( 29) and ( 30) into Eq. ( 3), the resonant frequencies of a laminated beam can be expressed as:

; ( ) = 2 1 U %% (;) g(( (;) (31)
The frequency ratio becomes:

@A ((;) B = 12 ℎ n 1 U %% (;) g(( (;) o (32)
Much research has already been performed using the CLBT model in order to determine the elastic modulus of thin film [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF][START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF]. By comparing their model to a finite element model (FEM), López-Puerto et al. [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF] found that the CLBT model is limited to a film thickness of 250 nm (Rh1 < 0.002). In our previous study [START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF], we found that the optimum error zone associated with the CLBT model is centered at Young's modulus ratio equal to 1. This is due to the assumption of symmetry on which is based the CLBT model. We also found that Pautrot's model presents the closest results to those of the finite element model. The next two sections (2.3 and 2.4) present two developed models based on the theories presented above. These models describe the behavior of multilayer beams in flexural vibrations.

Extended Pautrot's model (Ext-PM)

As Pautrot's model was the best used to determine the Young's modulus of single layer coatings, an extension is applied to it, where the neutral axis will shift after each layer deposited. This shift can be generalized for any number "N" of isotropic layers using a generalized equation of equilibrium of the axial forces as follows:

= p q > U > = ; >? Q T UT + = Q > T UT r s r Y ; >?% g i ZD k [g i = 0 (33)
With:

t % = d ; + u > = ℎ P >Z% P?% t = t % + ℎ > u > = \ 0 ] $ = 1 1 ] $ v 1 ($ = 1, … , a) (34)
By integrating Eq. ( 33) and applying the static equilibrium of a beam in pure bending, the generalized shift eN becomes:

d ; = ℎ -∑ O ℎ O wℎ O + 2u O ∑ ℎ PZ% O P? x ; O?% 2 ∑ O ℎ O ; O? (35)
Developing Eq. ( 8) by taking into consideration the shift (e2) of a bi-coated beam (N=2) in the expressions of second moments of inertia of both substrate and coating of Eq. ( 9), we obtain the new analytical expression developed herein, expressing the ratio of the frequencies of the substrate and the bi-coated substrate:

@A ( B = 1 + y (A C% ) + z (A C ) + 2{A C% A D% + 2NA C A D (1 + A D% A F% + A D A F )(1 + A C% A D% + A C A D ) (36)
Where:

A D = ℎ ℎ (37)
A F = (38)
A C = (39)
A ( = (40)
y = A D% & (41) z = A D & (42)
{ = 2 + 6A D% (A D% + A D + 1) + A D (3 + 2A D ) (43)
N = A C A D (2A D% + 3A D% A D + 2A D ) + 2A D% + 3A D% + 2 (44)
Knowing the thicknesses and, mass densities of the substrate and of the two films, the Young's modulus of the substrate and of the first film, and by inverting Eq. ( 36), the Young's modulus of the second film is determined using the following equation:

= -| % + }| % -4~% % 2~% (45)
Where:

~% = A D | % = 4A D E (1 + A C% A D% ) + A D H6 -A F @A ( B + 6A D% (2 + A C% A D% )I + A D L4 -@A ( B + 4A D% (3 + A C% A D% ) + A D% H12 -A F% @A ( B IM (47)
% = 1 -(1 + A C% A D% )@1 + A F% A D% + A F A D B@A ( B + A C% A D% (4 + 6A D% + 4A D% + A C% A D% E ) (48)

Enhancement of the CLBT model (Dev-CLBT)

As in the case of single layer coated beams, the CLBT model can be used to determine the Young's modulus of the k th layer (Ek) if the elastic modulus of the (k-1) th , (k-2) th , … , 1 st layers are known. For instance, in a bi-coated isotropic beam structure, the 3x3 bending stiffness matrix Dij is defined in Eq. ( 18) with N=2. An enhancement of the CLBT model is developed herein for "N" dissimilar layers; assuming that the neutral axis will shift after deposition using the shift of Eq. ( 35) and the new 3x3 bending stiffness matrix:

N OP • = Q R S OP ( ) T UT g i ZD k [g i + = Q R S OP (>) T UT r s r Y ; >?% (], ^= 1, 2 _ 6) (49)
Where Z1, Z2 are expressed in Eq. [START_REF] Foraboschi | Three-layered plate: elasticity solution[END_REF].

Inverting the new sub-matrix D ' ij of Eq. ( 49) for the case of a bi-coated (N=2) isotropic beam and applying the expression of d11 from Eq. ( 24), the new R0, R1 and R2 functions of the thicknesses and the shift e2, are determined as:

U %% ( ) = A 1 -. + % A % 1 -. % + A 1 -. L A 1 -. + % A % 1 -. % + A 1 -. M -L . A 1 -. + . % % A % 1 -. % + . A 1 -. M (50)
With:

A = ℎ E 3 -d ℎ + d ℎ (51)
A % = ℎ % E 3 + d ℎ % + d ℎ % (52)
A = ℎ E 3 + (d + ℎ % )(d + ℎ % + ℎ )ℎ (53)
The frequency ratio of a laminated beam is expressed using Eq. ( 32) with the new expressions of D ' ij and d11 from Eqs. ( 49) and [START_REF] Buckle | L'essai de microdureté et ses applications[END_REF], respectively. This expression is an implicit function. Therefore, in order to determine the solution, an iteration loop is applied using bisection method [START_REF] Walter | Méthodes numériques et optimisation, un guide du consommateur[END_REF]. The unknown elastic modulus of the second film (E2) is then found.

For bilayer coatings, López-Puerto et al. [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF] proved that the solution depends on the stacking sequence selected (Au/Al/Kapton or Al/Au/Kapton) because of the lack of symmetry that can lead to larger errors (>1.3%) in the prediction of the frequency of a multi-coated beam using CLBT. Therefore, the developed models, based on the two different theories (FRCB and CLBT) will be examined by comparing them to a numerical model based on finite element analysis (FEA) in order to analyze quantitatively the difference between them. An outline of the analytical models proposed in the literature and the models developed in the present paper is presented in Table 1.

Table 1

Summary of the analytical models.

Experimental details

Thin film deposition

High purity (99.99%) titanium (Ti) and niobium (Nb) thin films were deposited on stainless steel AISI 316 (sample 1), glass (sample 2) substrates and silicon (Si) wafers by magnetron sputtering in DC pulsed mode. The substrates were ultrasonically cleaned in acetone and ethanol for 10 min. Before deposition, they were cleaned by an Argon plasma etching at 0.38 Pa and 200 W (Ubias=21 V) during 30 min to remove impurities and surface oxides that could alter the quality of the adhesion of the films. In this phase, the Ti and Nb targets were pre-sputtered for 15 min. A DEPHIS4 Physical Vapor Deposition (PVD) machine was used with a 600 mm diameter cylindrical deposition chamber of 400 mm in height that was pumped down with a turbomolecular pump to less than 10 -6 Pa before filling it with argon (Ar) gas. An argon flow rate of 50 sccm led to a constant working pressure of 0.2 Pa. The target-substrate distance was kept constant at 10 cm.

Using a DC-pulsed power supply, the intensity applied to the targets was 3 A of a 275 V discharge voltage (frequency = 50 kHz; time off = 4 µs) for the first film of Ti and 228 V (frequency = 100 kHz; time off = 4 µs) for the second film of Nb. The substrates were placed on a rotating holder (10 rpm) to ensure a perfect thickness homogeneity of the films. Under these conditions, the deposition of the first (Ti) and second (Nb) films was carried out with a deposition rate of 16.13 nm/min and 18.90 nm/min, respectively. The stainless steel (AISI 316) and glass substrates were used for IET and nanoindentation characterizations and Si (100) wafers for the Scanning Electron Microscopy (SEM) images.

Dimensions and Density

The coating has the same length and width as the substrate with different thicknesses. The dimensions of each substrate were measured ten times in ten different locations. The average dimensions and their uncertainties are represented in Table 2. Samples 1 and 2 correspond to Nb/Ti/AISI 316 and Nb/Ti/Glass configurations, respectively. The standard uncertainties on the measured dimensions were evaluated by a type A method by computing the standard deviation [START_REF]Evaluation of Measurement Data -Guide to the Expression of Uncertainty in Measurement[END_REF]. The mass of the substrate was measured using a Sartorius precision balance and then its density was calculated. The uncertainties on the substrate masses were evaluated following a type B procedure [START_REF]Evaluation of Measurement Data -Guide to the Expression of Uncertainty in Measurement[END_REF]. According to technical specifications of the precision balance used, the uncertainty is equal to 0.1 mg. After deposition, the mass of the composite beam was measured. The bilayer beam corresponds to the Ti film deposited on the substrate. The second film was deposited on the previous beam in order to obtain a three-layered beam. The corresponding values are presented in Table 3. The difference in mass measured before and after each deposited film corresponds to the mass of the film. Its density was estimated by considering the mass and the dimensions of the film.

Morphology

Using a Hitachi S3500N SEM-FEG operated at 15 kV, the thickness of each bilayer film was measured ten times in ten different positions in order to determine the average thicknesses and their uncertainties (Table 2). Fig. 4 shows the fractured cross-sectional SEM image of Nb/Ti/Si multilayer sample. No significant variation appears in the film thickness. It can clearly be seen that the two films present a porous microstructure with a relatively columnar growth. This can be due to the change in energies of incident ions that are transferred to adatoms during the deposition [START_REF] Achache | Gum Metal thin films obtained by magnetron sputtering of a Ti-Nb-Zr-Ta target[END_REF][START_REF] Musil | Recent advances in magnetron sputtering technology[END_REF]. The deposition rate and pressure [START_REF] Liu | The structural characterizations of Ti-17 alloy films prepared by magnetron sputtering[END_REF][START_REF] Xu | Potential of niobium-based thin films as a protective and osteogenic coating for dental implants: The role of the nonmetal elements[END_REF] affect these energies. Besides, the absence of bias (polarization) can also favor this columnar film growth [START_REF] Achache | Gum Metal thin films obtained by magnetron sputtering of a Ti-Nb-Zr-Ta target[END_REF]. IET experiments The Impulse Excitation Technique (IET) is used to measure the natural frequencies of a specimen impacted by a striker. A beam can be excited in longitudinal, flexural and torsional vibrational modes [START_REF] Astm E | Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson[END_REF]. The sample is placed on a support with negligible interaction (nylon wires placed along the nodal lines of the considered mode) in order to adopt a free-free (FF) boundary condition. These conditions are the simplest to adopt because they represent the configuration with the least interaction between the support system and the specimen compared with the other configurations. An RFDA professional signal analysis system (Resonance Frequency and Damping Analysis) from IMCE Company (Genk, Belgium) was used to measure the resonant frequencies. It is equipped with an RFDA transducer, an acoustic microphone with a frequency range up to 100 kHz, a universal wire support, an automatic excitation unit and a computer system equipped with RFDA software. The microphone detects the mechanical vibrations produced during the excitation of the sample. The detected vibrations are transformed by a transducer into an electrical signal that determines the vibration amplitudes as a function of time. Using the Fast Fourier Transform (FFT), the amplitudes are transformed as a function of frequency defining the resonant frequencies of the sample at different vibration modes. The sample's support used is composed of wires located at the nodal points of the beam where the displacement is equal to zero. More detailed explanations of the procedure can be found in the literature [START_REF] Astm E | Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson[END_REF]. The measurement reliability can be characterized by its trueness and its accuracy, which were investigated in our previous paper [START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF].

The first four resonant frequencies of substrates (316L, Glass), bilayer (Ti/316L, Ti/Glass) and three-layer composite beams (Nb/Ti/316L, Nb/Ti/Glass) were measured. The vibration measurements were conducted sequentially, i.e., firstly, the resonant frequencies of the substrates were determined; then the titanium film was deposited on each substrate and the resonant frequencies of each bilayer beam were measured again; finally, the third layer (Nb) was deposited and the resonant frequencies of the three-layered beams were measured. As seen from Fig. 5, the frequency is shifted 2.58 Hz when the Ti film is deposited, and a new additional shift of 0.78 Hz is observed when the Nb film is deposited on the Ti/316L beam. These frequency shifts were used to calculate the substrate and films Young's moduli using the corresponding analytical model.

Nanoindentation

The contact reduced elasticity moduli of the films were also measured by nanoindentation (NI) tests using a TriboIndenter TI 980-Hysitron with continuous stiffness measurement (CSM) option. For the measurements, a Berkovich diamond tip was used with elasticity constants Eind =1140 GPa and νind = 0.07. Before performing the measurements, the equipment was calibrated using a reference polycarbonate glass specimen.

The films' reduced moduli were determined by taking the average of ten indents and the uncertainty of measurements was determined from their standard deviation. The tests were carried out under controlled load. In order to minimize the influence of the substrate or of previous layers, the indenter penetration depth was kept lower than 10 % of the coating thickness [START_REF] Buckle | L'essai de microdureté et ses applications[END_REF]. The corresponding penetration depth was found to be about 270 nm (8000 µN) for titanium and 300 nm (8000 µN) for niobium. The contact reduced modulus of the sample and the indenter can be expressed as [START_REF] Doerner | A method for interpreting the data from depth-sensing indentation instruments[END_REF][START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF][START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology[END_REF]:

€ • = √ 2ƒ √y ( 54
)
Where β is a constant that depends on the geometry of the indenter, S is the contact stiffness and A is the projected contact area.

Thus, the Young's modulus of the indented material can be calculated from the contact reduced stiffness and the elastic properties of the tip as [START_REF] Doerner | A method for interpreting the data from depth-sensing indentation instruments[END_REF][START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF][START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology[END_REF][START_REF] Sneddon | The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile[END_REF]:

= (1 -. ) G 1 € • - 1 -. O " O " K Z% (55)
Where Er c is the contact reduced modulus, Eind and νind are respectively the Young's modulus and the Poisson's ratio of the tip, E and ν are respectively the Young's modulus and Poisson's ratio of the indented material. More detailed explanations of the procedure can be found in the literature [START_REF] Doerner | A method for interpreting the data from depth-sensing indentation instruments[END_REF][START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF][START_REF] Oliver | Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology[END_REF][START_REF] Sneddon | The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile[END_REF].

3.4.3.

X-ray Diffraction X-ray diffraction (XRD) was employed to identify the crystallographic structure of the films using a Bruker D8 Advance diffractometer equipped with a CuKα (λ=0.15418 nm) radiation operated at 40 kV and 40 mA. Different spectra were acquired at different inclination angles Ψ. The data, in the range of 2ϴ between 20° and 150°, were collected. Analysis of the diffraction patterns was performed using DIFFRAC.EVA software.

Numerical simulation

Finite element analysis (FEA) was carried out using the commercial finite element code ABAQUS [55] in order to determine the frequencies in the flexural mode of the naked substrate, the bilayer and the three-layered beams. Fig. 6 represents the first four vibration modes of a FF multilayered beam. The beam has a length L of 70 mm, width b of 20 mm and thickness h0 of 1 mm with a rectangular cross-section. Two different finite element models were constructed by bonding the coatings on the substrate already created through one of two different functions: tie or partition; that give no significant differences in the "n" modes of frequency extracted. The first model presents a single layer coated beam with a film thickness h1 that was varied from 0 to 1 mm. The second one represents a bi-coated beam with a first film thicknesses h1 of 4 µm (Rh1 = 4x10 -3 ) and 100 µm (Rh1 = 0.1) and a second film with a thickness h2 that varied from 0 to 0.

The colors indicate the vertical displacement (mm).

To mesh the structure, C3D20 quadratic element was used since it presents excellent behavior for linear elastic calculations. A mesh convergence study of the first four frequency modes was performed to determine the mesh density at which the values of the first four resonant frequencies converge. By varying the number of elements in all three directions, the element size of 0.35x0.27x0.5 mm 3 was chosen for the substrate that led to invariant frequency values even with the smallest elements. Considering the beam dimensions mentioned in the first paragraph, it gives 30,000 elements: 200 in the length direction, 75 in the width and 2 in the thickness. For a multilayer beam with Rh1 > 0.5, one element through the film thickness was used (Fig. 7). For beam geometries whose substrate was significantly thicker than the first film (Rh1 ≤ 0.5), a variable element size through thickness was applied as shown in Fig. 7 to avoid the alteration of the frequency values. This mesh principle takes into account the effect of abrupt change in the size of the element at the film/substrate interface avoiding the cutoffs of the propagated waves that disturb the solution [START_REF] Massabò | Cut-off frequencies and correction factors of equivalent single layer theories[END_REF]. The resonant frequencies of the composite beam were extracted using the ABAQUS\Implicit Lanczos eigensolver. These configurations were chosen as a reference to validate the analytical models presented in section 2.

Modal analysis

A parametric comparison of the analytical models with the finite element model (FEM) was done with a combination of different Young's modulus and density ratios. The following values represent different configurations chosen to operate at the extreme border where the former models remain invalid [START_REF] López-Puerto | A vibrational approach to determine the elastic modulus of individual thin films in multilayers[END_REF][START_REF] Mazot | Détermination du module d'Young de dépôts par flexion dynamique: Application aux systèmes bicouche et tricouche[END_REF][START_REF] Berry | Vibrating reed internal friction apparatus for films and foils[END_REF].

For a bilayer beam, the evolution of the frequency ratio Rf1 as a function of the thickness, density and Young's modulus ratios, is represented in Fig. 8 with the thickness ratio Rh1 varying from 0 to 1. The ratios were chosen to cover a large range of materials used for surface coatings. A good agreement can be noted between the developed model (Dev-CLBT), Pautrot's model (Eq. ( 11)) and the FEM, for any Rh1, RE1 and Rρ1 ratios. The good reliability of Pautrot's model has already been shown in our previous paper [START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF]. As can be seen, the good agreement of Dev-CLBT model shows that the influence of symmetry becomes negligible when taking into account the shift of the neutral axis after the deposition of the first film. For thickness ratios < 0.1, all models give approximately the same result. For thicker films (Rh1 > 0.1), the discrepancies between the different models increase due to the different assumptions on which each model is based. The three models (Ext-PM, Dev-CLBT and FEM) predict very similar resonant frequencies for bilayer and three-layered beams. FEM is a numerical approach, which takes into account the slenderness of the beam, and thus it can be considered as a reference to check analytical models. The difference between FEM and the two analytical models remains lower than 0.2 % in the range of the tested ratios (Rh, RE and Rρ), which is negligible. Consequently, the two models are validated and can be used for the prediction of the resonant frequencies. In the next sections, the two models will be used to determine the Young's modulus of each film in a multilayer beam.

Uncertainty analysis

The uncertainty analysis was performed using the ISO standard guidelines [START_REF]Evaluation of Measurement Data -Guide to the Expression of Uncertainty in Measurement[END_REF]. The uncertainty on a quantity x is defined as the standard uncertainty calculated u(x) which is the standard deviation of the distribution of x values. In the present study, the quantitative values of the uncertainties of different sources related to the frequency measurements were imported from a previous work [START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF] where the experimental study was performed using the same IET set-up. Thus, other components of the uncertainty were calculated using the uncertainty propagation equation [START_REF] Bullough | The determination of uncertainties in Dynamic Young's Modulus, Standards Measurement and Testing Project No[END_REF].

Frequency Uncertainty

A series of experimental measurements were performed to identify and estimate the different sources of uncertainty on the measured frequencies. The sources of uncertainty were determined on the IET [START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF] and they led to a significant influence on the frequency measurement. Table 4 represents the uncertainty of the first four resonance frequencies.

Table 4

Uncertainty on the first four resonance frequencies.

The global uncertainties of the resonant frequencies can be calculated from the following expression [START_REF] Slim | On the determination of Young's modulus of thin film with Impulse Excitation Technique[END_REF]:

…( ) = †… ‡ˆ7 + … ‰ŠO‹ + … €g ‡ + … 7OE ‡ ‡ (56)
Where upos, ualign, urep and usupp are the uncertainties caused by the position of the microphone, the misalignment between the sample nodal lines and the supporting wires, the repeatability and the nature of the support, respectively.

It was found that the measurements are mostly affected by the nature of the support and the misalignment. The support and misalignment uncertainties present the highest contributions. This can be due to the approximate realization of the free-free condition.

Uncertainty (Hz) Mode I

Mode II Mode III Mode IV 0.117 0.147 0.250 0.201

Uncertainty on the substrate and coating Young's modulus

Using the equation of uncertainty propagation, the global uncertainty of the substrate Young's modulus was calculated by developing all the uncertainties that appear in Eqs. ( 4) and [START_REF] Tiron | Beryllium thin films deposited by thermionic vacuum arc for nuclear applications[END_REF]. The equation of uncertainty propagation is expressed by assuming that the covariance between the different parameters is null [START_REF]Evaluation of Measurement Data -Guide to the Expression of Uncertainty in Measurement[END_REF].

Using the dimensions uncertainties (Table 2) and the mass uncertainties (Table 3) discussed in section 3.2, the values of the uncertainty on the substrate density are calculated and presented in Table 3. Using the uncertainty of each quantity, the global uncertainty on the substrate Young's modulus was calculated. In a similar way, the uncertainty of the coating Young's modulus was calculated using the equation of uncertainty propagation for Ext-PM and Dev-CLBT. The uncertainty on Poisson's ratio was assumed negligible in the present calculation.

Results and discussion

Microstructure

Fig. 10 presents the X-ray diffraction patterns of the deposited titanium and niobium films. The presence of two phases: A body-centered cubic (bcc) metastable phase, known as β-Ti and a hexagonal closed packed (hcp) phase, known as α-Ti, can be clearly seen in the structure of the titanium film (Fig. 10a). These phases are classically observed on titanium as film [START_REF] Liu | The structural characterizations of Ti-17 alloy films prepared by magnetron sputtering[END_REF] and as bulk material [START_REF] Fréour | Determination of Single-Crystal Elasticity Constants in a cubic phase within a multiphase alloy -X-Ray Diffraction measurements and inverse scale transition modeling[END_REF]. The presence of the metastable phase in the Ti film could be due to the deposition parameters [START_REF] Achache | Gum Metal thin films obtained by magnetron sputtering of a Ti-Nb-Zr-Ta target[END_REF][START_REF] Chawla | Microstructural characterizations of magnetron sputtered Ti films on glass substrate[END_REF][START_REF] Chawla | Morphological study of magnetron sputtered Ti thin films on silicon substrate[END_REF][START_REF] Savaloni | Residual stress and structural characteristics in Ti and Cu sputtered films on glass substrates at different substrate temperatures and film thickness[END_REF]. Contrariwise, the niobium film structure (Fig. 10b) exhibits a bcc single-phase α structure, similar to several studies on Nb thin films [START_REF] Xu | Potential of niobium-based thin films as a protective and osteogenic coating for dental implants: The role of the nonmetal elements[END_REF][START_REF] Seifried | Structure, morphology and selected mechanical properties of magnetron sputtered (Mo, Ta, Nb) thin films on NiTi shape memory alloys[END_REF][START_REF] Gontad | Nanomechanical and electrical properties of Nb thin films deposited on Pb substrates by pulsed laser deposition as a new concept photocathode for superconductor cavities[END_REF][START_REF] Okolo | Stress, texture, and microstructure in niobium thin films sputter deposited onto amorphous substrates[END_REF][START_REF] Goldmann | 2.11.7 Nb (Niobium) (Z=41)[END_REF]

Measurement of resonant frequencies in multilayers

Table 5 lists a summary of the resonance frequencies measured (average value) and the uncertainties for each layered system. The table presents the values measured on two consecutive coatings of Ti and Nb deposited on AISI 316 (sample 1) and glass (sample 2) substrates under the same deposition parameters. An increase in f (n) is observed when each film is added. We can clearly see that the difference Δf in the measured frequencies before and after each deposition, is higher than their corresponding uncertainties, which gives the confidence to use them to determine the Young's modulus of each film. We can clearly see that the contribution (in %) on the measured frequency decreases by taking a higher mode which presents a higher frequency.

Table 5

Flexural resonance frequencies of samples measured by IET before and after deposition.

Determination of Young's modulus in a three-layered beam

Table 6 gives the Young's moduli and uncertainties of different substrates obtained by IET using Eq. ( 5) for the fundamental frequency and Eq. ( 4) for the other frequency modes. The contribution of each uncertainty source is presented in Fig. 11 that corresponds to sample No. 1 (AISI 316). The measurements of the substrate's thickness and density present the main contributions on the prediction of the substrate Young's modulus E0. The global uncertainty and the various contributions are approximately the same for the first four frequency modes with percentages of 79 % and 20 % respectively, for the substrate thickness and density. Thus, a very precise and accurate measurement should be performed on the measurement of the thickness and the density of the substrate. In Tables 7 and8, the Young's modulus and uncertainty of the deposited Ti and Nb films were obtained using Pautrot's, Dev-CLBT and Extended Pautrot's (Ext-PM) models. The Young's moduli are also obtained using nanoindentation. From Eq. ( 55) and the reduced modulus Er measured, the corresponding Young's modulus and the uncertainty values were obtained. We can notice that the difference between the Young's moduli increases with the frequency mode. The difference between the modulus determined from the first frequency and the higher frequencies may be due to the effect of shear and inertia. They were taken into account in the frequency equation of the first mode (Eq. ( 5)) using the correction factor, which is not the case in the frequency equation of the other modes (Eq. ( 4)).

The Poisson's ratios used in CLBT and Dev-CLBT models for the glass and AISI substrates, were respectively 0.187 [START_REF] Slim | An enhanced formulation to determine the Young's and shear moduli of thin films by means of Impulse Excitation Technique[END_REF][START_REF] Jämting | Investigation of the elastic modulus of thin films using simple biaxial bending techniques[END_REF] and 0.265 [START_REF] Smithells | Smithells Metals Reference Book[END_REF]. For titanium and niobium films, the Poisson's ratios were respectively 0.36 [START_REF] Smithells | Smithells Metals Reference Book[END_REF] and 0.4 [START_REF] Zhang | Young's modulus surface and Poisson's ratio curve for cubic metals[END_REF], which were taken from the literature for bulk behavior properties. Figs 12 and 13 present the frequency ratio as function of the ratio Rʋi=ʋi/ʋ0. We can conclude from these figures that there is no significant influence of the Poisson ratio on the determination of the Young's modulus in flexural mode. For the first sample, the contribution on the Ti film Young's modulus is in the order of 4.6 GPa (4.4 %), 2.7 GPa (2.5 %), 2.5 GPa (2.3 %) and 2 GPa (1.8 %), respectively in mode I, II, III, and IV, approximately for the two models (Table 7). By passing from the first to the fourth mode, the resonant frequency of the system increases and its contribution decreases (Table 5). The frequency difference Δf increases by a factor of 10 from mode I to IV whereas the frequency contribution decreases by a factor of 5. Using the equation of uncertainty propagation for Pautrot's and Dev-CLBT models, the uncertainty of Ti film Young's modulus decreases. The same reasoning is applied to the second sample.

For a three-layered system, the uncertainty of the Nb film Young's modulus is in the order of 8 GPa (9 %), 5.4 GPa (5.8 %), 5 GPa (5.7 %) and 4 GPa (4.8 %) respectively in mode I, II, III, IV with a slight difference between the two models, which was not the case for the bilayer beam. This difference appears starting from the second layer because of the difference in theory and the symmetry assumption of the laminated theory that can relatively disturb the results even if it was corrected by the shift of the neutral axis after deposition. The uncertainty of the second film Young's modulus is higher than that of the first film. This can be due to the layering where the cumulating effect of the uncertainties of the two precedent layers (substrate and the first coating) contributes to the second film Young's modulus. Figs. 14 and 15 present the contribution of each uncertainty source on the uncertainty of the Ti and Nb film Young's moduli, respectively. They are calculated using the equation of uncertainty propagation for Ext-PM and Dev-CLBT models. As we can notice for both models, the measured density of the film, the substrate Young's modulus and the two frequencies represent a high percentage contribution, which are the first to be improved. The Ti film Young's modulus is added for the case of the three-layered system. The large contributions of the frequencies are related to the shift in the measured frequency before and after deposition. Furthermore, we can clearly see that by increasing in mode, the contribution of the frequency on the film Young's modulus decreases progressively. Once the contribution of the frequency decreases, the contribution of the other parameters (E0, E1, ρ1, ρ2, etc.) increases. These parameters are included in the equation of uncertainty propagation.

Especially for a three-layered beam, the first film Young's modulus uncertainty presents a significant effect for the determination of the second film Young's modulus. It can reach 44 % and 27 % respectively for Ext-PM and Dev-CLBT models. The contribution on the Young's modulus of a thin film becomes higher with the addition of several films. The measured Young's moduli of the substrates (Table 6) are coherent with the literature for AISI 316 [START_REF] Smithells | Smithells Metals Reference Book[END_REF] and glass substrates [START_REF] Gadaud | Bulk and coated materials shear modulus determination by means of torsional resonant method[END_REF][START_REF] Jämting | Investigation of the elastic modulus of thin films using simple biaxial bending techniques[END_REF]. The values of Young's moduli of the titanium film presented in Table 7 varied from 97 to 112 GPa. The values published in the literature are highly dispersed: the values found in the present work are consistent with some of them [START_REF] Tsuchiya | Young's modulus, fracture strain, and tensile strength of sputtered titanium thin films[END_REF][START_REF] Chinmulgund | Effect of Ar gas pressure on growth, structure, and mechanical properties of sputtered Ti, Al, TiAl, and Ti3Al films[END_REF][START_REF] Verkhovtsev | Molecular dynamics simulations of the nanoindentation process of titanium crystal[END_REF] and differ from some others [START_REF] Kim | Elastic modulus measurement of thin film using a dynamic method[END_REF][START_REF] Cho | Elastic modulus measurement of multilayer metallic thin films[END_REF]. Differences can be explained by microstructural features such as phase proportions [START_REF] Slim | Determination of single-crystal elasticity constants of the beta phase in a multiphase tungsten thin film using impulse excitation technique, X-ray diffraction and micro-mechanical modeling[END_REF] or the presence of pores or by the process itself like in reference [START_REF] Peraud | Determination of Young's modulus by a resonant technique applied to two dynamically ion mixed thin films[END_REF] where ions are implanted into the crystalline structure during the deposition process.

The measured values of the niobium film Young's moduli varied in the range of 77 ~ 92 GPa (Table 8). These values are close to those reported in previous studies for a single layer of niobium [START_REF] Xu | Potential of niobium-based thin films as a protective and osteogenic coating for dental implants: The role of the nonmetal elements[END_REF][START_REF] Seifried | Structure, morphology and selected mechanical properties of magnetron sputtered (Mo, Ta, Nb) thin films on NiTi shape memory alloys[END_REF] and they are lower than those given in other studies [START_REF] Czichos | Das Ingenieurwissen: Werkstoffe[END_REF]. This can also be due either to the presence of pores in the film or to the layering of different materials that can toggle the results and the prediction of the Young's modulus of individual thin film in a multilayer system. We can conclude that both models can be used to determine the Young's modulus of individual film in a multilayer system considering carefully the different uncertainty sources.

Conclusion

This work presents the development of new analytical models for the determination of the Young's moduli of thin films in multilayer structures by means of the Impulse Excitation Technique (IET). The proposed models are based on two different theories (FRCB and CLBT). The comparison with the finite element model was performed to validate the developed models and to check the validity range of the other models. It was shown that both Extended Pautrot's Model (Ext-PM) and the new one (Dev-CLBT) are recommended to determine the Young's moduli of thin films whatever the coating thickness and the elastic properties of the films and substrate. Titanium (4.84 µm) and niobium (3.97 µm) thin films were sputtered on AISI 316, glass substrates and silicon wafers. The Young's moduli of the deposited Ti and Nb films were determined using Pautrot's and Dev-CLBT models. A good agreement was found between the obtained moduli and the values reported in the literature. However, some previous studies present values different from the measured ones. This difference might be due to the presence of pores in the two films. However, the presence of the metastable β-Ti phase in the titanium film can also alter the Young's modulus of the titanium film.

In the case of the studied material, the standard uncertainty on the substrate Young's modulus comes mainly from the uncertainty on its thickness (79 % of the total uncertainty) and density (20 % of the total uncertainty). The main uncertainty sources on the first and second films Young's moduli are the density of the two coatings, the Young's modulus of the substrate, the Young's modulus of the precedent coating, and the frequency before and after deposition.

The measurement uncertainty of the IET represents about 1.1 % for the substrate Young's modulus. The measurement uncertainties on the first and second films Young's moduli are calculated as a function of the frequency mode. By increasing the vibrational mode, the measurement uncertainty decreases from about 4.4 % to 1.8 % for the first film and from about 9 % to 4.8 % for the second film. The increase of the measurement uncertainties with the addition of another film may be due to the accumulation of uncertainties from the successive films. The measured values of the film Young's modulus are lower than the bulk one; this might be due to the presence of pores in the as-deposited films.

We are currently extending our study to evaluate macroscopically the anisotropic behavior of thin films. Compared to Ext-PM, Dev-CLBT model can take into account the anisotropy and will allow us to develop the vibrational technique (IET) in order to determine the anisotropic moduli of coating in different directions.

Figure Captions

Fig. 1 .

 1 Fig. 1. Schematic representation of a multilayer beam. a) Free configuration of a multilayer beam. b) The z-coordinates of each layer in a cross-sectional view.

Fig. 2 .

 2 Fig. 2. Schematic representation of the shift of the neutral axis: a) cross-sectional view, b) stress distribution in three-layer composite beam.

Fig. 3 .

 3 Fig. 3. Schematic view of a symmetrical laminated beam.

Fig. 4 .

 4 Fig. 4. Cross-sectional SEM image of the titanium and niobium films deposited on silicon substrate.

Fig. 5 .

 5 Fig. 5. The natural frequency of the substrate (AISI 316), one layer coated substrate (Ti/AISI 316) and the multilayer comprising three materials (Nb/Ti/AISI 316).

Fig. 6 .

 6 Fig. 6. Deformed flexural vibrating beam: a) mode I, b) mode II, c) mode III and d) mode IV.

Fig. 7 .

 7 Fig. 7. Cross-section of FEM with the meshing configuration: a) Rh1 > 0.5, b) Rh1 ≤ 0.5.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Comparison between analytical and numerical models for bilayer isotropic beam (substrate + film) for different Young's moduli and density ratios: a) RE1=0.17, Rρ1=0.14, b) RE1=2.07, Rρ1=2.41, c) RE1=3.56, Rρ1=8.28, d) RE1=18.98, Rρ1=13.6.For a three-layered beam (substrate + bilayer films), another comparison was done with different Young's modulus RE2 and density Rρ2 ratios. Fig.9represents the evolution of the frequency ratio Rf2 as a function of the thickness ratio Rh2 that varies from 0 to 0.55 with two different values of Rh1 equal to 0.004 and 0.1. A good agreement is noticed between the developed model (Dev-CLBT), the extended Pautrot's model (Ext-PM) (Eq. (36)) and the FEM. As for the case of a bilayer beam, we can clearly see the divergence of the CLBT model. This difference between the models increases with the thickness ratio Rh2. a) b)

Fig. 10 .

 10 Fig. 10. X-ray diffraction patterns for different inclination angles: a) titanium film and b) niobium film.

Fig. 11 .

 11 Fig. 11. Contribution of each uncertainty source (in %) on the AISI 316 substrate Young's modulus uncertainty.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Comparison between analytical and numerical models for bilayer isotropic beam (substrate + film) with Rh1=0.3 and for different Young's moduli, and density ratios: a) RE1=0.53, Rρ1=1.07, b) RE1=1.65, Rρ1=1.76. a) b)

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Contribution of each uncertainty source (in %) on the first (Ti) film Young's modulus given by: a) Pautrot's model and b) Present model. a) b)

Fig. 1 .

 1 Fig. 1. Schematic representation of a multilayer beam. a) Free configuration of a multilayer beam. b) The z-coordinates of each layer in a cross-sectional view.

Fig. 2 .

 2 Fig. 2. Schematic representation of the shift of the neutral axis: a) cross-sectional view, b) stress distribution in three-layer composite beam.

Fig. 3 .

 3 Fig. 3. Schematic view of a symmetrical laminated beam.

Fig. 4 .

 4 Fig. 4. Cross-sectional SEM image of the titanium and niobium films deposited on silicon substrate.

Fig. 5 .

 5 Fig. 5. The natural frequency of the substrate (AISI 316), one layer coated substrate (Ti/AISI 316) and the multilayer comprising three materials (Nb/Ti/AISI 316).

Fig. 6 .

 6 Fig. 6. Deformed flexural vibrating beam: a) mode I, b) mode II, c) mode III and d) mode IV. The colors indicate the vertical displacement (mm).

Fig. 7 .

 7 Fig. 7. Cross-section of FEM with the meshing configuration: a) Rh1 > 0.5, b) Rh1 ≤ 0.5.

Fig. 8 .

 8 Fig. 8. Comparison between analytical and numerical models for bilayer isotropic beam (substrate + film) for different Young's moduli and density ratios: a) RE1=0.17, Rρ1=0.14, b) RE1=2.07, Rρ1=2.41, c) RE1=3.56, Rρ1=8.28, d) RE1=18.98, Rρ1=13.6.

Fig. 9 .

 9 Fig. 9. Comparison between analytical and numerical models for a multilayer beam comprising three isotropic materials for different Young's moduli and density ratios: a) Rh1=0.004, RE2=0.26, Rρ2=0.44, b) Rh1=0.1, RE2=0.26, Rρ2=0.44, c) Rh1=0.004, RE2=3.56, Rρ2=8.28, d) Rh1=0.1, RE2=3.56, Rρ2=8.28, e) Rh1=0.004, RE2=18.68, Rρ2=1.9, f) Rh1=0.1, RE2=18.68, Rρ2=1.9.

Fig. 10 .

 10 Fig. 10. X-ray diffraction patterns for different inclination angles: a) titanium film and b) niobium film.

Fig. 11 .

 11 Fig. 11. Contribution of each uncertainty source (in %) on the AISI 316 substrate Young's modulus uncertainty.

Fig. 12 .

 12 Fig. 12. Comparison between analytical and numerical models for bilayer isotropic beam (substrate + film) with Rh1=0.3 and for different Young's moduli, and density ratios: a) RE1=0.53, Rρ1=1.07, b) RE1=1.65, Rρ1=1.76.

Fig. 13 .

 13 Fig. 13. Comparison between analytical and numerical models for a multilayer beam comprising three isotropic materials with Rh1=0.3, Rh2=0.1 and for different Young's moduli and density ratios: a) RE2=0.53, Rρ2=1.07, b) RE2=5.7, Rρ2=7.5.

Fig. 14 .

 14 Fig. 14. Contribution of each uncertainty source (in %) on the first (Ti) film Young's modulus given by: a) Pautrot's model and b) Present model.

Fig. 15 .

 15 Fig. 15. Contribution of each uncertainty source (in %) on the second (Nb) film Young's modulus given by: a) Pautrot's model and b) Present model.

Table 2

 2 Average dimensions and measurement uncertainty.

			Length (mm)	Width (mm)	Thickness (mm)
			L	u(L)	b	u(b)	h	u(h)
		Substrate (AISI 316) 69.98 0.027 19.97	4.7x10 -3	0.505 2.6x10 -3
	Sample 1	Film 1 (Ti)	69.98 0.027 19.97	4.7x10 -3	4.84	0.013
		Film 2 (Nb)	69.98 0.027 19.97	4.7x10 -3	3.97	0.029
		Substrate (Glass)	75.33 0.036 25.38	9.4x10 -3	1.016 8.1x10 -3
	Sample 2	Film 1 (Ti)	75.33 0.036 25.38	9.4x10 -3	4.84	0.013
		Film 2 (Nb)	75.33 0.036 25.38	9.4x10 -3	3.97	0.029

Table 3

 3 Mass and density with the measurement uncertainty.

			Masse (g)	Density (kg/m 3 )
			m	ρ	u(ρ)
		Substrate (AISI 316)	5.561	7879.70	40.75
	Sample 1	Film 1 (Ti)	0.031	4583.15	148.37
		Film 2 (Nb)	0.044	7930.68	189.36
		Substrate (Glass)	4.774	2457.70	19.66
	Sample 2	Film 1 (Ti)	0.043	4646.90	108.82
		Film 2 (Nb)	0.063	8300.24	145.12

Table 6

 6 Substrate Young's modulus and its uncertainties.

			Flexural resonance frequency (Hz)				u(f)
	Mode shape	Sample	Before deposition	After first deposition	After deposition second	Δf1=f1-f0 (Hz)	Δf2=f2-f1 (Hz)	(Hz)	%
	I	1 (AISI 316) 2 (Glass)	534.54 980.84	537.12 986.93	537.90 987.22	2.58 6.09	0.78 0.29	0.117 0.117	0.02 0.01
	II	1 2	1481.98 2712.05	1489.38 2728.55	1491.70 2729.43	7.40 16.50	2.32 0.88	0.147 0.147 5.4x10 -3 0.01
	III	1 2	2907.49 5317.45	2922.14 5351.86	2925.87 5353.71	14.65 34.41	3.73 1.85	0.250 8.6x10 -3 0.250 4.7x10 -3
	IV	1 2	4807.60 8692.02	4832.67 8743.18	4839.57 8745.74	25.07 51.16	6.90 2.56	0.201 4.2x10 -3 0.201 2.3x10 -3

Table 7

 7 Young's modulus of the first film of Ti obtained by IET and NI.

						First film (Ti)
	Sample	Mode shape	E1	Pautrot u(E1)	E1	Dev-CLBT u(E1)	E1	NI	u(E1)
			GPa	GPa %	GPa	GPa %	GPa	GPa %
		I	105.22	4.69 4.46	104.56	4.66 4.46
	1 (AISI 316)	II III	108.74 109.55	2.72 2.5 2.54 2.32	108.07 108.87	2.75 2.54 2.55 2.34	-	-	-
		IV	112.11	2.06 1.84	111.41	1.98 1.77
		I	105.28	2.81 2.67	102.62	2.82 2.75
	2 (Glass)	II III	104.54 108.45	2.37 2.27 2.4 2.21	101.9 105.72	2.22 2.18 2.4 2.27	107.26	2.89 2.7
		IV	100.31	2.18 2.17	97.77	2.17 2.22

Table 8

 8 Young's modulus of the second film of Nb obtained by IET and NI.

					Second film (Nb)
	Sample	Mode shape	E2	Ext-PM u(E2)	E2	Dev-CLBT u(E2)	E2	NI	u(E2)
			GPa	GPa %	GPa	GPa %	GPa	GPa %
		I	89.53	8.31 9.28	88.21	7.68 8.71
	1 (AISI 316)	II III	92.37 87.74	5.3 5.74 5.01 5.71	91.01 86.45	5.42 5.95 4.91 5.68	-	-	-
		IV	90.42	4.38 4.84	89.09	4.09 4.59
		I	82.64	5.89 7.13	79.35	5.76 7.26
	2 (Glass)	II III	83.36 83.78	5.3 6.36 5.37 6.41	80.04 80.44	4.96 6.2 4.9 6.09	87.55	3.87 4.42
		IV	81.11	4.98 6.14	77.89	4.52 5.8

&[START_REF] Achache | Gum Metal thin films obtained by magnetron sputtering of a Ti-Nb-Zr-Ta target[END_REF]

Acknowledgments:

The authors would like to thank the co-founders of CERA project: The European Union (Fond Européen de Développement Régional) and the Doctoral School in Science and Technologies at the Lebanese University (Réseau UT/INSA-UL).

Table Captions Table 1

Summary of the analytical models.

Table 2

Average dimensions and measurement uncertainty.

Table 3

Mass and density with the measurement uncertainty.

Table 4

Uncertainty on the first four resonance frequencies.

Table 5

Flexural resonance frequencies of samples measured by IET before and after deposition.

Table 6

Substrate Young's modulus and its uncertainties.

Table 7

Young's modulus of the first film of Ti obtained by IET and NI.

Table 8

Young's modulus of the second film of Nb obtained by IET and NI.