Vincent Savaux
email: vincent.savaux@centralesupelec.fr

Faouzi Bader
email: faouzi.bader@supelec.fr

Jacques Palicot
email: jacques.palicot@centralesupelec.fr

Transactions on Signal Processing OFDM/OQAM Blind Equalization Using CNA Approach

Keywords: Blind equalization, OFDM/OQAM, Iterative Methods, CMA, CNA, impulse noise cancellation

This paper deals with blind equalization based on the constant norm algorithm (CNA) in orthogonal frequency division multiplexing/offset quadrature amplitude modulation (OFDM/OQAM). The CNA is designed to fit the complex square constellations with high order. As a consequence, the received signal is reshaped in order to change the real OQAM symbols into complex ones. Although it increases the intrinsic interferences of the OFDM/OQAM signal, it is proved by analysis and through simulations that the CNA with proposed reshaped signal achieves better mean square error (MSE) performance than CMA applied to the received OFDM/OQAM signal. Furthermore, a suboptimal initialization strategy is proposed, which ensures a high convergence speed. The behavior of the proposed blind equalization method is analyzed under synchronization mismatch as well as in time-varying propagation environment.

OFDM/OQAM Blind Equalization Using

CNA Approach

I. INTRODUCTION

Filter bank multicarrier (FBMC), and in particular the orthogonal frequency division multiplexing/Offset quadrature amplitude modulation (OFDM/OQAM) is a good alternative to the OFDM for systems coexistence and spectrum sharing as it achieves better spectral containment [START_REF] Floch | Coded Orthogonal Frequency Division Multiplex[END_REF], [START_REF] Farhang-Boroujeny | OFDM Versus Filter Bank Multicarrier[END_REF]. Furthermore, it provides higher useful bit rate as it does not require any cyclic prefix addition. However, the good properties of OFDM/OQAM are due to subcarrier functions that are only orthogonal in the real field. As a consequence, the presence of imaginary intrinsic interference and the complex channel frequency response lead to develop some equalization methods specific to OFDM/OQAM. As in any modulation scheme, the channel estimation and equalization techniques in OFDM/OQAM can be classified into two main categories: i) the data-aided methods, in which pilots are multiplexed in the data stream and the signal is equalized using the estimated channel (see [START_REF] Javaudin | Pilot-Aided Channel Estimation for OFDM/OQAM[END_REF]- [START_REF] Kofidis | Preamble-based channel estimation in OFDM/OQAM systems: A review[END_REF] for data aided estimation methods). The usual zero forcing (ZF) and minimum mean square error (MMSE) equalizer are presented in a FBMC context in [START_REF] Lin | On efficient equalization for OFDM/OQAM systems[END_REF], and the authors of [START_REF] Ihalainen | Channel Equalization in Filter Bank Based Multicarrier Modulation for Wireless Communications[END_REF] propose to split the phase and the modulus equalization. In [START_REF] Ihalainen | Channel Equalization for Multi-Antenna FBMC/OQAM Receivers[END_REF], the topic of channel equalization has been extended to FBMC modulation scheme in multiple antennas systems.

ii) the blind techniques, which use only few features of the signal to invert the channel (see [START_REF] Bölcskei | A Subspace-Based Approach to Blind Channel Identification in Pulse Shaping OFDM/OQAM Systems[END_REF]- [START_REF] Lin | Convergence analysis of blind equalizer in a cosine modulated filter bank-based multicarrier communication system[END_REF]).

Although data-aided methods are simple to implement and widely employed, they may reduce the spectral efficiency of the transmitted signal. On the other hand, blind techniques have not been so deeply studied in OFDM/OQAM context, and the topic remains a pending challenge.

In [START_REF] Bölcskei | A Subspace-Based Approach to Blind Channel Identification in Pulse Shaping OFDM/OQAM Systems[END_REF], the cyclostationarity induced by the use of the OFDM/OQAM pulse shape is exploited in order to perform a second order blind equalization. However, this technique has a high complexity, and very long data records have to be stored to obtain good estimates of the second-order statistics of the signal. The more "classical" constant modulus algorithm (CMA) has been adapted to FBMC (cosine modulated filter bank) in [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF], and its convergence behavior has been analyzed in [START_REF] Lin | Convergence analysis of blind equalizer in a cosine modulated filter bank-based multicarrier communication system[END_REF]. The idea behind CMA applied in OFDM/OQAM is to iteratively update the algorithm by comparing the real part of the output of the equalizer with a given real constant that depends on the constellation size. However, it is worth noting that CMA is equivalent to Sato's algorithm [START_REF] Sato | A Method of Self-Recovering Equalization for Multilevel Amplitude-Modulation Systems[END_REF], the multimodulus algorithm (MMA) [START_REF] Yang | The Multimodulus Blind Equalization and Its Generalized Algorithms[END_REF], and the constant norm algorithm (CNA) [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF], due to the use of real transmitted OQAM symbols. Therefore, this limits the choice of the cost function to those that are adapted to real constellation, whereas numerous cost functions adapted to different complex constellations have been proposed in the scientific literature [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF]- [START_REF] Barbarossa | Blind Equalization Using Cost Function Matched to the Signal Constellation[END_REF].

In this paper, it is proposed to reshape the received signal by summing two consecutive symbols such that, from the equalizer point of view, the received symbols are complex. This provides an additional degree of freedom regarding the choice of the cost function of the update algorithm. As a consequence, we take advantage of these reshaped complex symbols to carry out a blind equalization using the CNA proposed in [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF], which has proven to be particularly adapted to square constellations. This algorithm can be seen as a generalization of the CMA, where the modulus is substituted by a norm which is determined as a function of the considered constellation. The performance of the proposed CNA adapted to OFDM/OQAM is analyzed, and it is proved that the CNA achieves better performance than the CMA.

The rest of the paper is organized as follows: the blind OFDM/OQAM receiver structure is presented in Section II. The proposed CNA using a reshaped received signal is developed in Section III.

Section IV provides simulations results that show the performance of the proposed blind equalization techniques. Finally, we draw our conclusions in Section V.

II. BLIND OFDM/OQAM RECEIVER

The blind OFDM/OQAM receiver is presented in Fig. 1. The demodulator consists of a serial to parallel converter (denoted by S/P), and an analysis filter bank (AFB). The latter is composed of i) the polyphase network (PPN) that contains a set of digital filters, and ii) the fast Fourier transform (FFT) of size M (M is also the number of subcarriers).

In the following expression, the receiver is considered to be perfectly synchronized with the transmitter in both time and frequency domains, and the channel delay is assumed to be short compared to the length of the used prototype filter, in such a way that the output of the AFB described in Fig.

1-(a) can be written as

y m,n = H m,n x m,n + j (p,q)∈Ω H p,q x p,q < g > p,q m,n Im,n +w m,n , (1)
where the subscripts "m, n" denote the frequency-time position. H m,n is the channel frequency response (CFR), w m,n is the complex Gaussian noise sample with circular covariance matrix R w .

Note that for convenience, this matrix is commonly approximated by σ 2 I, where σ 2 is the noise level, and I is the identity matrix. x m,n is the real OQAM symbol that is carried by the prototype filter g which is given by

g m,n [i] = g(i -n M 2 )e 2jπm M (i- L f -1 2 ) e jφm,n , (2)
where the phase term is φ m,n = (π/2)(m + n) + mnπ as defined in [START_REF] Siohan | Analysis and Design of OFDM/OQAM Systems Based on Filterbank Theory[END_REF]. It should be noted that the real symbols x m,n are transmitted with a period M 2 twice shorter than that of OFDM signal where symbols are complex. Therefore, the data rate in OFDM/OQAM is the same as in OFDM. The specificity of the OFDM/OQAM modulation scheme lies in the presence of interferences denoted by

I m,n in (1)
, which is due to the used prototype filter g. The term j < g > p,q m,n in (1) is called "intrinsic interference" (according to the definition in [START_REF] Kofidis | Preamble-based channel estimation in OFDM/OQAM systems: A review[END_REF]), and is defined as the scalar product In this paper, the Bellanger's filter is employed, but other filters with self interference effect could be used. This filter is also known as "PHYDYAS filter", and its frequency response can be expressed as (see [START_REF] Bellanger | Specification and Design of a Prototype Filter for Filter Bank Based Multicarrier Transmission[END_REF] for details)

i g m,n [i]g * p,q [i]dt =            1, if (p, q) = (m, n) j < g > m,n p,q , else . (3
G[f ] = K-1 k=-(K-1) G k sin(π(f -k Lf )L f ) L f π(f -k Lf ) , (4)
where

G 0 = 1, G 1 = 0.971960, G 2 = 1/ √ 2,
and G 3 = 0.235147 according to [START_REF] Bellanger | Specification and Design of a Prototype Filter for Filter Bank Based Multicarrier Transmission[END_REF], and

G k = G -k .
L f is the length of the filter defined as L f = KM , where K is called overlapping factor. As mentioned in [START_REF] Kofidis | Preamble-based channel estimation in OFDM/OQAM systems: A review[END_REF]-eq. ( 13), the weights < g > p,q m,n of any well-chosen prototype filter have a symmetrical property as

(-1) p a -b (-1) p a -(-1) p c 1 (-1) p c (-1) p a b (-1) p a , (5)
where the weight 1 corresponds to the position (m, n), and where a, b, and c values depend on M and K. Note that, due to (5), the noise component w m,n cannot be white. However, the assumption of "whiteness" of the noise in OFDM/OQAM has proven to be accurate, such as in [START_REF] Savaux | A Joint MMSE Channel and Noise Variance Estimation for OFDM/OQAM Modulation[END_REF].

The pulse shape g allows the OFDM/OQAM signal to have a good spectral containment, but it is orthogonal only in the real field. therefore the real OQAM symbols can be recovered by xm,n = Re{y m,n }. However, the presence of the channel gain H m,n ∈ C induces complex interferences from all the symbols at positions (p, q) ∈ Ω, where Ω is the set of frequency-time positions that surround (m, n). As a consequence, the distortion due to the channel must be canceled in order to recover the real orthogonality between subcarriers and symbols.

It is worth noting that in practice, the received signal y m,n in ( 1) is also distorted by some residual channel frequency offset (CFO) and time synchronization mismatch [START_REF] Medjahdi | The Impact of Timing Synchronization Errors on the Performance of OFDM/FBMC Systems[END_REF]- [START_REF] Zeng | Joint Time-Frequency Synchronization and Channel Estimation for FBMC[END_REF]. In this paper, it is assumed that the misalignment between the carrier frequencies generated at the transmitter and the receiver can be neglected in comparison with the subcarrier distance. However, the time synchronization mismatch induces a phase rotation after the demodulation stage, in such a way that the received signal should be rewritten y m,n e -2jπ τ k m M where τ k is the time shift.

In this paper, the signal is blindly equalized, which maximize its spectral efficiency since no pilot is used. Some techniques have been proposed in the literature for both single-carrier as in [START_REF] Oh | Modified constant modulus algorithm: blind equalization and carrier phase recovery algorithm[END_REF], [START_REF] Godard | Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems[END_REF], [START_REF] Johnson | Blind Equalization Using the Constant Modulus Criterion: A Review[END_REF] and [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF] and in multicarrier systems as OFDM/OQAM [START_REF] Bölcskei | A Subspace-Based Approach to Blind Channel Identification in Pulse Shaping OFDM/OQAM Systems[END_REF]- [START_REF] Lin | Convergence analysis of blind equalizer in a cosine modulated filter bank-based multicarrier communication system[END_REF]. The basic idea behind the blind equalization is to iteratively update the equalizer coefficient F m,n by using only few features of the signal. In the OFDM/OQAM modulation scheme, this iterative process is carried out on each subcarrier m ∈ 0, M -1 , as described in Fig. 1-(b). The corresponding output of the equalizer can be expressed as

a m,n = Re(F m,n y m,n ). (6)
The optimal coefficient F opt m,n can be obtained on each subcarrier by solving the optimization problem defined as

F opt m,n = min Fm,n J(a m,n ), (7)
where J is a given cost function that needs to be defined. In the rest of the paper, the minimization step in [START_REF] Lin | On efficient equalization for OFDM/OQAM systems[END_REF] is achieved by means of the usual stochastic gradient method as in [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF], [START_REF] Godard | Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems[END_REF] and [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF], and which can be expressed as

F m,n+1 = F m,n -µφ(a m,n ), (8)
where µ is the step-size parameter of the update algorithm, and

φ(a m,n ) = ∂J(a m,n ) ∂F * m,n . (9)
It should be noticed that a m,n = Re(F m,n y m,n ) is not an holomorphic function. Therefore, the Wirtinger's derivation [START_REF] Bouboulis | Wirtingers Calculus in general Hilbert Spaces[END_REF] is used in (9) instead of the usual complex derivation. It has been demonstrated in [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF] that the constant modulus algorithm (CMA) proposed by Godard in [START_REF] Godard | Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems[END_REF] is suitable to OFDM/OQAM. Furthermore, note that for the same reason, the modified CMA (MCMA)

in [START_REF] Oh | Modified constant modulus algorithm: blind equalization and carrier phase recovery algorithm[END_REF] (also called multimodulus algorithm (MMA) in [START_REF] Yang | The Multimodulus Blind Equalization and Its Generalized Algorithms[END_REF]) is exactly the same as CMA in the OFDM/OQAM modulation scheme. The cost function of CMA is written as

J(a m,n ) = E{(|a m,n | 2 -γ) 2 }, (10)
where γ is a real-valued constant that is defined hereafter 1 . The update algorithm can be rewritten by substituting [START_REF] Bölcskei | A Subspace-Based Approach to Blind Channel Identification in Pulse Shaping OFDM/OQAM Systems[END_REF] into [START_REF] Ihalainen | Channel Equalization in Filter Bank Based Multicarrier Modulation for Wireless Communications[END_REF] as

F m,n+1 = F m,n -2µ(|a m,n | 2 -γ) ∂|a m,n | 2 ∂F * m,n . (11)
The modulus |a m,n | 2 can be rewritten as

|a m,n | 2 = 1 4 (F m,n y m,n + (F m,n y m,n ) * ) 2 = 1 4 (F m,n y m,n ) 2 + 1 4 ((F m,n y m,n ) * ) 2 + 1 2 F m,n y m,n (F m,n y m,n ) * . (12)
Furthermore, the Wirtinger's derivative leads to ∂z ∂z * = ∂z * ∂z = 0 for any z ∈ C (see Theorem 2.5 in [START_REF] Bouboulis | Wirtingers Calculus in general Hilbert Spaces[END_REF]), then the substitution of ( 12) into [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF] yields

F m,n+1 = F m,n -2µy * m,n a m,n (|a m,n | 2 -γ). (13)
As indicated in [START_REF] Godard | Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems[END_REF], the optimal value of the constant γ must be chosen as

γ = E{|x| 4 } E{|x| 2 } . ( 14
)
In the following, we propose two alternatives to CMA for blind equalization in OFDM/OQAM.

Based on the results obtained for single-carrier modulation scheme in [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF], we adapt the CNA to OFDM/OQAM systems in hereafter Section III.

III. ADAPTING THE CNA IN OFDM/OQAM

The basic idea behind the CNA is to use a k-norm, k > 2, instead of the modulus in the cost function of the equalizer [START_REF] Bölcskei | A Subspace-Based Approach to Blind Channel Identification in Pulse Shaping OFDM/OQAM Systems[END_REF]. It has been proved in [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF] that CNA achieves better performance than CMA for square constellations as 16-QAM, since CMA is adapted for constant modulus constellations.

Furthermore, it has been shown in [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF] that for each square constellation corresponds a given knorm that maximizes the performance of the equalizer. However, the transmitted symbols in (1) are real, therefore the use of CNA in OFDM/OQAM requires some adaptations. Principally, the received signal in (1) must be redesigned and the symbols combined in order to highlight complex transmitted symbols. Since CNA is suitable for high order constellations, we consider throughout this section that the real OQAM symbols x m,n are randomly chosen among the set {-3, -1, 1, 3}, which corresponds to the real part of 16-QAM. The aims of this section are twofold: i) presenting the reshaping of the input signal y m,n in (1) in order to highlight complex-valued symbols xm,n , and ii) proving that carrying out the CNA using the reshaped inputs xm,n lieds to better performance than using CMA with real inputs x m,n .

A. Rewriting the Received Signal with Complex Input Symbols

Since the transmitted symbols x m,n are real, the first step before applying the CNA consists of converting the received symbols into complex data. To achieve this, the following procedure is carried out:

1) Build y c m,n defined as the sum of two consecutive received symbols y m,n and y m,n+1 as

y c m,n =jy m,n + y m,n+1 =H m,n jx m,n + H m,n+1 x m,n+1 + jI m,n + I m,n+1 + jw m,n + w m,n+1 . (15)
Note that H m,n = H m,n+1 in [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF], which leads to further developments.

2) y c m,n can be rewritten by substituting ( 1) into [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF] as

y c m,n =H m,n jx m,n (1+ < g > m,n m,n+1 ) + x m,n+1 (1-< g > m,n+1 m,n ) - (p,q)∈Ω q =n+1
H p,q x p,q < g > p,q m,n

Ĩm,n + j (p,q)∈Ω q =n H p,q x p,q < g > p,q m,n+1
Ĩm,n+1

+ jw m,n + w m,n+1 . (16)
It must be emphasized that ( 16) holds since the channel is assumed to be static. The effect of channel variations on the accuracy of ( 16) will be investigated in Section IV. The symmetry property of weights < g > p,q m,n given in (5) yields < g > m,n m,n+1 = -< g >

1+ < g > m,n m,n+1 = H m,n jx m,n + x m,n+1 xm,n -Ĩm,n + Ĩm,n+1 1+ < g > m,n m,n+1 Ĩr m,n + jw m,n + w m,n+1 1+ < g > m,n m,n+1 wm,n , (17)
where xm,n is a 16-QAM symbol.

It is worth noticing that the redesigned received signal in ( 17) is exactly equivalent to the transmission of complex 16-QAM symbols xm,n over the channel H m,n , with additive interference Ĩr m,n and non-white Gaussian noise wm,n . Therefore, the CNA can now be employed as blind equalizer whose input z m,n is defined as

z m,n = ja m,n + a m,n+1 , (18)
where a m,n and a m,n+1 are the outputs of the equalizer given by [START_REF] Kofidis | Preamble-based channel estimation in OFDM/OQAM systems: A review[END_REF]. The specific update algorithm for F m,n is described in the hereafter Section III-B. The overall principle of the blind equalizer using CNA adapted to OFDM/OQAM is depicted in Fig. 2.

B. Deriving CNA in OFDM/OQAM

The cost function of the CNA can be formulated in a general expression as

J(z m,n ) = 1 ab |||z m,n || a -γ| b , (19)
where ||.|| is a norm defined on C, and a and b are two parameters that give two degrees of freedom to the algorithm. In the following, it will be assumed that2 a = b = 2. Since x m,n is taken from a 16-QAM constellation, the real positive constant γ should be rewritten as

γ = E{||x m,n || 4 } E{||x m,n || 2 } = 13.2. (20)
The k-norm family ||.|| considered in this paper can be formulated for any complex z on the plane C as

||z|| k = k |Re{z}| k + |Im{z}| k . (21)
Note that the CNA reduces to the CMA when k = 2. It has been demonstrated in [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF] that the case k = 6 (denoted by CNA-6) is very suitable for 16-QAM constellation. According to this result, we hereafter employ the CNA-6 whose update algorithm can be adapted to OFDM/OQAM using [START_REF] Barbarossa | Blind Equalization Using Cost Function Matched to the Signal Constellation[END_REF] as

F m,n+1 =F m,n -µ(||z m,n || 2 6 -γ) × Re{z m,n } 5 + jIm{z m,n } 5 ||z m,n || 4 6 ỹ * m,n . (22)
The steady-state performance of any constant norm-based equalizers can be characterized by the average distance E{l} between the constellation symbols and the ball defined in [START_REF] Bölcskei | A Subspace-Based Approach to Blind Channel Identification in Pulse Shaping OFDM/OQAM Systems[END_REF] 17) should be lower than that of CMA using y m,n in (1). However, the distance E{l} does not take into account the interferences, and therefore a deeper analysis must be undertaken in order to show the influence of the interferences on the performance of the CMA and CNA-6.

C. Performance Analysis of CNA-6 Adapted to OFDM/OQAM

The proposed development leading to ỹm,n in (17) induces some changes on the noise variance and on the energy of the interferences. Thus, the variance σ2 of the noise wm,n can be expressed as

σ2 = E{| wm,n | 2 } = 2σ 2 |1+ < g > m,n m,n+1 | 2 . ( 23
)
It should be emphasized that σ2 in (23) depends on the chosen filter g, and thus, the noise level could be either higher or lower than the original noise variance σ 2 . In the considered case of PHYDYAS's filter, the noise variance is reduced as σ2 = 0.8172σ 2 .

E{|x m,n | 2 } σ 2 I < E{|x m,n | 2 } σ 2 Ĩ . ( 24
)
We can deduce from (24) that the signal used for the CNA-6 should be less sensitive to interference than that used for the CMA, even if σ 2 Ĩ > σ 2 I .

In addition to [START_REF] Godard | Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems[END_REF], the performances of the equalizers can be compared in terms of excess mean square error (EMSE), which assesses the noise of the equalizer in the steady-state. From [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF], the EMSEs of CMA and CNA-6 denoted by Γ CM A and Γ CN A6 can be approximated as in [START_REF] Johnson | Blind Equalization Using the Constant Modulus Criterion: A Review[END_REF] and ( 26)

1053-587X (c
Γ CM A ≈ µP y E{|a op | 6 -2γ|a op | 4 + γ 2 |a op | 2 } E{4|a op | 2 } -2γ , (25)
where a op and z op are the perfectly equalized symbols (i.e. when F opt m H m = 1 is assumed) in ( 6) and ( 18), respectively, and P y and P ỹ are the powers of the corresponding input signals y m,n and ỹm,n , respectively.

It is worth noting that Γ CM A in ( 25) and Γ CN A6 in ( 26) are linear functions of P y and P ỹ respectively. Since these inputs powers are functions of the interferences, it is possible to compare the achieved Γ CM A and Γ CN A6 values in function of the interference power, as depicted in Fig. 5.

The step-size µ value has been arbitrarily set equal to 4.10 -4 (in practice, the step-size parameter of CMA is larger than CNA's, as described in Section IV). Note that different µ values would only lead to a change in the slope of the curves. It can be observed that Γ CN A6 < Γ CM A for any interference power value. It reflects the fact that the CNA-6 with the proposed solution described in Fig. 2 achieved better performance than CMA using the real OQAM symbols as in [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF]. This theoretical performance results will be confirmed through simulations in Section IV.

Beyond the application of the blind equalizer using CNA-6 in OFDM/OQAM, the reshape of complex symbols as described in Fig. 2 opens new perspectives for blind equalization in filter bankbased modulations, as specific cost functions (not limited to the real field) can now be used and investigated.

D. Proposed Initialization Strategy

Since the proposed blind equalizer avoids the convergence of the coefficients F m,n toward local minima, it is not mandatory to choose a very low initialization value F m,0 as proposed in [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF], [12]. This solution may especially appear not adapted for channels with deep fading as the optimal coefficients |F opt m,n | have high values, which should lead to a long convergence time. It has been demonstrated in [START_REF] Savaux | Sub-Optimal Initialization for Blind Equalization with Fast Convergence in OFDM/OQAM Modulation[END_REF] that a sub-optimal initialization value (in the mean square error sense) can be assessed in order to take into account the channel value for each frequency position m. To achieve this, the coefficients can be defined for n = 0 and for any m ∈ {0, 1, .., M -1} as

F s-opt m,0 = E{x 2 m,0 } |y m,0 | 2 + α , ( 27
)
where α is a real constant that avoids the divergence of F m,0 when y m,0 is very close to zero. In practice, α is chosen such that α << 1.

The sub-optimal nature of the initialization F s-opt m,0 is shown hereafter, and compared with the [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF], [START_REF] Lin | Convergence analysis of blind equalizer in a cosine modulated filter bank-based multicarrier communication system[END_REF]. The mean error between the square moduli of the inverse of F s-opt m,0 and H m,0 is defined as

ǫ F = E 1 |F s-opt m,0 | 2 -|H m,0 | 2 . ( 28
)
Note that the noise samples w m,0 have a variance σ 2 , and are uncorrelated with x m,0 , H m,0 , and I m,0 . Therefore, the mean error ǫ F in ( 28) can be rewritten by substituting ( 1) into ( 27) and ( 28) as

ǫ F =E |y m,0 | 2 + α E{x 2 m,0 } -|H m,0 | 2 =E |x m,0 H m,0 + I m,0 + w m,0 | 2 + α E{x 2 m,0 } -|H m,0 | 2 = σ 2 HI E{2Re(x m,0 H m,0 I * m,0 )} +σ 2 + α E{x 2 m,0 } = σ 2 + α E{x 2 m,0 } , (29)
since Re(x m,0 ) = x m,0 is a real zero-mean variable, which is independent of H m,0 and I m,0 . Thus, a qualitative comparison with the solution |F m,0 | << 1 can be made. In fact, it should be noticed 28) instead of F s-opt m,0 , whereas the error ǫ F using the proposed initialization in ( 29) is inversely proportional to the symbol power to noise power ratio. Therefore, we conclude that the sub-optimal F s-opt m,0 solution in [START_REF] Goupil | Constant norm algorithms class[END_REF]

that ǫ F >> 1 if |F m,0 | << 1 is substituted in (

IV. SIMULATIONS RESULTS

In this section, simulations results show the performance of the different blind equalization methods proposed in this paper.

A. Simulations Parameters

Since we compare the solution proposed in Sections III with the CMA in [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF] and [START_REF] Lin | Convergence analysis of blind equalizer in a cosine modulated filter bank-based multicarrier communication system[END_REF], the same simulation parameters described in [START_REF] Farhang-Boroujeny | Multicarrier Modulation With Blind Detection Capability Using Cosine Modulated Filter Banks[END_REF] is used, we take α = 0.01. Furthermore, a 16-QAM constellation is used, i.e. OQAM symbols are randomly chosen among {-3, -1, 1, 3}, and the CMA in ( 13) is carried out with p = 2 and µ = 0.0033. This configuration is also used to test the equalizer using the CNA-6 in OFDM/OQAM.

The performance of the different blind equalizers is assessed by using the mean square error MSE defined as

MSE = E{(|a m,n | -|x m,n |) 2 }. ( 30
)
The achieved MSEs of the blind equalizers are also compared with that of the data-aided estimation using interference approximation method (IAM) proposed in [START_REF] Lélé | Channel estimation methods for preamble-based OFDM/OQAM modulations[END_REF]. The preamble of IAM is designed as follows: for any pilot x m,n = ±P, then the symbols at positions n -1 and n + 1 are zero, and

x m-1,n = -x m+1,n . Therefore the transmultiplexer gain described in (5) yields a received pilot with an energy larger than |P| 2 . This property makes IAM one of the most robust channel estimator in OFDM/OQAM, and we denote by ĤIAM m,n the estimated channel coefficients. The channel is then inverted with the zero-forcing (ZF) equalizer such as F IAM m,n = 1/ ĤIAM m,n . Note that the energy of the pilot is the same as the transmitted symbols, i.e. |P| 2 = 5. It must be emphasized that the results presented for the estimator using IAM are only indicative. In fact, a fair comparison would induce to test different channel estimation methods (IAM-R [START_REF] Kofidis | Preamble-based channel estimation in OFDM/OQAM systems: A review[END_REF], MMSE [START_REF] Savaux | A Joint MMSE Channel and Noise Variance Estimation for OFDM/OQAM Modulation[END_REF]), pilot energy, or equalizers [START_REF] Ihalainen | Channel Equalization in Filter Bank Based Multicarrier Modulation for Wireless Communications[END_REF]. A more exhaustive comparison with pilot-aided methods is out of the scope of this paper, but numerous estimation and equalization techniques can be found [START_REF] Kofidis | Preamble-based channel estimation in OFDM/OQAM systems: A review[END_REF], [START_REF] Ihalainen | Channel Equalization in Filter Bank Based Multicarrier Modulation for Wireless Communications[END_REF], [START_REF] Savaux | A Joint MMSE Channel and Noise Variance Estimation for OFDM/OQAM Modulation[END_REF].

B. Performance of CNA-6 in OFDM/OQAM

First series of simulations have been performed to analyze the performance of blind equalization using CNA-6 in OFDM/OQAM. Fig. 6 shows the trajectories of the CMA and the CNA-6 versus the OFDM/OQAM symbols, for a signal-to-noise ratio (SNR) of 30 dB. Two initialization values F m,0 = 0.01 and F s-opt m,0 have been tested. Furthermore, the achieved MSE of IAM is given as reference. It can be seen that the steady-state performance of CNA-6 has 1 dB gain compared with the MSE floor using CMA. This result is consistent with the analysis undertaken in Section III-C.

However, the CNA-6 using F m,0 = 0.01 requires at least 1000 OFDM/OQAM symbols more than CMA to achieve a given MSE value. This loss is reduced by half when the algorithms are initialized with F s-opt m,0 . This delay in the convergence is due to the intrinsic phase adjustment when using CNA-6. The basic idea behind this phenomenon can be described by using the balls of the equalizers in Fig. 3. In fact, it can be observed in Fig. 3-(a) that CMA is phase-independent since the balls of CMA is a circle. However, the optimal equalizer coefficient value depends on the phase of F m,n (see Fig. 3-(b)), therefore the phase adjustment slows down the convergence of CNA-6. This phenomenon has been described in detail in [START_REF] Goupil | New Algorithms for Blind Equalization: The Constant Norm Algorithm Family[END_REF]. It is also worth noticing that the use of the sub-optimal F s-opt m,0 value allows the algorithm to gain 2000 iterations compared with F m,0 = 0.01 for both CMA and CNA-6. Finally, Fig. 6 highlights that a gain of 1 dB and 2 dB is achieved by CMA and CNA-6

respectively compared with IAM. This shows the capability of the blind equalization to outperform data-aided techniques. 13) or ( 22) are less disturbed in noisy conditions than the ZF coefficient F IAM m,n using IAM. However, it is worth noticing that these results are obtained after 3000 iterations, whereas no convergence delay is required for data-aided methods (see Fig. 6 for instance).

1053-587X (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Furthermore, the MSE value of the IAM also depends on the pilot energy |P| 2 . Otherwise, it can be also observed that CNA-6 achieves better performance than CMA for SNR>20 dB. As indicated in Section II, the blind receiver may suffer from synchronization mismatch. The effect of such a time synchronization mismatch on the performance of CMA and CNA-6 is depicted in Fig. 9. The time shift between the transmitter and the receiver is given as a percentage of the prototype filter length L f . The performance of the system using IAM is shown as well, but it is only indicative. It can be observed that time synchronization mismatches of 1%, 3%, and 5% induce MSE losses of 11, 13, and 16.5 dB, respectively, compared with the results given in Fig. 6. Such a MSE loss is due to the interferences induced by the synchronization mismatch. Furthermore, additional simulations revealed that a time shift larger than 5% leads to instability, i.e. the blind equalizer diverges. These results show that both CMA and CNA-6 require a very accurate synchronization process in order to work at their best performance. Fig. 10 shows the MSE performance of CNA-6, CNA-2 (CMA) versus OFDM/OQAM symbols for three receiver speeds: (a) 1 km/h, (b) 3 km/h, and (c) 10 km/h. Two different initialization strategies have been considered: i) using F s-opt m,0 , and ii) using a preamble such that F m,0 = 1/ ĤIAM m,n (referred to "pilot init." on the figures). It can be observed that the equalizers performances tend to the MSE level whatever the initialization is (for 1 km/h, this limit is outside the OFDM/OQAM symbols range). However, the MSE starts at 4 dB where F s-opt m,0

is used, whereas it starts at -8.5 dB when F m,0 = 1/ ĤIAM m,n is used. Thus, it can be noted that the CMA and CNA-6 equalizer using F m,0 = 1/ ĤIAM m,n outperforms those using F s-opt m,0 from 0 to 5000 iterations and from 0 to 2500 iteration for 3 km/h and 10km/h respectively. Furthermore, it can be also noted in (a) and (b) that CMA achieves 0.5 MSE gain compared with CNA-6 where the steady state is reached the steady-state. This is mainly due to the approximation in [START_REF] Oh | Modified constant modulus algorithm: blind equalization and carrier phase recovery algorithm[END_REF], which does not hold anymore when time selectivity of the channel increases.

C. Discussion

The practical use of blind equalizers such as CMA and CNA-6 is discussed in this section. We deduce from Figs. 6 and8 that, even if the use of F s-opt m,0 and CNA-6 reduces the convergence delay, blind equalizers still require more than 1000 OFDM/OQAM symbols to converge. As a consequence, the use of these methods requires is limited by the length of the signal packets. Thus, broadcast applications could be a possible context to operate blind receivers. Note that the reuse of the received 1053-587X (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. Transactions on Signal Processing symbols as input of the equalizers seems to be a good solution to further reduce the convergence delay [START_REF] Savaux | Ultra-Fast Blind Equalization for OFDM: Principle and Steps Towards Implementation[END_REF].

It has been shown in Fig. 9 that the blind equalizers are very sensitive to time synchronization mismatch. Therefore, an accurate synchronization process is required. A usual solution is to undertake this task thanks to a preamble [START_REF] Cassiau | Time and frequency synchronization for CoMP with FBMC[END_REF]. Furthermore, this preamble could be used for channel estimation.

Fig. 10 revealed that an initialization of the blind equalizers thanks to an accurate estimate of the channel allows to hold lower MSE values much more longer than when using F s-opt m,0 in slowly time-varying environments. Thus, instead of using full-blind equalizers as in Figs. 6 and8, one can deduce from Figs. 9 and 10 that the use of regular preamble (e.g. at the beginning of each frame) could tackle the issue of synchronization mismatch, as well as allowing blind equalizers to operate in weakly time-selective channels.

V. CONCLUSION

In this paper, we have presented an adaptation of the CNA blind equalizer for OFDM/OQAM modulation. In Section III, the CNA, which can be seen as a generalized CMA, has been developed.

To be effective, the CNA requires a signal using a complex constellation. Therefore, it has been proposed to reshape the "real" received OQAM symbols to obtain "complex" 16-QAM symbols. The performance of the proposed method has been analyzed, and it has been proved that, despite the increase of the interference level due to the reshape, the CNA applied to the new signal achieves better performance than the CMA applied to the OFDM/OQAM signal. Furthermore, the principle of changing the shape of the received signal opens new perspectives for the blind equalization in the FBMC-based modulations, as specific cost functions (not limited to the real field) can now be investigated. In addition, a sub-optimal initialization strategy, in the sense of the mean square error, has been presented. Numerical results revealed that the CNA and the proposed initialization method allows to reduce by 2000 the number of iterations of the blind equalizer. Moreover, the performances of CMA and CNA-6 have been studied in conditions where the receiver is not well-synchronized with the transmitter and the channel is time-varying. Results have shown that this degrades the performance. In order to overcome these deteriorations, a relevant solution consists of using preamble, from which the synchronization and accurate initialization can be obtained.

Fig. 1 .

 1 Fig. 1. Blind OFDM/OQAM receiver structure with (a) AFB, and (b) blind equalizer.

Fig. 2 .

 2 Fig. 2. Proposed blind equalizer using CNA, adapted to OFDM/OQAM system.

Fig. 3 .Fig. 3 that

 33 Fig. 3. Balls of (a) CNA-2 (CMA) in (10) compared with {-3, -1, 1, 3}, and (b) CNA-6 in (22) compared 16-QAM.

Fig. 4 Fig. 4 .

 44 Fig. 4 depicts the probability density functions (PDFs) of the interferences Im(I m,n ), Re( Ĩr m,n ),and Im( Ĩr m,n ) in (1) and (17), respectively. These have been obtained thanks to an histogram over

Fig. 5 .

 5 Fig. 5. EMSE versus interference power, comparison between CMA and CNA-6.

  should achieve better performance than |F m,0 | << 1. This behavior will be numerically verified in Section IV. Furthermore, the computation of F s-opt m,0 in (27) requires one multiplication for |y m,0 | 2 , one addition, and one division. The value of E{x 2 m,0 } only depends on the constellation size, and therefore it can be computed off-line.

  are considered. The path gains of the constant multipath channel are h = [1, -0.2, 0.3, 0.2, 0.1, 0.2, 0.35, -0.2], and the phase of the channel is uniformly chosen in [-π 2 , π 2 ]. Each transmitted OFDM/OQAM symbol is composed of M = 128 subcarriers of 15 kHz width. The system operates at a carrier frequency of 400 MHz. According to [11] the constant initialization is set equal to F m,0 = 0.01 for any m ∈ {0, 1, .., M -1}. In order to avoid the divergence of the algorithm when the proposed initialization F s-opt m,0

Fig. 7

 7 Fig. 7 depicts the MSE-floor values of CMA, CNA-6, and IAM when the steady-state is reached

Fig. 6 .

 6 Fig.6. MSE performance versus OFDM/OQAM symbols. Comparison between CNA-6, CNA-2 (CMA) using Fm,0 = 0.01 and F s-opt m,0 , and pilot-aided method (IAM).

Fig. 8 Fig. 7 .

 87 Fig.8presents the results of simulations undertaken to compare the convergence speed of CMA

Fig. 8 .

 8 Fig. 8. MSE performance versus OFDM/OQAM symbols. Comparison between CNA-6, CNA-2 (CMA), and pilot-aided method.

Fig. 9 .

 9 Fig. 9. MSE performance of CNA-6, CNA-2 (CMA), and pilot-aided method versus OFDM/OQAM symbols in presence of time synchronization mismatch. The synchronization mismatch is given as a percentage of L f = {1%, 2%, 5%}.

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  3 km/h (b) 3 km/h receiver speed.

  10 km/h receiver speed.

Fig. 10 .

 10 Fig. 10. MSE performance of CNA-6, CNA-2 (CMA) versus OFDM/OQAM symbols using the initialization F s-opt m,0 compared with the initialization using a preamble, for three receiver speeds: (a) 1 km/h, (b) 3 km/h, and (c) 10 km/h.

APPENDIXA. 8 . 2 .2 1 / 6 .

 8216 Obtaining the Average Distances E{l 6 } and E{l 2 } The ball corresponding to the CMA is generated by |a m,n | 2 -γ = 0, i.e. it is a circle of radius √ γ = √ Since P(x m,n = ±1) = P(x m,n = ±3) = 0.5, the average distance E{l 2 } can be expressed asE{l 2 } = P(x m,n = ±1)( √ γ -1) + P(x m,n = ±3)(3 -√ γ) to the CNA-6 is defined by ||z m,n || 2 6 -γ = 0,and it is depicted in Fig.3-(b). The radius of this ball along the X axis is√ γ = √ 13.2, while its radius along the axis y = x is given by √ 2γ Since P(x m,n = ±1 ± j) = P(x m,n = ±3 ± 3j) = P(x m,n = ±1 ± 3j) = P(x m,n = ±3 ± j) = 0.25, the average distance E{l 6 } can be expressed asE{l 6 } = P(x m,n = ±1 ± j)( √ γ -1) + P(x m,n = ±3 ± 3j)( x m,n = ±1 ± 3j)( √ γ -3) + P(x m,n = ±3 ± j)( √ γ -

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  Thus, the interference term I m,n is purely imaginary if H m,n = 1, and This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2016.2519000, IEEE Transactions on Signal Processing

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	ỹm,n =	y c m,n

m,n+1 m,n , and therefore we can define ỹm,n from y c m,n in (16) as 1053-587X (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2016.2519000, IEEE Transactions on Signal Processing Γ CN A6 ≈ µP ỹ E{(||z op || 2 6 -γ) 2 (Re(z op ) 10 + Im(z op ) 10 )/||z op || 8 6 } E{ (4γ -2||z op || 2 6 )(Re(z op ) 10 + Im(z op ) 10 ) + 5||z op || 6 6 (||z op || 2 6 -γ)(Re(z op ) 4 + Im(z op ) 4 ) /||z op || 10

) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. 6

[START_REF] Bouboulis | Wirtingers Calculus in general Hilbert Spaces[END_REF]

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	0.45
	0.4
				PHYDYAS
	0.35
	0.3
	EMSE
	0.25
				CMA
	0.2
					CNA-6
	0.15
	0.1	0	1	2	3	4	5	6	7	8	9	10
					Interferences power

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  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TSP.2016.2519000, IEEE Transactions on Signal Processing solution |F m,0 | << 1 proposed in

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Note that the CMA can be written in a normalized manner, by factorizing by γ.

This choice is motivated, once again, by the difficulty of setting a proper step-size µ value when a and b increase.

This work is supported by the project PROFIL (Evolution of the Wideband Professional Mobile Radio Based on the Filter Bank MultiCarrier Modulation) funded by the French national research agency with grant agreement code: ANR-13-INFR-0007-03, and ICT EC-funded project Newcom# with code: FP7-IC-318306. EDICS: SPC-CEST

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M Yetna N'

F Roudet

M Idriss

O Bartier

D Chicot

Michel Yetna N'jock
email: michel.yetna-njock@insa-rennes.fr

Work-of-indentation coupled to contact stiffness for calculating elastic modulus by instrumented indentation

Keywords: Work-of-indentation, Elastic modulus, Finite Elements analysis method

Elastic modulus of a large variety of materials: low-carbon steel, rolled and rapid prototyping stainless steels, aluminum, brass, beta tricalcium phosphate (β-TCP) bioceramic and the TiB 2 -60% B 4 C composite ceramic were determined by Vickers instrumented indentation tests. A relationship the ratio of elastic recovery energy to total work-of-indentation and the product of the maximum applied load and the square of the contact stiffness is proposed to assess the elastic modulus of materials. A proportionality factor defining pile-up or sink-in behavior has been validated by both the Finite Element (FE) analysis and inverse method for a material presenting an intermediate mechanical behavior. Values obtained for elastic modulus agree satisfactorily with theoretical values found in literature. The main advantage of the proposed relationship in the determination of the elastic modulus is that it does not require the computation of the contact area between the indenter and the material which is often at the origin of uncertainties.

Introduction.

Instrumented indentation technique is widely used for the determination of the mechanical properties of materials for which many methodologies exist to assess all these properties. From a load-displacement curve (Zeng and Chiu, 2001), two distinct methods for calculating hardness and elastic modulus can be applied, i.e. the conventional method and work-of-indentation method. From a general point of view, the conventional method allows the computation of the hardness from the maximum load and a representative contact area between the indenter and the material [START_REF] Collin | An experimental method to determine the contact radius changes during a spherical instrumented indentation[END_REF]Alcala et al., 2000;[START_REF] Bartier | Theoretical and experimental analysis of contact radius for spherical indentation[END_REF] while the elastic modulus is determined from the

A C C E P T E D M

A N U S C R I P T 3 analysis of the unloading part of the load-displacement curve [START_REF] Bulychev | Determination of Young's modulus according to the indentation diagram[END_REF][START_REF] Loubet | Vickers indentation curves of magnesium-oxides[END_REF][START_REF] Doerner | A method for interpreting the data from depth-sensing indentation instruments[END_REF]. The work-of-indentation method bases the mechanical properties determination on the energy required to create a unit volume of deformation [START_REF] Sakai | Energy principle of the indentation-induced inelastic surface deformation and hardness of brittle materials[END_REF][START_REF] Malzbender | Comment on the determination of mechanical properties from the energy dissipated during Indentation[END_REF]. The hardness is then defined by the ratio between the total or plastic workof-indentation and the total or the residual impression volume obtained after the complete withdrawal of the indenter, while the elastic modulus depends on the hardness to elastic recovery energy ratio. Both deformation modes around the indent can occur for Vickers indentation of materials. In some cases, there is an upward extrusion of displaced material (pile-up mode). In the other situations, there is a tendency for the material to be depressed around the indentation (sink-in mode). These phenomena depend on the elastic modulus to the yield stress ratio and on the strain hardening exponent of the indented material (Alcala et al., 2000), on the friction coefficient [START_REF] Mata | The role of friction on sharp indentation[END_REF] and on the residual stress into the material [START_REF] Bolshakov | Influences of stress on the measurement of mechanical properties using nanoindentation: Part II. Finite element simulations[END_REF]. Usually, the conventional method necessitates the knowledge of the deformation mode around the indent as well as the contact area between the indenter and the material which must be well-defined for a precise determination of the properties. Indeed, the definition of the contact depth is connected to the deformation mode, i.e. the methodology of Oliver and Pharr (1992) must be applied when the sink-in mode of deformation is observed whereas the methodology of Loubet et al. (1993) is adequate when pile-up is formed. Moreover in the two cases, the influence of the tip defect must be taken into account for the computation of the contact area. For that, different calibration functions have been developed for which their accuracy depends on the scale of measurement. For example, the relation of [START_REF] Loubet | Nanoindentation with a Surface Force Apparatus[END_REF] that consists in adding the length of the truncated indenter tip to the contact depth is applicable for indenter displacements higher than approximately 100 -200 nm depending on the magnitude of the tip defect. On the other hand, the complex polynomial function proposed by Oliver and [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] which is without any doubt the most precise function is commonly applied in nanoindentation with the continuous stiffness measurement mode. An alternative function has been proposed by [START_REF] Chicot | A contact area function for Berkovich nanoindentation: Application to hardness determination of a TiHfCN thin film[END_REF] to be relevant from 10 nm in-depth with only one fitting parameter which is the truncated tip defect length.

The objective of this paper is to propose an alternative method based on the work-of-indentation which does not require the computation of the contact area when the deformation mode around the indenter is known. To validate the proposed methodology, instrumented microindentation tests using a Vickers indenter were performed on several materials presenting different mechanical behaviors: low-carbon steel, rolled and rapid prototyping stainless steels, aluminum, brass, beta

A C C E P T E D M A N U S C R I P T 4
tricalcium phosphate (β-TCP) bioceramic and the TiB 2 -60% B 4 C composite ceramic.

Theoretical background.

Conventional method.

The general definition of the hardness, H, is the ratio between the maximum load, P m , and a representative area of the contact between the indenter and the material, A:

A P H m (1)
Where A can be denoted by A R when considering the true contact area and A C when it designates the projected one in the plane of the indented surface.

The reduced elastic modulus E R is calculated from the slope of the unloading part of a load-depth curve, 1/C, and the contact area A C by applying the methodology developed by Oliver and Pharr (1992) as follows:

C f T R A 1 C C 1 1 2 E ( 2
)
Where of the load frame of the instrument, C f . E R includes both the indenter parameters (E i , i ) and the investigated material parameters (E, ) in the following relation:

β
i 2 i 2 R E 1 E 1 E 1 (3)
It can be noted from Eq. (1) and Eq. ( 2) that for a precise determination of hardness and elastic modulus parameters, the contact area must be accurately determined. This is achievable from Compared to the contact area calculated for a perfect indenter tip, this model only consists in adding the length of the truncated indenter tip to the contact indenter displacement:

2 b c C h h 56 . 24 A (4)
Where h c is the contact depth and h b the truncation length of the tip defect.

In nanoindentation when using the continuous stiffness measurement mode, Oliver and [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] suggest the use of the following complex area function:

128 / 1 c 8 4 / 1 c 3 2 / 1 c 2 1 c 1 2 c C h C ... h C h C h C h 56 . 24 A (5)
Where C 1 through C 8 are constant fitting parameters.

The leading term describes a perfect pyramidal indenter which can be sometimes also considered as a free parameter; the others describe deviations from the conical geometry due to the bluntness of the indenter tip. To reduce the number of constants and to provide them a physical meaning, numerous contact area functions have been proposed based on polynomials of second degree depending on the According to the methodology, the contact depth is expressed as follows:

-For sink-in, based on the residual indenter depth to maximum indentation displacement ratio. When this ratio is lower than 0.83, the sink-in mode is predominant whereas the pile-up deformation mode occurs when this ratio is higher than this limit value of 0.83.

As a conclusion for obtaining consistent values for the mechanical properties, a rigorous methodology must be followed since it is necessary to take into account the frame compliance of the instrument, the bluntness of the indenter tip and the deformation mode around the indent because these parameters can affect in a great extent the computation of the mechanical properties. 'Neil, 1951) which affects the calculation of the contact area. This has motivated this study on the determination of the elastic modulus from the work-of-indentation in order to circumvent the problems mentioned here-above.

Work-of-indentation.

Considering the energetic approach, Stillwell and Tabor (1961) express the hardness as the work-ofindentation divided by the impression volume. Sakai (1993) specifies this definition accordingly to the mechanical behavior of the material. For rigid plastic materials, this is the total hardness, H T , which is defined by the total energy, U T , divided by the total volume of the impression, V T . For elastic/plastic materials, the author suggested the computation of a plastic hardness, H P , which is defined as the plastic work-of-indentation, U P , and the residual impression volume, V P , ratio obtained after the complete withdrawal of the indenter. Where P L and P U relate to the force of the loading and unloading paths, respectively. B, D, n, m and h r are fitting parameters.

Based on these equations, the total work-of-indentation, U T , caused by the indenter in deforming the material, is calculated from the whole area under the loading path of a load-displacement curve. U T is then defined by the following relation:

1 n m h 0 n T h 1 n B dh h B U m (9)
For the computation of the plastic hardness, the plastic work-of-indentation U P is calculated from the areal difference between the total work-of-indentation, U T , and the elastic recovery energy, U E , corresponding to the area located under the unloading path of the load-displacement curve:

1 m r m 1 n m h h m r T E T p h h 1 m D h 1 n B dh h h D U U U U m r (10) A C C E P T E D M A N U S C R I P T 8
For rigid plastic materials, V T is equivalent to the volume of a cone calculated at the maximum indentation depth. The half-angle of the equivalent conical indenter for a Vickers indenter is θ = 70.3°. Thus, V T can be calculated by:

3 m 2 m T h tan 3 3 h A V (11)
For the computation of the plastic volume V P , Sakai (1993) simply suggested that V P and V T are directly proportional to the residual depth, h r , to the maximum depth, h m , ratio as follows: show that depends on the value of the ratio (U E /U T ) as it is clearly visible in Fig. 2. It is observed in Figure 2 that the parameter is equal to 5.17 when (U E /U T ) is higher than approximately 0.25 whereas it is equal to 7.30 for values of (U E /U T ) lower than 0.15. Choi et al. (2004) add that this change of the value of is related to the competitive phenomenon between pile-up formation and elastic deflection of the materials. equals 7.30 when pile-up occurs and it equals 5.17 for the sink-in deformation mode. Nevertheless, Figure 2 shows no experimental data in the middle region, i.e. when (U E /U T ) is between 0.15 and 0.25, consequently no information is given to estimate the value of between 7.30 and 5.17.

A C C E P T E D M A N U S C R I P T

On the other hand, it is clear that the computation of the reduced modulus from Eq. ( 14) greatly depends on the accuracy of the parameter . Moreover, no clear definition is given about the hardness that must be used in this calculation. This is an important point because, for example, Tan (2006) found a difference between the hardnesses HT or HP deduced from the work-of-indentation and the hardness HIT calculated by the methodology of Oliver and Pharr (1992) with an amplitude which seems to be dependent on the mechanical properties of the material. The present work

proposes to implement a similar approach in order to compare the reduced modulus calculated from Eq. ( 14) to the literature one when the classical hardness, HIT, and the plastic hardness, HP, are considered. The plastic hardness HP relates the plastic volume directly proportional to the impression volume, V T , proposed by Tan (2006) and the ratio between the residual indentation depth h r and the total depth at the maximum load h m as follows:

A C C E P T E D M A N U S C R I P T 10 m r 3 b c 2 P h h h h 3 tan U HP (15)
To avoid any confusion in the rest of this work, the reduced modulus in Eq. ( 14) is denoted as E IT when the hardness considered is HIT whilst it is denoted E P when considering HP. The compliance term of the instrument was determined for each series of indentation test and afterwards introduced into the indentation data for obtaining accurate values of the mechanical properties.

Experiments and simulations process.

Finite Elements Analysis.

An axisymmetric two-dimensional finite-element model was constructed to simulate the indentation response of elastoplastic rapid prototyping stainless steel (RPSS) using the commercial software

ABAQUS (2003).

The material properties used in the simulation have been identified by inverse analysis method described in the following section. A total of 9178 nodes and 8876 axisymmetric linear quadrilateral elements of type CAX4R were used. A fine mesh near the contact region becoming gradually coarser further from the contact region was designed to ensure numerical accuracy (Fig. 3).

The simulation was performed under friction contact conditions (μ = 0.1) using the large strain elastic-plastic feature of the ABAQUS finite element code. The constitutive model of the indented

A C C E P T E D M A N U S C R I P T
11 material was taken to follow the well-known J2-associated flow theory with rate-independent deformation and isotropic hardening. Yielding occurs according to the Von Mises's criterion and the stress-strain relationship follows the piecewise linear/power-law:

1 Y n n n Y Y E if E if (16)
Where is the total strain, is the stress, E is the Young modulus, Y is the yield stress and n is the strain hardening exponent. The indenter was modeled as a non-rigid cone of a semi-vertical angle 70.3° with properties 1141 GPa and 0.07 for the elastic modulus and Poisson's ratio, respectively. As it was mentioned previously, the indenter tip presents a tip defect h b of 150 nm. Therefore, the cone tip has been modeled as a smoothed tip by a sphere of radius, R equals to 4215 nm. A displacement was imposed on the indenter, causing it to be pushed into the surface of the material to 2248.2 nm maximum depth to simulate the indentation process. The loading and unloading curves were obtained directly from the ABAQUS output of the total reaction force in the normal direction on the non-rigid indenter as a function of the vertical top indenter displacement. Real contact area was obtained directly from ABAQUS output after the unloading step.

Inverse analysis technique

A C C E P T E D M A N U S C R I P T 12
For identifying the work hardening law of the RPSS material, the mode FRONTIER software has been used for the inverse analysis calculation. This technique consists in the use of a cost functional to quantify the difference between the finite element simulation and the experimental measurements in order to find out the required parameters. The proposed method is divided into three stages. First the experiment is performed and then modeled by the finite element method using initial mechanical values of σy and n which were randomly chosen for the material model (Fig. 4). The initial mechanical parameters are implemented in the finite element model using user subroutine UHARD. The simplex minimization algorithm is used in order to minimize the difference between the experimental curve and the numerical one. The identification problem consists in determining σy and n. Finally, material parameters are determined when the minimized value of the cost functional is reached.

Figure 4. Schema of the inverse analysis technique applied to the instrumented indentation test.

The cost functional is written in the following form:

m h 0 2 num ref m m 1 dh F F h 1 R h E (17)
Where h is indenter displacement, h m is the maximum penetration, F ref is the experimental load obtained for the tested material and F num is the numerical load obtained with the FE code.

The calculation stops when one of the convergence criteria is reached. The convergence criterion of the inverse analysis calculation is the value of the variation of the cost functional. Whenever this last value is lower than 10 -8 N 2 , the calculation stops and it is considered that the computation has converged to an optimal solution.

Experimental results.

The load-indenter displacement curves using a Vickers indenter obtained for the aluminum 6061-T6 are shown in Figure 5 as an example. Note that the same reproducibility during the loading are obtained for each tested sample. for pile-up phenomenon have been applied to the same sets of measurement. The contact area is calculated using Eq. ( 4) and Eqs. (7a and 7b) according to the mode of deformation around the indent. The corrective factor has been calculated for each material. Since the compliance term has not a fixed value, the total compliance is plotted versus the inverse of the square root of the contact area to determine this term. Indeed accordingly to Eq. ( 2), a linear representation must be obtained between the total compliance and the reciprocal square root of the contact area:

C R f T A 1 E 1 1 2 C C (18)
Consequently a linear regression is applied to the experimental data for determining C f which is the

A C C E P T E D M A N U S C R I P T 14
intercept at the origin of the straight line. Note that the reduced modulus is determined from the slope of this straight line (Fig. 6).

From Figure 6, it is perceived that C f has the same value whatever the use of the relationship due to [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] and (b) [START_REF] Loubet | Nanoindentation with a Surface Force Apparatus[END_REF].

A C C E P T E D M A N U S C R I P T

In a previous investigation, the authors determined the deformation mode by using a depth criterion Δ representing the ratio of the final depth to the corrected maximum indentation depth (Yetna et al.,

2015)

. It was found that ∆ is greater than 0.83 for materials having only the pile-up deformation mode while it is lower than 0.83 for the ones with sink-in and it was suggested that when neither pileup nor sink-in are dominant in the deformation mode, ∆ equal to 0.83 could then be considered as a criterion to precise the limiting situation between a governing sink-in and a predominant pile-up. A

summary of the values of obtained for Vickers indenter are presented in Table 1.

Following this methodology, it is possible to distinguish the deformation mode as it is indicated in 1. As it can be seen, when the ratio U E /U T is lower than 0.15, the deformation mode is pile-up whereas for higher values than 0.25, it is sink-in. This result agrees with the results presented in Fig. 2 due to Choi et al. ( 2004) and, more largely, with the results obtained for Vickers indentation of common materials. Indeed, sink-in is commonly observed for high values of (UE/UT) ratio. Contrarily, except for few materials like annealed pure metals, pile-up is commonly observed for low values of (UE/UT) ratio. Consequently, it can be reasonably assumed that the parameter mentioned in Eq. ( 14) takes the value of 7.30 for the aluminum, brass, low-carbon steel and the rolled stainless steel samples and the value of 5.17 for the two ceramics. In contrast, for the rapid prototyping stainless steel (RPSS) for which no predominant deformation mode were observed, it is not possible to select one value compared to the other, i.e.

5.17 or 7.3, and this is why in the following, it is suggested the use of these two coefficients.

The elastic reduced modulus of the tested materials is then calculated by applying Eq. ( 14)

considering the hardness HIT (by the methodology of Oliver and Pharr) and the plastic hardness HP (by work-of-indentation). Table 2 gives the contact hardness HIT and the reduced modulus E IT when applying the methodology of Oliver and Pharr and the plastic hardness HP and the elastic modulus E P by work-of-indentation. For information, Table 2 also collects the values of the Poisson's ratio and the reduced modulus, E RTh calculated from Eq. (3) using the standard elastic modulus value of the tested materials.

A C C E P T E D M

A N U S C R I P T From Table 2, it is noted that the elastic modulus is different when it is considered the contact hardness HIT with the methodology of Oliver and Pharr or the plastic hardness HP using the work-ofindentation approach and also it slightly differs from the theoretical values of the elastic modulus. It is thought that this discrepancy comes from the definition of the hardness and the influence of the tip defect on the computation of the plastic hardness as it is demonstrated in (Chicot et al., 2015). To avoid this mismatch in the determination of the elastic modulus, it is suggested expressing the elastic energy to total work-of-indentation ratio only as a function of the elastic modulus.

An improved relation for computing reduced modulus.

It was mentioned above that the relationship between the elastic recovery energy to the total workof-indentation ratio and the hardness to reduced modulus ratio was based on numerical simulations under the assumption of perfectly rigid indenter and instrument. From the combination of Eq. ( 1) and Eq.

(2), it can be possible to express the hardness as a function of the reduced modulus, the maximum applied load and of the two compliance terms, C T and C f , related to the whole indentation system and the instrument frame, respectively, as follows:

2 R 2 f T max 2 2 E C C P 4 H (19)
The relationship ( 19) is very interesting since it allows the computation of the hardness without the determination of the contact area which often requires the calibration of the indentation test in order to take into account the influence of the tip defect. By introducing Eq. ( 19) into Eq. ( 14), the ratio (U E /U T ) is expressed only as a function of the reduced modulus and some parameters which can be easily deduced from the analysis of the unloading part of a load-depth curve. Finally, it is obtained the

following equation: R 2 f T max 2 2 T P T E E C C P 4 U U 1 U U ( 20
)
Where is equal to 1.05. is only Poisson's coefficient dependent [START_REF] Hay | A critical examination of the fundamental relations used in the analysis of nanoindentation data[END_REF]. is equal to 5.17 when (U E /U T ) is higher than 0.25 and equals to 7.3 when (U E /U T ) is lower than 0.15.

For the rapid prototyping stainless steel (RPSS) for which no predominant deformation has been observed and for which (U E /U T ) is close to the limit value of 0.2, the value of is undetermined according to [START_REF] Choi | Analysis of sharp-tip-indentation load-depth curve for contact area determination taking into account pile-up and sink-in effects[END_REF]. In order to circumvent any arbitrary choice, the microindentation of the RPSS material is simulated for determining the exact value of . The material properties used in the simulation have been identified by inverse analysis method described in the previous section. 7 shows the comparison between the experimental load-depth curve and the simulated response deduced from finite element simulation obtained for the rapid prototyping stainless steel. A good agreement was found between the predictive curve by FEM analysis and the experimental data. From the load-depth curve obtained numerically, the ratio (U E /U T ) and (HIT/E IT ) was calculated.

The results obtained have been resumed in Table 4. A value of 6.6 was found for the ratio between (U E /U T ) and (HIT/E R ) thus confirming the theory since this value is clearly located between 5.17 and 7.3. Finally, Eq. ( 20) is applied to compute the reduced modulus and afterwards the elastic modulus of the tested materials using Eq. ( 2). given according to the value of the ratio (U E /U T ) compared to the limit value of 0.2 and the deformation mode, that is to say 7.30 when (U E /U T ) is lower than 0.15 and 5.17 when (U E /U T ) is higher than 0.25. Moreover, since the rapid prototyping stainless steel (RPSS) has a critical value close to this limit value, the calculated value of the elastic modulus is given for the three different conditions of , i.e. 7.3, 5.17 and 6.6 found by finite elements simulation. It is observed that from Eqs. ( 20) and (2) allow the determination of the values for the elastic modulus which are in a very good agreement with the values found in literature. For the rapid prototyping stainless steel, it has been considered the three values for since its value of the ratio (U E /U T ) is close to the limit value of 0.2. As it was expected, the best result is given for the intermediate value of which has been validated by the Finite Element (FE) simulation. However, it was observed a relatively great difference between the elastic modulus determined from Eqs. ( 20) and ( 2) and the theoretical value for the β-TCP bioceramic. This result can be due to the formation of cracks which appear along the diagonals of the indent due to the brittleness of this type of material. In addition, according to Table 6 it was found that the elastic modulus thus determined from Eqs. ( 20) and ( 2 20) and ( 2), using the methodologies of [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] and [START_REF] Loubet | Nanoindentation with a Surface Force Apparatus[END_REF]

Conclusions

Vickers instrumented indentation tests have been performed on a large variety of materials presenting different mechanical behaviors in terms of elastic modulus to hardness ratio and in terms of deformation mode (pile-up or sink-in mode). For the determination of the elastic modulus, we suggest the application of a relationship where the knowledge of the contact area is not required for energy to the total work-of-indentation as well as the maximum load and the total compliance corrected with the frame compliance. For the studied materials, the values obtained for elastic modulus with the proposed relationship were satisfactory except for the material which began to cracking during the indentation process and the results are comparable with the values determined by the methodologies of [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] or [START_REF] Loubet | Nanoindentation with a Surface Force Apparatus[END_REF]. Concerning the material presenting cracks, part of the work-of-indentation is used to create and propagate new surfaces owing the cracks along the diagonals of the indent. Consequently, this part of energy is consumed by the initiation and propagation of the cracks and then it influences the computation of the elastic recovery energy to total work-of-indentation ratio. In addition, the value of the proportionality factor of 7.30 for low ratio of energies and 5.17 for higher values was validated and a value of 6.6 was been found by a finite elements method analysis for an intermediate value of the ratio of energies between 0.15 and 0.25.

A C C E P T E D M

A N U S C R I P T 20) and the theoretical one for the tested materials.

Table 6. Elastic modulus calculated from Eqs. ( 20) and (2), using the methodologies of [START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF] and [START_REF] Loubet | Nanoindentation with a Surface Force Apparatus[END_REF], compared to the theoretical values for the tested materials.

  is a corrective factor introduced by Bulychev et al. (1975). A three dimension analysis of Vickers indentation performed on various materials ranging a large area of mechanical properties (yield stress, work hardening exponent and modulus of elasticity) leads to a value of β close to 1.05. γ is a factor depending on the Poisson's ratio introduced by Hay et al. (1999) to consider the elastic radial displacement neglected in the formulation of Sneddon. C T is the total compliance taking into account both the compliance of the tested sample and the compliance

  numerical and experimental approaches (Hay et al., 1999). To take into account the influence of the tip defect and for indenter displacement higher than a value approximately around 100 -200 nm, the model of Loubet et al. (1993) and confirmed by Troyon and Huang (2006) is consistent enough.

  contact depth (Chicot et al., 2014). To avoid the introduction of fitting parameters or to reduce them, Chicot et al. (2014) proposed the following relation: h b can be determined by regression analysis of the experimental data or estimated from microscopic observations at very high magnifications. Additionally to take into account the deformation mode around the indent, i.e. the sink-in deformation mode or the pile-up deformation mode, the contact depth can be calculated using the methodology of Oliver and Pharr (1992) or the methodology of Loubet et al. (1993), respectively.

  h m is the maximum indentation depth and P m the maximum indentation load. C T is the total compliance equals to the reciprocal slope of the unloading part of the load-depth curve. To identify the mode of deformation around the indent, Yetna et al. (2015) have proposed a criterion

Figure 1

 1 schematically represents the regions of the total workof-indentation and the elastic energy on a load-depth curve considered in the calculations.

Figure 1 .

 1 Figure 1. Identification of the plastic and elastic work-of-indentation on a load-displacement curve obtained from instrumented indentation test.

  take into account simultaneously the influence of the deformation mode around the indent and the influence of the tip defect into the calculation of V T , Tan (2006) proposed to modify Eq. (11) by introducing the contact area model of Malzbender et al. (2000), i.e. the truncated length of the rounded indenter tip, h b , is added to the indenter displacement, and the maximum depth is replaced by the contact indentation depth defined by Oliver and Pharr (1992) or Loubet et al. (1993) according to the mode of deformation, sink-in or pile-up, respectively: simulations realized between the elastic recovery energy to the total work-ofindentation ratio and the hardness to the reduced elastic modulus ratio, numerous studies (Malzbender, 2005 ; Alkorta et al., 2006 ; Yang et al., 2008 ; Chen and Bull, 2009 ; Yang et al., 2010) propose the following linear relationship: has been reported as a single value of around 5.0 independently of the work-hardening behavior from indentation when it is performed with a conical indenter having a half-included angle of 70.3° (Malzbender, 2005 ; Alkorta et al., 2006 ; Yang et al., 2008 ; Chen and Bull, 2009 ; Yang et al., 2010). However, FEM (finite element method) achieved by Choi et al. (2004)

Figure 2 .

 2 Figure 2. Relation between U E /U T and H/E R in finite elements analysis (Choi et al., 2004).

3. 1

 1 Experiments. Instrumented indentation tests have been performed using the CSM2-107 microhardness tester. Seven materials presenting different behavior: low-carbon steel, aluminum, brass, beta tricalcium phosphate (β-TCP) bioceramic, stainless steel (SS) and rapid prototyping stainless steel (RPSS) and ceramic composite TiB 2 -60% B 4 C have been tested. The manufacturing conditions and additional details on these materials can be found in (Yetna et al., 2015). Vickers indenter was used for each sample analysis with maximum loads ranging from 50 mN to 15 N. At least 20 tests were performed in this range. In addition, a dwell-time of 15 s was imposed at the maximum applied load according to the standard CSM test procedure. Loading and unloading rates (in mN/min) have been set up at twice the value of the maximum applied load according to Quinn et al. (2002). The tip defect of the indenter was evaluated using the HITACHI field emission scanning electron microscope of type S-4300 SE/N. The measurement of the tip defect gave 150 nm for the Vickers indenter (Chicot et al., 2013).

Figure 3 .

 3 Figure 3. Finite Element model for Vickers instrumented indentation simulation.

Figure 5 .

 5 Figure 5. Loading-unloading curves obtained from instrumented indentation test using a Vickers indenter for aluminum 6061-T6.

  Oliver and Pharr (C f = 0.042) or due to Loubet et al. (C f = 0.043). The same result is observed for all the tested materials. However as it was expected the value of C f differs from each set of the tested materials. The obtained values which are collected inTable 1 are afterwards introduced in Eq. (2) for calculating the reduced modulus.

Figure 6 .

 6 Figure 6. Contact stiffness of brass in function of the inverse of square root of the contact area for Vickers indenter (a, b) indenter following (a) the methodologies of[START_REF] Oliver | An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments[END_REF]

Figure 7 .

 7 Figure 7. Comparison between the results obtained from the FEM analysis and from the experimental analysis of indentation performed on the rapid prototyping stainless steel.

Figure 8 Figure 8 .

 88 Figure 8 confirms the existence of cracks in the extension of the diagonals of the indent and the chipping along the edges of the indent.

  ) seems to be more accurate than the values found in Yetna et al. (2015) when both Oliver and Pharr (1992) and Loubet et al. (1993) methodologies were applied respectively when sinkin and pile-up mode of deformation occurs.

  The proposed relationship only takes into account the ratio of elastic recovery

Indeed, Alcala et al. (2000) mentioned

	that errors up to 30 % can be introduced in the computation of the contact
	area if the deformation mode is not taken into account. Moreover, when the compliance term is not
	introduced in Eq. (2), the elastic modulus varies as a function of the contact area. If the compliance
	term is considered as a constant value in nanoindentation, this term has been found dependent on
	the indentation conditions in microindentation. Consequently, it must be absolutely determined for
	each series of indentation tests in this range of micro-loads (

Chicot et al., 2011). However

  , even if corrections have been carefully made in order to take into account the bluntness of the indenter tip

(Hochstetter et al., 1999) and the frame compliance (Oliver and Pharr, 2004), the

	mechanical
	properties can slightly differ in comparison to the theoretical values due to the modes of deformation
	(O

Table 1 .

 1 Frame

	Materials	Cf (μm/N)	Mode of deformation	U E /U T	∆
	Low-carbon steel	0.003 ± 0.006	Pile-up	0.05 0.02	0.94 ± 0.02
	Brass	0.04 ± 0.015	Pile-up	0.11 0.02	0.90 ± 0.03
	Aluminum	0.2 ± 0.01	Pile-up	0.10 0.01	0.90 ± 0.02
	Stainless Steel	0.49 ± 0.02	Pile-up	0.11 0.02	0.88 ± 0.02
	RPSS	0.12 ± 0.01	---	0.17 0.02	0.83 ± 0.02
	β-TCP	0.26 ± 0.03	Sink-in	0.47 0.05	0.72 ± 0.01
	T i B 2 -60% B 4 C	0.11 ± 0.01	Sink-in	0.52 0.04	0.44 ± 0.1

-compliance, C f in μm/N, the elastic recovery energy to the total work-ofindentation ratio ,U E /U T , mean values of the depth ratio, ∆, and the deformation mode for the tested materials under Vickers instrumented indentation test.

Table 1 .

 1 The knowledge of the deformation mode allows to select the adequate methodology for calculating the hardness and the elastic modulus, i.e. the methodology of Oliver

and Pharr (1992) for

sink-in and the methodology of Loubet et al. (1993) for pile-up. In order to compare with the results obtained by applying the methodology of Choi et al. (2004), the elastic recovery energy to the total work-of-indentation ratio (U E /U T ) has been calculated and given in Table

Table 2 .

 2 Contact hardness H IT and reduced modulus E IT , plastic hardness HP and reduced modulus E P for the tested materials. The theoretical values of E Rth are given for comparison (http://www.engineeringtoolbox.com/young-modulus-d_417.html;

Yetna et al., 2015; Chicot et al., 2013).

	Methodology	Oliver and Pharr	Work-of-indentation		E RTh
	Materials	E IT (GPa) HIT (GPa)	E P (GPa)	HP (GPa)		(GPa)
	Low-carbon steel	192	1.3	207	1.4	0.30	184
	Brass	74	1.2	49	0.8	0.33	102
	Aluminum	96	1.3	88	1.2	0.34	73
	Stainless Steel	207	2.8	192	2.6	0.30	192
	RPSS ( = 7.30)	236	5.5	253	5.9	0.30	165
	RPSS ( = 5.17)	167	5.5	179	5.9	0.30	165
	β-TCP	90	8	191	17	0.30	152
	T i B 2 -60% B 4 C	447	45	596	60	0.24	358

Table 3

 3

	Material	Elastic modulus E (GPa)	Poisson's ratio Yield stress Y (MPa)	Work-hardening Exponent n
	Rapid prototyping stainless steel	175	0.3	540	0.39
	Figure

collects the yield stress, Y , and the work-hardening exponent, n, which have been determined from an inverse analysis while elastic modulus E and Poisson's ratio was taken from the literature (Yetna et al., 2015).

Table 3. Mechanical properties of the rapid prototyping stainless steel (RPSS) used in the finite elements method analysis, i.e. the elastic modulus E, the Poisson's ratio , the yield stress Y and the work hardening exponent n.

Table 4 .

 4 Reduced elastic modulus, hardness, elastic recovery energy, total work-of-indentation and

	numerical value of the proportionality factor	obtained for the Rapid Prototyping
	Stainless Steel.
	Material	E R (GPa) HIT (GPa) HIT/E R U E (nJ) U T (nJ) U E / U T
	Rapid prototyping stainless steel	165	5.25	0.032	97	460	0.21	6.6

Table 5

 5 collects the values of the Poisson's ratio which is used to calculate the corrective coefficient introduced by Hay et al. (1999). The experimental data of the elastic recovery energy to the total work-of-indentation ratio (U E /U T ) and the product of the maximum applied load and the compliance term (C T -C f )² are also shown. Finally the coefficient is

Table 5 .

 5 Poisson's ratio , Hay's corrective factor , elastic to total work ratio (U E /U T ), maximum load multiplied by the compliance term Pm.(C T -C f )², factor , elastic modulus calculated from Eq. (20) and the theoretical one for the tested materials.

	Materials	(U E /U T )	Pm.(C T -C f )² (10 -6 GPa -1 )		E W (GPa)	E th. (GPa)
	Low-carbon steel 0.30 1.067 0.05 0.02	23	7.30	203	200
	Brass	0.33 1.059 0.11 0.02	90	7.30	104	100
	Aluminum	0.34 1.056 0.10 0.01	120	7.30	69	69
	Stainless Steel	0.30 1.067 0.11 0.02	51	7.30	201	200
	RPSS	0.30 1.097 0.17 0.02	96	7.30	149	175
	RPSS	0.30 1.097 0.17 0.02	96	6.60	168	175
	RPSS	0.30 1.097 0.17 0.02	96	5.17	224	175
	β-TCP	0.30 1.067 0.47 0.05	471	5.17	123	160
	T i B 2 -60% B 4 C	0.24 1.076 0.52 0.04	172	5.17	494	490

Table 6 .

 6 Elastic modulus calculated from Eqs. (

  , compared to the theoretical values for the tested materials.

	Materials	Low-carbon steel Brass Aluminum SS RPSS β-TCP T i B 2 -60% B 4 C
	Loubet et al. (1993)	208	92	81	212 157	147	481
	Oliver and Pharr (1992)	257	111	99	258 190	154	514
	Eqs. (20) and (2)	203	104	69	201 168	123	494
	Theoretical values	200	100	69	200 175	160	490

Table 1 .

 1 Frame-compliance, C f in μm/N, the elastic recovery energy to the total work-ofindentation ratio ,U E /U T , mean values of the depth ratio, ∆, and the deformation mode for the tested materials under Vickers instrumented indentation test.

Table 2 .

 2 Contact hardness H IT and reduced modulus E IT , plastic hardness HP and reduced modulus E P for the tested materials. The theoretical values of E Rth are given for comparison (http://www.engineeringtoolbox.com/young-modulus-d_417.html; Yetna

et al., 2015; Chicot et al., 2013).Table 3 .

 3 Mechanical properties of the rapid prototyping stainless steel (RPSS) used in the finite elements method analysis, i.e. the elastic modulus E, the Poisson's ratio , the yield stress Y and the work hardening exponent n.

Table 4 .

 4 Reduced elastic modulus, hardness, elastic recovery energy, total work-of-indentation and numerical value of the proportionality factor obtained for the Rapid Prototyping Stainless Steel.

Table 5 .

 5 Poisson's ratio , Hay's corrective factor , elastic to total work ratio (U E /U T ), maximum load multiplied by the compliance term Pm.(C T -C f )², factor , elastic modulus calculated from Eq. (

Table 1 .

 1

	Materials	Cf (μm/N)	Mode of deformation	U E /U T	∆
	Low-carbon steel	0.003 ± 0.006	Pile-up	0.05 0.02	0.94 ± 0.02
	Brass	0.04 ± 0.015	Pile-up	0.11 0.02	0.90 ± 0.03
	Aluminum	0.2 ± 0.01	Pile-up	0.10 0.01	0.90 ± 0.02
	Stainless Steel	0.49 ± 0.02	Pile-up	0.11 0.02	0.88 ± 0.02
	RPSS	0.12 ± 0.01	---	0.17 0.02	0.83 ± 0.02
	β-TCP	0.26 ± 0.03	Sink-in	0.47 0.05	0.72 ± 0.01
	T i B 2 -60% B 4 C	0.11 ± 0.01	Sink-in	0.52 0.04	0.44 ± 0.1

Table 2 .

 2

	Methodology	Oliver and Pharr	Work-of-indentation		E RTh
	Materials	E IT (GPa) HIT (GPa)	E P (GPa)	HP (GPa)		(GPa)
	Low-carbon steel	192	1.3	207	1.4	0.30	184
	Brass	74	1.2	49	0.8	0.33	102
	Aluminum	96	1.3	88	1.2	0.34	73
	Stainless Steel	207	2.8	192	2.6	0.30	192
	RPSS ( = 7.30)	236	5.5	253	5.9	0.30	165
	RPSS ( = 5.17)	167	5.5	179	5.9	0.30	165
	β-TCP	90	8	191	17	0.30	152
	T i B 2 -60% B 4 C	447	45	596	60	0.24	358

Table 3 .

 3

	Material	Elastic modulus E (GPa)	Poisson's ratio Yield stress Y (MPa)	Work-hardening Exponent n
	Rapid prototyping stainless steel	175		0.3	540		0.39
	Table 4.
	Material	E R (GPa) HIT (GPa) HIT/E R U E (nJ) U T (nJ) U E / U T
	Rapid prototyping stainless steel	165	5.25	0.032	97	460	0.21	6.6

Table 5 .

 5

	Materials	(U E /U T )	Pm.(C T -C f )² (10 -6 GPa -1 )		E W (GPa)	E th. (GPa)
	Low-carbon steel 0.30 1.067 0.05 0.02	23	7.30	203	200
	Brass	0.33 1.059 0.11 0.02	90	7.30	104	100
	Aluminum	0.34 1.056 0.10 0.01	120	7.30	69	69
	Stainless Steel	0.30 1.067 0.11 0.02	51	7.30	201	200
	RPSS	0.30 1.097 0.17 0.02	96	7.30	149	175
	RPSS	0.30 1.097 0.17 0.02	96	6.60	168	175
	RPSS	0.30 1.097 0.17 0.02	96	5.17	224	175
	β-TCP	0.30 1.067 0.47 0.05	471	5.17	123	160
	T i B 2 -60% B 4 C	0.24 1.076 0.52 0.04	172	5.17	494	490

Table 6 .

 6 MaterialsLow-carbon steel Brass Aluminum SS RPSS β-TCP T i B 2 -60% B 4 C

	Loubet et al. (1993)	208	92	81	212 157	147	481
	Oliver and Pharr (1992)	257	111	99	258 190	154	514
	Eqs. (20) and (2)	203	104	69	201 168	123	494
	Theoretical values	200	100	69	200 175	160	490