Imen Sfar
email: imen.sfar@voila.fr

Marjorie Grzeskowiak
email: grzeskow@univ-mlv.fr

Lotfi Osman
email: lotfi.osman@supcom.rnu.tn

Ali Gharsallah
email: ali.gharsallah@fst.rnu.tn

Odile Picon
email: odile.picon@univ-mlv.fr

Limitation of a Five-Port Reflectometer Using Planar Elliptic Couplers for UWB Applications

Keywords: Elliptic coupler, six-port circuit, five-port reflectometer, ultra wideband applications. I

In this paper, we present the design of ultra wideband (UWB) planar Five-Port Reflectometer using the elliptic coupler which has the advantage to offer a wide bandwidth compared to the hybrid one. The elliptic coupler and the five-port circuit were designed and simulated in the 1.5 to 3.5 GHz band using ADS software of Agilent. An analytical calculus and simulation results of a five-port circuit using elliptic coupler demonstrate limitations of this structure.

INTRODUCTION

During recent years, there has been a great interest in six-port circuits due to their broad range of applications. Initially, a six-port was applied for flexible measurements of complex reflection coefficients in microwave network analysis [START_REF] Engen | The six-port reflectometer: An alternative network analyzer[END_REF]. Many new applications using the six-port technology have been proposed over the last ten years, such as direct conversion receivers [START_REF] Li | Computer and Measurement Simulation of a New Digital Receiver Operating Directly Millimeter-Wave Frequencies[END_REF]- [START_REF] Abe | Direct Conversion Receiver MMIC Based on Six-port Technology[END_REF], which reduce circuit complexity and allow a higher level of integration than conventional heterodyne receivers [START_REF] Abidi | Direct-conversion radio transceivers for digital communications[END_REF], phase and frequency discrimination in a radar system [START_REF] Miguelez | A new automobile radar based on the six-port phase/frequency discriminator[END_REF] and beam-direction-finding circuits [START_REF] Tatu | Direction-of-arrival estimation method based on six-port technology[END_REF].

A typically six-port circuit is composed of a passive linear device of four 90° hybrid couplers, it has two inputs and four outputs connected to power detectors in order to obtain the baseband signals. One of the four power detectors is used as a reference to normalize any variation in the source power.

In the same perspective, a multiband six-port circuit using micro-strip rounded couplers has been built [START_REF] Serioja | Analysis of a New Multiband Multiport Circuit for Phase Detection Applications[END_REF]. However, these designs are characterized by low bandwidth, which is not sufficient for the high speed communication system. Recently, a new six-port based on elliptic coupler is proposed to overcome this inconvenience [START_REF] Traii | Design of a Six-Port Junction Based on Single Layer Technology For UWB Applications[END_REF]. The circuit allows performing low losses and wide bandwidth. During the development of the six-port technique, the reference detector may not be required and the six-port circuit can be replaced by simplified five-port architecture, thus reducing the number of analog-digital converters and low-pass filters. The possibility of using a relatively simple five-port circuit like a vectorial reflectometer had been discussed by Engen [START_REF] Engen | The six-port reflectometer: An alternative network analyzer[END_REF].

Homodyne five-port receiver configurations were proposed in [START_REF] Ratni | Design considerations for direct conversion receiver using five-port circuit[END_REF] and [START_REF] Neveux | Wide-band rf receiver using the five-port technology[END_REF], and some experimental five-port reflectometers have been published [START_REF] Riblet | A compact waveguide 'resolver' for the accurate measurement of complex reflection and transmission coefficient using the 6-port measurement concept[END_REF]- [START_REF] Martin | An automatic network analyzer using a slotted line reflectometer[END_REF]. This structure is very suitable for software-defined radio applications [START_REF] Mirzavand | A New Wide-Band Five-Port Receiver Based on Metamaterial (MTM) Structures[END_REF]. Lately, some direct-conversion receivers with different architectures, based on five-port technology, were proposed and studied [START_REF] De Sousa | Full Flexible Five-Port Receiver[END_REF]- [START_REF] Pérez-Lara | Broadband Five-Port Direct Receiver Based on Low-Pass and High-Pass Phase Shifters[END_REF]. This technique was particularly interesting due to the use of power detectors instead of mixers and hence provided simpler circuits in comparison with its heterodyne counterpart. The approach has many advantages. First of all, the broadband specifications can be easily obtained using passive elements. Also, they have very low power consumption. There are various methods to implement the five-port architecture but it is important to choose an efficient one to provide suitable performance. In this paper, we present the general theory and prototype of five-port reflectometer, the quadrature hybrid using the elliptic disk is designed. The ambiguity problem and design consideration of the new fiveport reflectometer will be discussed and some experimental results regarding low losses and wide bandwidth will be reported.

II. THE FIVE-PORT REFLECTOMETER

The conventional five-port reflectometer is a linear device with two inputs corresponding respectively to the radiofrequency (RF) and local oscillator and three outputs. Power detectors are connected to these outputs in order to obtain the baseband signals. Usually, the power detector consists of a Schottky diode and a low-pass filter. The voltage measured at the filter output is linearly related to the power of the RF signal present at the diode's input terminal only for small values. This condition cannot always be guaranteed and a linearization procedure must be performed on the measured voltage values [START_REF] Potter | Non linearity correction of microwave diode detectors using a repeatable attenuation step[END_REF].

The five-port reflectometer performs a direct conversion but using a redundant mixer and a 120 degrees basis [START_REF] Souza | Five-Port Junction: In the way of General Public Application[END_REF] instead of a cartesian basis. This redundancy makes the five-port system more robust to the phase and amplitude.

The relation between the incident (a i ) and reflected (b i ) pseudo-waves at the interferometer ports are completely described by the junction's scattering parameters S :

b ∑ S a with i ∈ {1,2,3,4,5} (1)
Assuming the power detectors permanently connected at the reflectometer's output ports, one may write:

a = b Γ with j ∈ {3,4,5} (2)
where Γ is the reflection coefficient of the power detectors at port j. The eight-equation linear system formed by ( 1) and ( 2) may be rewritten in function of two of the ten variables. It can be solved as function of two free variables a 1 and a 2 and we may write:

b = A a + B a with i ∈ {3,4,5} (3)
where Ai and Bi are constants of the reflectometer. These constants are determined by a calibration [START_REF] De Sousa | A new method for automatic calibration of 5-port reflectometers[END_REF]. It is important to notice that, contrary to a six-port system, the power level at the reference input port of a five-port system must be the same for system calibration. The power level detected at the reflectometer outputs may be expressed by:

P i =|b | (1 -|Γ | ), i ∈ {3,4,5} (4)
It is possible to write:

|ρ -q | = | | P , i ∈ {3,4,5} (5)
where ρ = a 2 a 1

, q and α are constants of the junction.

By means of a straightforward mathematical development equation [START_REF] Miguelez | A new automobile radar based on the six-port phase/frequency discriminator[END_REF], it is possible to write the complex ratio between the input pseudo power waves as a linear combination of the

     out out out v v v    (6)
where the complex values α,  and  are three calibration constants and

3  out v , 4  out v and 5  out v
are proportional to P 3 , P 4 and P 5 . The calibration procedure [START_REF] Neveux | Wide-band RF receiver using the Five-port technology[END_REF] determines three complex constants α,  and  to regenerate the complex envelope from the three output voltages using equation [START_REF] Tatu | Direction-of-arrival estimation method based on six-port technology[END_REF].

We use the same kit for calibrating both six-port and fiveport reflectometers. A new adapted procedure for the diode linearization is suggested. The calibration of the five-port is done using a similar approach as in six-port case where an intermediate variable w is introduced [START_REF] Engen | Calibrating the six-port reflectometer by means of sliding terminations[END_REF].

The used system is a part of a six-port reflectometer where the reference power data is not used. This simplification provides a considerable reduction in the time processing usually required by six-port based systems. In comparison to the six-port, the five-port reduces the circuit size.

A photograph of the implemented circuit is shown in Fig. 1. This circuit, inscribed in a rectangle of 10 cm in length and 8 cm in width, was designed to operate at 2.4 GHz with a bandwidth of 1 GHz. The five-port reflectometer is fabricated on TMM4 substrate with = 4.5 and thickness of 1.524 mm.

III. ELLIPTIC COUPLER DESIGN

The design of the directional coupler can follow different implementations among which, lumped elements [START_REF] Caloz | A Novel Composite Right-/Left-Handed Coupled-Line Directional Coupler With Arbitrary Coupling Level and Broad Bandwidth[END_REF], microstrip branch-line couplers [START_REF] Anada | New Synthesis Method for Branch Line 3dB Hybrid: A Hybrid Approach Comprising Planar and Transmission Line Circuit Concepts[END_REF], multilayered structures [START_REF] Hoffmnann | Microstrip-Slot Coupler Design -Part 1: S-Parameters of Uncompensated and Compensated Couplers[END_REF], ring couplers [START_REF] Kim | Broad-Band Design of Improved Hybrid-Ring-3dB Directional Couplers[END_REF], and circular or elliptical couplers [START_REF] Chan | A Quadrature-Hybrid Design Using a Four-Port Elliptic Patch[END_REF] are the most used. The elliptic patch design method is based on [START_REF] Bialkowski | Analysis and Design of a Circular Disc 3 dB Coupler[END_REF], where short-circuit stubs are introduced at the periphery of a metallic printed disk to function as dummy ports, in addition to the matching networks at the physical ports. The approach in [START_REF] Bialkowski | Analysis and Design of a Circular Disc 3 dB Coupler[END_REF] allows changing the overall impedance matrix of the structure by a suitable choice of the stub parameters. The Green's function of an elliptic patch consists of even and odd parts. So from [START_REF] Chan | A quadrature hybrid design using a four-port elliptic patch[END_REF], the even and odd elements of the impedance matrix Z for an elliptic patch with ports at the periphery, as shown in Fig. 2, are given as illustrated in the following equations:

= (ℎ, ℎ ) ℎ, , ∆ , (ℎ, , ∆ , ) (ℎ) (ℎ, ℎ ) (7) = (ℎ, ℎ ) ℎ, , ∆ , (ℎ, , ∆ , ) (ℎ) (ℎ, ℎ ) (8)
where (ℎ, cosh u) and (ℎ, cosh u) are even and odd first-kind radial Mathieu functions, (ℎ, cosh ), (ℎ, cosh ) are the even and odd first-kind circumferential Mathieu functions, respectively.

By the introduction of the effective parameters, the fringing fields are accounted. The effective relative permittivity [START_REF] Nakajima | A simplified formula for the characteristic impedance of microstrip lines[END_REF] of the elliptic patch can be taken as:

= + 1 2 + -1 2 1 + 10 2 (9)
The effective semiminor axis be of the elliptic patch can be taken as [START_REF] Chew | Effects of fringing fields on the capacitance of the circular microstrip disk[END_REF]:

= 1 + 2 2
+ (1.41 + 1.77) + (0.268 + 1.65) [START_REF] Neveux | Wide-band rf receiver using the five-port technology[END_REF] b represents the semiminor axis, given by b=l.sinh u and d is the thickness of the substrate.

The S-parameters of the quadrature hybrid are evaluated using the expression = ( -)( -) , where is the characteristic impedance of the transmission lines connected to the device.

The elliptic directional coupler was proposed by [START_REF] Chan | A quadrature hybrid design using a four-port elliptic patch[END_REF] to enhance the bandwidth of hybrid coupler. While introducing modifications on the structure original, a scaling procedure based on the following relation is performed:

= ( 11
)
where f 1 and f 2 are the central frequencies of the two bands (existed and desired), a 1 and a 2 are the corresponding characteristic dimensions of the patch (semi-major elliptic axis), and are the dielectric constants of the substrates.

The initial parameters are found after scaling and the final dimensions are obtained by using Momentum which is part of Advanced Design System (ADS). An optimization was carried out to determine the optimal values of the elliptic coupler dimensions.

The procedure of optimization is based on the scanning of a four-dimensional space by varying the length of the semimajor elliptic axis, the eccentricity, and the width and length of the impedance steps. The port angle is kept constant and in each circle of optimization, the flatness and the bandwidth of the coupler are evaluated. In the final step, the resultant geometry matches the desired characteristics. This geometry is given in the third column of Table I.

The actual implementation of the coupler is fabricated on a TMM4 substrate with relative dielectric constant = 4.5 and thickness h=1.524 mm. The implemented elliptic coupler is shown in Fig. 2, while the simulated and measured responses of the reflection and transmission scattering S parameters are compared in Fig. 3,4, 5 and 6. The elliptic coupler has been measured on network analyzer (E8361C). The power of the coupled ports, S 13 and S 14 , is equal to -( 3 ± 0.1) dB and presented in Fig . 3. The simulated return loss, S 11 , and the insertion loss, S 12 , of the elliptic coupler are shown in Fig. 4 and5.

The measured responses confirm the coupler's compliance with the bandwidth requirements, both the return loss and the isolation of better than 15 dB (at the center frequency). In terms of phase shift, Fig. 6 shows that the coupler produces a phase shift of about 90.

IV. CONFIGURATION OF THE PROPOSED FIVE-PORT CIRCUIT

The standard six-port junction consists of four 90° -3dB hybrid couplers and one phase shifter. This circuit doesn't provide any high performances in terms of low losses and wide bandwidth. To overcome these problems, a new six-port based on elliptic couplers is achieved [START_REF] Traii | Design of a Six-Port Junction Based on Single Layer Technology For UWB Applications[END_REF]. Performances concerning magnitude and phase shift of this coupler are better than the hybrid one. With these features, the six-port circuit based on elliptic couplers is suitable for wideband wireless applications system.

We have thought that we can adapt this optimization to achieve a UWB five-port circuit. Fig. 7 shows the desired layout of the wideband five-port configuration based on three elliptic couplers.

The five-port junction presented in this section is designed and simulated at a center frequency of 2.4 GHz. The proposed five-port circuit was simulated using Momentum of ADS [START_REF]Advanced Design System[END_REF]. It is a passive circuit composed of 3 four-port microstrip elliptic couplers and a phase shifter connected in port 7. The five-port circuit is designed with TMM4 having a thickness h of1.524 mm and a relative dielectric constant ε = 4.5. In this case, the size of the proposed five-port (159.6 mm*118.7 mm) is bigger compared with classic structure of the five-port. As perspective, a multi layer structure is considered to reduce the size. This technology needs an accurate fabrication process and alignment between the layers. Figures 8 and9 show simulation results of scattering parameters of the proposed five-port as well as those the classic five-port. From these results, it can be shown that the coupling is almost constant over the frequency band of (1-3 GHz) with a value of 6 dB. Contrary to the classic five-port, it can be seen that the coupling is about 6 dB for the central frequency to 2.4 GHz, which is suitable to build the proposed circuit or other microwave networks. In term of phase shift as illustrated in figure 10, the five-port ring provides a phase of approximately 120 ° between two ports. During simulation, we have noticed that the phase differences of the novel structure are not consistent for all ports for which the theoretical values are 120°. A mathematical treatment has been developed to justify the insufficiency of this technique, which allowed us to conclude that this structure had limits since we did not get good phase differences of the classic five-port.

V. CONCLUSION

In this paper, a new wideband five-port circuit based on the use of elliptic coupler has been presented. This elliptic coupler, which is also a part of the directional coupler family, is particularly suitable for low cost and UWB applications.

The performances of the proposed five-port in terms of magnitude are very promising, but this structure has limits in the phase shift. A five-port circuit composed of four elliptic couplers, which lead to sufficient degrees of liberty to optimize the 5-port circuit, is being studied, in despite of the size of this system. We have the possibility to realize a multilayer system in order to reduce the size.

  low-pass filter outputs after dc-offset cancellation, as shown below:

Figure 1 .

 1 Figure 1. Microstrip five-port reflectometer.

Figure 3 .

 3 Figure 3. Simulated and experimental results of the coupling port S13 of the proposed elliptic coupler.

Figure 4 .

 4 Figure 4. Simulated and experimental responses S11 of the elliptic coupler.

Figure 5 .

 5 Figure 5. Simulated and experimental responses S12 of the proposed elliptic coupler.

Figure 6 .

 6 Figure 6. Phase difference between S23 and S24.

Figure 7 .

 7 Figure 7. Layout of the five-port circuit.

Figure 8 .

 8 Figure 8. Scattering parameters of the proposed five-port magnitude.

Figure 9 .

 9 Figure 9. Scattering parameters of the five-port ring magnitude.

Figure 10 .

 10 Figure 10. Scattering parameters of the five-port ring phase difference.

TABLE I

 I

	.	DESIGN OPTIMIZATION OF THE PROPOSED ELLIPTIC
		COUPLER
	Parameter	Initial Values	Optimum Values
	Semimajor elliptic axis	18 mm	19.5 mm
	Eccentricity		0.58	0.64
	W (width of the impedance steps)	2.8 mm	3.8 mm
	Ds (length of the impedance step	16.9 mm	17 mm
	~ λ/4 transformers)
	Φ (the port angle)	51°	51°

Figure 2. Implemented elliptic coupler.

This work was supported by the Programme PHC-Utique 18521YK managed by CMCU (Mixed Committee of University Cooperation).

ACKNOWLEDGMENT

The authors would like to express their sincere gratitude to David Delcroix with ESYCOM EA2552, University Paris-Est, Marne-la-Vallée, France, for his technical assistance, helpful guidance and support.


I Serre

R M Pradeilles-Duval

N Celati

Tribological and corrosion experiments of graphite ring against Ti-6Al-4V disk: Influence of electrochemical and mechanical parameters

Keywords:

HAL is

Introduction

When used in a marine environment, planar contacts and ring-on-disk systems such as boat pumps or packing glands are expected to ensure their sealing capacity. The problem considered here consists in two rings in contact and in rotating with respect to each other. The friction phenomenon between the both rings induces a material wear, which can alter the waterproofing and the sealing capacity of the system. Tribocorrosion experiments can be used to analyze the mechanical and electrochemical wear modes, in view of the determination of the material wear laws, thanks to numerical simulation. Coupling between experimental data and numerical simulations can be involved to determine the system durability [START_REF] Lederer | Modélisation tribomécanique du frottement en milieu agressif[END_REF][START_REF] Serre | Contribution àl ' étude des phénomènes d'usure par frottement en milieu marin[END_REF].

Usually tribocorrosion experiments are carried out by using a pin-on-disk tribocorrosimeter with an alumina pin [START_REF] Mischler | A round robin on combined electrochemical and friction experiments on alumina/stainless steel contacts in sulfuric acid[END_REF][START_REF] Jemmely | Tribocorrosion behaviour of the Fe-17Cr stainless steel in acid and alkaline solutions[END_REF][START_REF] Lizandier | Characterization of porosity of ceramic coatings assigned to applications in sea water with the evolution of their electrochemical behaviour under friction[END_REF]. Alumina is chosen due to its neutral electrochemical properties: thus, the electrochemical phenomena occurring at the surface of the other material can be more easily explained by the measured electrochemical current. Moreover, because of its higher mechanical properties, no loss of material or significant degradation is observed for the alumina pin. Then any variation of the contact geometry during the test can only derive from a loss of matter of the non-alumina disk. In addition, it is well known that the wear behavior depends on the contact geometry, and the time evolution of its configuration (steady-state or periodic contact) [START_REF] Georges | Frottement, usure et lubrification[END_REF][START_REF] Kato | Wear in relation to friction-a review[END_REF]. So, using systematically an alumina pin is restrictive. That is why a ring-on-disk tribocorrosion device electrochemically and mechanically instrumented was developed in order to study the wear behavior of two materials in contact.

The purpose of this paper is to analyze and to understand the influence of the electrochemical and mechanical parameters on tribocorrosion wear. The paper is organized as follows: we first describe and analyze the tribocorrosion experiments and then the studied materials: a graphite and a titanium alloy. In Section 3, we present the results and the influence of various parameters on friction coefficient and electrochemical parameters: respective configuration (ring or disk) of the materials, applied load, test duration and global geometry of the contact. Finally, suggestions are made on the test duration and on the contact conditions, in connection with the industrial case under study.

Experimental device

Description of the tribocorrosimeter

The tribometer is a ring-on-disk tribocorrosimeter developed by a technical centre of the French Armament Procurement Agency ("centre technique d'Arcueil", CTA-DGA). The device consists in a fixed disk and a rotating ring (Fig. 1). A normal pressure applied by a servo-hydraulic tension/compression machine maintains the contact between the samples. An electrochemical cell is used to perform corrosion tests in artificial seawater under fixed electrochemical conditions. The ring has an average radius of 8 mm and the width of the contact zone is 3 mm. In order to maintain a plan/plan contact with the ring, the fixed disk is assembled on a spherical kneecap.

All the experiments were performed at room temperature in artificial seawater, (NaCl solution to 30 g/l buffered at pH 8 according to NFA 91 411 standards). A three-electrode set-up (Figs. 1 and2a)) was used to impose an electrochemical potential on the exposed surface of both samples and to measure the current. The given values of the electrochemical potential are referred to the Ag/AgCl saturated in KCl electrode. A circulation of the electrolyte was put in place (Fig. 1) for stability of the artificial seawater characteristics during the tests.

Experimental outputs 2.2.1. The friction coefficient

Numerical boundary element method (BEM) simulations [START_REF] Serre | Modeling an abrasive wear experiment by the boundary element method[END_REF] have been involved with the materials presented hereafter. The distribution of the contact pressure is as expected not uniform. It is higher near the two edges of the contact zone. This can explain that the loss of matter is likely to be initiated under these edges. The same qualitative observation was made by Water et al. [START_REF] Walter | The ring-on-disk method: clinical significance of a wear-screening experiment of biomaterials for hip joint alloplasty[END_REF] from ring-on-disk tribocorrosion experiments.

The measured friction coefficient µ(t) is a global Coulomb friction coefficient, i.e., the ratio between the actual tangential force and the actual applied normal force. Moreover, the radial tangential force can be neglected with respect to the orthoradial tangential one. As defined by Fig. 3, the average value on the track of the radial tangential stress σ rz is about 0.01 MPa while the orthoradial tangential stress σ z and the normal stress σ zz are Fig. 2. Electrochemical set-up (electric circuit). 0.75 and 5 MPa, respectively. So, in the following, the measured global Coulomb friction coefficient µ(t) is given by the ratio of the mean orthoradial tangential force (F t ) and the applied normal force (F n ):

F t = 2π R 2 R 1 σ θz dr and F n = 2π R 2 R 1 σ zz dr
A force sensor C (Fig. 3) was used to block the disk rotation and to measure a tangential force (F m ). Equilibrium moment in the vertical z-direction induces a relation between F m and F t :

F t = 3 2 (R 2 2 -R 2 1 ) (R 3 2 -R 3 1 )
RF m where R 1 , R 2 and R are, respectively, the internal, external radii of the track and the force sensor radius measured from the apparatus axis (Fig. 3). In the previous, the dependence of the zmoment versus the radial position is taken into account. Finally, it leads to the coefficient:

µ(t) = F t (t)
F n (t) .

Electrochemical measurements

As mentioned above, the choice of the materials, motivated by a given structural application, is such that both of them are electrochemically active. Several questions show up: is there a galvanic coupling between the two materials? Do the samples in contact have the same potential during the tests? What is the signification of the measured current during the tribocorrosion tests?

First, the risk of a galvanic coupling depends on the nature of the materials but also on the ratio of both areas in contact with the electrolyte. For our experiments, the area of the ring in contact with the electrolyte is approximately four times larger than the area of the disk in contact with the artificial seawater. But, due to the respective positions of the ring and of the counter electrode, a part of the surface of the ring (especially the inner one) is electrochemically masked. It follows directly from these remarks that the risk of the galvanic coupling cannot be followed from separate electrochemical studies on each material. Consequently, the two configurations (ring in one material, disk in the other and vice versa) had to be studied (see Section 3.1), in order to evaluate the influence of the materials location on galvanic coupling.

By using the standard electrochemical set-up reported in Fig. 2a, either the potential is fixed and a current is measured or the free potential is measured. The working electrode is assumed to be constituted of the two samples and the connection between the working electrode and the electric circuit usually occurs on the disk. Nevertheless, due to the motion of the samples with respect to each other, a difference of the potential can exist between the disk and the ring. To check this point, we use the electrochemical set-up reported in Fig. 2b. A connection with the electric circuit is established for each sample to measure the difference of potential between the disk and the ring during free potential experiments. With the studied materials (graphite and titanium alloys), we measure a difference of potential equal to 5 mV.

Finally, when a potential is imposed, the measured current is the sum of the contribution of each material; the contribution of each material is unknown. To evaluate the respective contributions, some specific experiments are carried out with an electrochemical set-up based on the current/potential converters [START_REF] Deslouis | Multi-channel potentiostatic control in electrochemistry: localized mass-transfer rates at the rotating disk electrode[END_REF] (see Fig. 4). In Fig. 5, we see the electrochemical contributions of the both studied materials (Ti-6Al-4V and graphite) in two configurations: in contact without friction, and in con- tact with friction. The interpretation of the current is not easy because of the evolution of the surfaces of the samples due to the friction contact. But we observe the important decrease of the corrosion potential for the Ti-6Al-4V during friction. This phenomenon in connection with a wear of Ti-6Al-4V according to the applied potential during tribocorrosion experiments [START_REF] Serre | Tribological and corrosion wear of graphite ring against Ti-6Al-4V disk in artificial seawater[END_REF] is associated with depassivation/repassivation phenomena of the Ti-6Al-4V. So, this set-up is a help for the understanding of the electrochemical phenomena during tribocorrosion tests.

Because it is quite difficult to maintain the connection with the moving sample during 6-or 50-h experiment (set-up reported in Fig. 2b), in the following, unless specified, the standard electrochemical set-up of Fig. 2a is used, with the assumption of perfect contact.

Wear measurement

After each test, the total wear of the disk is evaluated by profilometry, and the height of the ring is measured before and after each test. This results in the evaluation of the total loss of matter for each sample. To determine the wear law from the comparison between the experimental results and the numerical simulations, the measured evolution of the samples wear rate is needed. That is why the displacement of the crosshead of the tensile machine is continuously measured. Since no evolution of the elastically deformation of the apparatus (ring, disk and system around the samples) has been observed during the experiment, the evolution of the measured displacement corresponds to the relative axial displacement between the disk and the ring. It provides useful informations about the wear rate evolution and the presence of wear debris in the contact zone as underlined in Section 3.3.

Experimental results and discussion

The average normal pressure between the two materials is controlled thanks to a servo-hydraulic machine. The pressure is time-independent and less than 10 MPa. The rotational speed of the ring is fixed and equal to 40 rpm, which corresponds to a mean linear velocity of 0.033 m/s. The experiments are 6 h long, unless specified otherwise.

Studied materials

One of the samples is cut from a cube of Hot Isostatically Pressed (HIP) sintered graphite. The material is elastic, brittle and isotropic [START_REF] Carbone-Lorraine Society | Carbons and graphites for mechanical applications[END_REF]. The other sample is made of Ti-6Al-4V or of TiC-reinforced Ti-6Al-4V coating on a Ti-6Al-4V substrate. The TiC-reinforced Ti-6Al-4V coating width is about 2 mm. Due to the low magnitude of the applied load, only the properties of the coating influence the mechanical response of the sample when coated. According to the loading conditions and the mechanical behavior of these titanium alloys, they may be assumed as elastic.

The electrochemical behavior is derived from the measurement of the free potential after 1-h experiment in artificial seawater (Table 1). The free potential of graphite is higher than the one of the titanium alloys. This result provides us with the elec- Open circuit, load 800 N 2.5 ± 1.9 1.1 ± 0.8 trochemical coupling through the corrosion of titanium alloys by graphite, as analyzed below.

Influence of geometry

To understand the influence of geometry, two configurations have been studied: configuration 1 refers to the case where the ring is made of graphite and the disk is made of Ti-6Al-4V, and configuration 2 to the reverse situation. No potential is imposed. The free potential of the working electrode is measured.

For both configurations, the same value of the friction coefficient is found: its average value is 0.16 ± 0.04 after 6 h. The free potential of the two materials is about the same order for both configurations too: between -0.110 and -0.125 V. Ti-6Al-4V wear mainly corresponds to an increase of roughness in both configurations. The loss of graphite is more important in the configuration 1 (Table 2). The wear of the ring is almost uniform across the width of the contact zone. The height variations are less than 1 m for an average value of more than 10 m. On the contrary, the wear of the graphite disk is not uniform: it is higher on the external edge of the contact zone (Fig. 6). These differences can be explained by the evolution of the contact geometry, which differs according to the configuration. This might be due, at least in part, to the evacuation of the graphite debris out of the friction zone. As shown in Fig. 7, in configuration 1, the graphite debris can be ejected out of the contact area more easily than in configuration 2 where they are maintained inside the track. In [START_REF] Stemp | The effect of contact configuration on the tribocorrosion of stainless steel in reciprocating sliding under potentiostatic control[END_REF], Stemp et al. observed comparable wear volume with a comparable wear volume a metal pin sliding on a ceramic disk and the reverse situation despite the different wear mechanisms. This work and our results raise up that the geometry of the samples, their respective configuration and the wear behavior are coupled.

In the following, configuration 1 only is considered.

Influence of mechanical loading

The average contact pressure is either 2.65 or 5.3 MPa, for an applied normal force of 400 and 800 N, respectively. The influ-Fig. 6. Profilometry of the wear track for the graphite sample (free potential; applied load: 800 N, without artificial sea water circulation). ence of the applied load is studied for two imposed potentials: -1.010 and 0.140 V. For the experiments with a potential equal to -1.010 V, there is no influence of the loading on the mean value of the friction coefficient and on the measured current was observed. But, with a potential equal to 0.140 V, the larger the applied force (400 or 800 N), the smaller the friction coefficient. In addition, the measured current is smaller with a normal force of 400 at 0.140 V and not at -1.010 V. As underlined in [START_REF] Landolt | Electrochemical methods in tribocorrosion-a critical appraisal[END_REF], the electrochemical current increases with the mechanical loading.

At the end of the experiment for -1.010 V, no loss of matter of Ti-6Al-4V is measured (by profilometer). But an increase of roughness of the contact zone of Ti-6Al-4V is observed. It is 0.2 m after a 400 N experiment and 0.4-0.5 m with an 800 N load. The disk is more damaged in these last cases. For experiments at 0.140 V, a loss of matter from the Ti-6Al-4V disk is measured; it is 0.210 ± 0.062 mm 3 at 800 N compared to 0.103 ± 0.052 mm 3 at 400 N. The difference is readily explained by the local depassivation/repassivation phenomena. It is more active under a larger load (Fig. 8). For a potential of -1.010 V and a normal force of 400 N, the loss of the graphite matter is more important without circulation of artificial seawater (Table 3). In contrast, by coupling a circulation of artificial sea water and an anodic potential (0.140 V), opposite results are obtained (Table 3). These variations are probably due to several factors such as the properties of the Ti-6Al-4V surface, which vary with the potential, the evacuation of the wear debris that depends on load and on the circulation of artificial seawater. For a given potential of -1.010 V, Fig. 9 gives the axial displacement of the crosshead of the tensile machine. The displacement results from the wear of the graphite ring and the third body. The Ti-6Al-4V disk does not suffer any loss of matter. For the experiments under 400 N, this displacement does not evolve linearly in time but rather stepwise. The steps testify that wear debris are accumulated in the contact area (no axial displacement) before they are ejected out of the friction zone (stepwise evolution of the displacement). The contrast between a constant friction coefficient and an evolving axial displacement is associated to the existence of graphite transferred on the disk. Graphite/Ti-6Al-4V contact and graphite/graphite contact are simultaneously present in the track. At 800 N, the displacement curve is smoother, except during the initial phase of the experiment. The higher local pressure is probably crucial for the destruction of debris and their ejection.

Test duration

The main goal of the longer experiments is to observe the time evolution of wear. The 50-h long experiments are carried out at a given potential equal to -0.060 V, which corresponds to the value of the industrial application (friction contact between graphite and titanium alloy during 10,000 h).

The comparison between 6-h and 50-h-experiments shows a more important roughness of the Ti-6Al-4V track for the longest tests. Furthermore, the corrosion products are observed not only near the edges of the friction zone as observed for 6-h experiment, but also in the middle of the wear track. This phenomenon is due to the time evolution of the local electrochemical potential and of the local properties of the electrolyte. The loss of the graphite ring corresponds to a height decrease equal to 20 ± 2 m after 6 h and to 90 ± 10 m after 50 h. Assuming the wear rate obtained after 6-h experiments remains constant till 50 h, the extrapolated loss of height would have been up to 165 m for 50-h experiments, which is nearly twice the measured value. It suggests that the wear does not reach a steadystate regime after 6 h. The lifetime cannot be extrapolated solely from the results of 6-h experiments. In order to validate this conclusion, some experiments have been performed in the same conditions with another couple of materials: graphite and TiCreinforced Ti-6Al-4V coating on Ti-6Al-4V substrate. The industrial problem suggests that the smaller the loss of graphite matter is, the more relevant the couples of materials are. After the 6-h experiments, the couple graphite/TiC-reinforced Ti-6Al-4V is better than graphite/Ti-6Al-4V. The 50-h experiments lead to the opposite conclusion. This result confirms the limitations of short experiments. As a conclusion, the test duration should be chosen carefully in view of sound predictions for the industrial lifetime.

Influence of the contact configuration

For the purpose of comparison between two contact configurations (ring-on-disk, i.e., steady-state and permanent plan/plan contact, and pin-on-disk, i.e., periodic contact), experiments have been carried out with a pin-on-disk tribocorrosimeter [START_REF] Lederer | Modélisation tribomécanique du frottement en milieu agressif[END_REF]. The disk and the pin are, respectively, made of Ti-6Al-4V and of graphite. The pins are 10 mm diameter cylinders. Their surface of contact is plane with a chamfer of 1 mm. The plane surface is polished; its roughness is approximately 0.1 m. The average contact pressure between the two samples is 3 MPa and the rotational speed is 40 rpm, which corresponds to a linear velocity of 0.015-0.052 m/s across the friction track.

For the pin-on-disk experiments, a matter loss of the Ti-6Al-4V disk is observed for all studied electrochemical conditions, whereas for the ring-on-disk experiments, it is observed only with an anodic potential of 0.140 V. Moreover, no corrosion product is found and the oxygen concentration is nearly homogeneous on the whole disk surface subjected or not to friction during pin-on-disk experiments which contrasts with that of the ring-on-disk experiments. This means that the degradations of the Ti-6Al-4V are different for these two configurations.

Nonetheless, for both set-ups (ring-on-disk or pin-on-disk), the most significant wear is the graphite wear. We observe transfer of graphite on the Ti-6Al-4V disk, in both cases. The experiments underline the influence of the contact. More precisely, the wear response of the both materials depends on the permanent contact regime (ring-on-disk experiment) or the periodic one (pin-on-disk experiment) between the samples.

Conclusion

A new tribocorrosion experimental set-up has been developed to study materials wear for a ring in friction with a disk in seawater. The observations made at the end of the experiments allow us to determine the process of the loss of matter and of the damage of the ring and disk [START_REF] Serre | Tribological and corrosion wear of graphite ring against Ti-6Al-4V disk in artificial seawater[END_REF]. From the mechanical analysis of the contact conditions, the importance of the contact geometry is underlined and analyzed. The preferential matter loss on the edge of the friction zone is explained by the presence of high stresses in this part of the track. It is critical on the evolution of the contact conditions (distribution of the contact pressure), on the wear debris evacuation and on the evolution of the wear. Because of the electrochemical activity of both materials and of the geometry of the antagonists, the interpretation of the measured current is not direct. Consequently, it is necessary to use a specific electrochemical set-up to study the electrochemical conditions in the contact zone more accurately.

The influence of different parameters -such as the position of the materials in the structure, the applied load, the experiment time duration and the contact configuration -has been studied. The position of the materials in the ring-on-disk experiment changes quantitatively the graphite wear and the ejection of debris. The coupling of the electrochemical prescribed potential and the applied load on the wear has been focused on. It suggests further local analyses of the electrochemical conditions in the contact area including occlusive zones. The test duration appears to be crucial to evaluate a major parameter to estimate the wear law of the couple of materials because the cumulated loss of matter is nonlinear function of time.

Fig. 1 .

 1 Fig. 1. The principle of the ring-on-disk tribometer.

Fig. 3 .

 3 Fig. 3. Measure of the friction forces.

Fig. 4 .

 4 Fig. 4. Electrochemical set-up with current/potential converters (electric circuit).

Fig. 5 .

 5 Fig. 5. Contributions of each material to the elecrochemical current: potentiodynamic polarisation curve during tribocorrosion experiments with and without friction (test conditions: see Section 3.3; applied load: 400 N).

Fig. 7 .

 7 Fig.7. Influence of the configuration on the evacuation of the graphite debris (emphasized scale).

Fig. 8 .

 8 Fig. 8. Influence of the applied load. Profilometry of the wear track for the Ti-6Al-4V sample (applied potential: 0.140 V, with artificial sea water circulation).

Fig. 9 .

 9 Fig.9. Influence of load on the displacement of the lower crosshead of the tensile machine (applied potential: -1.010 V; without artificial sea water circulation).

Table 1

 1 Free potential of the materials in artificial sea water (after 1 h)

	Material		Free potential (mV)
	Graphite		95 ± 50
	Ti-6Al-4V		-265 ± 145
	10% TiC-reinforced Ti-6Al-4V		-105 ± 75
	15% TiC-reinforced Ti-6Al-4V		-140 ± 80
	Table 2
	Volume loss of graphite (mm 3 ) for the two configurations
	Conditions of experiments	Geometry
		Configuration 1	Configuration 2
	Imposed potential of	4.5 ± 0.8	2.6 ± 1.1
	-1.010 V, load 400 N

Acknowledgements

The authors wish to thank Mr. Louchart of Centre technique d'Arcueil (France) for his technical assistance. They gratefully acknowledge the help provided by M. Keddam and M. Takenouti from the laboratory "Physique des Liquides et Electrochimie" (Paris VI University, France) about the electrochemical set-up with the current/potential converters and by P. Ponthiaux of Ecole Centrale de Paris (France), for the pin-on-disk experiments.


Alexander M Korsunsky
email: alexander.korsunsky@eng.ox.ac.uk

Andrei Constantinescu

Work of indentation approach to the analysis of hardness and modulus of thin coatings

Keywords: Contact, Modulus, Hardness, Coating, Property identification

The problem of accurate identification of mechanical properties, primarily hardness and modulus, invariably arises whenever very thin coatings are considered that consist of single or multiple layers, e.g. in the context of fitness-for-purpose characterisation of coated systems, monitoring in-service degradation, component lifetime prediction, etc. Rapid developments in the areas of nano-fabrication, nano-manipulation and nanotechnology lead to the increased importance of reliable characterisation of mechanical properties of progressively thinner coatings. Instrumented small scale (nano-) indentation is particularly well-suited to surface-engineered and thin-coated systems. The present study presents a review and refinement of the interpretation techniques for instrumented (nano) indentation for reliable property de-convolution of coated systems. Particular attention is devoted to contact modulus and hardness, as the properties that play the key role in controlling the deformation response of any surface, and affect such service properties as impact and erosion resistance, wear and fretting fatigue resistance, resistance to crack initiation and propagation, etc. A flexible multi-scaling power law functional description is introduced and discussed, and its application to various example systems is illustrated.

Introduction

Motivation

In many situations of industrial manufacture one of the key problems is that of achieving optimal service properties of engineering components. Often these problems are best solved by using surface engineering techniques, e.g. physico-chemical surface treatment, or deposition of thin coating layers over the existing substrate. This approach minimises the expense of using high performance materials, such as very hard diamond-like carbon films for cutting tool applications, or highly heat-resistant yttria-stabilized zirconia ceramics for thermal barrier coating on aeroengine turbine blades. It also often provides the best balance between the properties of the coating (hardness, low thermal conductivity) and those of the substrate (e.g. the ductility and toughness of a load bearing metallic alloys).

In many such situations the problem of property identification arises for the thin single or multiple coating layers. Initially the requirement may arise in the context of fitness-for-purpose characterisation of the coating-substrate system, which is followed by the need to carry out process optimisation in order to achieve the best performance. Finally, property identification is important in the context of monitoring the in-service degradation, and for predicting component lifetime reliably.

The conventional approach to the task of system characterisation often relies on full or partial simulation of the in-service conditions. With this purpose in mind large scale tests of equipment are performed, often to destruction. Clearly experiments of this kind are extremely expensive and very time-consuming. Scaled-down laboratory rigs, such as wear testers, are very useful in capturing some aspects of the process. However, they often require preparation of specimens from small coupons, and thus cannot be considered as non-destructive.

The alternative to these approaches that is particularly wellsuited to surface-engineered and coated systems is small scale surface indentation. When an indenter made from a hard material such as diamond is brought into contact with the sample surface, the contact initially is purely elastic. This is reflected in the same load-displacement trace being repeatedly passed during unloading. The information contained in the load-displacement curve in principle allows the contact modulus of the coating alone to be determined, provided the range of loads considered is so small that the substrate deformation can be disregarded. However, in practice the measurement errors for displacement and load become progressively larger as the contact load is reduced, making accurate determination of mechanical properties of the coating difficult.

Considering the importance of reliable characterisation requirement for progressively thinner coatings finding widespread use in industry, this study aims to review and refine the techniques using instrumented indentation for reliable property de-convolution of coated systems. We focus our attention on two properties of most relevance to the mechanical performance of coated systems, namely, contact modulus and hardness. These two parameters play a key role in controlling the deformation response of any surface, and affect such service properties as impact and erosion resistance, wear and fretting fatigue resistance, resistance to crack initiation and propagation, etc.

Article layout

This paper reviews the construction and application of an approach that aims to identify the underlying relationship between the observed contact response (displacement of the indenter tip under given applied load), the coating thickness and the mechanical properties of the substrate/coating combination. The purpose of the study is to provide fast and efficient means of extracting material property data from indentation tests.

When a suitably sharp indenter is used (i.e. the tip radius of curvature is several orders of magnitude smaller than the contact radius and the coating thickness), then the response of the system may be assumed dominated by inelastic processes, such as fracture or plasticity. In this case detailed modelling of the elasto-plastic deformation process is required to predict the load-displacement curve. However, as a first step we present a qualitative analysis based on the work of indentation that allows some predictions to be made. In particular, it is possible to use the work of indentation approach to identify the functional form of the dependence of the apparent system hardness on the indentation depth.

The function identified in this way is termed the indentation response function. It is used extensively to interpret a series of displacement-dependent hardness measurements in coated systems, The reliability of the fit to experimental data that is obtained using the indentation response function provides some confidence in its utility. In order to provide further confirmation for the use of the indentation response function its predictions are compared with both experimental measurements and the result of elasto-plastic finite element modelling of the indentation process.

It is interesting to pose a question about the cases when purely elastic response is observed. In such cases the solution is not sensitive to the peculiarities of the material non-linear response. In the present paper, a semi-analytical model is applied, based on the axially symmetric integral equation relating the stress distribution within the contact area, on the one hand, and the vertical displacement on the other. The model is implemented in a highly efficient numerical procedure, and the results used to formulate a special indentation response function for the coating/substrate composite. The function parameters can be tuned up to the observed response, and the ultimate coating-only properties be extracted in an easy way. In fact, the response to any applied load can then be routinely predicted.

It is remarkable that the indentation response function suitable for describing the hardness variation can also be used for predicting the apparent elastic contact modulus. The principal utility of this function is in the ability to de-convolve the unknown properties without invoking complex and computationally expensive inverse method procedures.

Hardness response

Background

Understanding the contact response of coated systems to the point whereby reliable quantitative models can be constructed both to explain and predict performance is a critical step in the selection and optimisation of coatings for particular substrates and applications. However, since the mechanical response of coated systems will vary with contact severity and scale, it is important to develop soundly-based models that allow the performance over a suitably wide range of scales to be successfully predicted. In this context, there is general agreement that at contact scales of dimensions less than the coating thickness (t), the coating dominates the system response, while at scales which are very large compared to t, the substrate dominates with a mixed response occurring at intermediate scales. However, many previous attempts to model this behaviour quantitatively have largely failed, generally because they encountered one of the two problems. The first has been the difficulty of obtaining good experimental data at contact scales less than the coating thickness -something necessary to enable models to be both constructed and tested over the necessary wide range of scales involved. The second problem has been the selection of an appropriate model that can be applied at all contact scales. Previous models have been of two types, system models which separate the measured contact response into contributions from coating and substrate without detailed treatment of the deformation mechanisms in each [START_REF] Buckle | Science of Hardness Testing and its Research Applications[END_REF][START_REF] Jonsson | [END_REF][START_REF] Vingsbo | Microindentation Techniques in Materials Science and Engineering[END_REF][START_REF] Sargent | [END_REF][START_REF] Burnett | [END_REF][7][8][9][10][11][12], and mechanistic models which consider the effect of a given deformation mechanism on the measured contact response [13][14][15]. System models become invalidated when major changes in deformation occur (e.g. cracking of coatings) that conflict with assumptions on which the model is based. Similarly, mechanistic modes have significant shortcomings when used to fit experimental data originating from systems displaying other modes of deformation. In general, most coated systems show mixtures of deformation modes and thus there is a clear need for a model which, while having soundly-based physical origins, is capable of dealing with such generalised responses. There is a further need for any such model to be sufficiently mathematically tractable as to allow straightforward fitting procedures to be used.

In the present overview we only provide short references to some of the most significant previous attempts at describing the contact response of coated systems. For a more detailed discussion of the historical aspect of the problem the reader is referred to the earlier paper (Korsunsky et al. [1]). Buckle [START_REF] Buckle | Science of Hardness Testing and its Research Applications[END_REF] thought of the observed hardness as the sum of the varying hardness values within the plastic zone (or "influence zone") at various depths beneath the indentation multiplied by some weighting factor. The model could explain work-hardening effects, but critically depended on the choice of weighting factors. Using area law-of-mixtures approach, Jonsson and Hogmark [START_REF] Jonsson | [END_REF] expressed composite hardness H c as follows:

H c = A f A H f + A s A H s ( 1
)
where A f and A s are the load-supporting areas of the film and the substrate, respectively, A the total projected contact area (A = A s + A F ) and H f and H s are the hardness values for the coating and substrate. Coating fracture in response to indentation is implied, i.e. that load support in the coating arises from unfractured material at the rim of the contact. Geometric consideration of the rim size led Johnsson and Hogmark to express composite hardness as

H c = H s + 2c t d -c 2 t d 2 (H f -H s ) ( 2
)
where t is the coating thickness, d the indentation diagonal, and c is a constant dependent on the indenter geometry. The model does not capture the indentation size effect (ISE) at small sizes, although ISE behaviour was later included in the form

H(d) = H 0 + k d ( 3
)
where k is a constant and H 0 is the hardness at very large loads was subsequently added to increase the applicability of the model [START_REF] Vingsbo | Microindentation Techniques in Materials Science and Engineering[END_REF].

The volume law-of-mixtures model originally suggested by Sargent [START_REF] Sargent | [END_REF] assumed plastic deformation response of both coating and substrate, and was subsequently extended variously by Burnett, Page, Rickerby and Bull [START_REF] Burnett | [END_REF][7][8][9]. Deforming volumes within coating and substrate, V f and V s , respectively, contribute to composite hardness (for H f > H s )as

H c = V f V H f + V s V χ 3 H s ( 4
)
where V is total volume (V = V f + χ 3 V s ) and H f and H s both include ISE behaviour of the form

H(d) = H 1 d n-2 (5)
where H 1 and n are the constants, H 1 a hardness value at standard indentation size, and n is the ISE index (equal to 2 if no ISE occurs) [10]. Here χ is a dimensionless factor which represents the modification to the deforming volume of the soft substrate due to the constraint caused by the presence of the coating. This is expected to be a strong function of the difference between the plastic zone radii in the coating and substrate. Based on an expression due to Lawn et al. [11], Burnett and Rickerby expected χ to take the form of a power law

χ = E f H s E s H f m (6)
where E f and E s are the Young's moduli of coating and substrate, H f and H s the hardness values of coating and substrate at the scale of the contact (i.e. incorporating any indentation size effects) and m is a constant which can be determined by fitting to experimental data. This approach is successful at separating hardness of coating from composite hardness when plasticity dominates, i.e. there is little or no fracture.

Thomas [12] attempted to use the fact that empirical data for H c often fits the form

H c = A + B d (7)
where A and B are constants, similarly to the form of indentation size effect equation used by Vingsboo et al., Eq. ( 3), and effectively treats the coating as a contribution to the ISE. Despite dubious physical basis this approach gives a convenient and simple formula

H f = A + B c -B s t (8)
where B c and B s are the constants measured for composite and substrate, respectively and A is the large-depth hardness (i.e. of the substrate). Note that Eq. ( 8) only provides a value for H f , rather than describing the system response over a range of scales, that is critically dependent on the empirical constant B c . McGurk and co-workers [13,14] proposed a model applicable coating cracking that identifies the dimensionless ratio:

H H o ∝ E c h 3 d 3 H o (9)
as being a controlling parameter for describing hardness enhancement. Here H is the increase in hardness resulting from the presence of the coating, H o the substrate hardness, E c the Young's modulus of the coating, h the coating thickness and d is the characteristic diagonal length of indentation.

In the next section we describe the reasoning behind an energy-based model that is suitable for different coating response types (cracking and plasticity), is applicable at a wide range of contact scales, is easy to fit, relies on few empirical fitting parameters, has some basis in physical reality and has the potential to be developed into a predictive design tool.

Work of indentation model

In an indentation experiment with a sharp pyramidal or conical indenter, the highest applied load P is almost invariably found to relate to the maximum penetration depth, δ, (measured after the removal of the load and thus after elastic recovery of any surface flexure but ignoring any small elastic recovery of the depth of the indentation itself) by

P = Hδ 2 κ , ( 10
)
where H is the measured hardness and is a parameter describing the indenter geometry. For Vickers indenters δ = d/7 where d is the indentation diagonal. Similar expressions can be derived for other indenter geometries. The total energy required to produce an indentation of depth δ is then given by

W tot = δ o P dx = Hδ 3 3κ (11)
This is termed the 'work of indentation' and, if measured (as can now readily be achieved with continuous recording indentation techniques (CRIT)) can be used to define an effective value of H which usefully describes the resistance to deformation over the penetration δ. The following derivation of the model will be based on the application of an 'inversion' of this formula, whereby the hardness is expressed in terms of this energy, i.e.

H = 3κW tot δ 3 (12)
Note that no assumptions need to be made about the way the energy was expended. The above expression may therefore be justifiably applied both to the coated system and its substrate separately. It is the comparison between these two situations that will be important.

It is clear that the total energy dissipated in deforming the composite will contain contributions from both the substrate and the coating. However, the partitioning of the energy expended between deformation of the coating and the substrate will vary depending on the scale of indentation. It is convenient to start by considering the case when the indenter penetration depth is very large compared with the coating thickness. Under these conditions the energy term relating to the plastic deformation of the substrate dominates. The energy contribution from the coating is very small in comparison, and one can expect to recover the substrate hardness from Eq. (12). Examining the case for progressively smaller indentation depths, it is clear that while a substrate term (given by Eq. ( 10)) will remain (but become increasingly smaller), a further term due to the coating will start to become increasingly significant and the energy balance will favour the coating more and more, i.e. the share of its contribution to the total energy will increase.

Total energy expenditure is composed of two parts, the plastic work of deformation in the substrate (W s ), and deformation and fracture energy in the coating (W f ), i.e.

W tot = W s + W f ( 13
)
Extensive experimental evidence exists on the indentation of hard-coated systems. In developing the present approach, we have reviewed the similarities between deformation and fracture modes observed in a variety of materials used for substrate and coating, and over a range of coating thicknesses and deposition techniques. These observations suggest, that while plastic deformation usually occurs in both coating and substrate, and is primarily responsible for the permanent impression left by the indenter, fracture also occurs at larger loads and indenter displacements (typically, exceeding one tenth of the coating thickness where substrate deformation starts to assume significance [15]). Certain similarities also exist for all hard-coated systems, namely, that coating deformation is most often dominated by cracking, which occurs at the indentation apex, along the edges of the indenter pyramid, and in the form of circumferential cracks around the indentation perimeter. At low loads, fracture is localised along the indenter edges, with possibly one or more peripheral cracks being present. With increasing load, multiple cracking starts to dominate, producing a web pattern concentric around the indenter tip.

In the course of model development, plasticity and fracture must be considered separately, although the resulting models show some convergence, and are combined into a single master formula. This formula was found to be as follows [16]

H c = H s + H f -H s 1 + (β/β 0 ) X (14)
where the fitting parameters H s , H f -H s , β 0 and X are determined by fitting to the experimentally determined variation of H c with β.

Experimental validation

A series of hardness measurements was carried out using conventional Vickers pyramidal testers within the load range approximately 10-200 N (1-20 kg), microhardness testing (down to 0.5 N) and nanoindentation (down to 5 mN). The materials studied included: diamond-like carbon coatings, arcdeposited single and multi-layer nitrides and carbo-nitrides of titanium, vanadium, zirconium and niobium, and electroplated and electro-less plated nickel coatings. The substrates used were tool steels and copper [17].

Fig. 1(a) contains hardness plotted against the indenter displacement for sample M053 of McGurk [14] of 2.8 m NbN on M304. It is clearly unsuitable for interpreting the data and validating any models. Fig. 1(b) re-plots the same hardness data for M053 in partially non-dimensional form. Now, the relative indentation depth (β = δ/t) normalises the indentation depth with respect to the coating thickness, and logarithmic axis is used for this parameter.

It is now apparent from Fig. 1(b) that there is a transition in behaviour which occurs approximately between the points β ∼ 1 and β ∼ 0.1, i.e. for indentation depths between the full coating thickness and down to one tenth of the coating thickness. At depths approximately equal to the coating thickness the hardness reduces asymptotically to the near-substrate value of hardness, whereas for depths shallower than one tenth of the coating thickness the coating hardness "levels off" at a high hardness value, which is likely to be characteristic of the coating itself.

In order to test the capability of the indentation response function, Eq. ( 14), to describe the full range of hardness variation with indentation depth, a systematic study was undertaken of hardness of nickel coatings of different thicknesses on copper substrates. Copper blanks were used as cathode of a deposition bath with the anode made from 99.9% commercial purity nickel. A conventional Watts deposition bath formulation was used, with the solution consisting of 300 g nickel sulphate, 28 g of sodium chloride and 40 g of boric acid, dissolved in 1 l of distilled water. The deposition was conducted at a temperature of 55 ± 1 • C. The current density was regulated using a constant current power supply, so that the thickness of deposition could be accurately controlled for each specimen. The electrodes were mounted a fixed distance apart, and a non-conductive shield was used to improve the directionality of the current flow between them, since the uniformity of the coating thickness depends on the current density distribution.

The coatings were deposited to the nominal thickness of 10, 20, 30, 40 and 50 mm. The resulting thickness differed slightly from the target values due to the slight variations in the deposition conditions. The actual thickness was measured using ball cratering and metallographic cross sectioning.

The data for each individual coating was first plotted as composite hardness versus indentation depth. The plots clearly possessed a common shape, so in attempt to find a master curve all data was placed on a single graph with the logarithm of relative indentation depth, β, used as abscissa. The resulting combined plot is shown in Fig. 2. It clearly demonstrates that Fig. 2. A combined plot of apparent system hardness vs. the relative indentation depth for a series of Watts electroplated nickel coatings on copper substrates. The coating thickness ranged between 10 and 50 m.

the overall behaviour of the systems containing coatings of different thickness covering a wide range between 10 and 50 m can nevertheless be very well represented by a single function. The fit curve using indentation response function, Eq. ( 14),i s included in the plot.

Finite element simulation

Finite element analysis was used to gain an insight of the distribution of the mechanical fields in the coated sample during the indentation process. The indenter was considered as a cone with a tip angle of 70.3 • and both the indenter and the layered substrate have been treated as axially symmetric. The choice of this angle is justified by the fact that it gives an equivalent displaced volume equal to that for the Vickers four-sided or Berkovich three-sided pyramid.

The indentation process was simulated using CASTEM2000 FE package, and the analysis was performed under the assumption of large displacements and small rotations coupled with mesh updating after each load increment.

From the point of view of interpretation it is convenient to use the term 'small indentation' to refer to the range of relative indentation depths between zero and unity, and to talk about 'large indentation' for relative indentation depths exceeding unity.

To attain suitable fidelity of FE simulation results at all depths of indentation, from shallow to deep, and to cover at least a decade of scales, three different ranges of relative indentation depth were considered, and different specific meshes were generated for each case ('shallow indentation mesh' for relative indentation depths smaller than 0.1-0.2, 'medium indentation mesh' for relative indentation depths between 0.1-0.2 and 0.4-0.5, and 'deep indentation mesh' for relative indentation depth larger than 0.5). Meshes consisted of approximately 4500 linear elements. The choice of the type of element is imposed by the contact modelling routine used by CASTEM2000. It has been shown that this choice does not influence the accuracy of the final results. In order to improve the speed of the computation, an elastic super-element was used to represent the region far removed from the contact zone; the elasticity assumption for this zone was verified a posteriori. The super-element reduced the computational time by about 30%. The indenter was considered as rigid and the layer and the substrate were considered to be elasto-plastic and obey the von Mises yield criterion with non-linear isotropic hardening.

The accumulated plastic strain predicted by the model for the substrate and the coating is illustrated in Fig. 3. A series of finite element calculations was used to determine the variation of the composite hardness as a function of the relative penetration depth, in order to provide a comparison between the predictions of the model illustrated in Fig. 3, and the observed response shown in Fig. 2. The value of apparent hardness was computed from the definition, i.e. by averaging the vertical stress over the entire contact patch, and is shown in Fig. 4.

Contact modulus response

Background

Similarly to the hardness, the apparent contact stiffness of the coating/substrate combination undergoes a transition between the coating-only value at very low loads and the substrate-dominated value under high loads. Determination of the behaviour at intermediate values of the relative indenta-tion depth β requires detailed analysis of the complete problem, taking account of the specific constitutive behaviour of the components. This task of considerable difficulty usually requires the application of laborious numerical methods. It is possible to prove, however, that in the elastic case the solution must depend solely on the combination of the elastic properties of the coating and substrate, the relative indentation depth, and the indenter shape. It then possible to pose the question about finding the transition function which describes the variation with indentation depth of the apparent elastic modulus of the system. This parameter is deduced from the load-displacement curve using the formulae for homogeneous substrate. The solution is required for arbitrary indenter shape, but more specifically we are particularly interested in the important cases of a flat punch, and of conical and parabolic indenter profiles. Analysis can further include the consideration of punch shape imperfections, such as a blunted (rounded) tip of a conical indenter.

The methods of the classical theory of elasticity employed for the solution of elastic contact problems are reviewed in many monographs. The procedure usually involves two essential steps: the derivation of the solution for a concentrated force on the boundary, and inversion of the pair of dual integral equations for the boundary tractions and displacements, written on the basis of this solution.

This approach is employed in the present paper in order to study the dependence of the apparent elastic modulus of a coated system subjected to elastic indentation on the indenter displacement, indenter shape, and the coated system properties. For the purposes of analysis, it is convenient to consider the variation of apparent modulus as a function of the relative indentation depth (contact depth normalised with respect to the coating thickness). The analysis demonstrates that the predicted variation can be very well described by a function belonging to the same family parametric functions introduced for in the previous paragraph. This result allows the response at any load and indenter displacement to be predicted without repeating the calculation for any combination of the layer/substrate properties.

Indenter shape imperfections are well known to impose limitations on the resolution with which the substrate and coating properties can be determined. The present approach allows these limits to be explored quantitatively, with important implications for experimental data collection and interpretation.

The material parameter that directly governs the indention response of elastic homogeneous substrates is referred to as the contact modulus, also known as the plain strain modulus, and is defined as E * = E/(1 -ν 2 ). It can be expressed directly as a function of the applied load P, the punch shape parameter f, the displacement of the indenter d, and the contact radius a H (d)as E * = P/2fda H (d).

The general solution techniques for frictionless elastic contact mechanics problems for homogeneous substrates were addressed by Sneddon using integral transform methods. The functional dependence of the contact radius a on the indenter displacement for important indenter geometries, such as the cone and sphere, has the form of a power law. This dependence, a = a H (d), can be used to eliminate the contact radius from the expression for the contact modulus. Consequently, E * can be defined in terms of the applied load and indenter displacement. The contact mod is seen to play in the case of the elastic substrates a role similar to that of the hardness in the case of plastic substrates. Thus, for a coated system the apparent contact modulus is defined as:

E * = P 2fda H (d) (15)
Coated systems display some important differences from the behaviour of homogeneous substrates. Firstly, for coated systems the apparent contact modulus varies with the indentation displacement d. Secondly, the solution for the contact radius as a function of the applied force or indenter displacement is not available for the case of coated systems in closed form, and expressions for the apparent contact modulus cannot be obtained.

Integral equation solution

To analyse a series of indentation problems and to check the validity of the proposed indentation response function, we compute the solutions of elastic indentation problems using a boundary integral approach proposed by Yu et al. [18]. The fundamental advantage of this approach is that both the pressure distribution under the indenter and the displacement profile are assumed to be unknown a priori, but are related to each other through a pair of coupled boundary integral equations. By the use of an elegant transformation, the pair can be turned into a single Fredholm integral equation for a single unknown potential density function. The resulting integral equation is regular, and can be inverted very efficiently using collocation technique.

The numerical implementation of the system of equations described above has been programmed in Mathematica. The solution was found through an iterative procedure. For any chosen contact radius a, a Gauss-Legendre quadrature rule was used, and the roots of Legendre polinomials were also employed as collocation points for internal representation of functions and the kernel. This approach allows the boundary integral equation to be transformed into a linear algebraic system and easily solved.

For smooth punches and incomplete contact conditions, the correct value of the contact radius was found by imposing the condition of vanishing contact pressure at the edge of the contact region. Once again, an iterative procedure was used with the dichotomic method, starting with an initial value corresponding to the contact radius in the case of a homogenous substrate.

In order to speed up the numerical computations, one of the memory control options of Mathematica was used (the command form was f[x ]:=f[x]=...). This avoids repeated evaluation of the same expression, for example, encountered in the integral equation kernel. We also used the automatic compilation procedure in order to speed up the evaluation of functional calls. The accuracy of the solution procedure was tested by comparison with the well-known analytical solutions for homogenous substrates.

Fig. 5(a) illustrates the variation predicted by the model with a conical indenter of the apparent contact modulus for a coated system with the following parameters: ν s = 0.3, ν f = 0.2, E s = 100 GPa, E f = 500 GPa. It is clear that the indentation response function (continuous lines) provides an accurate representation of the observed variation. At the same time, once the fit parameters are determined, prediction of the apparent contact modulus under any load requires very little calculation compared with the detailed numerical procedure described above.

A further utility of the indentation response function approach is in determining the limits of applicability for various assumptions routinely made during indentation experiments. For example, the tip of a pyramidal (Vickers or Berkovich) or conical indenter is often assumed to be sharp. However, the real observed response may be strongly affected by tip blunting. Interpretation of the results assuming the tip is sharp is likely to lead to significant errors. Using the modelling approach described above we have demonstrated that large inaccuracies arise unless the tip radius is less than about 1/50th of the coating thickness. If this requirement is not observed, the coating stiffness may be overestimated by many tens of percent. This warning is particularly relevant for the cases when the coating is very thin, as is often the case for diamond-like carbon deposits and other films used for hardness and stiffness improvement. Fig. 5(b) serves as the final illustration of the success of the model in describing the experimentally observed variation of contact modulus. The markers show the data of McGurk [14] for Berkovich nanoindentation of 2.8 m NbN coating on M304 stainless steel, and the curve shows the model fit to the measurements, exhibiting excellent agreement.

Discussion

In this paper an overview has been presented of an approach to the property determination of coated systems through the systematic use of the indentation response function.

Both plasticity-dominated and elastic response cases were considered, and experimental data from a very wide range of industrially relevant coated systems was described. In many systems coating cracking was observed. The presence of this deformation mode was found to affect the parameters of the indentation response function, in particular, the exponent X in Eq. ( 14) [16]. However, even in the presence of cracking the overall response of most systems was still sufficiently well described by the indentation response function. It may be expected, however, that if a drastic change in the deformation mechanism were to occur at a particular load, it will be highlighted by a deviation of the plot from the expected curve.

The presented consideration demonstrates the usefulness of the proposed approach for efficient analysis of coated system properties.

Fig. 1 .

 1 Fig. 1. (a) A conventional plot of apparent system hardness as a function of indenter displacement for a thin ceramic coating on M304 tool steel. (b)The same data re-plotted using a logarithmic scale for the normalised indentation depth β = δ/t, and displaying the best fit curve obtained using the indentation response function, Eq. (14).

Fig. 3 .

 3 Fig. 3. A map of accumulated plastic strain in the coating and substrate during indentation by an axially symmetric cone punch.

Fig. 4 .

 4 Fig. 4. A comparison between the experimental hardness measurements (grey circles) and the predictions of elasto-plastic finite element analysis (black circles) for Watts electroplated nickel coatings (upper curves) and electro-less nickel coatings (lower curves) on copper substrates.

Fig. 5 .

 5 Fig. 5. (a) Predicted variation of apparent contact modulus using a cone indenter, where cases of perfect bonding and frictionless interface between coating and substrate were both considered, as indicated. Curves indicate best fit obtained using the indentation response function. (b) Experimentally measured contact modulus data, (markers) and indentation response function fit (curve) for Berkovich nanoindentation of 2.8 m NbN coating on M304 stainless steel. Data courtesy of McGurk [14].