Christophe Noël
email: christophe.noel@inrs.fr

phalanges in vibration -Test bench validation and parameter study

Keywords: hand-arm vibration, energy dissipation, Raynaud's phenomenon, phalanx dynamic stiffness, soft tissue damping

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Sustained exposure to excessive vibration levels may cause a number of pathological disorders of a vascular, osteo-articular, or musculo-tendinic order [START_REF] Bovenzi | Medical aspects of the Hand -Arm vibration syndrome[END_REF][START_REF] Griffin | Handbook of human vibration[END_REF][START_REF] Matoba | Pathophysiology and clinical picture of hand-arm vibration syndrome in Japanese workers[END_REF]. These pathologies are generically referred to as vibration syndrome or hand-arm vibration syndrome (HAVS).

Multiple epidemiological studies [START_REF] Åström | Hand-arm vibration syndrome (HAVS) and musculoskeletal symptoms in the neck and the upper limbs in professional drivers of terrain vehicles-A cross sectional study[END_REF][START_REF] Chetter | The hand arm vibration syndrome: A review[END_REF][START_REF] Kattel | The effects of rivet guns on hand-arm vibration[END_REF][START_REF] Narini | Une stratégie de modélisation multi-échelles pour la prédiction de certains effets des vibrations sur le réseau vasculaire digital[END_REF] supported by physiological and histological observations of the acute effects of hand-arm vibration [START_REF] Griffin | Frequency-dependence of Psychophysical and Physiological Responses to Hand-transmitted Vibration[END_REF][START_REF] Krajnak | Frequency-dependent Effects of Vibration on Physiological Systems: Experiments with Animals and other Human Surrogates[END_REF] have shown that these disorders are partly related to the frequencies emitted by the machine held by a worker and to other co-factors, such as working posture and pushing and gripping forces.

Measurements of the biodynamic responses of the hand-arm system enable the analysis of the influence of such factors, with the aim of better understanding the mechanisms that induce these vibration disorders. For several years, multiple research studies have focused on estimating the biodynamic responses of the upper limb; most of these have involved selecting mechanical impedance (Driving Point Mechanical Impedance, DPMI) as the biodynamic response of subjects gripping a vibrating handle [START_REF] Aldien | Influence of hand-arm posture on biodynamic response of the human hand-arm exposed to zh-axis vibration[END_REF][START_REF] Burström | Comparison of different measures for hand -arm vibration exposure[END_REF][START_REF] Gurram | Mechanical impedance of the human hand-arm system subject to sinusoidal and stochastic excitations[END_REF][START_REF] Marcotte | Effect of handle size and hand-handle contact force on the biodynamic response of the hand-arm system under zh-axis vibration[END_REF]. These types of measurement very often provide overall information on the hand-arm system.

They are better suited to the study of bone-joint or muscle-tendon pathologies than to that of vascular disorders situated in the finger (digital) blood vessels. Therefore, in order to focus more specifically on vascular pathologies, such as Vibration White Finger or Raynaud's syndrome, it is necessary to collect accurate knowledge of the biodynamic response of finger-transmitted vibration [START_REF] Griffin | Handbook of human vibration[END_REF]. There have been several attempts to obtain more local experimental data with the help of an instrumented handle by measuring, for example, the distributed impedance in the palm-fingers system [START_REF] Dong | Frequency weighting derived from power absorption of fingers-hand-arm system under z(h)-axis vibration[END_REF](Dong et al., , 2005a)), or by plotting vibration transmissibility diagrams on the dorsal face of hands [START_REF] Noel | Acceleration mapping and local biodynamic response of hand gripping a vibrating handle by using laser scanning vibrometer[END_REF]. These methods are likely to accurately provide overall information on the actual use of hand-held rotary machines. However, it has proven difficult to fully determine the relative contribution of each influencing factor (contact area, push/grip forces, posture, input vibration amplitude, etc.) for a high frequency condition, and some findings even contradicted each other [START_REF] Burström | The influence of biodynamic factors on the mechanical impedance of the hand and arm[END_REF][START_REF] Gurram | Mechanical impedance of the human hand-arm system subject to sinusoidal and stochastic excitations[END_REF][START_REF] Marcotte | Effect of handle size and hand-handle contact force on the biodynamic response of the hand-arm system under zh-axis vibration[END_REF]. However, a number of studies [START_REF] Adewusi | On the discrepancies in the reported human hand-arm impedance at higher frequencies[END_REF]Dong et al., 2008a) concluded that the biodynamic responses of the hand-arm-finger system measured from instrumented handles are often subject to errors at high frequencies (> 300 Hz). Alternatively, the results are not reproducible between laboratories owing to the measuring bias introduced by the vibrating handle itself at a high frequency (producing problems of mass cancellation at high frequencies, loss of signal coherence, vibration noise related to the handle assembly, etc.).

To potentially overcome the previous drawbacks, an alternative might consist of directly measuring the local biodynamic responses at the fingertips by using specific devices. For a number of years, the assessment of the local mechanical impedance of the fingers has been the objective of a number of different research fields, such as of course the pathological effects of finger-transmitted vibration [START_REF] Lundström | Local vibrations-mechanical impedance human hand's glabrous skin[END_REF][START_REF] Mann | Effect of contact conditions on the mechanical impedance of the finger[END_REF], but also the field of human tactile sensitivity [START_REF] Moore | Measurement of Specific Mechanical Impedance of the Skin: Effects of Static Force, Site of Stimulation, Area of Probe, and and Presence of a Surround[END_REF][START_REF] Moore | A Survey of the Mechanical Characteristics of Skin and Tissue in Response to Vibratory Stimulation[END_REF][START_REF] Wiertlewski | Mechanical behavior of the fingertip in the range of frequencies and displacements relevant to touch[END_REF], the design of haptic systems [START_REF] Chai-Yu | Direct Measurement of Index Finger Mechanical Impedance at Low Force[END_REF][START_REF] Kern | Studies of the Mechanical Impedance of the Index Finger in Multiple Dimensions[END_REF], and determining mathematical models of the skin dynamic response [START_REF] Gulati | Determination of mechanical properties of the human fingerpad in vivo using a tactile stimulator[END_REF][START_REF] Hajian | Identification of the mechanical impedance at the human finger tip[END_REF]. The common procedure was to locally vibrate the fingertip with a small-sized probe in contact with the skin for two main phalanx boundary conditions: i) clamped fingernail [START_REF] Gulati | Determination of mechanical properties of the human fingerpad in vivo using a tactile stimulator[END_REF][START_REF] Lundström | Local vibrations-mechanical impedance human hand's glabrous skin[END_REF][START_REF] Moore | Measurement of Specific Mechanical Impedance of the Skin: Effects of Static Force, Site of Stimulation, Area of Probe, and and Presence of a Surround[END_REF], and ii) free fingernail [START_REF] Hajian | Identification of the mechanical impedance at the human finger tip[END_REF][START_REF] Mann | Effect of contact conditions on the mechanical impedance of the finger[END_REF][START_REF] Wiertlewski | Mechanical behavior of the fingertip in the range of frequencies and displacements relevant to touch[END_REF]. When measuring the local finger biodynamic response, it was easier to carefully control the influencing parameters, such as the probe surface contact and the static-preload, than when assessing the distributed handfinger system with an instrumented handle. The disadvantage of this method is that it does not reproduce the use of hand-held rotary machines accurately. However, at a highfrequency condition, it is reasonable to expect the trends of the biodynamic responses to be comparable for whichever test method used, because these trends reflect the mechanical behaviour of the local soft tissues, which may be weakly dependent on those of the entire hand-finger system [START_REF] Wu | Analysis of the point mechanical impedance of fingerpad in vibration[END_REF]. Several studies have been carried out to determine in a nearly comprehensive way the influence of a number of parameters on the local finger biodynamic response, but some of these studies have produced contradictory findings, especially regarding the preloading effect [START_REF] Lundström | Local vibrations-mechanical impedance human hand's glabrous skin[END_REF][START_REF] Moore | Measurement of Specific Mechanical Impedance of the Skin: Effects of Static Force, Site of Stimulation, Area of Probe, and and Presence of a Surround[END_REF].

Moreover, some researchers pointed out certain potential measurement inconsistencies leading to an overestimation of the inertial effects (Dong et al., 2004b;[START_REF] Wu | Analysis of the point mechanical impedance of fingerpad in vibration[END_REF].

The purpose of our study was to describe, validate, and use in parameter analysis form a new experimental vibration test bench for measuring the local biodynamic responses of the distal phalanx. This test bench was designed to provide accurate, reliable, reproducible results at frequencies between 20 Hz and 500 Hz. We aimed at characterizing the mechanical dynamic behaviour of fingertip soft tissues at a high-frequency (> 50 Hz) condition by reducing the effects of other disturbing factors, such as muscle activity and inertial forces produced by the bones or fingernail, as much as possible. Inertial effects were minimized by clamping the fingernail [START_REF] Wu | Analysis of the point mechanical impedance of fingerpad in vibration[END_REF]. To avoid muscle activity as much as possible, the static preloading was performed in a passive way by compressing the fingertip between its support and the probe instead of actively pressing the indenter by using the muscles and tendons of the finger. We first describe the experimental setup, its related metrological equipment, and the signal processing parameters. The deconvolution method for driving the electrodynamic shaker in acceleration is presented. A phase calibration procedure was applied to cancel phase mismatch due to electronic devices. The phase angle accuracy was quantified. Then, the measurement bench was validated by comparison with the results obtained from a tension-compression machine and a reference dynamic mechanical analyser. Finally, a measurement study was performed on 20 subjects. The study parameters were the vibration amplitude and quasi-static preloading. The influence of the vibration frequency on the dynamic stiffness or the mechanical power dissipated by the distal phalanx was analysed before focusing on the effects of the preloading and level of excitation vibration. These results were then examined from a rheological standpoint in an attempt to provide a number of additional physiological explanations.

Method

Apparatus

The experimental setup for measuring the static and dynamic stiffnesses of preloaded distal phalanges is shown in Fig. 1. The seated subjects place their right forearm on an aluminium platform (Fig. 1 item k). This holder of adjustable height is fitted with four lateral stops with adjustable positions at the elbow and wrist. The purpose of these stops is to ensure appropriate positioning of the subject's forearm and prevent intentional or unintentional movements that could potentially disrupt the measurement. The middle, ring, and little fingers grip a 30 mm diameter plastic cylinder screwed to the holder and adjustable to the subject's morphology.

The index finger distal phalanx is adhered (using double sided tape) to the inside face of a curved steel support (Fig. 1 item i). This support is screwed to a piezoelectric force sensor (model B&K 8200) (Fig. 1 item c) rigidly fixed to a steel block. The force sensor is used to estimate the output effort between the fingernail and its support. A piezoelectric accelerometer (model B&K 4705B002) (Fig. 1 item b) with a low measurement range (70 m.s-2 sine wave peak) is positioned near the force sensor to check that the vibration level is sufficiently low so as not to disturb the output force measurement. The distal phalanx is vibrated using an electrodynamic shaker system (model B&K 4809) (Fig. 1 item e) generating a 10 to 20 kHz bandwidth signal and sustaining sine wave peak dynamic forces of a maximum of 45 N. The electrodynamic shaker is fixed to a support micrometer that is controlled manually (Fig. 1 item h). This allows the quasi-static preloading of the distal phalanx by moving the 7 mm diameter cylindrical indenter (Fig. 1 item j). The indenter is screwed to an impedance head (model B&K 8001) rigidly mounted to the vibratory shaker shaft (Fig. 1 item d). The impedance head sensor enables the measurement of both the input force applied by the cylindrical probe to the skin and the input acceleration. The compression distances used to estimate the static stiffnesses from the force/displacement curves were measured using a laser telemeter (model ILD 1700-200 from Micro-Epsilon) with a high resolution of 12 m (Fig. 1 item a). The telemeter laser beam impacted a 4 mm thick aluminium disk (Fig. 1 item f) fixed to the shaker moving shaft.

The data acquisition, control, and computing operations were performed using MATLAB software with specific modules. A conceptual diagram of the metrological equipment with the different interconnections is shown in Fig. 2.

Biodynamic responses

The dynamic behaviour of the distal phalanges was characterized by the input and output dynamic stiffnesses, ( ) and ( ) computed conventionally using the cross-spectrum method:

( ) ( ) ( ) (1) ( ) ( ) ( ) (2)
in which ( ) is the cross-power spectral density between the input force ( ) and the acceleration ( ), both measured with the impedance head;

( ) is computed from the output force ( ) acquired with the back force sensor and acceleration ( ). ( ) is the power spectral density of acceleration ( ). was the frequency and t the time. The crossspectra and auto-spectra were calculated for a 2048 Hz sampling frequency using Welch's averaged periodogram method and Hanning windowing, a 50% overlap and a 4 Hz spectral resolution. The recorded time signal duration was 5 s. Time signal coherences were measured to ensure that their values were close to unity in the frequency band used in the study. For the input dynamic stiffness, the mass cancellation method [START_REF] Håkansson | Bias errors in mechanical impedance data obtained with impedance heads[END_REF] was implemented in the spectral domain to overcome added mass effects induced by the indenter and the sensor seismic mass. The stiffnesses were chosen to characterize the biodynamic response instead of the more conventional mechanical impedance, because it is possible to explain the findings in the framework of experimental rheology, in which the stiffness is often used to characterize materials. This is also the reason for the output stiffness to be preferentially selected in the analysis of the findings.

The fingernail was clamped so that the test bench would simulate classical rheological experimental devices as closely as possible. We therefore adopted the vocabulary used in rheology. The real part of the dynamic stiffnesses was termed the storage modulus and the imaginary part the loss modulus.

In addition to the input and output dynamic stiffnesses, the mechanical power exchanged between the cylindrical indenter and distal phalanx could be estimated. If we assume that the force exerted by the indenter on the phalanx is exerted at a point, the mean power generated by this force and applied to the distal phalanx can be conventionally expressed by frequency-based integration of the cross-spectral power density ( ) between the force and velocity at the point of application:

∫ ( ) (3)
The expression shown in Eq. (3) can easily be rewritten with the input dynamic stiffness ( ) and acceleration autospectrum ( ):

∫ [ ( ) ( ( ))] ( ) (4)
In the special case of pure harmonic excitation with a frequency and an amplitude , the mean power generated by the external forces becomes:

[ ( ) ( ( ))] (5)
Since the kinetic energy variation is zero over a vibration cycle, the mean power generated by the external forces is, in absolute terms, equal to the mean power of the internal forces. Eq. ( 5) thus represents the mean power dissipated by the phalanx during a vibration cycle. This dissipated power is, of course, controlled not only by the excitation amplitude, but also by the imaginary component of the input dynamic stiffness, or similarly by the real component of the input DPMI, since these two quantities are deduced from each other by division by a factor of √ . The influence of different parameters (frequency, amplitude, preloading) on energy dissipation was investigated in previous studies by the real component of the DPMI (or equivalently by the imaginary component of the stiffness) either for the local [START_REF] Moore | Measurement of Specific Mechanical Impedance of the Skin: Effects of Static Force, Site of Stimulation, Area of Probe, and and Presence of a Surround[END_REF][START_REF] Wiertlewski | Mechanical behavior of the fingertip in the range of frequencies and displacements relevant to touch[END_REF] or the global [START_REF] Besa | Characterisation of the mechanical impedance of the human hand-arm system: The influence of vibration direction, handarm posture and muscle tension[END_REF][START_REF] Burström | Comparison of different measures for hand -arm vibration exposure[END_REF] biodynamic responses of the fingers-hand-arm system. However, we introduced the so-called acceleration power kernel written ( ) and expressed as:

( ) ( ) ( ( )) (6)
for characterizing the properties of energy dissipation. The acceleration power kernel was chosen by analogy with other studies (Dong et al., 2008b[START_REF] Dong | Frequency weighting derived from power absorption of fingers-hand-arm system under z(h)-axis vibration[END_REF], where it was used to propose a vibration-power-absorption-based weighting alternative to the existing ISO-5349 filter (ISO 5349-1:2001(ISO 5349-1: , 2001)). Indeed, Eqs. 4 and 5 clearly show that power absorption is proportionally linked to the square of acceleration through the acceleration power kernel.

As for the real component of the DPMI or the imaginary component of the stiffness, the acceleration power kernel enables the characterization of the power dissipation independently of the acceleration spectrum applied to the fingertip. However, it explains the dissipative properties of the fingertip assuming the same level of input acceleration.

Based on previous studies (Dong et al., 2005c(Dong et al., , 2004a)), where a new ISO-5349 filter was proposed from the vibration power absorption density in the whole finger, we could use the acceleration power kernel measured locally at the fingertip as a new candidate for ISO-5349 weighting under high frequency.

Generation of required target input acceleration

The shaker and its power amplifier (Fig. 2) were assumed to behave as a linear system characterized by impulse response ( ) between voltage ( ) and acceleration ( ). The aim was to establish a method of computing the voltage ( ) that would produce the required target acceleration ( ) at the system output. To solve this problem, we implemented a filtering method based on a restricted least squares regression, also termed the regularization method [START_REF] Norcross | Evaluation and alternative regularization of audio equalization techniques[END_REF]. The Tikhonov regularization method was applied to compute the voltage ̃( ), which is expressed as:

̃( ) ( ) {‖ ( ) ( ) ( )‖ ‖ ( ) ( )‖ } (7)
in which is the argument of the minimum, ‖ ‖ is the Euclidean standard, is a positive scalar parameter, and ( ) is a function to be defined.

Phase angle measurement accuracy

To minimize the measurement errors of both the complex dynamic stiffness and mechanical dissipated power, it is necessary to reduce the phase angle mismatch generated by the acquisition system by as much as possible. These phase angle errors are linked to the sensors (negligible in this case owing to the high resonance frequency of the sensors), the channel-to-channel phase mismatch of the input acquisition module, and the phase shift caused by the charge amplifiers (intrinsic electronic, digital, or analogue filtering, etc.). The phase calibration was a two-step procedure (Fig. 4) consisting of: i) measuring the channelto-channel phase mismatch, and ii) evaluating the phase shift of the charge amplifier by using a standard capacitor to simulate an ideal charge input. (Noel, 2015).

Test bench validation

To validate the measurements of the static and dynamic stiffnesses, the INRS test bench results were compared with those provided by reference experimental systems: a quasistatic tension-compression machine (Wolpert Testatron 20 TTZ) and a viscoanalyzer (Metravib VA 2000). The tested sample comprised a 15 mm diameter, 14 cm high polyurethane foam cylinder with mechanical characteristics approximating those of the index finger phalanx. Tests were conducted using the tension-compression machine at a displacement speed of 1 mm.min -1 (quasi-static load can be considered). The forcedisplacement curves subsequently obtained are shown in Fig. 6. The two forcedisplacement curves exhibited similar shapes. The INRS test bench results were higher than those obtained from the reference tension-compression machine from a compression of 1 mm upwards. The relative error between these two measurements was 9.2%, 6.5%, and 8.5% at compressions of 2 mm, 4 mm, and 6 mm, respectively. The difference between the two devices was linked to the fact that, for the tension-compression machine, the compression speed was very low (it can be considered to exhibit quasi-static behaviour), while for the INRS test bench, the compression speed ranged between 20 and 40 mm.min -1 (caused by the time taken by the experimenter to compress the finger). Therefore, because of the relaxation phenomena associated with the viscoelasticity properties of the finger and the higher compression speed of the INRS test bench than that of the tension-compression machine, the stress measured did not reach its quasi-static low limit. This could explain why the force-displacement curves measured with the two different devices were not identical. The dynamic stiffness of the polyurethane foam was then measured using a commercial viscoanalyzer. The material had previously been preloaded at a 15% compression ratio (ratio of compressed length to initial length), then vibrated by a sine wavefront generated in 10 Hz increments between 20 and 100 Hz. This viscoanalyzer was coupled with a 0. with commercial dynamic mechanical tester.

Subjects and experimental set-up

In accordance with the French law, the study protocol was reviewed and approved by the French state organism allowing biomedical research on human subjects.

Description of subject cohort

Static and dynamic stiffnesses were measured on 20 healthy, non-smoking, subjects (8 women and 12 men) aged between 19 and 39 years, and with a normal skin sensitivity threshold for their age (monofilament size 3.61 in Semmes-Weinstein test). Their distal phalanx thicknesses were measured using the laser telemeter (Fig. 1 item a). Their distal phalanx widths and index finger lengths were measured from three-dimensional images obtained by scanning each subject's hand with a laser scanner. Anthropometric markers common to all subjects and easily identifiable on these images were established in order to standardize the dimensional measurement. The distal phalanx width was defined as the width in the frontal plane (parallel to the palm of the hand) of the distal interphalangeal joint, and the index finger length was defined as the distance between the end of the finger and the metacarpophalangeal joint. The forearm volume was evaluated by simply measuring the quantity of water displaced when the subject placed his/her forearm up to the elbow in a water-filled container. Table 1 summarizes the different average anthropometric characteristics recorded for the subjects.

Experimental procedure

The quasi-static stiffnesses were computed as the derivatives of the output forces with respect to the cylindrical indenter displacements measured with the laser telemeter.

The dynamic stiffnesses (Eq. (1) and Eq. ( 2)) were estimated for the following two experimental parameters: the rms values of the cylindrical indenter input acceleration and the quasi-static preloading. The input accelerations were generated as described in Section 2.3. Their spectral shape is shown in Fig. 3 and their rms values are listed in Table 2.

The static preloading state can generally be defined in a variety of ways, such as: i) the cylindrical probe iso-displacement, ii) the iso-force applied by the probe on the fingertip (this is the definition most often selected in other studies (Dong et al., 2005b;[START_REF] Lundström | Local vibrations-mechanical impedance human hand's glabrous skin[END_REF][START_REF] Lundström | Mechanical impedance of the human hand-arm system[END_REF])), and iii) the iso-compression rate relative to the fingertip thickness. The methods that would involve specifying the preloading in relation to absolute compressions or forces were not selected, because a similar compression or force would produce, in two different subjects, two distinct forces or compressions, whose difference was not necessarily due to variations in mechanical properties, but to initial geometrical differences. For example, an 18 mm thick phalanx compressed by 3 mm would not be in the same static preloading state as a similarly compressed 11 mm thick phalanx.

Moreover, a level of preloading expressed as a degree of compression with respect to the total phalanx thickness again would not define a reference state of preloading common to all subjects (for a given fingertip thickness, the flesh and bone thicknesses could be different for different subjects). Therefore, as the reference state of preloading we adopted the operating points ( ) on the force -displacement curve, which provided the same static stiffness for all subjects . The static stiffness is the macroscopic resultant of the load and deformation fields. Hence, we considered it a relevant indicator for defining a reference state of preloading. We could therefore compare the dynamic stiffnesses measured for each subject for an equivalent static load in an iso-stiffness sense.

Table 2 summarizes the 20 parameters (5 accelerations × 4 states of preloading) used for the dynamic stiffness measurements.

Results

Static stiffness

We plotted static stiffnesses as a function of compression for five subjects in Fig. 8 (the two extremes and three other representatives-the curves of the other subjects are similar but are not shown for a better visualization). The static stiffness curves for all subjects have the same general shape that can be divided into two main parts. At low compression, the stiffness is small with a low slope. Beyond a compression of 2.5-3.5 mm, the stiffness increases quickly with higher compression values. This behaviour is similar to that observed in certain elastomers or some polyurethane cellular foams [START_REF] Riande | Polymer viscoelasticity, stress and strain in practice[END_REF].

The results of the present study are in good agreement with those reported in other works [START_REF] Serina | Force response of the fingertip pulp to repeated compression-Effects of loading rate, loading angle and anthropometry[END_REF][START_REF] Wu | Dynamic interaction between a fingerpad and a flat surface: experiments and analysis[END_REF] in which high dispersion existed over the subjects and the stiffness had the same order of magnitude. Hz where the stiffness is approximately 20 N.mm -1 and has moderate variation (the gradient is 2 N.mm -1 per octave). The second zone lies between 125 and 315 Hz where the storage modulus increases quickly from 21 to 33 N.mm -1 with an average gradient of 9 N.mm -1 per octave. The third zone lies between 315 and 550 Hz (the spectral limit of our measurements), where the stiffness increases rapidly (the gradient is 16 N.mm -1 per octave). The shapes of the stiffness curves in Fig. 9 are similar to those plotted using viscoanalyzer measurements for amorphous polymers (Menard, 2008 p. 26 andLakes, 2009 p. 228). By analogy with rheology, the phalanx stiffness can be considered to be composed of a rubbery plateau of low stiffness up to 125 Hz, followed by a region of steep rise in stiffness between 125 and 315 Hz, called the glassy transition, and finally, a glassy state of higher stiffness (more than twice that of the rubbery plateau). The dynamic stiffness phase angle has two local maxima at 150 and 450 Hz that are associated with two points of inflection in the real part of the stiffness. In the field of rheology, the first local maximum is called the 𝛼 peak or relaxation 𝛼 and is associated with the point of inflection in the glassy transition zone. The second local maximum is called the 𝛽 peak or relaxation 𝛽 and is related to the point of inflection in the stiffness rise in the glassy state.

Input and output stiffnesses exhibit a rubbery plateau followed by a glassy transition zone before entering an ultimate glassy state (Fig. 10 We quantified the inter-individual variation using the 95% confidence interval to gain an understanding of the output stiffness measurement dispersion. The mean output stiffness and 95% confidence interval are shown in Fig. 11 for the following experimental conditions: an acceleration level of 5 m.s -2 rms and a quasi-static preloading of 8 N.mm -1 . In this case, the half-width of the 95% confidence interval is 20%, and it has the same order of magnitude under all other experimental conditions. The acceleration power kernel decreases quasi-linearly with respect to frequency. The interpolated straight line slopes at approximately -7.6 dB/octave. In other words, the acceleration power kernel (and hence, the dissipated power for harmonic excitation) is divided by six, when the frequency doubles. As described in Sections 4.4 and 4.5 of this paper, this behaviour is exhibited regardless of the excitation amplitude and static preloading. We can therefore establish the following empirical rule: when a phalanx is preloaded and harmonically vibrated by a cylindrical indenter (diameter: 7 mm) under isoacceleration, doubling of the frequency leads to a six-fold reduction in the power dissipated by the distal phalanx.

When the vibration excitation is sinusoidal, the power dissipated by the distal phalanx is related to both the acceleration amplitude and frequency, as shown in Eq. ( 5). Based on this relation, iso-power dissipation charts can be plotted with respect to the rms sinusoidal acceleration level and its frequency. Examples of these charts are provided in Fig. 13. The acceleration power kernel used for plotting these charts corresponded to that of the following experimental condition: acceleration level = 5 m.s -2 rms and quasi-static preloading = 8 N.mm -1 . In the following sections, we show that this acceleration power kernel depends on the quasi-static preloading but depends weakly on excitation amplitude at frequencies below 80 Hz. Hence, these abacuses are suitable only above this frequency.

Charts of the same type but different from those shown in Fig. 13 can be plotted for each static preloading. The dissipated power was computed from the acceleration power kernel based on the following experimental condition: acceleration 5 m.s -2 rms and preloading 8 N.mm -1 .

These charts are especially of interest because they provide an overall view of the dissipated power variation with respect to the acceleration level and frequency. They also enable quick selection of an acceleration level or frequency combination that yields the given dissipated power. For example, it is easy to find which acceleration level or frequency combination is associated with a dissipated power of less than 0.1 mW. Under the assumption that we can link vibration pathology development to a dissipated power threshold that must not be exceeded, such charts will enable us to select the threshold amplitude or frequency combinations for vibration-induced white finger pathology at frequencies higher than 80 Hz.

Effects of preloading for a fixed vibration level

The impact of the quasi-static preloading at a fixed acceleration level can be analysed with regard to the following four quantities: storage modulus, loss modulus, acceleration power kernel, and output stiffness phase angle. These four quantities are shown in Fig. 14 (a), (b), (c), and (d) for an acceleration level of 5 m.s -2 rms. The effect of the static preloading on the storage and loss moduli is clearly shown in Fig. 14 (a) and (b). These moduli increase with the static preloading. The preloading influences for the rubbery plateau and the glassy transition zone are not identical. On the rubbery plateau, in addition to the overall increase in stiffness, when the preloading increases, the slope of the interpolated straight line on this plateau increases more moderately that that of the interpolated straight line in the glassy transition zone (an example of these straight lines is provided in Table 3 with the values of the gradients). The consequence of an increase in preloading is, therefore, an overall rise in dynamic stiffness with more pronounced effects in the glassy transition frequency zone (the ratio of the slopes for the maximum and minimum preloadings is 6) than on the rubbery plateau (the corresponding ratio is 2.5). This synergy between frequency and static preloading was also confirmed by the results of a variance analysis test in which the frequency and preloading combined term had a p-value much less than 0.05. In addition to the increased stiffness, the loss modulus increases with increase in the preloading; in this case too, the effect is more pronounced in the glassy transition zone. As showed in Fig. 14 (c), the acceleration power kernel is small when the preloading is low. The decrease in the frequency of the acceleration power kernel remains virtually identical regardless of the level of quasi-static pre-load. There is no synergetic effect between the level of preloading and frequency. The acceleration power kernel curves are simply translated when the preloading varies (except at approximately 400 Hz, where the curves differ slightly). We quantified the relationship between the static preloading variation and the power dissipation variation under iso-acceleration. On average, over the frequency band, doubling the static preloading causes twice as much energy dissipation (this ratio varies between 1.8 and 2.3 over the spectral bandwidth). Moreover, the output stiffness phase angles shown in Fig. 14 (d) exhibit two paradoxical effects of the static preloading on frequency mismatch in relaxations 𝛼 and 𝛽: the lower the preloading, the lower are the frequencies of 𝛼 and the higher are the frequencies of 𝛽. In other words, decreasing the static preloading effectively reduces the spectral bandwidth of the rubbery plateau and increases the bandwidth of the glassy transition zone, even when these variations are relatively low, especially in the case of 𝛽.

To gain a statistical understanding of the results, we plotted the mean storage and loss moduli and the corresponding lower and upper bounds of the 95% confidence interval in Fig. 15 (a) and (b). For both storage and loss moduli, the 95% confidence intervals are well separated for low static preload (2 N.mm -1 and 5 N.mm -1 ). In contrast, for higher static compressions (8 N.mm -1 and 10 N.mm -1 ), the 95% confidence intervals overlap. We also developed a general model for the analysis of variance (ANOVA) to determine the significance of the static preloads on the storage and loss moduli (differences were considered significant at the p < 0.05 level). Statistical analysis with the data from all subjects at a fixed vibration amplitude (here, 5 m.s -2 rms) shows that both frequency (p < 0.0001) and static preload (p < 0.0001) are statistically significant factors for dynamic stiffness. Furthermore, there exists a statistically significant interaction between frequency and static preload (p < 0.03). In fact, as shown in Fig. 15 (a) and (b), beyond 125 Hz, the slope of the dynamic stiffness in the leathery state depends on both the compression level and frequency. In addition, a Tukey's honest significant difference (HSD) multiple comparison test was performed to determine the means of the storage and loss modulus that are significantly different from each other. We found that the means of storage and loss modulus for each compression rate are statistically significantly different from each of the other three. The statistical analyses for the acceleration power kernel and phase angle are not presented here because these two quantities are determined from storage and loss moduli, and hence, the same conclusions as those for the latter quantities can be drawn. To summarize, the higher the static preloading, the higher is the dynamic stiffness and the more intense is the vibration energy dissipation. upper bound of the 95% confidence interval; lower bound of the 95% confidence interval.

Effects of the vibration level for a fixed preloading

The output stiffness loss moduli are shown in Fig. 16 for the four preloadings and five acceleration levels listed in Table 2. For the two smallest preloadings (2 N.mm -1 and 5 N.mm -1 ), only the first four acceleration amplitudes are plotted in Fig. 16 (a) and (b). This because when the excitation amplitude is 20 m.s -2 rms, the coherence between acceleration and force is too low, and the transfer function cannot be calculated accurately under such conditions. Fig. 16 (a) to (d)

clearly show a decrease in the distal phalanx stiffness when the excitation amplitude

increases. An analysis of variance highlighted that both frequency (p < 0.0001) and vibration amplitude (p < 0.0003) are statistically significant factors for the storage modulus.

There is no synergic interaction between frequency and amplitude (p > 1). Indeed for each vibration amplitude, the storage moduli are simply translated (like an offset effect) without any modification of the shape of the associated curves. A Tukey's HSD test was performed to identify the amplitudes for which the means of the storage modulus are significantly different. It shows significant differences between the means of the storage modulus at high acceleration amplitudes and low levels. The results are listed in Table 4.

Table 4

Statistical multi-comparisons of the significant differences in the mean storage modulus for several vibration amplitudes; the grey cells indicate no significant differences, and the white cells indicate significant differences.

2 m.s -2 rms 5 m.s -2 rms 10 m.s -2 rms 15 m.s -2 rms 20 m.s -2 rms 2 m.s -2 rms 5 m.s -2 rms 10 m.s -2 rms 15 m.s -2 rms 20 m.s -2 rms Fig. 17 (a), (b), (c), and (d) (both linear and logarithmic scales were used) show that the variation in vibration excitation amplitude has an insignificant influence on the acceleration power kernel for all frequencies for 2 N.mm -1 and 5 N.mm -1 preloadings and above 80 Hz for 8 N.mm -1 and 10 N.mm -1 preloadings. The acceleration excitation amplitude affects distal phalanx softening or stiffening, but has limited influence on the dissipated vibration power kernel at high frequencies (>80 Hz). In other words, at high frequencies (>80 Hz), the input vibration amplitude does not intrinsically affect the dissipation of the mechanical power of the phalanx regardless of the compression. At low frequencies, the input vibration amplitude has a small effect only for large preloadings.

Discussions

Main potential measurement errors

When measuring the output force, the fingertip is placed on a steel shell support. The weight of this phalanx support (11 g) was added to that of the top seismic mass of the piezoelectric element, thereby reducing the resonance frequency of the sensor. However, by using the sensor characteristics in its data sheet and by assuming a mechanical secondorder system behaviour, we theoretically found that the modified resonance frequency (>10 kHz) remained much higher than the maximum frequency of the spectral band considered in our study. Hence, the weight added by the phalanx support caused little disturbance in the output force measurements. Furthermore, the impedance of the foundation on which the force sensor is fixed can lead to discrepancies between the true and measured forces if it is not considered in the force assessment [START_REF] Braender | High-Frequency-Response to Force-Transducers[END_REF]. However, in the case of a very heavy foundation (i.e. very low foundation velocity), the measurement error becomes insignificant except near the resonant frequency of the sensor. For each measurement, we conducted a check with the control accelerometer (Fig. 1 item b) placed near the force sensor to ensure that the foundation velocity remains negligible.

With regard to the static stiffness assessment, both the fingertip compression i.e. the cylindrical probe displacement and the output static force should be accurately estimated.

However, phalanx compression may be inaccurate if the just-in-contact condition between the probe and phalanx is improperly evaluated. Indeed, the probe displacement should be measured with reference to the position at which the indenter and phalanx barely touch in order to prevent initial deformation of the phalanx, which can cause a measurement error.

Given the very low stiffness of soft tissues with respect to minor compressions, the just-incontact condition is extremely difficult to estimate and reproduce by the experimenter.

Therefore, a simple electronic system that operates at very low currents (<2 mA, which is safe for humans) was developed. When the indenter makes contact with the distal phalanx, an electroluminescent diode lights up.

Assessing the quasi-static effort with a piezoelectric sensor is a great challenge.

Actually, the measuring system only provided an estimate of the DC component of the force response because of both the metrological limitations caused by the combination of the charge amplifier and piezoelectric sensor and the physical stress relaxation phenomena.

The stresses measured dynamically at a given moment were different from those obtained under quasi-static conditions [START_REF] Wu | Dynamic interaction between a fingerpad and a flat surface: experiments and analysis[END_REF]. Forces were measured at a low loading speed to minimize viscous effects. Conversely, if the measuring time increases, the electronic discharge process (low frequency limitation) and charge amplifier output voltage drift (zero drift) strongly disrupt the force measurement [START_REF] Gatti | Applied structural and mechanical vibrations -theory, methods and measuring instrumentation[END_REF]. The final measuring time reflects a balance between the discharge time constant (100 s), charge amplifier output voltage drift (no effect if the measurement time is less than 20 s), and relaxation effect (weak if the measuring time exceeds 10 s based on phalanx tests). Based on a large number of measurements synthesised in the technical report (Noel, 2015), we concluded that if the measuring time is between 10 s to 15 s, the relative error between true static force and the quasi-static force measured with the piezoelectric force output sensor is less than 10%. This is in good agreement with the results of the validation tests as indicated by comparison with a reference compression test machine (Section 2.5). The measuring time selected was, therefore, between 10 s and 15 s.

Moreover, the aluminium disc with the telemeter for displacement measurement, the screw for fixing the impedance head, the cylindrical probe, and the fingertip support were designed such that their first resonance frequencies were much higher than the upper limit of the spectral range analysed in this study. This ensured that no noise was transmitted by these elements to the fingertip because the measured coherence functions between the different mechanical quantities were almost unity.

Rheological standpoint and physiological analogy

The dynamic stiffnesses of the distal phalanxes exhibit similar behaviour as those of the measured stiffnesses of certain elastomers (Fig 9). This indicates that the macromolecular structure of the conjunctive tissues of the distal phalanx is likely similar to that of amorphous polymers. More specifically, the viscoelastic properties of distal phalanxes are mainly induced by the extracellular matrix of the conjunctive tissues [START_REF] Vicente | Viscoelasticity -From Theory to Biological Applications[END_REF]. These tissues occupy the space between the cells, nourish, and support the cells. These tissues are composed of a complex network of macromolecules that can be grouped into three main types (Callen and Perasso, 2005 p. 432-443): i) very voluminous fibrous proteins, i.e.

collagen and elastin; ii) less voluminous glycoproteins that ensure the adhesion of the different matrix constituents with the cells or among themselves; and iii) complex carbohydrates (polysaccharides) and glycosaminoglycans that are often attached to a protein to form proteoglycans. They form a hydrated gel that fills the water-retaining matrix. The adhesive glycoproteins and polysaccharides form a gel substance called the ground substance. The most important polysaccharide in the ground substance of the distal phalanx is hyaluronic acid [START_REF] Feng | Introduction to skin biothermomechanics and thermal pain[END_REF][START_REF] Vicente | Viscoelasticity -From Theory to Biological Applications[END_REF]. The hyaluronic acid chains have secondary proteoglycan chains (the versican in the dermis: a special type of proteoglycan) that are linked by adhesion glycoproteins to the constituents of the extracellular matrix, such as collagen and elastin. Because of this complex assembly of macromolecules, the distal phalanxes exhibit a viscoelastic behaviour similar to that of elastomers. The vibration energy dissipation properties are therefore explained [START_REF] Purslow | Collagen orientation and molecular spacing during creep and stress-relaxation in soft connective tissues[END_REF][START_REF] Shen | Viscoelastic properties of isolated collagen fibrils[END_REF] either by molecular rearrangements of different components of the extracellular matrix (more specifically in the collagen and ground substance) or by movements that cause viscous shear forces between the fluid (ground substance) and solid (collagen and elastin) phases.

The softening phenomenon related to an increase in the excitation amplitude (Fig. 16) is known in certain amorphous polymers, especially those with large carbon black content.

The phenomenon is called the Payne effect or the Fletcher-Gent effect. This dependence of distal phalanx stiffness on the vibration excitation amplitude can be considered in the context of the thixotropic nature of the ground substance (Feng and Lu, 2011 p.14). The ground substance effectively has the consistency of a gel with the two defining properties of thixotropy [START_REF] Barnes | Thixotropy -A review[END_REF]: i) the gel is rheofluidifying (shear rate thinning), i.e. its viscosity decreases with increasing mechanical stress. In other words, this gel behaves more as a solid at low stresses and more as a fluid at higher stresses. The macroscopic consequence of this characteristic may be a reduction in stiffness under increased amplitude (relaxation associated with a more fluid nature of the ground substance at high stresses). ii) the effects of this fluidification are not out of phase with stress application, but occur later. Similar to viscoelasticity, this behaviour reflects the dissipative nature of the ground substance.

Physical interpretation of results and comparison with other works

A number of works [START_REF] Reynolds | Hand-Arm Vibration, Part 2: Vibration Transmission Characteristics of the Hand and Arm[END_REF][START_REF] Sörensson | Transmission of Vibration to the Hand[END_REF][START_REF] Wu | Analysis of the point mechanical impedance of fingerpad in vibration[END_REF] have highlighted that low-frequency vibrations can propagate from the fingers to the whole hand-arm system, while high-frequency vibrations (>100 Hz) are more localized in the finger contact area because of the mechanical damping abilities of soft tissues [START_REF] Wu | Analysis of the point mechanical impedance of fingerpad in vibration[END_REF]. Hence, findings may be compared in terms of the trend of highfrequency vibrations, regardless of the test methods even if some differences are present.

Thus, our results may be considered either from the perspective of local finger biodynamic responses or the responses distributed in the hand-finger system.

The biodynamic response of the fingertip is expected to be controlled by the tissue elasticity up to a frequency of approximately 125 Hz (almost constant stiffness-segment

[AB] Fig. 9-with a phase less than 15°). The same behaviour was observed in a number of studies where the frequency of slope failure (point B Fig. 9) varied between 100 Hz and 200

Hz. In Lundström's study [START_REF] Lundström | Local vibrations-mechanical impedance human hand's glabrous skin[END_REF], this frequency is approximately 200 Hz, while it is approximately 100 Hz for Kern and Werthschützky [START_REF] Kern | Studies of the Mechanical Impedance of the Index Finger in Multiple Dimensions[END_REF], [START_REF] Mann | Effect of contact conditions on the mechanical impedance of the finger[END_REF], and Wiertlewski and Hayward [START_REF] Wiertlewski | Mechanical behavior of the fingertip in the range of frequencies and displacements relevant to touch[END_REF] and 130 Hz for Moore and Mundie [START_REF] Moore | Measurement of Specific Mechanical Impedance of the Skin: Effects of Static Force, Site of Stimulation, Area of Probe, and and Presence of a Surround[END_REF][START_REF] Moore | A Survey of the Mechanical Characteristics of Skin and Tissue in Response to Vibratory Stimulation[END_REF]. The latter two studies highlighted that the failure frequency at 130 Hz occurs not only at the fingertip, but also in the thenar eminence, the volar forearm, and the thigh. This observation led to the hypothesis that this failure frequency is not specific to the finger but is related to the mechanical behaviour of the human skin. Dong and co-workers (Dong et al., 2004c) reported a stiffening of the fingers biodynamic response as measured with an instrument handle at approximately 80 Hz. The leathery transition (segment [BC] Fig. 9)

and the glassy state (segments [CD,EF] Fig. 9) were not emphasized by Lundström [START_REF] Lundström | Local vibrations-mechanical impedance human hand's glabrous skin[END_REF], but Wu and co-workers [START_REF] Wu | Analysis of the point mechanical impedance of fingerpad in vibration[END_REF] found some inconsistency in Lundström's measurements at high frequencies. However, the leathery and glassy states were thoroughly measured by Moore and Mundie [START_REF] Moore | Measurement of Specific Mechanical Impedance of the Skin: Effects of Static Force, Site of Stimulation, Area of Probe, and and Presence of a Surround[END_REF] who found the glassy state at approximately 400 Hz, which is in good agreement with our findings. Dong and co-workers (Dong et al., 2004c) also reported changes in the dynamic response at 300

Hz and 500 Hz. On a separate note, Liang and Boppart [START_REF] Liang | Biomechanical properties of in vivo human skin from dynamic optical coherence elastography[END_REF] measured the Young's modulus of the human skin in the palm and forearm by using dynamic optical coherence elastography. They concluded that the Young's modulus of the skin is almost constant between 100 Hz and 200 Hz, and then increases up to 300 Hz and remains constant up to 400 Hz, beyond which, it rises again. This material behaviour is very similar to those of the storage modulus measured in this study.

Output and input stiffnesses differ from 125 Hz (Fig. 10). Indeed, the input stiffness is mainly controlled by the soft tissues of the fingertip, whereas the output stiffness is influenced by stiffer anatomical elements such as the bone and the nail (because of measurement localisation). Similarly, the input stiffness phase angle is greater than the output stiffness phase angle because the bones and the nails have less damping ability than the soft tissues.

The effect of the static preloading on biodynamic responses was studied by Lundström [START_REF] Lundström | Local vibrations-mechanical impedance human hand's glabrous skin[END_REF] who found a slight increase in the dynamic stiffness with doubling of the pre-stress, but the preloading level considered in the previous study was much less than that in the present study. In the study of [START_REF] Moore | Measurement of Specific Mechanical Impedance of the Skin: Effects of Static Force, Site of Stimulation, Area of Probe, and and Presence of a Surround[END_REF], the preloading was only significant for frequencies less than 130 Hz, whereas in the highfrequency region, the effect of static loads was measured. However, Dong and co-workers (Dong et al., 2004c) and [START_REF] Kern | Studies of the Mechanical Impedance of the Index Finger in Multiple Dimensions[END_REF] found that the gripping forces or static loads have considerable effects, but they do not cause dissimilar behaviour between the rubbery plateau and the leathery or glassy state, as we found in the present work. However, the boundary conditions of the fingers in their works were different from those in our study. Their works incorporated inertial effects largely, and this may be the cause of the observed differences.

Lundström [START_REF] Lundström | Local vibrations-mechanical impedance human hand's glabrous skin[END_REF] studied the effects of input acceleration amplitude on biodynamic responses. He found that the input acceleration amplitude exerts an influence only in low-frequency (<100 Hz) region; however, for the same reason as explained earlier, no definitive conclusions can be draw at high frequencies. In the study of Gurram and coworkers [START_REF] Gurram | Mechanical impedance of the human hand-arm system subject to sinusoidal and stochastic excitations[END_REF], the input acceleration amplitudes do not modify the handarm-system biodynamic response regardless of the frequency range. On the other hand, Marcotte and co-workers [START_REF] Marcotte | Effect of handle size and hand-handle contact force on the biodynamic response of the hand-arm system under zh-axis vibration[END_REF] found that the level of input acceleration could change the mechanical impedance at low frequencies only (<100 Hz). In the present study, we showed that the input acceleration amplitude may have a major influence on the storage modulus at all frequencies. We further showed that it affects the dissipated mechanical power slightly only at low frequencies (<50 Hz).

Conclusions

In this study, an experimental apparatus was built for measuring static and dynamic stiffnesses in preloaded distal phalanxes. The experimental set-up and test bench were validated by comparing their results with those obtained using a tension-compression machine and a viscoanalyzer. Twenty subjects participated in the test, to provide the statistical data representative of an average index finger. Analysis of the measured dynamic stiffnesses highlighted that the phalanx exhibited mechanical behaviour similar to that of certain amorphous polymers. The phalanx stiffnesses in fact increase with the vibration frequency, exhibiting a rubbery plateau, a glassy transition zone, and a glassy state. Under iso-excitation amplitude and iso-static preloading, the power dissipated in the phalanx is divided by six when the frequency is doubled. The influence of the static preloading and vibration excitation amplitude was subsequently investigated. The higher the preloading state, the higher is the dynamic stiffness with a very pronounced frequency and preloading synergy effect below 125 Hz. The dissipated power is also affected by the level of preloading because doubling this level leads to twice as much vibration dissipation. Additionally, an increase in vibration excitation amplitude causes a decrease in the phalanx stiffness, similar to the Payne effect or the Fletcher-Gent effect encountered in a number of rubbers. The thixotropic nature of the extracellular matrix may be the cause of this decrease in phalanx stiffness. Our measurements also revealed that the acceleration power kernel was not influenced by the vibration excitation level at high frequencies (clearly, the dissipated power is directly linked to the vibration amplitude). Using these kernels, the iso-dissipated power charts were plotted for each static preloading and were used for selecting the frequency and amplitude combinations to realize a target dissipated power. These experimental results will be used for adjusting and validating finite element models of preloaded, vibrated phalanxes.

Fig. 1 .

 1 Fig. 1. Experimental apparatus: a) laser telemeter, b) control accelerometer, c) force sensor, d) impedance head, e) shaker, f) aluminium disk, g) safety arm, h) moving support, i) phalanx support, j) contact indenter, k) hand-arm rest, l) overview of the subject forearm positioning.

Fig. 2 .

 2 Fig. 2. Experimental test bench layout with equipment and interconnections; source output voltage; measured on NI 9234; measured on NI 9239.

Fig. 3

 3 Fig. 3 illustrates an example of implementation of this methodology and shows the measured acceleration generated by the shaker input voltage computed from the desired target acceleration. The rms acceleration measured in this way was 10.3 m.s -2 , which closely approximated the expected value (10 m.s -2 ).

Fig. 3 .

 3 Fig. 3. Power spectral density of input acceleration transmitted by the cylinder indenter to the phalanx; 10 m.s -2 rms target acceleration; measured acceleration (10.3 m.s -2 rms) generated by shaker input voltage computed from desired target acceleration.

Fig. 4 .

 4 Fig. 4. Sketches showing procedure used for system phase recalibration: (a) channel-to-channel phase angle mismatch measurement; (b) phase angle mismatch of charge amplifier measurement.

Fig. 5 .

 5 Fig. 5. Phase angle of dynamic output stiffness for control spring; INRS test bench; theoretical spring.

Fig. 6 .

 6 Fig. 6. Quasistatic test bench validation: force versus displacement of reference polyurethane foam sample; INRS test bench; reference tension-compression machine.

Fig. 7 .

 7 Fig. 7. Dynamic test bench validation: (a) dynamic stiffness magnitude of reference polyurethane foam sample; (b) phase angle of reference polyurethane foam sample; with INRS test bench;

Fig. 8 .

 8 Fig. 8. Example of static stiffness for five subjects; subject with high stiffness from low compression; subject with high stiffness from high compression

Fig. 9 .

 9 Fig. 9. Three basic ways to present the dynamic output stiffness for the experimental condition: acceleration 5 m.s -2 rms and preload 8 N.mm -1 ; the left axis is assigned to the stiffness modulus, the right axis to the phase angle; storage modulus; loss modulus; phase angle; linear interpolation of the storage modulus with an offset of -5 N.mm -1for better visualization.

Fig. 10 .

 10 Fig. 10. Comparison of input and output dynamic stiffness for the following experimental conditions: acceleration 5 m.s -2 rms and preload 8 N.mm -1 ; (a) storage modulus; (b) phase angle; storage modulus of input stiffness; storage modulus of output stiffness.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Dispersion of the output storage modulus under the following experimental conditions: acceleration 5 m.s -2 rms and preloading 8 N.mm -1 ; mean value; 95% confidence interval.

Fig. 13 .

 13 Fig. 13. Charts of dissipated iso-power under pure harmonic vibration for a preloading of 8 N•mm -1 .

Fig. 14 .

 14 Fig. 14. Effects of preloading for a 5 m.s -2 rms vibration level; (a) output storage modulus; (b) output loss modulus; (c) acceleration power kernel; (d) phase angle of the output stiffness; 2 N.mm -1 ; 5 N.mm -1 ; 8 N.mm -1 ; 10 N.mm -1 .

Fig. 15 .

 15 Fig. 15. Variability of subjects for a 5 m•s -2 rms vibration level; (a) output storage modulus; (b) output loss modulus; 2 N.mm -1 ; 5 N.mm -1 ; 8 N.mm -1 ; 10 N.mm -1 ;

Fig. 16 .

 16 Fig. 16. Effects on changes in the storage modulus of a fixed preloading vibration level; preloading: (a) 2 N.mm -1 ; (b) 5 N.mm -1 ; (c) 8 N.mm -1 ; (d) 10 N.mm -1 ; 2 m.s -2 rms; 5 m.s -2 rms; 10 m.s -2 rms; 15 m.s -2 rms; 20 m.s -2 rms.

Fig. 17 .

 17 Fig. 17. Effects of a fixed preloading vibration level on acceleration power kernel; preloading: (a) 2 N.mm -1 ; (b) 5 N.mm -1 ; (c) 8 N.mm -1 ; (d) 10 N.mm -1 ; 2 m.s -2 rms; 5 m.s -2 rms; 10 m.s -2 rms; 15 m.s -2 rms; 20 m.s -2 rms.

Table 1

 1 Overall and forefinger specific anthropometric characteristics for cohort subjects; = mean and = standard deviation; BMI = Body Mass Index.

	Overall characteristics			Forefinger characteristics
						Distal	Distal
					Forearm			Index Finger
	Age	Size	Weight	BMI		Phalanx	Phalanx
					volume			Length
	[year]	[m]	[kg]	[-]		Thickness	Width
					[l]			[mm]
						[mm]	[mm]
	22.9	1.7	66.3	22.4	1.2	13.5	17.2	78.1
	5.5	0.1	8.4	2.6	0.3	1.5	1.5	5.5

Table 2

 2 Summary of setup parameters used for the dynamic stiffness measurements.

	Indenter shape and dimensions	Acceleration level			Quasi-static stiffness
		[m.s -2 rms]				[N.mm -1 ]
	Cylinder with diameter of 7 mm	2	5	10	15	20	2	5	8	10

Table 3

 3 Slope of the storage modulus on the rubbery plateau and in the glassy transition zone for various

	preloadings.
	Quasi-static stiffness		2	5	8	10
	[ N.mm -1 ]
	Slope	Rubbery plateau	1	1.5	2	2.5
	[N.mm -1 per octave]	Glassy transition	2	5	9	12

Acknowledgements

The author would like to thank his INRS colleagues, who actively participated in the work described in this paper: Dr. Jean-Pierre Meyer (research physician), Isabelle Manet, Dominique Bruant, Alexandre Klingler, and Richard Poirot.


L F Zagonel

E J Mittemeijer

F Alvarez

The Microstructure of Tool Steel after Low Temperature Ion Nitriding

Keywords: ion beam nitriding, surface nitrogen concentration, phase transformation, precipitation. Microstructure of tool steel after low temperature ion nitriding

The microstructural development in H13 tool steel upon nitriding by an ion beam process was investigated. The nitriding experiments were performed at a relatively low temperature of about 400 ºC and at constant ion beam energy (400 eV) of different doses in a high-vacuum preparation chamber; the ion source was fed with high purity nitrogen gas. The specimens were characterized by X-ray photoelectron spectroscopy (XPS), electron probe microanalysis (EPMA), scanning and transmission electron microscopy (SEM and TEM), and grazing incidence and Bragg-Brentano X-ray diffractometry. In particular, the influence of the nitrogen surface concentration on the development of the nitrogen concentration-depth profile and the possible precipitation of alloying element nitrides were discussed.

Introduction

Nitriding is a thermo-chemical process which can lead to pronounced improvements of the mechanical and chemical properties, as the fatigue endurance and the corrosion resistance of steels. 1,2 The process basically consists of nitrogen incorporation through the surface and its subsequent diffusion into the material bulk. Even though nitriding processes are widely used in industrial applications, many fundamental aspects of the process are still not fully understood. Among these, one can single out the role of the (possibly varying with treatment time) nitrogen concentration at the material surface and the nitride precipitation kinetics of iron and alloying elements such as chromium, molybdenum and vanadium, which profoundly influence the nitriding "effect". [START_REF] Mittemeijer | Thermodynamics, kinetics, and process control of nitriding[END_REF]4,5 If local equilibrium prevails at the surface, the nitrogen concentration in the solid at the surface is determined by the chemical potential of nitrogen in the nitriding atmosphere. This thermodynamic parameter is crucial for (control of) the efficiency of the nitriding process, also under non-equilibrium conditions. In the gaseous nitriding process, at a non-beginning stage of nitriding, the N concentration at the material surface may indeed be given by the chemical potential of nitrogen in the gas atmosphere. [START_REF] Mittemeijer | Thermodynamics, kinetics, and process control of nitriding[END_REF]6 Ion nitriding is a process where such equilibrium conditions at the surface are not expected to occur. Ionic nitriding bombardment takes place at energies larger than the bonding energies of the atoms in the solid to be nitrided, thereby breaking bonds, and knocking-in and sputtering atoms away, emitting secondary electrons, exciting atoms, inducing dislocations, phonons, heat spike, texturization, etc., can occur. This (incomplete) list shows the complexity of the ion nitriding process. Thus, systematic and extensive experimental and theoretical investigation of ion nitriding is imperative in order to control key parameters as the nitrogen concentration at the material surface.

This study contributes to achieve understanding of the effect of nitriding at low temperature (400 ºC; usual nitriding temperatures with well-known, other nitriding processes, as gaseous nitriding, are in the range 500 ºC -600 ºC); attention is especially paid to the nitrogen concentration at the material surface, and the effects of ion current. The experiments were realized by using a broad Kaufman ion source at constant ion irradiation energy. The Kaufman cell allows a fine control of the beam species, energies and current applied to the sample in a virtually oxygen free nitriding process. 7

Experimental procedures

Rectangular samples (2x15x20mm³) were cut from a single tempered AISI-H13 block quenched in air from 1025 °C and tempered at 580 °C for two hours. This heat treatment causes partial precipitation of alloying elements as carbides and sets the bulk hardness to 7.5±0.4 GPa. Table 1 displays the material composition as determined by chemical analysis. The samples were polished up to 1 m diamond paste and cleaned in an acetone ultrasonic bath. One by one, the samples were introduced into the high vacuum (< 10 -5 Pa) system equipped with the Kaufman cell for ion beam nitriding. The deposition system is attached to an ultra high vacuum chamber (UHV) system (< 10 -7 Pa) for X-ray photoemission spectroscopy (XPS). Thus, immediately after irradiation, the samples can be transferred to the UHV system without atmospheric contamination. The nitriding times and ion current densities used in the experiments are displayed in Table 2. The ions from the Kaufman cell were fixed at 400 eV. The ion source was fed with 8 sccm (standard cubic centimetre per minute, i.e. 8 cubic centimetres at 1 atmospheric pressure) of high purity nitrogen gas (99.999%). A constant pressure of ~ 10 -2 Pa was maintained in the preparation chamber during the nitriding. The ion current densities were determined by measuring the collected ions by a Faraday cup. The sample temperature was maintained at (400±5)°C and the distance between the ion source and the sample was ~25 cm. More experimental details of the implantation system can be found elsewhere. 8 To determine the elemental surface composition just after the ion implantation, XPS measurements were performed in a UHV chamber attached to the nitriding chamber by a VG-CLAM-2 electron energy analyzer using a non-monochromatic Al K radiation and following the procedure described by Hüfner. [START_REF] Hüfner | Photoelectron Spectroscopy: principles and applications[END_REF] The photoemission cross-sections were taken as given by Lindau, the elastic means free path is considered to be proportional to E -0.29 , and the electron energy analyzer transmission is considered proportional to E -1 . [START_REF] Yeh | Atomic Data and Nuclear Data Tables[END_REF] Further material constituents chemical states analysis were performed by ex situ XPS experiments after Ar + sputtering cleaning (VG Thetaprobe system using monochromatic Al K radiation).

Samples cross-sections were analyzed ex situ by electron probe microanalysis (EPMA) employing a Cameca SX100 apparatus using a 10kV focused electron beam with an electron current of 100 nA. For this study, the samples cross sections were covered, before EPMA, by a protective electroplated nickel layer and polished (last step 0.25 µm diamond paste). The composition of the nitrided samples was determined from the emitted characteristic X-ray intensities by their comparison with the corresponding intensities recorded from appropriated standards such as pure metals and compounds (Fe 3 C, Fe 4 N). The final element content was calculated applying the z method. [START_REF] Pouchou | [END_REF] The estimated error from the EPMA measurements is lower than 0.5at % for all reported results. The morphology of the nitrided microstructure as revealed in specimen cross-sections was investigated by Scanning Electron Microscopy (SEM) using a Jeol JMS-5900LV apparatus. To probe the nanometric precipitates, a Carl Zeiss CEM 902 Transmission Electron Microscope (TEM) operating at 80 kV was employed. The TEM is equipped with a Castaing-Henry energy filter spectrometer for electron energy loss spectrometry (EELS) and a Proscan high-speed slow-scan CCD camera. Suitable TEM samples were prepared in planar view by mechanical polishing and dimpling to the depth of interest, followed by ion milling with 4keV Ar + ions at an angle of 6°. Crystalline phase analysis was obtained by grazing incidence X-ray diffraction using monocromatized Cu K radiation with an incidence angle of 3º (corresponding to an effective penetration/information depth of ~0.1 µm). A larger effective penetration/information depth (~1.4µm) was obtained by using a standard Bragg-Brentano diffraction geometry also using monocromatized Cu K radiation (for definition of penetration/information depth, see Ref. 12 ).

Results and discussion

Nitrogen surface concentration; the sticking probability

For the X-ray photon energy used in the XPS experiments, the information obtained from this technique originates from depths under the surface up to ~ 3 nm. [START_REF] Briggs | Surface Analysis by Auger and X-Ray Photoelectron Spectroscopy[END_REF] The surface concentration of relevant elements for the investigated specimens is shown in Fig. 1a as function of the ion current density and for constant irradiation time (300 minutes). A plateau level for the element concentrations is observed for ion current densities higher than a characteristic value. By irradiating the sample with a fixed ion current density (1.5 mAcm -2 ), the nitrogen concentration at the surface reaches a plateau before 75 minutes (see Fig. 1b). [START_REF] Abrasonis | [END_REF]15,16 This nitrogen concentration corresponds to a stationary state, i.e. a state where a balance is reached between incoming nitrogen and nitrogen losses by self-sputtering or bulk diffusion. The presence of epsilon phase nuclei forming a thin layer could be a possible interpretation of the experimental results. However, the diffractograms corresponding to samples 4 and 6 do not shown any trace of epsilon phase nuclei (see section 3.2). Therefore, one can conclude that the epsilon phase is formed in the stationary state after the accumulation of a relatively high nitrogen concentration.

For completeness, in Fig. 1 the concentrations obtained for a un-nitrided sample are also shown: a virtually oxygen free sample representing the material bulk (data at ion current and time equal to zero, respectively). Usually a thin native oxide layer is always present at the sample surface. However, Fig. 1 shows that nitrogen ions remove oxygen by physical/chemical sputtering mechanism. 17 The nitrogen sticking probability, i.e., the probability of the nitrogen-containing ions to remain attached to the surface upon bombardment, depends on kinetic ion-surface interactions, backscattering, sputtering, and nitrogen compound desorption. In studies performed by Rabalais et al., concerning N 2 + ion beam bombardment of a pure iron target, the sticking factor was determined as ~ 0.11 at 400 eV. 18,19 This rather low sticking probability indicates that, at the studied ion energies, it is unlikely that the ion upon hitting the surface can loose all its kinetic energy without been backscattered to the vacuum. 20 For the present experiments, the number of nitrogen atoms retained (i.e. incorporated into the sample) per unit of charge may also be smaller than two (atoms per electronic unit charge), which implies that not only N 2 + molecules arrive at the sample surface, but also N + , N°, N 2 °, etc. 21,22 It is noted that the release of nitrogen from the iron nitride surface, which strongly depends on temperature, have been studied by thermal desorption spectroscopy and is not important at 400ºC for iron surfaces. 23 Assuming the sticking probability to be constant during the nitriding process, the sticking probability can be estimated from the ratio of the ion dose and the total nitrogen incorporated in the material. 24 The assumption of a constant sticking probability is valid provided that the nitrogen concentration at the surface is low, i.e. no significant nitrogen self-sputtering takes place. The total amount of nitrogen taken up by the specimen can be estimated by integrating the nitrogen concentration-depth profile (cf. Fig 5) and assuming that the density remains close to the value of the original material. Sample 4 is particularly suitable for the estimation of the sticking probability since no iron nitrides were formed (Fig. 3) and the measured surface nitrogen concentration is below ~6 at. % (i.e. self-sputtering was not significant). The total nitrogen content in sample 4 (diffusion zone of thickness 10 µm) is ~3.0×10 18 at cm -2 . Therefore the sticking probability is ~ 0.15 (nitrogen atoms per electronic unit charge arriving at the sample surface). These quantities have been listed in Table 2 for all the samples investigated.

The sticking probability is an important parameter for understanding the evolution of the nitrogen surface concentration eventually leading to the stationary state at the surface. Indeed a stationary state at the surface of the sample has been reached after a process time which depends on the ion current density (see Fig. 1). Higher sticking probabilities would decrease both time and ion current density needed to reach a stationary state at the surface. After a stationary state at the surface has been achieved, the nitriding process becomes diffusion controlled, i.e. the ionic current does no longer play an important role. 25

Effect of sputtering on the ion nitriding process: specimen thickness

The impact of energetic nitrogen ions influences and modifies the surface by physical and chemical sputtering, adsorption, and chemical reactions. Sputtering involves loss of material from the specimen: decrease of specimen thickness. On the other hand, nitrogen uptake during the ion beam nitriding can lead to an increase of the specimen thickness as a consequence of stress development and phase transitions. Depending on the experimental conditions, the specimen thickness can either increase or decrease, as experimentally shown by profilometry sweepings across the nitrided and the (properly protected during nitriding) un-nitrided parts of the specimen surface. The height of the step created by the irradiation process as a function of the ion current density and nitriding time is shown in Fig. 2. The error bars approximately correspond to twice the surface roughness, which is mainly induced by the nitriding process but is low at such process temperature. 26 It follows (Fig. 2a) that, up to ~0.5 mAcm -2 the nitrided part of the specimen is thicker than the unnitrided part, which is in its original state. At higher ion current densities, sputtering dominates and the surface is eroded, thereby decreasing the specimen thickness. The same is exhibited in Fig. 2(b) showing the evolution of the specimen thickness, at a constant ion current density of 1.5 mAcm -2 , as a function of irradiation time. This plot shows that erosion dominates already before ~75 minutes irradiation time. Roughly linear relations can be adopted for the dependences of the specimen thickness on sputtering time and on ion current density (Figs. 2a andb), after a stationary state for the surface composition has been reached (Fig. 1), say, after 40 minutes (see Figs. 1b and2b ). The total eroded depth, vertical axis in Fig. 2, is much smaller then the nitrided case depth. Therefore, the error introduced by the sputtering erosion on the depth profiles as shown in Fig. 5 can be neglected. The surface composition and structure influence the sputtering yield. Therefore, in the nitriding process the native oxide layer is expected to have an effect on the sputtering yield.

As observed, the erosion caused by nitrogen ions after distinct nitrogen concentrations have been reached is higher than the one occurring for the initial material. For samples 1, 2, 5 and 6, the sputtering rate can be estimated by the ratio between the eroded length (from Fig. 2) and the implanted dose (from Table 2), giving approximately 0.05 atoms sputtered away per incident ion.

The initial thickness increment, shown in Fig. 2a, is explained by the development of residual stress upon nitriding. The -Fe 110 reflection (specimen 4 in Fig. 3) is shifted with respect to the unnitrided sample. This effect could be interpreted as due to the internal compressive macrostress parallel to the surface induced by the volume expansion in the nitrided zone due to the presence of nitrogen trapped as alloying-element nitride precipitates (Fig. 4). The dimension of the 110 lattice planes determined in the direction perpendicular to the surface for the ferrite matrix in Bragg-Brentano geometry (i.e. with the diffraction vector perpendicular to the specimen surface) are 292.1 pm and 288.5 pm for specimen 4 in the nitrided condition and the unnitrided condition, respectively. Assuming a ferrite-lattice expansion proportional to the nitrogen concentration and using the nitrogen profile given in Fig. 5, an increase of specimen thickness of 75±5 nm is estimated. This value for the increase of specimen thickness agrees well with the experimental thickness difference between nitrided and non-nitrided sample regions of 90±30 nm (Fig. 2a).

Crystalline phase analysis of the nitrided surface; iron-nitride formation at the surface

Grazing-incidence X-ray diffractograms recorded from the surface of specimens investigated are shown in Fig. 3. Specimens 1 to 3 gave rise to diffraction peaks indicative of the presence of the -Fe 2-3 N phase, i.e. an iron-nitride phase of a relatively high nitrogen concentration. At the lowest ion current density studied (specimen 4) a peak due to ferrite appears, compatible with a low "nitriding potency" of the ion nitriding process under such conditions. Specimen 5

shows that at intermediate nitriding times, iron nitrides are formed. For the lowest studied nitriding time (specimen 6) ferrite appears to be the only phase at the surface. This observation need not be in contradiction with the higher nitrogen concentration obtained from XPS, a technique probing only the first ~3 nm, whereas the grazing-incidence diffractograms shown in Fig. 3 correspond to a penetration/information depth of the order of about 0.1 μm (see section 2). Diffractograms recorded in Bragg-Brentano geometry (compatible with penetration/information depths of about 1.4 μm) indicate that the layer containing -Fe 2-3 N phase is thin (1 µm) for all studied samples, as expected for the low process temperature applied (400°C).

Chemical state of the alloying elements; alloying element nitride formation

Besides providing values for the concentrations of the various elements in the surface region of the analysed specimen, XPS gives information on the chemical state of these elements. This information is obtained by analysing the peak positions (binding energy values) associated with the studied element core level electrons. The XPS spectra of nitrided samples show that the positions and shapes of the bands associated with chromium, molybdenum, and vanadium differ from those observed for an untreated clean (Ar+ sputtered) reference sample. As observed in Figs. 4 (b) and (c), the bands associated with vanadium and molybdenum electron core levels shift towards high binding energies for samples 2 and 3 (E=+0.2 eV). However, for sample 4 the result is similar to the one found for the unnitrided reference sample. The band associated with chromium, on the other hand, displays only a slight increase of the high-binding energy tail (samples 2 and 3). The absolute amount of the alloying elements was similar for all samples, including those with an iron nitride phase at the surface.

These shifts in binding energy are ascribed to the formation of alloying element nitrides, occurring as precipitates in the nitrided matrix, such as CrN, VN, and MoN. The results indicate that up to an uptake corresponding to ~5at.% nitrogen, the binding energies of the alloying elements remain unchanged and therefore precipitation does not occur substantially up to this level of nitrogen uptake (see Fig. 7). For higher nitrogen concentration values, as holds for samples 2 and 3, the 2p levels of V and Mo reveal a different chemical state for V and Mo, which is considered to be indicative for the presence of alloying element nitride precipitates. On the other hand, the Cr 2p level spectra indicate that chromium carbides are present in the virginal material and that these are not dissolved upon nitriding to form chromium nitride (see section 3.5). (c) Mo 2p

Nitrogen concentration-depth profiles; precipitate particles in the nitrided zone

Nitrogen concentration-depth profiles as determined by EPMA from sample cross sections (see section 2) are shown in Fig. 5 for: a) specimens of the same nitriding time and treated at different ion current densities (Fig. 5a) and, b) for specimens treated at constant ion current density for different nitriding times (Fig. 5b).

Fig. 5(a) shows that the nitrogen penetration depth is about the same for all samples. In first order approximation this may be expected for equal nitriding time treatments, if the same concentration of dissolved nitrogen would occur at the surface, which however is not truly the case (see sections 3.1 and 3.4), and if solid state diffusion of nitrogen would be the dominantly rate controlling process. The amount of nitrogen in the sample increases monotonically from samples 4 to 1, i.e., on increasing ion nitrogen current density. This behaviour is ascribed to the earlier attainment of a stationary state (cf. section 3.1) at the sample surface. The nitrogen profiles observed for samples 1 and 2 are more or less similar and this suggests that the ~ 60% increase of the ion-beam current applied for sample 1 with respect to sample 2 (cf. Table 2) does not produce pronounced changes of the nitrogen depth profile. Indeed, after the establishment at the surface of a stationary state, as appears to hold for samples 1 and 2 (see Fig. 5a), the magnitude of the ion-beam current is not important for the progress of nitriding in the solid substrate. In the case of sample 4a a stationary state at the surface has evidently not been realized (Fig. 1 and Fig. 3). Near to the surface, the nitrogen concentrations as obtained from EPMA in the specimen cross section are smaller than those determined by XPS (Fig. 1). This is expected, since XPS, applied to the sample surface, and EPMA, applied to the sample cross section, probe up to a depth of ~ 3-5 nm and up to a volume of diameter about 1 µm, respectively. Therefore, EPMA results are quantitatively more reliable. However, it should be realized that EPMA, on the sample cross section, cannot be performed very close to the surface, because of the size of the analysed specimen volume by EPMA in a single measurement, whereas XPS applied to the surface of the nitrided sample provides data pertaining to the very surface adjacent region (see small depth of analysis indicated above). In particular for sample 6, the rapid decrease of the nitrogen concentration across the first micrometers leads to a considerable different between the EPMA and XPS nitrogen concentration values.

The highest value found for the concentration of nitrogen in the diffusion zone (substrate with ferritic matrix) is about 10 at.%, recognizing that the iron-nitride layer at the surface has a thickness less than about 1 µm (cf. section 3.3); see Fig. 5 and Fig. 3. Because the solubility for nitrogen dissolved in the ferrite matrix is much less (even considering the possibility of dissolved "excess" nitrogen (see Ref. 27), this concentration value also suggests that the predominant part of the nitrogen must be present as alloying element nitride precipitates. The possible presence of excess nitrogen, due to the misfit-strain fields surrounding the tiny, coherent, alloying element nitride particles, 28,29 may be supported by the difference on the amount of nitrogen present (10at.%) and the amount of nitrogen that could be combined to form alloying elements precipitates (maximum of about 5at.%). The nitrogen concentration difference, about 5at.%, would be present as excess nitrogen, even if a part of these atoms could be forming fine iron nitride precipitates. Nevertheless, the latest were not observed by XRD, ruling out this possibility.

The presence of fine nitrides particles was confirmed by TEM analysis. Two electron transparent foils were prepared from the nitrided sample 2. The first specimen was taken at ~5-10µm depth, i.e. within the diffusion zone/nitrided region; the second specimen was taken at depth ~50µm, i.e. in the nonnitrided core region of the sample. In the former specimen, the TEM and EELS measurements revealed only the presence of carbide particles (not shown). In particular, TEM analysis of the first specimen demonstrated the presence of vanadium nitride precipitates in the diffusion zone (see Fig. 6). Indeed, the EELS spectra of these precipitates showed relatively high concentrations of nitrogen and vanadium. An EPMA line scan across the nitrided case and into the unnitrided core is shown in Fig. 7 for sample 1. The coincidence of the carbon and chromium concentration peaks suggests that a chromium-carbide particle occurs in the transition region of the nitrided zone and the unnitrided material core. The conversion of chromium carbide into chromium nitride (as observed for gas nitrided Fe-Cr-C alloy [START_REF] Hekker | Excess nitrogen and discontinuous precipitation in nitrided iron-chromium alloys[END_REF]31 ) is probably kinetically hindered in the present experiments due to the relatively low nitriding temperatures applied in the current experiments (~400°C). This result is in agreement with the XPS data presented and as discussed in section 3.2 (XPS Cr spectrum). The presence of a chromium-carbide particle is revealed in the transition region of the nitrided case and the unnitrided core material.

Conclusions

-Upon ion nitriding the nitrogen surface concentration increases gradually until a stationary value has been attained, as the outcome of balancing of the reacting nitrogen ions arriving at the sample surface and the nitrogen removed by self-sputtering and by diffusion into the material bulk. A typical value for the sticking probability is ~ 0.15 N atoms per electronic unit charge.

-Upon ion nitriding the specimen thickness can decrease due to the effect of the accompanying sputtering and the specimen thickness can increase due to the uptake of nitrogen and the associated development of alloying element nitrides and the corresponding occurrence of a compressive macrostress parallel to the surface. At current densities above, say, 1 mA/cm 2 the sputtering effect dominates and the emergence of a stationary state for the nitrogen concentration at the surface (see above) parallels the occurrence of roughly linear Chromium Carbide relations between increase of treatment time and decrease of specimen thickness, and increase of current density and decrease of specimen thickness.

-Upon ion nitriding quenched and tempered tool steel at 400 ºC: (i) at the surface a thin iron-nitride (epsilon nitride) layer of thickness less than 1μm can develop;

(ii) the highest value found for the concentration of nitrogen in the diffusion zone (substrate with ferritic matrix), i.e. near to the interface with the thin epsilon-nitride surface layer, is about 10 at.%. The largest part of this nitrogen is incorporated in alloying element nitride precipitates;

(iii) the alloying elements V and Mo precipitate as tiny nitride particles; the chromium carbides present initially in the matrix are not converted to CrN at this low nitriding temperature, in contrast with nitriding at elevated temperature as with gaseous nitriding treatments.

Tables:

Figure 1 :

 1 Figure 1: Surface concentrations of nitrogen, iron and oxygen (a) samples nitrided during 300 minutes at different ion current densities and (b) samples nitrided with constant ion current density of 1.5mAcm -2 at different times. The solid lines are guides to the eyes.

Figure 2 :

 2 Figure 2: Height difference between nitrided and un-nitrided parts of the specimen for (a) specimens nitrided at constant time (300 minutes) and different ion current densities; and (b) for specimens nitrided at constant ion current density (1.5mAcm -2 ) and different times. The solid lines are guides to the eyes.

Figure 3 :

 3 Figure 3: Small angle X-ray diffractograms recorded from specimens nitrided under different conditions, as indicated.

Figure 4 :

 4 Figure 4: X-ray photoemission spectra of various samples (nonnitrided (reference) and three nitrided (see text)). Spectra are shown for the electron binding energy ranges as indicated in the figures for the major alloying elements: chromium (a), vanadium (b) and molybdenum (c). The arrows indicate the observed binding energy shift from the unnitrided reference sample to sample 2 (high nitrogen content).

Fig. 5 (

 5 Fig.5(a) shows that the nitrogen penetration depth is about the same for all samples. In first order approximation this may be expected for equal nitriding time treatments, if the same concentration of dissolved nitrogen would occur at the surface, which however is not truly the case (see sections 3.1 and 3.4), and if solid state diffusion of nitrogen would be the dominantly rate controlling process. The amount of nitrogen in the sample increases monotonically from samples 4 to 1, i.e., on increasing ion nitrogen current density. This behaviour is ascribed to the earlier attainment of a stationary state (cf. section 3.1) at the sample surface. The nitrogen profiles observed for samples 1 and 2 are more or less similar and this suggests that the ~ 60% increase of the ion-beam current applied for sample 1 with respect to sample 2 (cf. Table2) does not produce pronounced changes of the nitrogen depth profile. Indeed, after the establishment at the surface of a stationary state, as appears to hold for samples 1 and 2 (see Fig.5a), the magnitude of the ion-beam current is not important for the progress of nitriding in the solid substrate. In the case of sample 4a a stationary state at the surface has evidently not been realized (Fig.1and Fig.3). Fig.5(b) displays the effect of nitriding time for samples with identical (final) nitrogen surface concentrations, as follows from Fig.1(b). Sample 6 has a shallow nitrogen profile as a consequence of the short nitriding time even though the surface stationary state concentration has already been established.

Figure 5 :

 5 Figure 5: Nitrogen concentration-depth profiles (EPMA results) (a) for specimens nitrided for 300 minutes, for variable ion current density and (b) for specimens nitrided at constant ion current density (1.5mAcm -2 ) for variable nitriding time. Lines shown are guides to the eye.

Figure 6 :

 6 Figure 6: TEM micrograph (bright field) of vanadium-nitride precipitate within the diffusion zone of sample 2.

Figure 7 :

 7 Figure7: Concentration-depth profiles obtained by an EPMA line scan in the sample 1 cross section. The presence of a chromium-carbide particle is revealed in the transition region of the nitrided case and the unnitrided core material.

Table 1 :

 1 AISI H13 steel composition; data in atomic percent.

	Element	Fe	C	Mn	Si	Cr	Mo	V
	Content (±0.1)	87.3	2.5	0.4	2.1	5.9	0.8	1

Table 2 :

 2 Summary of applied ion nitriding parameters. Grey cells highlight constant-time and constant-ion-current-density specimen sets. The nitrogen beam dose, amount of retained nitrogen (i.e. the nitrogen uptake) and calculated nitrogen sticking probability (see section 3.6) are also displayed. All nitriding experiments were performed at 400 ºC.

						Sticking probability
		Ion current	Irradiation	Dose	Retained	(number of N atoms
	Sample	density	Time	(10 19 electronic	nitrogen	retained per
		(mAcm -2 )	(min)	charge units cm -2 )	(10 18 at. cm -2 )	electronic unit
						charge)
	1	2.43	300	273	4.7	0.017
	2	1.53	300	172	4.4	0.026
	3	0.37	300	4.2	3.6	0.086
	4	0.17	300	1.9	3.0	0.155
	5	1.53	179	10.3	4.2	0.0410
	6	1.53	75	4.3	0.71	0.0165

Acknowledgments

Part of this work was performed during a stay of LFZ at the Max Planck Institute for Metals Research in Stuttgart, Germany. The authors are grateful to Mrs. S. Haug and Mr. W.-D. Lang from the Max Planck Institute for Metals Research for assistance with the electron probe microanalysis measurements and TEM sample preparation, and to C. A. Piacenti from Unicamp for his technical assistance. This work is part of LFZ´s Ph.D thesis and was partially supported by FAPESP, project # Nº 97/12069-0. FLZ is a FAPESP and DAAD fellow. FA is CNPq fellow.