Driven phase transformations: A useful concept for wear studies?

C. R. Acad. Sci. Paris, t. 2, Série IV, p. 749–759, 2001 Surfaces, interfaces, films/Surfaces, interfaces, films (Physique appliquée/Applied physics)

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Driven phase transformations: A useful concept for wear studies? Laurent CHAFFRON, Yann LE BOUAR 1 , Georges MARTIN Service de recherches de métallurgie physique, Commissariat à l’énergie atomique, centre de Saclay, DEN/DMN, 91191 Gif-sur-Yvette cedex, France E-mail: [email protected] (Reçu le 23 février 2001, accepté le 23 février 2001)

Abstract.

The concept of driven alloys is introduced and examples are given for alloys under irradiation or under high energy ball milling. Both real and computer experiments show that the stationary configuration of alloys under external forcing depends on the overall temperature, on the ratio of the ballistic to the thermally activated atomic jump frequency, and on the space and time correlation of the ballistic jumps. As well as temperature, the description of driven phase transformations requires a new control parameter: the intensity of forcing. The latter is shown to be the irradiation flux for alloys under irradiation and the momentum transferred per unit time to an elementary volume of matter, under milling. We show how to use these concepts to address the wear rate of swift train wheels (TGV): it is found that the wear rate is proportional to the intensity of forcing.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS driven alloys / irradiation / ball milling / train wheels / Lyapunov functions / stochastic potentials / dislocations / phase separation

Transformations de phases dans les alliages forcés : Un concept utile pour les études d’usure Résumé.

On introduit le concept d’alliage forcé et on donne des exemples d’application aux alliages sous irradiation et sous broyage à haute énergie. Des expériences systématiques sur échantillons naturels et sur ordinateurs montrent que, sous sollicitation, l’alliage peut acquérir une configuration stationnaire qui dépend de la température, du rapport entre la fréquence des sauts atomiques balistiques et activés thermiquement, et de la corrélation spatio–temporelle des sauts balistiques. En plus de la température, la description des transitions de phase dans les alliages forcés nécessite un nouveau paramètre de contrôle : l’intensité de la sollicitation. Cette dernière est le flux d’irradiation pour les alliages sous irradiation, ou la percussion (quantité de mouvement transférée) par unité de temps dans un élément de volume représentatif pour les alliages sous broyage. Nous montrons comment tirer parti de ces notions pour un problème d’usure de roues de trains à grande vitesse (TGV) : on trouve que les vitesses d’usure et d’ovalisation des roues sont proportionnelles à l’intensité définie ci-dessus.  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS alliages forcés / irradiation / broyage / roue de train / fonction de Lyapunov / potentiels stochastiques / dislocations / séparation de phase

Note présentée par Guy L AVAL. S1296-2147(01)01217-3/FLA  2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés

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1. Introduction Phase transformations in the near contact region are often claimed to play a key role in the overall wear process: amorphization, second phase precipitation, carbide dissolution and martensitic transformations are currently reported. One usually ascribes the origin of such phase transformations to the temperature and pressure excursion at the contact points between the two bodies. However the magnitude of the flash temperature is a matter of debate (at least in metals) and seems to have been overestimated [1]. Moreover, the precise effect of a temperature transient is very sensitive to the details of the associated thermal history: e.g. depending on its time structure, a heat pulse may trigger either amorphization or crystallization, a fact well known in high storage density rewritable optical discs. The question then arises to identify which parameters control the occurrence of a phase transformation under the operating conditions. In this paper, we draw attention to two fields where a similar question has been successfully answered: the phase transformations in alloys under irradiation and the formation of new phases by high energy ball milling (‘mechanical alloying’). Finally, we show how to take advantage of these concepts for addressing the question of phase transformations in high speed train wheels and related questions. It is worth mentioning that in both cases of alloys under irradiation and of mechanical alloying, the effect of high transient temperatures (thermal spike) has long been (and sometimes still is) advocated, but with a rather low predicting power. On the contrary, the concepts presented below, explain in some detail and with real predicting power, many intriguing features of phase transformations driven by irradiation or by ball milling. Before presenting these concepts, we summarize below the basic idea of the theory of driven phase transformations common to all types of external forcing. 2. Basics of the theory of driven alloys Under standard thermodynamical conditions (e.g. isothermal, isochore) the alloy explores its configuration space according to a rule such that the stable configuration (solid solution, two-phase alloy or ordered compound, etc.) is the most frequently visited one. In the simple case of diffusion controlled transformations, the above rule results from the way the jump frequency of atoms (activation energy, frequency factor) depends on their environment, which evolves due to atomic jumps. Under irradiation or under repeated shearing, atoms change their environment because of the thermally activated atomic jumps, as above, but also because of external forcing: nuclear collisions under irradiation, dislocation glide under plastic deformation, fracture and welding under more severe milling, etc. The latter atomic movements (hereafter named ‘ballistic jumps’) occur at a frequency that does not depend on the temperature (as do the ‘thermal jumps’), but on some intensity of the external forcing which remains to be identified. The question is now to identify which is the most stable configuration of an alloy when the configuration space is explored according to the above two mechanisms, operating in parallel: ballistic and thermally activated jumps. A phase change will be observed when the relative weight of both mechanisms is sufficiently altered. This way of phrasing the problem is inspired from the kinetic Ising model with competing dynamics [2]. However, several features of importance for materials science have not yet been addressed by the statistical physics community and are dealt with here. In particular, the ballistic jumps, unlike the thermal jumps, occur in a correlated manner, in space and time: e.g. all atoms above a slip plane are shifted by one Burgers vector with respect to their neighbors below the plane, every time a dislocation sweeps through the plane. This space–time correlation of the ballistic jumps (which can be viewed as a structured external noise) has important consequences on phase stability under forcing. Reference [3] gives a detailed recent account of the state of the art for driven alloys.

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That some phases are destroyed and other are nucleated in alloys under irradiation is a well documented fact. But few systematic studies have been done to assess if a given phase is stable under certain irradiation conditions and not stable under other conditions. If so the concept of equilibrium phase diagrams could be extended for alloys under irradiation, and one should identify which control parameter should be used along side temperature and concentration to draw such diagrams. In the simplest case, the collisions of energetic particles (1 MeV electrons, 1 MeV neutrons, few 100 keV ions) with the atoms of a metallic alloy result in two elementary processes: • the production of point defects, at a rate proportional to the irradiation flux: the latter rate is measured in dpa/s (number of displacements per atom per second); • the intermixing of atoms on the lattice (ballistic mixing) at a rate proportional to the flux, and expressed in rpa/s (number of replacements per atom per second). The point defects so created diffuse towards elimination centers (extended defects or recombination centers): as a result, after some transient, a steady defect supersaturation (which enhances atomic mobility) and steady defect fluxes settle in the alloy under irradiation; the latter fluxes induce solute redistribution due to inverse Kirkendall effect. To be more precise, such states are quasi stationary only: indeed at a larger time scale, the microstructure evolves and the defect supersaturation and fluxes as well. Ballistic mixing, enhanced atomic mobility and local defect fluxes are the elementary processes at work in alloys under irradiation. It has been experimentally established that under irradiation, alloys may achieve a (quasi)stationary configuration (e.g. single- or two-phase configuration, with ordered or disordered compounds, etc.) which depends on the irradiation conditions: one may therefore introduce dynamical equilibrium phase diagrams. Few such diagrams have been experimentally established: three of them, shown in figure 1, reveal important features of phase equilibria under irradiation (for details, cf. [3–5]). As can be seen on figure 1: (a) the stability of a given phase under irradiation (here the ordered compound FeAl with the B2 structure) depends on the irradiation temperature; at low temperature, the stationary value of the long range order parameter is close to zero, i.e. the compound is disordered, while at higher temperature, the stationary state is ordered. The transition between the disordered and the ordered state under 1 MeV electron flux of 7·1019 e− ·cm−2 ·s−1 occurs at  60 ◦ C; the transition is of the first kind (i.e. with an abrupt jump in the degree of order and with hysteresis) while the thermal transition is of the second kind. (b) the solubility (here of Zn in Al) is a function of the irradiation temperature and of the irradiation flux. Notice that under irradiation, GP zones form at temperatures much greater than the thermal solubility limit; the higher the irradiation flux Φ, the higher the limit temperature for the formation of GP zones. (c) for a given irradiation flux and temperature, the stability of a phase (here γ  Ni3 Si in a Ni-6 at% Si solid solution) also depends on the way the displacements are produced: an electron irradiation which produces the displacements in a dilute manner stabilizes the γ  phase in a broader temperature range than a Ni ion irradiation which produces large displacement cascades, where many atoms are replaced at once in a small volume (few nm3 ) of the crystal. Cascade size is one more control parameter of the stability of phases under irradiation. All the above features and a few others have been modeled by various means described in reference [3]. As an example, we summarize below the theory of driven order–disorder transition since it will be used also for alloys under ball milling (see Section 4). As suggested by figure 1a, an ordered alloy under irradiation experiences disordering processes (forced by the ballistic atomic jumps) which compete with the reordering process resulting from thermally activated jumps [7,8]. Ignoring all the microstructural details, one defines a mean ballistic jump frequency Γbal , while the frequency of thermally activated atomic jumps is Γt . The rate of change of the degree of long range order S is given by [7,10]:

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3. Alloys under irradiation

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In situ measurements of LRO

(a) The degree of long-range order (S) under 1 MeV irradiation with an electron flux of 7·1019 e− ·cm−2 ·s−1 in the FeAl stoechiometric compound, as a function of the time, reaches a steady value (left-hand figure); the latter (S) depends on the temperature (right-hand figure); an order–disorder transition under irradiation is observed at  60 ◦ C [3]. GP zones in situ TEM

Post mortem γ  TEM, Ni 6at.%Si

−1 MeV · e− −150 keV Ni+

Cascade size effect (b) The coherent solubility limit of Zn in Al under 1 MeV electron irradiation; (· · ·) coherent solvus outside irradiation, (− −) coherent solvus under irradiation flux Φ = 2.5·1019 e− ·cm−2 ·s−1 , (—) idem with flux 10·Φ [4].

(c) The coherent solubility limit of Si in Ni under irradiation: in a Ni-6at% Si solid solution, γ  precipitates (Ni3 Si with the L12 structure) form under irradiation in a well defined field of the flux–temperature diagram (while the solubility limit is about 10 at% in the absence of irradiation in the temperature range under consideration). This field is broader for e− irradiation (1 MeV·e− , - -) than for ion irradiation (150 keV Ni+ , —) [5].

Figure 1.

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    dS ∂L(S) = 4Γt  (1 − S)2 exp(2S/τ ) + γ − (1 + S)2 exp(−2S/τ ) + γ = − dt ∂S Γbal T with γ = and τ = Γt  Tc

(1a) (1b)

Tc in equation (1b) is the critical temperature for the order–disorder transition (in the absence of irradiation). Indeed if Γbal is set to zero, one recovers the classical mean field kinetic model for the order–disorder transition [3]. In equation (1a), a Lyapunov function L(S) has been introduced in the last right hand side. L(S) is the negative of the primitive of the first right hand side of equation (1a). As a consequence, the extrema of L(S) (∂L(S)/∂S = 0) give the possible stationary values of S: the maxima correspond to unstable stationary states, and the minima to locally stable stationary states. As seen in equation (2), by construction, L(S) always decreases with increasing time:  2 ∂L(S) ∂L ∂L ∂S = × =− ∂t ∂S ∂t ∂S

(2)

The first row of figure 2 displays the only four possible shapes of L(S), depending on the values of the reduced control parameters, γ = Γb /Γt  and τ = T /Tc . Column (d) corresponds to a stable disordered state (S = 0); if varying γ and τ , L(S) transforms into the shape depicted in column (a), the disordered state becomes unstable (L(0) is a maximum) to the benefit of a partially ordered state (L(0.8) is a minimum) and the transition is of the second kind. If the change in γ and τ is such that L(S) transforms into the shape depicted in column (b), the transition is of the first kind, since both S = 0 and S = 0.7 are locally stable stationary states. The kinetic pathway towards the stationary value S¯ of the degree of long range order, also depends on γ and τ as shown by the second row of figure 2. However, L(S) has not all the properties of a free energy, as discussed in [6]. In the same spirit, a Lyapunov functional has been constructed for the case of precipitation–dissolution. In the simple case of a regular solution model and with some simplifying assumptions (in particular, neglecting

Figure 2. The four possible shapes of L(S) (first row) and the associated kinetic pathways (second row) as a function of the control parameters of the model (γ, τ ). Each column corresponds to one of the four distinct cases.

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the inverse Kirkendall effect), the Lyapunov functional is nothing but the free energy functional evaluated at an effective temperature Teff = T (1 + Dbal /Dchem) where Dbal and Dchem are respectively the ballistic and chemical diffusion coefficients (the latter being enhanced by the point defect supersaturation). Taking the inverse Kirkendall effect into account, reveals more subtle effects such as, for instance, an increase of the critical point, yielding a decrease of solubility (as exhibited in figures 1b and 1c). The effective temperature concept, although very crude, often yields a good rule of the thumb to estimate the effect of irradiation on the stability of phases. As an example, at low temperature, Dchem is small while Dbal remains large; Teff may thus achieve very high values, e.g. above the melting point: the stable phase would be an amorphous phase. More sophisticated theoretical approaches are necessary to account for the effect of the size of the cascade. Stochastic potentials have been introduced which correctly predict this type of effect. Also, kinetic Monte Carlo simulations of the configuration of an alloy on a rigid lattice where atoms change site both because of ballistic jumps and of thermally activated point defect jumps are of great use and are discussed in [3]. 4. Alloys under high-energy ball milling Several studies show that in a given mill, for a given adjustment of the milling parameters and after a sufficiently long milling time, the material achieves some sort of steady state, at least at the scale of standard characterization techniques. This is the case for the grain size and internal strain in pure metals, for the level of long range order and the domain size in ordered compounds, for the single or two phase state in alloys and for the amorphized fraction in crystalline compounds (cf. [9] for a review). The above mentioned steady states depend on the milling conditions. For a given material and a given milling device, two studies at least show that, the steady state which is achieved, depends on two parameters: the milling temperature and the milling intensity. The experiments prove that the milling intensity is the momentum transferred by the ball to the unit mass of powder per unit time: I=

Mb · Vb · f  2 m/s Mp

(3)

where Mb and Vb are respectively the mass of the ball and its impact velocity on the powder in a fully plastic collision regime, Mp is the mass of the powder and f is the frequency of the impacts. Figure 3 shows that a NiZr compound reaches a fully amorphous steady state only if milling is done with an intensity, as defined above, beyond some threshold value. The very same milling intensity allows us to rationalize the steady states which are achieved by a FeAl compound at stoechiometry under milling [7,10].

Figure 3. End state (amorphous or crystal + amorphous) for the Ni10 Zr7 compound under ball milling as a function of the impact frequency and of the momentum of the ball.

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Figure 4. (a) Disordering kinetics of FeAl at 3 distinct temperatures, with a milling intensity I = 2000 ms−2 ; (b) Time evolution of the degree of order for two distinct milling intensities at 373 K; for a given intensity, the same steady state is reached independently of the initial state, fully ordered (#6) or fully disordered (#18).

For a given milling temperature, the higher the intensity, the lower the stationary degree of order; for a given milling intensity, the lower the milling temperature, the lower the degree of order ( figure 4a). At a low enough temperature and a high enough intensity, the steady state is fully disordered. Finally, at given milling temperature and intensity, the steady state degree of order which is achieved does not depend on the initial state of the material (fully ordered or fully disordered) ( figure 4b). A slightly intuitive explanation of the milling intensity is as follows: the momentum transferred by the ball to the powder, Mb Vb , is a measure of the product of the force applied by the ball onto the powder by the duration of the interaction τ0 ; τ0 is the time for the ball to stop moving:  Mb · (Vb − 0) = −

τ0

Mb · a(t) · dt

(4)

0

In (4), a(t) is the acceleration of the ball and Mb · a(t) is the force applied by the ball onto the powder. One may accept the idea that, the larger the force and the duration of its application on a given grain, the larger the plastic strain, which is at the origin of the changes of microstructure under study. The second ingredient of the intensity is the ratio f /Mp : i.e. the frequency of the collisions between ball and powder, for a unit mass of powder. For a milling device which produces a perfect mixing of the powder, the latter ratio scales with the frequency at which a given grain of powder interacts with the ball. A more exact formulation of this idea is to write the collision frequency for a given grain as f · (Ma /Mp ) where Ma is the mass of powder affected by a single collision. The expression we get for the intensity appropriate to milling is: Ibm = Mb · Vb · f · (Ma /Mp ) (N )

(5)

Notice that all the experiments performed on FeAl under ball milling were done with a single ball size which makes Ma a constant: I, used in figure 4 is thus simply Ibm divided by the constant Ma . Since the steady state depends on the impact frequency, the steady state must result from a competition between damaging and restoring processes: the latter are less effective, the shorter the time between two impacts on the grain under consideration (i.e. the higher the frequency). In the opposite case where the steady state would be independent of the frequency, one would conclude that the material has developed a shock resistant microstructure, at least for the impacts under consideration.

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Figure 5. Diagram (γ, τ ) of the dynamical equilibrium states for the A2 ↔ B2 transition as computed by the driven alloy model described by equations (1a) and (1b). Each point corresponds to one ball milling experiment: plain symbols correspond to the experiments where the observed and computed kinetics are identical; the two open symbols point to computed kinetic pathways which do not agree with the observed ones.

This is the case neither in the works on amorphization, nor in the works on the order–disorder transition which helped in defining the milling intensity I (3); in both cases, the effect of the collision frequency has been established experimentally. As in the case of irradiation, one may therefore speak of a dynamical equilibrium between phases and establish dynamical equilibrium phase diagrams, where the stability fields of various phases are sketched as a function of the two control parameters: the milling temperature and the milling intensity. The analogy with alloys under irradiation is straightforward: in the case of the order–disorder transition in FeAl under ball milling, the damaging process (destruction of the long-range order) is a result of the plastic shear of the grains, e.g. one dislocation gliding along its slip plane leaves a planar stacking fault behind; alternatively, one pair of jogged dislocations on a single slip plane, leaves behind a tube of stacking faults, etc. Each atom along the fault has changed neighbors; this happens every time a dislocation glide passes close to that atom. As for the restoring mechanisms, differential thermal analysis of a powder which has been disordered by ball milling, shows that the thermal reordering process takes place prior to recrystallization, and in a temperature range similar to the milling temperature. Moreover, shear induced vacancies enhance the atomic mobility, hence speed up the reordering process in the time interval between two collisions. Based on this simple scheme, the reduced forcing parameter γ can be expressed as a function of the intensity and of the temperature [10]. The ballistic jump frequency is found to be directly proportional to the milling intensity I; the thermal jump frequency contains a term proportional to I which can be interpreted as resulting from the injection of vacancies by non conservative dislocation movements. Varying the milling temperature and the milling intensity changes γ and τ ; following figure 2, S varies either smoothly or abruptly when crossing transition lines in the dynamical equilibrium phase diagram (γ, τ ). The transition line, as deduced from equation (1a), is shown on figure 5, together with the points representative of all experiments done. Among the 8 experiments, 6 (marked by plain symbols) exhibit behavior in agreement with the theory. The two odd results (experiments 1 and 7) are obtained close to transition lines, the theory of which is beyond the scope of the paper [3]. The above theory ignores all the microstructural features and drastically caricatures the contribution of ballistic jumps to the disordering process: their contribution is introduced as an uncorrelated jump frequency of individual atoms, while ‘ballistic jumps’ occur in a collective manner along the shear planes, in the course of the deformation of the crystallites. The effect of this latter space and time correlation of ballistic jumps has been studied using atomic scale computer simulations by a kinetic Monte Carlo technique [7].

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The main experimental findings are reproduced, provided we assume that the disordering process is created by APB tubes and that non equilibrium vacancies, produced by non conservative dislocation glide, are eliminated at the boundaries of the nano-grains. The microstructures, which are generated by the simulation, exhibit many interesting features which are worth investigating experimentally.

High-speed train wheels are known to develop localized damage which results in a modulation of the wheel radius (e.g. with a period 2π/3 on figure 6). The amplitude of the modulation (h) increases approximately exponentially with the mileage as shown by figure 7. Each point of the surface contact of the rolling strip experiences a stress burst whenever it comes into contact with the rail. The classical theory of hertzian contact [11] gives the stress level in the vicinity of the contact and the dimensions of the affected volume. In other words, each point of the wheel is periodically strained and eventually restores between two straining events. We recognize the basic feature of driven alloys and we introduce the ‘intensity’ of the forcing in the spirit explained above: Iw (w stands for wheel) is the force applied on the Hertz volume of affected materials times the duration of straining, times the frequency of the process. All these terms can be deduced from the elastic theory of contact and from classical mechanics [12]. We find:  Iw =

N 3 (θ) E ∗ · R · br

1/2 (N )

(6)

θ defines a fixed angular position on the wheel, N (θ) is the force acting at point θ when the latter is in contact with the rail, E ∗ is an effective modulus, R is the mean radius of the wheel and br is the width of the rolling strip in contact with the rail. For a perfectly circular wheel, Iw is the same all around the wheel; for a deformed wheel, Iw varies with the angular position around the wheel. It is found [12] that, above some cross-over velocity the intensity is maximum at those points where the radius is minimum. From the master curve of the wear of many wheels as a function of distance traveled, it is found that the rate of wear is proportional to the intensity as estimated from equation (5). Since the higher the intensity the faster the wear, any local variation of the radius is expected to be amplified at high speed. An order of magnitude estimate yields Iw in the range 100 to 150 N, for an amplitude of modulation of 0.6 mm at 200 km/h. Figure 8 shows that ball milling a XC48 steel (wheel material) at an intensity

Figure 6. Profile of a TGV wheel showing the periodic modulation of the radius. (Courtesy of SNCF.)

Figure 7. Amplitude of the radius modulation as a function of total mileage (in 103 km), according to SNCF.

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Figure 8. Neutron diffraction profiles of the XC50 powder ball milled at 4000 m·s−2 and 303 K for increasing times (from 1 to 92 h). The insert shows the decrease of the cementite fraction for increasing milling times.

(Ibm · Ma ) of 4000 m·s−2 and at room temperature induces the dissolution of cementite in the steel, as is known to occur in rails [1,13]. Work is in progress to make a quantitative link between Iw and Ibm . 6. Conclusion The occurrence of phase transformations in driven alloys with a given composition is controlled by three parameters: the overall temperature, the intensity of the forcing and the space and time correlation of the latter. The forcing can be measured by the ratio of the frequencies of ballistic and thermally activated atomic jumps. Under irradiation, the intensity is proportional to the flux of particles; under repeated shearing, the intensity is the momentum transferred to the strained volume of material per unit time; this concept when applied to train wheels shows that the wear rate is proportional to the intensity. The effect of the space and time correlation of the forcing is identified experimentally under irradiation, where it corresponds to the effect of the size of the displacement cascades. Under repeated shearing, a similar effect has been identified on computer simulations but is not yet demonstrated experimentally. The present concepts are worth extending to fatigue studies: along this line, a fatigue test should be characterized by the value of the applied load, the duration of application of the load, and the period of loading. 1

Now at ONERA, LEM, BP72-29, avenue de la Division Leclerc, 92322 Châtillon, France.

Acknowledgements. The technical assistance of G. André (LLB Saclay) and S. Poissonnet is gratefully acknowledged. We also thank J.J. Viet (AEF/SNCF) and F. Demilly (Valdunes) for their interest in this work and for providing material from train wheels.

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[4] Soisson F., Dubuisson P., Bellon P., Martin G., in: W.C. Johnson, J.M. Howe, D.E. Laughlin, W.A. Soffa (Eds.), Solid Phase Transformations, Vol. 981, TMS, 1984. [5] Cauvin R., CEA Report R5105, Gif-sur-Yvette, France, 1981; Cauvin R., Martin G., Phys. Rev. B 23 (1981) 3322 and 3333; Cauvin R., Martin G., Phys. Rev. B 25 (1982) 3385. [6] Barbu A., Martin G., Chamberod A., J. Appl. Phys. 51 (1980) 6192. [7] Pochet P., thesis, CEA Report R 5753, 1997. [8] Bellon P., Averback R.S., Phys. Rev. Lett. 74 (1995) 1819; Bellon P., Averback R.S., in: G. Ananthakrishna, L. Kubin, G. Martin (Eds.), Solid State Phenomena, Non-Linear Phenomena in Materials Science III, Vol. 42–43, Transtech, Aedermannsdorf, 1995, p. 69. [9] Pochet P., Chaffron L., Bellon P., Martin G., Ann. Chim. Sci. Mat. 22 (1997) 363. [10] Pochet P., Tominez E., Chaffron L., Martin G., Phys. Rev. B 52 (1995) 4006. [11] Johnson K.L., Contact Mechanics, Cambridge University Press, 1985. [12] Le Bouar Y., Chaffron L., Martin G., in preparation. [13] Sunwoo H., Fine M.E., Meshii M., Stone D.H., Metal. Trans. A 13A (1982) 203.

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