Modelling of the Ultrasonic Shot Peening Process. C. Pilé, M. François, D. Retraint, E. Rouhaud and J. Lu

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1 Materials Science Forum Online: ISSN: 1-975, Vols. 9-91, pp 7-7 doi:1.8/ 5 Trans Tech Publications, Switzerland Modelling of the Ultrasonic Shot Peening Process C. Pilé, M. François, D. Retraint, E. Rouhaud and J. Lu LAboratoire des Systèmes mécaniques et d Ingénierie Simultanée (LASMIS, FRE CNRS 719) Université de Technologie de Troyes BP - 11 Troyes Cedex France manuel.francois@utt.fr Keywords: Ultrasonic shot peening, shot speed distribution Abstract. The aim of this work is to reach a better understanding of the ultrasonic shot-peening process and, in particular, the evolution of the shot speed distribution. A simple 1D modelling of the interaction between the shots and the sonotrode is carried out. The impact is considered as inelastic with an energy absorption that depends on the speed of the shot. It is found that after about 1 interactions ( 1s) the speed distribution in the chamber follows a Maxwell-Boltzmann distribution, which is the distribution found in a perfect gas at equilibrium. The influence of various process parameters such as the sonotrode amplitude, the vibration frequency on the average speed and on the Almen intensity is studied. Introduction Shot peening is a cold working process usually performed to improve the fatigue life of metallic components by generating compressive superficial residual stresses. In this study, we focus on a mechanical treatment derived from conventional shot peening, called ultrasonic shot peening [1]. It consists of a piezoelectric generator exciting the vibration of a sonotrode by means of a converter. The chamber is made up of the sonotrode and a cover. It contains spherical steel shots with diameters of 1 or mm. In contact with the sonotrode, shot is projected and move randomly. The part is placed inside the chamber. Due to the high frequency of the system (khz), the treated sample is peened with a very high number of impacts during a very short time over its entire surface. Various parameters make it possible to vary shot peening intensity: the specimen-sonotrode distance, the diameter of the shot, the shot quantity, the volume and shape of the chamber, the amplitude and the frequency of vibration of the sonotrode as well as the treatment time. In order to optimise the process performances, it is necessary to model the operating details. We thus studied the shot-sonotrode interaction in order to know the shot speed distribution induced by this interaction. Description of the model It concerns a one dimensional model which describes the interaction between the shot and the sonotrode. The impact is considered as inelastic with an absorbed energy which depends on the speed of the shot. The following assumptions are posed: - the contact is instantaneous, Indeed, the contact duration was estimated at approximately 1µs, which is small compared to the vibration period of the sonotrode (5µs). - shot arrives randomly on the sonotrode, - if V is the initial speed, then the interaction speed is V +W (t) where W (t) is the sonotrode speed at the moment t of the contact, All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-1/5/1,1:7:)

2 8 Residual Stresses VII, ICRS7 - to take into account dissipations, it was assumed that the rebound speed is V=α(V +W (t)) where α is a restitution coefficient that decreases with the interaction speed. It varies between (perfectly soft shock) and 1 (perfectly elastic shock). In the present study we considered it as a global coefficient that takes into account all the interactions which take place in the chamber (shot-sonotrode, shot-shot, shot-specimen, shot-chamber walls). To simplify the variation obtained in a previous study [], α is taken equal to 1 when speed is null and equal to when it is higher than a given speed and, in between, it varies linearly (Fig. 1). A parametric study varying the slope was carried out to obtain the best fit with experimental results [3]. The limit speed obtained is 13 m/s. This parameter is the only one that is adjusted in the model. Restitution coefficient 1, 1,8,,, Shot speed (m/s) Fig. 1 : Variation of the restitution coefficient versus the shot speed. The shot movement in the chamber are assumed to be chaotic, i.e. the memory of the system is lost after several interactions of a given shot with other shot or with the specimen. The mean free path of a shot in the chamber is smaller than the size of the chamber. Thus it was assumed that the arrival probability of a shot on the sonotrode is random and uncorrelated with its previous interaction with the sonotrode. The shot movement is described by x(t)=x -V t, where axis x is the vibration axis of the sonotrode. The value of x is taken randomly distributed with a constant probability density. The variation range of x is chosen so that the arrival time remains within a vibration period of the sonotrode (see figure ). The movement of the sonotrode is described by X(t)=X cos(πf t), where X is the amplitude of the sonotrode vibration (known) and f is the frequency (known). x(t) x(t) Shot trajectories x B V Sonotrode trajectory X A interaction no interaction C High interaction probability (speed gain) Low interaction probability (speed loss) Fig. : Representation of the shot-sonotrode interaction with low shot speed (left) and high shot speed (right) On figure the shot and sonotrode positions are reported versus time. It can be seen that interaction between the shot and the sonotrode can only occur between A and B, when the sonotrode speed is

3 Materials Science Forum Vols positive. Thus, the interaction speed V +W (t) will be larger than V and the shot will gain speed. Between B and C, when the sonotrode speed is negative and larger than the shot speed, interaction is impossible so, the probability of a shot to loose speed is equal to zero. When the speed V increases, the BC range disappears and shot can loose speed. However, the probability to loose speed is much lower than the probability to gain speed, except when V tends towards infinity. This schematic description is, of course, to be modulated by the influence of the restitution coefficient, in particular when it depends on the interaction speed. Figure 3 represents two examples of shot speed distributions obtained with the model. These distributions were obtained after 1 iterations when the system has reached equilibrium. In the first case, the restitution coefficient is equal to 1. The shocks are thus completely elastic. It can be noted that the speed distribution follows a Maxwell-Boltzmann distribution. In the second case, the restitution coefficient depends on the shot speed. The speed distribution keeps a similar shape but it is not exactly a Maxwell-Boltzmann distribution. 8 Shot quantity (%) 1 8 Shot quantity (%) Speed (m/s) 8 1 Speed (m/s) 8 1 Fig. 3 : Shot speed distribution. The points represent the model response. The curve represents the Maxwell-Boltzmann distribution fitted on the model results. Left : elastic shocks. Right : inelastic shocks. Figure shows the evolution of mean shot speed versus the iterations number. Whatever the initial shot speed, the system reaches equilibrium with a constant mean speed after few iterations. The average duration between two shocks is of 1 ms approximately. This duration is very weak compared with treatment times which are carried out (about a few minutes). It can be considered that the system is at equilibrium during the whole treatment duration. Mean shot speed (m/s) v=1m/s v=5m/s 8 1 Number of iterations Fig. : Evolution of the mean shot speed versus the iteration number.

4 7 Residual Stresses VII, ICRS7 Results The average shot speed was studied as a function of the vibration frequency and amplitude. When the restitution coefficient is taken equal to 1, the average speed tends towards infinity when the number of iterations increases. With a constant restitution coefficient lower than one, the speed distribution stabilizes at finite values. The average speed (or similarly the quadratic average speed) increases monotonously with the frequency f or with the displacement amplitude X of the sonotrode. In fact, these two factors are not independent, they are connected to the speed amplitude W through the relation W = π f X. If the variations of the average speed with X or f are plotted versus W, it can be seen (figure 5) that they belong to the same curve. Practically, this point is not important because a sonotrode is built to work at a given frequency and only the vibration amplitude X can be varied. With our equipment, two values of X are available : 5 and 5 µm. On figure 5, it can be seen that the curve increases and then decreases with the speed amplitude of the sonotrode. This is due to the competition between two phenomena : an increasing amplitude tends to give more kinetic energy to the shot but, as it increases the interaction speed, the dissipated energy is larger. Wether this evolution is physically justified remains to be investigated with a more realistic evolution of the restitution coefficient with speed. Shot mean speed (m/s) 3,5 3,5 1,5 1,5 Xo = 5µm f = khz y = -,x +,93x 3 -,1773x + 1,189x +, Amplitude speed of the sonotrode Fig. 5 : Evolution of the mean shot speed with the amplitude speed of the sonotrode. For a fixed frequency, the amplitude X is varied (stars) and for a fixed amplitude, the frequency f is varied (diamond shapes) As no experimental results of the speed distribution on our equipment is available yet, the model was validated on bibliographic results. To our knowledge, the only available results are those given by Chardin [3]. He machined a special specimen in which a piezoelectric accelerometer was fitted. Each time a shot arrived on the accelerometer surface, an electric impulse was created by the sensor. The amplitude of the impulse was related to the shot speed. The relation between the two quantities was calibrated by dropping shot from a given height on to the accelerometer. In Chardin s case, the sonotrode amplitude was µm and the frequency khz. These values were fed into the model and the speed distribution was computed. The slope of the restitution coefficient was adjusted in order to obtain the best possible fit with Chardin s results. The experimental and calculated values are given on figure. It can be seen that the agreement between the two distributions is good.

5 Materials Science Forum Vols Shot quantity (%) Model Experimental,7851,1879 5,95 7,989 Shot speed (m/s) Fig. : Comparison between experimental and model results. The distribution is given with speed classes only to cope with Chardin s results. However, it gives a coarse idea of the actual distributions which can be seen, for example, on figures 3 and 7. Then, assuming that the dissipation in our equipment is equivalent to that in Chardin s equipment, the parameters of our sonotrode were fed into the model. Two amplitude values X = 5 µm and X = 5 µm, corresponding to speed amplitudes W = 3.1 m/s and W =.8 m/s respectively, were studied and the corresponding distributions are plotted on figure 7. It can be seen that the overall shapes of the two distributions are very close. However, for W =.8 m/s, % of the shot have a speed higher than m/s while, for W = 3.1 m/s, only 1% of the shot exceed a speed of m/s. This means that there are more shot with high energy for W =.8 m/s than for W = 3.1 m/s. Low energy shot gives a quasi elastic rebound while high energy shot leads to significant plastification of the peened material. Shot frequency (%) Wo = 3,1 m/s Wo =,8 m/s 8 1 Shot speed (m/s) Fig. 7 : Shot speed distributions for two speed amplitudes of the sonotrode. Almen strips were shot peened using the two sonotrode amplitudes and give 7A and 5A for W = 3.1 m/s and W =.8 m/s respectively. This confirms that the distribution with higher speeds gives higher residual stresses than the other. Discussion The model developed in the present study, although very simple, can give interesting information on the speed distribution of the shot, in particular on the average speed or energy of the shot. Its main advantage is that it contains only one adjustable parameter. All the others are known. However, it does not take into account some process parameters that can have an influence such as : - shot diameter [], - shot quantity,

6 7 Residual Stresses VII, ICRS7 - volume and shape of the chamber, - hardening of the shot, the sonotrode, the specimen and the chamber walls. The latter parameter can be taken into account through the restitution coefficient, but its evolution with time may be difficult to model. The restitution coefficient was selected linear versus speed to simplify the model. Actually, the restitution coefficient for a single shock decreases exponentially with speed and seems not to reach a zero value, even for high speeds [5]. A shot experiences many interactions within the chamber (shot-wall, shot-shot, shot-specimen, shot-sonotrode) and it may be oversimplifying to reduce it to a single coefficient. In the present model, the shocks are normal to the sonotrode surface which is not true in reality. There is a distribution of the incidence angle that can only be modeled with a fully 3-dimensional model of the shot trajectories. The present model is more elaborate than the one developped by Chardin who considered a purely elastic interaction between the shot and the sonotrode. Thus, he could not compare quantitatively his numerical results with the experiments. The speed range that he obtains with his model is much higher than the experimental values. Furthermore, the speed distributions that he obtains are not equilibrium distributions : as mentionned above, for purely elastic interactions, the speed tends towards infinity. Conclusion In the present study, a simple model of the interaction between the shot and the sonotrode has been developed to predict the speed distribution during a ultrasonic shot-peening process. It correlates well with experimental results. However, these were obtained on a different equipment than ours. To remedy this and get more detailed information about the actual speed distribution, we intend to perform measurements with a high speed camera. The variations of the restitution coefficient can be described more accurately with the help of the finite element model developed in a previous study []. This will allow a better exploitation of the model in order to correlate the process parameters with the residual stress levels. If necessary, it is intended to develop a 3D model to take into account the incidence angles of the shocks. The financial support of this study by Région Champagne-Ardenne is gratefully acknowledged. References [1] J. Lu, P. Peyre, C. Oman Nonga, A. Benamar and J.F. Flavenot, Residual stress and mechanical surface treatments, Current trends and future prospects, in Proceedings of the th International Conference on Residual Stresses, Baltimore, p. 115, SEM (199). [] E. Rouhaud and D. Deslaef, Materials Science Forum, Vols. -7, (), pp [3] H. Chardin,Etude de la densification par grenaillage ultrasons d un matériau métallique poreux élaboré par métallurgie des poudres, PhD thesis, Ecole des Mines de Paris, december 199. [] C. Pilé, D. Retraint, M. François, J. Lu, Effect of very high stress levels on the fatigue life of a TiAl based alloy, present conference. [5] A. H. Kharaz and D. A. Gorham, A study of the restitution coefficient in elastic-plastic impact, Philosophical Magazine Letters, Vol. 8, No. 8, (), pp

7 Residual Stresses VII, ICRS7 1.8/ Modelling of the Ultrasonic Shot Peening Process 1.8/ DOI References [] E. Rouhaud and D. Deslaef, Materials Science Forum, Vols. -7, (), pp / [5] A. H. Kharaz and D. A. Gorham, A study of the restitution coefficient in elastic-plastic impact, Philosophical Magazine Letters, Vol. 8, No. 8, (), pp /