1. Introduction

Shot peening process has been widely used as one of the most common surface treatments for mechanical and structural components. In this process, a large number of hard shot particles are projected by compressive air at high speed and they impinge on the surface of processed materials. At that time, compressive residual stress and strain hardening are induced in the near surface of treated materials and this treatment improves the fatigue strength of mechanical components.

Shot peening process is very complex phenomena, since a large number of particles hit the specimen surface in a short time. Therefore, the surface modification is difficult to estimate. Recently, numerical approach has been taken to predict the surface modification for shot peening processes. Numerical studies about compressive residual stress induced by shot peening process have been conducted by theoretical based analysis [1] and finite element analyses [2-12]. They focused on the collision behaviour between shot particles and treated materials. The study about shot velocities has been carried out by Ogawa et al. [13], and the analyses of modelling particle behaviours by discrete element method have been developed by Bhuvaraghan et al. [14] and Hong et al. [15]. However, they did not consider the forces from compressive air, which strongly affect the particle behaviour before collision. For the application of erosion testing and abrasive air jet machining, the particle collision and particle streams has been simulated by Ciampini et al. [16], Papini et al. [17] and Li et al. [18]. It is noted that the velocity and the size of shot particle in those application is different from the shot peening process.

The purpose of this paper is to model shot particles before collision and implement the model into multi-body dynamics simulator. The particle behaviour before collision are modelled by considering all forces acting on shot particles, those are drug force from compressive air, the interactive forces between particles. Then, the motion of shot particles are implemented and calculated by multi-body dynamic simulator.

The organization of this paper is followings. To investigate particle behaviours during shot peening in detail, shot peening experiments were carried out. Then, the effects of particle diameter, pressure and the number of particles on impingement behaviours were studied. The developed models were implemented in multi-body dynamics frame work and particle behaviours were simulated and compared with the experimental results.

2. Experimental

2.1. Experimental Set-up

Shot peening experiments were carried out to investigate the effect of shot conditions such as particle diameter, air pressure and the number of particles on the particle behaviours.

The specimen is JIS A5052 (ASTM 5052) aluminium alloy plate. The size of the specimen was 100 mm x 100 mm in plane, and the thickness of the specimen was 10 mm. Shot particles were JIS SUJ-2 (ASTM 52100) high carbon-chromium bearing steel and the sphericity of shot particles is 0.5. The experimental apparatus is shown in Figure 1 [19]. Pressure gauge was set at the upper side of compressor tank. The acceleration length in the nozzle, l_n, is 40 mm, and its diameter, d, is 6 mm as shown in Figure 2. Distance from the nozzle tip to the specimen, l, is 100 mm. Shot particles are supplied from the supply inlet of the nozzle. The limitation of the experimental set-up is treated time because the particles are supplied from the inlet by hand. Usually, commercially available shot peening machine shot the particle continually for a long time. However, for this model apparatus, the number of particles in one shot is limited.

Figure 1. Experimental setup for shot peening process

Figure 2. Sectional view of the nozzle

Table 1 shows the experimental conditions. Particle diameter, air pressure and the number of particles at one experiment were changed. From preliminary experiments, the numbers of shot particles were determined. From the indentation of particle impingement on the specimen surface, the spread of shot particles were measured as the distance from the centre of the specimen. The centre of the specimen is set just below the nozzle exit.

Table 1. Experimental conditions Test No. Particle diameter, D [mm] Pressure, P [MPa] Number of Particles, N 1 0.8 0.6 30 2 0.8 0.6 40 3 0.8 0.3 30 4 0.5 0.6 30 5 0.8 0.6 1

2.2. Experimental Results

The distance from the indentation of the particle impingement to the centre of specimen was measured by image analyses. The experimental results were shown by histogram of 5 mm intervals from Figure 3 to Figure 5. Figure 3 shows the effect of air pressure on the impact location. Most particles hit the specimen surface within 20 mm from the centre of specimen. When the air pressure is lower, the surface impact location is broader. Figure 4 shows the effect of shot diameter on the impact location. When the diameter of the shot particle is smaller, the indentation of the particle impingement is broader. Figure 5 also shows the experimental results that single particle experiments and multi-particles experiments are compared. For multi-particles, the impact location is broader due to the collision between particles.

Figure 3. The effect of air pressure on the impact location on specimen

Figure 4. The effect of shot diameter on the impact location on specimen

Figure 5. The effect of particle collision on the impact location on specimen

3. Modeling of Particle Behavior

To make detail modelling about particle behaviours, the jet flow from the nozzle is considered. After blasting air from the nozzle, turbulent jets form three types of regions that are potential core region, mixing region and developed region [18, 20-22]. Particles are accelerated by the jet flow when they pass through these regions. We also focus on the interactive force between particle and specimen and between particles. The interactive force was modelled by using Hertzian contact theory [23, 24].

3.1. Equation of Motion for Shot Particles

The structure of the turbulent jet flows is described in Figure 6 [20, 21]. In its initial stage of development, a region near the nozzle exit and along the center of the jet is called as the potential core region. In this region, mean velocity was assumed to be constant. The region that the potential core region is surrounded with is the mixing region. Over some distance from the nozzle, the jet becomes a fully developed and forms the developed region. The distance at the end of potential core region from the nozzle exit can be described as 5d, where d is the diameter of the nozzle [21].

Figure 6. The structure of turbulent jet flow

When shot particles are passing through these regions, they are accelerated by the drag force from the jet and the equation of motion of particles may be described as [13, 25],

(1)

where m is the particle mass and C_D is the coefficient of drag. For the spherical shape in turbulent flow, C_D is 0.47 [13, 25]. ρ_a(y) is the air density. D is the particle diameter. u(x, y) is the air velocity and v(x, y) is the particle velocity. The term related to velocities means the relative velocity between air and shot particle. Air density, ρ_a(y), and air velocity, u(x, y), depend on the region where each shot particle is.

Air density in atmosphere, ρ_ab, is assumed to be 1.205 Kg/m³ [25], and air density in the compressor, ρ_at, is

(2)

where P is the pressure of the compressor and P_b is atmospheric pressure [13]. The air density in the nozzle, ρ_as, is obtained as,

(3)

Air density in potential core region, ρ_ap(y), is equal to air density at compressor, ρ_at, as,

(4)

Air density in mixing region, ρ_am(y), can be derived from the conservation of mass. This relation can be described as,

(5)

Air density in development region, ρ_ad(y), can be described as,

(6)

By substituting Eq. (3) into Eq. (6),

(7)

3.1.1. Air Velocity in Each Region

If a₀ is the speed of sound and the flow is isentropic, air velocity at the nozzle exit, u_i, can be written as,

(8)

where κ is specific-heat ratio of the compressive air, P is the pressure of compressor and P_e is the pressure at the nozzle exit [25]. If the flow is assumed to be a perfect gas and the normal temperature is 293 K, air velocity at the nozzle exit u_i is 313.47 m/s.

In the potential core region, air velocity, u(x,y), can be preserved. Thus, air velocity in the potential core region, u_p(x, y), is,

(9)

In the mixing region, linear approximation can be applied and the air velocity in the mixing region, u_m(x, y), can be written as,

(10)

The air velocity in the developed region, u_d(x, y), can also be written as,

(11)

3.1.2. Governing Equations in Each Region

The governing equation of particle motion in the potential core region can be obtained by substituting Eqs. (4) and (9) into Eq. (1) like,

(12)

The governing equation of particle motion in the mixing region can also be obtained by substituting Eqs. (5) and (10) into Eq. (1) like,

(13)

The governing equation of particle motion in the developed region is also obtained by substituting Eqs. (7) and (11) into Eq. (1) like,

(14)

3.2. Horizontal Force Acting on Shot Particles

Horizontal force acting on shot particles was considered after shot particles pass through the nozzle. Streamline of the jet is tilted toward horizontal direction. Angle αin Figure 7, which is the angle between path line and vertical line, means divergence angle. The magnitude of tanα can be obtained from the ratio of air velocity of horizontal direction u_x divided by air velocity of vertical direction u_y, and this is equal to 0.10.

Figure 7. The divergence of the jet

Weight functions are also set in the jet flow. At the center of the jet, the horizontal force is equal to 0, while the horizontal force at the edge of the jet is equal to 1. Figure 7 shows the weight function. The weight function in the potential core region and mixing region, f_t(x,y), can be described as,

(15)

The weight function in the developed region, f_d(x,y), can also be described as,

(16)

From these relations, horizontal force in the potential core region and mixing region can be written as,

(17)

where C_L is the coefficient of lift. For the spherical shape in turbulent flow, the magnitude of lift coefficient C_L varies from 0.03 to 0.3 irregularly [26, 27]. Therefore, lift coefficient C_L is assumed to be 0.15.

Horizontal force in the developed region can also be written as,

(18)

3.3. Interactive Force between Shot Particle and Specimen

As mentioned above, interactive force between shot particle and specimen can be modelled by Hertzian contact theory [23, 24]. In Hertzian contact theory, contact force can be modelled by non-linear dashpot and non-linear spring as shown in Figure 8. The equation of motion for particles during collision can be described as,

(19)

where X is the penetration depth from the contact point, K_PS is the coefficient of non-linear spring, and η_PS is the damping coefficient.

Figure 8. Modelling of Hertzian contact between particle and specimen

Non-linear spring coefficient K_PS can be written as,

(20)

where R_P is particle radius, ν_P is the Poisson’s ratio of particle, E_P is its elastic modulus.ν_S is the Poisson’s ratio of specimen, and E_S is its elastic modulus . The damping coefficient η_PS can be taken as a fraction of the critical damping η_critical, thus it can be written as,

(21)

by Tsuji et al. [28]. Fraction coefficient γ_PS is the function of the coefficient of restitution between shot particle and specimen. The coefficient of restitution between shot particle and specimen e_PS can be measured from the freefall test and it was 0.70 in our experimental result, which is similar with the result of Wong et al. [29].

3.4. Interactive Force between Shot Particles

Interactive force between shot particles can also be modelled by Hertzian contact theory [23, 24]. Colliding force between shot particles can also be modeled by dashpot and spring. The equation of motion for particles during collision can be described as,

(22)

where X is the penetration depth from the contact point, K_PP is the coefficient of non-linear spring, and η_PP is the damping coefficient.

Non-linear spring coefficient K_PP can be written as,

(23)

where R_P is the particle radius, ν_P is the Poisson’s ratio of shot particle and E_P is its elastic modulus. The coefficient of non-linear dashpot, η_PP, can also be written as,

(24)

Fraction coefficient γ_PP is also the function of the coefficient of restitution between shot particles. The coefficient of restitution between shot particles was referred as 0.90 [29, 30].

3.5. Numerical Model and Conditions

The above forces acting on particles were implemented in Multi-body dynamics simulator, MotionSolve (Altiar Engineering) and the equation of motion was solved. Figure 9 shows the numerical model of particle projection. 30 particles set in the side of nozzle randomly. Because there assumed to be the effect of jet entrainment [31], particles were set in the x-direction from 1.5 to 3.0 randomly. The initial velocity of each particle is zero at the nozzle inlet. Governing equation of motion for particles in the nozzle is equal to Eq. (12). Parameters used in numerical simulation are shown in Table 2. Calculations were conducted more than 5 times by changing the initial location of particles.

Figure 9. (a) Initial distribution of shot particles in the nozzle. (b) Overview of turbulent jet

Table 2. Parameters used in numerical simulations Analysis time, ms 10 Analysis time interval, ms 0.01 Pressure P , MPa 0.60 Atmospheric pressure Pb , MPa 0.10 Particle diameter D ,mm 0.8 Particle radius Rs, mm 0.4 Projection length l, mm 100 Nozzle diameter d, mm 6 Nozzle length ln, mm 40 Poisson’s ratio of shot particle νs 0.3 Elastic modulus of shot particles Es, GPa 210 Poisson’s ratio of specimen νw 0.3 Elastic modulus of specimen Ew, GPa 210 Number of shot particles 30

After projecting compressive air from the nozzle, turbulent jets form the three types of regions as shown in Figure 9 (b). Potential core region comes into effect in the x, y region that meets following conditions.

(25)

(26)

Mixing region comes into effect in the x, y region that meets following conditions.

(27)

(28)

Developed region also comes into effect in the x, y region that meets following conditions.

(29)

4. Results and Discussion

4.1. Particle Projection

Figure 10 shows particle projection. Due to the interaction between particle and compressive air, particles were accelerated by compressive air depending on where particles pass through in the compressive air region. Figure 11 focuses on the velocity variation from the nozzle inlet to the specimen surface. This is the verification of velocity variation in each region. Similar velocities has been experimentally observed by Maeda et al. [32]. In this figure, the case of non-collision between particles and the case of collision among particles are also compared. Velocity of collision particles are varied by the collision between particles.

Figure 10. Particle projection at (a) 1 ms and (b) 1.5 ms

Figure 11. Particle velocity variation

4.2. Particle Velocity and Impingement Location

The particle velocities for y-direction just before impingement are shown by histogram of 5 m/s intervals in Figure 12. About 65% of particles have the velocities between 60 and 80 m/s. But, about 16% of particles were slower than 60 m/s or higher than 80 m/s. Another 19% of particles did not collide with specimen, and their trajectories varied dramatically. This is the effect of the collision among particles. From the calculation of single particle shot, the impingement velocity was about 65 to 70 m/s. When the collisions between particles occur, the particle velocity was changed. Thus, it can be said that the distribution of impingement velocity is due to the collision between particles. Similar trend has been experimentally observed by Shipway and Hutchings [33].

Figure 12. Histogram of impingement velocities of particles

The impact locations of particles were also described by histogram of 5 mm intervals in Figure 13. The peak of the impingement rate is the same as those in Figures 3 to 5. Thus, the numerical model can catch the trend of particle behavior before collision correctly.

Figure 13. Histogram of impact location of particles

5. Conclusions

In this paper, the numerical simulations of shot peening process by multi-body dynamics were carried out. Special contribution of this paper is the detail modelling about the interaction between shot particles and compressive air. Particle interactions are also modeled by Hertzian contact theory. The numerical results were compared with the experimental results. Results obtained in this paper are summarized as the followings.

1. Considering acting forces on the shot particle from the jet flow of compressive air, equation of motion was derived and implemented in multi-body dynamics simulation.

2. Distribution of impingement velocity can be found. This phenomenon is caused by the collision between particles.

3. Distribution of particle impingement location can also be observed. The peak of the impingement rate is 5 to 10 mm from the center of the nozzle. This result is identical to the experimental results.

The particle velocities and the location of impingement obtained by the proposed approach are useful for the initial condition of the following collision analyses, such as finite element analyses.

The future work of this study is the direct comparison between experimental and numerical results. The shot particles can be observed by a high speed camera and the results will be reported elsewhere [34].

ACKNOWLEDGEMENTS

The authors acknowledge the Academic Open Program of Altair Engineering Inc. for using HyperWorks software.