Analytical modelling for ultrasonic surface mechanical attrition treatment

The grain refinement, gradient structure, fatigue limit, hardness, and tensile strength of metallic materials can be effectively enhanced by ultrasonic surface mechanical attrition treatment (SMAT), however, never before has SMAT been treated with rigorous analytical modelling such as the connection among the input energy and power and resultant temperature of metallic materials subjected to SMAT. Therefore, a systematic SMAT model is actually needed. In this article, we have calculated the averaged speed, duration time of a cycle, kinetic energy and kinetic energy loss of flying balls in SMAT for structural metallic materials. The connection among the quantities such as the frequency and amplitude of attrition ultrasonic vibration motor, the diameter, mass and density of balls, the sample mass, and the height of chamber have been considered and modelled in details. And we have introduced the one-dimensional heat equation with heat source within uniform-distributed depth in estimating the temperature dist...


I. INTRODUCTION
In traditional engineering treatments, shot peening by using steel balls to bombard onto metal surfaces has been adopted to leave compressive residual strains within the affected region in promoting the fatigue properties. 1,2These balls have typical diameters of 0.1 ∼ 2 mm and gain their speed by compressed air.Normally, these balls bombard the metal surfaces in the frequency range of 20 ∼ 100 H z and speed range of 50 ∼ 100 m/s.
A new physical treatment named ultrasonic surface mechanical attrition treatment (SMAT) was firstly introduced in 1999. 3,4The balls in the SMAT experiment are accelerated and bombarded by the ultrasonic motor on the bottom of a chamber, as shown in Fig. 1(a).The diameter and speed of flying balls and the bombarding frequency are in the range, 1 ∼ 10 mm, 1 ∼ 20 m/s, and 10 ∼ 100 k H z, respectively. 5The most important fact is that the incident direction onto the metal surface can be designed to vary with time lapse in making smaller grain size of metallic materials.This will lead to promising properties of metals such as grain refinement and gradient structure.][11][12] Although SMAT has gradually developed into a matured engineering surface treatment, never before has SMAT been treated with rigorous analytical modelling.Therefore, a systematic SMAT model is actually needed.The kinetic energy loss of flying balls in the SMAT chamber can be mainly converted into three parts as indicated in Fig. 1(b).4][15] Secondly, it is the heat energy of sample, in which the heat flow and its temperature distribution are both important factors for the resulting metal micro-structure. 16Recrystallization might be taken place in the sample while the experienced temperature reaches some critical values.Finally, it would be the sonic energy and heat energy in the chamber originated from the collisions between flying balls.In this paper, we have established connections among the parameters of flying balls, the ball size, flying speed, the bombarding frequency and amplitude of motor, the height of chamber, and the energy and power of sample.During the SMAT processing time, we can find the input energy and power of sample through these connections.The condition for the frequency of flying ball reaching a stead speed can also be obtained in this approach.For the heat energy of sample, we have introduced the one-dimensional heat equation with the heat source to estimate the heat flow and temperature distribution of sample which are hard to be measured in the SMAT experiment.With the temperature distribution of sample, we can connect its strain rate, hardness, and the grain size of sample.With these connections and modelling, one can find an optimized approach to the mechanical performance of metal surface via the SMAT experiment.

A. The average kinetic energy of balls
The harmonic longitudinal vibration motion can be characterized by the equation where ω is the angular velocity and ν is the vibration frequency of motor.The velocity of ultrasonic motor top, v m0 , can be expressed as The average ratio of horizontal speed to vertical speed is 0.16 ∼ 25, 5 namely, the average angle of both motor and sample surfaces is about 80 • .The kinetic energy ratio of horizontal component to vertical component is about 0.05, which is ≪ 1.Thus, we can assume the angle of both impact surfaces are normal without loss of generality.
We can calculate the kinetic energy of flying balls based on the assumption of elastic collision between the flying balls and motor surface.The initial speed of flying ball at the motor top before second collision can represented as where m b is the mass of a flying ball, m m is the motor mass, and φ is the phase of harmonic vibration.A flying ball will have a collision against the material under the condition where h is the chamber height, D is the diameter of a flying ball, g is the gravitational acceleration, and M and θ 0 are parameters defined by Eq. (1).Since the collisions between the chamber top and flying balls will only take place over the phase range we can do the integration over the range with respect to φ to obtain the averaged initial speed of flying balls Since the function of sinφ is symmetrical to φ = π 2 , this integration can be carried out over first quadrant.
For common SMAT experiments, we are allowed to use the following conditions and approximations Then the averaged initial speed of these balls turn immediately to The flying time of these balls from the motor top to the sample bottom are given by where α and η are both parameters defined in Eq. (3).In addition, these identical-flying balls can be regarded as a system interacting with the sample bottom while ignoring the collisions between these flying balls.For this reason, we can consider the time-averaged value of single representative ball.With the expansion of flying time of these balls from the motor top to the sample bottom, the averaged flying time of these balls can be acquired by carrying out the integration with respect to φ τ ≈ αM 2g  For common SMAT experiments, the scale of A, ω, h, and D are 40 ∼ 80 µm, 40π krad/s, 20 mm, and 1 ∼ 3 mm respectively.The corresponding value of parameters, α and sin θ 0 , are from 8.242 × 10 −4 ∼ 3.685 × 10 −3 and 2.871 × 10 −2 ∼ 6.070 × 10 −2 respectively, so the higher order terms can be neglected without loss of generality.The relative errors of approximation in Eq. ( 2) are less than 0.1%.The τ ranges from 2.413 ∼ 4.570 ms.The averaged initial speed of these balls ranges from 6.645 ∼ 13.033 m/s.The time-averaged speed of these balls before second collision can be obtained by which gives the averaged-speed of these balls, varying from 6.588 ∼ 12.693 m/s in good agreement with the speed of flying balls in the SMAT experiment measured by high-speed cameras. 5The variation trends of averaged flying ball speed as a function of SMAT vibration amplitude and frequency are presented in Fig. 2. It can be seen that the ball speed would increase in proportional to the SMAT amplitude and frequency. 5hus, the total-averaged kinetic energy of these flying balls before second collision between the sample bottom and flying balls is where N is the total number of balls and ρ b is the density of flying balls.For common SMAT experiments, the balls will cover about 25% area of chamber surface, namely, the higher the diameter of ball is, the lower the total number of balls would be.Thus, the N is proportional to 1 D 2 .

B. The kinetic energy loss of flying balls
On colliding with the sample surface, the kinetic energy of flying balls will not be conserved due to the inelastic collision between flying balls and sample.Considering the inelastic collision, we can introduce the coefficient of restitution (the act of recovering to a former state), where v b is the ball speed and v s is the sample speed before collision and v b ′ is the ball speed and v s ′ is the sample speed after collision.e = 1 and e = 0 corresponds to the elastic and total inelastic collision respectively.For a common SMAT experiment, the sample is initially at rest, namely, v s = 0 and the e is approximately 0.25 in previous experiment. 5The coefficient of restitution can be simplified as By the conservation of momentum, we can calculate their velocity after collision, i.e.
to acquire the formulas It is obvious that the averaged loss of kinetic energy of sample and flying balls are This loss will be converted into the heat energy of sample and flying balls and the internal energy (or so-called the strain energy) of sample and flying balls.And the kinetic energy of sample will be assumed to convert almost into the internal energy of sample.Figure 3 presents the variation trends of kinetic energy loss of flying balls, sample, and total (the sum of those for balls and sample).It is apparent that the kinetic energy loss will increase with increasing m b m s .Similarly, the average time, τ ′ , of flying balls from the sample bottom to the motor top between second collision of the sample bottom and flying balls and third collision of the motor top and flying balls can be obtained by the following equations where A s is the cross-sectional area of sample, L is the sample thickness, ρ s is the sample density, and η ′ and χ assumed to be smaller than e are parameters defined in Eq. ( 8).Similarly, the average time of these flying balls from the sample bottom to the motor top between second and third collision, τ ′ , can be obtained by the expansion of η ′ Thus, the averaged time period of flying balls (going up and down between second and third collision) and total-averaged power loss of kinetic energy of balls are It's not hard to discover that the incidental speed of flying balls is steady under the condition And so does the total-averaged power loss of kinetic energy of sample Figure 4 illustrates the variation trend of averaged time period of flying balls with respect to the SMAT vibration amplitude and frequency.It is obvious that the time period decreases with increasing amplitude and frequency.In addition, Fig. 5 shows the variation trends of power loss of flying balls, sample, and total (the sum of those for balls and sample).It is apparent that with increasing m b m s the energy loss will increase, but with different trends.

A. The heat equation in one dimension
Considering the one-dimensional heat equation, where the sample temperature, T, is described by the partial differential equation (PDE) where z is the distance from the sample bottom, t is the time, k 0 is the thermal conductivity of sample at 300 K, the heat source, q, is the heat energy generation per unit volume per unit time, and C is the specific heat of sample.

B. The heat source
We can count the collisions between the sample bottom and flying balls as the heat source.Usually, the balls in SMAT acts on a effective depth within 100 µm. 9With this idea, the heat source can be thought as uniformly distributed in some effective depth of sample and it can be described as where q is heat coming from the power loss of averaged kinetic energy of sample and flying balls, l is the effective depth with heat source within it, and u(z) is the unit step function since it has the following property

C. The steady state solution
On reaching the steady state, the changing rate of sample temperature with respect to time will become zero, ∂T ∂t = 0, the Eq. ( 12) can be simplified as To solve Eq. ( 13), we can integrate it with respect to z ′ from 0 to z to obtain where T b is the equilibrium temperature of bottom surface of sample and T t is the equilibrium temperature of top surface of sample.The temperature drop predicted by Eq. ( 14) from the surface (subject to SMAT bombarding) to the inner portion is presented in Fig. 6 for pure Cu and 304 stainless steel, for the region near the surface (less than 0.05 mm) and the overall depth (up to 1 mm).Near the surface region, there exists a small hump, and then the temperature continues to drop all the way into the inner portion.With the known temperature distribution, we can calculate the heat energy of sample by integration where T 0 is the initial temperature of sample with isothermal-distributed temperature.Normally, T 0 can be set to room temperature, 300 K. Now we turn to consider the case in which the thermal conductivity has temperature dependence, i.e.
All  where β is a constant.The integration of Eq. ( 13) in the change of variables, T v = T − T 0 , should be re-written as Then we can immediately obtain The temperature drop predicted by Eq. ( 16) from the surface (subject to SMAT bombarding) to the inner portion is presented in Fig. 7 for pure Cu and 304 stainless steel, for the region near the surface (less than 0.1 mm) and the overall depth (up to 1 mm).With small changes in the temperature of sample and small β for general metal, the solution will be returning to the solution of Eq. (13).Similarly, the heat energy of sample can be obtained by Eq. ( 15) with

D. The internal energy of sample
The internal energy (or so-called the strain energy) power of sample is denoted as where ∆U int, s is the change of internal energy of sample during the time.We can use the energy conservation to estimate the internal energy of sample acquired per unit time in SMAT.
where we have ignored the sonic energy and heat energy of speeding balls because the volume and mass of chamber is much higher than those of balls.Since the collision probability is proportional to their cross section area, the probability ratio of ball-ball to ball-sample should be proportional to 4A s .For the common ball size of 2 mm in diameter and SMAT sample area of 40 × 20 mm 2 , the ratio value is less than 1 %.In this case, the collisions between the balls is not so frequent and the heat energy of balls is also small compared to that of chamber and sample.

A. Experimental methods
In order to compare with the proposed model, an AISI 304 stainless steel was adopted as the tested material, with chemical compositions of (in wt%): 0.049 C,18.20 Cr, 8.66 Ni, 0.58 Si, 1.04 Mn, 0.021 P, 0.007 S and the balanced Fe.A plate measuring 40 × 20 × 1 mm was set on the top of the SMAT chamber, with a cylindrical chamber measuring 70 mm in diameter and 20 mm in height.The SUJ2 bearing steel balls with smooth surface and high hardness in the R C scale of 62 are applied as the energy deliverer and are placed in a reflecting chamber that is vibrated by a vibration generator with a fixed vibration frequency ν = 20 k H z. The vibration amplitude, A, was chosen to vary in three levels, 40, 60 and 80 µm.Three sizes of the balls were selected, namely, 1, 2 and 3 mm in diameter.The density of these balls of different sizes, ρ b , is all fixed to be 7.8 g/cm 3 .In order to maintain the fixed ball coverage area of 25% inside the chamber, the 1 mm ball case would install 5 g of the total ball weight, the 2 mm case for 10 g and the 3 mm case for 15 g, respectively.Throughout the SMAT experiment, the working temperature is controlled and traced to be below 150 • C, which is about 0.2 T m (melting temperature) of the 304 stainless steel and is considered to be relatively low for the 304 stainless steel samples.
After careful mechanical grinding and polishing of the cross section of the SMAT samples, the sample surface roughness and the morphology level were sufficient for nano-indentation to extract the hardness variation from the free surface (subject to SMAT) into the inner portion.The SEM observations were performed using a Zeiss Supra 55 field-emission scanning electron microscope.With a low acceleration energy at 5 kV and a low working distance at 5 ∼ 7 mm, it is able to visualize the distinguishable grains from the back-scattering images (BEIs).The cross-sectional transmission electron microscopy (TEM) foils of the SMAT samples were fabricated using the dual-beam focused-ion-beam (FIB) system (Seiko, SMI3050) with an operating voltage of 30 kV and an ion beam current of 1 pA.The TEM foils were examined by the Tecnai G20 field emission transmission electron microscopy with an operating voltage of 200 kV .The hardness of the SMAT specimens from the cross-sectional surface was measured by the MTS Nano Indenter XP System.The tests were operated with a displacement rate about 10 nm/s, and the allowable vibration drift of the environment was controlled under 0.05 nm/s.The indented depth limit was set to be 1200 nm.

B. Relating strain rate and temperature with sample micro-structure in SMAT
For metallic materials, it is almost a universal rule that the sample microstructure would be related to the processing parameters by the Zener-Holloman relationship. 8In general, the average grain sizes would decrease with decreasing working temperature and increasing working strain rate.The Zener-Holloman Z parameter is defined as where ε is the strain rate, Q the activation energy, T the absolute temperature, and R the gas constant.The accumulative strain ε by the successive bombarding cycles can be approximately expressed by where ε i is each strain by each ball bombarding incident, ∆x is each compressed depth by each ball bombarding incident, and x is the sample depth experiencing the bombarding impact.The precise strain is difficult to be calculated since the bombarding can be induced by the flying balls from various directions and the induced strain would be different for bombarding from different directions.The average is estimated to be about 0.2 in Eq. (20).But in general larger balls are expected to induce a higher degree of strain per bombarding, it is thus postulated that the 1, 2 and 3 mm balls would induce an average strain about 0.15, 0.20, and 0.25, respectively.The overall strain accumulated by numerous ball bombarding events is simply expressed by where n is a statistic evaluation of the overall ball bombarding events during the SMAT time duration t.Since the balls in the chamber can fly randomly in all 3D directions, the probability P that one ball will bombard on the sample can be rationalized by the sample flat surface divided by the total surface area including the chamber wall and sample surfaces.This probability can be varied for different SMAT machine system designs.If the sample surface occupies 10% of the overall surface area, then P is assumed to be 0.1.Given a vibration frequency ν and overall SMAT time t, the bombarding event onto the sample surface will be Thus, the strain rate is equal to the accumulative strain divided by the SMAT time duration t, The SMAT working temperature T may or may not be measured with reasonable accuracy, depending on the chamber design.Also, even the temperature can be measured from the sample surface, the temperature should be a gradient profile from the outer surface to the inner portion of the sample.Since the grain sizes in many SMAT metals or alloys are in the nano-to micro-scale with no pronounced grain growth, the experienced temperature is thought to be around or less than 0.2 T m , where T m is the melting temperature of the metallic sample.With the activation energy Q for the involved major diffusion species, Z can be calculated.And then it is hoped that the grain size can be related to the Z parameter.

V. DISCUSSION AND COMPARISON WITH THE SMAT EXPERIMENT
Experimentally, it is observed that the grain size is appreciably refined by SMAT, from the initially about 20 µm down to less than 100 nm, as shown in Fig. 8(a) with a gradient trend as viewed from the sample cross-section. 17In parallel, the hardness increases from the initial about 2.7 GPa up to about 6.0 GPa, as shown in Fig. 8(b). 17The kinetic energy from the flying balls appear to effectively induce substantial internal or strain energy into the sample surface, increasing the dislocations and other defects, refining the grain size, and raising the hardness.
The ball speed can be estimated from Eq. ( 4) to be within the range of 5 ∼ 10 m/s, and the kinetic energy for all the flying balls can also be estimated to be about 10 ∼ 120 mJ.From the experimental results in Fig. 8, coupled with the estimated values based on the current analytical model, it appears that the optimum speed for the 304 stainless steel might be around 8 ∼ 10 m/s and the optimum kinetic energy might be around 70 ∼ 75 mJ.The adjustment of the SMAT parameters will influence accordingly the speed (in Fig. 2), kinetic energy (in Fig. 3), flying time period (in Fig. 4), power (in Fig. 5), and temperature profiles within the experienced range of the samples (in Figs. 6 and 7).In this model, we have made efforts in evaluating the temperature profile from the bombarded surface to the sample inner portion (Figs. 6 and 7).This profile can be used as a reference in assessing the experienced temperature at the particular sample depth.For example, based on the calculated temperature in Fig. 7 for the 304 stainless steel, the temperature at the depth of 200 µm from the surface would be 365 K or 92 • C.
The other parameter left would be the strain rate.In accordance with Eq. ( 23), the strain rate would vary from 3 × 10 2 ∼ 5 × 10 2 s −1 .Taking the 4 × 10 2 s −1 as the mean value, and 92 • C as the experienced temperature, we can incorporate into Eq.( 19) to extract the Zener-Holloman Z parameter, which is useful for estimate the materials microstructure properties.For 304 stainless steel, the governing activation energy Q should be related to the Fe diffusion, and Q ∼ 220 k J/mol is a logical value. 18,19With the above information and the gas constant R = 8.3 J/K, Z can be calculated to be 1.4 × 10 34 s −1 .With the same calculation manner, we can estimate all values for various cross-sectional positions of the SMAT sample, and plot the measured grain size and Zener-Holloman Z parameter, as presented in Fig. 9.
Thus, for SMAT researchers, we can first design the SMAT working parameters (based on the needs), and can calculate the resulting speed, temperature, strain rate, and energy based on this model in Figs. 2 to 7. With all the information, we can estimate the grain size from the Zener-Holloman Z parameter based on Fig. 9.The current approach and modeling nicely establish the link between the fundamental solid state physics and the engineering material surface modifications.

077126- 2 Huang
FIG. 1.(a) The desiged-dimension of chamber in the SMAT experiment.(b) The schematic drawing showing that the sample material gains the heat and strain energy from the kinetic energy loss of sample and flying balls.

FIG. 3 .
FIG. 3. (a) The variation trend of ∆E k, l o s s, b predicted by Eq. (5) as a function of m b ms for the parameters H = 20 mm, A = 60 µm, ω = 40π k r ad/s, m b mm = 10 −6 , and e = 0.25, (b) the variation trend of ∆E k, l o s s, s predicted by Eq. (6) as a function of m b ms for the parameters H = 20 mm, A = 60 µm, ω = 40π k r ad/s, m b mm = 10 −6 , and e = 0.25, (c) the variation trend of total energy loss, i.e., the sum of ∆E k, l o s s, b and ∆E k, l o s s, s as a function of m b ms for the parameters H = 20 mm, A = 60 µm, ω = 40π k r ad/s, m b mm = 10 −6 , and e = 0.25.

FIG. 5 .
FIG. 5. (a) The variation trend of P l o s s, b predicted by Eq. (10) as a function of m b ms for the parameters H = 20 mm, A = 60 µm, ω = 40π k r ad/s, m b mm = 10 −6 , and e = 0.25.(b) The variation trend of P l o s s, s predicted by Eq. (11) as a function of m b ms for the parameters H = 20 mm, A = 60 µm, ω = 40π k r ad/s, m b mm = 10 −6 , and e = 0.25.(c) The variation trend of the total power loss, i.e., the sum of P l o s s, b and P l o s s, s as a function of m b ms for the parameters H = 20 mm, A = 60 µm, ω = 40π k r ad/s, m b mm = 10 −6 , and e = 0.25.

FIG. 6 .
FIG.6.The temperature distributions for pure Cu predicted by Eq. (14) for narrow region near the surface in (a) and for wider region in (b) with the parameters k 0 = 401 W /m • K , q = 0.772 × 10 3 ∼ 2.28 × 10 3 W /mm 3 , A s = 800 mm 2 , L = 1 mm, l = 5 µm, T b = 398 K , and T t = 358 K .The temperature distributions for 304 stainless steel predicted by Eq. (14) for narrow region near the surface in (c) and for wider region in (d) with the parameters k = 14.9 W /m • K , q = 0.773 × 10 3 ∼ 2.28 × 10 3 W /mm 3 , A s = 800 mm 2 , L = 1 mm, l = 5 µm, T b = 378 K , and T t = 318 K .The different colored lines correspond to various percentages of kinetic energy loss which is converted into the heat energy of sample.

FIG. 7 .
FIG.7.The temperature distributions for pure Cu predicted by Eq. (16) for narrow region near the surface in (a) and for wider region in (b) with the parameters k 0 = 401 W /m • K , q = 0.772 × 10 3 ∼ 2.28 × 10 3 W /mm 3 , A s = 800 mm 2 , L = 1 mm, l = 5 µm, T b = 398 K , and T t = 358 K .The temperature distributions for 304 stainless steel predicted by Eq. (16) for narrow region near the surface in (c) and for wider region in (d) with the parameters k = 14.9 W /m • K , q = 0.773 × 10 3 ∼ 2.28 × 10 3 W /mm 3 , A s = 800 mm 2 , L = 1 mm, l = 5 µm, T b = 378 K , and T t = 318 K .The different colored lines corresponds to various percentages of kinetic energy loss which is converted into the heat energy of sample.

FIG. 8 .
FIG. 8. (a) The cross-sectional SEM micrograph taken from the sample subject to SMAT with the 2 mm flying balls and 40 µm SMAT amplitude.(b) The gradient variation trend of hardness of selected SMAT 304 SS samples.