Materials Processing
Peen Forming of Anisotropic Double-Curved Surface by Vibration Peening Using Rectangular Solid Pin
2020 年 61 巻 1 号 p. 162-168
詳細
2020 年 61 巻 1 号 p. 162-168
In this study, a new peen forming technique by means of vibration peening using a rectangular solid pin is developed. It is possible to produce a smaller x-direction (short side of the pin) curvature radius Rx than the z-direction (long side of the pin) curvature radius Rz. Vibration peening, which projects the pin into the specimen using the reciprocating motion of the electric hammer, is applied. In the experiment, Rz/Rx increases as the pin-tip radius r decreases. The motion of the pin is analyzed by the dynamic explicit finite-element method (FEM). The pin reciprocates between the hammer and the specimen and collides with the specimen 17 times in 0.2 s. The number of collisions agrees with the experimental result of 10 to 18 times in 0.2 s. The plastic strain distributions are determined using the maximum velocity of the pin. Peen forming is analyzed by the following method. In Step 1, the initial velocity is set for multiple pins, and plastic strain distributions generated by collisions are analyzed by the dynamic explicit FEM. In Step 2, plastic strain distributions analyzed in Step 1 are input to the specimen to analyze the deformation by the static implicit FEM. The analysis results indicating the increase in Rz/Rx with decrease in r agree with the experimental results.
This Paper was Originally Published in Japanese in J. JSTP 60 (2019) 33–38. Figures 5, 6, 9 and 10 were slightly changed.
Peen forming is a method of shaping a metal plate by applying plastic strain near its surface. Plastic strain is generated when steel shots collide with the plate surface. Peen forming is primarily used in forming the outer skins of the main wings of aircraft.1–11) In recent years, methods using a pin for ultrasonic peening have been applied to peen forming.4)
The outer skins of the main wings of an aircraft are formed with complicated shapes such as those resembling a spherical surface, saddle, and single-curved surface. However, the plastic strain generated by the conventional peen forming using sphere shots is axially symmetric, and the deformations also tend to become sphere curved surfaces. Therefore, to deal with the various shapes of main wings, an elaborate forming preparation is required, such as masking (to limit the peening area) and using shot sizes that vary by location.1–3) Stress peen forming, in which stress is applied to the skins of main wings in advance, is used for shapes that require single-curved surfaces.5–7) Stress peen forming requires considerable time for the positioning of the instruments used, which results in the increased forming time.
In peening that uses a rectangular solid pin, the plastic strain that occurs in the x-direction (direction along the short edge of the rectangular solid pin) and the z-direction (direction along the long edge of the rectangular solid pin) can be changed to obtain a shape with anisotropy (Rx < Rz), such that the x-direction curvature radius Rx of the formed shape is different from the z-direction curvature radius Rz. A method has been proposed to control the curvature radius of a double-curved surface by peening using an elliptical pin attached to an air hammer.12) Additionally, a dropping test of multiple rectangular solid pins was performed to study the anisotropy of the curvature radius.13) However, previous studies did not evaluate the behavior of the plastic strain generated by repeated collisions with a vibrating hammer, and plastic strain when peening is performed on the entire surface of plates. Therefore, predicting the shapes that would form was difficult.
Many studies have been conducted on analytical methods of peen forming.7–11) Xiao et al. analyzed the residual stress distribution generated by collision with a single shot and calculated the stretch forces and bending moments from the residual stress to analyze the deformation.7) Takahashi et al. analyzed the residual plastic strain when a single shot was pressed into a plate using the static implicit method, and applied the obtained plastic strain to the surface of the shell element to calculate the deformation.8) Faucheux et al. calculated the inherent strain from the experimental deformation and residual stress.9) Gariépy et al. analyzed shot collision using the dynamic explicit method and applied the obtained residual stress to the surface of the shell element to calculate the deformation using the static implicit method.10) Xiao et al. used the dynamic explicit method to analyze the residual stress distribution of the randomly colliding shot and calculated the stretch forces and bending moments from the residual stress to analyze the deformation.11) However, there are no reports on examples of quantitative analyses for peen forming using vibration peening, which injects a pin using a hammer with a reciprocating motion.
In this study, peen forming using a rectangular solid vibration pin projected onto the specimen was experimentally confirmed to yield the anisotropy of the curvature radius (Rx < Rz), and a means of predicting the formed shape using quantitative analysis was studied.
In this study, an electric hammer (Hasegawa Machine NPF-95) with a stable frequency and amplitude was used. The hammer was connected to a motor, and had a frequency of 50 Hz and an amplitude of 10 mm. The hammer and the rectangular solid pin it strikes were not fixed. The pin began to move upon collision with the hammer and reciprocated between the specimen and the hammer. Along the guide holes, the pin collided with the specimen at a right angle. In this way, peening was performed by using only the kinetic energy of the pin. Therefore, this process could be controlled by adjusting the pin shape and hammer position. Large reaction forces were not generated on the hammer, and therefore the holding forces did not need to be controlled. The experimental device is shown in Fig. 1. The pin shape had a width of 2.8 mm, length of 6 mm, and height of 70 mm. At the tip, a 1 mm-radius chamfering was performed in the z-direction (long edge direction), and a tip radius r was provided in the x-direction (short edge direction). The three values of r were 4 mm, 8 mm, and 12 mm. The pin was made of pre-hardened steel (Daido Steel NAK55, HRC 37-43). The distance between the pin and the specimen when the hammer was raised the furthest was 10 mm. The mass of the pin was 8.94 g when r = 4 mm, 8.96 g when r = 8 mm, and 8.98 g when r = 12 mm. The specimen was made of aluminum alloy A7075-T6 (yield stress = 506 MPa), with two thicknesses, h, of 5 mm and 10 mm, width of 50 mm, and length of 50 mm. To avoid affecting the deformation, a rubber plate was placed behind the specimen, and the specimen edges parallel to the z-axis were fastened by bolting.
Experimental equipment and definition of parameter. (a) Appearance, (b) Shape of pin, (c) Definition of specimen directions.
A load cell (Kyowa Electronic Instruments: LUX-B-5KN-ID) was placed on the back of the specimen, and the impact force of the pin was measured. The measurement results using the pin with a radius of 8 mm are shown in Fig. 2. The measurement was performed for 1 s at intervals of 10 µs. The pin collided with the specimen 73 times per second. The collision rate was higher than the hammer frequency of 50 Hz. The collision time intervals were not uniform, and the impact force varied with each collision (Fig. 2). A sufficient collision rate per second was deemed to have been achieved at one operation per second. After the specimen was fixed and peened for 1 s, it was moved at 1 mm pitch in the x-direction, and a single-line construction was performed on the full length of the specimen. Next, the specimen was repeatedly moved at 4 mm pitch in the z-direction and 1 mm pitch in the x-direction, and nearly the entire surface was peened. The portion without peening (the part that was held for fixation) was cut off, and a laser-based shape measurement device was used to measure the deformed shape.
History of impact load for r = 8 mm.
The measurement results for the specimen shape after peen forming are shown in Fig. 3, when r = 8 mm and h = 5 mm. To measure the deformation, the specimen shown in Fig. 1 was turned upside down. There were convexities and concavities on the specimen surface due to peening indentations, and the deformation was larger in the x-direction than in the z-direction. To calculate Rx and Rz, the experimental results were fitted to the circle equation to minimize the sum of squares of errors. The circle to which the fitting was performed is illustrated in Fig. 3. The relationships between r and Rx, and r and Rz are shown in Fig. 4. The vertical axis indicates the curvature. When r decreases, Rx and Rz decrease simultaneously. Therefore, for all values of r, Rx < Rz is established.
Measured profiles of specimen for r = 8 mm.
Effect of pin-tip radius r and thickness h on curvature of specimens after peening.
When comparing the results when h = 5 mm with those when h = 10 mm, the same trend was observed. For all values of Rx when h = 10 mm was approximately 2.8 times that when h = 5 mm. Examining the anisotropy (Rz/Rx) of the curvature radius when h = 5 mm, I observed that Rz/Rx is 1.76 when r = 4 mm, 1.69 when r = 8 mm, and 1.58 when r = 12 mm implying that Rz/Rx increases as r decreases. On the other hand, when h = 10 mm, the Rz/Rx was 2.04 when r = 4 mm, 1.80 when r = 8 mm, and 1.56 when r = 12 mm implying that Rz/Rx increases as r decreases, similar to when h = 5 mm. At r = 4 and 8 mm, Rz/Rx was larger when h = 10 mm.
These results demonstrate that changing r enables to change in Rx and Rz, even when peen forming is performed using a pin of the same shape and a single electric hammer.
In the vibration peening method used in this study, a hammer vibrating at a frequency f of 50 Hz and an amplitude A of 10 mm accelerates a pin, and the specimen is struck. The displacement δ of the hammer at time t is expressed by the sine curve of eq. (1). The velocity of the hammer VH is shown in eq. (2), and the maximum velocity is 3.14 m/s.
| \begin{equation} \delta = A\cdot \sin(2\pi ft) \end{equation} | (1) |
| \begin{equation} V_{H} = 2\pi fA\cdot \cos(2\pi ft) \end{equation} | (2) |
To identify the collision properties of the pin shown in Fig. 2, I used the dynamic explicit finite-element method (FEM) to analyze how the pin would move in a 0.2 s period, using LS-DYNA ver. R800. The analysis model simulating the experiment shown in Fig. 1 is illustrated in Fig. 5. It is an upside-down model, in which the force of gravity was not considered. The specimen dimensions were 5 × 10.8 × 10.8 mm. For the pin, r = 8 mm was targeted. The mass of the pin was 8.96 g. Considering the geometric symmetry in the x- and z-directions, the model was made with 1/4. The element is an 8-node, hexahedral reduction integration element. The dimensions of the elements near the surface were 0.125 mm in the thickness direction and 0.166 mm in the x- and z-directions. The pin was an elastic body. The hammer and the base were rigid bodies. The specimen was fixed to the base to prevent vibration. Proceeding half of the wavelength of the sine curve defined in eq. (1) in the hammer, the value of δ that was set to move toward the bottom (minus side) at t = 0 s was input as the forced displacement. The hammer was 60 × 60 × 20.6 mm with a mass of 578 g. The specimen was an elastic plastic material simulating A7075-T6 (Young’s modulus E = 70 GPa, Poisson’s ratio ν = 0.3, density ρ = 2.7 × 10−6 kg/mm3). The material model used the kinematic hardening law of linear hardening. The contact friction was Coulomb friction, and the representative value of the friction coefficient when a lubricant was not used was 0.2.
Analytical model of vibration peening for r = 8 mm. (a) Analysis model, (b) Details of specimen model.
The displacements of the pin and hammer are shown in Fig. 6. The hammer underwent forced displacement along the sine curve of f = 50 Hz and A = 10 mm. Meanwhile, the pin began to move on the minus side (specimen side) on collision with the hammer at t = 0 ms. The specimen surface was −20 mm on the vertical axis and the movement direction of the pin was changed by collision. The hammer moved downward during this time; therefore, the pin and hammer collided again at approximately 5.95 ms, and the pin changed its movement direction downward. However, when the pin and hammer collide at approximately 9.65 ms, the pin did not begin to move downward but continued to move upward because the hammer moved upward. The pin collided with the hammer at approximately 19.7 ms when the hammer moved downward, and the pin began moving downward. In this manner, the pin repeated the reciprocating motion between the hammer and the specimen and collided with the specimen at non-uniform time intervals. The collision history changed because of the initial position of the hammer. In this model, the pin and the specimen collided 17 times in 0.2 s. This collision rate was higher than the 50 Hz frequency of the hammer. In the experimental measurement results shown in Fig. 2, 73 collisions per second (10–18 collisions in 0.2 s) were recorded; therefore, the measurement results agreed with the analytical results.
Histories of displacements of pin and hammer.
Figure 7 shows the change in the pin velocity. The velocity in the negative direction indicates the velocity when the pin collides with the specimen. At 0–3.5 ms, the velocity was approximately −6.26 m/s, and the maximum velocity was −7.01 m/s at approximately 186 ms. This is higher than the −3.14 m/s maximum velocity of the hammer calculated using eq. (2). Similar results have been reported for the pin movement velocity in ultrasonic peening treatment.14) The pin velocity changed significantly with time, and these results correspond to the change in the impact force shown in Fig. 2. Figure 7 shows the change in the x-direction plastic strain εx at the specimen surface below the pin collision position. When the pin collision velocity was greater than that in previous history, εx tends to rise.
Histories of pin velocity and x-plastic strain at surface.
Figure 8 shows the plastic strain distribution in the thickness direction at the central axis after the pin and specimen initially collide (after 3.5 ms), and after their final collision (after 187 ms). The figure shows an expanded view from the surface to a depth of 3 mm. The plastic strain was 0 at a depth of 3–5 mm. When the collision velocity increased, the introduced plastic strain increased after 187 ms, in comparison with that after 3.5 ms, and the plastic strain was generated from the surface to near 1.2 mm. When the plastic strain in the x-direction εx and the plastic strain in the z-direction εz were compared, εx was larger except at the surface, thereby matching the curvature radius anisotropy and trends shown in Fig. 4.
Plastic strain distributions along centerline.
Analyzing the movement of the hammer and pin enabled the determination of the change mechanism in the collision velocity and the impact force measured by experiment. The pin was also found to move faster than the hammer.
In peen forming by vibration peening, it is necessary to analyze the entire surface while moving the specimen in the model as illustrated in Fig. 5. This method is time consuming and not realistic. As shown in Fig. 7, the plastic strain distribution in the specimen is almost entirely determined by the maximum collision velocity. Therefore, inputting an initial velocity to the pin, an analysis was performed to simulate the plastic strain distribution at a single collision. The analytical model is shown in Fig. 9. In Step 1, the specified region (6 mm × 13.5 mm), where multiple pins were made to collide once each, was analyzed using the dynamic explicit FEM (LS-DYNA), as shown in Fig. 9(a). The base of the specimen was fixed to minimize the vibration. The element was an 8-node, hexahedral reduction integration element. The dimensions of the elements near the surface were 0.0625 mm in the thickness direction and 0.125 mm in the x- and z-directions. The shape of the pin was the same as that shown in Fig. 1. A total of 18 pins was arranged: 6 rows at 1 mm pitch in the x-direction and 3 columns at 4 mm pitch in the z-direction. The initial position was set such that there would be no pins simultaneously colliding with the specimen. The input conditions, such as material properties, were the same as those shown in Fig. 5. The analysis was performed with the initial velocity of the pin at 5–10 m/s. In Step 2, the free-state deformation of the base of the specimen at the wide region (80 × 80 mm) was analyzed, and the distribution of εx and εz in the thickness, calculated using LS-DYNA, was input as the inherent strain. An elastic analysis was performed using the static implicit FEM (Marc 2015), as illustrated in Fig. 9(b). To apply the inherent strain, a different linear expansion coefficient was inputted for each element at the same depth from the surface (considering the anisotropy of the x- and z-directions). The analysis was performed under conditions in which the overall temperature increased to 100°C. The material properties were the same as those shown in Fig. 5. Considering the geometric symmetry in the x- and z-directions, the model was made with 1/4. The element division in the thickness matched the LS-DYNA model. Figure 9(b) shows the analysis model with h = 5 mm. The model with h = 10 mm was made by increasing the elements on the back side of the model with h = 5 mm. The εx and εz obtained by LS-DYNA when h = 5 mm were also used for the case when h = 10 mm.
Analytical models for deformation of peen forming. (a) Analytical model for plastic strain distributions (LS-DYNA), (b) Analytical model for deformation (Marc).
Figure 10 shows the contours of εx and εz in the evaluation region at the second layer from the surface when v = 6.5 m/s and r = 8 mm. As shown in Fig. 10, εx and εz for each element differed. For εx, the mean was 0.0049, maximum was 0.036, and minimum was −0.014. A large tensile plastic strain and a compressive plastic strain were locally generated around the indentation, but the tensile plastic strain was generated at the mean. Considering that the mean plastic strain affected the deformation, the mean of εx and εz for the 3840 elements with the same depth from the surface in the evaluation area was adopted. The effect of the initial velocity v on the distribution of εx and εz in the thickness is shown in Fig. 11. The mean of εx and εz was plotted. εx increased as v increased, and the depth at which the plastic strain was generated increased. εz also followed a similar trend as εx, but εz was smaller than εx at any depth.
Plastic strain distributions with v = 6.5 m/s and r = 8 mm. (a) εx, (b) εz.
Effect of pin-velocity v on plastic strain distributions.
Figure 12 shows the effect of r on the distribution of εx and εz when v = 6.5 m/s. The peak value of εx was almost the same when r = 8 mm and 4 mm, and the introduced plastic strain depth was greater when r = 4 mm. εz also followed a similar trend as εx, but εz was smaller than εx at any depth. Meanwhile, when r = 12 mm, the peak value of εx was small, and the introduced plastic strain depth was shallow, and almost no difference was observed between εx and εz.
Effect of pin-tip radius r on plastic strain distributions.
Figure 13 shows the analytical results for effects of r and v on Rx and Rz when h = 5 mm. The vertical axis indicates the curvature. As in the case of Fig. 3, Rx and Rz were determined by fitting node displacements to the circular equation that minimizes the sum of squares of errors. At the same r, Rx decreased as v increased. At the same v, Rx decreased as r decreased. Rz was larger than Rx under all conditions. When r was smaller, the difference between Rx and Rz was larger. Figure 13 shows the experimental results. The analytical results agreed with the experimental results. v was estimated to be slightly slower than 6.5 m/s when r = 4 mm and 8 mm. As shown in Fig. 7, in the vibration peening analysis, when r = 8 mm, the initial velocity of the pin was approximately 6.26 m/s, and the maximum velocity was approximately 7.01 m/s, nearly matching the v estimated in Fig. 13. When v = 6.5 m/s, the Rz/Rx was 1.51 when r = 4 mm, 1.45 when r = 8 mm, and 1.00 when r = 12 mm. At r = 12 mm, the anisotropy of the curvature radius was lost. The results of the analysis of anisotropy of the curvature radius when r = 4 mm and 8 mm were smaller than the experimental results shown in Fig. 4. This may be attributed to the anisotropy of the material and the effect of the fixed method of the specimen in the experiment.
Effects of pin-tip radius r and pin velocity v on curvature of specimens when h = 5 mm.
Figure 14 shows the analytical results for effects of r and v on Rx and Rz when h = 10 mm. The trend was the same as for h = 5 mm shown in Fig. 13. When h = 10 mm, the distribution of εx and εz in the thickness analyzed by using LS-DYNA when h = 5 mm was used as an input value. The trend of the analytical result was in good agreement with that of the experimental result. When v = 6.5 m/s at h = 10 mm, Rz/Rx was 1.56 when r = 4 mm, 1.53 when r = 8 mm, and 1.01 when r = 12 mm. The results of the anisotropy of the curvature radius were also smaller than the experimental results shown in Fig. 4.
Effects of pin-tip radius r and pin velocity v on curvature of specimens when h = 10 mm.
Figure 15 shows the comparison of the analytical and experimental results of residual stress distribution in the thickness when h = 10 mm, r = 8 mm, and v = 6.5 m/s.
Comparison of residual stress distributions between experimental and analytical results.
The residual stress was measured using an X-ray residual stress analyzer (µ-X360, Pulstec Inc.) by the cos α method. The diffraction peak of the (311) plane of aluminum was measured using the Cr Kα ray. The residual stress measurement range was approximately 2 mm in diameter. The residual stress in the thickness was measured by performing electropolishing. In the analysis, both the residual stress σx in the x-direction and the residual stress σz in the z-direction were approximately −300 MPa near the surface. The depth of the compressive residual stress was 1 mm from the surface. The experimental results also tended to be approximately the same as the analytical results, however, the depth of the compressive residual stress was deeper in the experimental results, between 1.0 and 1.5 mm. In the analysis, at 1 mm or less, σz was slightly larger than σx, but no difference was observed in the experimental results. It may be assumed that the initial residual stress existed in the specimen.
These results demonstrate that the effect of r on the peen forming shape can be predicted by combining the dynamic explicit FEM and static implicit FEM. The vibration peening analysis can also be used to predict the pin velocity; thus, the analytical methods used in this study were shown to be effective for predicting peen forming.
Anisotropy of the curvature radius was experimentally confirmed in peen forming using vibration peening in which a rectangular solid pin was projected onto the specimen using an electric hammer, and the methods of predicting the formed shape using finite element analysis were studied.
This study was performed using a research grant from the Amada Foundation. I would like to express our gratitude to Mr. Mizuki Ihara, a student of the Engineering Department of Tokai University, who collaborated with me on this experiment.
