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Numerical research on stress peen forming with prestressed regular model
Accepted Manuscript Title: Numerical research on stress peen forming with prestressedregular model Author: X.D. Xiao Y.J. Wang W. Zhang J.B. Wang S.M. Wei PII: DOI: Reference:
S0924-0136(15)00051-5 http://dx.doi.org/doi:10.1016/j.jmatprotec.2015.02.008 PROTEC 14281
To appear in:
Journal of Materials Processing Technology
Received date: Revised date: Accepted date:
22-6-2014 29-11-2014 7-2-2015
Please cite this article as: X.D. Xiao, Y.J. Wang, W. Zhang, J.B. Wang, S.M. Wei, Numerical research on stress peen forming with prestressedregular model, Journal of Materials Processing Tech. (2015), http://dx.doi.org/10.1016/j.jmatprotec.2015.02.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical research on stress peen forming with prestressed regular model X.D. Xiaoa , Y.J. Wang a,∗, W. Zhangb , J.B. Wanga , S.M. Weia School of Mechanical Engineering, Northwestern Polytechnical University, 127 Youyixi Road, Xi’an Shaanxi 710072, China b Xi’an Aircraft International Corporation, Xi’an Shaanxi 710089, China
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Abstract
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Stress peen forming is widely used in aerospace industry to form the thin components with complex geometry. In the stress peen forming, the prestressing condition is a critical factor that need to be precisely controlled. In this study, the effects of the prestress on the peening deformations are investigated by using static and dynamic simulations. A regularly parked four-shot model is used to study the interactions of adjacent shot treatments. To verify the finite element model with precisely controlled parameters, regularly indenting tests are adopted. The analytical results reveal that the tensile prestress enhances the shot dimple and the plastic region sizes. The contributions of the prestress to the induced bending moments and stretching forces are indicated with two scale factors. Comparison of the singleshot model and regular parked four-shot model shows the peening effects can be equivalent within certain averaging regions. The static and dynamic simulations are compared on dimple sizes and forming effects. Finally, the indenting and peening tests are carried out to verify the analytical results.
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Keywords: Stress peen forming, Indentation, Finite element simulation, Residual stress, Experiment
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1. Introduction
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Shot peen forming is a dieless process for forming metallic components. It is especially suitable for forming large panel contours where the bend radii are within the metal’s elastic range. In the forming process, numerous shots impact the surface of the component. Each impact induces a elastically-plastically deformed dimple which results in a compressive stress field near the treated surface. The compressive stress cases the material to develop a convex shape to the treated surface. The conventional peen forming process induces isotropic compressive stresses resulting in identical curvatures in all directions. To form the component such as a wing panel that has a larger curvature in chordwise direction, stress peen fixtures are usually used. Li (1981) applied the stress peening forming process on forming integrally stiffened wing panels. Meyer et al. (1987) formed stringer-reinforced workpieces with the stress peen forming. The determination of the stress forming parameters is mainly dependent on trial and error. Many experimental works on the stress peen forming have been carried out. Barrett and Todd (1984) measured the residual stresses by x-ray diffraction while peen formed plate remained in curved condition. Xie et al. (2012) measured the residual stress profiles along different directions in titanium matrix composites treated by stress peening with various prestresses. Miao et al. (2010) investigated the relations between the prebending moment and the resulting arc height. Al-Hassani (1981) expressed the peening induced stress profile with a cosine function. Li et al. (1991) developed a simplified analytical model to calculate the induced stress for 100% peening coverage. Xiao et al. (2014) developed a cavity model to calculate the induced stress filed. However, it is difficult to involve the prestress in the analytical models mentioned above. Finite element (FE) method can be used to investigate the stress peen forming. For example, Miao et al. (2011) utilized the FE simulation to obtain the induced stresses with various prebending moments. Gari´epy et al. (2013b) experimentally and numerically investigated the influence of initial stresses in the plate on the resulting curvatures. author. Tel.: +86 029 88493717 Email addresses: [email protected] ( X.D. Xiao), [email protected] (Y.J. Wang )
∗ Corresponding
Preprint submitted to JMPT
February 12, 2015
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2. Stress peen forming and regularly parked four shots model
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2.1. Stress peen forming process
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In the FE simulation, the condition of unsaturated peening should be taken into account. Baughman (1986) advised that the shot peening coverage should be selected at a low value to leave room to increase contour with increased coverage. At a low peening coverage, a shot may solely acting on a small region of the target while the stress field is influenced by adjacent shot impacts. Meguid et al. (1999) studied the influence of the separation distance between the co-indenting shots upon the residual stress field. To simplify the analytical model, the random distribution of the shot impacting locations are usually idealized as a regularly parked shots model. Majzoobi et al. (2005) investigated the effect of impact multiplicity on the residual stress profile with a regularly parked multiple shots model. Kim et al. (2010) used a symmetry-cell FE model to obtain the area-averaged peening residual stress. Kirk and Hollyoak (2005) investigated the influence of coverage on the surface residual stress with regularly indenting tests. In the present work, static and dynamic simulations as well as experiments are performed to research the stress peen forming . First, the involved stresses and forming forces of stress peen forming are discussed. Then, a regularly parked four shots model is used to study the influence of peening coverage on the stress field. The acting effects of four shots are compared with of one shot to study the interactions of adjacent impacts. To precisely control the input parameters for verifying the analytical results, a prestressing indentation testing method is proposed. Finally, the analytical results will be compared with both indentation tests and shot peen forming tests.
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In conventional peen forming, the workpieces are usually kept in initial shape or is hanged freely during the forming process. In the stress peen forming, the workpiece is elastically prebended along a certain direction before and during the peening process, as shown in Fig. 1. The workpiece will obtain a larger resulting curvature in the prebending direction.
Figure 1: Stress peen forming process and related stresses and forces.
Since the peening deformations of square plates are influenced by secondary factors such as the rolling direction (Kulkarni et al., 1981), which leads to a basic instability of curvature, a narrow plate is taken into research in this work. When a narrow flat-plate is bent in the longitudinal direction to a curvature ρ xpre , a bending moment should be applied at both sides. Considering the narrow plate is freely restrained in the transverse direction, the applied bending moment per unit can be calculated as: M xpre = EIρ xpre (1) where I = h3 /12 is the moment of inertia per unit width, h is the thickness of plate. Then the transverse bending moment per unit Mypre = 0, and curvature ρypre = νρ xpre . E and ν are the elastic modulus and Poisson’s ratio of the material, respectively.
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The bending deformation induces a prestress field in the plate. The transverse and normal prestresses are approximated to zero. The longitudinal prestress at depth z is expressed as: σ xpre (z) = M xpre (h/2 − z) /I = Eρ xpre (h/2 − z)
(2)
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In order to indicate the magnitude of the prestress, a parameter λ is introduced here. The λ is defined as the ratio of the prestress on the peening surface to the yield stress σ s of the material as Eq. (3). The pre-deformation is usually controlled in the elastic range. The values of λ vary from 0 to 1 which induces tensile stress under the peening surface of the plate. λ = σ xpre (0)/σ s (3)
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In the stress peening process, the plate is maintained in the prebending configuration by external fixtures. The peening treatment changes the prestress field near the treated surface. The induced stresses under prebending σipre ind are not self-equilibrated, where i is used to indicated either x or y axes. After releasing the external fixing forces, the plate will be bent and stretched by σipre ind to a equilibrium state from the prebending configuration. With the assumption that the deformation during this process is in the elastic range, the residual stresses σres i can be calculated as Al-Hassani (1981): pre ind + σis + σipos b (4) σres i = σi
0
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The actions of the stress σipre ind can be equivalent to a bending moment per unit Mipre ind and a stretching force per unit Fipre ind as: Z h pre ind = (5) Mi σipre ind (h/2 − z) dz Fipre ind =
0
σipre ind dz
(6)
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The Mipre ind is a resultant of Miind induced by peening treatment and partial prebending moment (−αi Mipre ) as Eq. (7) (Gari´epy et al., 2014). The parameter α indicates the ratio of residual prebending moment to the initial prebending moment. (7) Mipre ind = Miind − αi Mipre
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The peening stress induces a stretching force Fiind while residual prestress also produces a additional stretching force. A dummy initial stretching force is defined as the force induced by the prestress within the half plate. A parameter βi is used to indicate the ratio of the stretching force corresponding to the residual prestress to the dummy initial stretching force. So the Fipre ind can be expressed as a resultant as: Fipre ind
=
Fiind
+ βi
Z 0
h/2
σipre (z) dz
(8)
In Eq. (4), the stretching stress σis = Fipre ind /h. The post-bending (from prebending configuration) stress σipos b (z) = (h/2 − z) /I. So the total bending (from the flat configuration) stress σbi = σipre + σipos b . Neglecting the influence of the stretching force on the resultant curvature, the post-bending curvatures can be calculated from the Kirchhoff-Love theory of plates as: pre ind pre ind M − νM x y ρ xpos = (9) EI pre ind My − νM xpre ind pos ρy = (10) EI Then, the final residual forming curvature can be calculated as: Mipre ind
pre ρres + ρipos i = ρi
(11)
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2.2. Regularly parked four shots model
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A regularly parked four-shot model is used to study the interactions of adjacent shot impacts and to determine the discussed parameters in stress peen forming, such as αi and βi . As shown in Fig. 2, assuming four impacts located at the four corners of a square area Ac with side length of d, the stress fields that relatively induced by the four impacts interact each other in the Ac . The impacting sequence of the four shots is ignored here. Single shot-impact produces a dimple with projected radius of a and a residual stress field. The peening coverage in Ac can be calculated as: πa2 /d2 for d ≥ 2a p 2 d 2 2 − d 2 /4 /d 2 for d ≥ 2a a πa − 4a arccos + 2d (12) C= 2a √ 1 for d ≤ 2a
(b)
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Figure 2: (a) Single shot impacting target. (b)Four shots parked regularly and residual stress fields overlapping in the square area.
Averaging the peening stresses within A s and Ac , the integral forces on the regions can be obtained. The relationships between the peening forces and the coverage can be established. Then the relations would be used to calculate the resultant forming shape. The average stresses also can be used to further analyses as Gari´epy et al. (2012). The resultant deformation of a plate is only related to the residual stresses in the material. For the convenience of precisely controlling the input parameters, the stresses also can be introduced by other equivalent methods such as indentation (Kirk and Hollyoak, 2005). The indentation method is used in this paper to impose precise coverage percentages. The indentation are also compared with shot peen forming. 3. Finite element simulation models Four FE models are proposed: static simulation model (SSM) of prestressed single shot indenting, dynamic simulation model (DSM) of prestressed single shot impacting, SSM of regularly parked four-shot indenting, and DSM of regularly parked four-shot impacting, as shown in Fig. 3.
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Figure 3: One-quarter of discretized geometries of target for (a) single shot indenting and impacting simulations, (b) four-shot indenting simulation, and (c) four-shot impacting simulation.
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3.1. Material properties The target material under study is aluminum alloy 2024-T351. The Johnson Cook (JC) model is used to represent the response of the material. As the shot peen forming is a cold-working process , the thermal properties of the material is neglected here, and the JC model can be reduced as: σ = A + Bεn 1 + C ln ε˙ ∗
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where σ is the effective stress, ε is the effective plastic strain, ε˙ ∗ is the normalized effective plastic strain rate (typically normalized to a strain of 1.0 s−1 ), n is the work hardening exponent and A, B, C, and m are constants. The constants and other parameters of the material (Lesuer, 2000) are listed in Tab. 1. Table 1: Material parameters of Al alloy 2024-T351. ν
σs
73 GPa
0.33
343 MPa
A
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E
C
n
Elongation
ρ
684 MPa
0.0083 MPa
0.73
17%
2770 kg/m3
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369 MPa
B
3.2. SSM of prestressed single shot indenting The simulation procedure is carried out by two static/general steps: applying prebending to obtain the prestress distribution and the pre-deformation of the target, then indenting the target with one shot by loading and unloading a force. The FE model shown in Fig. 3(a) is used for the simulation. A shot (Radius = 1.5875 mm) is normally loaded at the symmetrical position of the one-quarter model of target. The dimensions of h = 5 mm and L = 6 mm are selected. A refined mesh is made at the contact region. Considering the relatively high yield and hardness values of the indenter compared to the target material, the shot is assumed to be rigid. In the first step, the symmetry boundary conditions are imposed on the yOz and zOx planes. The nodes on the line of x = 0 and z = h are fixed in the z direction. Linearly distributed pressure with an amplitude of λσ s (2z − h) /h along z direction is applied on the x = L surface of the target. The prestress is: σ xpre (z) = −λσ s (2z − h) /h
(14)
The values of λ under study are 0, 0.2, 0.4, 0.6, 0.8 and 1. In the second step, the normal displacements of the x = 0, x = L and y = 0 surfaces of the target are fixed at current position with symmetric boundary conditions. The bottom surface of the target is fixed at current position in 5
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z direction. Then, the target is subjected to a shot indenting with a gradually increased load F at normal incidence. After reaching maximum values, the load is gradually unloaded again. The maximum values of the load under study are 1839/4 N, 1226/4 N and 613/4 N. 3.3. DSM of prestressed single shot impacting
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This simulation has the same procedure, boundaries, FE model and mesh dimensions as the SSM. Two static/general - dynamic/implicit steps are carried out. The difference is that, in the second step, the rigid shot impacts the target at normal incidence with an initial velocity V. The values of V under study are 30, 40 and 50 m/s. One quarter of the lumped mass of the shot 0.01595 g is given to the shot model. The lumped mass of the shot was determined by measuring the real APB1/8 shot.
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3.4. SSM of regularly parked four shots indenting
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3.5. DSM of regularly parked four shots impacting
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The study is further extended to investigate the overlapping effects of the adjacent indenting stress fields. A FE model, as shown in Fig. 3(b), is used to simulate the process of four shots (Radius = 1.5875 mm) successively indenting the prestressed target. The four shots are respectively located at the four corners of Ac with side length of d. The values of d under study are 1, 2, 3, 4, and 6 mm. The target dimensions are 8(L) × 8(L) × 5(h) mm. A refined mesh is made at the contact region. The simulation procedure and the boundary constraints are the same as the single-shot model. The values of λ under study are 0, 0.375, and 0.75. In the second step, the four shots successively indent the target in order of 1-2-3-4 marked in Fig. 3(b). The indenting force F under study is 1226 N. In the FE model, shot marked 1 is loaded 1839/4 N, shot marked 2 and 4 loaded 1839/2 N, and shot marked 3 loaded 1839 N.
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This simulation has the same procedure, boundaries, FE model and mesh dimensions as the above static fourshot model. Two static/general - dynamic/implicit steps are used to simulate the process of four shots successively impacting the prestressed target with a initial velocity V, as shown in Fig. 3(c). The V under study is 40 m/s. In the FE model, shot marked 1 is given lumped mass of 0.01595 g, shot marked 2 and 4 given 2 × 0.01595 g, and shot marked 3 given 4 × 0.01595 g. The impacting order is also 1-2-3-4.
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4. Experimental processes 4.1. Prebending device and specimen
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A metallic holder shown in Fig. 4(a) was used to bend and maintain the specimen in the experiments. The predeformation of the specimen is controlled by the prebending mold. Three prebending molds, as shown in Fig. 4(b), are used with radii of curvature 626 mm, 1252 mm, and infinite, respectively. Rectangular strips with dimensions of 100(L s ) × 30(Bs ) × 5(h s ) mm were cut from an Al 2024-T351 plate. The longitudinal direction of the strip is the same as the rolling direction of the plate. The pre-deformations of the strips bent on the holder were simulated with FE method. In order to shorten the paper the models are not shown here. The prestress ratios were determined that λ = 0.75, 0.375 and 0 corresponding to R pre = 626, 1252 and ∞. The transverse and normal stresses under prebending are approximated to zero. 4.2. Regularly indenting tests
In order to obtain precise coverage and indenting locations, a set of static indenting tests was performed by using a developed hardness tester. A manual cross slide was fixed on the work table of the hardness tester to precisely locate the indenting positions, as shown in Fig. 5(a). The indenter installed a carbide ball with diameter of 3.175 mm. Three strips were respectively bent to R pre = ∞, 1252 and 626 mm. Each strip was indented with loads of 613 N and 1839 N at different locations. The indentation size is determined by measuring two diagonals of the indentation by an optical microscope. Under the load of 1226 N, regularly distributed indentations with indenting gaps of 2, 3 and 4 mm in association with prebending radii of ∞, 1252 and 626 mm were produced on nine test strips, respectively. The indenting locations 6
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Figure 4: (a) Prebending device and (b) prebending molds with radii of curvature R pre = ∞, 1252 and 626 mm.
(b)
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Figure 5: (a) Regularly indenting device and (b) indentation locations and sequence.
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and sequence are plotted in Fig. 5(b). The related parameters for different indenting gaps are listed in Tab. 2. The treating sequence influences the resultant forming shape (Gari´epy et al., 2013a). However, the size of the testing strip is relatively small, so the effects of the treating sequence can be neglected. Only one type of treating sequence was used. The resultant curvatures in the transverse and longitudinal directions of the test strips were calculated from the measured arc heights. Table 2: Dimensions of indentation distributions. d (mm) Bc (mm) Lc (mm) Total indentations
2 3 4
24 24 24
80 78 80
13 × 41 9 × 27 7 × 21
4.3. Stress peen forming tests Stress peen forming tests were carried out on the Wheelabrator MP20000 Aircraft Wing Peening System. Hardened steel peening ball APB1/8 with nominal diameter of 3.175 mm was used. One peening pass was along the longitudinal centerline of the strip. The average shot velocity can be measured with a high-speed photographic setup attached to the MP20000. The shot velocity of 40 m/s is selected. The distance between the peening nozzle and the plate is 500 mm. The rate of shot 7
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flow is 10 kg/min. Prestress ratio of 0, 0.375 and 0.75 in association with nozzle moving speeds of 4, 6, 8 m/min were imposed on nine test strips, respectively. The shot dimple diameters and the resultant curvatures were measured. The peening coverage was measured by the optical inspection (Miao et al., 2010). 5. Results and discussion
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5.1. Effects of prestress on the indentation and shot dimple sizes By the simulations, the indenting diameters under different F and λ as well as the impacting diameters with various V and λ are obtained by detecting the maximum indenting depth in the loading process. The dimple diameter D is calculated from the depth δ as: 1/2 D = 2δR − δ2 (15)
(a)
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The simulated dimple diameters are compared with the experimental values as shown in Fig. 6. The simulated values are consistent with the experimental values, which verifies the simulation model. The dynamically simulated results have the same tendency as the static simulation. The dimple size increases with the increase of the prestress ratio corresponding to increasing the tensile stress beneath the treated surface. It may be explained that the shot compresses the target perpendicularly to the surface, the tensile stress in the target increases the shear stress beneath the shot resulting in larger deformations (Simes et al., 1984).
(b)
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Figure 6: Dimple diameters measured in the experiments and (a) static simulations as well as (b) dynamic simulations..
The indentation diameter D st is fitted as a function of the indenting force F and the prestress ratio λ as: D st = 0.4786 + 5.871 × 10−4 F + 0.05798λ + 3.609 × 10−6 Fλ − 8.921 × 10−8 F 2 + 8.625 × 10−3 λ2
(16)
In the same way, the shot dimple size Ddy is fitted as a function of the shot velocity V and the prestress ratio λ as: Ddy = 0.5173 + 0.02256V + 0.04117λ + 3.487 × 10−5 Vλ − 1.108 × 10−4 V 2 + 6.995 × 10−3 λ2
(17)
The coefficients of determination (R2 ) of Eq. (16) and Eq. (17) are 0.99997 and 0.99998, respectively. According to Eq. (16) and Eq. (17), a dimple with a certain size may be obtained either by static indenting or by dynamic impacting with appropriate treating parameters. Tab. 3 shows the matched indenting forces and shot velocities to obtain the same dimple diameter. For a certain indenting force, the matched shot velocities with different λ vary in a small range. The variances (var.) of the V are relatively small no matter what indenting forces. 5.2. Effects of prestress on the plastic region size The sizes of the plastic region of one shot treating were detected in the simulations. As shown in Fig. 7, the plastic deformation is examined with the equivalent plastic strain at integration points greater than 0.002. The plastic 8
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Table 3: Matched indenting forces and impacting velocities with the same dimple diameter. F=613 N
F=1226 N
F=1839 N
λ
D st = D = Ddy mm
V m/s
D st = D = Ddy mm
V m/s
D st = D = Ddy mm
V m/s
0 0.375 0.75
0.80 0.83 0.85
13.67 14.04 14.45
1.06 1.09 1.12
28.13 28.63 29.17
1.26 1.28 1.31
41.03 41.70 42.42
14.05 0.1524
28.64 0.2698
41.72 0.4826
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Aver. Var.
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region size is denoted as r x , ry and rz , which respectively indicate the half plastic region size in the x and y axises and the plastic region depth. Fig. 8 shows the values of r x , ry and rz under different treating parameters. It can be seen that greater indenting load or impacting velocity produces larger plastic deformation. Especially, the λ has an obvious influence on the value of ry . The r x and rz increase slightly with the increase of the λ. With a relative small λ, the plastic region depth is greater than its half widths. But the ry will exceed the rz with increasing λ. The difference between r x and ry can be explained that the tensile prestress in the x direction makes the material near the x axis yield later than the material near the y axis.
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Figure 7: Plastic deformation examined with the equivalent plastic strain at integration points greater than 0.002.
5.3. Effects of prestress on the indenting and impacting stresses The prestress influences the plastic deformation of the material as well as the induced stress σipre ind . Fig. 9 shows and σypre ind along the center line beneath the shot in the one-shot model with λ = 0.6. Fig. 9(a) depicts the stress profiles under three indenting forces and Fig. 9(b) under three impacting velocities. It can be seen from the figures that under prebending: σ xpre ind
(1) The compressive σipre ind at the surface increases with increasing either the indenting force or the shot velocity. (2) The differences of the σipre ind of different treating parameters decrease with the depth from the surface. When the compressive σipre ind is up to the maximum values beneath the surface, the differences become negligible. (3) The depth of the induced compressive σipre ind increases with increasing either the indenting force or the shot velocity. (4) The induced compressive σ xpre ind and σypre ind have the same profile near the treated surface, while significant differences are in the bottom half of the target. Fig. 10 compares the induced stresses with and without prestress. The prestress profiles are also plotted in the figures. Comparison of the stress profiles leads to the following observations: 9
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Figure 8: Plastic region sizes of (a) static simulations with various loads and prestress ratios, and (b) dynamic simulations with various shot velocities and prestress rations.
(b)
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Figure 9: σi (a) in static simulation with λ = 0.6 and various indenting forces, (b) and in dynamic simulation with λ = 0.6 and various impacting velocities.
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(1) Along the center line, the profiles of the compressive σipre ind have not obvious changes with the increase of the λ. The maximum values of the compressive σipre ind slightly decrease with increasing the λ. (2) In the bottom half of the target, the σipre ind is approximately equal to the σipre . (3) In the middle region, the σipre ind is tensile. The tensile σ xpre ind is obviously greater than the tensile σypre ind and σipre . The stress field of one shot treating various from the core region. The average stresses of each layer with projected region A s are calculated from the stresses at the integration point and the current volume of each element. Fig. 11 shows the static and dynamic average induced stress profiles under prestress ratio of 0.4 within A s with various values of l . From the figures, it can be seen that: (1) The average σipre ind within the bottom half of the target does not have changes with the variation of the size of As . (2) Within the top half of the target, the compressive σipre ind decreases with increasing the size of A s . (3) When the A s is relatively large, the compressive σ xpre ind is lower than the σypre ind . Within a smaller A s , the compressive σ xpre ind is approximately equal to σypre ind . The plastic deformation can reduce the difference between 10
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(a)
(c)
(d)
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Figure 10: Prestress profiles and comparison of induced stresses without prebending and with prebending when (a) F = 1226 N and λ = 0.2, (b) F = 1226 N and λ = 0.8, (c) V = 40 m/s and λ = 0.2 (d) V = 40 m/s and λ = 0.8.
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σ xpre and σypre even eliminate the difference with large plastic strains. So the prestresses have little influence on the induced stresses within the central plastic region.
(a)
(b) pre ind
Figure 11: (a) Statically and (b) dynamically simulated average σi
with λ = 0.4 within A s with various values of l .
5.4. Overlapping of adjacent stress and strain fields The stress field induced by single shot have been discussed. For multiple shots treating, there are interactions of the stress fields respectively induced by adjacent shots. Fig. 12 shows the Mises stress fields induced by single shot 11
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and four shots with V = 40 m/s, d = 2 mm and λ= 0, 0.375. Comparison of the single-shot and four-shot impacts shows that the overlapping of the stress fields mainly takes place in the middle region Ac . Comparison of the stress fields induced with and without prebending shows there are not significant difference of stress field near the impacting position.
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(d)
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Figure 12: Mises stress fields induced by (a, c) one shot and (b, d) four shots with V = 40 m/s and d = 2 mm.
Averaging the σipre ind within A s and Ac with l = 1 and d = 2, the average stress profiles are plotted in Fig. 13. It can be seen that: (1) The stress profiles of static and dynamic simulations have the same tendency. (2) The σipre ind averaged in A s with l = 1 and in Ac with d = 2 have the similar profiles. The depths of the compressive stresses are equal. (3) In the surface layer, the σipre ind of four shots are greater than those of single shot. The stresses are overlapped at the surface. (4) The stress profiles of four shots are wholly higher than those of single shot, which cased by the stenching forces of four treats on the whole FE model greater than those of single treat. If the treating locations are near enough, the plastic regions will be in connection, as shown in Fig. 14. When the treating locations is far enough, the plastic regions are in separation. Between the two adjacent plastic regions, the deformation of the material is in the elastic range. 5.5. Bending and stretching forces from induced stresses The equivalent bending moment Mipre ind and stretching force Fipre ind can be used to represent the acting effects of the average σipre ind with Eqs. (5) and (6). Fig. 15 shows both Mipre ind and Fipre ind with λ = 0. Four curves are 12
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pre ind
in A s with l = 1 and in Ac with d = 2 under (a) F = 1226 N and (b) V = 40 m/s.
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Figure 13: Comparison of the average σi
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Figure 14: Equivalent plastic strain fields with (a) d = 1 mm, (b) d = 2 mm and (c) d = 3 mm. The shot velocity is 40 m/s and the prestress ratio λ = 0.375.
The curves corresponding to the dynamic and static simulations have the same tendency. The equivalent forces decrease sharply in the beginning then approach to zero with increasing d. The FACL is close to the FACD. The deviations are cased by the interactions of the stress fields. The FASD is lower than the FACD while the FFASD is close to the FACD when the d is greater than a critical value. The critical value is approximately equal to the plastic region width (2 ×r x (ry )) shown in Fig. 8.
Ac ce
(1) (2) (3) (4)
pt
compared in each figure: equivalent force in Ac versus d (FACD), equivalent force in A s versus l where l = d/2 (FASL), equivalent force in A s versus l where l = d (FASD), and four times equivalent force in A s versus l where l = d (FFASD). From the figures, it can be seen that:
So that In the condition of without prebending, the peening effects of single shot can be considered as that of four shots. The FACD can be represented by FFASD for d > 2 × r x otherwise by FACL for d ≤ 2 × r x in the range of permitted errors . In the stress peen forming, the σipre ind involves a part of σipre . The FASD and FFASD can not compared with the FACD. Fig. 16 shows the comparison of the FASL and the FACD of stress peening. The FACD with λ = 0 shown in FIg. 15 is also plotted in Fig. 16. It can be seen that: (1) The FASL is close to the FACD no matter what λ. (2) The influence of the magnitude of λ on the Mipre ind is greater than the influence on the Fipre ind . (3) The FACD with prestress are very close to the FACD without prestress in the y direction. The effects of prestress is mainly on the FACD in the x direction. (4) The values of Fipre ind and Mypre ind are greater than zero while the values of M xpre ind may be positive or negative. When M xpre ind = 0, the final forming curvature could be the same as the prebending curvature in the x direction. 13
Page 13 of 22
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(b)
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(c)
(d)
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Figure 15: Comparison of (a) equivalent bending moments vs. separation distance d and (b) equivalent stretching forces vs. d in dynamic simulations and (c, d) in static simulations.
pt
In Eqs. (7) and (8), the αi and βi indicate the effects of the prestress on the induced stresses. Smaller αi and greater βi indicate the greater contributions of the prestress on the Mipre ind and Fipre ind . The Miind and Fiind are hard to be determined under prebending. If the Mipre ind and Mipre ind with λ = 0 are used as universal Miind and Fiind , the αi and βi will be determined. As shown in Fig. 16, in the y direction the FACD with prestress are very close to the FACD without prestress, so that αy = 0 and βy = 0. Fig. 17 shows the α x and β x corresponding to various λ and d. It can be seen from the figures that:
Ac ce
(1) In the dynamic and static simulations, the curves of α x vs. d and β x vs. d have the same tendency. (2) The variation of the λ has little influence on the α x and β x . The α x and β x are mainly dependent on the d and the treating intensity, such as F and V. (3) When d = 1 corresponding to about full coverage, the α x and β x are approximated to 0.5. When the d is relatively great corresponding to a very low intensity of treating, the α x tends to 1 and β x tends to 0. 5.6. Resultant curvature and experimental verification The post-bending curvatures ρipos are calculated with Eqs. (9) and (10). Fig. 18 shows the ρipos with λ = 0.375, 0.75 under impacting and indenting. It can be seen that 1. The ρypos are greater than zero while the ρ xpos may be a negative values. After removing the fixing loads on the plate, springback of the plate will take place in the x direction if ρ xpos < 0. If ρ xpos = 0, the ρ xpre would be leaved. 2. The λ has a significant influence on the ρ xpos while a slight influence on the ρypos . The measured curvature of experimental specimen is the residual curvature ρres i . Fig. 19 shows the indentation strips with indenting gaps of 2, 3 and 4 mm. Fig. 20 shows the shot peening strips with nozzle moving speed of 4, 6 res and 8 m/s. The experimental ρres i are obtained by measuring the final arc heights. The analytical ρi can be calculated 14
Page 14 of 22
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(b)
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(a)
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(c)
(d)
Ac ce
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ed
Figure 16: Comparison of (a, c) equivalent bending moment vs. d and (b, d) equivalent stretching force vs. d in dynamic simulation with different λ.
(a)
(b)
Figure 17: Ratios of (a) prebending moment and (b) stretching force in static and dynamic simulations.
res with Eq. (11). Fig. 21 shows the comparison of the experimental ρres i and the analytical ρi . The analytical results are in good agreement with the experimental values. Fig. 21(a) depicts the relationship between ρres i and peening coverage. The analytical coverage is calculated with Eq. (12). Fig. 21(b) depicts the relationship between ρres and the d of indentation. Comparison of the ρres with i i res prebending and the ρres without prebending shows that the increase of the λ leads to the increase of ρ while the x i . The same tendency were also obtained by Miao et al. (2010) and Li (1981). decrease of ρres y
15
Page 15 of 22
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Figure 18: Post-curvatures with different λ in (a) dynamic simulations and (b) static simulations.
Ac ce
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Figure 19: Indenting specimens with different indenting gaps and radii of prebending curvature.
Figure 20: Stress peen formed specimens with different nozzle moving speed and radii of prebending curvature.
5.7. Indenting force vs. impacting velocity The connections between the dynamic and static analyses can be established with the equivalent forming forces. Under a certain conditions of the target and shot, the same Mipre ind can be obtained either by shot peening with an impacting velocity V or by indentation with an indenting force F. The relationships between the Mipre ind and the F( or V) as well as λ are obtained with the response surface quadratic model. The values of the R-Squared are all close to 1. Fig. 22 shows the both relationships of F vs. Mipre ind and V vs. Mipre ind with l = 1. Starting from a certain F, a corresponding V can be found with the same Mipre ind , and vice versa. Tab. 4 lists the matched indenting forces and impacting velocities with the same Mipre ind . The variances of the predicted values of V with different λ are relatively small. The variation of the λ has little influence on the predicted values. The predicted average V (Tab. 4) within A s with different sizes are listed in Tab. 5. The variances of the V are also relatively small except the values corresponding to F = 613 N. When F = 613 N, the predicted V is about 16.5 m/s 16
Page 16 of 22
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an
(a)
Ac ce
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Figure 21: Comparisons of (a) dynamically analytical residual curvatures and experimental values as well as (b) statically analytical residual curvatures and experimental values with different λ.
(a)
(b) pre ind
Figure 22: Indenting force vs. impacting velocity with the same (a) M x
pre ind
and (b) My
under different prestress ratios.
which is far away from the range of V(30, 50) m/s studied in the simulation. The deviation between the fitting and real values increases when the input value is far away from the fitting region. In Tab. 6, the predicted values of V in the Tab. 5 and the values of V in the Tab. 3 are compared. There is a direct correlation between the dimple size and the forming force. With the same shot and target, if the sizes of the dimples produced by different methods are the same, the produced forming forces are approximately equal under the conditions presented in this paper. So the indenting experiments can be used to precisely introducing the induced stresses to study the characteristics of the shot peening by monitoring the size of the macroscopic dimple. In this work, the connections between the static and dynamic analysis ae established on the material Al alloy 2024-T351 whose strain rate sensitivity is relatively low. For the material with high sensitivity of strain rate , a further 17
Page 17 of 22
development on the connections may be needed. pre ind
Table 4: Indenting force vs. indenting velocity with the same Mi
0 0.425 0.85
F=1839 N N
V m/s
pre ind Mx
N
V m/s
pre ind My
N
V m/s
pre ind Mx
N
V m/s
pre ind My
N
V m/s
73.62 -407.96 -889.53
19.10 18.01 17.17
80.72 44.41 8.11
17.96 18.36 18.79
275.94 -150.43 -576.80
29.62 29.53 29.47
261.76 216.57 171.38
28.37 28.64 28.93
470.28 99.10 -272.07
41.11 42.14 42.89
18.09 0.933
18.37 0.1687
29.54 0.0054
N
V m/s
474.64 420.57 366.51
41.01 41.23 41.47
28.65 0.0779
42.05 0.7963
41.24 0.0542
cr
Aver. Var.
pre ind
My
ip t
λ
pre ind Mx
averaging in A s with l=1.0 mm.
F=1226 N
F=613N
F=613 N
Aver. Aver. Var.
) m/s
F=1226 N pre ind
V(My
) m/s
pre ind
V(M x
F=1839 N
pre ind
) m/s
V(My
14.28 18.09 17.18
14.19 18.37 15.36
28.42 29.54 28.37
16.52 16.25 3.5374
15.97
28.78 28.39 0.4982
) m/s
pre ind
) m/s
V(M x
pre ind
V(My
) m/s
27.50 28.65 27.84
40.36 42.05 40.89
39.82 41.24 40.65
28.00
41.10 40.83 0.5855
40.57
an
0.5 1 1.5
pre ind
V(M x
M
l mm
us
Table 5: Predicted average indenting velocities with different sizes of averaging region.
ed
Table 6: Comparison of shot velocities respectively predicted with shot dimple size and bending moment. F = 1226 N
F = 1839 N
14.05 16.25 14.48%
28.64 28.39 0.89%
41.72 40.83 2.14%
Ac ce
6. Conclusion
F = 613 N
pt
Corresponding V (m/s) with the same dimple diameter Corresponding V (m/s) with the same bending moment Relative error
The main conclusions of this study are as follows: 1) The effects of prestress on the stress peen forming were studied with both static and dynamic simulations. The contributions of the prestress to the forming effects are indicated with two ratios αi and βi . Then the forming effects under different prestresses can be predicted directly from the results of conventional peen forming and the ratios. 2) A regularly parked four shots model was used to investigate the interactions of adjacent shot treatments. The results shows that the forming effects of four shot impacts within a square area are very close to the forming effects of single shot impact within the quarter area. So the analytical model of single shot impact can be used to predict the peening effects of multiple shot impacts. 3) The effects of static indenting are nearly equivalent to the effects of dynamic impacting with the same size of macroscopic dimple. The regularly indenting test can be used in the researches to introducing the peening stress with precisely controlled parameters. This work focused on the deformations of the narrow strip. The prebending was only applied in the longitudinal direction in analytical model. Considering the complexity of the practical prebending conditions with respect to large scale components, further works are required to study the influence of complex fields of prestress on the resulting curvatures. 18
Page 18 of 22
Acknowledgement The authors would like to acknowledge the financial supports by the National Natural Science Foundation of China under Grant No. 512754 20, and by the corresponding research project from AVIC Xian Aircraft Industry (Group) Company Ltd..
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References
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us
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Al-Hassani, S., 1981. Mechanical aspects of residual stress development in shot peening, in: Proceedings of the 1st international conference on shot peening, pp. 583–602. Barrett, C., Todd, R., 1984. Investigation of the effects of elastic pre-stressing technique on magnitude of compressive residual stress induced by shot peening of thick aluminum plates, in: Proceedings of the 2nd international conference on shot peening, pp. 15–21. Baughman, D., 1986. An overview of peen forming technology. Pergamon Press, Advances in Surface Treatments. Technology–Applications– Effects. 3, 209–214. Gari´epy, A., Cyr, J., Levers, A., Perron, C., Bocher, P., Lvesque, M., 2012. Potential applications of peen forming finite element modelling. Advances in Engineering Software 52, 60–71. Gari´epy, A., Larose, S., Perron, C., Bocher, P., L´evesque, M., 2013a. On the effect of the peening trajectory in shot peen forming. Finite Elements in Analysis and Design 69, 48–61. Gari´epy, A., Larose, S., Perron, C., Bocher, P., Lvesque, M., 2013b. On the effect of the orientation of sheet rolling direction in shot peen forming. Journal of Materials Processing Technology 213, 926–938. Gari´epy, A., Miao, H., Lvesque, M., 2014. 3.12 - peen forming, in: Yilbas, S.H.F.B. (Ed.), Comprehensive Materials Processing. Elsevier, Oxford. Comprehensive Materials Processing, pp. 325–326. Kim, T., Lee, J.H., Lee, H., Cheong, S.k., 2010. An area-average approach to peening residual stress under multi-impacts using a three-dimensional symmetry-cell finite element model with plastic shots. Materials & Design 31, 50–59. Kirk, D., Hollyoak, R.C., 2005. Relationship between coverage and surface residual stress, in: Proceedings of the 9th international conference on shot peening, pp. 373–378. Kulkarni, K.M., Schey, J.A., Badger, D.V., 1981. Investigation of shot peening as a forming process for aircraft wing skins. Journal of Applied Metalworking 1, 34–44. Lesuer, D., 2000. Experimental investigation of material models for Ti-6Al-4V and 2024-T3. Technical Report. FAA Report DOT/FAA/AR-00/25. Li, J., Mei, Y., Duo, W., Renzhi, W., 1991. Mechanical approach to the residual stress field induced by shot peening. Materials Science and Engineering: A 147, 167–173. Li, K., 1981. Using stress peen-forming process for integrally stiffened wing panels, in: Proceedings of the 1st international conference on shot peening, Paris. pp. 555–563. Majzoobi, G.H., Azizi, R., Alavi Nia, A., 2005. A three-dimensional simulation of shot peening process using multiple shot impacts. Journal of Materials Processing Technology 164165, 1226–1234. Meguid, S., Shagal, G., Stranart, J., Daly, J., 1999. Three-dimensional dynamic finite element analysis of shot-peening induced residual stresses. Finite Elements in Analysis and Design 31, 179–191. Meyer, R., Reccius, H., Schulein, R., 1987. Shot peen-forming of nc-machined parts with integrated stringers using large balls.(retroactive coverage). Deutsche Gesellschaft fur Metallkunde e. V, , 327–334. Miao, H., Demers, D., Larose, S., Perron, C., Lvesque, M., 2010. Experimental study of shot peening and stress peen forming. Journal of Materials Processing Technology 210, 2089 – 2102. Miao, H., Larose, S., Perron, C., L´evesque, M., 2011. Numerical simulation of the stress peen forming process and experimental validation. Advances in Engineering Software 42, 963–975. Simes, T., Mellor, S., Hills, D., 1984. A note on the influence of residual stress on measured hardness. The Journal of Strain Analysis for Engineering Design 19, 135–137. Xiao, X.D., Wang, Y.J., Gao, G.Q., Wang, J.B., Wei, S.M., 2014. An analytical approach to relate shot peen forming parameters to resulting curvature with expanding cavity model, in: Proceedings of the 12th international conference on shot peening. Xie, L., Jiang, C., Lu, W., Chen, Y., Huang, J., 2012. Effect of stress peening on surface layer characteristics of (TiB + TiC)/Ti6Al4V composite. Materials & Design 33, 64 – 68.
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List of Figures
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Fig. 1. Stress peen forming process and related stresses and forces. Fig. 2. (a) Single shot impacting target. (b)Four shots parked regularly and residual stress fields overlapping in the square area. Fig. 3. One-quarter of discretized geometries of target for (a) single shot indenting and impacting simulations, (b) four-shot indenting simulation, and (c) four-shot impacting simulation. Fig. 4. (a) Prebending device and (b) prebending molds with radii of curvature R pre = ∞, 1252 and 626 mm. Fig. 5. (a) Regularly indenting device and (b) indentation locations and sequence. Fig. 6. Dimple diameters measured in the experiments and (a) static simulations as well as (b) dynamic simulations.. Fig. 7. Plastic deformation examined with the equivalent plastic strain at integration points greater than 0.002. Fig. 8. Plastic region sizes of (a) static simulations with various loads and prestress ratios, and (b) dynamic simulations with various shot velocities and prestress rations. Fig. 9. σipre ind (a) in static simulation with λ = 0.6 and various indenting forces, (b) and in dynamic simulation with λ = 0.6 and various impacting velocities. Fig. 10. Prestress profiles and comparison of induced stresses without prebending and with prebending when (a) F = 1226 N and λ = 0.2, (b) F = 1226 N and λ = 0.8, (c) V = 40 m/s and λ = 0.2 (d) V = 40 m/s and λ = 0.8. Fig. 11. (a) Statically and (b) dynamically simulated average σipre ind with λ = 0.4 within A s with various values of l . Fig. 12. Mises stress fields induced by (a, c) one shot and (b, d) four shots with V = 40 m/s and d = 2 mm. Fig. 13. Comparison of the average σipre ind in A s with l = 1 and in Ac with d = 2 under (a) F = 1226 N and (b) V = 40 m/s. Fig. 14. Equivalent plastic strain fields with (a) d = 1 mm, (b) d = 2 mm and (c) d = 3 mm. The shot velocity is 40 m/s and the prestress ratio λ = 0.375. Fig. 15. Comparison of (a) equivalent bending moments vs. separation distance d and (b) equivalent stretching forces vs. d in dynamic simulations and (c, d) in static simulations. Fig. 16. Comparison of (a, c) equivalent bending moment vs. d and (b, d) equivalent stretching force vs. d in dynamic simulation with different λ. Fig. 17. Ratios of (a) prebending moment and (b) stretching force in static and dynamic simulations. Fig. 18. Post-curvatures with different λ in (a) dynamic simulations and (b) static simulations. Fig. 19. Indenting specimens with different indenting gaps and radii of prebending curvature. Fig. 20. Stress peen formed specimens with different nozzle moving speed and radii of prebending curvature. Fig. 21. Comparisons of (a) dynamically analytical residual curvatures and experimental values as well as (b) statically analytical residual curvatures and experimental values with different λ. Fig. 22. Indenting force vs. impacting velocity with the same (a) M xpre ind and (b) Mypre ind under different prestress ratios.
20
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List of Tables
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Material parameters of Al alloy 2024-T351. Dimensions of indentation distributions. Matched indenting forces and impacting velocities with the same dimple diameter. Indenting force vs. indenting velocity with the same Mipre ind averaging in A s with l=1.0 mm. Predicted average indenting velocities with different sizes of averaging region. Comparison of shot velocities respectively predicted with shot dimple size and bending moment.
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Tab. 1. Tab. 2. Tab. 3. Tab. 4. Tab. 5. Tab. 6.
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Research Highlights
Highlights • Prestressed regular model are used to study stress peen forming. • The forming parameters are related to peen forming forces and resulting shape.
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• The simulation results are verified by indenting and peening tests.
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• The relations between one-shot and regular four-shot treating are established.
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S0924-0136(15)00051-5 http://dx.doi.org/doi:10.1016/j.jmatprotec.2015.02.008 PROTEC 14281
To appear in:
Journal of Materials Processing Technology
Received date: Revised date: Accepted date:
22-6-2014 29-11-2014 7-2-2015
Please cite this article as: X.D. Xiao, Y.J. Wang, W. Zhang, J.B. Wang, S.M. Wei, Numerical research on stress peen forming with prestressedregular model, Journal of Materials Processing Tech. (2015), http://dx.doi.org/10.1016/j.jmatprotec.2015.02.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Numerical research on stress peen forming with prestressed regular model X.D. Xiaoa , Y.J. Wang a,∗, W. Zhangb , J.B. Wanga , S.M. Weia School of Mechanical Engineering, Northwestern Polytechnical University, 127 Youyixi Road, Xi’an Shaanxi 710072, China b Xi’an Aircraft International Corporation, Xi’an Shaanxi 710089, China
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a
Abstract
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Stress peen forming is widely used in aerospace industry to form the thin components with complex geometry. In the stress peen forming, the prestressing condition is a critical factor that need to be precisely controlled. In this study, the effects of the prestress on the peening deformations are investigated by using static and dynamic simulations. A regularly parked four-shot model is used to study the interactions of adjacent shot treatments. To verify the finite element model with precisely controlled parameters, regularly indenting tests are adopted. The analytical results reveal that the tensile prestress enhances the shot dimple and the plastic region sizes. The contributions of the prestress to the induced bending moments and stretching forces are indicated with two scale factors. Comparison of the singleshot model and regular parked four-shot model shows the peening effects can be equivalent within certain averaging regions. The static and dynamic simulations are compared on dimple sizes and forming effects. Finally, the indenting and peening tests are carried out to verify the analytical results.
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Keywords: Stress peen forming, Indentation, Finite element simulation, Residual stress, Experiment
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1. Introduction
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Shot peen forming is a dieless process for forming metallic components. It is especially suitable for forming large panel contours where the bend radii are within the metal’s elastic range. In the forming process, numerous shots impact the surface of the component. Each impact induces a elastically-plastically deformed dimple which results in a compressive stress field near the treated surface. The compressive stress cases the material to develop a convex shape to the treated surface. The conventional peen forming process induces isotropic compressive stresses resulting in identical curvatures in all directions. To form the component such as a wing panel that has a larger curvature in chordwise direction, stress peen fixtures are usually used. Li (1981) applied the stress peening forming process on forming integrally stiffened wing panels. Meyer et al. (1987) formed stringer-reinforced workpieces with the stress peen forming. The determination of the stress forming parameters is mainly dependent on trial and error. Many experimental works on the stress peen forming have been carried out. Barrett and Todd (1984) measured the residual stresses by x-ray diffraction while peen formed plate remained in curved condition. Xie et al. (2012) measured the residual stress profiles along different directions in titanium matrix composites treated by stress peening with various prestresses. Miao et al. (2010) investigated the relations between the prebending moment and the resulting arc height. Al-Hassani (1981) expressed the peening induced stress profile with a cosine function. Li et al. (1991) developed a simplified analytical model to calculate the induced stress for 100% peening coverage. Xiao et al. (2014) developed a cavity model to calculate the induced stress filed. However, it is difficult to involve the prestress in the analytical models mentioned above. Finite element (FE) method can be used to investigate the stress peen forming. For example, Miao et al. (2011) utilized the FE simulation to obtain the induced stresses with various prebending moments. Gari´epy et al. (2013b) experimentally and numerically investigated the influence of initial stresses in the plate on the resulting curvatures. author. Tel.: +86 029 88493717 Email addresses: [email protected] ( X.D. Xiao), [email protected] (Y.J. Wang )
∗ Corresponding
Preprint submitted to JMPT
February 12, 2015
Page 1 of 22
2. Stress peen forming and regularly parked four shots model
an
2.1. Stress peen forming process
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In the FE simulation, the condition of unsaturated peening should be taken into account. Baughman (1986) advised that the shot peening coverage should be selected at a low value to leave room to increase contour with increased coverage. At a low peening coverage, a shot may solely acting on a small region of the target while the stress field is influenced by adjacent shot impacts. Meguid et al. (1999) studied the influence of the separation distance between the co-indenting shots upon the residual stress field. To simplify the analytical model, the random distribution of the shot impacting locations are usually idealized as a regularly parked shots model. Majzoobi et al. (2005) investigated the effect of impact multiplicity on the residual stress profile with a regularly parked multiple shots model. Kim et al. (2010) used a symmetry-cell FE model to obtain the area-averaged peening residual stress. Kirk and Hollyoak (2005) investigated the influence of coverage on the surface residual stress with regularly indenting tests. In the present work, static and dynamic simulations as well as experiments are performed to research the stress peen forming . First, the involved stresses and forming forces of stress peen forming are discussed. Then, a regularly parked four shots model is used to study the influence of peening coverage on the stress field. The acting effects of four shots are compared with of one shot to study the interactions of adjacent impacts. To precisely control the input parameters for verifying the analytical results, a prestressing indentation testing method is proposed. Finally, the analytical results will be compared with both indentation tests and shot peen forming tests.
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In conventional peen forming, the workpieces are usually kept in initial shape or is hanged freely during the forming process. In the stress peen forming, the workpiece is elastically prebended along a certain direction before and during the peening process, as shown in Fig. 1. The workpiece will obtain a larger resulting curvature in the prebending direction.
Figure 1: Stress peen forming process and related stresses and forces.
Since the peening deformations of square plates are influenced by secondary factors such as the rolling direction (Kulkarni et al., 1981), which leads to a basic instability of curvature, a narrow plate is taken into research in this work. When a narrow flat-plate is bent in the longitudinal direction to a curvature ρ xpre , a bending moment should be applied at both sides. Considering the narrow plate is freely restrained in the transverse direction, the applied bending moment per unit can be calculated as: M xpre = EIρ xpre (1) where I = h3 /12 is the moment of inertia per unit width, h is the thickness of plate. Then the transverse bending moment per unit Mypre = 0, and curvature ρypre = νρ xpre . E and ν are the elastic modulus and Poisson’s ratio of the material, respectively.
2
Page 2 of 22
The bending deformation induces a prestress field in the plate. The transverse and normal prestresses are approximated to zero. The longitudinal prestress at depth z is expressed as: σ xpre (z) = M xpre (h/2 − z) /I = Eρ xpre (h/2 − z)
(2)
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In order to indicate the magnitude of the prestress, a parameter λ is introduced here. The λ is defined as the ratio of the prestress on the peening surface to the yield stress σ s of the material as Eq. (3). The pre-deformation is usually controlled in the elastic range. The values of λ vary from 0 to 1 which induces tensile stress under the peening surface of the plate. λ = σ xpre (0)/σ s (3)
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In the stress peening process, the plate is maintained in the prebending configuration by external fixtures. The peening treatment changes the prestress field near the treated surface. The induced stresses under prebending σipre ind are not self-equilibrated, where i is used to indicated either x or y axes. After releasing the external fixing forces, the plate will be bent and stretched by σipre ind to a equilibrium state from the prebending configuration. With the assumption that the deformation during this process is in the elastic range, the residual stresses σres i can be calculated as Al-Hassani (1981): pre ind + σis + σipos b (4) σres i = σi
0
h
M
Z
an
The actions of the stress σipre ind can be equivalent to a bending moment per unit Mipre ind and a stretching force per unit Fipre ind as: Z h pre ind = (5) Mi σipre ind (h/2 − z) dz Fipre ind =
0
σipre ind dz
(6)
ed
The Mipre ind is a resultant of Miind induced by peening treatment and partial prebending moment (−αi Mipre ) as Eq. (7) (Gari´epy et al., 2014). The parameter α indicates the ratio of residual prebending moment to the initial prebending moment. (7) Mipre ind = Miind − αi Mipre
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The peening stress induces a stretching force Fiind while residual prestress also produces a additional stretching force. A dummy initial stretching force is defined as the force induced by the prestress within the half plate. A parameter βi is used to indicate the ratio of the stretching force corresponding to the residual prestress to the dummy initial stretching force. So the Fipre ind can be expressed as a resultant as: Fipre ind
=
Fiind
+ βi
Z 0
h/2
σipre (z) dz
(8)
In Eq. (4), the stretching stress σis = Fipre ind /h. The post-bending (from prebending configuration) stress σipos b (z) = (h/2 − z) /I. So the total bending (from the flat configuration) stress σbi = σipre + σipos b . Neglecting the influence of the stretching force on the resultant curvature, the post-bending curvatures can be calculated from the Kirchhoff-Love theory of plates as: pre ind pre ind M − νM x y ρ xpos = (9) EI pre ind My − νM xpre ind pos ρy = (10) EI Then, the final residual forming curvature can be calculated as: Mipre ind
pre ρres + ρipos i = ρi
(11)
3
Page 3 of 22
2.2. Regularly parked four shots model
pt
ed
M
an
us
cr
ip t
A regularly parked four-shot model is used to study the interactions of adjacent shot impacts and to determine the discussed parameters in stress peen forming, such as αi and βi . As shown in Fig. 2, assuming four impacts located at the four corners of a square area Ac with side length of d, the stress fields that relatively induced by the four impacts interact each other in the Ac . The impacting sequence of the four shots is ignored here. Single shot-impact produces a dimple with projected radius of a and a residual stress field. The peening coverage in Ac can be calculated as: πa2 /d2 for d ≥ 2a p 2 d 2 2 − d 2 /4 /d 2 for d ≥ 2a a πa − 4a arccos + 2d (12) C= 2a √ 1 for d ≤ 2a
(b)
Ac ce
(a)
Figure 2: (a) Single shot impacting target. (b)Four shots parked regularly and residual stress fields overlapping in the square area.
Averaging the peening stresses within A s and Ac , the integral forces on the regions can be obtained. The relationships between the peening forces and the coverage can be established. Then the relations would be used to calculate the resultant forming shape. The average stresses also can be used to further analyses as Gari´epy et al. (2012). The resultant deformation of a plate is only related to the residual stresses in the material. For the convenience of precisely controlling the input parameters, the stresses also can be introduced by other equivalent methods such as indentation (Kirk and Hollyoak, 2005). The indentation method is used in this paper to impose precise coverage percentages. The indentation are also compared with shot peen forming. 3. Finite element simulation models Four FE models are proposed: static simulation model (SSM) of prestressed single shot indenting, dynamic simulation model (DSM) of prestressed single shot impacting, SSM of regularly parked four-shot indenting, and DSM of regularly parked four-shot impacting, as shown in Fig. 3.
4
Page 4 of 22
ip t cr
(b)
(c)
us
(a)
Figure 3: One-quarter of discretized geometries of target for (a) single shot indenting and impacting simulations, (b) four-shot indenting simulation, and (c) four-shot impacting simulation.
M
an
3.1. Material properties The target material under study is aluminum alloy 2024-T351. The Johnson Cook (JC) model is used to represent the response of the material. As the shot peen forming is a cold-working process , the thermal properties of the material is neglected here, and the JC model can be reduced as: σ = A + Bεn 1 + C ln ε˙ ∗
(13)
ed
where σ is the effective stress, ε is the effective plastic strain, ε˙ ∗ is the normalized effective plastic strain rate (typically normalized to a strain of 1.0 s−1 ), n is the work hardening exponent and A, B, C, and m are constants. The constants and other parameters of the material (Lesuer, 2000) are listed in Tab. 1. Table 1: Material parameters of Al alloy 2024-T351. ν
σs
73 GPa
0.33
343 MPa
A
pt
E
C
n
Elongation
ρ
684 MPa
0.0083 MPa
0.73
17%
2770 kg/m3
Ac ce
369 MPa
B
3.2. SSM of prestressed single shot indenting The simulation procedure is carried out by two static/general steps: applying prebending to obtain the prestress distribution and the pre-deformation of the target, then indenting the target with one shot by loading and unloading a force. The FE model shown in Fig. 3(a) is used for the simulation. A shot (Radius = 1.5875 mm) is normally loaded at the symmetrical position of the one-quarter model of target. The dimensions of h = 5 mm and L = 6 mm are selected. A refined mesh is made at the contact region. Considering the relatively high yield and hardness values of the indenter compared to the target material, the shot is assumed to be rigid. In the first step, the symmetry boundary conditions are imposed on the yOz and zOx planes. The nodes on the line of x = 0 and z = h are fixed in the z direction. Linearly distributed pressure with an amplitude of λσ s (2z − h) /h along z direction is applied on the x = L surface of the target. The prestress is: σ xpre (z) = −λσ s (2z − h) /h
(14)
The values of λ under study are 0, 0.2, 0.4, 0.6, 0.8 and 1. In the second step, the normal displacements of the x = 0, x = L and y = 0 surfaces of the target are fixed at current position with symmetric boundary conditions. The bottom surface of the target is fixed at current position in 5
Page 5 of 22
z direction. Then, the target is subjected to a shot indenting with a gradually increased load F at normal incidence. After reaching maximum values, the load is gradually unloaded again. The maximum values of the load under study are 1839/4 N, 1226/4 N and 613/4 N. 3.3. DSM of prestressed single shot impacting
ip t
This simulation has the same procedure, boundaries, FE model and mesh dimensions as the SSM. Two static/general - dynamic/implicit steps are carried out. The difference is that, in the second step, the rigid shot impacts the target at normal incidence with an initial velocity V. The values of V under study are 30, 40 and 50 m/s. One quarter of the lumped mass of the shot 0.01595 g is given to the shot model. The lumped mass of the shot was determined by measuring the real APB1/8 shot.
cr
3.4. SSM of regularly parked four shots indenting
M
3.5. DSM of regularly parked four shots impacting
an
us
The study is further extended to investigate the overlapping effects of the adjacent indenting stress fields. A FE model, as shown in Fig. 3(b), is used to simulate the process of four shots (Radius = 1.5875 mm) successively indenting the prestressed target. The four shots are respectively located at the four corners of Ac with side length of d. The values of d under study are 1, 2, 3, 4, and 6 mm. The target dimensions are 8(L) × 8(L) × 5(h) mm. A refined mesh is made at the contact region. The simulation procedure and the boundary constraints are the same as the single-shot model. The values of λ under study are 0, 0.375, and 0.75. In the second step, the four shots successively indent the target in order of 1-2-3-4 marked in Fig. 3(b). The indenting force F under study is 1226 N. In the FE model, shot marked 1 is loaded 1839/4 N, shot marked 2 and 4 loaded 1839/2 N, and shot marked 3 loaded 1839 N.
ed
This simulation has the same procedure, boundaries, FE model and mesh dimensions as the above static fourshot model. Two static/general - dynamic/implicit steps are used to simulate the process of four shots successively impacting the prestressed target with a initial velocity V, as shown in Fig. 3(c). The V under study is 40 m/s. In the FE model, shot marked 1 is given lumped mass of 0.01595 g, shot marked 2 and 4 given 2 × 0.01595 g, and shot marked 3 given 4 × 0.01595 g. The impacting order is also 1-2-3-4.
pt
4. Experimental processes 4.1. Prebending device and specimen
Ac ce
A metallic holder shown in Fig. 4(a) was used to bend and maintain the specimen in the experiments. The predeformation of the specimen is controlled by the prebending mold. Three prebending molds, as shown in Fig. 4(b), are used with radii of curvature 626 mm, 1252 mm, and infinite, respectively. Rectangular strips with dimensions of 100(L s ) × 30(Bs ) × 5(h s ) mm were cut from an Al 2024-T351 plate. The longitudinal direction of the strip is the same as the rolling direction of the plate. The pre-deformations of the strips bent on the holder were simulated with FE method. In order to shorten the paper the models are not shown here. The prestress ratios were determined that λ = 0.75, 0.375 and 0 corresponding to R pre = 626, 1252 and ∞. The transverse and normal stresses under prebending are approximated to zero. 4.2. Regularly indenting tests
In order to obtain precise coverage and indenting locations, a set of static indenting tests was performed by using a developed hardness tester. A manual cross slide was fixed on the work table of the hardness tester to precisely locate the indenting positions, as shown in Fig. 5(a). The indenter installed a carbide ball with diameter of 3.175 mm. Three strips were respectively bent to R pre = ∞, 1252 and 626 mm. Each strip was indented with loads of 613 N and 1839 N at different locations. The indentation size is determined by measuring two diagonals of the indentation by an optical microscope. Under the load of 1226 N, regularly distributed indentations with indenting gaps of 2, 3 and 4 mm in association with prebending radii of ∞, 1252 and 626 mm were produced on nine test strips, respectively. The indenting locations 6
Page 6 of 22
ip t cr
(a)
(b)
M
an
us
Figure 4: (a) Prebending device and (b) prebending molds with radii of curvature R pre = ∞, 1252 and 626 mm.
(b)
ed
(a)
pt
Figure 5: (a) Regularly indenting device and (b) indentation locations and sequence.
Ac ce
and sequence are plotted in Fig. 5(b). The related parameters for different indenting gaps are listed in Tab. 2. The treating sequence influences the resultant forming shape (Gari´epy et al., 2013a). However, the size of the testing strip is relatively small, so the effects of the treating sequence can be neglected. Only one type of treating sequence was used. The resultant curvatures in the transverse and longitudinal directions of the test strips were calculated from the measured arc heights. Table 2: Dimensions of indentation distributions. d (mm) Bc (mm) Lc (mm) Total indentations
2 3 4
24 24 24
80 78 80
13 × 41 9 × 27 7 × 21
4.3. Stress peen forming tests Stress peen forming tests were carried out on the Wheelabrator MP20000 Aircraft Wing Peening System. Hardened steel peening ball APB1/8 with nominal diameter of 3.175 mm was used. One peening pass was along the longitudinal centerline of the strip. The average shot velocity can be measured with a high-speed photographic setup attached to the MP20000. The shot velocity of 40 m/s is selected. The distance between the peening nozzle and the plate is 500 mm. The rate of shot 7
Page 7 of 22
flow is 10 kg/min. Prestress ratio of 0, 0.375 and 0.75 in association with nozzle moving speeds of 4, 6, 8 m/min were imposed on nine test strips, respectively. The shot dimple diameters and the resultant curvatures were measured. The peening coverage was measured by the optical inspection (Miao et al., 2010). 5. Results and discussion
cr
ip t
5.1. Effects of prestress on the indentation and shot dimple sizes By the simulations, the indenting diameters under different F and λ as well as the impacting diameters with various V and λ are obtained by detecting the maximum indenting depth in the loading process. The dimple diameter D is calculated from the depth δ as: 1/2 D = 2δR − δ2 (15)
(a)
pt
ed
M
an
us
The simulated dimple diameters are compared with the experimental values as shown in Fig. 6. The simulated values are consistent with the experimental values, which verifies the simulation model. The dynamically simulated results have the same tendency as the static simulation. The dimple size increases with the increase of the prestress ratio corresponding to increasing the tensile stress beneath the treated surface. It may be explained that the shot compresses the target perpendicularly to the surface, the tensile stress in the target increases the shear stress beneath the shot resulting in larger deformations (Simes et al., 1984).
(b)
Ac ce
Figure 6: Dimple diameters measured in the experiments and (a) static simulations as well as (b) dynamic simulations..
The indentation diameter D st is fitted as a function of the indenting force F and the prestress ratio λ as: D st = 0.4786 + 5.871 × 10−4 F + 0.05798λ + 3.609 × 10−6 Fλ − 8.921 × 10−8 F 2 + 8.625 × 10−3 λ2
(16)
In the same way, the shot dimple size Ddy is fitted as a function of the shot velocity V and the prestress ratio λ as: Ddy = 0.5173 + 0.02256V + 0.04117λ + 3.487 × 10−5 Vλ − 1.108 × 10−4 V 2 + 6.995 × 10−3 λ2
(17)
The coefficients of determination (R2 ) of Eq. (16) and Eq. (17) are 0.99997 and 0.99998, respectively. According to Eq. (16) and Eq. (17), a dimple with a certain size may be obtained either by static indenting or by dynamic impacting with appropriate treating parameters. Tab. 3 shows the matched indenting forces and shot velocities to obtain the same dimple diameter. For a certain indenting force, the matched shot velocities with different λ vary in a small range. The variances (var.) of the V are relatively small no matter what indenting forces. 5.2. Effects of prestress on the plastic region size The sizes of the plastic region of one shot treating were detected in the simulations. As shown in Fig. 7, the plastic deformation is examined with the equivalent plastic strain at integration points greater than 0.002. The plastic 8
Page 8 of 22
Table 3: Matched indenting forces and impacting velocities with the same dimple diameter. F=613 N
F=1226 N
F=1839 N
λ
D st = D = Ddy mm
V m/s
D st = D = Ddy mm
V m/s
D st = D = Ddy mm
V m/s
0 0.375 0.75
0.80 0.83 0.85
13.67 14.04 14.45
1.06 1.09 1.12
28.13 28.63 29.17
1.26 1.28 1.31
41.03 41.70 42.42
14.05 0.1524
28.64 0.2698
41.72 0.4826
ip t
Aver. Var.
pt
ed
M
an
us
cr
region size is denoted as r x , ry and rz , which respectively indicate the half plastic region size in the x and y axises and the plastic region depth. Fig. 8 shows the values of r x , ry and rz under different treating parameters. It can be seen that greater indenting load or impacting velocity produces larger plastic deformation. Especially, the λ has an obvious influence on the value of ry . The r x and rz increase slightly with the increase of the λ. With a relative small λ, the plastic region depth is greater than its half widths. But the ry will exceed the rz with increasing λ. The difference between r x and ry can be explained that the tensile prestress in the x direction makes the material near the x axis yield later than the material near the y axis.
Ac ce
Figure 7: Plastic deformation examined with the equivalent plastic strain at integration points greater than 0.002.
5.3. Effects of prestress on the indenting and impacting stresses The prestress influences the plastic deformation of the material as well as the induced stress σipre ind . Fig. 9 shows and σypre ind along the center line beneath the shot in the one-shot model with λ = 0.6. Fig. 9(a) depicts the stress profiles under three indenting forces and Fig. 9(b) under three impacting velocities. It can be seen from the figures that under prebending: σ xpre ind
(1) The compressive σipre ind at the surface increases with increasing either the indenting force or the shot velocity. (2) The differences of the σipre ind of different treating parameters decrease with the depth from the surface. When the compressive σipre ind is up to the maximum values beneath the surface, the differences become negligible. (3) The depth of the induced compressive σipre ind increases with increasing either the indenting force or the shot velocity. (4) The induced compressive σ xpre ind and σypre ind have the same profile near the treated surface, while significant differences are in the bottom half of the target. Fig. 10 compares the induced stresses with and without prestress. The prestress profiles are also plotted in the figures. Comparison of the stress profiles leads to the following observations: 9
Page 9 of 22
ip t cr (b)
us
(a)
(a) pre ind
ed
M
an
Figure 8: Plastic region sizes of (a) static simulations with various loads and prestress ratios, and (b) dynamic simulations with various shot velocities and prestress rations.
(b)
pt
Figure 9: σi (a) in static simulation with λ = 0.6 and various indenting forces, (b) and in dynamic simulation with λ = 0.6 and various impacting velocities.
Ac ce
(1) Along the center line, the profiles of the compressive σipre ind have not obvious changes with the increase of the λ. The maximum values of the compressive σipre ind slightly decrease with increasing the λ. (2) In the bottom half of the target, the σipre ind is approximately equal to the σipre . (3) In the middle region, the σipre ind is tensile. The tensile σ xpre ind is obviously greater than the tensile σypre ind and σipre . The stress field of one shot treating various from the core region. The average stresses of each layer with projected region A s are calculated from the stresses at the integration point and the current volume of each element. Fig. 11 shows the static and dynamic average induced stress profiles under prestress ratio of 0.4 within A s with various values of l . From the figures, it can be seen that: (1) The average σipre ind within the bottom half of the target does not have changes with the variation of the size of As . (2) Within the top half of the target, the compressive σipre ind decreases with increasing the size of A s . (3) When the A s is relatively large, the compressive σ xpre ind is lower than the σypre ind . Within a smaller A s , the compressive σ xpre ind is approximately equal to σypre ind . The plastic deformation can reduce the difference between 10
Page 10 of 22
ip t
(b)
M
an
us
cr
(a)
(c)
(d)
ed
Figure 10: Prestress profiles and comparison of induced stresses without prebending and with prebending when (a) F = 1226 N and λ = 0.2, (b) F = 1226 N and λ = 0.8, (c) V = 40 m/s and λ = 0.2 (d) V = 40 m/s and λ = 0.8.
Ac ce
pt
σ xpre and σypre even eliminate the difference with large plastic strains. So the prestresses have little influence on the induced stresses within the central plastic region.
(a)
(b) pre ind
Figure 11: (a) Statically and (b) dynamically simulated average σi
with λ = 0.4 within A s with various values of l .
5.4. Overlapping of adjacent stress and strain fields The stress field induced by single shot have been discussed. For multiple shots treating, there are interactions of the stress fields respectively induced by adjacent shots. Fig. 12 shows the Mises stress fields induced by single shot 11
Page 11 of 22
us
cr
ip t
and four shots with V = 40 m/s, d = 2 mm and λ= 0, 0.375. Comparison of the single-shot and four-shot impacts shows that the overlapping of the stress fields mainly takes place in the middle region Ac . Comparison of the stress fields induced with and without prebending shows there are not significant difference of stress field near the impacting position.
(b)
(c)
pt
ed
M
an
(a)
(d)
Ac ce
Figure 12: Mises stress fields induced by (a, c) one shot and (b, d) four shots with V = 40 m/s and d = 2 mm.
Averaging the σipre ind within A s and Ac with l = 1 and d = 2, the average stress profiles are plotted in Fig. 13. It can be seen that: (1) The stress profiles of static and dynamic simulations have the same tendency. (2) The σipre ind averaged in A s with l = 1 and in Ac with d = 2 have the similar profiles. The depths of the compressive stresses are equal. (3) In the surface layer, the σipre ind of four shots are greater than those of single shot. The stresses are overlapped at the surface. (4) The stress profiles of four shots are wholly higher than those of single shot, which cased by the stenching forces of four treats on the whole FE model greater than those of single treat. If the treating locations are near enough, the plastic regions will be in connection, as shown in Fig. 14. When the treating locations is far enough, the plastic regions are in separation. Between the two adjacent plastic regions, the deformation of the material is in the elastic range. 5.5. Bending and stretching forces from induced stresses The equivalent bending moment Mipre ind and stretching force Fipre ind can be used to represent the acting effects of the average σipre ind with Eqs. (5) and (6). Fig. 15 shows both Mipre ind and Fipre ind with λ = 0. Four curves are 12
Page 12 of 22
pre ind
in A s with l = 1 and in Ac with d = 2 under (a) F = 1226 N and (b) V = 40 m/s.
M
an
us
Figure 13: Comparison of the average σi
ip t
(b)
cr
(a)
(a)
(b)
(c)
ed
Figure 14: Equivalent plastic strain fields with (a) d = 1 mm, (b) d = 2 mm and (c) d = 3 mm. The shot velocity is 40 m/s and the prestress ratio λ = 0.375.
The curves corresponding to the dynamic and static simulations have the same tendency. The equivalent forces decrease sharply in the beginning then approach to zero with increasing d. The FACL is close to the FACD. The deviations are cased by the interactions of the stress fields. The FASD is lower than the FACD while the FFASD is close to the FACD when the d is greater than a critical value. The critical value is approximately equal to the plastic region width (2 ×r x (ry )) shown in Fig. 8.
Ac ce
(1) (2) (3) (4)
pt
compared in each figure: equivalent force in Ac versus d (FACD), equivalent force in A s versus l where l = d/2 (FASL), equivalent force in A s versus l where l = d (FASD), and four times equivalent force in A s versus l where l = d (FFASD). From the figures, it can be seen that:
So that In the condition of without prebending, the peening effects of single shot can be considered as that of four shots. The FACD can be represented by FFASD for d > 2 × r x otherwise by FACL for d ≤ 2 × r x in the range of permitted errors . In the stress peen forming, the σipre ind involves a part of σipre . The FASD and FFASD can not compared with the FACD. Fig. 16 shows the comparison of the FASL and the FACD of stress peening. The FACD with λ = 0 shown in FIg. 15 is also plotted in Fig. 16. It can be seen that: (1) The FASL is close to the FACD no matter what λ. (2) The influence of the magnitude of λ on the Mipre ind is greater than the influence on the Fipre ind . (3) The FACD with prestress are very close to the FACD without prestress in the y direction. The effects of prestress is mainly on the FACD in the x direction. (4) The values of Fipre ind and Mypre ind are greater than zero while the values of M xpre ind may be positive or negative. When M xpre ind = 0, the final forming curvature could be the same as the prebending curvature in the x direction. 13
Page 13 of 22
ip t
(b)
an
us
cr
(a)
M
(c)
(d)
ed
Figure 15: Comparison of (a) equivalent bending moments vs. separation distance d and (b) equivalent stretching forces vs. d in dynamic simulations and (c, d) in static simulations.
pt
In Eqs. (7) and (8), the αi and βi indicate the effects of the prestress on the induced stresses. Smaller αi and greater βi indicate the greater contributions of the prestress on the Mipre ind and Fipre ind . The Miind and Fiind are hard to be determined under prebending. If the Mipre ind and Mipre ind with λ = 0 are used as universal Miind and Fiind , the αi and βi will be determined. As shown in Fig. 16, in the y direction the FACD with prestress are very close to the FACD without prestress, so that αy = 0 and βy = 0. Fig. 17 shows the α x and β x corresponding to various λ and d. It can be seen from the figures that:
Ac ce
(1) In the dynamic and static simulations, the curves of α x vs. d and β x vs. d have the same tendency. (2) The variation of the λ has little influence on the α x and β x . The α x and β x are mainly dependent on the d and the treating intensity, such as F and V. (3) When d = 1 corresponding to about full coverage, the α x and β x are approximated to 0.5. When the d is relatively great corresponding to a very low intensity of treating, the α x tends to 1 and β x tends to 0. 5.6. Resultant curvature and experimental verification The post-bending curvatures ρipos are calculated with Eqs. (9) and (10). Fig. 18 shows the ρipos with λ = 0.375, 0.75 under impacting and indenting. It can be seen that 1. The ρypos are greater than zero while the ρ xpos may be a negative values. After removing the fixing loads on the plate, springback of the plate will take place in the x direction if ρ xpos < 0. If ρ xpos = 0, the ρ xpre would be leaved. 2. The λ has a significant influence on the ρ xpos while a slight influence on the ρypos . The measured curvature of experimental specimen is the residual curvature ρres i . Fig. 19 shows the indentation strips with indenting gaps of 2, 3 and 4 mm. Fig. 20 shows the shot peening strips with nozzle moving speed of 4, 6 res and 8 m/s. The experimental ρres i are obtained by measuring the final arc heights. The analytical ρi can be calculated 14
Page 14 of 22
ip t
(b)
an
us
cr
(a)
M
(c)
(d)
Ac ce
pt
ed
Figure 16: Comparison of (a, c) equivalent bending moment vs. d and (b, d) equivalent stretching force vs. d in dynamic simulation with different λ.
(a)
(b)
Figure 17: Ratios of (a) prebending moment and (b) stretching force in static and dynamic simulations.
res with Eq. (11). Fig. 21 shows the comparison of the experimental ρres i and the analytical ρi . The analytical results are in good agreement with the experimental values. Fig. 21(a) depicts the relationship between ρres i and peening coverage. The analytical coverage is calculated with Eq. (12). Fig. 21(b) depicts the relationship between ρres and the d of indentation. Comparison of the ρres with i i res prebending and the ρres without prebending shows that the increase of the λ leads to the increase of ρ while the x i . The same tendency were also obtained by Miao et al. (2010) and Li (1981). decrease of ρres y
15
Page 15 of 22
ip t
(b)
cr
(a)
M
an
us
Figure 18: Post-curvatures with different λ in (a) dynamic simulations and (b) static simulations.
Ac ce
pt
ed
Figure 19: Indenting specimens with different indenting gaps and radii of prebending curvature.
Figure 20: Stress peen formed specimens with different nozzle moving speed and radii of prebending curvature.
5.7. Indenting force vs. impacting velocity The connections between the dynamic and static analyses can be established with the equivalent forming forces. Under a certain conditions of the target and shot, the same Mipre ind can be obtained either by shot peening with an impacting velocity V or by indentation with an indenting force F. The relationships between the Mipre ind and the F( or V) as well as λ are obtained with the response surface quadratic model. The values of the R-Squared are all close to 1. Fig. 22 shows the both relationships of F vs. Mipre ind and V vs. Mipre ind with l = 1. Starting from a certain F, a corresponding V can be found with the same Mipre ind , and vice versa. Tab. 4 lists the matched indenting forces and impacting velocities with the same Mipre ind . The variances of the predicted values of V with different λ are relatively small. The variation of the λ has little influence on the predicted values. The predicted average V (Tab. 4) within A s with different sizes are listed in Tab. 5. The variances of the V are also relatively small except the values corresponding to F = 613 N. When F = 613 N, the predicted V is about 16.5 m/s 16
Page 16 of 22
ip t cr us (b)
an
(a)
Ac ce
pt
ed
M
Figure 21: Comparisons of (a) dynamically analytical residual curvatures and experimental values as well as (b) statically analytical residual curvatures and experimental values with different λ.
(a)
(b) pre ind
Figure 22: Indenting force vs. impacting velocity with the same (a) M x
pre ind
and (b) My
under different prestress ratios.
which is far away from the range of V(30, 50) m/s studied in the simulation. The deviation between the fitting and real values increases when the input value is far away from the fitting region. In Tab. 6, the predicted values of V in the Tab. 5 and the values of V in the Tab. 3 are compared. There is a direct correlation between the dimple size and the forming force. With the same shot and target, if the sizes of the dimples produced by different methods are the same, the produced forming forces are approximately equal under the conditions presented in this paper. So the indenting experiments can be used to precisely introducing the induced stresses to study the characteristics of the shot peening by monitoring the size of the macroscopic dimple. In this work, the connections between the static and dynamic analysis ae established on the material Al alloy 2024-T351 whose strain rate sensitivity is relatively low. For the material with high sensitivity of strain rate , a further 17
Page 17 of 22
development on the connections may be needed. pre ind
Table 4: Indenting force vs. indenting velocity with the same Mi
0 0.425 0.85
F=1839 N N
V m/s
pre ind Mx
N
V m/s
pre ind My
N
V m/s
pre ind Mx
N
V m/s
pre ind My
N
V m/s
73.62 -407.96 -889.53
19.10 18.01 17.17
80.72 44.41 8.11
17.96 18.36 18.79
275.94 -150.43 -576.80
29.62 29.53 29.47
261.76 216.57 171.38
28.37 28.64 28.93
470.28 99.10 -272.07
41.11 42.14 42.89
18.09 0.933
18.37 0.1687
29.54 0.0054
N
V m/s
474.64 420.57 366.51
41.01 41.23 41.47
28.65 0.0779
42.05 0.7963
41.24 0.0542
cr
Aver. Var.
pre ind
My
ip t
λ
pre ind Mx
averaging in A s with l=1.0 mm.
F=1226 N
F=613N
F=613 N
Aver. Aver. Var.
) m/s
F=1226 N pre ind
V(My
) m/s
pre ind
V(M x
F=1839 N
pre ind
) m/s
V(My
14.28 18.09 17.18
14.19 18.37 15.36
28.42 29.54 28.37
16.52 16.25 3.5374
15.97
28.78 28.39 0.4982
) m/s
pre ind
) m/s
V(M x
pre ind
V(My
) m/s
27.50 28.65 27.84
40.36 42.05 40.89
39.82 41.24 40.65
28.00
41.10 40.83 0.5855
40.57
an
0.5 1 1.5
pre ind
V(M x
M
l mm
us
Table 5: Predicted average indenting velocities with different sizes of averaging region.
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Table 6: Comparison of shot velocities respectively predicted with shot dimple size and bending moment. F = 1226 N
F = 1839 N
14.05 16.25 14.48%
28.64 28.39 0.89%
41.72 40.83 2.14%
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6. Conclusion
F = 613 N
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Corresponding V (m/s) with the same dimple diameter Corresponding V (m/s) with the same bending moment Relative error
The main conclusions of this study are as follows: 1) The effects of prestress on the stress peen forming were studied with both static and dynamic simulations. The contributions of the prestress to the forming effects are indicated with two ratios αi and βi . Then the forming effects under different prestresses can be predicted directly from the results of conventional peen forming and the ratios. 2) A regularly parked four shots model was used to investigate the interactions of adjacent shot treatments. The results shows that the forming effects of four shot impacts within a square area are very close to the forming effects of single shot impact within the quarter area. So the analytical model of single shot impact can be used to predict the peening effects of multiple shot impacts. 3) The effects of static indenting are nearly equivalent to the effects of dynamic impacting with the same size of macroscopic dimple. The regularly indenting test can be used in the researches to introducing the peening stress with precisely controlled parameters. This work focused on the deformations of the narrow strip. The prebending was only applied in the longitudinal direction in analytical model. Considering the complexity of the practical prebending conditions with respect to large scale components, further works are required to study the influence of complex fields of prestress on the resulting curvatures. 18
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Acknowledgement The authors would like to acknowledge the financial supports by the National Natural Science Foundation of China under Grant No. 512754 20, and by the corresponding research project from AVIC Xian Aircraft Industry (Group) Company Ltd..
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References
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Al-Hassani, S., 1981. Mechanical aspects of residual stress development in shot peening, in: Proceedings of the 1st international conference on shot peening, pp. 583–602. Barrett, C., Todd, R., 1984. Investigation of the effects of elastic pre-stressing technique on magnitude of compressive residual stress induced by shot peening of thick aluminum plates, in: Proceedings of the 2nd international conference on shot peening, pp. 15–21. Baughman, D., 1986. An overview of peen forming technology. Pergamon Press, Advances in Surface Treatments. Technology–Applications– Effects. 3, 209–214. Gari´epy, A., Cyr, J., Levers, A., Perron, C., Bocher, P., Lvesque, M., 2012. Potential applications of peen forming finite element modelling. Advances in Engineering Software 52, 60–71. Gari´epy, A., Larose, S., Perron, C., Bocher, P., L´evesque, M., 2013a. On the effect of the peening trajectory in shot peen forming. Finite Elements in Analysis and Design 69, 48–61. Gari´epy, A., Larose, S., Perron, C., Bocher, P., Lvesque, M., 2013b. On the effect of the orientation of sheet rolling direction in shot peen forming. Journal of Materials Processing Technology 213, 926–938. Gari´epy, A., Miao, H., Lvesque, M., 2014. 3.12 - peen forming, in: Yilbas, S.H.F.B. (Ed.), Comprehensive Materials Processing. Elsevier, Oxford. Comprehensive Materials Processing, pp. 325–326. Kim, T., Lee, J.H., Lee, H., Cheong, S.k., 2010. An area-average approach to peening residual stress under multi-impacts using a three-dimensional symmetry-cell finite element model with plastic shots. Materials & Design 31, 50–59. Kirk, D., Hollyoak, R.C., 2005. Relationship between coverage and surface residual stress, in: Proceedings of the 9th international conference on shot peening, pp. 373–378. Kulkarni, K.M., Schey, J.A., Badger, D.V., 1981. Investigation of shot peening as a forming process for aircraft wing skins. Journal of Applied Metalworking 1, 34–44. Lesuer, D., 2000. Experimental investigation of material models for Ti-6Al-4V and 2024-T3. Technical Report. FAA Report DOT/FAA/AR-00/25. Li, J., Mei, Y., Duo, W., Renzhi, W., 1991. Mechanical approach to the residual stress field induced by shot peening. Materials Science and Engineering: A 147, 167–173. Li, K., 1981. Using stress peen-forming process for integrally stiffened wing panels, in: Proceedings of the 1st international conference on shot peening, Paris. pp. 555–563. Majzoobi, G.H., Azizi, R., Alavi Nia, A., 2005. A three-dimensional simulation of shot peening process using multiple shot impacts. Journal of Materials Processing Technology 164165, 1226–1234. Meguid, S., Shagal, G., Stranart, J., Daly, J., 1999. Three-dimensional dynamic finite element analysis of shot-peening induced residual stresses. Finite Elements in Analysis and Design 31, 179–191. Meyer, R., Reccius, H., Schulein, R., 1987. Shot peen-forming of nc-machined parts with integrated stringers using large balls.(retroactive coverage). Deutsche Gesellschaft fur Metallkunde e. V, , 327–334. Miao, H., Demers, D., Larose, S., Perron, C., Lvesque, M., 2010. Experimental study of shot peening and stress peen forming. Journal of Materials Processing Technology 210, 2089 – 2102. Miao, H., Larose, S., Perron, C., L´evesque, M., 2011. Numerical simulation of the stress peen forming process and experimental validation. Advances in Engineering Software 42, 963–975. Simes, T., Mellor, S., Hills, D., 1984. A note on the influence of residual stress on measured hardness. The Journal of Strain Analysis for Engineering Design 19, 135–137. Xiao, X.D., Wang, Y.J., Gao, G.Q., Wang, J.B., Wei, S.M., 2014. An analytical approach to relate shot peen forming parameters to resulting curvature with expanding cavity model, in: Proceedings of the 12th international conference on shot peening. Xie, L., Jiang, C., Lu, W., Chen, Y., Huang, J., 2012. Effect of stress peening on surface layer characteristics of (TiB + TiC)/Ti6Al4V composite. Materials & Design 33, 64 – 68.
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List of Figures
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Fig. 1. Stress peen forming process and related stresses and forces. Fig. 2. (a) Single shot impacting target. (b)Four shots parked regularly and residual stress fields overlapping in the square area. Fig. 3. One-quarter of discretized geometries of target for (a) single shot indenting and impacting simulations, (b) four-shot indenting simulation, and (c) four-shot impacting simulation. Fig. 4. (a) Prebending device and (b) prebending molds with radii of curvature R pre = ∞, 1252 and 626 mm. Fig. 5. (a) Regularly indenting device and (b) indentation locations and sequence. Fig. 6. Dimple diameters measured in the experiments and (a) static simulations as well as (b) dynamic simulations.. Fig. 7. Plastic deformation examined with the equivalent plastic strain at integration points greater than 0.002. Fig. 8. Plastic region sizes of (a) static simulations with various loads and prestress ratios, and (b) dynamic simulations with various shot velocities and prestress rations. Fig. 9. σipre ind (a) in static simulation with λ = 0.6 and various indenting forces, (b) and in dynamic simulation with λ = 0.6 and various impacting velocities. Fig. 10. Prestress profiles and comparison of induced stresses without prebending and with prebending when (a) F = 1226 N and λ = 0.2, (b) F = 1226 N and λ = 0.8, (c) V = 40 m/s and λ = 0.2 (d) V = 40 m/s and λ = 0.8. Fig. 11. (a) Statically and (b) dynamically simulated average σipre ind with λ = 0.4 within A s with various values of l . Fig. 12. Mises stress fields induced by (a, c) one shot and (b, d) four shots with V = 40 m/s and d = 2 mm. Fig. 13. Comparison of the average σipre ind in A s with l = 1 and in Ac with d = 2 under (a) F = 1226 N and (b) V = 40 m/s. Fig. 14. Equivalent plastic strain fields with (a) d = 1 mm, (b) d = 2 mm and (c) d = 3 mm. The shot velocity is 40 m/s and the prestress ratio λ = 0.375. Fig. 15. Comparison of (a) equivalent bending moments vs. separation distance d and (b) equivalent stretching forces vs. d in dynamic simulations and (c, d) in static simulations. Fig. 16. Comparison of (a, c) equivalent bending moment vs. d and (b, d) equivalent stretching force vs. d in dynamic simulation with different λ. Fig. 17. Ratios of (a) prebending moment and (b) stretching force in static and dynamic simulations. Fig. 18. Post-curvatures with different λ in (a) dynamic simulations and (b) static simulations. Fig. 19. Indenting specimens with different indenting gaps and radii of prebending curvature. Fig. 20. Stress peen formed specimens with different nozzle moving speed and radii of prebending curvature. Fig. 21. Comparisons of (a) dynamically analytical residual curvatures and experimental values as well as (b) statically analytical residual curvatures and experimental values with different λ. Fig. 22. Indenting force vs. impacting velocity with the same (a) M xpre ind and (b) Mypre ind under different prestress ratios.
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List of Tables
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Material parameters of Al alloy 2024-T351. Dimensions of indentation distributions. Matched indenting forces and impacting velocities with the same dimple diameter. Indenting force vs. indenting velocity with the same Mipre ind averaging in A s with l=1.0 mm. Predicted average indenting velocities with different sizes of averaging region. Comparison of shot velocities respectively predicted with shot dimple size and bending moment.
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Tab. 1. Tab. 2. Tab. 3. Tab. 4. Tab. 5. Tab. 6.
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Research Highlights
Highlights • Prestressed regular model are used to study stress peen forming. • The forming parameters are related to peen forming forces and resulting shape.
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• The simulation results are verified by indenting and peening tests.
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• The relations between one-shot and regular four-shot treating are established.
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