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Abstract

To simulate the deformations of the strip peen formed plate more realistic, and using low computational resources, a strategy combining analytical and finite element methods is proposed in this article. First, the internal stresses in the target induced by single shot impact are calculated with expanding cavity model. Second, the stress field of single shot impact is used to derive the stress field of multiple shot impacts by considering the overlaps of adjacent shot impacts. Third, the calculated stress field is introduced to the finite element model to obtain the resultant shape of the plate. The shot dimple distribution in reality is detected and fitted with normal distribution function. The random distribution of the positions of shot impacts is involved in the simulation to make the simulation more realistic. In the finite element model, the plate is modeled with shell element to reduce the demand of the computational resources. The simulated shapes of the plate under different peen forming parameters are compared with the scanned three-dimensional experimental shapes with the same forming parameters. The comparison shows that the simulated shapes are in good agreement with the experiments.

Keywords

Peen forming finite element simulation residual stress analytical model peening coverage

Introduction

Shot peen forming is a cold working process to shape curved metal parts. It is primarily used in the aerospace industry for shaping large sections such as wing panels.^1,2 In the forming process, numerous shot impacts on the component surface with high speed cause the surface to deform plastically. The plastic deformation results in a residual compressive stress distribution in the surface layer and a curvature deformation of the component.^3,4

To control the process, many control parameters related to both shot peening machine and target are monitored. The key parameters that determine the resulting shape are shot type, pneumatic pressure for driving the shots, relative moving speed of the nozzle, shot flow from the nozzle, target material properties, peened region on the target and so on. These parameters are usually indicated with shot velocity, peening coverage, Almen intensity and so on. The relationships between the forming parameters and the resulting shape are usually obtained by a large number of costly experiments.

Several theoretical works are developed to understand the process. Al-Hassani⁵ expressed the induced stress distribution with a cosine function. Tan et al.⁶ empirically modeled the relationships between residual stresses and peening parameters with sinusoidal decay function. Li et al.⁷ developed a simplified analytical model to calculate the residual stress field due to shot peening with assumptions that, after 100% coverage, the plastic deformation is steady and continuous and the work piece is still flat. Miao et al.⁸ followed Li’s models to calculate the induced stress in a semi-infinite body, and used Guagliano’s⁹ model to calculate the residual stress in the Almen strip. Zhang et al.¹⁰ developed a cross-sectional linear indentation coverage method by considering regular indentation and coverage.

Finite element (FE) simulation is an important method to predict the peening stresses under different peening parameters. Bagherifard et al.¹¹ made a review of different multiple impacting patterns. However, those simulations are limited to a few shot impacts. Wang et al.¹² presented a FE analysis for modeling up 1000 random impacts. In the real-life shot peen forming, a large component may be peened with millions of shots.

Several numerical tools were developed to obtain the formed shape of the large components using equivalent loading methods. Grasty and Andrew¹³ proposed an equivalent method to simulate the impacting effect of a large number of shots in which the surface layers of the plates were subjected to a squeeze pressure to produce a small plastic deformation similar to shot impacting. Levers and Prior¹⁴ proposed a static analysis method to simulate the peening deformation by introducing the material coefficient of thermal expansion and temperature profiles into the elements. Wang et al.¹⁵ developed the thermal loading model by applying a loading unit to induce an equivalent plastic layer in a plate. Han et al.¹⁶ proposed that, in the first stage, the residual stress/strain profiles under the particular set of peening parameters were identified by simulating the peening process for a small-scale sample, and then in the second stage, the obtained stress/strain profiles were applied to the entire component to obtain the final shape. Gariépy et al.¹⁷ and Faucheux et al.¹⁸ obtained the residual stress profiles by simulating the practical peening processes, and then assigned the stress profiles to plates modeled with composite shell element. In the above equivalent numerical methods, the stress and strain distributions induced by shot impacts are averaged locally. Bhuvaraghan et al.¹⁹ used a discrete and FE method to obtain the more realistic residual stress and plastic strain distributions induced by a large number of shot impacts. The simulation time in FE model has been reduced by avoiding contact detection.

In this article, aiming to predict the peen formed shapes of plates under more realistic peening conditions and costing less computational time, a combined method is proposed. First, an analytical model is used to calculate the residual stress field of single shot impact and multiple shot impacts. Then, the calculated stress field is assigned to FEs to obtain the resulting shape. The numerical predictions are compared with experimental results.

Experiment

In the shot peen forming process, the wing skin hanged vertically on carrier is peened within strip areas by passing through the peening enclosure.²⁰ The process also can be implemented by controlling the peening nozzles moving over the target surface.²¹ Varying peening intensities are selected on the strip areas depending on the desired curvatures.²²

To investigate the effect of strip peening treatment on the plate deformation, experiments were performed on the aluminum alloy 2024-T351 (AA2024-T351) plates with dimensions of 200 × 200 × 5 mm, as shown in Figure 1(a). The strip peening treatment is imposed in the center strip area aligned with the rolling direction of the plate. The experiments are performed on the Wheelabrator MP20000 Aircraft Wing Peening System with standard shot of APB1/8.

Figure 1.

(a) Schematic view of experimental specimen and (b) shot dimple distribution in peening strip.

The distance between the peening nozzle and the plate is 200 mm. The rate of shot flow is 10 kg/min. The pneumatic pressure is 0.3 MPa. The average shot velocity when leaving the nozzle is measured with the high-speed photographic setup attached to the MP20000, which is 40 m/s. The relative moving speeds of the nozzle of four experiments are 2, 4, 6 and 8 m/min, respectively.

The target material is aluminum alloy 2024-T351, which is usually used for aerospace application requiring high strength to weight ratio, as well as good damage tolerance. The true stress–true strain curves of AA2024-T351 were fitted using the power-law hardening model in equation (1) from conventional tensile test by Heerens et al.²³ The material parameters of AA2024-T351 are listed in Table 1

σ = {\begin{matrix} E ε, ε \leq \frac{σ_{s}}{E} \\ K ε^{n}, ε > \frac{σ_{s}}{E} \end{matrix}

(1)

Table 1.

Material parameters of AA2024-T351.

$E$	$ν$	$σ_{s}$	$K$	$n$	$ρ$
$73 GPa$	$0.33$	$343 MPa$	$804 MPa$	$0.159$	$2780 kg / m^{3}$

The resulting shape of the plate with striped peening area is not simple spherical shape as the uniformly peened shape. The complex resulting shape is hard to be detected with limited measurements of local arc height. The shape of the plate is measured with three-dimensional scanning device HandyScan300 of CREAFORM.

The shot dimple distribution in the $x$ direction within the peening strip (Figure 1(b)) is approximate normal distribution.²⁴ In the $y$ direction, the shot dimple distribution is uniform random distribution if the peening nozzle is moving uniformly.

In order to detect the dimple distribution in $x$ direction, the dimples in an area of 80 mm ( $x$ ) × 50 mm ( $y$ ) in the peening strip are counted in square grids with side length of 10 mm. The distribution in $x$ direction is fitted with normal distribution function

f (x) = A \frac{1}{\sqrt{2 π} σ} \exp (- \frac{{(x - μ)}^{2}}{2 σ^{2}})

(2)

where $f (x)$ is the dimple number in a square grid with side length of 10 mm. $A$ , $σ$ and $μ$ are fitting constants.

Expanding cavity model

Single shot impact model

The process of one shot with velocity $V$ impacting onto a target can be treated as a quasi-static process.²⁵ The impacting direction is approximately vertical to the target surface considering low probability of inter-collision owing to low density of shots blasted out from peening nozzle, and the similar effects of oblique impact with small oblique angle compared to vertically impact effects on peen forming. The peening stress field is calculated with expanding cavity model. As shown in Figure 2, the cavity radius $a$ is equal to the projected radius of the shot dimple. The outer radius of the plastic zone and the elastic zone are $r_{c}$ and $r_{o}$ , respectively.

Figure 2.

Schematic view of shot indentation with cavity radius $a$ and plastic zone outer radius $r_{c}$ .

The stress components in the target during loading for power-law hardening plastic material are derived from the work of Gao et al.,²⁶ given in equations (3)–(7). In the spherical coordinate system, the radial stress $σ_{r}$ in the plastic zone ( $a \leq r \leq r_{c}$ ) shown in Figure 2 is

σ_{r} = \frac{2 σ_{s}}{3} ((\frac{1}{n} - 1) - \frac{1}{n} {(\frac{r_{c}}{r})}^{3 n} + \frac{r_{c}^{3}}{r_{o}^{3}})

(3)

where $n$ is the strain-hardening exponent and $σ_{s}$ is the yield stress of the target material.

The hoop stresses $σ_{θ}$ and $σ_{φ}$ are the same and are expressed in the plastic zone ( $a \leq r \leq r_{c}$ ) as

σ_{θ} = σ_{φ} = \frac{2 σ_{s}}{3} ((\frac{1}{n - 1}) + (\frac{3}{2} - \frac{1}{n}) {(\frac{r_{c}}{r})}^{3 n} + \frac{r_{c}^{3}}{r_{o}^{3}})

(4)

The radial stress $σ_{r}$ in the elastic zone ( $r_{c} \leq r \leq r_{o}$ ), shown in Figure 2, is

σ_{r} = \frac{2 σ_{s}}{3} \frac{r_{c}^{3}}{r_{o}^{3}} (1 - \frac{r_{o}^{3}}{r^{3}})

(5)

The hoop stresses $σ_{θ}$ and $σ_{φ}$ in the elastic zone ( $r_{c} \leq r \leq r_{o}$ ) are

σ_{θ} = σ_{φ} = \frac{2 σ_{s}}{3} \frac{r_{c}^{3}}{r_{o}^{3}} (1 + \frac{r_{o}^{3}}{2 r^{3}})

(6)

The mean contact pressure between the shot and the target is

p = \frac{2}{3} σ_{s} (1 + \frac{1}{n} {(\frac{Ea}{4 R σ_{s}})}^{n} - \frac{1}{n})

(7)

Particularly, in shot peening process, the initial kinetic energy of one shot is mostly converted into elastic-plastic work during the impact. The parameters about the shot can be related to the shot dimple as

\frac{1}{2} km V^{2} = \int_{0}^{h^{*}} π p a^{2} dh

(8)

where $k$ is an efficiency coefficient related to elastic and thermal dissipation and taken as 0.8,²⁷ $m$ is the mass of one shot and $h^{*}$ is the finial depth of the shot dimple. The integral results are

\frac{π σ_{s}}{3 kmR} \frac{n - 1}{n} a^{4} + \frac{4 π σ_{s}}{3 kmR} {(\frac{E}{4 R σ_{s}})}^{n} \frac{a^{n + 4}}{n (n + 4)} - V^{2} = 0

(9)

The shot dimple radius for certain peen forming parameters and conditions can be obtained numerically with equation (9). Then, the outer radius of plastic zone is

r_{c} = {(\frac{E a^{4}}{4 R σ_{s}})}^{1 / 3}

(10)

Substituting $r_{c}$ to equations (3)–(6), the stress distribution for one-shot loading is obtained. The stress components after shot unloading is calculated by considering that unloading is an elastic process and the hydrostatic stress does not introduce plastic deformation. The equilibrium equation of transverse residual stresses is

\begin{matrix} π r^{2} σ_{r}^{t} (r) - π {(r + d r)}^{2} σ_{r}^{t} (r + d r) \\ + \frac{π}{2} (σ_{θ}^{t} (r) + σ_{θ}^{t} (r + d r)) ({(r + d r)}^{2} - r^{2}) = 0 \end{matrix}

(11)

where $σ_{r}^{t}$ and $σ_{θ}^{t}$ are the transverse residual stresses, $σ_{r}^{t} (r) = σ_{r} (r) - Δ σ_{r} (r)$ , $σ_{r}^{t} (r + d r) = σ_{r} (r + d r) - Δ σ_{r} (r + d r)$ , $σ_{θ}^{t} (r) = σ_{θ} (r) - Δ σ_{θ} (r) = σ_{θ} (r) + Δ σ_{r} (r) / 2$ and $σ_{θ}^{t} (r + d r) = σ_{θ} (r + d r) - Δ σ_{θ} (r + d r) = σ_{θ} (r + d r) + Δ σ_{r} (r + d r) / 2$ . The transverse residual stresses are obtained with iterative solution of equation (11) with a tiny discretization value and initial condition $Δ σ_{r} (a) = σ_{r} (a)$ .

Multiple shot impacts model

For two shots impacting on the target with a separation distance $d$ , when $d < 2 r_{c}$ , there are three types of overlapping zones: elastic-elastic zone (EEZ), elastic-plastic zone (EPZ) and plastic-plastic zone (PPZ), as shown in Figure 3(a). When $2 r_{c} \leq d \leq 2 r_{o}$ , there are two types of overlapping zones: EEZ and EPZ. When $d > 2 r_{o}$ , there is no overlapping zone.

Figure 3.

Schematic view of (a) two shots impacting onto the target with a separation distance $d$ and (b) regularly distributed four shot dimples.

Assuming multiple shot impacts are imposed at the regular positions as shown in Figure 3(b), the stress fields induced by adjacent impacts would overlap each other. Since the plastic zone of each impact is relatively small and subsequent impact is hard to yield the former hardened material, the overlapping of adjacent stress fields is assumed to only take place in EEZ. In EEZ, the stress field is the sum of adjacent stress fields. In the EPZ and PPZ, the stress fields remain in the state of single shot impacting.

Numerical simulation

In the previous section, the peening stresses of single shot impact are calculated with expanding cavity model in the polar coordinates. To investigate the deformation of a plate peened by many shots, FE simulation is needed. In the cavity model, a symmetry constraint is applied on the peened surface of the target. First, a FE model of single shot impact is used to simulate the internal stresses of the target when removing the symmetry constraints on the peened surface. Second, the deformations of a shell strip and a solid strip involving the calculated peening stresses are compared. Finally, the deformations of plates simulated with realistic parameters are compared with the experimental results. The material under study of the numerical simulation is the same as the material of the experiments.

A clear distinction is made at this point between stresses in the target material. The stresses after loading of the expanding spherical model is called expanding stresses and the stresses after unloading is called expanding residual stresses. The stresses after removing the symmetry constraint on the peened surface is called transresidual stresses.⁸ The stresses after shot peening treatments and before the release of the rigid constraints on the component is called induced stresses.

Single shot peening model

In the simulation procedure, the expanding residual stresses are introduced to the FE model to obtain the transresidual stress field after removing the symmetry constraint on the peening surface. The expanding residual stresses are calculated from equations (3)–(11) in software MATLAB 7.10.²⁸ The calculation parameters are the same as the parameters of the experiments in the “Experiment” section. Then, FE simulation is performed in software ABAQUS 6.10.²⁹

The FE model shown in Figure 4 is used for the single shot peening simulation. The following dimensions are selected for the FE model: height $h = 5 mm$ , length and width $L = 10 mm$ . These dimensions are determined by considering the dimensions of the expanding residual stress field. Coordinate $(0, 0, 0)$ is located at a corner point on the top surface of the target. A higher element density is meshed in the core region with elements of minimum dimensions of $0.04 \times 0.04 \times 0.04 mm$ . The dimensions of the elements gradually increase from the core region to the sides of the model with maximum values of $0.2 \times 0.2 \times 0.2 mm$ .

Figure 4.

Discretized geometries of target in the single shot peening model.

The expanding stresses and the expanding residual stresses were calculated in the spherical-polar coordinates. To introduce the expanding residual stresses to the FE model as initial stresses, the stresses in the spherical-polar coordinates need be transformed to rectangular coordinates with

[\begin{matrix} σ_{x} & σ_{xy} & σ_{xz} \\ σ_{yx} & σ_{y} & σ_{yz} \\ σ_{zx} & σ_{zy} & σ_{z} \end{matrix}] = T [\begin{matrix} σ_{r} \\ σ_{θ} \\ σ_{φ} \end{matrix}] T^{T}

(12)

where $T = [\begin{matrix} \sin θ \cos φ & \cos θ \cos φ & - \sin φ \\ \sin θ \sin φ & \cos θ \sin φ & \cos φ \\ \cos θ & - \sin θ & 0 \end{matrix}]$ , $θ$ is the polar angle measured from the $z$ axis of the rectangular coordinates, $φ$ is the polar angle measured from the $x$ axis and $T^{T}$ is the transpose of $T$ .

On the FE model, the symmetry boundary conditions are imposed on the $x = 0$ , $y = 0$ and $y = L$ planes; the nodes on the $z = h$ plane are fixed in $z$ direction. The deformations of the material after removing the symmetry constraint are considered in the elastic range. The transresidual stresses are obtained and compared with the initial expanding residual stresses.

Model of strip peened regularly

The deformations of rectangular strips peened regularly on the top surface are simulated. As shown in Figure 5, the strip dimensions are $100 (L_{s}) \times 30 (B_{s}) \times 5 (h_{s}) mm$ . Three cases with regular indenting gaps ( $d$ ) of 2, 3 and 4 mm are simulated. The dimensions of indentation distribution are plotted, and the corresponding parameters for different indenting gaps are listed in Table 2. The simulation of the strip is undertaken with two types of FE models: solid element model and shell element model.

Figure 5.

Shot distribution and dimensions of peened strip with $L_{s}$ = 100 mm and $B_{s}$ = 30 mm.

Table 2.

Dimensions of shot distribution.

$d$ (mm)	$B_{c}$ (mm)	$L_{c}$ (mm)	$N$
2	24	80	533
3	24	78	243
4	24	80	147

$N$ denotes the total number of shots.

Strip model with solid element

The strip is modeled with solid element C3D8R in Abaqus with element dimensions of 0.2 × 0.2 × 0.2 mm. The initial stresses in the elements of the strip are the expanding residual stresses calculated from the expanding cavity model. The overlapping of the stress fields of adjacent shot impacts are considered as proposed in the “Multiple shot impacts model” section. The simulation procedure is carried out by two static/general steps. In the first step, the symmetry boundary conditions are imposed on the $x = 0$ , $y = 0$ , $x = L_{s}$ and $y = B_{s}$ planes; the nodes on the $z = h_{s}$ plane are fixed in the $z$ direction. In this step, the initial stresses involved in the model are released in the $z$ direction, for the $z = 0$ plane of the strip model is free contrast to the boundaries of the expanding cavity model. The internal stresses after released in the $z$ direction is the induced stresses. In the second step, the displacements of node (0, 0, $h_{s}$ ) are fixed, the node ( $L_{s}$ , 0, $h_{s}$ ) is fixed in the $y$ and $z$ direction and the node (0, $B_{s}$ , $h_{s}$ ) is fixed in the $z$ direction. In this step, the strip is bent and elongated elastically by the induced stresses.

The resulting shapes of the strips under different indenting gaps $d$ are measured along the longitudinal centerline on the $z = h_{s}$ plane.

Strip model with shell element

The strip is further modeled with shell element S4R in Abaqus with element dimensions of 0.2 × 0.2 mm. The shell has a constant thickness of 5 mm and 11 Simpson integration points through thickness. The reference surface of the shell element is taken to be the bottom surface ( $z = h_{s}$ ) of the strip. The simulation procedure is carried out by only one static/general step. The introduced initial stresses to the shell element is the induced stresses. To calculate the induced stresses of multiple shot impacts, first, the transresidual stresses need be calculated from the expanding residual stresses. With equation (12), the expanding residual stresses are transformed from the spherical-polar coordinates to the rectangular coordinates. Now, the expanding residual stresses in the rectangular coordinates are transformed to the cylindrical-polar coordinates with equation (13)

[\begin{matrix} σ_{rr} & σ_{r φ} & σ_{rz} \\ σ_{φ r} & σ_{φ φ} & σ_{φ z} \\ σ_{zr} & σ_{z φ} & σ_{zz} \end{matrix}] = C [\begin{matrix} σ_{x} & σ_{xy} & σ_{xz} \\ σ_{yx} & σ_{y} & σ_{yz} \\ σ_{zx} & σ_{zy} & σ_{z} \end{matrix}] C^{T}

(13)

where $C = [\begin{matrix} \cos φ & \sin φ & 0 \\ - \sin φ & \cos φ & 0 \\ 0 & 0 & 1 \end{matrix}]$ , $φ$ is the polar angle measured from the $x$ axis of the rectangular coordinates and $C^{T}$ is the transpose of $C$ .

In the cylindrical-polar coordinates, removing the symmetry constraint on the peened surface of the expanding model, the expanding residual stress in the $z$ direction is partially relaxed. The relaxation value of the stress in the $z$ direction can be written as

σ_{zz}^{rel} = λ σ_{zz}

(14)

where values of $λ$ from 0 to 1 induce non-relax and total-relax states, respectively.

In accordance with Hooke’s law, the relaxation values in the radial and hoop directions in the cylindrical-polar coordinates can be calculated as

σ_{rr}^{rel} = σ_{φ φ}^{rel} = \frac{ν}{1 - ν} σ_{zz}^{rel}

(15)

Then, the transresidual stresses of one shot impact can obtained as

σ_{zz}^{tr} = (1 - λ) σ_{zz}

(16)

σ_{rr}^{tr} = σ_{rr} - σ_{rr}^{rel} = σ_{rr} - \frac{ν}{1 - ν} σ_{zz}^{rel}

(17)

σ_{φ φ}^{tr} = σ_{φ φ} - σ_{φ φ}^{rel} = σ_{φ φ} - \frac{ν}{1 - ν} σ_{zz}^{rel}

(18)

Neglecting the relaxation of the shear stresses, the in-plane transresidual stresses in the rectangular coordinates can be calculated from the stresses in the cylindrical-polar coordinates as

\begin{matrix} [\begin{matrix} σ_{xx}^{tr} & σ_{xy}^{tr} \\ σ_{xy}^{tr} & σ_{yy}^{tr} \end{matrix}] = & [\begin{matrix} \cos φ & - \sin φ \\ \sin φ & \cos φ \end{matrix}] [\begin{matrix} σ_{rr}^{tr} & σ_{r φ} \\ σ_{φ r} & σ_{φ φ}^{tr} \end{matrix}] \\ [\begin{matrix} \cos φ & \sin φ \\ - \sin φ & \cos φ \end{matrix}] \end{matrix}

(19)

Then, the induced stresses are calculated by considering the overlapping of the stress fields of adjacent shot impacts proposed in the “Multiple shot impacts model” section. The induced stresses are introduced to the section points of the shell element along thickness. In the simulation procedure, the displacements of node (0, 0, 0) are fixed, the node ( $L_{s}$ , 0, 0) is fixed in the $y$ and $z$ direction and the node (0, $B_{s}$ , 0) is fixed in the $z$ direction. To equilibrate the induced stress, the strip is bent and elongated elastically.

The resulting shapes of the shell element strip under different indenting gaps (Table 2) are measured along the longitudinal centerline of the strip. The values of the relaxation parameter $λ$ are taken as 0.9 and 1. The resulting shapes of the shell element strip are compared with the resulting shapes of the solid element strip.

Model of plate peened in reality

In this section, a FE model is proposed to simulate the deformation of a plate peened with the real peening parameters as the experiments in the “Experiment” section. In reality, the locations of the shot impacts are random as shown in Figure 1. In the experiments, the shot dimple distribution under different peening parameters are fitted with normal distribution function in the transverse direction of the peening strip. Based on the values of the fitting constants of the experiments, the simulating positions of the shot impacts on the plate are obtained numerically with the random functions $random$ and $randi$ in the software Matlab 7.10. The $x$ coordinates of the shot impact positions are produced with the function $random$ and the $y$ coordinates with the function $randi$ .

Figure 6 shows the discretized geometries of the plate modeled with shell element S4R in the software Abaqus 6.10. The dimensions of the elements in the middle region are 0.4 × 0.4 mm. A low element density is meshed in the both sides of the plate with elements of maximum dimensions of 5 × 5 mm. Boundary conditions (Figure 6) are chosen to present the rigid body motion of the plate.

Figure 6.

Discretized geometries of the plate with side length of 200 mm.

The induced stresses calculated with the algorithm mentioned in the “Strip model with shell element” section are introduced to the plate model as initial stresses of the FE element. The assignment of the initial stress is carried out by the subroutine SIGINI of Abaqus 6.10. The resulting shape of the plate is obtained by a static/general simulation step with nonlinear solutions. The resulting shapes under different peening parameters are compared with the scanned experimental shapes.

Results and discussion

Internal stress profiles

In the “Single shot impact model” section, the expanding cavity model was proposed to calculate the expanding stresses and the expanding residual stresses of single shot impact. With the peening parameters and the material properties of the experiments in the “Experiment” section, the radial and hoop stresses of loading (expanding stresses) and the radial and hoop stresses of unloading (expanding residual stress) in the spherical-polar coordinates are calculated as shown in Figure 7(a). The equivalent stresses of loading and unloading are shown in Figure 7(b). From the expanding cavity model, there are no values of the stresses in the region of the internal cavity with radius of $a$ .

Figure 7.

(a) Radial and hoop stresses and (b) equivalent stresses of loading and unloading in the cavity model.

Comparing the stresses of loading and unloading, it can be seen that the radial stress decreases after unloading while the hoop stress increases. Since the reverse yielding when unloading is neglected in the expanding cavity model, the unloading equivalent stress exceeds the values of loading in the surface layer with depth of $s$ . To correct the calculated hoop stress of unloading, the hoop stress in the reverse yielding region is set to be equal to the stress of the reverse yield point, as shown in Figure 7(a).

In the “Single shot peening model” section, the expanding residual stresses of single shot impact are introduced to the FE model to obtain the transresidual stresses. In the region of the internal cavity, the expanding residual stresses are set to be equal to the stresses on the internal cavity surface. Figure 8(a) shows the initial Mises stress field and initial stress in the $z$ direction on the $y = L / 2$ plane. The initial stresses are transformed from then expanding residual stresses. Figure 8(b) shows the residual Mises stress field and the stress in the $z$ direction of the simulation. The residual stresses are called the transresidual stresses of one-shot impact. It can be seen that the stress in the $z$ direction is relaxed largely.

Figure 8.

(a) Initial Mises stress field and initial stress in $z$ direction and (b) residual Mises stress field and residual stress in $z$ direction on the $y = L / 2$ plane.

Figure 9(a) and (b) compare the initial stresses and the residual stresses along the vertical centerline of the $y = L / 2$ plane of the single shot impact model. The initial stress in the $z$ direction is largely relaxed while the relaxations of the stress in the $x$ and $y$ directions are relatively small. Figure 9(c) and (d) compare the initial stresses and the residual stresses along the centerline on $z = 0$ plane in the $x$ direction. The stress in the $z$ direction is relaxed totally. The stress in the $y$ direction is relaxed largely. The relaxation of the stress in the $x$ direction is relatively small. So the relaxations of the stresses in the simulation mainly take place in the $z$ direction.

Figure 9.

Initial and residual stresses (a) in $z$ direction and (b) in $x$ and $y$ directions along the centerline beneath the cavity core. Initial and residual stresses (c) in $y$ and $z$ directions and (d) in $x$ direction along the $x$ direction line on peening surface.

To simplify the analysis, the deformations of the material in the simulation were considered in the elastic region, which leads to irrational tensile stresses in a tiny region near the peening point, as shown in Figures 8(b) and 9. Since the region involving the irrational tensile stresses is very small compared with the target, its influence on the macro-deformation of the target can be neglected.

Resulting shape of regularly peened strip

In the “Strip model with solid element” section, the resulting shapes of the strips under different peening parameters are obtained by introducing the overlapped expanding residual stresses to the solid elements. In the “Strip model with shell element” section, the resulting shapes of the strips corresponding to different peening parameters are obtained by introducing the overlapped transresidual stresses to the shell elements.

Figure 10(a) shows the initial Mises stress field on the top surface ( $z = 0$ ) of the solid element strip and resulting displacement of the bottom surface ( $z = h_{s}$ ) in the $z$ direction with indenting gap of 4 mm. Figure 10(b) shows the initial Mises stress field on the top surface of shell element strip and resulting displacement in the $z$ direction with indenting gap 4 mm and relaxation ratio $λ = 1$ .

Figure 10.

(a) Initial Mises stress field on the top surface ( $z = 0$ ) of the solid element strip and resulting displacement of the bottom surface ( $z = h_{s}$ ) in the $z$ direction with indenting gap 4 mm. (b) Initial Mises stress field on the top surface of shell element strip and resulting displacement in the $z$ direction with indenting gap 4 mm and relaxation ratio $λ = 1$ .

Along the longitudinal centerline on the bottom surface of the solid element strip and the longitudinal centerline of the shell element strip, the resulting shapes are compared between the two different element models. Figure 11(a) shows the comparisons of the resulting shapes between the solid element strip and shell element strip with relaxation ratio $λ = 1$ . Figure 11(b) shows the comparisons of the resulting shapes between the solid element strip and shell element strip with relaxation ratio $λ = 0.9$ .

Figure 11.

Comparison of the resulting deflections of the shell element strip and the solid element strip along the longitudinal centerline. The values of the relaxation ratio $λ$ of the shell element model are (a) 1 and (b) 0.9.

It can be seen that under different indenting gaps, the resulting shapes of the solid element model and the shell element model agree with each other when $λ = 1$ . When $λ = 0.9$ , the resulting shapes of the shell element model are slightly larger than the resulting shapes of the solid element model. When $λ = 0.9$ , the expanding residual stress in the $z$ direction is not totally relaxed in the calculation while the expanding residual stresses introduced to the solid element are almost totally relaxed in the $z$ direction.

The above comparisons show that the shell element model combined with the algorithm proposed in the “Strip model with shell element” section can be used to simulate the deformations of shot peen formed plate.

Shot dimple distribution

The mean mass of one APB1/8 shot was measured on METTLER TOLEDO analytical balances. The mean mass of one shot is 0.12975 g. According to the shot flow rate and the nozzle moving speed, the total number of the shot per 10 mm length of peening trajectory can be calculated, as shown in Table 3. The dimple number per 10 mm length of peening trajectory on the peened plate was also counted and compared with the theoretical number. The counted numbers are smaller than the theoretical number. The deviation between the counted number and the theoretical number increases with the decrease of the nozzle moving speed. The differences might be attributed to the following:

The peening coverage increases with the decrease of the nozzle moving speed, which leads to the number of shot repeatedly impacting on the same region increase.

The selected representative region for counting the shot dimple number is limited and some shot maybe impacted out of the region.

There are some deviations on the calculation of the theoretical shot number and on the realistic shot number ejected out of the nozzle.

Table 3.

Parameters of shot distribution with different nozzle moving speeds.

Nozzle velocity m/min	Counted shot number/10 mm length	$A$ /10 mm length	$σ$	$μ$ (mm)	Theoretical shot number/10 mm length	Shot number deviation (%)
2	323	3332	15.97	1.4	385	16.23
4	169	1729	15.82	0.5	193	12.81
6	118	1214	14.86	−0.5	128	7.98
8	93	932.5	14.91	0.7	96	3.67

The shot dimple distributions of different peening parameters are fitted with the normal distribution function in the experiments. The values of the fitting parameters are listed in the Table 3. The standard deviation $σ$ determines the shape of the distribution. With the increase of the nozzle velocity, the width of the peening strip decreases. The non-zero values of the mean $μ$ of the distribution show that there was an offset between the peening nozzle and the centerline of the plate in the experiments.

Based on the values of the fitting parameters, the numerical shot impacting positions are obtained. Figure 12 compares the shot dimple distribution between the experiment and numerical result with nozzle moving speed of 8 m/min. In the numerical shot distributions, the values of the mean $μ$ of the distribution are set to be zero.

Figure 12.

(a) Experimental and (b) numerical shot dimple distribution in one peening trajectory with peening nozzle moving speed of 8 m/min.

With the numerical shot impacting positions, the induced stresses are calculated and introduced to the section points of the shell element of the plate. Figure 13 shows the initial Mises stress distributions with relaxation ratio $λ = 0.9$ on the top surface of the plate under different nozzle velocities. The region of high Mises stress expands with the decrease of the nozzle velocity.

Figure 13.

Numerical shot distributions and initial Mises stress fields with relaxation ratio $λ = 0.9$ under different nozzle moving speed.

Resulting shape of simulated and experimental plate

Under different peening parameters, the resulting shapes of the plate were obtained by simulations. The peen forming shapes of the plate in experiments were scanned in three dimensions (3D). Figure 14 shows the comparison of the simulated resulting shapes with relaxation ratio $λ = 0.9$ and the experimental shapes under four different nozzle moving speeds. The simulated shapes are in good agreement with the experimental shape.

Figure 14.

Comparison of simulated and experimental contours with nozzle moving speeds of (a) 2, (b) 4, (c) 6 and (d) 8 m/min. The relaxation ratio $λ$ is 0.9.

The shape of the plate peened in strip region is aspheric. In the peening region, the forming curvature in the transverse direction is obviously larger than the curvature along the peening strip. In the blank region, the curvature deformations are relatively small.

Another comparison between the simulated resulting shapes with relaxation ratio $λ = 1$ and the experimental shapes shows that deflections of the simulated shapes are larger than the experimental deflections. This means that, in reality, the shot impacting stress in the $z$ direction is not relaxed totally. The plastic deformations of the material around the shot dimple would partly prevent the relaxation of the stress in the $z$ direction, while in the solid element simulation in the “Strip model with solid element” section, there are no shot dimple to prevent the relaxation.

The geometry of peened plate is dependent on the impact locations and plate size. With uniform location of impacts and narrow plate, the peened plate tends to be spherical in geometry as shown in Figure 10. Peening treatment applied on local regions results in non-spherical geometry as shown in Figures 13 and 14. In addition, uniform location of impacts combined with a square plate may lead to a non-spherical geometry owing to instability of geometry deformation.¹⁸ With the methods proposed in this article, the resultant geometry of peened plate can be determined by FE simulation with nonlinear solutions.

Computational resources

The shot peening forming deformations of a plate peened in reality can be numerically simulated with the above combined methodology. In the procedure, first, the transresidual stress field of single shot impact is calculated with the expanding cavity model. Second, the induced stress field of a large number of shot impacts is calculated by considering the overlapping of the stress fields of adjacent shot impacts. Third, the induced stress field is introduced to the FE model to obtain the resulting shape of the plate.

The first and second steps were carried out in the software Matlab 7.10. The third step was achieved in the software Abaqus 6.10. Compared with the conventional simulations of shot peen forming processes with a large number of shot impacts, such as Wang et al.’s¹² work, the computation time in FE simulation has been significantly reduced by avoiding contact detection between the shots and the target. The demand of computational resources in terms of disk space is reduced by modeling the plate with shell element instead of solid element.

The methodology proposed in this article has the merit of the finite/discrete element method¹⁹ that the random distribution of the shot impacts is considered. In the FE simulation process, the solution of the problem is only to balance the induced stresses.

Conclusion

A methodology combining analytical methods and FE simulation was proposed to make the simulation of the shot peen forming process more realistic. The computational time and the demand of the disk space are significantly reduced by avoiding the contact detection between the surfaces of the shots and the target and by modeling the plate with shell element. The combined methodology provides a tool for shot peen forming simulation with a large number of shot impacts. The simulation can be used to predict the relationships between the shot peen forming parameters and the formed shapes, thus saving the cost of the physical experiments to establish the relationships.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the support for this research work from Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2018JQ5084), Natural Science Foundation of Shaanxi Provincial Department of Education (Grant No. 18JK0571), National Natural Science Foundation of China (Grant No. 51805432) and China Postdoctoral Science Foundation (Grant No. 2018M633541).

ORCID iD

Xudong Xiao

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Prediction of peen forming stress and plate deformation with a combined method

Abstract

Keywords

Introduction

Experiment

Expanding cavity model

Single shot impact model

Multiple shot impacts model

Numerical simulation

Single shot peening model

Model of strip peened regularly

Strip model with solid element

Strip model with shell element

Model of plate peened in reality

Results and discussion

Internal stress profiles

Resulting shape of regularly peened strip

Shot dimple distribution

Resulting shape of simulated and experimental plate

Computational resources

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References