Numerical and experimental investigations on the effect of particle properties on the erosion behavior of aluminum alloy during abrasive air jet machining process

In this study, experimental and numerical investigations have been done to explore the effect of the particle properties on the erosion behavior of aluminum alloy during the abrasive air jet machining (AAJM) process by using the novel medium-hard amino thermoset plastic (ATP) and conventional super-hard alumina (Al2O3) particles. In the numerical simulation, a novel linear elastic material model with the failure standard was proposed to define the ATP particle and the conventional rigid material model was used to define the Al2O3 particle. Meantime, the smooth particle hydrodynamics (SPH) interpolant with the moving-least-squares method was used to establish the impact target model. Then, a multi-particle impact model based on the SPH and finite element coupling method (SPH-FEM) was further developed to investigate the particle impact process. It indicates that the SPH-FEM method can be used to simulate the erosion behaviors of the aluminum alloy during the AAJM process by using not only the super-hard Al2O3 particle but also the medium-hard ATP particle, and the simulation results are fundamentally consistent with the experimental ones. The results demonstrate that the effect of particle hardness on erosion behavior is much greater than that of compressive air pressure. Furthermore, there exists an optimal impact angle where the surface material can be removed by chip formation resulting in the maximum material removal rate, and the surface erosion behavior can be accurately predicted by simulation. Moreover, with the particle hardness increasing, the optimal impact angle would be reduced accordingly.


Introduction
In the abrasive air jet machining (AAJM) process, due to the relatively smaller size of the abrasive particles causing a small amount of material removal, the erosion process of the impact target is hard to be determined by the experiment. While according to the recent literature, the finite element method (FEM) may provide an effective way to resolve the above issue. In the early FEM research, two-dimensional (2D) single-particle impact models based on the assumption of plane strain and plane stress were established [1,2]. However, due to the 2D simulations being impossible to obtain the real impact condition, the simulated results are quite different from the actual observations, and the issues such as multi-particle collision and collision surface overlap cannot be thoroughly analyzed. To solve this problem, Aquaro et al. [3,4] proposed a three-dimensional (3D) FEM model based on Euler's formula and Lagrange's formula to simulate the particle impact process. Unfortunately, such 3D FEM model is only suitable for analyzing erosion behavior at relatively large impact angles (≥ 60°). For this reason, Takaffoli et al. [5] used the smooth particle hydrodynamics and finite element coupling method (SPH-FEM) to simulate the impact of a single angular particle on the aluminum alloy.

3
The results indicated that the SPH-FEM method can be used to analyze the erosion behavior under most of the impact angles; furthermore, the relevant damage phenomena (such as crater and fragmentation) can be also simulated. Based on the above research, Azimian and Schmitt [6] performed the simulation of the single and several particle impacts on the copper surface and pointed out that the damage phenomena including pits and debris can be stimulated. Moreover, Yaer et al. [7] conducted the simulation of the simultaneous impact of several spherical particles on the wrought superalloy materials and declared that the SPH-FEM method can be used to investigate the correlation mechanism between the erosion failure of the surface and the elastoplastic behavior of the subsurface.
However, although the SPH-FEM method can be used to simulate the erosion behavior of the impact target during the AAJM process, there still exist some problems. For example, the real abrasive particles are usually with angular shapes, while the spherical shape of the particles is usually numerically modeled; thus, the FEM results are not so accurate to satisfy the real event. For this reason, Papini and Takaffoli [8,9] established the particle model according to the real particle geometric structure, and physical and mechanical properties; furthermore, in the simulation of the particle impact process, not single or several particles but massive amounts of particles simultaneously impacting on the target were developed. According to this view, Arani et al. [10] further developed the SPH-FEM method and successfully simulated the material removal behaviors of the impact target. Moreover, Liu and Wan [11] conducted FEM simulations by using both the Johnson-Cook constitutive model and the shear failure model to explore the effects of particle shapes on the erosion mechanism. The obtained results indicate that the erosion mechanism was greatly influenced by the particle shapes. Di and Wang et al. [12] confirmed the above results and declared that the shapes of the particles have a great effect on the erosion rate.
On the other hand, due to the conventional AAJM process using the super-hard particles (e.g., Al 2 O 3 (Mohs hardness of 9.0), SiC (Mohs hardness of 9.5)), the defeats such as the abrasive embedding and ploughing-grooves are likely to generate, especially for the targets made of aluminum alloy, titanium alloy, and other nonferrous metals [13,14]. To deal with this issue, Zhu et al. [15] fabricated a novel medium-hard (Mohs hardness of about 4.0) amino thermoset plastic (ATP) particle for AAJP of aluminum alloy. The results indicate that due to the medium hardness of the ATP particle, the defeats such as the abrasive embedding and ploughing-grooves can be resolved. Since the ATP particle is expected to be a promising alternative to the conventional super-hard particles for AAJM of aluminum alloy, the mechanism of the ATP-AAJM of aluminum alloy needs to be investigated urgently. Although Zhu et al. [16] established a 2D theoretical model based on the particle micro-cutting mechanism to investigate the mechanics of surface formation for ATP-AAJP of aluminum alloy, the surface erosion process has not been explored yet. While according to the above research, it can be inferred that the SPH-FEM method can provide a promising way to solve the above issue. If so, some new challenges would be raised accordingly, for instance, how to establish a suitable material model for the novel medium-hard ATP particle, and how to define an appropriate SPH-FEM model to investigate the particle impact process.
To solve the above problems, in this study, a novel linear elastic material model with the failure standard was proposed to define the medium-hard ATP particle, and the conventional rigid material model was applied to define the super-hard Al 2 O 3 particle. After that, the SPH interpolant with the moving-least-squares method was used to establish the impact target (aluminum alloy 7075 (AA7075)) model. Then, a multi-particle impact model based on the SPH-FEM method was developed to investigate the erosion behavior of the aluminum alloy. Furthermore, the corresponding experiments were also conducted. Combined with the simulation and experimental results, the feasibility of the SPH-FEM method for the numerical simulation of the erosion behavior of aluminum alloy during AAJM by using the novel ATP particle was demonstrated, and the effects of the particle hardness and impact angle on the erosion behaviors of aluminum alloy were further investigated, as well.
2 Finite element model of multi-particle impact

Modeling of multi-particle impact
To simulate the real particle impact, the microstructures of the initial Al 2 O 3 and ATP particles with the size of 0.6-0.7mm (shown in Fig. 1) were measured and are shown in Fig. 1. From Fig. 1, it is found that although the shapes of the ATP and Al 2 O 3 particles are almost the same (showing a stable polyhedral structure with sharp edges and vertices), the surface topographies of the above particles are greatly different. For the Al 2 O 3 particle, the smooth surface is obvious (Fig. 1a), implying a typical brittle fracture feature. While for the ATP particle (formaldehyde and melamine as the raw main materials), the rough surface with the approximate dimple structure is presented (Fig. 1b), indicating a typical feature of brittle fracture. According to the above SEM observations, five types of particle models with relatively different sizes and structures were established (Fig. 2a). Then, the particles were divided into parallel groups (shown in Fig. 2b and c), and each particle group including 20-25 particles was set as a circular area with a diameter of 12 mm (Fig. 2a); furthermore, the position of the particles is random and the distance between them is about 5-10 mm. The centers of the particle groups are assumed to be on a straight line (red line in Fig. 2b and c), which is parallel to the impact direction. θ Impact direction Impact direction Fig. 2 Particle impact model for a particle distribution in one of the groups. b and c are the multi-particle impact model for Al 2 O 3 particle and ATP particles, respectively Then, a multi-particle impact model based on the SPH-FEM method was developed and is shown in Fig. 2b and c. As illustrated in Fig. 2b and c, the impact target model (14mm×14mm×4mm) is assembled into the particle group model, and its upper and bottom surfaces are parallel to the particle groups. In this way, the angle between the impact direction and the surface of the impact target is assumed to be the impact angle θ. Meanwhile, the impact target model is established by using SPH interpolant with the movingleast-squares method, in that the SPH particles without fixed connectivity were used to discretize the target.

Meshing of the model
Meanwhile, to satisfy the tradeoff between the computation time and solution accuracy, for the Al 2 O 3 particle impact model, due to the higher particle density and hardness (listed in Table 1) inclining to cause a relatively larger impact force (Eqs. (6) and (7)), the Al 2 O 3 particle model was divided by one mesh and the spacing between the SPH particles of the impact target was set to 0.15mm. While for the ATP particle impact model, owing to the relatively smaller particle density and hardness, a relatively smaller impact force is inclined to generate; thus, the ATP particle model was discretized by the hexahedral mesh elements with an average element size of 0.2mm, and the spacing between the SPH particles was set to 0.1 mm.

Boundary conditions and contact settings
The node-to-surface contact model is used to define the contact between the impact particles and the target surface (here, the contact between impact particles is not considered). Furthermore, the dynamic friction coefficient between the impact particle and the target surface is assumed to be equal to their static friction coefficient, the value of which is assumed to be about 0.2. Moreover, the boundary conditions are set as follows: the bottom surface of the target is fixed, the particle velocities are set to vary from 40 to 220m/s, and the impact angles are set at 15°, 30°, 45°, 60°, 70°, and 90°, respectively.

Material model
According to the SEM result in Fig. 1a that the Al 2 O 3 particle shows a typical brittle fracture feature, the conventional rigid material model was used to define the Al 2 O 3 particle; that is, during the Al 2 O 3 particle impact process, the Al 2 O 3 particles show no fragmentation and fracture, causing some SPH particles directly removed from the target surface (Fig. 3a). While due to the ATP particle presenting an approximate tough fracture feature (Fig. 1b), a novel elastic material model with the failure standard was developed to define the ATP particle, in that, during the ATP particle impact process, particles that reach the failure standard will automatically delete the failed mesh elements by the element elimination method, resulting in particle splitting (Fig. 3b). Meanwhile, in such conditions, the SPH particles may be not  3 Simulations of the particle impact for a Al 2 O 3 particle and b ATP particle directly removed from the surface, while inclining to bring about the plastic deformation (this issue would be discussed in detail in Section 4.2).
In addition, Vahid Hadavi et al. [14] pointed out that the Johnson-Cook (J-C) failure model can be used to predict the distribution of the surface craters during the Al 2 O 3 particle impacting aluminum alloy. Walid Jomaa et al. [18] demonstrated that the J-C constitutive and damage equations are suitable for simulating the serrated chip formation in high-speed machining of AA7075 alloy. Thus, in this study, due to the erosion behaviors such as chip formation or material deformation largely occurring in the particle impact on aluminum alloy [16], J-C constitutive and damage equations have been used to simulate the erosion behavior of the impact target AA7075. Besides that, the thermo-mechanical behavior of the impact target AA7075 is modeled by the J-C constitutive equation, which is expressed as [18] where σ is the flow stress, is the equivalent strain rate, 0 is the reference strain rate, A is the yield strength, B is the hardening modulus, C is the strain rate sensitivity coefficient, m is the thermal softening index, n is the strain-hardening index, and / 0 is the normalized equivalent plastic strain rate. T * is the dimensionless temperature, which is defined as follows: where T is the current temperature of the material, T r is room temperature, and T m is the melting point of the material. Meanwhile, according to Reference [18], the above J-C material parameters and the main physical properties of the AA7075 can be obtained and respectively listed in Table 2.
The damage behavior of the impact target AA7075 is modeled by the JC failure model, which is based on the equivalent plastic strain at failure ε f and used as a criterion for damage initiation [18]. Here, ε f can be written as where σ * is the mean stress normalized by the effective stress, and D 1 -D 5 are the damage constants, which also can be obtained by Reference [18] and listed in Table 3. Furthermore, in Eq. (3), the first bracketed term corresponds to the average pressure, the second bracket term corresponds to the strain rate, and the third bracket term corresponds to the temperature. Moreover, based on a cumulative damage law, the damage initiation threshold can be presented as where ∆ε is the increase in the equivalent plastic strain. Furthermore, if D were equal to 1, failure behavior would occur (evidenced by the surface material removal or surface material deformation), and the stress for the SPH particle is assumed to be zero.

State equation
Due to that the impact of the solid particle on the metal material is a transient process, when the dynamic pressure of the particles was greater than the yield stress of the impact target material, the erosion behavior of the impact target is inclined to generate. In such conditions, the dynamic response process of the impact target needs to be fully described. Meantime, owing to that the state equation can define the relationship between hydrostatic pressure, local density, and local specific energy [19], in this study, the GRUNEISEN state equation with cubic shock velocity and the particle velocity (v s -v p ) is used to define pressure for compressed material and for expanded material  where C is the intercept of the v s -v p curve; s 1 , s 2 , and s 3 are the unitless slope coefficients of the v s -v p curve; γ 0 is the unitless factor of the GRUNEISEN equation of state; a is the first-order unitless volume correction to γ 0 ; E is the internal energy; μ is the compressibility factor, which can be written as where ρ and ρ 0 are the particle density and initial air density, respectively. Meanwhile, according to Reference [19], the parameters used in the GRUNEISEN state equation can be obtained and listed in Table 4. By entering the above parameters, the dynamic response process of the impact target AA7075 can be described.

Experimental procedure
The experiments of AAJM are shown in Fig. 4. Specimens (40 mm × 40 mm) were cut from the rolled plates of AA7075 with a thickness of 4 mm. The hardness of the AA7075 is about 160HV. When the AAJM is conducted (Fig. 4a), the boron carbide nozzle is fixed on a rotating support, where the impact angle and the stand-off-distance (SOD, centerline distance between the nozzle exit and the target) can be modified (Fig. 4b). The parameters of the particle impact are set as follows: the compressive air pressure is 0.1-0.5 MPa, the impact angle is 15-90°, and the SOD is 10-20 cm. Each experiment was conducted for 3 min and repeated three times [15]. Particle velocities are measured by using high-speed 3D photography equipment (green dotted square mark in Fig. 4a), where the high-power LED lamp (100 watts) is placed perpendicular to the high-speed camera equipped with the high-magnification zoom lens so that the axis of the LED lamp and the lens of the high-speed camera can be vertically aligned. According to the calibrated scale factor of the camera, the particle displacement can be determined by two consecutive image distances as the particle jet passes through the high-speed camera (Fig. 4c). Furthermore, due to the image time interval being known, the particle velocity can be calculated (Fig. 4d). Such measurements are repeated three times, and the averages of these measurements are determined. The measured Al 2 O 3 and ATP particle velocities as a function of the compressed air pressure are shown in Fig. 4e and f, respectively.

Microstructural characterization
The surface topography of the AA7075 specimen after the particle impact was examined by using FlexSEM1000 scanning electron microscope (SEM) and Nanovea PS50 3D optical profilometer. The microstructures of the ATP and Al 2 O 3 particles were examined by FlexSEM1000 SEM, as well.

Determination of the material removal rate
The calculations of the experimental and simulation material removal rates are given in Eqs. (8) and (9), respectively. In Eq. (8), the weight of the AA7075 specimens before and after the particle impact process was measured by the BSA124S electronic analytical and precision balance (the least count of 0.1 mg), and the obtained results are used to determine the material removal or weight loss of the impact target (WL impact target , mg). Meantime, the weight of the impact particle (W impact particle-experiment , g) was measured by the CN-LPC20001 electronic balance.
While in Eq. (9), the SPH particle weight is obtained by dividing the total weight of the impact target model (W impact target model , g, which is assumed to be the density of AA7075 multiplying the volume of the impact target model) by the number of SPH particles; meantime, the weight of the removed SPH particle (W removed SPH particle , mg) is determined by multiplying the SPH particle weight and the number of the removed SPH particles. Furthermore, the total weight of the particles is calculated by multiplying the bulk density of the particle (listed in Table 1) and the volume of the particle group model in the multi-particle impact model (W impact particle-simulation , g)

Determination of the cross-section profile of the simulated surface topography
To achieve better results, besides the comparison of the simulated and experimental 3D surface topography of the specimen surfaces, the cross-section profiles have been also analyzed in this study. The detailed steps are as follows: firstly, according to the simulated 3D topography (Fig. 5a), a 2D contour map was achieved by using the Matlab software (Fig. 5b); then, a novel MATLAB program (listed in (8) Experimental material removal rate = WL impact target W impact particle-experiment × 10 −3

(9)
Simulation material removal rate = W removed SPH particle W impact particle-simulation × 10 −3  (e) (f) Fig. 4 Experimental device for AAJM process for a apparatus. b Scheme of the particle impact. c Captured particle trajectory. d Measured ATP particle velocity. e and f illustrate the ATP and Al 2 O 3 particle velocities vary with air pressure, respectively Table 5) based on the principle of 2D interpolation was developed to achieve the profile on any cross-section of the simulated surface topography. The obtained result is presented in Fig. 5c. In addition, the roughness profiles of the experimental specimens are used to compare the corresponding cross-section profiles of the simulated surface topographies, which were measured by the PS50 profilometer. Figure 5d shows the roughness profile of the AA7075 specimen before the AAJM process. Figure 6 shows the comparison of the cross-section profiles of the simulation and experimental surface topography treated by Al 2 O 3 and ATP particles after 10,000 particle impacts, respectively. Here, the compressive air pressure is 0.3MP, the impact velocity of the particle is 141m/s, and the impact angles are 15° and 90°, respectively. From Fig. 6, it can be found that compared with the cross-section profile of workpiece before AAJM process (Fig. 5d), the maximum height of the profiles after AAJM process is obviously enhanced. Furthermore, when the impact angle is 15° (Fig. 6a, b, and e), although the maximum height (about 70μm) of the experimental waviness profile of the Al 2 O 3 particle impact is larger than that of the simulated one (about 40μm), the skewness and kurtosis of the experimental profile (which are 0.38 and 1.36, respectively) are approximately the same as those of the simulated one (which are − 0.02 and 1.56, respectively). While for the ATP particle impact (Fig. 6c, d, and f), the simulation results (the maximum  (1),xx (2),N); y1=linspace(yy(1),yy (2),N); z1=interp2(X,Y,Z,x1,y1 ); if abs((yy(2)-yy(1))/(xx(2)-xx(1)))<tan(89*2*pi/360) Fig Fig. 6d and f), respectively) are more consistent with the experimental ones (the maximum height, skewness, and kurtosis of the experimental waviness profile are about 16μm, − 0.49, and − 0.15, respectively). Furthermore, when the impact angle was increased to 90° (Fig. 6g-l), no matter the ATP particle impact or the Al 2 O 3 particle impact, the FEM simulation results are largely in agreement with the experimental ones. It demonstrates that the SPH-FEM method can be largely used to simulate the erosion behaviors of the aluminum alloy during AAJM by using the super-hard Al 2 O 3 particle and the medium-hard ATP particle. In addition, due to the simulated and experimental kurtosis values of the waviness profiles of the Al 2 O 3 particle impact being close to 1.5 and the corresponding values of the ATP particle impact being less than 0.6, a Gaussian distribution of peak height is more obvious than that of ATP particle impact. The reason for this is as follows: owing to the higher hardness and density of the Al 2 O 3 particle, the impact force of the Al 2 O 3 particle is larger than that of the ATP particle (according to Eqs. (6) and (7)). As a result, the relatively larger impact depth and more material removal tend to generate by the Al 2 O 3 particle impact; thus, the higher maximum height of the waviness profile coupled with an obvious Gaussian distribution of peak height is inclined to produce. In addition, for this reason, it can be inferred that for the AAJM of the aluminum alloy, a relatively better surface quality tends to achieve by using ATP particle.

Effect of particle hardness on the surface erosion
To further verify the above results, the experimental and simulation material removal rates (MRRs) have been calculated and are shown in Fig. 7. As shown in Fig. 7, three significant observations can be found. First, it is clear that the MRR increased linearly with the compressive air pressure. The second observation is that the simulation results are consistent with the experimental ones: for the Al 2 O 3 particle impact, the error between them is less than 1% (Fig. 7a); while for the ATP particle impact (Fig. 7b), the error between them is less than 15%. The above results demonstrate once again that the SPH-FEM method is suitable for simulating the AAJM process of aluminum alloy. The third observation is that the effect of the particle hardness on the erosion behavior is more significant than that of the compressive air pressure. For instance, when the compressive air pressure is increased about 2 times (from 0.3 to 0.5MPa), the MRRs of the Al 2 O 3 and ATP particle impacts are enhanced about 2 times, respectively. However, when the abrasive material is changed from Al 2 O 3 to ATP (the hardness of the particle increases about 2 times), the MRR of Al 2 O 3 particle impact is about 200 times higher than that of ATP particle impact.

Effect of impact angle on the surface erosion
Meanwhile, besides the compressive air pressure and the particle hardness, Papini and Takaffoli [8,9] pointed out that the erosion behavior of the impact target during AAJM process is also affected by the impact angle. They performed the simulation and experimental investigations to explore the mechanism of the effect of the impact angle on erosion behavior, and the rules of the effect were obtained. However, the above research is aimed at the super-hard particle impact; for the medium-hard particle impact, few FEM researches have been done. Therefore, in this study, the SPH-FEM coupling simulation was carried out to explore the effect of the impact angle on the surface erosion of the aluminum alloy, and the obtained results are shown in Fig. 8. Combined with Fig. 8 and Fig. 6c, d, f, i, j, and l, two remarkable observations can be found. First, the maximum height of the waviness profile is enhanced with the increment of the impact angle. The second observation is that, when the impact angle is 70°, the simulation cross-section profile is most similar to the experimental one (Fig. 8l), coupled with almost the same values of the experimental and simulation kurtosis and skewness (Fig. 8m). To verify the above observations, the corresponding microstructures of the impacted surfaces and the MRRs at different impact angles were examined. Here, due to the detailed investigation of the surface microstructures of aluminum alloy impacted by ATP particles having been reported in our previous work [15], in this study, the microstructures of the corresponding impacted surfaces were analyzed briefly and are shown in Fig. 9.
From Fig. 9, it can be seen that when the impact angle is in the range of 15°-60°, the rolled grooves (Fig. 9a) on the initial surface disappeared and are replaced by the relatively slender deep grooves with material accumulation on the edges (Fig. 9b-e); furthermore, with the impact angle increasing, the length of the groove is relatively decreased. It indicates that particle ploughing is the main reason for the surface formation in such conditions; furthermore, with the impact angle increasing, the ploughing effect of the particle would be possibly enhanced. Meantime, when the impact angle is enhanced to 70°, the morphological feature of the impact surface is transformed again; that is, the previous ploughing-grooves disappeared, leaving a relatively large plastic deformation coupled with obvious material displacement (Fig. 9f). It implies that, in this condition, particle cutting plays an important role in surface formation [15]. However, when the impact angle is further increased to 90°, the localized material deformations evidenced by some indented dents with highly deformed "lips" around edges are obvious (Fig. 9g), implying that particle indentation is the main reason for the surface formation.
To help understand the above results, the corresponding simulations of the material removal by single-particle impact have been done and are shown in Figs. 10 and 11. As shown in Fig. 10, considering the particle backward flipping [21], the fundamental simulation of the ATP particle cutting process was performed. In this case, due to the chip formation by the particle cutting being well simulated by the SPH particle disconnecting and removing, the erosion behavior of aluminum alloy can be predicted accurately by the SPH-FEM simulation. While for the particle ploughing or sliding process (shown in Fig. 11), the situation is relatively complicated. For example, if the particle impact (Fig. 11a) caused the failure behavior of the SPH particle (marked No. 1 with the red dotted circle), it would be removed from the surface (Fig. 11b, c). However, if the failure behavior of the SPH particle did not occur by the particle impact, it would undergo elastic-plastic deformation and remain on the surface (marked No. 2 with the blue dotted oval circle). Due to the uncertainties of the SPH particle disconnection and removal in such conditions, the corresponding MRRs at different impact angles have been further investigated and are shown in Fig. 12. Figure 12 shows the simulation and experimental MRRs at different impact angles by using the ATP and Al 2 O 3 abrasives, respectively. From Fig. 12, two important observations can be found. First, during the AAJM process, there possibly exists an optimal impact angle, where the surface material would be removed by the particle cutting and the maximum material removal rate can be obtained; furthermore, the surface erosion behavior can be accurately predicted by the numerical simulation. For example, for the AAJM by using ATP abrasive, when the impact angle is enhanced from 15° to 70°, the MRRs are increased (which can be also referenced by the green region in Fig. 6c and Fig. 8a, d, g, and j). Especially for the impact angle of 70°, the maximum values of the simulation and experimental MRRs have been achieved and are almost the same. When the impact angle further increases to 90°, the MRR is shown to be relatively decreased (it can be referenced by Fig. 6i). While for the Al 2 O 3 particle impact, the same trend can be also found at the impact angle of 30°. The second observation is that the effect of the particle hardness on the optimal impact angle is obvious; furthermore, with the hardness of the particle increasing, the optimal impact angle would be decreased. Thus, based on the above analysis results, the effect of impact angle on the surface erosion can be explained that, during the ATP-AAJM of aluminum alloy, when the surface formation was dominated by the particle ploughing process (impact angle is less than 70°), with the impact angle increasing the normal force of the particle would be increased (evidenced by Fig. 2a), causing the ploughing effect of the particle to be enhanced; thus, the erosion behavior would be increased leading to the increment of the maximum height of the waviness profile. However, when the surface formation was dominated by the particle cutting (impact angle is up to 70°), due to the improvement of the normal force of the particle causing surface material to be removed by chip formation, a further increment of the erosion can be achieved, causing a further enhancement of the maximum height of the waviness profile. In addition, when the impact angle is close to 90°, due to the surface formation being dominated by the particle indentation, the localized material deformation would be enhanced with the impact angle increasing; consequently, the maximum height of the waviness profile can continue to increase. However, in such conditions, due to surface material removal being hard to occur, the erosion behavior of the aluminum alloy would be decreased, resulting in a decrement of the MRRs. Moreover, since the high particle density and hardness largely cause large particle impact force (Eqs. (6) and (7)), although Al 2 O 3 particle impact was performed at a relatively small impact angle, the impact force can be still large enough to cause the material removed by chip formation. Thus, the optimal impact angle for the Al 2 O 3 particle impact is lower than that of the ATP particle impact.

Conclusions
For the AAJM of aluminum alloy, although the SPH-FEM method is useful for the prediction of the erosion behavior of the impact target, it only aims at the superhard particle impact. For the AAJM process by using the medium-hard ATP particle, it is required to develop the FEM-SPH method to predict such erosion behavior. This research work, therefore, proposed a novel linear elastic material coupled with failure criteria to define ATP particles; then, a multi-particle impact model based on the FEM-SPH method was further developed. It is shown that this FEM-SPH model helps estimate the erosion behavior of aluminum alloy accurately. The following conclusions presented here are: 1. The SPH-FEM method can be used to simulate the erosion behaviors of aluminum alloy impacted by not only the super-hard Al 2 O 3 particle but also the medium-hard ATP particle, and the simulation results are fundamentally in agreement with the experimental results; the error between them is less than 15%. 2. Although the surface erosion behavior is affected by not only the compressive air pressure but also the particle hardness, the effect of the particle hardness on the erosion behavior is more significant. Thus, for the AAJM of aluminum alloy, better surface quality tends to be achieved by using medium-hard ATP particle than that by using super-hard Al 2 O 3 particle. 3. During the AAJM of aluminum alloy, there possibly exists an optimal impact angle, where the surface material can be removed by chip formation causing the maximum material removal rate and the surface erosion behavior to be accurately predicted by the numerical simulation; furthermore, with the particle hardness increasing, the optimal impact angle would be reduced accordingly.

Data availability
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

Code availability Not applicable
Authors' contributions Yansong Zhu: project administration, funding acquisition, writing-review and editing. Xiang Yang: investigation, data curation. Shen Wang: visualization. Wen Zhuang Lu: mechanism analysis. Tae Jo Ko: data analysis.

Fig. 12
Simulation and experimental material removal rates versus impact angle for a ATP particle impact and b Al 2 O 3 particle impact 10  (a) ( b) APT particle impact A l2O3 particle impact