Simulating a high-speed abrasive particle impacting on a tensile block using SPH-FEM

In recent years, more and more researches have been carried out on the erosion mechanism of abrasive particles on target materials in the abrasive waterjet cutting process. However, the effect of material property factors on the target erosion damage is rarely studied systematically. In this work, a 3D smoothed particle hydrodynamics-finite elements model is established for the simulation. The controlled variable method is used to study how each material property factor affects the erosion process of single abrasive particle and to find out the key material property factors of Al6061-T6 and Ti-6Al-4V. The influence of the interaction of the key material property factors on the target erosion damage is further evaluated using the orthogonal test method.


Introduction
Erosion wear caused by solid particles is a phenomenon commonly happening in the industries of mechanical processing, mining, and constructing. Based on the idea that utilizes solid particles erosion instead of eliminating it, one of the promising and innovative machining technologies called abrasive waterjet (AWJ) cutting is proposed by Hashish in early 1980s [1]. As the only cold high-energy beam processing technology in the world today, AWJ cutting has numerous benefits like no thermal distortion, high level of mobility, narrow kerf width, negligible heat effected zone, and small cutting forces [2][3][4]. Almost all engineering materials can be cut by AWJ, especially the advanced difficult-to-cut materials such as ceramics [5]. Due to the specific characteristics of AWJ cutting, this processing technology has been used extensively in various industries in the past two decades.
In the AWJ cutting process, the erosion process is induced by the impact of the abrasive particles accelerated by highspeed water, which is regarded as a carrier. When the plastic deformation caused by the cumulative erosion of the abrasive particles on ductile target material reaches the tensile limit of the material, the removal of the workpiece material by the abrasive particles is realized.
A lot of work has been done to reveal the mechanism of hard particles eroding the target. Based on the theoretical analysis and experiments, two empirical erosion analytical mathematical models for ductile materials are developed by Finnie [6] and Bitter [7,8]. It is found that the erosion rate is related to the impact angle, impact velocity and material property factors such as the flow stress, and the density of the target. However, these models only coincide with the low-impact angle experiments of ductile materials [9]. Hutchings finds that the critical strain and dynamic hardness are also the main erosion parameters that influence the erosion rate [10]. Several experiments are conducted by Sundararajan, who also finds that particle shape and specific heat capacity of the target have an effect on erosion of ductile materials [11,12]. Li WY et al. use finite element method to simulate the impacts of ultrahigh velocity micro-particles on steels and find that the material removal process is associated with the magnitude of the plastic strain, the flow stress and thermal conditions [13]. Mieszala M et al. report that the erosion rate of the target material is positively correlated with the grain size and negatively correlated with the hardness. And it is related to the Young's modulus of the target material [14]. Barabas SA et al. notice that the machinability is influenced by other factors like the hardness and the microstructure of the removed material [15]. However, although the relationship between the erosion mechanism of ductile materials and certain material property factors has been studied by a lot of theoretical analysis and experiments, it is still not clear.
In the AWJ cutting process, the cutting quality and accuracy are directly affected by the material property factor of the workpiece material. However, the existing cutting model evades the crucial point and the combined effect of all material property factors is replaced by a single parameter named as machinability [16]. Therefore, in the current study, by studying how each material property factor affects the erosion process of single abrasive particle, the key material property factors could be found out for the ductile materials (Al6061-T6 and Ti-6Al-4V) by establishing a 3D smoothed particle hydrodynamics-finite elements model (SPH-FEM) [17,18] in ANSYS/LS-DYNA software. The study of the mechanisms of single abrasive eroding the target is of critical importance in the advancement of the AWJ cutting technique.

Smoothed particle hydrodynamics-finite elements model
The distortion and tangling of Lagrange mesh of the impacted area, which can cause calculation errors, are caused by the great kinetic energy of a high-speed abrasive particle during large deformation. A large amount of time is spent on the calculation process using SPH method, and smooth particles are not easily added with boundary conditions. As a result, SPH-FEM mothed is proposed to take advantages of each mothed in this paper. The coupling method is used to study the dynamic behavior when abrasive particles impact the target, which contributes to a better understanding erosion mechanism of the target materials.

Material model and equation of state
The Johnson-Cook (J-C) constitutive model [19] with Gruneisen equation of state (EOS) [20] is employed to describe the response of the target materials to large deformation and high strain rate. In the J-C model, the functional relationship among the von Mises flow stress σ f for ductile materials, yield stress constant A, strain hardening constant B, effective plastic strain ε P , strain rateε * , and homologous temperature T * is given by where n, c, and m are input constants with arbitrary unit; T * = (T − T room )/(T melt − T room ); T, T room , and T melt are the workpiece temperature, the room temperature, and the melting point of the workpiece, respectively.
In our work, the J-C shear failure model is adopted to study the damage behavior of the target materials. The failure criterion is described as the fracture occurs when the value of the damage parameter D ¼ ∑Δε P =ε f reaches 1. The strain ε f at fracture is defined as where D 1~D5 are failure parameters and σ * is the ratio of pressure divided by effective stress. The Gruneisen EOS is widely used to characterize the relation between the pressure of ductile materials and the rate of change of density under dynamic loading [21]. According to the different metal crystal structures, common metal crystal structures can be classified into three categories: bodycentered cubic lattice, face-centered cubic lattice, and closepacked hexagonal lattice. The body-centered cubic lattice and the face-centered cubic lattice belong to the cubic crystal system. Since the plasticity of the body-centered cubic lattice is lower than that of the face-centered cubic lattice, the material with face-centered cubic is studied in this work. In order to fully understand the response of different metal crystal structures to a high-speed abrasive particle impacting, the material with close-packed hexagonal is also studied in our work. The material constants of Al6061-T6 and Ti-6Al-4V are listed in Table 1.

SPH-FEM and boundary conditions
In the AWJ cutting process, the abrasive particles are entrained into the mixing chamber under the Venturi effect and are broken in the cutting head. After passing through the focusing tube, the average size of the abrasive particles is reduced by 35% [5]. In this work, the 80-mesh garnet abrasive is used as the impacting particle. In order to improve the computing efficiency, the deformation and fragmentation of the impacting particle during the erosion process was not considered. The actual shape of the particle is not a sphere in fact. The particle shape is irregular. However, the content of this paper is the influence of target material properties on particle erosion process, and the difference of particle shape has no influence on this study. Therefore, in order to simplify the particle shape, it is set as a sphere. The impacting particle is modeled using rigid solid-164 tetrahedral elements, with properties listed in Table 2.
The target material size should not be too large under the condition of meeting the calculation accuracy and efficiency. In this study, the target material is modeled as a rectangular block with the size of 400 * 400 * 200 μm 3 . Before solving the calculation, the "NON_REFLECTING" condition is applied to the ground and the side of the target material to simulate unlimited boundary. As mentioned above, the impacting area of the target material is modeled by SPH particles, and the rest is discretized by using the eight-node brick hexahedral elements with one integration point to reduce integration and hourglass control. The impacting area of the target is set to be no less than 1.5 d A by 1.5 d A [25]. And it also should not be too large, since the computing efficiency will be too low. Therefore, the SPH particles are distributed in a box with the dimension of 200 * 200 * 100 μm 3 near the impact location.
Considering the calculation time and accuracy, an appropriate spacing of SPH particles needs to be determined. The von Mises stress under different SPH particle spacing (10μm, 5μm, 4μm, and 3μm) are tested, and the difference between the results of von Mises stress with the spacing of SPH particles under 5μm and 3μm is only 0.6%. Thus, the spacing between the SPH particles is determined as 5μm to compromise between the computation efficiency and relatively high accuracy.
The erosion damage mechanism of ductile materials includes extrusion, cutting, melting, and deformation wear. In the AWJ cutting process, various erosion damage mechanisms are caused by various impact angles of abrasive particles. The plastic deformation of material surface caused by extrusion when the AWJ just touches the target material and the abrasive particles impact the target material at 90-degree impact angle is studied in this paper. Since the eroding process of single abrasive particle at the impact angle of 90°is axisymmetric, 1/ 4 geometric model is used to simplify the calculation in this numerical simulation, as illustrated in Fig. 1. Totally, 8,000 SPH particles and 8,166 solid elements are generated by the erosion model.
SPH and FEM are coupled by the master-slave contact algorithm, where the SPH particles are defined as slave segment and the finite elements are set as master segment. The contact coupling algorithm is shown in Fig. 2. In this model, the SPH and FE models are coupled together through the contact type of "TIED_NODES_TO_SURFACE_OFFSET" in LS-DYNA. Similarly, the "EROSING_NODES_TO_SURFACE" is established between the SPH section and the impacting particle. Since only 1/4 model is analyzed, the XZ and YZ planes are set as symmetry planes and constraints are imposed on the symmetry planes. And the unconstrained boundary conditions are   prescribed on the erosion surface of the target material. The "NON_REFLECTING" condition is adopted on the ground and the side of the target material to eliminate the reflection of stress waves, reduce its computational domain, and simulate unlimited boundary [26,28].
In the AWJ cutting process, the three-phase flow of water, abrasive, and air in the cutting head is a fully developed turbulent flow. The laws of conservation of mass, energy, and momentum must be followed in the flow process. According to the method proposed by Liu D et al. [29], the impact velocity of single abrasive particle can be estimated as where ν ra is the real impact velocity of single particle (m/s); ν a is the theoretical impact velocity of single particle (m/s); ν o is the velocity of waterjet (m/s); and Ρ i is the fluid pressure at the orifice inlet (MP a ). The working pressure range of abrasive water jet cutting process is between 320MP a and 380MP a . And the incident velocity of abrasive particle is calculated according to formula (3). Finally, 630m/s is determined as the incident velocity of the impacting particle.

Process simulation
In the AWJ cutting process, the erosion damage of ductile materials is affected by many factors including material performance parameters, process parameters, and equipment parameters. Among them, the material performance parameters are regarded as the most relevant and essential factor to determine the target damage. Based on the works of Finnie and Bitter et al., whether flow stress, density, specific heat, shear modulus, and Young's modulus have effects on erosion damage of the target material from the perspective of material performance parameters is investigated in this work. The effect of the interaction of the material property factors on the erosion damage of the material is further studied.

Simulation analysis of single factor
In the LS-DYNA software, some simulation experiments were done through the controlled variable method to study the influence of single material performance factor on the plastic strain of the target. Before the simulation analysis, it can be seen from the flow stress formula (1) that when the constant m increases, the flow stress increases. And the relationship between the flow stress and the target erosion damage can be obtained indirectly. Therefore, the constant m is chosen as an independent variable to explore the effect of flow stress on the erosion damage of target. In this work, one of our research contents is which properties of the target material affect the abrasive erosion process. Therefore, based on the real material properties, the material properties are changed to use the control variable method to study whether a material property affects the abrasive erosion process in this paper. The variation range of each material property parameter is detailed in Tables 3 and 4.
The erosion process of the high-speed abrasive particle is shown in Figs. 3 and 4. The initial contact state of the spherical abrasive particle impacting on the target is point contact, so the high-speed abrasive particle can make the surface of the target bear large extrusion force instantaneously. At the same time, part of the kinetic energy of the abrasive particle is  [26,27] transformed into the heat energy of the target, resulting in the softening of the surface of the target. Therefore, the abrasive particles can continue to act on the workpiece until the impact force of the high-speed abrasive particles is not enough to continue to penetrate into the target and rebound. As for the impact process, the crater is induced by the surface of the target which is extruded by the abrasive particle. Since Test 5 in Table 3 and Test 5 in Table 4 are the real material properties of the target materials, it is more representative to show the simulation process of these two sets of tests. Figure 3 shows the simulation process of Experiment 5 in Table 3.
Similarly, Fig. 4 shows the simulation process of Experiment 5 in Table 4. The simulation results are shown in Figs. 5 and 6. It is obvious from Figs. 5 and 6 that the plastic strain of the target is related to the flow stress, mass density, specific heat, and shear modulus, while with nothing to do with Young's modulus. On the whole, the maximum effective plastic strain of the target decreases with the increase of flow stress, mass density and specific heat, and increases with the increase of shear modulus.

Simulation analysis of multifactor
In order to reduce the complexity of experiments, the orthogonal experiment method is adopted in this paper to study the influence of the interaction of flow stress, mass density, specific heat, and shear modulus on the plastic strain of the target. In order to deeply explore the interrelationship among the material property factors, five experimental levels of each material performance factor are set. And the orthogonal experimental table, L 25 (5 6 ), is used as an experimental design scheme. Relevant experimental parameters and levels of Al6061-T6 are listed in Table 5, and the experimental results of orthogonal test are shown in Table 6.
In Table 6, A is the constant m, B is the mass density, C is the specific heat, D is the shear modulus, E and F are the blank variables, and S is the maximum effective plastic strain. Analysis of variance is carried out for the orthogonal experimental data. The analysis results are shown in Table 7.
Similarly, the analysis results of variance for Ti-6Al-4V are listed in Table 8.
According to the results, mass density has a significant influence on the erosion damage of the target materials, while the constant m, specific heat and shear modulus have a small influence.

Conclusion and verification
In this work, a new method called SPH-FEM is used to investigate the influence of the key material property factors of the target materials on the AWJ cutting. The advantage of the SPH-FEM is that it is easy to measure the plastic strain caused by the interaction of these factors under a high-speed abrasive particle impacting on the target materials, which is difficult to determine by the experiment. Utilizing this method, the relationship between the maximum effective plastic strain of the target material and flow stress, mass density, specific heat, shear modulus, and Young's modulus is analyzed. In this work, the target materials for experimental research must meet the requirements that one material property is different, and the other properties are exactly the same. Unfortunately, due to the particularity of materials, no material manufacturer can provide the required Fig. 3 Erosion process of Al6061-T6 by abrasive particle with the evolution of crater and strain contour materials. If we find a suitable target material in the future, we will do further experimental research to verify the simulation results. However, comparison against the theoretical analysis results and erosion models by Finnie and Sundararajan are reported. It is found that our simulation results highly match the research results obtained by Finnie and Sundararajan, which is helpful to better understand the fundamental mechanisms of erosion in the AWJ cutting process.
In summary, (1) This work proposed a new idea for studying the influence of material factors on the AWJ cutting performance; (2) Our simulation results indicated that the plastic strain of the target is related to the flow stress, the mass density, the specific heat, and the shear modulus, but not to the Young's modulus; (3) Mass density has a significant influence on the erosion damage of the target materials, whereas the specific heat, and the shear modulus have negligible influence.   Fig. 6 Relationship between maximum effective plastic strain and material property parameters of Ti-6Al-4V