Modeling method and experimental study on the random distribution of abrasive particles in the jet cutting process

The distribution of abrasive particles in fluids is an important research topic in the study of abrasive waterjet cutting processes. However, it is impossible to obtain the accurate distribution law and influencing factors by performing only experiments; therefore, it is necessary to study abrasive waterjet cutting processes with the help of numerical models. The existing numerical models usually adopt the form of artificial settings for the distribution of abrasive particles in fluid. This method cannot accurately simulate the random distribution of particles. In this paper, the random algorithm method is used to simulate the impact azimuth and the random distribution of abrasive particles in water. The smoothed particle hydrodynamics (SPH) method is used to simulate the distribution of abrasive particles and the process of jet impingement. The influence of the particle distribution on the simulation results is studied. Comparisons show that the dent formed by the jet impinging on the target with random abrasive particles is similar to the dent from the actual cut, and the contour distribution of the dent is more uniform than that of the cut. The simulation results obtained by the SPH method are accurate.


Introduction
An abrasive waterjet is a solid-liquid two-phase medium jet formed by mixing abrasive particles and high-speed water. The main difference between the abrasive waterjet and pure waterjet is that the cutting pressure of the abrasive waterjet is less than that of the pure waterjet, but the cutting efficiency is improved, and there is no thermal reaction or chemical reaction during the cutting process. This technology is widely used in the fields of oil exploitation, mechanical processing, mining, and tunnel excavation. The influence of abrasive particles is the most critical and complex factor affecting the abrasive waterjet cutting process. Clarifying the distribution and movement of particles in water is of great importance for understanding the cutting process. ElTobgy et al. [1] established a finite element analysis model of a micro abrasive erosion material, and the model was solved to simulate the erosion process of multiple particles. However, during modeling, the abrasive geometry was simplified to a spherical shape, and the sharp edges of the abrasive particles were ignored. Babets [2] used the volume of fluid (VOF) model to take the fluid medium as the continuous phase and the abrasive particles as the dispersed phase, and the Lagrangian method was used to track the particles to obtain the variation rules of parameters such as the velocity, turbulent kinetic energy, density, and particle trajectory. Ahmed et al. [3] considered both abrasive particles and water to be continuous phases while analyzing them using the Euler method, and determined the optimal incident angle of abrasive particles. Feng et al. [4] used the SPH method to numerically model the abrasive waterjet cutting process. In the model, abrasive particles were parameterized and expressed in the form of SPH particles. The model can simulate the movement and impact of particles in water. Liang et al. [5] established a multiphase flow model to quantitatively describe the flow characteristics of abrasive waterjets, clarify the actual performance of abrasive waterjets in the contact area, and establish a new fuzzy prediction system to predict the turbulence characteristics according to the processing parameters. Dong et al. [6] established a model of the impact of angular particles on different forms of materials and solved it with the SPH method. The impact morphology of different forms of angular particles was simulated, but the influencing factors of the water jet were not considered.
The following limitations exist in the research of abrasive waterjet cutting processes: (a) the influence of an abrasive particle shape is ignored in some models, so the effect of abrasive particles on the cutting performance cannot be studied. (b) The geometric modeling of abrasive particles is more difficult in conventional SPH numerical modeling than other modeling approaches, so the average density method of the SPH method is often used to simulate the impact process of abrasive waterjets. (c) Waterjet and abrasive particles are modeled separately, without considering the influence of coupling factors. Based on the reasons described above, the SPH algorithm is used to parameterize the model of abrasive particles, the random algorithm is introduced to simulate the random distribution of particles in fluid, and the influence of the particle distribution on the calculation model in the modeling process is studied. In this paper, when modeling the abrasive waterjet cutting process, the combination of the SPH modeling method and random algorithm cannot only highlight the advantages of the SPH method in determining the impact but also improve the simulation of the abrasive particle distribution state and improve the accuracy of the simulation. The other sections of this paper are summarized as follows. In the second section, the basic principle of SPH and the SPH modeling method of the abrasive waterjet cutting model are introduced. In the third section, random algorithm modeling of abrasive particles is carried out, and the numerical simulation results without the random algorithm and with the random algorithm are compared and analyzed. In the fourth section, the abrasive waterjet cutting process is experimentally studied, and the simulation results are compared with the experimental data to verify the effectiveness and accuracy of the simulation method. The fifth part summarizes the research conclusion.
2 Basic theory of the SPH algorithm and modeling process of abrasive water jets

Basic theory of the SPH method
The basic principles of SPH can be summarized as follows [7]: (a) first, the SPH algorithm is a meshless algorithm that uses a series of arbitrarily distributed particles instead of meshing when describing the problem domain. (b) In the SPH algorithm, the field function needs to be approximately solved, and the integral representation method is used during this process, which has the advantage of improving the stability of the solution because the integral method has a smooth effect and is equivalent to the weak form equation. The above process, called kernel approximation, is the first step in the construction of the SPH equation. (c) The second key step is called particle approximation. To replace the integral expression of the field function and its derivatives, the corresponding values of the neighboring particles in a certain region are added and summed. This process can produce a discrete system matrix, which can effectively improve the calculation efficiency. (d) The SPH algorithm is adaptive. Particle approximation is needed during every calculation step, and the particles in the process of particle approximation are the particles distributed in the current computational domain. (e) The Lagrangian characteristics of the SPH algorithm applies the particle approximation method to the field function correlation terms so that only the discretized ordinary differential equations related to time can be obtained. (f) The dynamic properties of the SPH algorithm are obtained using the display integral to calculate the ordinary differential equations, and the fastest time integration can be obtained. At the same time, the change in the field variables of all the particles with time can be obtained.

Kernel approximation process [8]
In the SPH method, the integral representation of function f(x) is shown in Eq. (1): where f is a function of x, Ω is the integral volume of x, and (x − x � ) is the Dirac function. The process of replacing the Dirac function with the smooth function W (x − x', H) is called the kernel approximation. Therefore, Eq. (1) can be written as follows: where the support domain and problem domain of smooth function W should satisfy the following conditions: a) The normalized condition is shown in Eq. (3). b) When the value of smooth length h tends to 0, the smooth function is approximately equal to the Dirac function. (1) The International Journal of Advanced Manufacturing Technology (2022) 121:3173-3191 3174 c) For the compact support condition, the following equation can be obtained when |x − x| ≤ h.
where k is the correlation constant with a smooth function at X.
The standard expression of the kernel approximation is obtained as given by Eq. (6): where m j and j are the mass and density of particle j, respectively.

Particle approximation process [9]
In another important calculation process (particle approximation), the particle in Eq. (6) can be approximated as follows: where m j is the mass of SPH particle j, and j is the density of particle j; therefore, m j j is the volume associated with particle j or node j.

SPH equation of the jet and target
The Navier-Stokes equations of fluid media can be expressed by the conservation of mass: The momentum conservation equation is expressed as follows: where is the density of the fluid, t is the time, v is the velocity component, x is the position vector of the fluid, f is the external force, and and are Cartesian coordinates x and y, respectively. The repeated superscripts in the above equations conform to Einstein's summation rule. In Eq. (11), is the total stress tensor component, which consists: where is the Kronecker tensor (if = , = 1; otherwise, = 0). In a Newtonian fluid, the viscous shear stress is directly proportional to the shear strain rate and is correlated by the proportional coefficient μ (viscosity coefficient).
where In the SPH approximation [10], the governing Navier-Stokes equations for the conservation of mass and momentum can be expressed as: Artificial viscosity is added to the equation to represent the effect of a high-speed water jet, so Eq. (14) can be transformed into the following equation [11]: where Π ij is the artificial viscosity. In this method, it is used to reduce the nonphysical oscillation in the numerical calculation process around impact area.
When the waterjet cuts the material, it is in a compression state, and the change in density produces pressure. The equation of state is in the Mie-Gruneisen form [12]: The material parameters and coefficients of the equation of state of the jet are shown in Table 1.
The SPH equation of the solid governing equation is [13]: In the numerical calculation of solid medium, the stress can be expressed as: The Jaumann rate is replaced by the stress deviator. The stress deviator can be expressed as: where G is the shear modulus and ̇ is the strain rate tensor, which can be expressed as: is the rate of the rotation tensor, and Two typical materials are selected as cutting materials, namely, ductile material and brittle material.
A Ductile material

Constitutive model
The solid target is modeled as titanium alloy (TC4), and the constitutive model of the ductile material is the Johnson and Cook constitutive model [14]: where y is the stress yield of the material; A, B, C, m, and n are the relevant parameters of the material; p is the equivalent plastic strain; ̇ * e is the dimensionless plastic strain rate; T * is the homologous temperature; p is the plastic strain; 0 is the reference strain rate; T is the actual temperature; T r is room temperature; and T m is the melting point of the material.

Failure model
The Johnson-Cook fracture criterion [15] is adopted for the failure fracture behavior of the target material. The expression of loss effect change f in the model is: where D 1 , D 2 , D 3 , D 4 , and D 5 are material performance parameters, m is the average value of the main stress, and eq is the equivalent stress.

State equation
To calculate the isotropic pressure of the material, the Mie-Gruneisen equation is used to simulate the effect of the impact between high-speed abrasive particles and ductile materials. The equation of state can be expressed as: where 0 is the reference density (kg/m 3 ); Γ 0 is the constant of the Gruneisen equation; S a is the linear Hugoniot coefficient; and E is the internal energy per unit mass. The specific parameters of TC4 are shown in Table 2. B Brittle material

Brittle material model
For brittle materials, Johnson-Holmquist (JH2) model is adopted in this paper. The failure of the model is defined as: where * is the equivalent strength (MPa), A is the cohesion strength (MPa), D is the damage factor, B is the pressure hardening coefficient, P * is the dimensionless HEL pressure, P * = P∕P HEL , N is the stress hardening index, C is the strain rate coefficient, and ̇ * is equal effect rate of change (s −1 ). The damage factor D represents the damage amount of the target material. When D = 0, it means that the target material is not damaged; when D = 1, it means that the target material begins to fracture. The expression is: where Δ p is equal plastic deviatoric strain increment, Δ p is the increment of plastic volumetric strain, f p is the equivalent plastic deviatoric strain, and f p is the plastic volumetric strain. The damage relationship of JH 2 model is shown in Fig. 1. In the figure, T represents the maximum tensile strength, D 1 and D 2 are damage constants, and E fmin represents the minimum plastic strain before fracture. The strength limit surface curve of JH 2 model is shown in Fig. 2. Where f c represents the static uniaxial compressive strength, while parameter A represents the normalized cohesive strength, B represents the normalized pressure hardening coefficient, C represents the strain rate coefficient, * represents the normalized equivalent strength, and P is the static pressure.
In the JH2 model, when impacted by abrasive particles, the target material is in compression first, and then in the limit state, it is stretched and peeled off. In the analysis, the two processes are analyzed independently. The compression stage is divided into elastic zone, plastic transition zone, and tight compression material zone, while in the process of tensile spalling, there is only damage factor elastic zone.
As shown in Fig. 3, according to the pressure and volume relationship curve of JH2 model, the unit volume change of brittle material in the elastic compression stage is linear with the pressure it is subjected to, that is: where K is the bulk modulus (MPa), is the volumetric strain, = ∕ 0 − 1 , is the density of brittle material (kg/m3), and 0 indicates the initial density of the brittle material.
In the plastic transition stage, some of the brittle materials have produced irreparable damage, that is, 0.014 D 5 3.87 where P c is the pressure at the impact point (MPa), P l is the pressure at the compaction point of the material (MPa), c is the volumetric strain at the impact point, and l represents the volumetric strain at the compaction point of the material. The time of the material compaction stage is very short. After the compaction stage, the micro cracks begin to expand, and then form staggered and complex cracks, resulting in the spalling of the material. In this process, the expression of pressure is: where K 1 , K 2 , and K 3 are the material constants.
The cutting material used in this paper is glass, and the target material parameters are shown in Table 3.

Equation of state for brittle materials
The equation of state expression of brittle materials can be expressed as: where K 1 , K 2 , and K 3 are the material related constant, which are measured by the plate impact test, = ∕ 0 − 1 . and 0 are the current density and initial density, respectively.

Abrasive particle modeling process
In this paper, the factors affecting the random distribution law of particles are mainly considered in the study of jet impact. In the modeling process, the random shape shown in Fig. 4 is taken as an example to simplify it into angular particles and circular particles.
In the previous study, it was found that the erosion efficiency of angular particles was much higher than that of circular particles. After sampling the abrasive particles (as shown in Fig. 5), it is found that the edges and corners of each particle are obvious, mostly angular impact.
In this paper, because the impact form of particles is not the main research object, on the premise of ensuring the accuracy, in order to improve the efficiency of impact simulation, equilateral triangular particles are approximately used for simulation.   The velocity of abrasive particles can be divided into the translation speed and rotation speed.
where i is the ith abrasive particle, V i and W i are the translation speed and rotation speed, X i is the position vector of the center of mass, M i is the mass of the ith particle, I i is the moment of inertia, F i is the total force acting on the ith particle, and T i is the total torque.
In this model, each particle is associated with a number of discrete SPH nodes, which carry the field variables including the mass and velocity. The equation of motion of the nodes is: where j is the jth node and X i is the position vector of the node.

Contact and fracture
In the process of impact, the jet is in contact with the target, and the target material is broken and fractured after being impacted. The contact and fixation between SPH particles is through the application of binding force. The radius of the support region of the SPH particles is twice as long as the smooth length. When the distance between two particles is twice the smooth length, the SPH particles will generate the corresponding contact force.
The contact potential function is defined as: where NCONT is the number of particles belonging to different objects in the support domain of particle i, Δd avg is the average distance of two SPH particles, and K and N are user-defined parameters. The value of K is related to the material properties and impact velocity. After being extruded, the material will fracture when it exceeds the compression limit. The SPH method can predict the contact between the jet and the target and accurately describe the "contact" and "stripping" of the two components. During abrasive waterjet (AWJ) cutting, the target material will be removed by a high-speed AWJ. Therefore, the possibility of fracture must be considered in the SPH model. To simulate the fracture of the target material, the smooth length of the SPH particles at the fracture is reduced. This reduces the interaction between the fractured particles and other particles. However, reducing the smoothing length of invalid particles will lead to a decrease in the time step and an increase in the calculation time. Therefore, when a critical value is reached, the particle is considered invalid, and fracture is assumed to occur at the position related to the particle. Then, the particles on the site are removed from the SPH calculation. However, to ensure the conservation of mass and momentum, the mass and momentum of invalid particles are still retained.

Artificial viscous force
In impact process, the artificial viscous force (artificial viscosity) is introduced into the solution method so that the algorithm can simulate the collision problem and eliminate any nonphysical oscillation. When the law of conservation of mass, momentum, and energy is applied to the shock wave surface, kinetic energy is converted into heat energy. In the process of calculation, the energy is transformed into viscous dissipation. In this article, it can be expressed as [16]: where Π and Π are the standard constants, and Π = 1 and Π = 1.
In the above equation, C is the velocity of sound and V is the velocity vector related to the particle.

Time integration mode
The leap-frogging (LF) algorithm [17], which has the advantages of low memory requirements and high efficiency, is used in this article. When the calculation of a time step is completed, the density, velocity, and internal energy advance a half step forward from the initial state, and the particle position advances by one step.
where Δt is the time step, is the density of the particles, V is the velocity vector of particle motion, x is the position vector of the particles, and the upper corner of each parameter represents the time step.
In the overall calculation, to ensure that the parameters such as the density, velocity, and internal energy of the particles can remain consistent with the position of the particles, the values of these parameters will advance half a time step forward at the beginning of each operation step to obtain the value of the integer step length.
where n represents the time step.

Modeling of an abrasive particle distribution based on a stochastic algorithm
Taking the modeling process of regular triangular abrasive particles as an example, a random distribution model of the angular particles is established in this paper. The characteristic parameters of abrasive particles include the particle shape and size, the incident angle and impact velocity, the reflection angle, the reflection velocity, and the reflection angular velocity. As shown in Fig. 6, the incident velocity of the regular triangle particle is v i , the angle between the incident velocity and the horizontal axis is the incident angle α i , the reflection velocity of the abrasive particles after rebound is v r , the angle between the reflection velocity and horizontal direction is reflection angle α r , the reflection angular velocity is ꞷ r , the height of the equilateral triangle is h, and the contact angle of the angular particles is A. The length of the target material is l and the height is b.
In the actual abrasive water jet cutting process, the abrasive particles are randomly distributed in the water. A random algorithm is used to generate the abrasive particle distribution in the water, which replicates the working conditions. The initial angle of the abrasive particles is also random. Therefore, two random algorithms are used in this process. According to the initial water column width d w , the distance between the water column and target l w is the initial coordinate, as shown in Fig. 7. To simplify the distribution of abrasive particles during modeling, the particles in the jet are distributed in layers, such as the particle layers shown in the figure. The distribution of the particles in each particle layer is random. The number of layers of particles depends on the initial input value (such as the size of the abrasive particles).
The maximum number of particles n max can be obtained according to the geometric parameters of the abrasive particles and the jet size parameters. As shown in Fig. 8, the single particle layer spacing is l o . When l o = 3 4 h , the number of particle layers is the highest.
The number of abrasive particles in each particle layer of the model is the same, assuming that the number of abrasive particles in each particle layer is N, the number of N abrasive particles from the left boundary to the right boundary is n 1 , n 2 , n 3 , …, n N , and the center point of the abrasive particles is located in the centerline of the particle layer. When the abrasive particles are randomly generated, the center coordinates of the abrasive particles are generated first. As shown in Fig. 8, the centerline of the particle layer is represented as the interval of [0,1], and the interval is divided into J (1), J (2), J (3), …, J(n).
According to the flow of the random algorithm, let s = , s = 0 , s = 1 , and I(s) = [ s , s ] , and a value a r is randomly obtained from the input centerline sequence. Then, a subinterval of I(s) is generated.
where P j = ∑ j k=1 p k , p k is the probability distribution, and P 0 = 0 . If J(i) belongs to the generated subdomain I(sa r ) , the distance between the generated center point and the boundary of the left and right sides of the jet should be determined. For regular triangular abrasive particles, if Δx 0 ≥ 2 3 h , the initial point can be generated, and its coordinates (x 1 , y 1 ) can be stored. The above process is repeated until the Nth central point (x N , y N ) is generated, and the distance between the two points is determined.
where x N is the abscissa of the center point of the Nth abrasive particle and (N − 1) the abscissa of the (N − 1)th abrasive particle. Since the center of the abrasive particle in each particle layer is located on the centerline, the influence of the ordinate can be ignored. To ensure that there is no interference between the particles in the layer of particles, Δx ≥ 4 3 h. When generating the coordinates of the geometric points of abrasive particles, as shown in Fig. 9, assume that the coordinates of center point A of the abrasive particles are (x 0 , y 0 ), and the impact angle of the particles is θ. According to the parameters previously assumed, A is the center point of the abrasive particles, and its coordinates are (x 0 , y 0 ); then, the coordinates of the impact vertex of the equilateral triangle are (x 1 , y 1 ). Therefore, For vertex a 2 , Therefore, Therefore, the coordinates of the other two vertices of the triangle abrasive particles are a 2 (x 2 , y 2 ) and a 3 (x 3 , y 3 ), namely, a 2 (x 1 − ( (66) input parameters include the abrasive particle shape, jet boundary and abrasive particle size Whether the central particle interferes with the jet boundary and other particles is analyzed initial particle position, azimuth and impact angle where the nodes adjacent to k + 1 are k and k + 2, and ⇀ x k+2 = ( x k+2 , y k+2 ) and ⇀ x k = ( x k , y k ). Regular triangular particles can be triangulated when solving the moment of inertia of angular particles. After the triangulation of the diagonal particles, the coordinate of the center of mass is where A n is the total area of angular particles and ⇀ x nc is the centroid coordinate of the subtriangle after subdivision.
Therefore, the mass and moment of inertia of the angular abrasive particles are as follows: where I z is the moment of inertia of the nth dissector triangle around its center of mass, m n is the mass of the nth subtriangle, and the three sides of the subtriangle are a n , b n , and c n . In a water jet column, the number of abrasive particles is also related to the initial abrasive concentration. In the initial setting, if the concentration of abrasive particles in the abrasive jet is set, the influence of the abrasive concentration on the generation of abrasive particles should be considered.
In the simulation of abrasive water jet cutting, the volume flow ratio of the abrasive particles to that of water is φ m .
where V p is the volume of abrasive particles distributed between the initial position of the jet and the target material and V j is the volume of the jet.
V pe is the volume of regular triangle particles, and it is expressed as the area in the 2D plane: where h is the height of an abrasive particle in the shape of an equilateral triangle, and a is the side length of an abrasive particle in the shape of a square. Therefore, the volume flow ratio of the abrasive particles to that of water is: where N all is the total number of abrasive particles in the whole jet column.
The number of abrasive particles in the whole jet column can be determined based on the volume flow ratio of the abrasive particles and water. In the initial setting, if the concentration of abrasive particles needs to be considered, the two-dimensional abrasive water jet cutting model can be ensured to be the same as the actual cutting scenario. The flow chart of the random algorithm for the whole abrasive particle generation is shown in Fig. 11.
Simulations, numerical results, and discussion

Simulation calculation
Taking two-dimensional triangular abrasive particles as an example, it is assumed that the abrasive particles are distributed along the centerline and the jet velocity ( v j ) is 170 m/s. The cutting times of (a), (b), (c), and (d) in Fig. 12 are 0.2 s, 0.4 s, 1 s, and 1.6 s, respectively. The corresponding cutting depths at different times are 0.15 mm, 0.32 mm, 1.20 mm, and 2.00 mm. The target material is a ductile material. When the jet with a single row of abrasive particles impinges on the extended target, the notch in the center of the slit is obvious at the initial position, and the notch on both sides of the slit is shallow. With the deepening of the slit, the gap between the center and both sides increases. When the jet completely penetrates the target material, the slit presents a "V" shape. It is assumed that the abrasive particles are asymmetrically distributed along the centerline; the jet velocity v j is 170 m/s; the cutting times are 0.2 s, 0.4 s, 1 s, and 1.6 s; and the corresponding cutting depth at different times is 0.07 mm, 0.45 mm, 1.31 mm, and 4.00 mm, respectively. The simulation results at different times are shown in Fig. 13. At the initial time, the notch is shallow, and the surface of the notch is uneven. When the target material is completely penetrated, the shape of the notch is similar to that of the single row of particles, but the slope is smaller than that of the single row of particles.
According to the actual volume ratio of abrasive particles, the abrasive particle distribution is introduced into the random algorithm, and the setting of jet parameters is the same as the former. The time detection points are also selected as 0.2 s, 0.4 s, 1 s, and 1.6 s, and the simulation results are shown in Fig. 14. Compared with the first two groups, the cutting marks are evenly distributed. At the initial time, the cutting marks are relatively flat, and the cutting depth has little difference between the cutting center and both sides. When the jet completely penetrates the target material, the slit formed presents a "V" shape, and the taper is less than that of single row particles and asymmetric particles.
According to the actual volume ratio of abrasive particles, the abrasive particles are evenly arranged in the center of the jet, as shown in Fig. 15, and the time detection points are 0.2 s, 0.4 s, 1 s, and 1.6 s. Compared with the simulation results using the random algorithm, the flatness of the dent at the initial time is higher than that of the surface formed by the jet with randomly distributed abrasives. The taper of the formed slit is smaller than that of the jet with a random distribution.
When cutting glass, the irregular shape of the cutting mark is caused by the breakage of the cutting target. The concave shape of the cutting process is obtained through simulation, as shown in Fig. 16.

Comparison of the experimental results
The TC4 sample is cut with a jet velocity of 170 m/s as the initial condition. The jet cutting test bench is shown in Fig. 17, and the whole cutting dynamic process is captured by a high-speed camera. The comparison diagram of simulation results is shown in Fig. 18; the slit shapes formed by the jet cutting with randomly distributed particles and uniformly   Figure 19 shows the shape of the cutting seam obtained after the actual cutting and the contour obtained by the point tracing method. The actual cutting thickness is 30 mm. After reducing the size of the cutting seam by ten times, the curve in Fig. 19 is obtained by comparing with the simulation results.
The results show that the slit shape formed by the jet with random abrasive particles is consistent with the actual cutting shape. A comparison of the dents formed by the jet on the surface at the initial time is shown in Fig. 20. The left side of the figure shows the dent formed on the material surface at the initial time of jet impact. A comparison between time dent and simulation results is also shown in Fig. 20. During the simulation, the extrusion accumulation of materials first occurs on both sides of the cutting area, and then a dent is formed, which is essentially consistent with the actual cutting process. Irregular cut marks will be produced in the cutting process of glass materials, and the shape of the cut is difficult to describe. Therefore, three cut-off sections are selected to trace the cut-off points of the glass. The three sections are the section with the largest broken area (Sect. 1), the section with the smallest broken area (Sect. 2), and the random section (Sect. 3), as shown in Fig. 21.
The area of Sect. 3 is between the maximum crushing area and the minimum crushing area. Therefore, the contour of Sect. 3 is selected and compared with the simulation results, as shown in Fig. 22. Through comparison, it is found that the fitting degree between the simulation results and the experimental results is very high.

Conclusion
This paper mainly studies the influence of the random distribution of abrasive particles on the numerical simulation of jet cutting. The SPH method is used in the modeling, and the random algorithm approach is incorporated to simulate the distribution of abrasive particles in water. Based on the results of numerical simulation and experimental study, the following conclusions are obtained: 1. The influence of abrasive particle distribution on cutting was studied by using the SPH model. The numerical results show that the slot formed by the jet with random abrasive particles has a small taper and high parallelism. 2. By comparing the experimental and simulation results of cutting ductile material (TC4) and brittle material (glass), it is found that the simulation results of cutting target material with jet containing randomly distributed abrasive particles are in good agreement with the actual slit shape. The model can well reflect the process of cutting the target material, and the random distribution model is suitably accurate in the study of the abrasive water jet cutting process.
Author contribution Not applicable.
Funding This research was funded by Shandong Provincial Natural Science Foundation, China, grant number: ZR2021QA071 and Qingdao applied research project funding.
Availability of data and material All data generated or analyzed during this study are included in this published article.

Code availability
The code used or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations
Ethics approval Not applicable.
Consent to participate Written informed consent was obtained from all authors.

Competing interests
The authors declare no competing interests.