An Improved Abrasive Flow Processing Method for Complex Geometric Surfaces of Titanium Alloy Artiﬁcial Joints

Featured Application: A new material removal theoretical model of abrasive flow is set up, and it can offer solution scheme of fluid-based precise processing for workpieces with irregular geometric structures. An improved abrasive flow processing method is proposed to provide direct technical supports for the manufacturing of titanium alloy artificial joints. Abstract: Precise processing for complex geometric surfaces of titanium alloy artiﬁcial joints has higher technical difﬁculties. This paper addresses the matter by proposing an improved abrasive ﬂow processing method. According to the micro-cutting principle, the processing mechanism on curvature surface of the titanium alloy workpiece by the abrasive ﬂow is analyzed. A new material removal model of abrasive ﬂow is proposed to reveal the processing regularities for complex geometric surfaces of titanium alloy artiﬁcial joints. Based on the model, in combination with the realizable k - ε turbulence model, the total force affecting on a wall region of constrained ﬂow passage is obtained to estimate the quantity of material removal. A multi-segment proﬁling constrained ﬂow passage is designed, and an optimized ﬂow passage scheme is provided. Numerical results show that the optimized ﬂow passage can improve the pressure/velocity proﬁle uniformities of abrasive particles; by the product of velocity and pressure, the cutting coefﬁcient for complex surface is obtained. A processing experimental platform is developed, and the processing experiment results indicate that the proposed material removal model can estimate the processing effects and removal regularities, and the size accuracy and surface quality of the titanium alloy surface are improved.


Introduction
Titanium alloy materials have been widely used in many engineering areas, especially for the artificial joints manufacturing, because of their high specific strength, good corrosion resistance and excellent biocompatibility [1][2][3][4][5][6]. However, the manufacturing technologies of titanium alloy artificial joints face two challenges. Firstly, titanium alloy has poor thermal conductivity and low modulus of elasticity, so the adhesion wear between finished surface and cutting tool reduces machining efficiency. Secondly, traditional cutting tools can hardly adapt the complex surfaces of artificial joints (shown in Figure 1), and easily conduct processing trails to cause negative influences of surface roughness and uniformity [7][8][9][10].
Related research works suggested that abrasive flow processing is an effective approach for the manufacturing of workpiece with complex geometric surfaces [11][12][13]. Based on the high viscoelasticity To address the technical problems of precision processing for titanium alloy artificial joints, this paper proposes an improved abrasive flow processing method for complex geometric surfaces. By the multi-segment profiling constraint modules, a circular constrained flow passage that can cover the machined curvature surface is constructed. Consequently, the machined surface becomes a part of the wall flow passage. If the abrasive flow is under the full-developed turbulence state, the microcutting effects are conducted by the disorder movement of abrasive particles, and the processing chippings will be carried off by the abrasive flow.
Based on the contacting kinematics regularities between particles and machined surface, this paper mainly studies the tangential velocity distribution of particles in abrasive flow, and the particle pressure to machined surface in the cutting process. For the processing mechanism of abrasive flow, the normal grinding force, particle movement and cutting depth coefficient are researched, and a new material removal model of abrasive flow is set up. Based on the above model, in combination with the realizable k-ε turbulence model, the total force affecting on a wall region of constrained flow passage is obtained to estimate the quantity of material removal, and to provide technical supports for the regulation of abrasive flow processing.

Multi-Segment Profiling Constrained Flow Passage
In this study, the abrasive flow consists of distilled water, electrostatic dispersant and silicon carbide particles. It is a kind of processing medium with low abrasive particle fraction and high motion velocity, and the abrasive particle fraction is generally less than 15%. Abrasive flow is a typical two-phase incompressible fluid, in which the distilled water is the carrier of abrasive particles, and the electrostatic dispersant can address the even particle distribution of abrasive flow [34][35][36][37][38][39].
To adapt the complex surface of artificial joint, a multi-segment profiling constrained flow passage is designed, as shown in Figure 2. The machined workpiece is a simplified artificial knee joint. There are seven segments of constrained modules covering the curvature surface of the workpiece. Accordingly, a circular constrained flow passage is formed, in which the machined surface is the lower wall of the flow passage. Firstly, the abrasive flow is driven by a diaphragm pump with the required inlet conditions. Then, the abrasive flow enters the profiling constrained flow passage and forms full-developed turbulence state. The micro-force/cutting is performed by the random motion of abrasive particles driven by the turbulence. Finally, it flows out with the processing chippings from the outlet, and forms a processing cycle. To address the technical problems of precision processing for titanium alloy artificial joints, this paper proposes an improved abrasive flow processing method for complex geometric surfaces. By the multi-segment profiling constraint modules, a circular constrained flow passage that can cover the machined curvature surface is constructed. Consequently, the machined surface becomes a part of the wall flow passage. If the abrasive flow is under the full-developed turbulence state, the micro-cutting effects are conducted by the disorder movement of abrasive particles, and the processing chippings will be carried off by the abrasive flow.
Based on the contacting kinematics regularities between particles and machined surface, this paper mainly studies the tangential velocity distribution of particles in abrasive flow, and the particle pressure to machined surface in the cutting process. For the processing mechanism of abrasive flow, the normal grinding force, particle movement and cutting depth coefficient are researched, and a new material removal model of abrasive flow is set up. Based on the above model, in combination with the realizable k-ε turbulence model, the total force affecting on a wall region of constrained flow passage is obtained to estimate the quantity of material removal, and to provide technical supports for the regulation of abrasive flow processing.

Multi-Segment Profiling Constrained Flow Passage
In this study, the abrasive flow consists of distilled water, electrostatic dispersant and silicon carbide particles. It is a kind of processing medium with low abrasive particle fraction and high motion velocity, and the abrasive particle fraction is generally less than 15%. Abrasive flow is a typical two-phase incompressible fluid, in which the distilled water is the carrier of abrasive particles, and the electrostatic dispersant can address the even particle distribution of abrasive flow [34][35][36][37][38][39].
To adapt the complex surface of artificial joint, a multi-segment profiling constrained flow passage is designed, as shown in Figure 2. The machined workpiece is a simplified artificial knee joint. There are seven segments of constrained modules covering the curvature surface of the workpiece. Accordingly, a circular constrained flow passage is formed, in which the machined surface is the lower wall of the flow passage. Firstly, the abrasive flow is driven by a diaphragm pump with the required inlet conditions. Then, the abrasive flow enters the profiling constrained flow passage and forms full-developed turbulence state. The micro-force/cutting is performed by the random motion of abrasive particles driven by the turbulence. Finally, it flows out with the processing chippings from the outlet, and forms a processing cycle.

Normal Grinding Force of Abrasive Particle
Since the particles have no certain geometric shape, a particle model is commonly set up with conicalness and pyramid. In this paper, the silicon carbide particle is taken as the abrasive particle, and is set as a regular hexahedron model [40][41][42].
As presented in Section 2, the abrasive flow enters the constrained flow passage and forms the fully-developed turbulent state. For the movement complexity of turbulence, we select a single abrasive particle in contact with complex surface as the study subject, in which the normal grinding force is a key parameter. Based on the above hypothesis, the material removal model of single particle can be described by Figure 3.

Normal Grinding Force of Abrasive Particle
Since the particles have no certain geometric shape, a particle model is commonly set up with conicalness and pyramid. In this paper, the silicon carbide particle is taken as the abrasive particle, and is set as a regular hexahedron model [40][41][42].
As presented in Section 2, the abrasive flow enters the constrained flow passage and forms the fully-developed turbulent state. For the movement complexity of turbulence, we select a single abrasive particle in contact with complex surface as the study subject, in which the normal grinding force is a key parameter. Based on the above hypothesis, the material removal model of single particle can be described by Figure 3. In this model, abrasive flow is turbulent so that the abrasive particles in the flow passage strike and cut the complex surface under the effect of normal force in vertical direction, where FN is the pressure of fluid driving particle. The cutting force of the single abrasive particle can be expressed by [43][44][45]: where cn is a constant of particle wearing, and determined by the particle shape and granulate hardness; p is the average normal stress of particles collide with surface; and Sjc is the projection of abrasive particle on the machined surface and can be described as where H is surface hardness of workpiece. Therefore, the indentation depth is determined by the normal force and the hardness of workpiece. The projection area is supposed to be an isosceles right triangle, and the waist length is given by According to micro-cutting wear mechanism [46][47][48], a small part of the abrasive particle is pressed into the machining surface by the normal force. Then, an abrasive particle is driven by the friction between the base fluid and abrasive particle. The furrow action is taken by abrasive particle In this model, abrasive flow is turbulent so that the abrasive particles in the flow passage strike and cut the complex surface under the effect of normal force in vertical direction, where F N is the pressure of fluid driving particle. The cutting force of the single abrasive particle can be expressed by [43][44][45]: where c n is a constant of particle wearing, and determined by the particle shape and granulate hardness; p is the average normal stress of particles collide with surface; and S jc is the projection of abrasive particle on the machined surface and can be described as where H is surface hardness of workpiece. Therefore, the indentation depth is determined by the normal force and the hardness of workpiece. The projection area is supposed to be an isosceles right triangle, and the waist length is given by

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According to micro-cutting wear mechanism [46][47][48], a small part of the abrasive particle is pressed into the machining surface by the normal force. Then, an abrasive particle is driven by the friction between the base fluid and abrasive particle. The furrow action is taken by abrasive particle on the machined surface, so the wear scars with trough shapes are appeared on the surface. For the turbulent abrasive flow processing, high accuracy finishing process is achieved by numerous irregular wear scars with trough shapes. In this research, the material of abrasive particle is silicon carbide, and the material of workpiece is titanium alloy. Because the hardness of silicon carbide is greater than the hardness of titanium alloy, the processing result is affected by the hardness of silicon carbide. If the slip distance of a particle is ds, the removal volume of different machined surface is described as: If the removal volume is maximum, the machined surface is with concave profile. Similarly, if the workpiece has a convex surface, the removal volume is minimum.

Abrasive Particle Movement in Abrasive Flow Processing
Abrasive flow is a wear particle group made up of many abrasive particles, so the average number of abrasive particles follows a certain law of mathematical statistics. This paper supposes that there is a random particle distribution in the constrained flow passage, as shown in Figure 4. on the machined surface, so the wear scars with trough shapes are appeared on the surface. For the turbulent abrasive flow processing, high accuracy finishing process is achieved by numerous irregular wear scars with trough shapes. In this research, the material of abrasive particle is silicon carbide, and the material of workpiece is titanium alloy. Because the hardness of silicon carbide is greater than the hardness of titanium alloy, the processing result is affected by the hardness of silicon carbide. If the slip distance of a particle is ds, the removal volume of different machined surface is described as: If the removal volume is maximum, the machined surface is with concave profile. Similarly, if the workpiece has a convex surface, the removal volume is minimum.

Abrasive Particle Movement in Abrasive Flow Processing
Abrasive flow is a wear particle group made up of many abrasive particles, so the average number of abrasive particles follows a certain law of mathematical statistics. This paper supposes that there is a random particle distribution in the constrained flow passage, as shown in Figure 4. This paper supposes that abrasive particles in the abrasive flow are evenly distributed, and the sizes of abrasive particles accord with the Gaussian distribution. In the volume whose floor space is a unit area with height dmax, the number of abrasive particles is described as: where ρmix is the density of the abrasive flow; ρsic is the density of the abrasive particle; dmax is the maximum diameter of abrasive particle; and da is average diameter of abrasive particle.
Since the geometric sizes and central locations of abrasive particles in abrasive flow are random, they are supposed to obey probability distribution. If the particle number for cutting and the particle number in boundary layer of workpiece surface are not equal, the abrasive particles contact surface in the fluid layer with height dmax. The share of abrasive particles participating in This paper supposes that abrasive particles in the abrasive flow are evenly distributed, and the sizes of abrasive particles accord with the Gaussian distribution. In the volume whose floor space is a unit area with height d max , the number of abrasive particles is described as: where ρ mix is the density of the abrasive flow; ρ sic is the density of the abrasive particle; d max is the maximum diameter of abrasive particle; and d a is average diameter of abrasive particle. Since the geometric sizes and central locations of abrasive particles in abrasive flow are random, they are supposed to obey probability distribution. If the particle number for cutting and the particle number in boundary layer of workpiece surface are not equal, the abrasive particles contact surface in the fluid layer with height d max . The share of abrasive particles participating in cutting is supposed as k a , so the number of affecting abrasive particles is N a [49,50], as expressed by Equation (6): where k a is a probability function related to the diameter and distribution of abrasive particles, and k a < 1.

Material Removal Model in Contact Length
The material removal depth in per contact length is defined as removal depth of workpiece surface contacting with abrasive particle, and known as linear removal rate [51][52][53]. As shown in Figure 5, an element of abrasive flow is cut out of boundary layer on the workpiece surface, in which dx, dy, and dz are the width, height and length, respectively. Because different positions have different curvatures, there is no a certain worn volume in the abrasive flow processing. cutting is supposed as ka, so the number of affecting abrasive particles is Na [49,50], as expressed by Equation (6): where ka is a probability function related to the diameter and distribution of abrasive particles, and ka < 1.

Material Removal Model in Contact Length
The material removal depth in per contact length is defined as removal depth of workpiece surface contacting with abrasive particle, and known as linear removal rate [51][52][53]. As shown in Figure 5, an element of abrasive flow is cut out of boundary layer on the workpiece surface, in which dx, dy, and dz are the width, height and length, respectively. Because different positions have different curvatures, there is no a certain worn volume in the abrasive flow processing. Cutting depth of abrasive particle is less than pressed depth on the workpiece surface during processing [54]. For the cutting process, the number of abrasive particles is very great, so the remove depth cannot simply be expressed by cutting depth. When many abrasive particles participate in cutting, the cutting depth will have an average distribution. In the model in Figure 4, dy is the cutting depth of abrasive flow. Consequently, referring to Equation (6), the number of abrasive particles in the area dxdz of micro-element is approximately dxdzNa. When a micro-element flows from z0 to z0 + ds, the normal pressure Fn of the abrasive flow on the abrasive particle is a certain value. According to micro-cutting wear mechanism, the material removal depth is described by: where Kv is the material removal coefficient. Therefore, the nonlinear material removal amount is proportional to the force of abrasive flow on the particle, and is related to the curvature of workpiece surface.

Cutting Depth Coefficient of Abrasive Flow Processing
According to the load analysis of single abrasive particle and the statistic distribution of abrasive particle in the flow passage, a material removal model of abrasive flow processing is proposed. For the material removal depth model of high viscoelastic abrasive flow, this paper defines a cutting depth coefficient for the curvature variations of different workpiece surface: Cutting depth of abrasive particle is less than pressed depth on the workpiece surface during processing [54]. For the cutting process, the number of abrasive particles is very great, so the remove depth cannot simply be expressed by cutting depth. When many abrasive particles participate in cutting, the cutting depth will have an average distribution. In the model in Figure 4, dy is the cutting depth of abrasive flow. Consequently, referring to Equation (6), the number of abrasive particles in the area dxdz of micro-element is approximately dxdzN a . When a micro-element flows from z 0 to z 0 + ds, the normal pressure F n of the abrasive flow on the abrasive particle is a certain value. According to micro-cutting wear mechanism, the material removal depth is described by: where K v is the material removal coefficient. Therefore, the nonlinear material removal amount is proportional to the force of abrasive flow on the particle, and is related to the curvature of workpiece surface.

Cutting Depth Coefficient of Abrasive Flow Processing
According to the load analysis of single abrasive particle and the statistic distribution of abrasive particle in the flow passage, a material removal model of abrasive flow processing is proposed. For the material removal depth model of high viscoelastic abrasive flow, this paper defines a cutting depth coefficient for the curvature variations of different workpiece surface: where the factors impacting on the coefficient of cutting depth are as follows: (1) surface hardness; (2) surface curvature; (3) surface roughness; (4) particle granularity; (5) particle mass fraction; (6) particle shape; and (7) turbulence intensity in flow passage. According to Equation (8), the equation of the material removal depth in unit time is: It is the material removal model of turbulent abrasive flow processing for complex surface. Apparently, Equation (9) is similar to Preston equation [55]. For the determination of the material removal coefficient of different machined surface, the distributions of the particle velocity and particle pressure to machined surface should be optimized to get a uniform flow field.

Turbulence Model
As presented in Section 1, abrasive flow is a typical two-phase engineering fluid media, and with the characteristics of high velocity, weak viscosity and turbulent state [56]. Based on the above characteristics of abrasive flow, the particle group is supposed as a pseudo-fluid interpenetrated with fluid.
Standard k-ε model is a commonly used classical turbulence model, but it will be distorted for the strongly swirling flow, curved wall flow and curving streamline flow. In the constrained flow passage, it could result in negative normal stresses. Different from the standard k-ε turbulence model, the expression for the turbulent viscidity in realizable k-ε model introduces rotation and curvature, so it will be able to adapt to avoid the negative normal stress [57][58][59]. Accordingly, a fluid dynamic model for abrasive flow processing is set up based on the realizable k-ε turbulence model. Based on the model, the profiles of pressure and velocity of abrasive flow on the workpiece surface are obtained, and the optimized processing conditions and constrained modules could be confirmed.
In the realizable k-ε turbulence model, the transport formulae for turbulent kinetic energy k and dissipation rating ε are described as: where ρ is fluid density; x i and x j are coordinate components; σ k and σ ε are turbulent Prandtl numbers of turbulent kinetic energy k and dissipation rate ε; G k is turbulent kinetic energy resulted from average velocity gradient; µ 1 is molecular viscosity coefficient; and µ t is turbulence viscosity coefficient.

Dynamic Characteristics of Abrasive Flow Processing
Based on the above hypothesis, the total force affecting on a wall region of constrained flow passage can be regarded as the dot product of the components of pressure and viscous force, and can be described as: where a, F p , and F v are the force vector, pressure vector and viscous vector, respectively [63].
Considering the actual physical parameters of abrasive flow field, the related force coefficients can be obtained for the target wall regions, by the reference values of velocity, pressure, viscosity and turbulent kinetic energy. Generally, the force coefficient can be expressed by 0.5ρv 2 S, in which ρ is the fluid density, v is the fluid velocity, and S is the contacting area. Accordingly, the force parameters of abrasive flow in constrained flow passage for different target wall regions can be acquired.
For a fluid micro-element of abrasive flow, the total moment vector of a wall contacting area can be expressed by the cross product of the components of pressure and viscous force, i.e., a vector generated by the moment center A to the force origin B, as shown in Figure 6, and the expression of total moment vector can be described as: where A is the moment center of a wall contacting area in flow passage; B is the force origin a fluid micro-element of abrasive flow; r AB is the fluid micro-element moment vector; and F p and F v are the force components of pressure and viscous force, respectively.

Dynamic Characteristics of Abrasive Flow Processing
Based on the above hypothesis, the total force affecting on a wall region of constrained flow passage can be regarded as the dot product of the components of pressure and viscous force, and can be described as: where a, Fp, and Fv are the force vector, pressure vector and viscous vector, respectively [63].
Considering the actual physical parameters of abrasive flow field, the related force coefficients can be obtained for the target wall regions, by the reference values of velocity, pressure, viscosity and turbulent kinetic energy. Generally, the force coefficient can be expressed by 0.5ρv 2 S, in which ρ is the fluid density, v is the fluid velocity, and S is the contacting area. Accordingly, the force parameters of abrasive flow in constrained flow passage for different target wall regions can be acquired.
For a fluid micro-element of abrasive flow, the total moment vector of a wall contacting area can be expressed by the cross product of the components of pressure and viscous force, i.e., a vector generated by the moment center A to the force origin B, as shown in Figure 6, and the expression of total moment vector can be described as: where A is the moment center of a wall contacting area in flow passage; B is the force origin a fluid micro-element of abrasive flow; rAB is the fluid micro-element moment vector; and Fp and Fv are the force components of pressure and viscous force, respectively. According to the above reduction, the moment vector direction can be determined by the right-hand rule. Similar to the force vector, by the actual physical parameters of abrasive flow field, the corresponding moment coefficients can be obtained for the target wall regions, by the reference values of velocity, pressure, viscosity and turbulent kinetic energy. Therefore, the moment coefficient can be expressed by 0.5ρv 2 SL, in which L is the fluid affecting length in constrained flow passage. Consequently, the total moment and related coefficients of abrasive flow in constrained flow passage for different target wall regions can be obtained.
By this computation method, according to fundamental physical parameters, force vector, moment vector and coefficients of abrasive flow field, all moments of the target wall region about different coordinate axes can be obtained, i.e., the dot product between a unit vector and the related axis direction. Subsequently, to restrict the rounding error, a reference pressure variable is adopted to normalize the pressure of abrasive flow micro-element, i.e., the pressure vector sum on the affecting interface: According to the above reduction, the moment vector direction can be determined by the right-hand rule. Similar to the force vector, by the actual physical parameters of abrasive flow field, the corresponding moment coefficients can be obtained for the target wall regions, by the reference values of velocity, pressure, viscosity and turbulent kinetic energy. Therefore, the moment coefficient can be expressed by 0.5ρv 2 SL, in which L is the fluid affecting length in constrained flow passage. Consequently, the total moment and related coefficients of abrasive flow in constrained flow passage for different target wall regions can be obtained.
By this computation method, according to fundamental physical parameters, force vector, moment vector and coefficients of abrasive flow field, all moments of the target wall region about different coordinate axes can be obtained, i.e., the dot product between a unit vector and the related axis direction. Subsequently, to restrict the rounding error, a reference pressure variable is adopted to normalize the pressure of abrasive flow micro-element, i.e., the pressure vector sum on the affecting interface: Sn (15) where n is the interface number of abrasive flow micro-element, S is the affecting area on the interface, andn is the unit normal to the interface. For the 2D flow field in constrained flow passage, the stress components of pressure and viscous force can be regarded as the line distribution, in which crossing point of the line and reference line is the pressure center point. Apparently, there is a zero moment on this point. Considering a 3D flow field, the stress components of pressure and viscous force can be described as the spatial force system in a fluid micro-element. In this hypothesis, the resultant moment is described as In the 3D flow flied system, we can define a reference plane for the target wall region, and make the two equations in Equations (16)-(18) be zero. Then, the crossing point of the axis and reference plane about the target wall region can be obtained. For a 2D instance, only Equation (18) should be considered by the user-specified reference line to determine the center of pressure.

Numerical Model and Grid Meshing
The profiling constrained flow passage for titanium alloy complex surface of artificial joint is composed of multi-segment constrained modules and workpiece surface, and the numerical model is shown as Figure 7. Firstly, a three-dimensional mechanical structure model is set up, and then it is imported into a finite element model. The radius of circular arc of upper surface from left to right are 10 mm, 32 mm, and 10 mm, respectively. The mesh dividing is completed and essential boundary condition is configured, in which the element is Tet/Hybrid, the type is TGrid, the interval size is 1, and the number of cells is 324,041. The mesh dividing schematic of the profiling constrained flow passage is shown in Figure 7b. For the essential boundary condition setting, the inlet is VELOCITY_INLET, the outlet is OUTFLOW, and other sides is WALL.
where n is the interface number of abrasive flow micro-element, S is the affecting area on the interface, and n is the unit normal to the interface. For the 2D flow field in constrained flow passage, the stress components of pressure and viscous force can be regarded as the line distribution, in which crossing point of the line and reference line is the pressure center point. Apparently, there is a zero moment on this point. Considering a 3D flow field, the stress components of pressure and viscous force can be described as the spatial force system in a fluid micro-element. In this hypothesis, the resultant moment is described as In the 3D flow flied system, we can define a reference plane for the target wall region, and make the two equations in Equations (16)-(18) be zero. Then, the crossing point of the axis and reference plane about the target wall region can be obtained. For a 2D instance, only Equation (18) should be considered by the user-specified reference line to determine the center of pressure.

Numerical Model and Grid Meshing
The profiling constrained flow passage for titanium alloy complex surface of artificial joint is composed of multi-segment constrained modules and workpiece surface, and the numerical model is shown as Figure 7. Firstly, a three-dimensional mechanical structure model is set up, and then it is imported into a finite element model. The radius of circular arc of upper surface from left to right are 10 mm, 32 mm, and 10 mm, respectively. The mesh dividing is completed and essential boundary condition is configured, in which the element is Tet/Hybrid, the type is TGrid, the interval size is 1, and the number of cells is 324,041. The mesh dividing schematic of the profiling constrained flow passage is shown in Figure 7b. For the essential boundary condition setting, the inlet is VELOCITY_INLET, the outlet is OUTFLOW, and other sides is WALL. This paper selects 3D single-precision steady implicitly pressure solver, and the solution is obtained by the SIMPLEC algorithm of pressure-linked equations. In this paper, the solid particle phase is silicon carbide particle, the density is 3170, the mean diameter of abrasive particle is 0.05 mm, and the volume fraction is 0.1; the fluid phase is distilled water. According to computed Reynolds number, the abrasive flow in the flow passage can form the turbulent state, and the turbulence intensity is 9.8%. Based on the features of flow passage, the solving conditions are configured for the Euler model and realizable k-ε turbulence model.  Table 1.  Velocities of particles in profiling constrained flow passage are shown in Figure 9. In Figure 9a, the X velocities in the L1 and L5 are close to 0. Figure 9b shows that the Y velocities in the L1 are This paper selects 3D single-precision steady implicitly pressure solver, and the solution is obtained by the SIMPLEC algorithm of pressure-linked equations. In this paper, the solid particle phase is silicon carbide particle, the density is 3170, the mean diameter of abrasive particle is 0.05 mm, and the volume fraction is 0.1; the fluid phase is distilled water. According to computed Reynolds number, the abrasive flow in the flow passage can form the turbulent state, and the turbulence intensity is 9.8%. Based on the features of flow passage, the solving conditions are configured for the Euler model and realizable k-ε turbulence model.  Table 1.  This paper selects 3D single-precision steady implicitly pressure solver, and the solution is obtained by the SIMPLEC algorithm of pressure-linked equations. In this paper, the solid particle phase is silicon carbide particle, the density is 3170, the mean diameter of abrasive particle is 0.05 mm, and the volume fraction is 0.1; the fluid phase is distilled water. According to computed Reynolds number, the abrasive flow in the flow passage can form the turbulent state, and the turbulence intensity is 9.8%. Based on the features of flow passage, the solving conditions are configured for the Euler model and realizable k-ε turbulence model.  Table 1.  Velocities of particles in profiling constrained flow passage are shown in Figure 9. In Figure 9a, the X velocities in the L1 and L5 are close to 0. Figure 9b shows that the Y velocities in the L1 are Velocities of particles in profiling constrained flow passage are shown in Figure 9. In Figure 9a, the X velocities in the L1 and L5 are close to 0. Figure 9b shows that the Y velocities in the L1 are obviously decreased, but the Y velocities in L5 are obviously increased. Because particles vertically enter L1 and leave L5, the X velocities in L1 and L5 are close to 0. Due to the particles moving parallel to the workpiece along with fluid after particles enter L1, the included angle between vectors of particles and vertical direction becomes larger, and the Y velocities in L1 are decreased. The velocity variation in L5 is similar. The X velocities of particles in L2, L3 and L4 are the same, but Y velocities of particles in L2, L3 and L4 are changed slowly. From the flow field profile results, we can conclude the following laws. (a) On the different machined surface, velocities direction of particles participating in cutting are changed. In the plane, the X velocities and Y velocities of particles are not change nearly. (b) On the curved surface, the amplitudes of X velocities and Y velocities are changed with the radius of the curved surface.  The X/Y velocities of particles can be obtained by CFD method, but the tangential velocity must be obtained by calculation for further study, as shown in Figure 10. The trend of the tangential velocity in L1 and L5 is similar to the Y velocity, and the maximum in L1 is bigger than in L5. In L2, the tangential velocity is gradually decreased, but, in L4, the tangential velocity is gradually decreased and the changing speed is smaller. In L3, the tangential velocity is gradually increased. On the curved surface, the bigger the radius of the curved surface is, the more quickly the velocity of the particle changes. The X/Y velocities of particles can be obtained by CFD method, but the tangential velocity must be obtained by calculation for further study, as shown in Figure 10. The trend of the tangential velocity in L1 and L5 is similar to the Y velocity, and the maximum in L1 is bigger than in L5. In L2, the tangential velocity is gradually decreased, but, in L4, the tangential velocity is gradually decreased and the changing speed is smaller. In L3, the tangential velocity is gradually increased. On the curved surface, the bigger the radius of the curved surface is, the more quickly the velocity of the particle changes. Based on the numerical simulation results of the tangential velocity, the following regularities can be acquired. Because of the linear loss, the tangential velocity is gradually decreased. On the impact face, the tangential velocity is gradually increased, but, on the non-impact face, the tangential velocity is gradually decreased, and the changing speed is much bigger than the changing speed caused by the linear loss.

Numerical Results Discussion
To obtain the affecting forces of particles on the workpiece, L1 and L5 are evenly divided into 14 planes, respectively. L3 is evenly divided into 12 planes. The X forces and Y forces of particles on the planes is obtained by the CFD method, and the total pressure force is shown in Figure 11. The forces of particles in L1 are obviously reduced; the forces of particles in L2 are almost as much as in L3; the forces of particles in L4 are much higher than in L2 and L3; the minimum forces of particles in L5 are on the right of L5 and they gradually become larger; the forces on the left of L1 are much higher than on the right of L5, and forces in L2 are much smaller than in L4; the force peaks in L1 and L5 and the low of L3 are almost equal. Because L2 and L4 are plane, the forces in L2 and L4 are bigger than in the peaks of L1, L5 and L3. Referring to the simulation results of pressure, the following laws can be obtained. The normal force of particle to the impact surface is higher than the non-impact surface. On the curved surface, Based on the numerical simulation results of the tangential velocity, the following regularities can be acquired. Because of the linear loss, the tangential velocity is gradually decreased. On the impact face, the tangential velocity is gradually increased, but, on the non-impact face, the tangential velocity is gradually decreased, and the changing speed is much bigger than the changing speed caused by the linear loss.
To obtain the affecting forces of particles on the workpiece, L1 and L5 are evenly divided into 14 planes, respectively. L3 is evenly divided into 12 planes. The X forces and Y forces of particles on the planes is obtained by the CFD method, and the total pressure force is shown in Figure 11. The forces of particles in L1 are obviously reduced; the forces of particles in L2 are almost as much as in L3; the forces of particles in L4 are much higher than in L2 and L3; the minimum forces of particles in L5 are on the right of L5 and they gradually become larger; the forces on the left of L1 are much higher than on the right of L5, and forces in L2 are much smaller than in L4; the force peaks in L1 and L5 and the low of L3 are almost equal. Because L2 and L4 are plane, the forces in L2 and L4 are bigger than in the peaks of L1, L5 and L3. Based on the numerical simulation results of the tangential velocity, the following regularities can be acquired. Because of the linear loss, the tangential velocity is gradually decreased. On the impact face, the tangential velocity is gradually increased, but, on the non-impact face, the tangential velocity is gradually decreased, and the changing speed is much bigger than the changing speed caused by the linear loss.
To obtain the affecting forces of particles on the workpiece, L1 and L5 are evenly divided into 14 planes, respectively. L3 is evenly divided into 12 planes. The X forces and Y forces of particles on the planes is obtained by the CFD method, and the total pressure force is shown in Figure 11. The forces of particles in L1 are obviously reduced; the forces of particles in L2 are almost as much as in L3; the forces of particles in L4 are much higher than in L2 and L3; the minimum forces of particles in L5 are on the right of L5 and they gradually become larger; the forces on the left of L1 are much higher than on the right of L5, and forces in L2 are much smaller than in L4; the force peaks in L1 and L5 and the low of L3 are almost equal. Because L2 and L4 are plane, the forces in L2 and L4 are bigger than in the peaks of L1, L5 and L3. Referring to the simulation results of pressure, the following laws can be obtained. The normal force of particle to the impact surface is higher than the non-impact surface. On the curved surface, Referring to the simulation results of pressure, the following laws can be obtained. The normal force of particle to the impact surface is higher than the non-impact surface. On the curved surface, the normal force of particle is changed, and the bigger the radius of the curved surface is, the more quickly the velocity of the particle changes. Based on Equation (9), a PV product graph is made to study the cutting depth coefficient K 0 , as shown in Figure 12. The products in L2, the left of L3, the right of L3 and L4 and the middle of L5 are higher than other areas. According to the numerical results of PV graph, the product variation regularities of force and velocity can be obtained. Consequently, an optimized constrained flow passage is designed, as shown in Figure 13.
Appl. Sci. 2018, 8, 1037 13 of 25 the normal force of particle is changed, and the bigger the radius of the curved surface is, the more quickly the velocity of the particle changes. Based on Equation (9), a PV product graph is made to study the cutting depth coefficient K0, as shown in Figure 12. The products in L2, the left of L3, the right of L3 and L4 and the middle of L5 are higher than other areas. According to the numerical results of PV graph, the product variation regularities of force and velocity can be obtained. Consequently, an optimized constrained flow passage is designed, as shown in Figure 13.  The velocities of particles in the optimized constrained flow passage are shown in Figure 14, and the tangential amplitude profile is shown in Figure 15. In Figure 14, we can find that the velocities in L1 and L5 of particles in the optimized constrained flow passage are similar to the velocity of particles in the original flow passage; the velocities in L2, L3 and L4 of the optimized constrained flow passage are more uniform than the velocities in the original flow passage. Appl. Sci. 2018, 8, 1037 13 of 25 the normal force of particle is changed, and the bigger the radius of the curved surface is, the more quickly the velocity of the particle changes. Based on Equation (9), a PV product graph is made to study the cutting depth coefficient K0, as shown in Figure 12. The products in L2, the left of L3, the right of L3 and L4 and the middle of L5 are higher than other areas. According to the numerical results of PV graph, the product variation regularities of force and velocity can be obtained. Consequently, an optimized constrained flow passage is designed, as shown in Figure 13.  The velocities of particles in the optimized constrained flow passage are shown in Figure 14, and the tangential amplitude profile is shown in Figure 15. In Figure 14, we can find that the velocities in L1 and L5 of particles in the optimized constrained flow passage are similar to the velocity of particles in the original flow passage; the velocities in L2, L3 and L4 of the optimized constrained flow passage are more uniform than the velocities in the original flow passage. The velocities of particles in the optimized constrained flow passage are shown in Figure 14, and the tangential amplitude profile is shown in Figure 15. In Figure 14     The forces of particles in the optimized flow passage are shown in Figure 16. The product profile of forced and velocity of particles in the optimized flow passage is shown in Figure 17. Figure 16 shows that the forces graph is more chaotic than the forces graph of the original flow passage, but Figure 17 shows that the PV graph is much smoother than the PV graph of the original flow passage. In consideration of precision finishing, the removal volumes in all area of workpiece should be almost equal, so the optimized flow passage is more applicable for precision finishing for complex surface.
Appl. Sci. 2018, 8, 1037 15 of 25 The forces of particles in the optimized flow passage are shown in Figure 16. The product profile of forced and velocity of particles in the optimized flow passage is shown in Figure 17. Figure 16 shows that the forces graph is more chaotic than the forces graph of the original flow passage, but Figure 17 shows that the PV graph is much smoother than the PV graph of the original flow passage. In consideration of precision finishing, the removal volumes in all area of workpiece should be almost equal, so the optimized flow passage is more applicable for precision finishing for complex surface.  According to the cutting depth coefficient and the curve of PV, the theoretical cutting coefficients of different position in the optimized constrained flow passage can be calculated. Because processing conditions of abrasive flow processing are very complicated, the accurate theoretical cutting coefficients are hard to calculate. Therefore, the theoretical cutting coefficients of plane (L2 and L4) are set as 1, and theoretical cutting coefficient curve of complex surface is shown in Figure 18. The forces of particles in the optimized flow passage are shown in Figure 16. The product profile of forced and velocity of particles in the optimized flow passage is shown in Figure 17. Figure 16 shows that the forces graph is more chaotic than the forces graph of the original flow passage, but Figure 17 shows that the PV graph is much smoother than the PV graph of the original flow passage. In consideration of precision finishing, the removal volumes in all area of workpiece should be almost equal, so the optimized flow passage is more applicable for precision finishing for complex surface.  According to the cutting depth coefficient and the curve of PV, the theoretical cutting coefficients of different position in the optimized constrained flow passage can be calculated. Because processing conditions of abrasive flow processing are very complicated, the accurate theoretical cutting coefficients are hard to calculate. Therefore, the theoretical cutting coefficients of plane (L2 and L4) are set as 1, and theoretical cutting coefficient curve of complex surface is shown in Figure 18. According to the cutting depth coefficient and the curve of PV, the theoretical cutting coefficients of different position in the optimized constrained flow passage can be calculated. Because processing conditions of abrasive flow processing are very complicated, the accurate theoretical cutting coefficients are hard to calculate. Therefore, the theoretical cutting coefficients of plane (L2 and L4) are set as 1, and theoretical cutting coefficient curve of complex surface is shown in Figure 18.  Figure 18 shows that cutting coefficient in L3 is biggest and the theoretical cutting coefficient curve is a line of binary linear equation. When the curvature radius of surface is increased, the cutting coefficient of the surface is closed to the plane. The cutting coefficient of the concave surface is higher than the plane. However, the cutting coefficient of the convex surface is smaller than the plane. The curvature radius of surface is bigger, and the morphology of the concave surface and the convex surface is more closed to the plane. Thus, the cutting coefficient of the concave surface and the convex surface is closed to the plane, with increasing of the curvature radius.

Processing Experiment Platform
To check the effectiveness of the improved abrasive flow processing method, a processing experiment platform for complex titanium alloy surfaces of artificial joints was developed, as shown in Figure 19.
(a) Figure 18. The theoretical cutting coefficient curve of complex surface. Figure 18 shows that cutting coefficient in L3 is biggest and the theoretical cutting coefficient curve is a line of binary linear equation. When the curvature radius of surface is increased, the cutting coefficient of the surface is closed to the plane. The cutting coefficient of the concave surface is higher than the plane. However, the cutting coefficient of the convex surface is smaller than the plane. The curvature radius of surface is bigger, and the morphology of the concave surface and the convex surface is more closed to the plane. Thus, the cutting coefficient of the concave surface and the convex surface is closed to the plane, with increasing of the curvature radius.

Processing Experiment Platform
To check the effectiveness of the improved abrasive flow processing method, a processing experiment platform for complex titanium alloy surfaces of artificial joints was developed, as shown in Figure 19.
Appl. Sci. 2018, 8, 1037 16 of 25 Figure 18. The theoretical cutting coefficient curve of complex surface. Figure 18 shows that cutting coefficient in L3 is biggest and the theoretical cutting coefficient curve is a line of binary linear equation. When the curvature radius of surface is increased, the cutting coefficient of the surface is closed to the plane. The cutting coefficient of the concave surface is higher than the plane. However, the cutting coefficient of the convex surface is smaller than the plane. The curvature radius of surface is bigger, and the morphology of the concave surface and the convex surface is more closed to the plane. Thus, the cutting coefficient of the concave surface and the convex surface is closed to the plane, with increasing of the curvature radius.

Processing Experiment Platform
To check the effectiveness of the improved abrasive flow processing method, a processing experiment platform for complex titanium alloy surfaces of artificial joints was developed, as shown in Figure 19. For the experimental platform, a TC4 titanium alloy workpiece (divided by L1, L2, L3, L4 and L5) was fixed to the base of constrained flow passage, and the multi-segment profiling constrained module was installed on the top of workpiece. The inlet was connected to a high-pressure pump, and the pump was placed in an abrasive flow storage tank. All these devices make up a circular processing system, and the procedures are described as follows: the abrasive flow is pumped from the storage tank by high-pressure pump; the abrasive flow enters the constrained flow passage under a certain high pressure and speed, and forms the fully-developed turbulent state; the workpiece is processed by unordered micro-cutting effects; and, after the abrasive flow finishes cutting work, it will flow back to storage tank with the processed chippings.
There is no nonlinear material removal coefficient in actual processing, so this experiment measures the mass difference of workpiece to validate the influence of curvature in the same conditions of processing equipment, abrasive flow and processing time. The related parameters of processing experiment are shown in Table 2.

Processing Experiments and Results Discussion
Because the surface appearance of all areas is different, Step 1 is used to eliminate the burrs by rough machining. The surface appearance is observed every 1 h, the material removal amount is measured every 0.5 h after Step 1, and the experimental effect is recorded. As shown in Figure 19, the workpiece has a curvature surface and is divided into five blocks. The workpiece surface quality is measured by the white-light interferometer. The material removal amount is measured by precision electronic scale, and the accuracy of scale is 0.1 mg. The comparative processing experiment results of surface roughness are shown in Figure 20. For the experimental platform, a TC4 titanium alloy workpiece (divided by L1, L2, L3, L4 and L5) was fixed to the base of constrained flow passage, and the multi-segment profiling constrained module was installed on the top of workpiece. The inlet was connected to a high-pressure pump, and the pump was placed in an abrasive flow storage tank. All these devices make up a circular processing system, and the procedures are described as follows: the abrasive flow is pumped from the storage tank by high-pressure pump; the abrasive flow enters the constrained flow passage under a certain high pressure and speed, and forms the fully-developed turbulent state; the workpiece is processed by unordered micro-cutting effects; and, after the abrasive flow finishes cutting work, it will flow back to storage tank with the processed chippings.
There is no nonlinear material removal coefficient in actual processing, so this experiment measures the mass difference of workpiece to validate the influence of curvature in the same conditions of processing equipment, abrasive flow and processing time. The related parameters of processing experiment are shown in Table 2.

Processing Experiments and Results Discussion
Because the surface appearance of all areas is different, Step 1 is used to eliminate the burrs by rough machining. The surface appearance is observed every 1 h, the material removal amount is measured every 0.5 h after Step 1, and the experimental effect is recorded. As shown in Figure 19, the workpiece has a curvature surface and is divided into five blocks. The workpiece surface quality is measured by the white-light interferometer. The material removal amount is measured by precision electronic scale, and the accuracy of scale is 0.1 mg. The comparative processing experiment results of surface roughness are shown in Figure 20.     In Figures 21-24, we can find that the scratches on the workpiece almost disappeared after 3 h, but pits were dug by particles after 3 h. L3 had the most pits, L1 and L5 had the fewest. It is obvious that the concave is more easily processed by particles than the convex.
The workpiece surface roughness is measured by a white-light interferometer, and the results are as shown in Figures 25 and 26. After 5 h, average surface roughness of whole workpiece decreased from 345.74 nm to 56.8 nm, and the surface roughness of best processing area even dropped to 49.46 nm. Many obvious scratches were on the workpiece before processing, while all scratches disappeared after processing. Meanwhile, some pits were made by particles, with even distribution. The material removal mass of five blocks are shown in Figure 27. Compared with results of five blocks, the material removal mass increased as the curvature increased. Over time, the material removal mass became smaller. To keep processing efficiency, the processing conditions should be adjusted in the processing. The cutting coefficient K0 of different positions is shown in Figure 28. In Figures 21-24, we can find that the scratches on the workpiece almost disappeared after 3 h, but pits were dug by particles after 3 h. L3 had the most pits, L1 and L5 had the fewest. It is obvious that the concave is more easily processed by particles than the convex.
The workpiece surface roughness is measured by a white-light interferometer, and the results are as shown in Figures 25 and 26. After 5 h, average surface roughness of whole workpiece decreased from 345.74 nm to 56.8 nm, and the surface roughness of best processing area even dropped to 49.46 nm. Many obvious scratches were on the workpiece before processing, while all scratches disappeared after processing. Meanwhile, some pits were made by particles, with even distribution. In Figures 21-24, we can find that the scratches on the workpiece almost disappeared after 3 h, but pits were dug by particles after 3 h. L3 had the most pits, L1 and L5 had the fewest. It is obvious that the concave is more easily processed by particles than the convex.
The workpiece surface roughness is measured by a white-light interferometer, and the results are as shown in Figures 25 and 26. After 5 h, average surface roughness of whole workpiece decreased from 345.74 nm to 56.8 nm, and the surface roughness of best processing area even dropped to 49.46 nm. Many obvious scratches were on the workpiece before processing, while all scratches disappeared after processing. Meanwhile, some pits were made by particles, with even distribution. The material removal mass of five blocks are shown in Figure 27. Compared with results of five blocks, the material removal mass increased as the curvature increased. Over time, the material removal mass became smaller. To keep processing efficiency, the processing conditions should be adjusted in the processing. The cutting coefficient K0 of different positions is shown in Figure 28. In Figures 21-24, we can find that the scratches on the workpiece almost disappeared after 3 h, but pits were dug by particles after 3 h. L3 had the most pits, L1 and L5 had the fewest. It is obvious that the concave is more easily processed by particles than the convex.
The workpiece surface roughness is measured by a white-light interferometer, and the results are as shown in Figures 25 and 26. After 5 h, average surface roughness of whole workpiece decreased from 345.74 nm to 56.8 nm, and the surface roughness of best processing area even dropped to 49.46 nm. Many obvious scratches were on the workpiece before processing, while all scratches disappeared after processing. Meanwhile, some pits were made by particles, with even distribution. The material removal mass of five blocks are shown in Figure 27. Compared with results of five blocks, the material removal mass increased as the curvature increased. Over time, the material removal mass became smaller. To keep processing efficiency, the processing conditions should be adjusted in the processing. The cutting coefficient K0 of different positions is shown in Figure 28. The material removal mass of five blocks are shown in Figure 27. Compared with results of five blocks, the material removal mass increased as the curvature increased. Over time, the material removal mass became smaller. To keep processing efficiency, the processing conditions should be adjusted in the processing. The cutting coefficient K 0 of different positions is shown in Figure 28.  The results have shown that the proposed abrasive flow processing method has a good practical processed effect. In addition, the result of this experiment shows that the coefficient of cutting depth of abrasive flow processing is changeable. As the curvature of workpiece increased, K0 increased. To keep processing efficiency, the hydraulic diameter of the flow passage should be adjusted according to the curvature of workpiece, which can reduce the processing effect differences on the positions with different curvatures.

Conclusions
To address the technical problems of precise processing for complex geometric surfaces of titanium alloy artificial joints, this paper sets up a material removal theoretical model of abrasive flow, and proposes an improved abrasive flow processing method. For the above research target, related research works have been performed, and the conclusions are as follows.
(1) A material removal model of abrasive flow is proposed to reveal the processing regularities for complex geometric surfaces of titanium alloy artificial joints. In combination with the realizable k-ε turbulence model, the total force affecting a wall region of constrained flow passage is obtained to estimate the quantity of material removal.  The results have shown that the proposed abrasive flow processing method has a good practical processed effect. In addition, the result of this experiment shows that the coefficient of cutting depth of abrasive flow processing is changeable. As the curvature of workpiece increased, K0 increased. To keep processing efficiency, the hydraulic diameter of the flow passage should be adjusted according to the curvature of workpiece, which can reduce the processing effect differences on the positions with different curvatures.

Conclusions
To address the technical problems of precise processing for complex geometric surfaces of titanium alloy artificial joints, this paper sets up a material removal theoretical model of abrasive flow, and proposes an improved abrasive flow processing method. For the above research target, related research works have been performed, and the conclusions are as follows.
(1) A material removal model of abrasive flow is proposed to reveal the processing regularities for complex geometric surfaces of titanium alloy artificial joints. In combination with the realizable k-ε turbulence model, the total force affecting a wall region of constrained flow passage is obtained to estimate the quantity of material removal. The results have shown that the proposed abrasive flow processing method has a good practical processed effect. In addition, the result of this experiment shows that the coefficient of cutting depth of abrasive flow processing is changeable. As the curvature of workpiece increased, K 0 increased. To keep processing efficiency, the hydraulic diameter of the flow passage should be adjusted according to the curvature of workpiece, which can reduce the processing effect differences on the positions with different curvatures.

Conclusions
To address the technical problems of precise processing for complex geometric surfaces of titanium alloy artificial joints, this paper sets up a material removal theoretical model of abrasive flow, and proposes an improved abrasive flow processing method. For the above research target, related research works have been performed, and the conclusions are as follows.
(1) A material removal model of abrasive flow is proposed to reveal the processing regularities for complex geometric surfaces of titanium alloy artificial joints. In combination with the realizable k-ε turbulence model, the total force affecting a wall region of constrained flow passage is obtained to estimate the quantity of material removal.
(2) To improve the processing effects, a multi-segment profiling constrained flow passage is designed, and an optimized flow passage scheme is provided according to the surface curvature variation.
(3) Numerical results show that the optimized flow passage can improve the pressure/velocity profile uniformities of abrasive particles. By the product of velocity and pressure, the cutting coefficient for complex surface is obtained. The cutting coefficient of the concave surface and the convex surface is closed to the plane, with increasing of the curvature radius.
(4) Processing experiment results show that the proposed material removal model is verified by the processing experiment results; the proposed abrasive flow processing method can perform an apparent processing result for the workpiece with complex surface; the scratches on the workpiece almost disappeared after 3 h, but pits were dug by particles after 3 h. As the curvature of workpiece increased, the cutting depth coefficient K 0 increased.
Subsequent research works will be conducted on the facets of processing efficiency optimization, surface biocompatibility of artificial joint and multi-phase abrasive flow processing method.