Investigating the Effect of Cutting Parameters of Ti–6Al–4V on Surface Roughness Based on a SPH Cutting Model

Weilong Niu; Rong Mo; Zhiyong Chang; Neng Wan

doi:10.3390/app9040654

Investigating the Effect of Cutting Parameters of Ti–6Al–4V on Surface Roughness Based on a SPH Cutting Model

Niu, Weilong;Mo, Rong;Chang, Zhiyong;Wan, Neng 2019-02-15 00:00:00 applied sciences Article Investigating the Effect of Cutting Parameters of Ti–6Al–4V on Surface Roughness Based on a SPH Cutting Model Weilong Niu , Rong Mo *, Zhiyong Chang and Neng Wan Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education, Northwestern Polytechnical University, 127 Youyi Road, Xi’an 710072, Shanxi, China; weilong0723@gmail.com (W.N.); Changzy@nwpu.edu.cn (Z.C.); wanneng@nwpu.edu.cn (N.W.) * Correspondence: morong@nwpu.edu.cn; Tel.: +86-029-88495354 Received: 16 January 2019; Accepted: 4 February 2019; Published: 14 February 2019 Abstract: This work establishes a 2D numerical model to simulate the cutting process of workpieces made of Ti–6Al–4V, by applying an improved Smoothed Particle Hydrodynamics algorithm together with a modiﬁed constitutive model based on the Johnson–Cook model known as Hyperbolic Tangent (TANH). The location information of the surface particles obtained by the SPH cutting model are used to evaluate the variation trend of surface roughness with different parameters. Parameters that affect the surface roughness are investigated in detail by using the Taguchi method and the SPH cutting model. The present work provides an efﬁcient and cost-effective approach to determine the optimal parameters for cutting processes for Ti–6AL–4V workpieces, through computer simulations in virtual environments, instead of expensive and time-consuming cutting experiments using actual workpieces. Keywords: smoothed particle hydrodynamics; hyperbolic tangent; taguchi method; Ti–6Al–4V; surface roughness 1. Introduction Ti–6AL–4V is one of most popular titanium alloys in many engineering applications. It is an important material for modern mechanical components and equipment, especially in biomedical and aerospace systems, since it not only has superb corrosion resistance, but also has a high strength to weight ratio. Machining of Ti–6AL–4V workpieces, however, is tricky, and it is very difﬁcult to produce the desired components with the required shape and high-quality surface ﬁnishing. Cutting is the most popular and important way to fabricate titanium components. As the technology advances, more and higher requirements are necessary for cutting the titanium alloy, especially for quality surfaces. Surface roughness, as an effective measure to evaluate the quality of a surface, is affected by different settings of cutting parameters such as different cutting speed, rake angle and feed [1]. Thus, investigation of the effects and subsequent optimization of the cutting parameters on surface roughness is crucial and, currently, the Taguchi method is one of the most popular methods used. [2–4]. It ﬁrst establishes orthogonal experiment to decrease the number of experiments and, further, to reduce experimental time. The surface roughness is then measured and recorded in these experiments, according to which the effectiveness of each combination of cutting parameters is evaluated and conﬁrmed by Taguchi S/N ratios and the variance analysis (ANOVA) [4]. Finally, an optimal set of cutting parameters is found to achieve the lowest surface roughness. Previous research has conducted experiments to optimize cutting parameters to minimize the surface roughness value based on the Taguchi method [5–13]. Sahoo and Pradhan studied the inﬂuence of process parameters using the Taguchi method and provided Appl. Sci. 2019, 9, 654; doi:10.3390/app9040654 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 654 2 of 23 optimized parameters by applying an uncoated tungsten carbide tool in the machining Al/SiCp metal matrix composite without adding cutting ﬂuid [5]. Rao and Padmanabhan applied the Taguchi method to study the inﬂuence of cutting parameters on the metal removal rate [6]. Motorcu studied the inﬂuence of cutting parameters on surface roughness, such as feed rate, cutting speed, and drill bit angle, and then the optimum levels of control factors are deﬁned to reduce the surface roughness by applying S/N ratios [7]. Measuring surface roughness by conducting experiments is the current practice to study the effects of cutting parameters on surface roughness. It is, however, costly as well as time-consuming, involving a substantial waste of materials. This problem is not negligible especially for some material which is very expensive and very hard to cut such as titanium alloy. Even though the number of experiments can be decreased substantially by applying an orthogonal array of the Taguchi method, the costs and time taken for the experiments may not be acceptable. With the rapid development of computer technology, the advantages of numerical simulation of cutting processes attracts a lot of attention and has become an important means for analyzing the mechanism of cutting processes. This paper proposes an approach to investigate the effects of cutting parameters on surface roughness by combining the numerical simulation of the cutting process and the Taguchi method. Compared to other experiments, our approach is applicable to investigate the effects of cutting parameters with substantially reduced costs and time. This paper gives an example of turning to show how our method works. First, a cutting model is simpliﬁed as the orthogonal cutting model. Then a numerical simulation based on the SPH method is applied to simulate the cutting process as an alternative to the real cutting experiment. Furthermore, the surface roughness of this simulation model is calculated and the variation trend of surface roughness with different cutting parameters is evaluated. After that, we employ the Taguchi method which involves establishing an orthogonal simulation array, applying an S/N ratio and ANOVA to investigate the effects of cutting parameters on surface roughness. Finally, we compare our result with experiments and make evaluations. There are two key components in our work. The ﬁrst one is to establish a reliable and accurate numerical cutting model, and the second one is to come up with a method to evaluate the variation trend of surface roughness with different cutting parameters. As for the ﬁrst component, most of the traditional numerical models apply the Finite Element Method (FEM) which bears problems in handling with large deformations and material fragmentations during the cutting process and sometimes leads to simulation breakdowns caused by excessive mesh distortion [8–11]. At the same time, the simulation result is sensitive to different ways of generating the mesh, which affect the accuracy of the simulation [14]. Another problem of FEM is that it is not capable of deriving the surface roughness from the simulated surface since FEM simulates the cutting process, including chips formation, through setting chip separation criteria and deleting elements [9]. To avoid these problems, we applied Smoothed Particles Hydrodynamic (SPH) method, a Lagrangian meshfree method, to establish the numerical cutting model. SPH divides material into individual particles with independent properties, such as density, mass and speed, in the process of simulation instead of generating mesh for materials and cutting tools. Compared with FEM, its adaptivity can be obtained at the early stage of approximation of ﬁeld variables, and its formula is not affected by the distribution of particles. Thus, SPH is capable of handling large deformation, which always occurs during the cutting process, and at the same time, naturally simulates the process of chip separation. Previous studies have proven that SPH can simulate cutting processes effectively [8–11,15,16] It is well known that the Johnson–Cook (JC) model [17] and Johnson–Cook damage [18,19] model are the most popular constitutive material laws in describing metal behavior. High strain and high strain rates, however, always occur in Ti–6AL–4V machining process, and under this condition, the dynamic mechanical properties of the material, such as strain softening and dynamic recrystallization mechanisms of Ti–6AL–4V materials, cannot be exactly described by JC law. This paper, hence, applies Appl. Sci. 2019, 9, 654 3 of 23 Hyperbolic Tangent (TANH) constitutive law, which was proposed by Calamaz [20] and later further developed by the Sima M and Özel T [21] to describe dynamic mechanical properties of Ti–6AL–4V material. The TANH constitutive model not only considers those problems which are mentioned above, but also describes the process of dynamic recrystallization mechanisms which also always occurs in the cutting Ti–6Al–4V process [22]. In addition, an improved SPH algorithm is adopted through adding modiﬁed schemes for approximating density (density correction) and kernel gradient approximation (kernel gradient correction) in this work, which has been proven to be very efﬁcient in improving the accuracy of a cutting model [15]. Since the traditional SPH cannot exactly reproduce linear functions in the entire problem domain, in its initial form particle summation formulation, the SPH does not exactly reproduce a constant near the boundary because of the loss of symmetry in the smoothing operation. The second component involves the evaluation of the variation trend of surface roughness with different cutting parameters. The SPH algorithm adopted in this paper naturally simulates the separation of chips from the workpiece. Each individual particle carries independent attributes, such as density, mass, speed, and location. After examining the location of each individual particle of the workpiece surface and incorporating it into the deﬁnition and calculation of surface roughness, this paper proposes a method, Surface Particles Method (SPM), to calculate the surface roughness of the SPH model. Then we evaluate the variation trend of surface roughness based on the surface roughness of the SPH cutting model by changing the cutting parameters during the simulation of the cutting process. To summarize, this paper (1) adopts an improved SPH algorithm and the TANH constitutive law to establish an improved SPH cutting model and make the simulation cutting model more accurate; (2) proposes a Surface Particles Method (SPM) to evaluate the variation trend of surface roughness with different parameters instead of the real experiments for cutting Ti–6AL–4V and substantially reduces work time and cost; (3) investigates the inﬂuence of cutting parameters on surface roughness during cutting by the Taguchi method and avoids the laborious experimental process; and 4) ﬁnally, compares our simulation results with experimental results to testify the effectiveness. The comparison shows that our method is reliable and effective to investigate the effects of cutting parameters on surface roughness. 2. SPH Modeling of Cutting Model 2.1. SPH Basic Principles Lucy came up with SPH method in 1977 and the original purpose of this method was to solve astrophysical problems [23,24]. It was then extensively used in many different areas, such as ﬂuid mechanics and solid mechanics [25,26]. Previous studies have proven that SPH can solve the ﬂuent problem effectively, such as free surface ﬂow [27,28] and bubble rising [29,30]. The cutting problem can also be regarded as a ﬂuent (“solid-ﬂow”) problem, since the solid deforms drastically when the solid materials are machined by a cutting tool at high speed, and thus it can be treated as a “ﬂow” of solid materials. Therefore, SPH should be an effective method for simulating cutting processes [11,15,31,32]. The kernel approximation and particles approximation are two very important steps in the SPH method: 2.1.1. Kernel Approximation: h f (x )i = f x W x x , h dx (1) i j i j j hr f (x )i = f x rW x x , h dx (2) i j i j j W Appl. Sci. 2019, 9, 654 4 of 23 where x is a three dimension coordinate vector; x and x are the positions of particles i and j; f is i j the function of x; f (x ) is the value of f (x) at particle j; and Kernel function W represent a weighted contribution of particle j to particle i. The gradient of kernel function rW can be expressed as x x i j ¶W rW = . h is the smooth length, which is used to deﬁne the inﬂuence domain of the smooth r ¶r ij ij kernel function. In this work, we choose the cubic spline function [33], which closely resembles a Gaussian function [25], as our smoothing function. It is expressed as following: 2 1 2 3 q + q , 0 q 1 3 2 W = a (3) ij d 1 3 (2 q) , 1 q 2 where a is the normalization factor that can be expressed as a = 15/ 7ph . It is always used to d d solve 2D problems. q is the normalized distance between particle i and j, which can be expressed as q = r /h. The value of the smoothing length is chosen based on actual problems, which means the ij larger value may affect the computational efﬁciency, while the smaller value may cause low accuracy of computation. In this study, we choose the 1.5 times particle spacing as the smoothing length [15]. 2.1.2. Particle Approximation In the SPH algorithm, the computation domain can be discretized with a set of particles by using DV instead of the dx in Equations (1) and (2): j i DV = (4) where m is the mass of particle j. r is the density of particle j. N is the total number of particles which j j are involved in the computational ﬁeld. Substituting Equation (4) into Equation (1) and Equation (2): h f (x )i = f x W x x , h (5) i å j i j j=1 hr f (x )i = f x rW x x , h (6) i å j i j j=1 From above, it should be noticed that some macroscopic variables (temperature, pressure, density, etc.) are expressed as integral form by a set of disorderly point values in the SPH method. In this set of points, the interaction between various points are described and a kernel estimate of a point is obtained by an interpolation function. In the SPH equation, the governing equation is the Lagrangian form of Navier–Stokes(NS) equations [14], which is used for governing the materials. The mass conservation law and momentum conservation laws can be expressed as following: dr ¶v = r (7) dt ¶x a ab dv 1 ¶s = + f (8) dt r ¶x where v is the velocity; t is the time; the coordinate direction can be denoted by a and b; f is the ab component of acceleration; and s is the total stress tensor which consists of two parts: ab ab ab s = Pd + t (9) ab ab where P is pressure; d is the Kronecker tensor; and t is the deviatoric shear stress. Appl. Sci. 2019, 9, 654 5 of 23 The shear stresses are generated and change with time when solids are in the dynamic velocity ﬁeld with spatial gradients. The stress t can be expressed as following [14]: ab dt . 1 . . . ab gg bg ag ab ag gb = 2G(# d # ) + t r + t r (10) i i i i i i dt 3 ab where # is the strain rate tensor; G is the shear modulus which can be obtained by an experimental test; ab and r is the rotation tensor. After elastic deformation, the material will enter the plastic deformation named plastic behavior and it can be identiﬁed by the critical point of plastic deformation [14]: f = p (11) 3J where J is the second stress tensor invariant. s is the ﬂow stress and its initial value is the shear stress 2 y ab ab (s = t t ). When the value of f is lower than 1 ( f < 1), the plastic deformation is said to occur. y y y As mentioned above, the traditional SPH method has low accuracy [34]. Therefore, in this study, we employed an improved SPH method where density and kernel gradient are corrected by Moving Least Squares and a correction matrix. It has been proven to be efﬁcient in improving the accuracy of the cutting model [15]. In the scheme of the improved SPH algorithm, we use the Moving Least Squares (MLS) method to correct the density ﬁeld periodically; the linear variation of the density ﬁeld can be exactly reproduced and a smoother pressure can be obtained [35]. The correction scheme is expressed as: N N MLS MLS hr i = r W = m W (12) i å j å j ij ij j j MLS where W is the MLS kernel; more details can be found in Reference [35]. ij The Kernel Gradient Correction (KGC) method is applied to improve the SPH approximation accuracy. The second order accuracy of SPH approximation can also be restored by this correction new scheme. The corrected value of r W for arbitrary particle i is expressed as follows: ij new r W = L(r ) r W (13) i ij i i ij where L(r ) is the correct matrix, which can be expressed as: 2 3 ¶W ¶W ij ij x y ij ji ¶x ¶x i i 4 5 L(r ) = V (14) i å j ¶W ¶W ij ij j x y ji ji ¶y ¶y i i Finally, the SPH computational domain is discretized by a limited number of particles which have their own attributes such as mass (m) and individual space. The continuity equation ﬁnally is expressed as following: m ¶W dr j b ij = r v (15) iå dt r ¶x j=1 The momentum equation is expressed as following: 2 3 ab ab a N t + t P + P ¶W dv i j i j ij i ab ab a 4 5 = m d + d + f (16) å j Õ dt r r r r i j i j ¶x j=1 ij where is called artiﬁcial viscosity and is used to reduce the unphysical oscillations in the numerical ij results around the shocked region. Appl. Sci. 2019, 9, 654 6 of 23 2.2. Material Constitutive Model Johnson–Cook (JC) is the most frequently adapted constitutive law to describe the dynamic mechanism properties (strain hardening, strain hardening rate and thermal softening) of mental material when the elastic deformation transforms to the plastic deformation stage. The JC constitutive law, however, does not take into account the strain-softening of the material under high strain conditions. Cutting Ti–6AL–4V, however, is a high stress and high stress rate process, and JC law cannot describe this process accurately. To solve the problems of JC law, Calamaz proposed a TANH constitutive law which modiﬁes the stress, strain and temperature components of JC law by adding a, b, c, d parameters and a tanh function [20]. Later, Sima M and Özel T introduced exponent r into the TANH constitutive model to further control the tanh function for thermal softening [21]. The TANH constitutive model not only considers the stress softening of the material as shown in Figure 1, but also considers the recrystallization phenomenon. Figure 1 is the ﬂow stress-strain curve with the TANH law. N # r e f f 1 TT 1 room s = A + B( ) 1 + Cln . 1 (D + (1 D)(tanh( )) ) (17) y a T T exp(# ) # room (#+S) melt e f f 0 D = 1 ( ) (18) S = (19) T = e/C + T (20) p room . . where # is the equivalent plastic strain; # is the equivalent plastic strain rate; # is the reference e f f e f f 0 plastic strain rate; A, B, C, N, and M are material-dependent constants which are the same parameters as in the JC equation and are obtained by SHPB tests [36]. T is the room temperature. T is the room melt melting temperature of Ti–6AL–4V. T is the current temperature which is computed by Equation (20). a, b, c, d and r are corrected parameters which are obtained by the control variable method. We vary one parameter while keeping the others constant to investigate the effects of the parameters on strain-stress curves and make the strain-curve of the TANH model be in good agreement with that of the JC model under the low strain (the strain can be considered low strain when the strain is lower than 0.5). As shown in Figure 2a, parameter a controls the strain-hardening part by decreasing ﬂow stress after a critical strain value, and a low value of the parameter will lead to a big difference between TANH and the SHPB test data. As a increases, the maximum stress will increase. Parameter b in (Equation (19)) controls the temperature-dependent ﬂow-softening effect and points out where the maximum ﬂow stress is attained. The lower the value of parameter b is, the lower the strain value of the peak ﬂow stress that occurs, as shown in Figure 2b. Parameter d is an exponent for the temperature that controls the degree of temperature dependency of parameter D as given in (Equation (18)). As shown in Figure 2c, parameter d has a strong impact on the value of ﬂow softening and it determines the minimum ﬂow stress value. As shown in Figure 2d, parameter r controls the softening trend. The higher the parameter r is, the faster the ﬂow stress-strain curve enters the softening region. With the strain increasing, the slope of strain softening curves will decrease and the maximum ﬂow stress remains unchanged. The c has similar effects to r on the ﬂow stress [16]. After observing the trend of the curve with different parameters, the best ﬁt parameters and best stress-strain curve are given in Table 1 and Figure 1. 2.2. Material Constitutive Model Johnson–Cook (JC) is the most frequently adapted constitutive law to describe the dynamic mechanism properties (strain hardening, strain hardening rate and thermal softening) of mental material when the elastic deformation transforms to the plastic deformation stage. The JC constitutive law, however, does not take into account the strain-softening of the material under high strain conditions. Cutting Ti–6AL–4V, however, is a high stress and high stress rate process, and JC law cannot describe this process accurately. To solve the problems of JC law, Calamaz proposed a TANH constitutive law which modifies the stress, strain and temperature components of JC law by adding a, b, c, d parameters and a tanh function [20]. Later, Sima M and Özel T introduced exponent r into the TANH constitutive model to further control the tanh function for thermal softening [21]. The TANH constitutive model not only considers the stress softening of the material as shown in Figure 1, but also considers the recrystallization phenomenon. Figure 1 is the flow stress-strain curve with Appl. Sci. 2019, 9, 654 7 of 23 the TANH law. x 10 3.5 TANH to a big difference between TANH 3 and the SHPB test data. As a increases, the maximum stress will increase. Parameter b in (Equation (19)) controls the temperature-dependent flow-softening effect and points out where the maximum flow stress is attained. The lower the value of parameter b is, the 2.5 lower the strain value of the peak flow stress that occurs, as shown in Figure 2b. Parameter d is an exponent for the temperature that controls the degree of temperature dependency of parameter D as given in (Equation (18)). As shown in Figure 2c, parameter d has a strong impact on the value of flow softening and it determines the minimum flow stress value. As shown in Figure 2d, parameter r 1.5 controls the softening trend. The higher the parameter r is, the faster the flow stress-strain curve enters the softening region. With the strain increasing, the slope of strain softening curves will decrease and the maximum flow stress remains unchanged. The c has similar effects to r on the flow 0 1 2 3 4 5 stress [16]. After observing the trend of the curve wi True strain th different parameters, the best fit parameters and best stress-strain curve are given in Table 1 and Figure 1. Figure 1. The stress-strain curve. Figure 1. The stress-strain curve. 8 8 x 10 x 10 5 5   ε   1 TT − 1 N eff r room 4.5 4.5 σ = AB++ ( ) 1 Cln  1− (D+ (1−D)(tanh( )) )  (17)    y a b= c 0.1 a=0.1 exp(εε ) TT −  ε + S  eff 0 melt room ()    b=0.5 4 4 a=1   b=1 a=2 3.5 3.5 b=5 a=5 JC JC D =− 1( ) 3 3 (18) 2.5 2.5 a=2; c=2; d=2; r=1 b=5; c=2; d=2; r=1 2 2  (19) S = 1.5 1.5   m 1 1 0 1 2 3 4 5 0 1 2 3 4 5 True strain True strain Te=+ /C T (20) 8 8 p room (a) (b) x 10 x 10 5 5 where 𝜀 is the equivalent plastic strain; 𝜀 is the equivalent plastic strain rate; 𝜀 is the 4.5 d=0.1 r=0.01 reference plastic strain rate; A, B, C, N, and M are material-dependent constants which are the same d=0.5 r=0.1 d=1 r=0.5 parameters as in the JC equation and are obtained by SHPB tests [36]. 𝑇 is the room temperature. d=5 r=1 3.5 JC 𝑇 is the melting temperature of Ti–6AL JC –4V. T is the current temperature which is computed by Equation (20). a, b, c, d and r are corrected parameters which are obtained by the control variable 2.5 method. We vary one parameter while keeping the others constant to investigate the effects of the a=2; b=5; c=2; r=1 parameters on strain-stress curves and make the strain-curve of the TANH model be in good agreement with that of the JC model under the low s 1.t5rain (the strain can be considered low strain a=2; b=5; d=2; c=1 when the strain is lower than 0.5). As shown in Figure 2a, parameter a controls the strain-hardening 0 1 0 1 2 3 4 5 0 1 2 3 4 5 part by decreasing ﬂow st True re sss trainafter a critical strain value, and a low val Tu rue e of strai t n he parameter will lead (d) (c) Figure 2. JC strain-stress curve and TANH strain-stress curve with varying value parameters: the Figure 2. JC strain-stress curve and TANH strain-stress curve with varying value parameters: the strain −1 strain rate 2000 s ((a): vary parameter a; (b): vary parameter b; (c): vary parameter d; (d): vary rate 2000 s ((a): vary parameter a; (b): vary parameter b; (c): vary parameter d; (d): vary parameter r). parameter r). Table 1. Material properties of Ti–6AL–4V for workpiece [36,37]. Physical Parameter Constant Work Material (Ti–6AL–4V) Density, ρ0 (Kg/m) 4430 Possion’s ratio v 0.342 Shear modulus (G) 110 Gpa Specific heat, Cp J/(Kg·K) 580 Thermal conductivity (W/mk) 7.3 T (℃) melt 1878 T (℃) 25 room A 968 Mpa TANH model (Equation (17)) B 380 Mpa C 0.02 Flow stress (MPa) Flow stress (MPa) Flow stress [MPa] Flow stress (MPa) Flow stress (MPa) Appl. Sci. 2019, 9, 654 8 of 23 Table 1. Material properties of Ti–6AL–4V for workpiece [36,37]. Physical Parameter Constant Work Material (Ti–6AL–4V) Density, r (Kg/m ) 4430 Possion’s ratio v 0.342 Shear modulus (G) 110 Gpa Speciﬁc heat, C J/(KgK) 580 Thermal conductivity (W/mk) 7.3 T ( C) 1878 melt T ( C) 25 room A 968 Mpa B 380 Mpa C 0.02 N 0.577 M 0.421 TANH model (Equation (17)) a 0.05 b 2 c 1 d 1 r 5 2.3. Contact Algorithms Figure 3a provides a conceptual explanation of the particle distribution and interaction and how the contact algorithm is used to calculate the contact force. In this work, the total number of particles -9 is 25,000 and the time step is 3.0 s. (When the number of particles is greater than 25,000 or the time -9 step is less than 3.0 s. The accuracy of the model does not change obviously, while the running time of the program will increase with the increase of particles and decrease of time step). The cutting tool and material are regarded as groups of particles in our improved SPH cutting model. We treat the cutting tool as a rigid body without carrying any attributes as the workpiece material, such as density and quality. The location information (x , y ) and velocity information (u , v ) are only needed to set k k k k up for the particles of the cutting tool, which is computed by the initial location and velocity of the cutting tool respectively. Then, this information is used to calculate surface vectors (unit tangent vector and unit normal vector). We can use them to determine the direction of the cutting force and contact location between the cutter and workpiece respectively. When the cutter contacts the workpiece, two different kinds of particles (i.e. particles of the workpiece and particles of the cutter) occur contact force. To acquire the contact force, we need to 1) set a threshold d to decide when the contact is considered to occur, and 2) calculate the contact force. Figure 3b illustrates how the cutting tool and workpiece are regarded as the contact. P is the vertical point from particle i to the surface of the cutting tool. d is the actual vertical distance between particle i and the cutting tool. d is the contact threshold for these two kinds of particles. When d is 0 p smaller than d , these two particles contact. The tangent vector t and normal vector n at point p are 0 p p calculated by the following Equation: x x y y k+1 k k+1 k t = t , t = , p x y jx x j jx x j k+1 k k+1 k (21) y y x x : k+1 k k+1 k n = t , t = , p y x x x x x j j j j k+1 k k+1 k where x and x are the coordinate values of cutting tool particle k and k + 1. k +1 k Appl. Sci. 2019, 9, 654 9 of 23 Regular Cutting tool Irregular Rigid body k+1 Workpiece (a) (b) Figure 3. (a) Particles distribution and (b) particles interaction. Figure 3. (a) Particles distribution and (b) particles interaction. The next step is to calculate the contact force between the particles of the workpiece and cutting The next step is to calculate the contact force between the particles of the workpiece and cutting tool. The forces on point i have two parts, the force along the normal direction and the force along the tool. The forces on point i have two parts, the force along the normal direction and the force along tangential direction. The forces are calculated as the following Equation: the tangential direction. The forces are calculated as the following Equation: < F =  d d n ni 2 0 p p (Dt)  (22) Fn =− dd ⋅ n () o ni 2 0 p p mF m ti i  Δt F = min ()jF j , v t t  ti ni pi p p  jF j Dt ti (22)   μF m  where m is the mass of particle i; v =v v , v is the velocity at point p. v is the velocity of particle τi i i pi p i p i FF=⋅ min , v ⋅ττ  ()   τini pipp  i. m = 0.3 is the friction coefﬁcient between the workpiece and cutting tool [15,35]. According to the F Δt τi     formulas above, the interaction forces between the cutting tool and workpiece are calculated [15]. v=v -v v m p v i pi p i p i i 3. where OptimalC is parameters the mass of particle ; , is the velocity at point . is the μ =0.3 velocity of particle i. is the friction coefficient between the workpiece and cutting tool [15,35]. 3.1. Evaluation of the Surface Roughness of the SPH Cutting Model According to the formulas above, the interaction forces between the cutting tool and workpiece are The surface roughness is the repeating series of peaks and valleys on the surface. It is calculated calculated [15]. as the average deviation from the ideal along the direction of the normal vector of the real surface. Figure 4 illustrates the surface proﬁle of a workpiece sample. The x axis is regarded as the central line. 3. OptimalC parameters In the real measurement, the surface of the workpiece is discretized into measurement points. The vertical distance from the point to the center line is measured, and then the average value of these 3.1. Evaluation of the Surface Roughness of the SPH Cutting Model vertical distances is calculated as surface roughness R . The more measurement points there are, the The surface roughness is the repeating series of peaks and valleys on the surface. It is calculated more accurate R is. The equation is expressed as: as the average deviation from the ideal along the direction of the normal vector of the real surface. Figure 4 illustrates the surface profile of a workpiece sample. The x axis is regarded as the central R = jy(x )j (23) a å i line. In the real measurement, the surface of the workpiece is discretized into measurement points. The vertical distance from the point to the center line is measured, and then the average value of these vertN icais l d the ista number nces is cof alc sampling ulated as sur points. face roughness R . The more measurement points there are, the more accurate R is. The equation is expressed as: 1 N Ry = ()x (23) ai  N is the number of sampling points. Appl. Sci. 2019, 9, 654 10 of 23 Ra Ra Roughness Roughness center line center line Figure 4. Surface roughness proﬁle of workpiece. Figure 4. Surface roughness profile of workpiece. Figure 4. Surface roughness profile of workpiece. Similar to calculating the surface roughness derived from the vertical distances of measurement Similar to calculating the surface roughness derived from the vertical distances of measurement Similar to calculating the surface roughness derived from the vertical distances of measurement points, we extract particles on the simulated surface from the improved SPH cutting model and points, we extract particles on the simulated surface from the improved SPH cutting model and points, we extract particles on the simulated surface from the improved SPH cutting model and transform their location information into a coordinate (The SPH cutting model is established in the transform their location information into a coordinate (The SPH cutting model is established in the transform their location information into a coordinate (The SPH cutting model is established in the coordinate system and each particle has the corresponding particle number and coordinate position coordinate system and each particle has the corresponding particle number and coordinate position coordinate system and each particle has the corresponding particle number and coordinate position information. When the surface particles are extracted, the particle number of surface particles are information. When the surface particles are extracted, the particle number of surface particles are information. When the surface particles are extracted, the particle number of surface particles are obtained as well as the coordinate information of the particles). Figure 5 demonstrates the particles of obtained as well as the coordinate information of the particles). Figure 5 demonstrates the particles obtained as well as the coordinate information of the particles). Figure 5 demonstrates the particles the simulated surface after cutting, and Figure 6 illustrates the surface proﬁle of the extracted particles of the simulated surface after cutting, and Figure 6 illustrates the surface profile of the extracted of the simulated surface after cutting, and Figure 6 illustrates the surface profile of the extracted under a coordinate system. We extract the coordinate information of these particles and calculate the particles under a coordinate system. We extract the coordinate information of these particles and particles under a coordinate system. We extract the coordinate information of these particles and surface roughness. calculate the surface roughness. calculate the surface roughness. Figure Figure 5. 5. Distribution of Distribution ofsurface partic surface particles les ofof ththe e workpiece a workpiecefter cutting ( after cutting cutting parameters are (cutting parameters ar v e = Figure 5. Distribution of surface particles of the workpiece after cutting (cutting parameters are v = v 240 m/min, = 240 m/min, f = 0.07 f = mm, 0.07 mm, γ = 0° ). = 0 ). 240 m/min, f = 0.07 mm, γ = 0°). -6 Appl. Sci. 2019, 9, 654 11 of 23 x 10 -3 -6 -4 x 10 -3 -5 -4 -6 -5 -7 -6 -8 -7 -9 0 50 100 150 200 250 -8 Particle number -9 Figure 6. The coordinate location of surface particles after cutting (cutting parameters are v = 240 0 50 100 150 200 250 Particle number m/min, f = 0.07 mm, γ = 0°, the red circles are invalid points). Figure 6. The coordinate location of surface particles after cutting (cutting parameters are Figure 6. The coordinate location of surface particles after cutting (cutting parameters are v = 240 Based on the coordinate, we calculate the vertical distance from each particle to the center line. v = 240 m/min, f = 0.07 mm, = 0 , the red circles are invalid points). m/min, f = 0.07 mm, γ = 0°, the red circles are invalid points). In Figure 4, the y coordinate value of the center line is the difference between workpiece height and Based on the coordinate, we calculate the vertical distance from each particle to the center line. cutting depth. Based on the coordinate, we calculate the vertical distance from each particle to the center line. In Figure 4, the y coordinate value of the center line is the difference between workpiece height and In Figure 4, the y coordinate value of the center line is the difference between workpiece height and f =− Hh centerline (24) cutting depth. cutting depth. f = H h (24) centerline where H is the high value of the workpiece and h is the cutting depth. What is worth noting is, since f =− Hh centerline (24) there is a small amount of random errors in the simulating process, a tiny number of particles deviate far from the center line, which does not conform to the real cutting process (When the distance where H is the high value of the workpiece and h is the cutting depth. What is worth noting is, since where H is the high value of the workpiece and h is the cutting depth. What is worth noting is, since between a surfac there is a smalle p amount articleof an random d the roughness center errors in the simulating line is gr process, eater tahtiny an 0.number 5 µm, we of cons particles ider t deviate his there is a small amount of random errors in the simulating process, a tiny number of particles deviate particle deviate far from the center line). Thus, we regard these particles as outliers and remove them far from the center line, which does not conform to the real cutting process (When the distance between far from the center line, which does not conform to the real cutting process (When the distance when a surface calculat particle ing sur and face the roughnes roughness s. Fig center ure 7 is line the is st greps of su eater than rface 0.5 roughne m, we consider ss calcul this ation particle . deviate between a surface particle and the roughness center line is greater than 0.5 µm, we consider this far from the center line). Thus, we regard these particles as outliers and remove them when calculating particle deviate far from the center line). Thus, we regard these particles as outliers and remove them surface roughness. Figure 7 is the steps of surface roughness calculation. when calculating surface roughness. Figure 7 is the steps of surface roughness calculation. Surface Invalid points Surface Invalid points Roughness center line Roughness center line Figure 7. The steps of surface roughness calculation. Figure 7. The steps of surface roughness calculation. Figures 8–10 show the distribution of surface particles extracted from the cutting model under Figures 8–10 show the distribution of surface particles extracted from the cutting model under different cutting parameters. These ﬁgures show that while the higher feed and smaller rake angle Figure 7. The steps of surface roughness calculation. different cutting parameters. These figures show that while the higher feed and smaller rake angle lead to a more disordered distribution of the surface particles (i.e., the more rough surface), the cutting lead to a more disordered distribution of the surface particles (i.e., the more rough surface), the Figures 8–10 show the distribution of surface particles extracted from the cutting model under different cutting parameters. These figures show that while the higher feed and smaller rake angle lead to a more disordered distribution of the surface particles (i.e., the more rough surface), the Y cutting speed does not exert a significant effect on the surface roughness since the order of particles did not change too much compared to the other two cutting parameters when cutting speed was Appl. cutting speed Sci. 2019, 9, 654 does not exert a significant effect on the surface roughness since the order of partic 12 of les 23 changed. did not change too much compared to the other two cutting parameters when cutting speed was cutting speed does not exert a significant effect on the surface roughness since the order of particles changed. speed did not cha does n not ge too much exert a signiﬁcant compared to effect the other on the surface two cutti roughness ng parameters when cutti since the order of particles ng speed was did not change changed too . much compared to the other two cutting parameters when cutting speed was changed. (c) (a) (b) (c) (a) (b) (c) (a) (b) Figure 8. Surface particles distribution with different feeds (v = 120 m/min, γ = 0°). (a) f = 0.07 mm; (b) f = 0.01 mm; (c) f = 0.13 mm. Figure 8. Surface particles distribution with different feeds (v = 120 m/min, γ = 0°). (a) f = 0.07 mm; (b) Figure 8. Surface particles distribution with different feeds (v = 120 m/min, = 0 ). (a) f = 0.07 mm; f = 0.01 mm; (c) f = 0.13 mm. (b) f = 0.01 mm; (c) f = 0.13 mm. Figure 8. Surface particles distribution with different feeds (v = 120 m/min, γ = 0°). (a) f = 0.07 mm; (b) f = 0.01 mm; (c) f = 0.13 mm. (b) (a) (c) (a) (b) (c) Figure 9. Surface particles distribution of rake angle (v = 240 m/min, f = 0.13 mm). (a) = 7 ; (b) Figure 9. Surface particles distribution of rake (a) angle (v = 2(b) 40 m/min, f = 0.13 mm). (a) γ ( = 7°; ( c) b) γ = 3°; ( = 3c) ;γ (c = 0 ) °= . 0 . Figure 9. Surface particles distribution of rake angle (v = 240 m/min, f = 0.13 mm). (a) γ = 7°; (b) γ = 3°; (c) γ = 0°. Figure 9. Surface particles distribution of rake angle (v = 240 m/min, f = 0.13 mm). (a) γ = 7°; (b) γ = 3°; (c) γ = 0°. (b) (a) (c) (a) (b) (c) Figure 10. Surface particles distribution of cutting speed ( = 3 , f = 0.15 mm/rev). (a) v = 60 m/min; Figure 10. Surface particles distribution of cutting speed (γ = 3°, f = 0.15 mm/rev). (a) v = 60 m/min; (b) v = 120 m/min; (c) v = 240 m/min. (a) (b) (c) (b) v = 120 m/min; (c) v = 240 m/min. Figure 10. Surface particles distribution of cutting speed (γ = 3°, f = 0.15 mm/rev). (a) v = 60 m/min; 3.2. The(Basic b) v = 120 Principle m/min; of (T caguchi ) v = 240 m/mi Method n. Figure 10. Surface particles distribution of cutting speed (γ = 3°, f = 0.15 mm/rev). (a) v = 60 m/min; Taguchi proposed a method [38] which considers the improvement of product quality. The (b) v = 120 m/min; (c) v = 240 m/min. 3.2. The Basic Principle of Taguchi Method product quality can be obtained by designing rather than inspecting. In this study, we employed the 3.2. The Basic Principle of Taguchi Method Taguchi method to investigate the effects of cutting parameters on surface roughness. The Taguchi 3.2. The Basic Principle of Taguchi Method method has two core analysis tools. The ﬁrst core is the orthogonal array. Since the orthogonal array Appl. Sci. 2019, 9, 654 13 of 23 Taguchi proposed a method [38] which considers the improvement of product quality. The can ensure the impartiality and uniformity of the experiment’s data, it is very representative and product quality can be obtained by designing rather than inspecting. In this study, we employed the can comprehensively reﬂect the impact of various factors and each level on indicators. Therefore, Taguchi method to investigate the effects of cutting parameters on surface roughness. The Taguchi compared to the traditional experimental design which has a large number of experiments and is too method has two core analysis tools. The first core is the orthogonal array. Since the orthogonal array complex, the orthogonal array can simplify the complex experimental tests and minimize the effects can ensure the impartiality and uniformity of the experiment’s data, it is very representative and can of the factors which are out of control otherwise. The second core is the signal-to-noise expecting comprehensively reflect the impact of various factors and each level on indicators. Therefore, (ined by design rather than the (S/N). The S/N ratio is calculated based on the experimental results compared to the traditional experimental design which has a large number of experiments and is too of the orthogonal array, which can indicate the loss of product quality [39]. Usually, there are three complex, the orthogonal array can simplify the complex experimental tests and minimize the effects performance of the factors which characteristics are out of contro in analyzing l otherwise. the S/NTratio. he second They coar re is the sign e the smaller al-to-noise ex -the-betterpecting characteristic, (S/N). The S/N ratio is calculated based on the experimental results of the orthogonal array, which the nominal-the-better characteristic and the larger-the-better characteristic. Neglecting the grouping can indicate the loss of product quality [39]. Usually, there are three performance characteristics in of the above performance characteristic, the greatest S/N value stands for the best performance analyzing the S/N ratio. They are the smaller-the-better characteristic, the nominal-the-better characteristic. Hence, the higher the S/N ratio is, the better the level of the cutting parameters is. In characteristic and the larger-the-better characteristic. Neglecting the grouping of the above addition, the variance analysis (ANOVA) is applied to determine which cutting parameters have a performance characteristic, the greatest S/N value stands for the best performance characteristic. signiﬁcant effect. By analyzing the S/N ratio and variance, we can not only investigate the effects of Hence, the higher the S/N ratio is, the better the level of the cutting parameters is. In addition, the cutting parameters, but also obtain the optimal cutting parameters. It should be noticed that we need variance analysis (ANOVA) is applied to determine which cutting parameters have a signiﬁcant to obtain the cutting parameters which achieve the lowest surface roughness value. The performance effect. By analyzing the S/N ratio and variance, we can not only investigate the effects of cutting characteristic in analyzing the S/N ratio is the smaller-the better. The equations are as following: parameters, but also obtain the optimal cutting parameters. It should be noticed that we need to obtain the cutting parameters which achieve the lowest surface roughness value. The performance characteristic in analyzing the S/N ratio is the smaller-the better. The equations are as following: M.S.D = y (25) å i n i=1 M.. SD = y (25)  i i =1 S/N = 10 log(M.S.D) (26) SN/1 =−0log(M.S.D) (26) where M.S.D is the mean-square deviation; n is the total number of experiments; y is the measured value of surface roughness; and i is the serial number of the experiment. where M.S.D is the mean-square deviation; n is the total number of experiments; yi is the measured In this paper, the turning process (Figure 11a) is approximated as the orthogonal cutting value of surface roughness; and i is the serial number of the experiment. process [40] (Figure 11b). The parameters in turning can be approximated by the orthogonal cutting In this paper, the turning process (Figure 11a) is approximated as the orthogonal cutting process [40] (Figure 11b). The parameters in turning can be approximated by the orthogonal cutting parameters as shown in Table 2. parameters as shown in Table 2. Cutti ng speed Rake angl e chip C utti ng to ol Work pi ece h1 Edge radius C utti ng speed C learanc e angl e Fc h2 f Chip Fc Ft Workpiece Workpiece Ft C utti ng to ol (a) (b) Figure 11. Cutting model. (a): Turning; (b): Orthogonal cutting Figure 11. Cutting model. (a): Turning; (b): Orthogonal cutting Table 2. Approximation of Turning by Orthogonal Cutting [40]. Table 2. Approximation of Turning by Orthogonal Cutting [40]. Turning Operation Orthogonal Cutting Turning Operation Orthogonal Cutting Feed f = Chip thickness before cut t Feed f= Chip thickness before cut t Cutting speed V= Cutting speed V Cutting speed V= Cutting speed V Rake angle = Rake angle Rake angle γ= Rake angle γ Clearance angle = Clearance angle Cutting force F = Cutting force F c c Feed force F = Feed force F t t Appl. Sci. 2019, 9, 654 14 of 23 As shown above, the three parameters (cutting speed, feed and rake angle) can be controlled by this simulating cutting model. They can also affect surface roughness signiﬁcantly in the actual turning process. Therefore, the reasonable choice of them would improve the surface quality and decrease the surface roughness. Though the cutting depth and clearance angle in turning are other parameters in the cutting process, the cutting depth cannot be controlled in the 2D simulation model and clearance angle has little effect on surface roughness. In this work, we only consider the above three factors on surface roughness and therefore we choose them as controllable factors in this work. In addition, three levels are also considered in this experiment, as shown in Table 3. Therefore, as shown in Table 4, we selected L orthogonal array as our experimental design [41]. It should be noticed that to select an appropriate orthogonal array for the experiments, the total degree of freedom needs to be computed. The degrees of freedom are deﬁned as the number of comparisons between the process parameters that need to be made to determine which level is better and speciﬁcally how much better it is. In this study, the L array which has four columns and nine rows has eight degrees of freedom and it can handle three-level process parameters (There are six degrees of freedom owing to there being three cutting parameters in the turning operations). Table 3. Three levels for cutting parameters. Symbol Cutting Parameters Level1 Level2 Level3 A Rake angle 0 3 7 B Cutting speed 240 m/s 120 m/s 60 m/s C Feed 0.07 mm 0.1 mm 0.13 mm Table 4. L orthogonal array in simulated experiments. Cutting Parameter Level A B C Experiment Number Rake Angle Cutting Speed Feed 1 1 1 1 2 1 2 2 3 1 3 3 4 2 1 3 5 2 2 1 6 2 3 2 7 3 1 2 8 3 2 3 9 3 3 1 To summarize, we use L orthogonal array to design the simulated experiment, and then we study the inﬂuence of different cutting parameters (cutting speed, feed and rake angle) on surface roughness by analyzing the S/N ratio and variance. Based on these results, we obtain the effects of cutting parameters on surface roughness and optimize the cutting parameters to achieve the lowest surface roughness [41,42]. 4. Results and Discussion 4.1. Experiment Test for Improved SPH Cutting Model We established an improved SPH cutting model which incorporated the improved SPH algorithm, TANH constitutive model and contact algorithm. The improved cutting model is validated by comparing results from Calamaz et al [37]. The following Figure 12 is the comparison between the simulated cutting process and the experiment (chips morphology formation is recorded by a camera in the experimental cutting process) both under the same cutting parameters (cutting depth, 0.1 mm; cutting speed, 235 m/min; and edge radius, 20 m, rank angle 0 , clearance angle 11 ). It Appl. Sci. 2019, 9, 654 15 of 23 depth, 0.1 mm; cutting speed, 235 m/min; and edge radius, 20 µm, rank angle 0°, clearance angle 11°). depth, 0.1 mm; cutting speed, 235 m/min; and edge radius, 20 µm, rank angle 0°, clearance angle 11°). shows that the simulation results, such as strain localization along a curved shear and the curvature It shows that the simulation results, such as strain localization along a curved shear and the curvature It shows that the simulation results, such as strain localization along a curved shear and the curvature change of a slipped shear band (new segment), match the experiments very well. change of a slipped shear band (new segment), match the experiments very well. change of a slipped shear band (new segment), match the experiments very well. Strain localization New segment Strain localization New segment Shearing Shearing (a) (b) (c) (d) (a) (b) (c) (d) Figure 12. Two stages of a numerical and experimental segmented chip formation: (a) the beginning Figure 12. Two stages of a numerical and experimental segmented chip formation: (a) the beginning Figure 12. Two stages of a numerical and experimental segmented chip formation: (a) the beginning of strain localization and shearing in the primary; (c) primary shear zone begins to slip [37]; (b,d) the of strain localization and shearing in the primary; (c) primary shear zone begins to slip [37]; (b,d) the of strain localization and shearing in the primary; (c) primary shear zone begins to slip [37]; (b,d) the simulation results. simulation results. simulation results. In addition, the accuracy of the cutting model can be veriﬁed by the segmented chip morphology In addition, the accuracy of the cutting model can be verified by the segmented chip morphology In addition, the accuracy of the cutting model can be verified by the segmented chip morphology and cutting force. We can compare three parameters of chip morphology between the simulation and and cutting force. We can compare three parameters of chip morphology between the simulation and and cutting force. We can compare three parameters of chip morphology between the simulation and experiments, such as segmentation width (W), maximum segmentation height (h1) and minimum experiments, such as segmentation width (W), maximum segmentation height (h1) and minimum experiments, such as segmentation width (W), maximum segmentation height (h1) and minimum segmentation height (h2), as shown in Figure 12. The comparisons of simulation and experimental segmentation height (h2), as shown in Figure 12. The comparisons of simulation and experimental segmentation height (h2), as shown in Figure 12. The comparisons of simulation and experimental chip morphology are presented in Figures 13–15. Table 5 shows the comparison of relative error in chip morphology are presented in Figures 13–15. Table 5 shows the comparison of relative error in chip morphology are presented in Figures 13–15. Table 5 shows the comparison of relative error in simulation and experimental data for segmentation characteristics. simulation and experimental data for segmentation characteristics. simulation and experimental data for segmentation characteristics. Exp Exp Simulation Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure 13. Maximum segmentation height (h1) under different cutting speeds. Figure 13. Maximum segmentation height (h1) under different cutting speeds. Figure 13. Maximum segmentation height (h1) under different cutting speeds. h1 (μm) h1 (μm) Appl. Sci. 2019, 9, 654 16 of 23 Exp Exp 30 Simulation Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure 14. Minimum segmentation height (h2) under different cutting speeds. Figure Figure 14. 14. Minimum Minimum segment segmentation ation height (h height (h2) 2) under under diff differ erent cu ent cutting tting spe speeds. eds. Exp Exp 30 Simulation Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure Figure 15. 15. Seg Segmentation mentation width under differ width under different ent cutting spe cutting speeds. eds. Figure 15. Segmentation width under different cutting speeds. Table 5. Comparison of errors in the simulation and experimental data for segmentation characteristics. Table 5. Comparison of errors in the simulation and experimental data for segmentation Table 5. Comparison of errors in the simulation and experimental data for segmentation characteristics. characteristics. Speed h1 (Relative Error %) h2 (Relative Error %) W (Relative Error %) 75 9.84% 8.70% 5.19% Speed h1 (Relative Error %) h2 (Relative Error %) W (Relative Error %) Speed h1 (Relative Error %) h2 (Relative Error %) W (Relative Error %) 94 8.96% 6.85% 2.41% 75 9.84% 8.70% 5.19% 75 9.84% 8.70% 5.19% 116 19.35% 13.48% 4.94% 94 8.96% -6.85% 2.41% 94 8.96% -6.85% 2.41% 150 2.80% 2.35% 14.29% 116 -19.35% -13.48% 4.94% 116 -1 188 9.35% 13.14% -1 8.45%3.48% 5.06%4.94% 233 12.90% 10.45% 11.11% 150 2.80% -2.35% 14.29% 150 2.80% -2.35% 14.29% 188 13.14% -8.45% 5.06% 188 13.14% -8.45% 5.06% 233 12.90% 10.45% 11.11% 233 12.90% 10.45% 11.11% The average cutting force and average thrust force are predicted under different cutting speeds by the improved SPH model. Figures 16 and 17 show the comparison between the simulated results and The average cutting force and average thrust force are predicted under different cutting speeds The average cutting force and average thrust force are predicted under different cutting speeds the experimental results [37]. Table 6 compares the quantitative differences in relative errors between by the improved SPH model. Figure 16 and Figure 17 show the comparison between the simulated by the improved SPH model. Figure 16 and Figure 17 show the comparison between the simulated the simulation and experimental results for cutting force and thrust force. results and the experimental results [37]. Table 6 compares the quantitative diﬀerences in relative results and the experimental results [37]. Table 6 compares the quantitative diﬀerences in relative errors between the simulation and experimental results for cutting force and thrust force. errors between the simulation and experimental results for cutting force and thrust force. Segm Segment ent wi widt dth h (μm (μm)) h2 h2 ((μm μm)) Appl. Sci. 2019, 9, 654 17 of 23 300 Exp 300 Exp Simulation 200 Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure 16. Average cutting force under different cutting speeds. Figure 16. Figure 16. Ave Average rage cutting force under cutting force under dif different cutting sp ferent cutting speeds. eeds. Exp Exp 150 Simulation 150 Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure 17. Figure 17. Average thrust force under Average thrust force under dif differ fere ent nt cutting spee cutting speeds. ds. Figure 17. Average thrust force under different cutting speeds. Table 6. Comparison of errors in the simulation and experimental data for cutting force and feed Table 6. Comparison of errors in the simulation and experimental data for cutting force and feed force Table 6. Comparison of errors in the simulation and experimental data for cutting force and feed force force values. values. values. Speed Cutting Force (Relative Error %) Thrust Force (Relative Error %) Speed Cutting Force (Relative Error %) Thrust Force (Relative Error %) Speed Cutting Force (Relative Error %) Thrust Force (Relative Error %) 75 7.49% 11.03% 75 -7.49% -11.03% 75 -7.49% -11.03% 94 3.87% 16.34% 94 -3.87% 16.34% 94 -3.87% 16.34% 116 7.24% 11.39% 116 -7.24% -11.39% 116 -7.24% -11.39% 150 10.26% 9.42% 150 -10.26% 9.42% 150 -1 188 0.26%8.67% 7.83%9.42% 233 7.73% 8.64% 188 -8.67% -7.83% 188 -8.67% -7.83% 233 -7.73% 8.64% 233 -7.73% 8.64% Regarding the veriﬁcation of the model, the h1, h2, and segment width and forces have Regarding the verification of the model, the h1, h2, and segment width and forces have been Regarding the verification of the model, the h1, h2, and segment width and forces have been been proven to be effective and efﬁcient factors for verifying cutting models [15,19,20,37]. From proven to be effective and efficient factors for verifying cutting models [15,19,20,37]. From the proven to be effective and efficient factors for verifying cutting models [15,19,20,37]. From the the comparison of the simulation and the experiment in terms of segmentation width, maximum comparison of the simulation and the experiment in terms of segmentation width, maximum comparison of the simulation and the experiment in terms of segmentation width, maximum segmentation height (h1) and minimum segmentation height (h2), cutting force and thrust force, we segmentation height (h1) and minimum segmentation height (h2), cutting force and thrust force, we segmentation height (h1) and minimum segmentation height (h2), cutting force and thrust force, we can conclude the improved SPH cutting model simulates the cutting process accurately and effectively. can conclude the improved SPH cutting model simulates the cutting process accurately and can conclude the improved SPH cutting model simulates the cutting process accurately and 4.2. Analysis of the S/N Ratio and Variance effectively. effectively. The surface particles are extracted and the coordinates of particles are obtained to calculate 4.2. Analysis of the S/N Ratio and Variance 4.2. Analysis of the S/N Ratio and Variance the surface roughness of the simulation model. Table 7 shows the simulation results of the surface roughness The surface p value and artic S/N les ratio are ex for tracted and t surface roughness. he coordinates of particles are obtained to calculate the The surface particles are extracted and the coordinates of particles are obtained to calculate the surface roughness of the simulation model. Table 7 shows the simulation results of the surface surface roughness of the simulation model. Table 7 shows the simulation results of the surface roughness value and S/N ratio for surface roughness. roughness value and S/N ratio for surface roughness. Thrust Thrust fforce orce (N) (N) Cu Cuttin tting g fo force(N) rce(N) Table 7. Simulation results for surface roughness and S/N ratio. Cutting Parameter Level Model Surface S/N Ratios for Appl. Sci. 2019, 9, 654 18 of 23 Experiment A B C −1 Roughness (10 Surface Number Rake Cutting µm) Roughness Feed Angle Table 7. Simulation Speed results for surface roughness and S/N ratio. 1 0° 240 m/s 0.07 mm/rev 2.251 −7.048 Cutting Parameter Level 2 0° 120 m/s 0.1 mm/rev 6.136 −15.758 Model Surface S/N Ratios for Experiment A B C Roughness (10 m) Surface Roughness Number 3 0° 60 m/s 0.13 mm/rev 12.256 −21.767 Rake Angle Cutting Speed Feed 4 3° 240 m/s 0.13 mm/rev 3.191 −10.079 1 0 240 m/s 0.07 mm/rev 2.251 7.048 5 3° 120 m/s 0.07 mm/rev 5.341 −14.553 2 0 120 m/s 0.1 mm/rev 6.136 15.758 3 0 60 m/s 0.13 mm/rev 12.256 21.767 6 3° 60 m/s 0.1 mm/rev 2.086 −6.387 4 3 240 m/s 0.13 mm/rev 3.191 10.079 7 7° 240 m/s 0.1 mm/rev 2.693 −8.605 5 3 120 m/s 0.07 mm/rev 5.341 14.553 6 3 60 m/s 0.1 mm/rev 2.086 6.387 8 7° 120 m/s 0.13 mm/rev 1.020 −0.172 7 7 240 m/s 0.1 mm/rev 2.693 8.605 9 7° 60 m/s 0.07 mm/rev 1.753 −4.876 8 7 120 m/s 0.13 mm/rev 1.020 0.172 9 7 60 m/s 0.07 mm/rev 1.753 4.876 Table 8 is the S/N response table for surface roughness; it gives each level a mean S/N ratio and the total mean S/N ratio. In addition, the different values of the S/N ratio between maximum and Table 8 is the S/N response table for surface roughness; it gives each level a mean S/N ratio and the minimum are also given in Table 8. It can be seen that the feed and rake angle have the highest total mean S/N ratio. In addition, the different values of the S/N ratio between maximum and minimum difference value, 10.3221 and 10.4239 respectively. According to the Taguchi method, the larger the are also given in Table 8. It can be seen that the feed and rake angle have the highest difference value, difference between the values of the S/N ratio are, the more effect there will be on surface roughness. 10.3221 and 10.4239 respectively. According to the Taguchi method, the larger the difference between Therefore, changing the feed and rake angle will change the value of surface roughness of the the values of the S/N ratio are, the more effect there will be on surface roughness. Therefore, changing simulation model significantly. the feed and rake angle will change the value of surface roughness of the simulation model signiﬁcantly. Table 8. S/N ratio response table. Table 8. S/N ratio response table. Mean S/N Ratio Mean S/N Ratio Symbol Cutting Parameter Cutting Parameter Symbol Level 1 Level 2 Level 3 Max-min Level 1 Level 2 Level 3 Max-min A Rake angle −14.8574 −8.5769 −4.5353 10.3221 A Rake angle 14.8574 8.5769 4.5353 10.3221 B Cutting speed −9.3391 −10.1607 −10.2373 0.8982 B Cutting speed 9.3391 10.1607 10.2373 0.8982 CC Feed Feed 4.5508 −4.5508 −11.0096 11.0096 −14 14.9747 .9747 10.423 10.42399 Total mean S/N ratio = −9.8046. Total mean S/N ratio = 9.8046. Figure 18 shows the S/N response graph for surface roughness. As mentioned in Equation (25) Figure 18 shows the S/N response graph for surface roughness. As mentioned in Equation (25) and Equation (26), a greater value of the S/N ratio will have a smaller variance of surface roughness and Equation (26), a greater value of the S/N ratio will have a smaller variance of surface roughness (the smaller the better). (the smaller the better). -2 -4 -6 -8 -10 -12 -14 -16 A1 A2 A3 B1 B2 B3 C1 C2 C3 Cutting parameter Level Figur Figure e 18. 18.C Cutting utting pa parameters rameters of of S/N S/N response response graph (A- graph (A-Rake Rake angle; angle; B- B- Cutting Cutting speed; speed; C-C- Feed). Feed). The variance analysis (ANOVA) is also a helpful and efﬁcient tool for testing and determining which cutting parameter have an obvious effect on surface roughness. As shown in Table 9, it shows the rake angle and feed have an obvious effect on the surface roughness, while the cutting speed compared Mean S/N ratio The variance analysis(ANOVA) is also a helpful and efficient tool for testing and determining Appl. Sci. 2019, 9, 654 19 of 23 which cutting parameter have an obvious effect on surface roughness. As shown in Table 9, it shows the rake angle and feed have an obvious effect on the surface roughness, while the cutting speed compared to other parameters has little effect on surface roughness, since their contributions in to other parameters has little effect on surface roughness, since their contributions in percentage percentage are 45.602, 46.721 and 1.0407 respectively. In addition, if we only consider these three are 45.602, 46.721 and 1.0407 respectively. In addition, if we only consider these three parameters parameters for lowering the surface roughness in cutting processing. The best of combinations of the for lowering the surface roughness in cutting processing. The best of combinations of the cutting cutting parameters (the level 3 of rake angle, the level 1 of cutting speed, and the level 1 of feed) can parameters (the level 3 of rake angle, the level 1 of cutting speed, and the level 1 of feed) can be be obtained according to analyze the S/N ratio and variance. obtained according to analyze the S/N ratio and variance. Table 9. Table 9. V Varia ariance nce analysis fo analysis forr surface roughness. surface roughness. Parameters Degree of Freedom Sum of Squares Mean Square F Ratio Contribution (%) Parameters Degree of Freedom Sum of Squares Mean Square F Ratio Contribution (%) Rake angle 2 160.15 80.073 28.493 45.602 Rake angle 2 160.15 80.073 28.493 45.602 Cutting speed 2 9.1471 4.5736 1.6275 1.0407 Cutting speed 2 9.1471 4.5736 1.6275 1.0407 Feed 2 163.94 81.969 29.168 46.721 Feed 2 163.94 81.969 29.168 46.721 Error 2 5.6202 2.8101 6.6345 Error 2 5.6202 2.8101 6.6345 Total 8 338.85 100 Total 8 338.85 100 4.3. Experimental Tests for Effect of Cutting Parameters on Surface Roughness 4.3. Experimental Tests for Effect of Cutting Parameters on Surface Roughness We apply the turning (dry cutting) and single-point cutting as our experimental test. Figure 19 We apply the turning (dry cutting) and single-point cutting as our experimental test. Figure 19 shows the experimental instruments. The cutting tool used carbide inserts and a tool holder. We shows the experimental instruments. The cutting tool used carbide inserts and a tool holder. We applied three different carbide inserts and a different tool holder. These inserts have the same nose applied three different carbide inserts and a different tool holder. These inserts have the same nose radius 0.8 mm and edge radius 20 m. After installation, the effective rake which is a combination of the radius 0.8 mm and edge radius 20 µm. After installation, the effective rake which is a combination of tool holders angle of inclination and the rake built into the insert, are about 0 , 3 and 7 . Figures 20–22 the tool holders angle of inclination and the rake built into the insert, are about 0°, 3° and 7°. Figures are the comparisons of surface roughness between the simulation results and experimental results 20–22 are the comparisons of surface roughness between the simulation results and experimental (cutting depth in turning is set 0.1 mm). From these ﬁgures, we can conclude that although there results (cutting depth in turning is set 0.1 mm). From these figures, we can conclude that although are differences between the surface roughness of the SPH cutting model calculated by SPM and there are differences between the surface roughness of the SPH cutting model calculated by SPM and the measured surface roughness from the experiment, they have the same trend under the same the measured surface roughness from the experiment, they have the same trend under the same changes of cutting parameters. The feed and rake angle have signiﬁcant effects on surface roughness. changes of cutting parameters. The feed and rake angle have significant effects on surface roughness. Figure 20 shows that the surface roughness will decrease as the rake angle increases. Figure 21 shows Figure 20 shows that the surface roughness will decrease as the rake angle increases. Figure 21 shows that the surface roughness will increase as the cutting feed increases. Figure 22 shows the cutting that the surface roughness will increase as the cutting feed increases. Figure 22 shows the cutting speed still affects the surface roughness; however, there is little change compared to the other two speed still affects the surface roughness; however, there is little change compared to the other two cutting parameters. cutting parameters. Workpiece Cutting tool Roughmeter Figure Figure 19. 19. Experimental Experimental i instr nstruments. uments. Appl. Sci. 2019, 9, 654 20 of 23 Simulation Simulation 6 Simulation 4 Experiment 4 Experiment Experiment 02 46 8 02 46 8 02 46 8 Rake angle (°) Rake angle (°) Rake angle (°) Figure 20. Comparison between the simulated results and experimental results under different rake Figure 20. Comparison between the simulated results and experimental results under different rake Figure Figure 20. 20. Comparison Comparison betw between een the the simulated simulated re results sults and and experimental experimental results results under under different rake different rake angles (v = 120 m/min, f = 0.1 mm/rev). angles (v = 120 m/min, f = 0.1 mm/rev). angles (v = 120 m/min, f = 0.1 mm/rev). angles (v = 120 m/min, f = 0.1 mm/rev). 6 Simulation 6 Simulation 6 Simulation Experiment Experiment 4 Experiment 0.05 0.07 0.09 0.11 0.13 0.15 0.05 0.07 0.09 0.11 0.13 0.15 0.05 0.07 0.09 0.11 0.13 0.15 Feed (mm) Feed (mm) Feed (mm) Figure 21. Comparison between the simulated results and experimental results under different feeds Figure 21. Comparison between the simulated results and experimental results under different feeds Figure 21. Comparison between the simulated results and experimental results under different feeds Figure 21. Comparison between the simulated results and experimental results under different feeds (v = 120 m/min, γ = 0°). (v = 120 m/min, g = 0 ). (v = 120 m/min, γ = 0°). (v = 120 m/min, γ = 0°). 6 Simulation 6 Simulation 6 Simulation Experiment Experiment 4 Experiment 0 100 200 300 0 100 200 300 0 100 200 300 Speed (m/min) Speed (m/min) Speed (m/min) Figure Figure 22. 22. Comparison Comparison bet between ween the s the simulated imulated resu results and lts experimental and experimental results under different results under different cutting Figure 22. Comparison between the simulated results and experimental results under different Figure 22. Comparison between the simulated results and experimental results under different speeds (f = 0.1 mm/rev, g = 0 ). cutting speeds (f = 0.1 mm/rev, γ = 0°). cutting speeds (f = 0.1 mm/rev, γ = 0°). cutting speeds (f = 0.1 mm/rev, γ = 0°). To summarize, the results of analyzing the S/N ratio and variance based on the SPH cutting model To summarize, the results of analyzing the S/N ratio and variance based on the SPH cutting To summarize, the results of analyzing the S/N ratio and variance based on the SPH cutting To summarize, the results of analyzing the S/N ratio and variance based on the SPH cutting are the same with the experimental results, because they have the same variation trend of surface model are the same with the experimental results, because they have the same variation trend of model are the same with the experimental results, because they have the same variation trend of model are the same with the experimental results, because they have the same variation trend of surface roughness value when the cutting parameters are changed. Therefore, the cutting parameters surface roughness value when the cutting parameters are changed. Therefore, the cutting parameters surface roughness value when the cutting parameters are changed. Therefore, the cutting parameters can be investigated and optimized by the SPH model. It should be known that the real cutting process can be investigated and optimized by the SPH model. It should be known that the real cutting process can be investigated and optimized by the SPH model. It should be known that the real cutting process -1 -1 -1 -1 -1 -1 -1 -1 -1 Surface roughness (10 µm) Surface roughness (10 µm) Surface roughness (10 µm) Surface Surface roughness roughness (10 (10 µm µm )) Surface Surface roughness roughness (10 (10 µm µm )) Surface Surface roughness roughness (10 (10 µm µm )) Appl. Sci. 2019, 9, 654 21 of 23 roughness value when the cutting parameters are changed. Therefore, the cutting parameters can be investigated and optimized by the SPH model. It should be known that the real cutting process is a highly complex process. All the existing models which simulate the cutting process simplify several factors that affect the cutting process, such as machine tool vibration and tool wear. Therefore, the simulation is not as accurate as experiments in the real world and it is hard to simulate the cutting model to predict the surface roughness accurately. We simulated the cutting process to predict the variance trend of surface roughness with different cutting parameters and then compared the simulation results with the real experimental results. From another perspective, it might add more errors if we compare the simulated results with another simulation (there will be an accumulation of errors which can lead to more inaccurate results). After serious consideration, we compared our simulated results with real experimental results to analyze the variation trend of surface roughness with different cutting parameters. Although we ignore a lot of factors (cutting edge angle, the vibration in actual cutting process et al.), we only consider the cutting parameters which have signiﬁcant inﬂuences on surface roughness and can also be controlled by the 2D SPH cutting model. In the experiment, all the parameters remain the same except the three parameters (cutting speed, rake angle and feed). 5. Conclusions We applied an improved SPH method to simulate the Ti–6AL–4V cutting process and proposed a Surface Particle Method (SPM) to evaluate the variance trend of surface roughness with different cutting parameters. Then we employed the Taguchi method to investigate the effects of cutting parameters on the variance trend of surface roughness. The investigation applied the S/N ratio and variance analysis to analyze the effect of different levels of cutting parameters. We found that the rake angle and feed have an obvious effect on surface roughness, while the variance of cutting speed had little inﬂuence on the surface roughness. Comparing our results with the experiments, we concluded that although the surface roughness calculated based on the simulated surface particles is different from the measured one in the experiment, they had the same variation trend. This means we can use this SPH model to investigate the effects of cutting parameters on surface roughness effectively. Our work proved there are merits to using the improved SPH cutting model to optimize cutting parameters. These merits are listed as: 1. The improved SPH method avoids the misconvergence caused by mesh distortion of FEM and increases the accuracy of the simulation. Meanwhile, the TANH constitutive law improves the accuracy of the description for the dynamic properties of the material under a high stress rate compared with JC constitutive law. 2. This paper proposed a Surface Particle Method (SPM) to evaluate the variance trend of surface roughness with different cutting parameters. Although there are differences between the surface roughness calculated by SPM and the measured surface roughness from the experiment, they have the same variation trend when the cutting parameters are changed. Thus, the surface roughness based on SPM is valid as an alternative for the measured variance trend of surface roughness with different cutting parameters for investigating and optimizing cutting parameters. 3. This paper applied the Taguchi method to test the inﬂuence of cutting speed and feed and rake angle on the surface roughness of a Ti–6AL–4V workpiece. The Taguchi method considerably reduces the number of experimental groups under different cutting parameters and provides a solid theoretical foundation for optimizing cutting parameters. 4. This work proves that the effects of cutting parameters on surface roughness can be investigated through simulation instead of real experiments for cutting Ti–6AL–4V. This avoids the laborious experimental process and substantially reduces work time and cost. Appl. Sci. 2019, 9, 654 22 of 23 Author Contributions: All authors contributed to the work. W.N. established the SPH cutting model and performed the data analysis; R.M. guided the research work and designed the experiment. Z.C. and N.W. edited/proof-read the paper. Acknowledgments: This research work is supported by the National Natural Science Foundation of China (No. 51775445, No.51875475). Xi’an science and technology project (201805042YD20CG26-(9)). Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. D’addona, D.M.; Raykar, S.J.; Narke, M.M. High speed machining of Inconel 718: Tool wear and surface roughness analysis. Procedia CIRP 2017, 62, 269–274. [CrossRef] 2. Kilickap, E. 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Publisher: Multidisciplinary Digital Publishing Institute
Copyright: © 1996-2019 MDPI (Basel, Switzerland) unless otherwise stated
ISSN: 2076-3417
DOI: 10.3390/app9040654
Publisher site: See Article on Publisher Site

Abstract

applied sciences Article Investigating the Effect of Cutting Parameters of Ti–6Al–4V on Surface Roughness Based on a SPH Cutting Model Weilong Niu , Rong Mo *, Zhiyong Chang and Neng Wan Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Ministry of Education, Northwestern Polytechnical University, 127 Youyi Road, Xi’an 710072, Shanxi, China; weilong0723@gmail.com (W.N.); Changzy@nwpu.edu.cn (Z.C.); wanneng@nwpu.edu.cn (N.W.) * Correspondence: morong@nwpu.edu.cn; Tel.: +86-029-88495354 Received: 16 January 2019; Accepted: 4 February 2019; Published: 14 February 2019 Abstract: This work establishes a 2D numerical model to simulate the cutting process of workpieces made of Ti–6Al–4V, by applying an improved Smoothed Particle Hydrodynamics algorithm together with a modiﬁed constitutive model based on the Johnson–Cook model known as Hyperbolic Tangent (TANH). The location information of the surface particles obtained by the SPH cutting model are used to evaluate the variation trend of surface roughness with different parameters. Parameters that affect the surface roughness are investigated in detail by using the Taguchi method and the SPH cutting model. The present work provides an efﬁcient and cost-effective approach to determine the optimal parameters for cutting processes for Ti–6AL–4V workpieces, through computer simulations in virtual environments, instead of expensive and time-consuming cutting experiments using actual workpieces. Keywords: smoothed particle hydrodynamics; hyperbolic tangent; taguchi method; Ti–6Al–4V; surface roughness 1. Introduction Ti–6AL–4V is one of most popular titanium alloys in many engineering applications. It is an important material for modern mechanical components and equipment, especially in biomedical and aerospace systems, since it not only has superb corrosion resistance, but also has a high strength to weight ratio. Machining of Ti–6AL–4V workpieces, however, is tricky, and it is very difﬁcult to produce the desired components with the required shape and high-quality surface ﬁnishing. Cutting is the most popular and important way to fabricate titanium components. As the technology advances, more and higher requirements are necessary for cutting the titanium alloy, especially for quality surfaces. Surface roughness, as an effective measure to evaluate the quality of a surface, is affected by different settings of cutting parameters such as different cutting speed, rake angle and feed [1]. Thus, investigation of the effects and subsequent optimization of the cutting parameters on surface roughness is crucial and, currently, the Taguchi method is one of the most popular methods used. [2–4]. It ﬁrst establishes orthogonal experiment to decrease the number of experiments and, further, to reduce experimental time. The surface roughness is then measured and recorded in these experiments, according to which the effectiveness of each combination of cutting parameters is evaluated and conﬁrmed by Taguchi S/N ratios and the variance analysis (ANOVA) [4]. Finally, an optimal set of cutting parameters is found to achieve the lowest surface roughness. Previous research has conducted experiments to optimize cutting parameters to minimize the surface roughness value based on the Taguchi method [5–13]. Sahoo and Pradhan studied the inﬂuence of process parameters using the Taguchi method and provided Appl. Sci. 2019, 9, 654; doi:10.3390/app9040654 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 654 2 of 23 optimized parameters by applying an uncoated tungsten carbide tool in the machining Al/SiCp metal matrix composite without adding cutting ﬂuid [5]. Rao and Padmanabhan applied the Taguchi method to study the inﬂuence of cutting parameters on the metal removal rate [6]. Motorcu studied the inﬂuence of cutting parameters on surface roughness, such as feed rate, cutting speed, and drill bit angle, and then the optimum levels of control factors are deﬁned to reduce the surface roughness by applying S/N ratios [7]. Measuring surface roughness by conducting experiments is the current practice to study the effects of cutting parameters on surface roughness. It is, however, costly as well as time-consuming, involving a substantial waste of materials. This problem is not negligible especially for some material which is very expensive and very hard to cut such as titanium alloy. Even though the number of experiments can be decreased substantially by applying an orthogonal array of the Taguchi method, the costs and time taken for the experiments may not be acceptable. With the rapid development of computer technology, the advantages of numerical simulation of cutting processes attracts a lot of attention and has become an important means for analyzing the mechanism of cutting processes. This paper proposes an approach to investigate the effects of cutting parameters on surface roughness by combining the numerical simulation of the cutting process and the Taguchi method. Compared to other experiments, our approach is applicable to investigate the effects of cutting parameters with substantially reduced costs and time. This paper gives an example of turning to show how our method works. First, a cutting model is simpliﬁed as the orthogonal cutting model. Then a numerical simulation based on the SPH method is applied to simulate the cutting process as an alternative to the real cutting experiment. Furthermore, the surface roughness of this simulation model is calculated and the variation trend of surface roughness with different cutting parameters is evaluated. After that, we employ the Taguchi method which involves establishing an orthogonal simulation array, applying an S/N ratio and ANOVA to investigate the effects of cutting parameters on surface roughness. Finally, we compare our result with experiments and make evaluations. There are two key components in our work. The ﬁrst one is to establish a reliable and accurate numerical cutting model, and the second one is to come up with a method to evaluate the variation trend of surface roughness with different cutting parameters. As for the ﬁrst component, most of the traditional numerical models apply the Finite Element Method (FEM) which bears problems in handling with large deformations and material fragmentations during the cutting process and sometimes leads to simulation breakdowns caused by excessive mesh distortion [8–11]. At the same time, the simulation result is sensitive to different ways of generating the mesh, which affect the accuracy of the simulation [14]. Another problem of FEM is that it is not capable of deriving the surface roughness from the simulated surface since FEM simulates the cutting process, including chips formation, through setting chip separation criteria and deleting elements [9]. To avoid these problems, we applied Smoothed Particles Hydrodynamic (SPH) method, a Lagrangian meshfree method, to establish the numerical cutting model. SPH divides material into individual particles with independent properties, such as density, mass and speed, in the process of simulation instead of generating mesh for materials and cutting tools. Compared with FEM, its adaptivity can be obtained at the early stage of approximation of ﬁeld variables, and its formula is not affected by the distribution of particles. Thus, SPH is capable of handling large deformation, which always occurs during the cutting process, and at the same time, naturally simulates the process of chip separation. Previous studies have proven that SPH can simulate cutting processes effectively [8–11,15,16] It is well known that the Johnson–Cook (JC) model [17] and Johnson–Cook damage [18,19] model are the most popular constitutive material laws in describing metal behavior. High strain and high strain rates, however, always occur in Ti–6AL–4V machining process, and under this condition, the dynamic mechanical properties of the material, such as strain softening and dynamic recrystallization mechanisms of Ti–6AL–4V materials, cannot be exactly described by JC law. This paper, hence, applies Appl. Sci. 2019, 9, 654 3 of 23 Hyperbolic Tangent (TANH) constitutive law, which was proposed by Calamaz [20] and later further developed by the Sima M and Özel T [21] to describe dynamic mechanical properties of Ti–6AL–4V material. The TANH constitutive model not only considers those problems which are mentioned above, but also describes the process of dynamic recrystallization mechanisms which also always occurs in the cutting Ti–6Al–4V process [22]. In addition, an improved SPH algorithm is adopted through adding modiﬁed schemes for approximating density (density correction) and kernel gradient approximation (kernel gradient correction) in this work, which has been proven to be very efﬁcient in improving the accuracy of a cutting model [15]. Since the traditional SPH cannot exactly reproduce linear functions in the entire problem domain, in its initial form particle summation formulation, the SPH does not exactly reproduce a constant near the boundary because of the loss of symmetry in the smoothing operation. The second component involves the evaluation of the variation trend of surface roughness with different cutting parameters. The SPH algorithm adopted in this paper naturally simulates the separation of chips from the workpiece. Each individual particle carries independent attributes, such as density, mass, speed, and location. After examining the location of each individual particle of the workpiece surface and incorporating it into the deﬁnition and calculation of surface roughness, this paper proposes a method, Surface Particles Method (SPM), to calculate the surface roughness of the SPH model. Then we evaluate the variation trend of surface roughness based on the surface roughness of the SPH cutting model by changing the cutting parameters during the simulation of the cutting process. To summarize, this paper (1) adopts an improved SPH algorithm and the TANH constitutive law to establish an improved SPH cutting model and make the simulation cutting model more accurate; (2) proposes a Surface Particles Method (SPM) to evaluate the variation trend of surface roughness with different parameters instead of the real experiments for cutting Ti–6AL–4V and substantially reduces work time and cost; (3) investigates the inﬂuence of cutting parameters on surface roughness during cutting by the Taguchi method and avoids the laborious experimental process; and 4) ﬁnally, compares our simulation results with experimental results to testify the effectiveness. The comparison shows that our method is reliable and effective to investigate the effects of cutting parameters on surface roughness. 2. SPH Modeling of Cutting Model 2.1. SPH Basic Principles Lucy came up with SPH method in 1977 and the original purpose of this method was to solve astrophysical problems [23,24]. It was then extensively used in many different areas, such as ﬂuid mechanics and solid mechanics [25,26]. Previous studies have proven that SPH can solve the ﬂuent problem effectively, such as free surface ﬂow [27,28] and bubble rising [29,30]. The cutting problem can also be regarded as a ﬂuent (“solid-ﬂow”) problem, since the solid deforms drastically when the solid materials are machined by a cutting tool at high speed, and thus it can be treated as a “ﬂow” of solid materials. Therefore, SPH should be an effective method for simulating cutting processes [11,15,31,32]. The kernel approximation and particles approximation are two very important steps in the SPH method: 2.1.1. Kernel Approximation: h f (x )i = f x W x x , h dx (1) i j i j j hr f (x )i = f x rW x x , h dx (2) i j i j j W Appl. Sci. 2019, 9, 654 4 of 23 where x is a three dimension coordinate vector; x and x are the positions of particles i and j; f is i j the function of x; f (x ) is the value of f (x) at particle j; and Kernel function W represent a weighted contribution of particle j to particle i. The gradient of kernel function rW can be expressed as x x i j ¶W rW = . h is the smooth length, which is used to deﬁne the inﬂuence domain of the smooth r ¶r ij ij kernel function. In this work, we choose the cubic spline function [33], which closely resembles a Gaussian function [25], as our smoothing function. It is expressed as following: 2 1 2 3 q + q , 0 q 1 3 2 W = a (3) ij d 1 3 (2 q) , 1 q 2 where a is the normalization factor that can be expressed as a = 15/ 7ph . It is always used to d d solve 2D problems. q is the normalized distance between particle i and j, which can be expressed as q = r /h. The value of the smoothing length is chosen based on actual problems, which means the ij larger value may affect the computational efﬁciency, while the smaller value may cause low accuracy of computation. In this study, we choose the 1.5 times particle spacing as the smoothing length [15]. 2.1.2. Particle Approximation In the SPH algorithm, the computation domain can be discretized with a set of particles by using DV instead of the dx in Equations (1) and (2): j i DV = (4) where m is the mass of particle j. r is the density of particle j. N is the total number of particles which j j are involved in the computational ﬁeld. Substituting Equation (4) into Equation (1) and Equation (2): h f (x )i = f x W x x , h (5) i å j i j j=1 hr f (x )i = f x rW x x , h (6) i å j i j j=1 From above, it should be noticed that some macroscopic variables (temperature, pressure, density, etc.) are expressed as integral form by a set of disorderly point values in the SPH method. In this set of points, the interaction between various points are described and a kernel estimate of a point is obtained by an interpolation function. In the SPH equation, the governing equation is the Lagrangian form of Navier–Stokes(NS) equations [14], which is used for governing the materials. The mass conservation law and momentum conservation laws can be expressed as following: dr ¶v = r (7) dt ¶x a ab dv 1 ¶s = + f (8) dt r ¶x where v is the velocity; t is the time; the coordinate direction can be denoted by a and b; f is the ab component of acceleration; and s is the total stress tensor which consists of two parts: ab ab ab s = Pd + t (9) ab ab where P is pressure; d is the Kronecker tensor; and t is the deviatoric shear stress. Appl. Sci. 2019, 9, 654 5 of 23 The shear stresses are generated and change with time when solids are in the dynamic velocity ﬁeld with spatial gradients. The stress t can be expressed as following [14]: ab dt . 1 . . . ab gg bg ag ab ag gb = 2G(# d # ) + t r + t r (10) i i i i i i dt 3 ab where # is the strain rate tensor; G is the shear modulus which can be obtained by an experimental test; ab and r is the rotation tensor. After elastic deformation, the material will enter the plastic deformation named plastic behavior and it can be identiﬁed by the critical point of plastic deformation [14]: f = p (11) 3J where J is the second stress tensor invariant. s is the ﬂow stress and its initial value is the shear stress 2 y ab ab (s = t t ). When the value of f is lower than 1 ( f < 1), the plastic deformation is said to occur. y y y As mentioned above, the traditional SPH method has low accuracy [34]. Therefore, in this study, we employed an improved SPH method where density and kernel gradient are corrected by Moving Least Squares and a correction matrix. It has been proven to be efﬁcient in improving the accuracy of the cutting model [15]. In the scheme of the improved SPH algorithm, we use the Moving Least Squares (MLS) method to correct the density ﬁeld periodically; the linear variation of the density ﬁeld can be exactly reproduced and a smoother pressure can be obtained [35]. The correction scheme is expressed as: N N MLS MLS hr i = r W = m W (12) i å j å j ij ij j j MLS where W is the MLS kernel; more details can be found in Reference [35]. ij The Kernel Gradient Correction (KGC) method is applied to improve the SPH approximation accuracy. The second order accuracy of SPH approximation can also be restored by this correction new scheme. The corrected value of r W for arbitrary particle i is expressed as follows: ij new r W = L(r ) r W (13) i ij i i ij where L(r ) is the correct matrix, which can be expressed as: 2 3 ¶W ¶W ij ij x y ij ji ¶x ¶x i i 4 5 L(r ) = V (14) i å j ¶W ¶W ij ij j x y ji ji ¶y ¶y i i Finally, the SPH computational domain is discretized by a limited number of particles which have their own attributes such as mass (m) and individual space. The continuity equation ﬁnally is expressed as following: m ¶W dr j b ij = r v (15) iå dt r ¶x j=1 The momentum equation is expressed as following: 2 3 ab ab a N t + t P + P ¶W dv i j i j ij i ab ab a 4 5 = m d + d + f (16) å j Õ dt r r r r i j i j ¶x j=1 ij where is called artiﬁcial viscosity and is used to reduce the unphysical oscillations in the numerical ij results around the shocked region. Appl. Sci. 2019, 9, 654 6 of 23 2.2. Material Constitutive Model Johnson–Cook (JC) is the most frequently adapted constitutive law to describe the dynamic mechanism properties (strain hardening, strain hardening rate and thermal softening) of mental material when the elastic deformation transforms to the plastic deformation stage. The JC constitutive law, however, does not take into account the strain-softening of the material under high strain conditions. Cutting Ti–6AL–4V, however, is a high stress and high stress rate process, and JC law cannot describe this process accurately. To solve the problems of JC law, Calamaz proposed a TANH constitutive law which modiﬁes the stress, strain and temperature components of JC law by adding a, b, c, d parameters and a tanh function [20]. Later, Sima M and Özel T introduced exponent r into the TANH constitutive model to further control the tanh function for thermal softening [21]. The TANH constitutive model not only considers the stress softening of the material as shown in Figure 1, but also considers the recrystallization phenomenon. Figure 1 is the ﬂow stress-strain curve with the TANH law. N # r e f f 1 TT 1 room s = A + B( ) 1 + Cln . 1 (D + (1 D)(tanh( )) ) (17) y a T T exp(# ) # room (#+S) melt e f f 0 D = 1 ( ) (18) S = (19) T = e/C + T (20) p room . . where # is the equivalent plastic strain; # is the equivalent plastic strain rate; # is the reference e f f e f f 0 plastic strain rate; A, B, C, N, and M are material-dependent constants which are the same parameters as in the JC equation and are obtained by SHPB tests [36]. T is the room temperature. T is the room melt melting temperature of Ti–6AL–4V. T is the current temperature which is computed by Equation (20). a, b, c, d and r are corrected parameters which are obtained by the control variable method. We vary one parameter while keeping the others constant to investigate the effects of the parameters on strain-stress curves and make the strain-curve of the TANH model be in good agreement with that of the JC model under the low strain (the strain can be considered low strain when the strain is lower than 0.5). As shown in Figure 2a, parameter a controls the strain-hardening part by decreasing ﬂow stress after a critical strain value, and a low value of the parameter will lead to a big difference between TANH and the SHPB test data. As a increases, the maximum stress will increase. Parameter b in (Equation (19)) controls the temperature-dependent ﬂow-softening effect and points out where the maximum ﬂow stress is attained. The lower the value of parameter b is, the lower the strain value of the peak ﬂow stress that occurs, as shown in Figure 2b. Parameter d is an exponent for the temperature that controls the degree of temperature dependency of parameter D as given in (Equation (18)). As shown in Figure 2c, parameter d has a strong impact on the value of ﬂow softening and it determines the minimum ﬂow stress value. As shown in Figure 2d, parameter r controls the softening trend. The higher the parameter r is, the faster the ﬂow stress-strain curve enters the softening region. With the strain increasing, the slope of strain softening curves will decrease and the maximum ﬂow stress remains unchanged. The c has similar effects to r on the ﬂow stress [16]. After observing the trend of the curve with different parameters, the best ﬁt parameters and best stress-strain curve are given in Table 1 and Figure 1. 2.2. Material Constitutive Model Johnson–Cook (JC) is the most frequently adapted constitutive law to describe the dynamic mechanism properties (strain hardening, strain hardening rate and thermal softening) of mental material when the elastic deformation transforms to the plastic deformation stage. The JC constitutive law, however, does not take into account the strain-softening of the material under high strain conditions. Cutting Ti–6AL–4V, however, is a high stress and high stress rate process, and JC law cannot describe this process accurately. To solve the problems of JC law, Calamaz proposed a TANH constitutive law which modifies the stress, strain and temperature components of JC law by adding a, b, c, d parameters and a tanh function [20]. Later, Sima M and Özel T introduced exponent r into the TANH constitutive model to further control the tanh function for thermal softening [21]. The TANH constitutive model not only considers the stress softening of the material as shown in Figure 1, but also considers the recrystallization phenomenon. Figure 1 is the flow stress-strain curve with Appl. Sci. 2019, 9, 654 7 of 23 the TANH law. x 10 3.5 TANH to a big difference between TANH 3 and the SHPB test data. As a increases, the maximum stress will increase. Parameter b in (Equation (19)) controls the temperature-dependent flow-softening effect and points out where the maximum flow stress is attained. The lower the value of parameter b is, the 2.5 lower the strain value of the peak flow stress that occurs, as shown in Figure 2b. Parameter d is an exponent for the temperature that controls the degree of temperature dependency of parameter D as given in (Equation (18)). As shown in Figure 2c, parameter d has a strong impact on the value of flow softening and it determines the minimum flow stress value. As shown in Figure 2d, parameter r 1.5 controls the softening trend. The higher the parameter r is, the faster the flow stress-strain curve enters the softening region. With the strain increasing, the slope of strain softening curves will decrease and the maximum flow stress remains unchanged. The c has similar effects to r on the flow 0 1 2 3 4 5 stress [16]. After observing the trend of the curve wi True strain th different parameters, the best fit parameters and best stress-strain curve are given in Table 1 and Figure 1. Figure 1. The stress-strain curve. Figure 1. The stress-strain curve. 8 8 x 10 x 10 5 5   ε   1 TT − 1 N eff r room 4.5 4.5 σ = AB++ ( ) 1 Cln  1− (D+ (1−D)(tanh( )) )  (17)    y a b= c 0.1 a=0.1 exp(εε ) TT −  ε + S  eff 0 melt room ()    b=0.5 4 4 a=1   b=1 a=2 3.5 3.5 b=5 a=5 JC JC D =− 1( ) 3 3 (18) 2.5 2.5 a=2; c=2; d=2; r=1 b=5; c=2; d=2; r=1 2 2  (19) S = 1.5 1.5   m 1 1 0 1 2 3 4 5 0 1 2 3 4 5 True strain True strain Te=+ /C T (20) 8 8 p room (a) (b) x 10 x 10 5 5 where 𝜀 is the equivalent plastic strain; 𝜀 is the equivalent plastic strain rate; 𝜀 is the 4.5 d=0.1 r=0.01 reference plastic strain rate; A, B, C, N, and M are material-dependent constants which are the same d=0.5 r=0.1 d=1 r=0.5 parameters as in the JC equation and are obtained by SHPB tests [36]. 𝑇 is the room temperature. d=5 r=1 3.5 JC 𝑇 is the melting temperature of Ti–6AL JC –4V. T is the current temperature which is computed by Equation (20). a, b, c, d and r are corrected parameters which are obtained by the control variable 2.5 method. We vary one parameter while keeping the others constant to investigate the effects of the a=2; b=5; c=2; r=1 parameters on strain-stress curves and make the strain-curve of the TANH model be in good agreement with that of the JC model under the low s 1.t5rain (the strain can be considered low strain a=2; b=5; d=2; c=1 when the strain is lower than 0.5). As shown in Figure 2a, parameter a controls the strain-hardening 0 1 0 1 2 3 4 5 0 1 2 3 4 5 part by decreasing ﬂow st True re sss trainafter a critical strain value, and a low val Tu rue e of strai t n he parameter will lead (d) (c) Figure 2. JC strain-stress curve and TANH strain-stress curve with varying value parameters: the Figure 2. JC strain-stress curve and TANH strain-stress curve with varying value parameters: the strain −1 strain rate 2000 s ((a): vary parameter a; (b): vary parameter b; (c): vary parameter d; (d): vary rate 2000 s ((a): vary parameter a; (b): vary parameter b; (c): vary parameter d; (d): vary parameter r). parameter r). Table 1. Material properties of Ti–6AL–4V for workpiece [36,37]. Physical Parameter Constant Work Material (Ti–6AL–4V) Density, ρ0 (Kg/m) 4430 Possion’s ratio v 0.342 Shear modulus (G) 110 Gpa Specific heat, Cp J/(Kg·K) 580 Thermal conductivity (W/mk) 7.3 T (℃) melt 1878 T (℃) 25 room A 968 Mpa TANH model (Equation (17)) B 380 Mpa C 0.02 Flow stress (MPa) Flow stress (MPa) Flow stress [MPa] Flow stress (MPa) Flow stress (MPa) Appl. Sci. 2019, 9, 654 8 of 23 Table 1. Material properties of Ti–6AL–4V for workpiece [36,37]. Physical Parameter Constant Work Material (Ti–6AL–4V) Density, r (Kg/m ) 4430 Possion’s ratio v 0.342 Shear modulus (G) 110 Gpa Speciﬁc heat, C J/(KgK) 580 Thermal conductivity (W/mk) 7.3 T ( C) 1878 melt T ( C) 25 room A 968 Mpa B 380 Mpa C 0.02 N 0.577 M 0.421 TANH model (Equation (17)) a 0.05 b 2 c 1 d 1 r 5 2.3. Contact Algorithms Figure 3a provides a conceptual explanation of the particle distribution and interaction and how the contact algorithm is used to calculate the contact force. In this work, the total number of particles -9 is 25,000 and the time step is 3.0 s. (When the number of particles is greater than 25,000 or the time -9 step is less than 3.0 s. The accuracy of the model does not change obviously, while the running time of the program will increase with the increase of particles and decrease of time step). The cutting tool and material are regarded as groups of particles in our improved SPH cutting model. We treat the cutting tool as a rigid body without carrying any attributes as the workpiece material, such as density and quality. The location information (x , y ) and velocity information (u , v ) are only needed to set k k k k up for the particles of the cutting tool, which is computed by the initial location and velocity of the cutting tool respectively. Then, this information is used to calculate surface vectors (unit tangent vector and unit normal vector). We can use them to determine the direction of the cutting force and contact location between the cutter and workpiece respectively. When the cutter contacts the workpiece, two different kinds of particles (i.e. particles of the workpiece and particles of the cutter) occur contact force. To acquire the contact force, we need to 1) set a threshold d to decide when the contact is considered to occur, and 2) calculate the contact force. Figure 3b illustrates how the cutting tool and workpiece are regarded as the contact. P is the vertical point from particle i to the surface of the cutting tool. d is the actual vertical distance between particle i and the cutting tool. d is the contact threshold for these two kinds of particles. When d is 0 p smaller than d , these two particles contact. The tangent vector t and normal vector n at point p are 0 p p calculated by the following Equation: x x y y k+1 k k+1 k t = t , t = , p x y jx x j jx x j k+1 k k+1 k (21) y y x x : k+1 k k+1 k n = t , t = , p y x x x x x j j j j k+1 k k+1 k where x and x are the coordinate values of cutting tool particle k and k + 1. k +1 k Appl. Sci. 2019, 9, 654 9 of 23 Regular Cutting tool Irregular Rigid body k+1 Workpiece (a) (b) Figure 3. (a) Particles distribution and (b) particles interaction. Figure 3. (a) Particles distribution and (b) particles interaction. The next step is to calculate the contact force between the particles of the workpiece and cutting The next step is to calculate the contact force between the particles of the workpiece and cutting tool. The forces on point i have two parts, the force along the normal direction and the force along the tool. The forces on point i have two parts, the force along the normal direction and the force along tangential direction. The forces are calculated as the following Equation: the tangential direction. The forces are calculated as the following Equation: < F =  d d n ni 2 0 p p (Dt)  (22) Fn =− dd ⋅ n () o ni 2 0 p p mF m ti i  Δt F = min ()jF j , v t t  ti ni pi p p  jF j Dt ti (22)   μF m  where m is the mass of particle i; v =v v , v is the velocity at point p. v is the velocity of particle τi i i pi p i p i FF=⋅ min , v ⋅ττ  ()   τini pipp  i. m = 0.3 is the friction coefﬁcient between the workpiece and cutting tool [15,35]. According to the F Δt τi     formulas above, the interaction forces between the cutting tool and workpiece are calculated [15]. v=v -v v m p v i pi p i p i i 3. where OptimalC is parameters the mass of particle ; , is the velocity at point . is the μ =0.3 velocity of particle i. is the friction coefficient between the workpiece and cutting tool [15,35]. 3.1. Evaluation of the Surface Roughness of the SPH Cutting Model According to the formulas above, the interaction forces between the cutting tool and workpiece are The surface roughness is the repeating series of peaks and valleys on the surface. It is calculated calculated [15]. as the average deviation from the ideal along the direction of the normal vector of the real surface. Figure 4 illustrates the surface proﬁle of a workpiece sample. The x axis is regarded as the central line. 3. OptimalC parameters In the real measurement, the surface of the workpiece is discretized into measurement points. The vertical distance from the point to the center line is measured, and then the average value of these 3.1. Evaluation of the Surface Roughness of the SPH Cutting Model vertical distances is calculated as surface roughness R . The more measurement points there are, the The surface roughness is the repeating series of peaks and valleys on the surface. It is calculated more accurate R is. The equation is expressed as: as the average deviation from the ideal along the direction of the normal vector of the real surface. Figure 4 illustrates the surface profile of a workpiece sample. The x axis is regarded as the central R = jy(x )j (23) a å i line. In the real measurement, the surface of the workpiece is discretized into measurement points. The vertical distance from the point to the center line is measured, and then the average value of these vertN icais l d the ista number nces is cof alc sampling ulated as sur points. face roughness R . The more measurement points there are, the more accurate R is. The equation is expressed as: 1 N Ry = ()x (23) ai  N is the number of sampling points. Appl. Sci. 2019, 9, 654 10 of 23 Ra Ra Roughness Roughness center line center line Figure 4. Surface roughness proﬁle of workpiece. Figure 4. Surface roughness profile of workpiece. Figure 4. Surface roughness profile of workpiece. Similar to calculating the surface roughness derived from the vertical distances of measurement Similar to calculating the surface roughness derived from the vertical distances of measurement Similar to calculating the surface roughness derived from the vertical distances of measurement points, we extract particles on the simulated surface from the improved SPH cutting model and points, we extract particles on the simulated surface from the improved SPH cutting model and points, we extract particles on the simulated surface from the improved SPH cutting model and transform their location information into a coordinate (The SPH cutting model is established in the transform their location information into a coordinate (The SPH cutting model is established in the transform their location information into a coordinate (The SPH cutting model is established in the coordinate system and each particle has the corresponding particle number and coordinate position coordinate system and each particle has the corresponding particle number and coordinate position coordinate system and each particle has the corresponding particle number and coordinate position information. When the surface particles are extracted, the particle number of surface particles are information. When the surface particles are extracted, the particle number of surface particles are information. When the surface particles are extracted, the particle number of surface particles are obtained as well as the coordinate information of the particles). Figure 5 demonstrates the particles of obtained as well as the coordinate information of the particles). Figure 5 demonstrates the particles obtained as well as the coordinate information of the particles). Figure 5 demonstrates the particles the simulated surface after cutting, and Figure 6 illustrates the surface proﬁle of the extracted particles of the simulated surface after cutting, and Figure 6 illustrates the surface profile of the extracted of the simulated surface after cutting, and Figure 6 illustrates the surface profile of the extracted under a coordinate system. We extract the coordinate information of these particles and calculate the particles under a coordinate system. We extract the coordinate information of these particles and particles under a coordinate system. We extract the coordinate information of these particles and surface roughness. calculate the surface roughness. calculate the surface roughness. Figure Figure 5. 5. Distribution of Distribution ofsurface partic surface particles les ofof ththe e workpiece a workpiecefter cutting ( after cutting cutting parameters are (cutting parameters ar v e = Figure 5. Distribution of surface particles of the workpiece after cutting (cutting parameters are v = v 240 m/min, = 240 m/min, f = 0.07 f = mm, 0.07 mm, γ = 0° ). = 0 ). 240 m/min, f = 0.07 mm, γ = 0°). -6 Appl. Sci. 2019, 9, 654 11 of 23 x 10 -3 -6 -4 x 10 -3 -5 -4 -6 -5 -7 -6 -8 -7 -9 0 50 100 150 200 250 -8 Particle number -9 Figure 6. The coordinate location of surface particles after cutting (cutting parameters are v = 240 0 50 100 150 200 250 Particle number m/min, f = 0.07 mm, γ = 0°, the red circles are invalid points). Figure 6. The coordinate location of surface particles after cutting (cutting parameters are Figure 6. The coordinate location of surface particles after cutting (cutting parameters are v = 240 Based on the coordinate, we calculate the vertical distance from each particle to the center line. v = 240 m/min, f = 0.07 mm, = 0 , the red circles are invalid points). m/min, f = 0.07 mm, γ = 0°, the red circles are invalid points). In Figure 4, the y coordinate value of the center line is the difference between workpiece height and Based on the coordinate, we calculate the vertical distance from each particle to the center line. cutting depth. Based on the coordinate, we calculate the vertical distance from each particle to the center line. In Figure 4, the y coordinate value of the center line is the difference between workpiece height and In Figure 4, the y coordinate value of the center line is the difference between workpiece height and f =− Hh centerline (24) cutting depth. cutting depth. f = H h (24) centerline where H is the high value of the workpiece and h is the cutting depth. What is worth noting is, since f =− Hh centerline (24) there is a small amount of random errors in the simulating process, a tiny number of particles deviate far from the center line, which does not conform to the real cutting process (When the distance where H is the high value of the workpiece and h is the cutting depth. What is worth noting is, since where H is the high value of the workpiece and h is the cutting depth. What is worth noting is, since between a surfac there is a smalle p amount articleof an random d the roughness center errors in the simulating line is gr process, eater tahtiny an 0.number 5 µm, we of cons particles ider t deviate his there is a small amount of random errors in the simulating process, a tiny number of particles deviate particle deviate far from the center line). Thus, we regard these particles as outliers and remove them far from the center line, which does not conform to the real cutting process (When the distance between far from the center line, which does not conform to the real cutting process (When the distance when a surface calculat particle ing sur and face the roughnes roughness s. Fig center ure 7 is line the is st greps of su eater than rface 0.5 roughne m, we consider ss calcul this ation particle . deviate between a surface particle and the roughness center line is greater than 0.5 µm, we consider this far from the center line). Thus, we regard these particles as outliers and remove them when calculating particle deviate far from the center line). Thus, we regard these particles as outliers and remove them surface roughness. Figure 7 is the steps of surface roughness calculation. when calculating surface roughness. Figure 7 is the steps of surface roughness calculation. Surface Invalid points Surface Invalid points Roughness center line Roughness center line Figure 7. The steps of surface roughness calculation. Figure 7. The steps of surface roughness calculation. Figures 8–10 show the distribution of surface particles extracted from the cutting model under Figures 8–10 show the distribution of surface particles extracted from the cutting model under different cutting parameters. These ﬁgures show that while the higher feed and smaller rake angle Figure 7. The steps of surface roughness calculation. different cutting parameters. These figures show that while the higher feed and smaller rake angle lead to a more disordered distribution of the surface particles (i.e., the more rough surface), the cutting lead to a more disordered distribution of the surface particles (i.e., the more rough surface), the Figures 8–10 show the distribution of surface particles extracted from the cutting model under different cutting parameters. These figures show that while the higher feed and smaller rake angle lead to a more disordered distribution of the surface particles (i.e., the more rough surface), the Y cutting speed does not exert a significant effect on the surface roughness since the order of particles did not change too much compared to the other two cutting parameters when cutting speed was Appl. cutting speed Sci. 2019, 9, 654 does not exert a significant effect on the surface roughness since the order of partic 12 of les 23 changed. did not change too much compared to the other two cutting parameters when cutting speed was cutting speed does not exert a significant effect on the surface roughness since the order of particles changed. speed did not cha does n not ge too much exert a signiﬁcant compared to effect the other on the surface two cutti roughness ng parameters when cutti since the order of particles ng speed was did not change changed too . much compared to the other two cutting parameters when cutting speed was changed. (c) (a) (b) (c) (a) (b) (c) (a) (b) Figure 8. Surface particles distribution with different feeds (v = 120 m/min, γ = 0°). (a) f = 0.07 mm; (b) f = 0.01 mm; (c) f = 0.13 mm. Figure 8. Surface particles distribution with different feeds (v = 120 m/min, γ = 0°). (a) f = 0.07 mm; (b) Figure 8. Surface particles distribution with different feeds (v = 120 m/min, = 0 ). (a) f = 0.07 mm; f = 0.01 mm; (c) f = 0.13 mm. (b) f = 0.01 mm; (c) f = 0.13 mm. Figure 8. Surface particles distribution with different feeds (v = 120 m/min, γ = 0°). (a) f = 0.07 mm; (b) f = 0.01 mm; (c) f = 0.13 mm. (b) (a) (c) (a) (b) (c) Figure 9. Surface particles distribution of rake angle (v = 240 m/min, f = 0.13 mm). (a) = 7 ; (b) Figure 9. Surface particles distribution of rake (a) angle (v = 2(b) 40 m/min, f = 0.13 mm). (a) γ ( = 7°; ( c) b) γ = 3°; ( = 3c) ;γ (c = 0 ) °= . 0 . Figure 9. Surface particles distribution of rake angle (v = 240 m/min, f = 0.13 mm). (a) γ = 7°; (b) γ = 3°; (c) γ = 0°. Figure 9. Surface particles distribution of rake angle (v = 240 m/min, f = 0.13 mm). (a) γ = 7°; (b) γ = 3°; (c) γ = 0°. (b) (a) (c) (a) (b) (c) Figure 10. Surface particles distribution of cutting speed ( = 3 , f = 0.15 mm/rev). (a) v = 60 m/min; Figure 10. Surface particles distribution of cutting speed (γ = 3°, f = 0.15 mm/rev). (a) v = 60 m/min; (b) v = 120 m/min; (c) v = 240 m/min. (a) (b) (c) (b) v = 120 m/min; (c) v = 240 m/min. Figure 10. Surface particles distribution of cutting speed (γ = 3°, f = 0.15 mm/rev). (a) v = 60 m/min; 3.2. The(Basic b) v = 120 Principle m/min; of (T caguchi ) v = 240 m/mi Method n. Figure 10. Surface particles distribution of cutting speed (γ = 3°, f = 0.15 mm/rev). (a) v = 60 m/min; Taguchi proposed a method [38] which considers the improvement of product quality. The (b) v = 120 m/min; (c) v = 240 m/min. 3.2. The Basic Principle of Taguchi Method product quality can be obtained by designing rather than inspecting. In this study, we employed the 3.2. The Basic Principle of Taguchi Method Taguchi method to investigate the effects of cutting parameters on surface roughness. The Taguchi 3.2. The Basic Principle of Taguchi Method method has two core analysis tools. The ﬁrst core is the orthogonal array. Since the orthogonal array Appl. Sci. 2019, 9, 654 13 of 23 Taguchi proposed a method [38] which considers the improvement of product quality. The can ensure the impartiality and uniformity of the experiment’s data, it is very representative and product quality can be obtained by designing rather than inspecting. In this study, we employed the can comprehensively reﬂect the impact of various factors and each level on indicators. Therefore, Taguchi method to investigate the effects of cutting parameters on surface roughness. The Taguchi compared to the traditional experimental design which has a large number of experiments and is too method has two core analysis tools. The first core is the orthogonal array. Since the orthogonal array complex, the orthogonal array can simplify the complex experimental tests and minimize the effects can ensure the impartiality and uniformity of the experiment’s data, it is very representative and can of the factors which are out of control otherwise. The second core is the signal-to-noise expecting comprehensively reflect the impact of various factors and each level on indicators. Therefore, (ined by design rather than the (S/N). The S/N ratio is calculated based on the experimental results compared to the traditional experimental design which has a large number of experiments and is too of the orthogonal array, which can indicate the loss of product quality [39]. Usually, there are three complex, the orthogonal array can simplify the complex experimental tests and minimize the effects performance of the factors which characteristics are out of contro in analyzing l otherwise. the S/NTratio. he second They coar re is the sign e the smaller al-to-noise ex -the-betterpecting characteristic, (S/N). The S/N ratio is calculated based on the experimental results of the orthogonal array, which the nominal-the-better characteristic and the larger-the-better characteristic. Neglecting the grouping can indicate the loss of product quality [39]. Usually, there are three performance characteristics in of the above performance characteristic, the greatest S/N value stands for the best performance analyzing the S/N ratio. They are the smaller-the-better characteristic, the nominal-the-better characteristic. Hence, the higher the S/N ratio is, the better the level of the cutting parameters is. In characteristic and the larger-the-better characteristic. Neglecting the grouping of the above addition, the variance analysis (ANOVA) is applied to determine which cutting parameters have a performance characteristic, the greatest S/N value stands for the best performance characteristic. signiﬁcant effect. By analyzing the S/N ratio and variance, we can not only investigate the effects of Hence, the higher the S/N ratio is, the better the level of the cutting parameters is. In addition, the cutting parameters, but also obtain the optimal cutting parameters. It should be noticed that we need variance analysis (ANOVA) is applied to determine which cutting parameters have a signiﬁcant to obtain the cutting parameters which achieve the lowest surface roughness value. The performance effect. By analyzing the S/N ratio and variance, we can not only investigate the effects of cutting characteristic in analyzing the S/N ratio is the smaller-the better. The equations are as following: parameters, but also obtain the optimal cutting parameters. It should be noticed that we need to obtain the cutting parameters which achieve the lowest surface roughness value. The performance characteristic in analyzing the S/N ratio is the smaller-the better. The equations are as following: M.S.D = y (25) å i n i=1 M.. SD = y (25)  i i =1 S/N = 10 log(M.S.D) (26) SN/1 =−0log(M.S.D) (26) where M.S.D is the mean-square deviation; n is the total number of experiments; y is the measured value of surface roughness; and i is the serial number of the experiment. where M.S.D is the mean-square deviation; n is the total number of experiments; yi is the measured In this paper, the turning process (Figure 11a) is approximated as the orthogonal cutting value of surface roughness; and i is the serial number of the experiment. process [40] (Figure 11b). The parameters in turning can be approximated by the orthogonal cutting In this paper, the turning process (Figure 11a) is approximated as the orthogonal cutting process [40] (Figure 11b). The parameters in turning can be approximated by the orthogonal cutting parameters as shown in Table 2. parameters as shown in Table 2. Cutti ng speed Rake angl e chip C utti ng to ol Work pi ece h1 Edge radius C utti ng speed C learanc e angl e Fc h2 f Chip Fc Ft Workpiece Workpiece Ft C utti ng to ol (a) (b) Figure 11. Cutting model. (a): Turning; (b): Orthogonal cutting Figure 11. Cutting model. (a): Turning; (b): Orthogonal cutting Table 2. Approximation of Turning by Orthogonal Cutting [40]. Table 2. Approximation of Turning by Orthogonal Cutting [40]. Turning Operation Orthogonal Cutting Turning Operation Orthogonal Cutting Feed f = Chip thickness before cut t Feed f= Chip thickness before cut t Cutting speed V= Cutting speed V Cutting speed V= Cutting speed V Rake angle = Rake angle Rake angle γ= Rake angle γ Clearance angle = Clearance angle Cutting force F = Cutting force F c c Feed force F = Feed force F t t Appl. Sci. 2019, 9, 654 14 of 23 As shown above, the three parameters (cutting speed, feed and rake angle) can be controlled by this simulating cutting model. They can also affect surface roughness signiﬁcantly in the actual turning process. Therefore, the reasonable choice of them would improve the surface quality and decrease the surface roughness. Though the cutting depth and clearance angle in turning are other parameters in the cutting process, the cutting depth cannot be controlled in the 2D simulation model and clearance angle has little effect on surface roughness. In this work, we only consider the above three factors on surface roughness and therefore we choose them as controllable factors in this work. In addition, three levels are also considered in this experiment, as shown in Table 3. Therefore, as shown in Table 4, we selected L orthogonal array as our experimental design [41]. It should be noticed that to select an appropriate orthogonal array for the experiments, the total degree of freedom needs to be computed. The degrees of freedom are deﬁned as the number of comparisons between the process parameters that need to be made to determine which level is better and speciﬁcally how much better it is. In this study, the L array which has four columns and nine rows has eight degrees of freedom and it can handle three-level process parameters (There are six degrees of freedom owing to there being three cutting parameters in the turning operations). Table 3. Three levels for cutting parameters. Symbol Cutting Parameters Level1 Level2 Level3 A Rake angle 0 3 7 B Cutting speed 240 m/s 120 m/s 60 m/s C Feed 0.07 mm 0.1 mm 0.13 mm Table 4. L orthogonal array in simulated experiments. Cutting Parameter Level A B C Experiment Number Rake Angle Cutting Speed Feed 1 1 1 1 2 1 2 2 3 1 3 3 4 2 1 3 5 2 2 1 6 2 3 2 7 3 1 2 8 3 2 3 9 3 3 1 To summarize, we use L orthogonal array to design the simulated experiment, and then we study the inﬂuence of different cutting parameters (cutting speed, feed and rake angle) on surface roughness by analyzing the S/N ratio and variance. Based on these results, we obtain the effects of cutting parameters on surface roughness and optimize the cutting parameters to achieve the lowest surface roughness [41,42]. 4. Results and Discussion 4.1. Experiment Test for Improved SPH Cutting Model We established an improved SPH cutting model which incorporated the improved SPH algorithm, TANH constitutive model and contact algorithm. The improved cutting model is validated by comparing results from Calamaz et al [37]. The following Figure 12 is the comparison between the simulated cutting process and the experiment (chips morphology formation is recorded by a camera in the experimental cutting process) both under the same cutting parameters (cutting depth, 0.1 mm; cutting speed, 235 m/min; and edge radius, 20 m, rank angle 0 , clearance angle 11 ). It Appl. Sci. 2019, 9, 654 15 of 23 depth, 0.1 mm; cutting speed, 235 m/min; and edge radius, 20 µm, rank angle 0°, clearance angle 11°). depth, 0.1 mm; cutting speed, 235 m/min; and edge radius, 20 µm, rank angle 0°, clearance angle 11°). shows that the simulation results, such as strain localization along a curved shear and the curvature It shows that the simulation results, such as strain localization along a curved shear and the curvature It shows that the simulation results, such as strain localization along a curved shear and the curvature change of a slipped shear band (new segment), match the experiments very well. change of a slipped shear band (new segment), match the experiments very well. change of a slipped shear band (new segment), match the experiments very well. Strain localization New segment Strain localization New segment Shearing Shearing (a) (b) (c) (d) (a) (b) (c) (d) Figure 12. Two stages of a numerical and experimental segmented chip formation: (a) the beginning Figure 12. Two stages of a numerical and experimental segmented chip formation: (a) the beginning Figure 12. Two stages of a numerical and experimental segmented chip formation: (a) the beginning of strain localization and shearing in the primary; (c) primary shear zone begins to slip [37]; (b,d) the of strain localization and shearing in the primary; (c) primary shear zone begins to slip [37]; (b,d) the of strain localization and shearing in the primary; (c) primary shear zone begins to slip [37]; (b,d) the simulation results. simulation results. simulation results. In addition, the accuracy of the cutting model can be veriﬁed by the segmented chip morphology In addition, the accuracy of the cutting model can be verified by the segmented chip morphology In addition, the accuracy of the cutting model can be verified by the segmented chip morphology and cutting force. We can compare three parameters of chip morphology between the simulation and and cutting force. We can compare three parameters of chip morphology between the simulation and and cutting force. We can compare three parameters of chip morphology between the simulation and experiments, such as segmentation width (W), maximum segmentation height (h1) and minimum experiments, such as segmentation width (W), maximum segmentation height (h1) and minimum experiments, such as segmentation width (W), maximum segmentation height (h1) and minimum segmentation height (h2), as shown in Figure 12. The comparisons of simulation and experimental segmentation height (h2), as shown in Figure 12. The comparisons of simulation and experimental segmentation height (h2), as shown in Figure 12. The comparisons of simulation and experimental chip morphology are presented in Figures 13–15. Table 5 shows the comparison of relative error in chip morphology are presented in Figures 13–15. Table 5 shows the comparison of relative error in chip morphology are presented in Figures 13–15. Table 5 shows the comparison of relative error in simulation and experimental data for segmentation characteristics. simulation and experimental data for segmentation characteristics. simulation and experimental data for segmentation characteristics. Exp Exp Simulation Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure 13. Maximum segmentation height (h1) under different cutting speeds. Figure 13. Maximum segmentation height (h1) under different cutting speeds. Figure 13. Maximum segmentation height (h1) under different cutting speeds. h1 (μm) h1 (μm) Appl. Sci. 2019, 9, 654 16 of 23 Exp Exp 30 Simulation Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure 14. Minimum segmentation height (h2) under different cutting speeds. Figure Figure 14. 14. Minimum Minimum segment segmentation ation height (h height (h2) 2) under under diff differ erent cu ent cutting tting spe speeds. eds. Exp Exp 30 Simulation Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure Figure 15. 15. Seg Segmentation mentation width under differ width under different ent cutting spe cutting speeds. eds. Figure 15. Segmentation width under different cutting speeds. Table 5. Comparison of errors in the simulation and experimental data for segmentation characteristics. Table 5. Comparison of errors in the simulation and experimental data for segmentation Table 5. Comparison of errors in the simulation and experimental data for segmentation characteristics. characteristics. Speed h1 (Relative Error %) h2 (Relative Error %) W (Relative Error %) 75 9.84% 8.70% 5.19% Speed h1 (Relative Error %) h2 (Relative Error %) W (Relative Error %) Speed h1 (Relative Error %) h2 (Relative Error %) W (Relative Error %) 94 8.96% 6.85% 2.41% 75 9.84% 8.70% 5.19% 75 9.84% 8.70% 5.19% 116 19.35% 13.48% 4.94% 94 8.96% -6.85% 2.41% 94 8.96% -6.85% 2.41% 150 2.80% 2.35% 14.29% 116 -19.35% -13.48% 4.94% 116 -1 188 9.35% 13.14% -1 8.45%3.48% 5.06%4.94% 233 12.90% 10.45% 11.11% 150 2.80% -2.35% 14.29% 150 2.80% -2.35% 14.29% 188 13.14% -8.45% 5.06% 188 13.14% -8.45% 5.06% 233 12.90% 10.45% 11.11% 233 12.90% 10.45% 11.11% The average cutting force and average thrust force are predicted under different cutting speeds by the improved SPH model. Figures 16 and 17 show the comparison between the simulated results and The average cutting force and average thrust force are predicted under different cutting speeds The average cutting force and average thrust force are predicted under different cutting speeds the experimental results [37]. Table 6 compares the quantitative differences in relative errors between by the improved SPH model. Figure 16 and Figure 17 show the comparison between the simulated by the improved SPH model. Figure 16 and Figure 17 show the comparison between the simulated the simulation and experimental results for cutting force and thrust force. results and the experimental results [37]. Table 6 compares the quantitative diﬀerences in relative results and the experimental results [37]. Table 6 compares the quantitative diﬀerences in relative errors between the simulation and experimental results for cutting force and thrust force. errors between the simulation and experimental results for cutting force and thrust force. Segm Segment ent wi widt dth h (μm (μm)) h2 h2 ((μm μm)) Appl. Sci. 2019, 9, 654 17 of 23 300 Exp 300 Exp Simulation 200 Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure 16. Average cutting force under different cutting speeds. Figure 16. Figure 16. Ave Average rage cutting force under cutting force under dif different cutting sp ferent cutting speeds. eeds. Exp Exp 150 Simulation 150 Simulation 50 100 150 200 250 50 100 150 200 250 Cutting speed(m/min) Cutting speed(m/min) Figure 17. Figure 17. Average thrust force under Average thrust force under dif differ fere ent nt cutting spee cutting speeds. ds. Figure 17. Average thrust force under different cutting speeds. Table 6. Comparison of errors in the simulation and experimental data for cutting force and feed Table 6. Comparison of errors in the simulation and experimental data for cutting force and feed force Table 6. Comparison of errors in the simulation and experimental data for cutting force and feed force force values. values. values. Speed Cutting Force (Relative Error %) Thrust Force (Relative Error %) Speed Cutting Force (Relative Error %) Thrust Force (Relative Error %) Speed Cutting Force (Relative Error %) Thrust Force (Relative Error %) 75 7.49% 11.03% 75 -7.49% -11.03% 75 -7.49% -11.03% 94 3.87% 16.34% 94 -3.87% 16.34% 94 -3.87% 16.34% 116 7.24% 11.39% 116 -7.24% -11.39% 116 -7.24% -11.39% 150 10.26% 9.42% 150 -10.26% 9.42% 150 -1 188 0.26%8.67% 7.83%9.42% 233 7.73% 8.64% 188 -8.67% -7.83% 188 -8.67% -7.83% 233 -7.73% 8.64% 233 -7.73% 8.64% Regarding the veriﬁcation of the model, the h1, h2, and segment width and forces have Regarding the verification of the model, the h1, h2, and segment width and forces have been Regarding the verification of the model, the h1, h2, and segment width and forces have been been proven to be effective and efﬁcient factors for verifying cutting models [15,19,20,37]. From proven to be effective and efficient factors for verifying cutting models [15,19,20,37]. From the proven to be effective and efficient factors for verifying cutting models [15,19,20,37]. From the the comparison of the simulation and the experiment in terms of segmentation width, maximum comparison of the simulation and the experiment in terms of segmentation width, maximum comparison of the simulation and the experiment in terms of segmentation width, maximum segmentation height (h1) and minimum segmentation height (h2), cutting force and thrust force, we segmentation height (h1) and minimum segmentation height (h2), cutting force and thrust force, we segmentation height (h1) and minimum segmentation height (h2), cutting force and thrust force, we can conclude the improved SPH cutting model simulates the cutting process accurately and effectively. can conclude the improved SPH cutting model simulates the cutting process accurately and can conclude the improved SPH cutting model simulates the cutting process accurately and 4.2. Analysis of the S/N Ratio and Variance effectively. effectively. The surface particles are extracted and the coordinates of particles are obtained to calculate 4.2. Analysis of the S/N Ratio and Variance 4.2. Analysis of the S/N Ratio and Variance the surface roughness of the simulation model. Table 7 shows the simulation results of the surface roughness The surface p value and artic S/N les ratio are ex for tracted and t surface roughness. he coordinates of particles are obtained to calculate the The surface particles are extracted and the coordinates of particles are obtained to calculate the surface roughness of the simulation model. Table 7 shows the simulation results of the surface surface roughness of the simulation model. Table 7 shows the simulation results of the surface roughness value and S/N ratio for surface roughness. roughness value and S/N ratio for surface roughness. Thrust Thrust fforce orce (N) (N) Cu Cuttin tting g fo force(N) rce(N) Table 7. Simulation results for surface roughness and S/N ratio. Cutting Parameter Level Model Surface S/N Ratios for Appl. Sci. 2019, 9, 654 18 of 23 Experiment A B C −1 Roughness (10 Surface Number Rake Cutting µm) Roughness Feed Angle Table 7. Simulation Speed results for surface roughness and S/N ratio. 1 0° 240 m/s 0.07 mm/rev 2.251 −7.048 Cutting Parameter Level 2 0° 120 m/s 0.1 mm/rev 6.136 −15.758 Model Surface S/N Ratios for Experiment A B C Roughness (10 m) Surface Roughness Number 3 0° 60 m/s 0.13 mm/rev 12.256 −21.767 Rake Angle Cutting Speed Feed 4 3° 240 m/s 0.13 mm/rev 3.191 −10.079 1 0 240 m/s 0.07 mm/rev 2.251 7.048 5 3° 120 m/s 0.07 mm/rev 5.341 −14.553 2 0 120 m/s 0.1 mm/rev 6.136 15.758 3 0 60 m/s 0.13 mm/rev 12.256 21.767 6 3° 60 m/s 0.1 mm/rev 2.086 −6.387 4 3 240 m/s 0.13 mm/rev 3.191 10.079 7 7° 240 m/s 0.1 mm/rev 2.693 −8.605 5 3 120 m/s 0.07 mm/rev 5.341 14.553 6 3 60 m/s 0.1 mm/rev 2.086 6.387 8 7° 120 m/s 0.13 mm/rev 1.020 −0.172 7 7 240 m/s 0.1 mm/rev 2.693 8.605 9 7° 60 m/s 0.07 mm/rev 1.753 −4.876 8 7 120 m/s 0.13 mm/rev 1.020 0.172 9 7 60 m/s 0.07 mm/rev 1.753 4.876 Table 8 is the S/N response table for surface roughness; it gives each level a mean S/N ratio and the total mean S/N ratio. In addition, the different values of the S/N ratio between maximum and Table 8 is the S/N response table for surface roughness; it gives each level a mean S/N ratio and the minimum are also given in Table 8. It can be seen that the feed and rake angle have the highest total mean S/N ratio. In addition, the different values of the S/N ratio between maximum and minimum difference value, 10.3221 and 10.4239 respectively. According to the Taguchi method, the larger the are also given in Table 8. It can be seen that the feed and rake angle have the highest difference value, difference between the values of the S/N ratio are, the more effect there will be on surface roughness. 10.3221 and 10.4239 respectively. According to the Taguchi method, the larger the difference between Therefore, changing the feed and rake angle will change the value of surface roughness of the the values of the S/N ratio are, the more effect there will be on surface roughness. Therefore, changing simulation model significantly. the feed and rake angle will change the value of surface roughness of the simulation model signiﬁcantly. Table 8. S/N ratio response table. Table 8. S/N ratio response table. Mean S/N Ratio Mean S/N Ratio Symbol Cutting Parameter Cutting Parameter Symbol Level 1 Level 2 Level 3 Max-min Level 1 Level 2 Level 3 Max-min A Rake angle −14.8574 −8.5769 −4.5353 10.3221 A Rake angle 14.8574 8.5769 4.5353 10.3221 B Cutting speed −9.3391 −10.1607 −10.2373 0.8982 B Cutting speed 9.3391 10.1607 10.2373 0.8982 CC Feed Feed 4.5508 −4.5508 −11.0096 11.0096 −14 14.9747 .9747 10.423 10.42399 Total mean S/N ratio = −9.8046. Total mean S/N ratio = 9.8046. Figure 18 shows the S/N response graph for surface roughness. As mentioned in Equation (25) Figure 18 shows the S/N response graph for surface roughness. As mentioned in Equation (25) and Equation (26), a greater value of the S/N ratio will have a smaller variance of surface roughness and Equation (26), a greater value of the S/N ratio will have a smaller variance of surface roughness (the smaller the better). (the smaller the better). -2 -4 -6 -8 -10 -12 -14 -16 A1 A2 A3 B1 B2 B3 C1 C2 C3 Cutting parameter Level Figur Figure e 18. 18.C Cutting utting pa parameters rameters of of S/N S/N response response graph (A- graph (A-Rake Rake angle; angle; B- B- Cutting Cutting speed; speed; C-C- Feed). Feed). The variance analysis (ANOVA) is also a helpful and efﬁcient tool for testing and determining which cutting parameter have an obvious effect on surface roughness. As shown in Table 9, it shows the rake angle and feed have an obvious effect on the surface roughness, while the cutting speed compared Mean S/N ratio The variance analysis(ANOVA) is also a helpful and efficient tool for testing and determining Appl. Sci. 2019, 9, 654 19 of 23 which cutting parameter have an obvious effect on surface roughness. As shown in Table 9, it shows the rake angle and feed have an obvious effect on the surface roughness, while the cutting speed compared to other parameters has little effect on surface roughness, since their contributions in to other parameters has little effect on surface roughness, since their contributions in percentage percentage are 45.602, 46.721 and 1.0407 respectively. In addition, if we only consider these three are 45.602, 46.721 and 1.0407 respectively. In addition, if we only consider these three parameters parameters for lowering the surface roughness in cutting processing. The best of combinations of the for lowering the surface roughness in cutting processing. The best of combinations of the cutting cutting parameters (the level 3 of rake angle, the level 1 of cutting speed, and the level 1 of feed) can parameters (the level 3 of rake angle, the level 1 of cutting speed, and the level 1 of feed) can be be obtained according to analyze the S/N ratio and variance. obtained according to analyze the S/N ratio and variance. Table 9. Table 9. V Varia ariance nce analysis fo analysis forr surface roughness. surface roughness. Parameters Degree of Freedom Sum of Squares Mean Square F Ratio Contribution (%) Parameters Degree of Freedom Sum of Squares Mean Square F Ratio Contribution (%) Rake angle 2 160.15 80.073 28.493 45.602 Rake angle 2 160.15 80.073 28.493 45.602 Cutting speed 2 9.1471 4.5736 1.6275 1.0407 Cutting speed 2 9.1471 4.5736 1.6275 1.0407 Feed 2 163.94 81.969 29.168 46.721 Feed 2 163.94 81.969 29.168 46.721 Error 2 5.6202 2.8101 6.6345 Error 2 5.6202 2.8101 6.6345 Total 8 338.85 100 Total 8 338.85 100 4.3. Experimental Tests for Effect of Cutting Parameters on Surface Roughness 4.3. Experimental Tests for Effect of Cutting Parameters on Surface Roughness We apply the turning (dry cutting) and single-point cutting as our experimental test. Figure 19 We apply the turning (dry cutting) and single-point cutting as our experimental test. Figure 19 shows the experimental instruments. The cutting tool used carbide inserts and a tool holder. We shows the experimental instruments. The cutting tool used carbide inserts and a tool holder. We applied three different carbide inserts and a different tool holder. These inserts have the same nose applied three different carbide inserts and a different tool holder. These inserts have the same nose radius 0.8 mm and edge radius 20 m. After installation, the effective rake which is a combination of the radius 0.8 mm and edge radius 20 µm. After installation, the effective rake which is a combination of tool holders angle of inclination and the rake built into the insert, are about 0 , 3 and 7 . Figures 20–22 the tool holders angle of inclination and the rake built into the insert, are about 0°, 3° and 7°. Figures are the comparisons of surface roughness between the simulation results and experimental results 20–22 are the comparisons of surface roughness between the simulation results and experimental (cutting depth in turning is set 0.1 mm). From these ﬁgures, we can conclude that although there results (cutting depth in turning is set 0.1 mm). From these figures, we can conclude that although are differences between the surface roughness of the SPH cutting model calculated by SPM and there are differences between the surface roughness of the SPH cutting model calculated by SPM and the measured surface roughness from the experiment, they have the same trend under the same the measured surface roughness from the experiment, they have the same trend under the same changes of cutting parameters. The feed and rake angle have signiﬁcant effects on surface roughness. changes of cutting parameters. The feed and rake angle have significant effects on surface roughness. Figure 20 shows that the surface roughness will decrease as the rake angle increases. Figure 21 shows Figure 20 shows that the surface roughness will decrease as the rake angle increases. Figure 21 shows that the surface roughness will increase as the cutting feed increases. Figure 22 shows the cutting that the surface roughness will increase as the cutting feed increases. Figure 22 shows the cutting speed still affects the surface roughness; however, there is little change compared to the other two speed still affects the surface roughness; however, there is little change compared to the other two cutting parameters. cutting parameters. Workpiece Cutting tool Roughmeter Figure Figure 19. 19. Experimental Experimental i instr nstruments. uments. Appl. Sci. 2019, 9, 654 20 of 23 Simulation Simulation 6 Simulation 4 Experiment 4 Experiment Experiment 02 46 8 02 46 8 02 46 8 Rake angle (°) Rake angle (°) Rake angle (°) Figure 20. Comparison between the simulated results and experimental results under different rake Figure 20. Comparison between the simulated results and experimental results under different rake Figure Figure 20. 20. Comparison Comparison betw between een the the simulated simulated re results sults and and experimental experimental results results under under different rake different rake angles (v = 120 m/min, f = 0.1 mm/rev). angles (v = 120 m/min, f = 0.1 mm/rev). angles (v = 120 m/min, f = 0.1 mm/rev). angles (v = 120 m/min, f = 0.1 mm/rev). 6 Simulation 6 Simulation 6 Simulation Experiment Experiment 4 Experiment 0.05 0.07 0.09 0.11 0.13 0.15 0.05 0.07 0.09 0.11 0.13 0.15 0.05 0.07 0.09 0.11 0.13 0.15 Feed (mm) Feed (mm) Feed (mm) Figure 21. Comparison between the simulated results and experimental results under different feeds Figure 21. Comparison between the simulated results and experimental results under different feeds Figure 21. Comparison between the simulated results and experimental results under different feeds Figure 21. Comparison between the simulated results and experimental results under different feeds (v = 120 m/min, γ = 0°). (v = 120 m/min, g = 0 ). (v = 120 m/min, γ = 0°). (v = 120 m/min, γ = 0°). 6 Simulation 6 Simulation 6 Simulation Experiment Experiment 4 Experiment 0 100 200 300 0 100 200 300 0 100 200 300 Speed (m/min) Speed (m/min) Speed (m/min) Figure Figure 22. 22. Comparison Comparison bet between ween the s the simulated imulated resu results and lts experimental and experimental results under different results under different cutting Figure 22. Comparison between the simulated results and experimental results under different Figure 22. Comparison between the simulated results and experimental results under different speeds (f = 0.1 mm/rev, g = 0 ). cutting speeds (f = 0.1 mm/rev, γ = 0°). cutting speeds (f = 0.1 mm/rev, γ = 0°). cutting speeds (f = 0.1 mm/rev, γ = 0°). To summarize, the results of analyzing the S/N ratio and variance based on the SPH cutting model To summarize, the results of analyzing the S/N ratio and variance based on the SPH cutting To summarize, the results of analyzing the S/N ratio and variance based on the SPH cutting To summarize, the results of analyzing the S/N ratio and variance based on the SPH cutting are the same with the experimental results, because they have the same variation trend of surface model are the same with the experimental results, because they have the same variation trend of model are the same with the experimental results, because they have the same variation trend of model are the same with the experimental results, because they have the same variation trend of surface roughness value when the cutting parameters are changed. Therefore, the cutting parameters surface roughness value when the cutting parameters are changed. Therefore, the cutting parameters surface roughness value when the cutting parameters are changed. Therefore, the cutting parameters can be investigated and optimized by the SPH model. It should be known that the real cutting process can be investigated and optimized by the SPH model. It should be known that the real cutting process can be investigated and optimized by the SPH model. It should be known that the real cutting process -1 -1 -1 -1 -1 -1 -1 -1 -1 Surface roughness (10 µm) Surface roughness (10 µm) Surface roughness (10 µm) Surface Surface roughness roughness (10 (10 µm µm )) Surface Surface roughness roughness (10 (10 µm µm )) Surface Surface roughness roughness (10 (10 µm µm )) Appl. Sci. 2019, 9, 654 21 of 23 roughness value when the cutting parameters are changed. Therefore, the cutting parameters can be investigated and optimized by the SPH model. It should be known that the real cutting process is a highly complex process. All the existing models which simulate the cutting process simplify several factors that affect the cutting process, such as machine tool vibration and tool wear. Therefore, the simulation is not as accurate as experiments in the real world and it is hard to simulate the cutting model to predict the surface roughness accurately. We simulated the cutting process to predict the variance trend of surface roughness with different cutting parameters and then compared the simulation results with the real experimental results. From another perspective, it might add more errors if we compare the simulated results with another simulation (there will be an accumulation of errors which can lead to more inaccurate results). After serious consideration, we compared our simulated results with real experimental results to analyze the variation trend of surface roughness with different cutting parameters. Although we ignore a lot of factors (cutting edge angle, the vibration in actual cutting process et al.), we only consider the cutting parameters which have signiﬁcant inﬂuences on surface roughness and can also be controlled by the 2D SPH cutting model. In the experiment, all the parameters remain the same except the three parameters (cutting speed, rake angle and feed). 5. Conclusions We applied an improved SPH method to simulate the Ti–6AL–4V cutting process and proposed a Surface Particle Method (SPM) to evaluate the variance trend of surface roughness with different cutting parameters. Then we employed the Taguchi method to investigate the effects of cutting parameters on the variance trend of surface roughness. The investigation applied the S/N ratio and variance analysis to analyze the effect of different levels of cutting parameters. We found that the rake angle and feed have an obvious effect on surface roughness, while the variance of cutting speed had little inﬂuence on the surface roughness. Comparing our results with the experiments, we concluded that although the surface roughness calculated based on the simulated surface particles is different from the measured one in the experiment, they had the same variation trend. This means we can use this SPH model to investigate the effects of cutting parameters on surface roughness effectively. Our work proved there are merits to using the improved SPH cutting model to optimize cutting parameters. These merits are listed as: 1. The improved SPH method avoids the misconvergence caused by mesh distortion of FEM and increases the accuracy of the simulation. Meanwhile, the TANH constitutive law improves the accuracy of the description for the dynamic properties of the material under a high stress rate compared with JC constitutive law. 2. This paper proposed a Surface Particle Method (SPM) to evaluate the variance trend of surface roughness with different cutting parameters. Although there are differences between the surface roughness calculated by SPM and the measured surface roughness from the experiment, they have the same variation trend when the cutting parameters are changed. Thus, the surface roughness based on SPM is valid as an alternative for the measured variance trend of surface roughness with different cutting parameters for investigating and optimizing cutting parameters. 3. This paper applied the Taguchi method to test the inﬂuence of cutting speed and feed and rake angle on the surface roughness of a Ti–6AL–4V workpiece. The Taguchi method considerably reduces the number of experimental groups under different cutting parameters and provides a solid theoretical foundation for optimizing cutting parameters. 4. 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Investigating the Effect of Cutting Parameters of Ti–6Al–4V on Surface Roughness Based on a SPH Cutting Model

Investigating the Effect of Cutting Parameters of Ti–6Al–4V on Surface Roughness Based on a SPH Cutting Model

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Investigating the Effect of Cutting Parameters of Ti–6Al–4V on Surface Roughness Based on a SPH Cutting Model

Investigating the Effect of Cutting Parameters of Ti–6Al–4V on Surface Roughness Based on a SPH Cutting Model

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