Prediction and optimization of machining forces using oxley's predictive theory and RSM approach during machining of WHAs

Chithajalu Kiran Sagar,Tarun Kumar,Amrita Priyadarshini,Amit Kumar Gupta

Mechanical Engineering Department,BITS-Pilani,Hyderabad Campus,Telangana,500078,India

ABSTRACT Tungsten heavy alloys have come up as one of the best alternatives for high density fragmenting devices and armor piercing ammunition.Machining is mandatory for obtaining the final shapes of such kind of ammunitions.However,due to high density and elastic stiffness of WHAs,cutting forces will be higher than for most of the metals and alloys; thus, making the machining operation challenging. The machining variable,namely,cutting force components are significantly influenced by the cutting parameters.This paper makes use of Oxley's predictive analytical model in conjunction with Johnson-Cook constitutive equation to predict forces under different speed and feed combinations during machining of 95 W tungsten heavy alloy.The cutting forces,so predicted by M1,are considered as input data for the optimization of cutting parameters(cutting speed and feed)using Response Surface Method(RSM).

Keywords:Tungsten heavy alloy Machinability Response surface method Oxley's method

1. Introduction

Tungsten heavy alloys(WHAs)are ideally suited to a wide range of density applications,because of exceptionally high values of density.WHAs provide a unique combination of density,mechanical strength,machinability,corrosion resistance,and economy.WHAs have come up as better alternatives to lead and depleted uranium with regards to corrosion,toxicity,and disposal issues.These factors make WHAs the material of choice for high density fragmenting devices and armor piercing(AP)ammunition[1].AP ammunition range from small caliber 5.56 mm rounds up to 120 mm anti-tank projectiles and beyond[2,3].WHAs are produced by a powder metallurgy(P/M)technique known as liquid phase sintering(LPS),in which completely dense,fully alloyed parts are formed from pressed metal powders at a temperature less than half the melting point of pure tungsten[4].Most parts however,by nature of size or geometric complexity,require some secondary machining and are fabricated as near net shape blanks.Machining is mandatory for the shapes with very tight dimensional requirements as in the case of APs.However,due to high density and elastic stiffness of WHAs,cutting forces will be higher than for most metals;thus,making the machining operation challenging[5].

Cutting forces have a major influence on the deformation of work piece being machined,its dimensional accuracy,machining system stability,tool wear progress and vibration and chatter[6].Sanchez et al.[7]had investigated tool wear monitoring on Ti alloys under dry machining,it was observed that cutting forces had major influence on tool wear under different feed rate and cutting velocity.Haddag et al.[8]conducted dry machining on AA2024-T351 alloy and observed that rake angle,coefficient of friction and cutting velocity had major influence on cutting forces which further effects the chip segmentation and built up edge formation on tool rake face.Hence,the present work focuses on the determination of right combination of cutting parameters to get the optimum cutting force during machining of WHAs.However,this requires extensive experimental tests which are expensive and time consuming.This paper makes use of Oxley's predictive analytical model in conjunction with Johnson-Cook constitutive equation to predict forces under different speed and feed combinations.The cutting forces,so predicted,are considered for the optimization of cutting parameters (cutting speed and feed) using Response Surface Methodology(RSM).

There are various cutting process models available such as Ernst and Merchant(1941),Lee and Shaffer(1951)and Kobayashi and Thomsen(1962)for analysis.But all of these are oversimplified thus making the models unrealistic[9].Whereas,Oxley(1989)predictive machining theory,also known as Oxley's theory is robust as compared to others because not only it is based on parallel sided shear zone theory but also takes into consideration the effect of strain,strain-rate,and temperature on material property[9,10].Several researchers have used Oxley's theory successfully for predicting forces,temperatures and stresses for various workpiece materials[11,12].Similarly,RSM is the most popular optimization method used in recent years which have been applied for optimization of machining operations[10].Makadia et al.[13]investigated the effect of cutting parameters on surface roughness of AISI 410 steel during turning and had identified optimum cutting parameters with minimum surface roughness using RSM method.It is observed that there are limited studies which focus on the prediction of forces and optimization of cutting parameters together,especially,during machining of WHAs.Hence,the work focuses on an approach that reduces the experimental tests to minimum possible by developing a methodology that uses Oxley's model to predict forces and RSM to predict optimum cutting parameters.Note that only a few experimental tests would be required to validate the predictive model based on Oxley's theory.Hence,the proposed methodology would help the defense manufacturing industries to come up with an optimal solution for manufacturing AP ammunitions economically and efficiently.

Nomenclature

σ flow stress(MPa)

Ayield strength(MPa)

Bhardening modulus(MPa)

Cstrain rate dependency coefficient

nstrain hardening exponent

mthermal softening coefficient

ε equivalent strain

Tmmelting temperature of work piece(°C)

Tcurrent temperature(°C)

Twinitial work piece temperature(°C)

neqequivalent strain hardening exponent

εABequivalent strain at shear plane

llength of shear plane(mm)

t1undeformed chip thickness(mm)

φ shear angle(°)

Vccutting speed(m/min)

VShvelocity of shear(m/min)

γ normal rake angle(°)

ηABshear strain along shear plane

C0ratio of shear plane length to thickness of primary shear zone

TABtemperature along shear plane(°C)

λ Sensible heat at shear plane(%)

ΔTSZtemperature rise in shear zone(°C)

ζ heat partition coefficient

rtchip thickness ratio

CPspecific heat of work piece material(J/Kg K)

mchipmass of chip per unit time(g/sec)

ETnon-dimensional thermal number

ρ density of work piece material(Kg/m3)

Kthermal conductivity of work piece material(W/mK)

qABshear flow stress at the shear plane(MPa)

σABflow stress at shear plane(MPa)

FShshear force along the shear plane(N)

wwidth of work piece(mm)

ψ angle between resultant cutting force and shear plane(°)

β average friction angle at tool-chip interface(°)

Fffriction force at the tool-chip interface(N)

Rresultant cutting force(N)

Nnormal force at tool-chip interface(N)

FCutcutting force(N)

Ftthrust force(N)

FNnormal force(N)

t2cut chip thickness(mm)

VChipchip velocity(m/min)

Linttool-chip interface length

τintshear stress at tool-chip interface(MPa)

Δintequivalent strain-rate at tool-chip interface

ηMtotal maximum shear strain at chip tool interface

ζ ratio of tool-chip interface plastic zone thickness to chip thickness

Tintaverage temperature along tool-chip interface(°C)

ΔTCaverage temperature rise in chip(°C)

ΔTMmaximum temperature rise in chip(°C)

θ temperature factor

qchipshear flow stress along the tool-chip interface(MPa)

Tintaverage temperature along tool-chip interface(°C)

δs1thickness of secondary deformation zone(mm)

δs2thickness of primary deformation zone(mm)

εABequivalent strain at shear plane

σNnormal stress at tool-chip interface calculated from resultant force(MPa)

qABshear flow stress along shear plane(MPa)

ffeed rate(mm)

ddepth of cut(mm)

C·Iconfidence interval

Zstandard normal distribution

sstandard deviation of each factor

Mnumber of sample data

Yoptoptimum value of output factor

2. Materials and methods

2.1. Experimental details

The workpiece material used in the present study is Fe-Ni based WHA with 95%tungsten content.The chemical composition of the material was determined using X-ray fluorescence(XRF)spectrometry.The density was measured using Mettler Toledo Density Kit MS-DNY-54.Vicker's hardness was estimated using Mitutoyo HM-200 Series 810-micro Vicker's hardness Tester.

Machining tests were conducted for experimental validation of the developed model based on Oxley's predictive theory.Turning operation was conducted on a CNC lathe(HMT PTC200)with spindle speed ranging from 100 to 4000 rpm.Uncoated carbide inserts with specification CNMG120408 were used for turning 95 WHA under dry cutting conditions.The initial dimensions of the workpiece material were taken as 11 mm diameter and 130 mm length.During turning operation,cutting speed and feed rates were varied while the depth of cut and length of cut were kept constant.Table 1 presents the test conditions taken for validating the developed model based on Oxley's machining theory.

Table 1Test conditions for experimental validation.

The cutting force components acting on the tool holder were measured using a three-component piezoelectric force dynamometer(NK instruments).The falnk wear was measured with the help of a Stereo microscope(make:METZER-219/7000 TZM)and surface roughness was measured using Taylor Hobson precision profilometer.Fig.1 shows the experimental set up used for conducting machining tests.

2.2. Oxley's predictive machining theory and J C constitutive equation

2.2.1. JC constitutive equation and determination of JC model constants

Johnson and Cook(1983)developed a material model that provides an empirical relationship to depict the flow stress behaviour of a material undergoing plastic deformation.It considers the effect of strain,strain-rate and temperature on the work piece material. Hence, the model is commonly employed for defining flow stress behaviour of work material during modeling of machining operations.The constitutive equation is given as:

The JC constantsA,B,C,nandmvalues used in the model are empirical material constants that can be found from different mechanical tests.Since very high values of strain rates are encountered in typical machining processes,accurate evaluation of these material constants is a very challenging task.Split Hopkinson Pressure Bar(SHPB)test is generally performed for determination of JC model constants at high strain rates.The SHPB test enables to perform material tests at strain rates up to 104s-1by exposing the sample to stress waves.

Fig.1.Photographic view of experimental setup(a)Force measurement Piezoelectric dynamometer along with data acquisition system(b)Surface roughness measurement profilometer.

In this paper,SHPB experimental data were taken from Woodward et al.[14]and utilized to determine JC model constants under wide range of strain rates(775,350,27 s-1)at room temperatures for 95%WHA.Experimental stress-strain curves obtained from the mentioned literature are converted to true stress-strain curves and data points were taken in plasticity region till Ultimate Tensile Strength(UTS).The final values of the constants were calculated by using these data points based on the principle of minimization of average absolute error between experimental and predicted flow stress.The obtained values of JC model constants are further finetuned using two evolutionary algorithms based optimization techniques namely,Genetic Algorithm(GA)and ABC algorithm i.e.M1 and M2,respectively.The values of JC constants obtained have already been presented in Ref.[15]where in a comparison of both the models have been made by calculating absolute error and coefficient of correlation.

2.2.2. Extension of Oxley's machining theory to JC constitutive equation

Initially,Oxley's model used the Power Law equation to depict the flow stress behaviour.Later Oxley's model got extended to Johnson Cook Material Model by introducing a modified strain hardening exponentneqin place of n.The exponentneqcan be calculated using the JC model constantsA,B&nusing the following equation[9]:

Fig.2.Parallel sided shear zone model.

The parallel-sided shear zone model for orthogonal machining by Oxley and his co-workers[9]is shown in Fig.2.

The basic purpose of Oxley's machining model is to evaluate the value of φ,C0and ζ from a given set of input parameters such as cutting speed,feed,depth of cut and rake angle using a computer program that runs on an iterative loop.The final value of shear angle is calculated based on the error between shear stress(τint)and shear flow stress(qchip)in the chip at the interface.Similarly,C0is decided based on minimum difference between normal stress at tool-chip interface and the stress derived using the stress boundary condition at pointThe value of ζ is decided based on the minimum force value obtained.The flow chart in Fig.3 explains the procedure to evaluate the desired output parameters.

The length of shear plane AB can be found from the undeformed chip thickness

Shear velocity is given by:

The equivalent strain and strain-rate at AB are derived using von Mises criterion:

The average temperature along AB is determined by considering the plastic work done in the primary shear zone and is given by:

For the present analysis,λ is assumed to be 0.9.The primary shear zone considering the plastic workFShVShdone in shear zone and the value of ξ is dependent on the non-dimensional thermal numberETand is calculated based on the following conditions:

If 0.04 ≤ETtan φ ≤10

ifETtan φ ≥10

The average flow stress in the primary shear zone(qAB)is calculated using the JC Material Model:

The shear forceFShalong the shear plane AB can be calculated fromqAB:

The angle ψ between shear plane AB and the resultant forceRis given by

Using the value of ψ,the average friction angle at tool-chip interface can be determined:

Resultant cutting force,Friction force and Normal force at the tool-chip interface are given by:

The cutting force and thrust force in the velocity direction can be calculated using the following equations:

t2can be calculated fromf,γ,φ using the equation:

Similarly,VChipcan be found from the cutting speed:

The tool-chip interface lengthLintcan be evaluated by considering the moments of the normal stresses about B.

Shear stress at tool-chip interface is derived from the friction force and is given by:

The maximum shear strain at the tool-chip interface is given by:

The equivalent strain-rate at the tool-chip interface is given by:

This is derived using von Mises criterion with an assumption that the plastic zone formed at the tool-chip interface is a rectangular zone of thickness ζt2.

The average temperature at the tool-chip interface is given by:

Fig.3.Flow chart for predicting cutting forces using Oxley's machining theory.

Table 2Experiments levels and cutting parameters output predicted by Oxley's model.

The value of ΔTMcan be evaluated using the following equation:

OnceTintis calculated,the shear flow stress along the tool-chip interfaceqchipcan be determined as:

2.3. RSM approach

In a turning operation,there are a large number of factors which can be considered as the cutting parameters that affect output variables.But,the review of literature shows that the cutting speed and feed rate are the most dominant cutting parameters while turning hard materials.In the present study these two parameters were selected as design factors keeping remaining parameters constant over the experimental domain.Two factorial design with six levels for each factor was taken,thus,making a total of 36 trails.The output variable considered for optimization in the present case was taken as cutting force.The cutting forces for the 26Full Factorial Design(FFD)were predicted using the developed model based on Oxley's theory.This approach reduces the need of actual experimental tests considerably,thus saving much of the time,money and material.The levels of cutting parameters for the experiments have been listed in Table 2.

RSM approach was used for optimization of machining 95WHA.RSM is a modeling tool that adapts both mathematical and statistical techniques to optimize the desired output by establishing the relationships between independent variables and response output variables within the design space[16,17].A regression equation,as shown below,was developed to demonstrate the effect of process parameters on cutting forces.Where b is regression coefficient

In order to identify the contribution of each cutting parameter,ANOVA table was developed and optimal combination was identified by plotting a normal distribution plot.The Optimum parameters were further validated by confirmation experiments.Confidence interval for optimal combination is given as:

Fig.4.Flow chart of the implemented strategy for RSM.

Fig.4 shows the flow chart of the implemented strategy for RSM.

Fig.5 shows the schematics of the overall methodology adopted for predicting the cutting forces using Oxley's model and optimization of the cutting forces using RSM.

3. Results and discussion

3.1. Experimental results

Table 3 lists the chemical composition,density and hardness of 95 WHA as well as AISI 1045 steel for reference.It is noted that density,hardness as well as yield strength in case of 95 WHA is considerably high as compared to that of AISI 1045.Such high values make 95 WHA,on one hand,one of the most appropriate materials for defense applications and on the other hand,one of the most difficult to machine material.

Turning operation is performed under dry conditions by varying cutting speed and feed rate at a constant depth of cut of 0.15 mm.Fig.6 shows effect of cutting speed and feed rate on three output variables namely,cutting force,surface roughness and flank wear during machining of 95 WHA.

Fig.6(a)shows that as the cutting speed increased the cutting force,surface roughness and flank wear decreased.At lower cutting speeds,the chips remain in contact with the rake face of the cutting tool for a longer time,thus exhibiting more amount of friction.This could be the probable reason for increased value of cutting force at lower cutting speed.It is a known fact that higher cutting forces exhibited on cutting tool lead to faster tool wear and poor surface finish[19].Similar trend is observed for the present case as well.In addition,at lower cutting speed,chances of getting built up edge are more obvious which gradually disappears as the cutting speed increases.Such phenomenon is clearly observed in Fig.7(a).Similarly,Fig.6(b)shows the effect of feed rate on cutting force,surface roughness and flank wear,taking a mid-value of cutting speed from the selected range as the test condition.In case of feed rate,its effect on cutting force is straight forward i.e.,with the increase in feed rate,uncut chip thickness increases which in turn increases the cutting force.Fig.7(b)shows that with the increase in feed rate,BUE becomes more prominent,even though the cutting speed taken is 50 m/min.The overall results prove that the output variables while machining 95WHA are considerably affected by cutting speed and feed rate.It can also be stated that cutting force is one of the critical output variables which is not only sensitive to the change in cutting parameters but also affects other output variables such as tool wear and surface finish.Hence,cutting force is often used as an indicative measure for understanding the tool condition and surface condition of machined surface during machining[20].Jose et al.[21]had performed CNC turning on D2 steel using acoustic emission and force sensors.It was observed that increase in cutting forces not only increased the flank wear but also the affected the quality of machined surface.

3.2. Predicted forces using Oxley's machining theory

The constants of the JC material model for flow stress behaviour of 95 W WHA are computed using GA and ABC algorithm as M1 and M2,respectively.The values of JC model constants using M1 and M2 are listed in Table 4.

Fig.5.Schematics of the methodology adopted to predict and optimize cutting forces.

The proposed methodology in the present work predicts the process variables in the primary and secondary deformation zones as well as the cutting forces.The predicted average strain,strainrate,shear flow stress,and temperature both on the shear planeAB (primary deformation zone)and at the tool-chip interface(secondary deformation zone)using M1 and M2 are given with respect to the cutting test conditions in Tables 5 and 6,respectively.

The computed values of process variables in primary and secondary shear deformation zones are finally used for predicting cutting force and thrust force.The predicted values of cutting force,thrust force and shear angle for both M1 and M2 are compared with that of the measured ones for experimental validation of the developed model.Fig.8(a)and Fig.8(b)show the predicted and the measured values of cutting force and thrust force at different cutting speeds and feed rates,respectively.

It is observed that the developed model using Oxley's theoryusing both M1 and M2 are able to predict the output variables reasonably well.Moreover,both the M1 and M2 could capture the effect of cutting speed and feed rate on forces fairly well.Shear angle is calculated from the experimentally measured chip thickness and rake angle,as given:

Table 3Comparison of chemical composition and physical properties of 95 WHA and AISI1045 steel.

Fig.6.Effect of(a)cutting speed and(b)feed rate on cutting force,surface roughness and flank wear.

Fig.9 shows the predicted and measured values of shear angles at different cutting speeds and feed rates.Shear angles predicted by both M1 and M2 follow the same trend as shown by the experimental values.However,predicted shear angles are not closer to the experimental ones.This may be attributed to two distinct facts.Firstly,inconsistent measurements of chip thickness values as the chips are highly fragmented and discontinuous. Secondly,inadequacy of Oxley's model to predict discontinuous chips accurately,thus predicting lower values of shear angles.

To give a better insight for validating as well as comparing the proposed model using M1 and M2,the absolute error percentage for the cutting force and thrust force have been plotted in Fig.10.As far as the cutting force is concerned,both the modules are capableof predicting the values satisfactorily with deviations being well below 15%.

Table 4JC material model constants[15].

Fig.7.Photographs of flank surface of cutting tool at various(a)cutting speeds(m/min)and(b)feed rates(mm/rev).

Table 5Predicted output variables using M1.

Table 6Predicted output variables using M2.

Fig.8.Predicted and measured cutting forces and thrust forces at(a)different velocities(b)feed rates.

Fig.9.Predicted and measured shear angles at(a)different velocities(b)feed rates.

Table 7 presents the mean error and standard deviation for M1 and M2. It can be stated that M1 comes out to be a better compromise as compared to M2 because M1 shows lowest mean error percentage and standard deviation for both cutting force and thrust force.Hence,M1 was chosen as the final input parameter to be used in Oxley's model for prediction of cutting forces required for optimization of machining parameters using RSM.

Fig.10.Measured versus predicted values of(a)cutting forces and(b)thrust forces.

Table 7Measured mean and standard deviation of Fc and Ft.

Table 8Predicted cutting forces from Oxley's model.

Table 9Summary of regression analysis.

3.3. Optimization using RSM approach

Cutting forces were predicted using Oxley's model for 26FFD and presented in Table 8.

In this work commercially,available package Mini Tab statistical software(Version 16)was used to develop the response equations and evaluate the coefficient values.The statistical empirical equation was developed for cutting force as output variable using only the significant coefficients,as shown below:

R2values are determined for the proposed model using RSM and presented in Table 9.It is observed that predictedR2is in good agreement with adjustedR2.Moreover the R2value is high and closer to unity which is desirable.

The analysis of variance(ANOVA)and theF-ratio test have been performed to check the adequacy of the models as well as the significance of the individual model coefficients.Table 10 presents the ANOVA table for the second order model proposed for cutting force in Eq.(34).From thePvalues of cutting force model,it can be observed that linear,square and interaction effect are significant.This indicates that the terms in the model have significant effect on the response.Further,the adjusted mean square for residual error came out to be unity which is desirable as it means there is no lackof fit.

Table 10Analysis of Variance for cutting force.

Fig.11 represents pareto chart of the standardized effects by plotting input parameters with linear,linear interaction and linear square versus standardized effect (cutting forces). It can be observed that feed rate had major influence on cutting forces as compared to that of cutting speed and interaction effects.

Fig.11.Pareto chart of the standardized effects.

Fig.14.Normal distribution plot for representing global optima for cutting force using RSM.

Fig.12.Residual plot for(a)Normal probability plot and(b)verses fits.

Fig.13.(a)3D surface graph(b)2D surface graph for cutting force with respect to cutting speed and feed rate.

Table 11Optimization results of cutting forces using RSM.

Table 12Confirmation tests.

The sufficiency of the model was further examined by inspection of residuals.The residuals are the difference between the measured values and predicted values of response.In order to examine the residuals,normal probability plots of the residuals as well as residuals versus the predicted response plots were plotted in Fig.12.

Fig.12(a)shows that the points on the normal probability plots of the residuals form a straight line.Fig.12(b)shows that there is no specific pattern for the plots of the residuals versus the predicted response.Since these are the two conditions essential for proving the adequacy of the model,the proposed model is acceptable and there is no reason for constant variance assumption[16].

Fig.13(a)and Fig.13(b)show the two-way interactive effects of cutting speed and feed rate on cutting force.

It is clear from 3D surface graph(see Fig.13(a))that cutting force decreases with decreasing feed rate and increasing cutting speed.But the decrease in cutting force could be found up to cutting speed of 80 m/min, thereafter not much change in cutting force is observed.However,in case of feed rate,the effect on cutting force is straightforward.The 2D surface graph,presented in Fig.13(b),can be used for determining the cutting force values for any specific combination of the input parameters namely,cutting speed and feed rate.The darker region represents higher cutting forces and lighter regions represent lower cutting forces.The graph clearly shows that higher cutting speed and lower feed rate can yield lower cutting forces.

In the present work,the goal is to achieve minimum possible value of cutting forces.Fig.14 and Table 11 show the optimization results of cutting forces using RSM.In Fig.14,it was observed that global optima of minimum cutting force was obtained at cutting speed of 96.12 m/min and feed rate of 0.05 mm/rev.

In order to verify the accuracy of the developed model,three confirmation tests were performed.The test conditions chosen for the confirmation tests were well within the range specified initially.The predicted values were compared with that of the experimentally measured values and error percentage was calculated.Table 12 shows the test conditions as well as the error percentage for three confirmation tests performed.It is observed that predicted forces shows slightly higher values than that of the experimental ones.However,error percentage is within the permissible limits.Such kind of deviation could be attributed to the limiting capability of Oxley's model to predict discontinuous chips.Further fine-tuning of Oxley's machining model or extending Oxley's machining theory to alternate constitutive equations(other than JC model)may improve the accuracy of the model.

4. Conclusions

Based on the findings in the present work,the conclusions can be summarized as follows:

·An analytical cutting force model for orthogonal cutting was developed by extending Oxley's predictive machining theory for difficult to machine alloy,namely,95 WHA.This was achieved by integrating JC constitutive equation.

·The developed model was verified by comparing the predicted cutting force and thrust force components with the corresponding experimental data,in which M1 had good agreement.

·The statistical empirical equation was developed for cutting force as output variable during machining of 95 WHA using RSM approach.

·Feed rate was found to be the most influencing process variable on cutting force as observed from Pareto chart.

·The developed surface graphs were useful in determining the optimum condition to obtain particular values of cutting forces in the manufacture of ammunitions for defense applications

·Optimization using RSM shows that the optimal combination of machining parameters is as follows:cutting speed=96 m/min and feed rate=0.05 mm/rev.