Inverse identification of constitutive parameters of Ti2AlNb intermetallic alloys based on cooperative particle swarm optimization

Linjiang HE,Honghua SU,Jiuhua XU,Liang ZHANG

College of Mechanical and Electrical Engineering,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

KEYWORDS

Abstract Ti2AlNb intermetallic alloy is a relatively newly developed high-temperature-resistant structural material,which is expected to replace nickel-based super alloys for thermally and mechanically stressed components in aeronautic and automotive engines due to its excellent mechanical properties and high strength retention at elevated temperature.The aim of this work is to present a fast and reliable methodology of inverse identification of constitutive model parameters directly from cutting experiments.FE-machining simulations implemented with a modified Johnson-Cook(TANH)constitutive model are performed to establish the robust link between observables and constitutive parameters.A series of orthogonal cutting experiments with varied cutting parameters is carried out to allow an exact comparison to the 2D FE-simulations.A cooperative particle swarm optimization algorithm is developed and implemented into the Matlab programs to identify the enormous constitutive parameters.Results show that the simulation observables(i.e.,cutting forces,chip morphologies,cutting temperature)implemented with the identified optimal material constants have high consistency with those obtained from experiments,which illustrates that the FE-machining models using the identified parameters obtained from the proposed methodology could be predicted in a close agreement to the experiments.Considering the wide range of the applied unknown parameters number,the proposed inverse methodology of identifying constitutive equations shows excellent prospect,and it can be used for other newly developed metal materials.

1.Introduction

With the development in the aerospace industry and higher requirements of aircraft performance,considerable efforts have been made to develop several kinds of lightweight hightemperature resistant materials to replace nickel-based super alloys for thermally and mechanically stressed components,such as low-pressure turbine blades and high-temperature compressor blades,in aviation and automotive engines.1,2Recently,a new class of titanium intermetallic alloys,based on the orthorhombic Ti2AlNb(O)phase,has been receiving great attention due to their low density,excellent fracture toughness,and high strength retention at elevated temperature.However,such properties along with low thermal conductivity,high chemical reactivity with tool materials,and strong tendency to hardening,make Ti2AlNb intermetallic alloys to be one of the most difficult-to-cut materials.3,4During machining these materials,serrated chips are normally generated,which are assumed to be the source of undesirable cutting force vibrations,excessive tool wear,poor surface quality,and poor dimensional accuracy of the machined feature.Unfortunately,most of the work on Ti2AlNb alloys mainly focuses on the material composition and structural properties,while the study on the machining processes of Ti2AlNb alloys is severely insufficient.5–7This largely limits the wide application and optimization of the machining operations of Ti2AlNb alloys.

Since chip formation is the heart of metal machining processes,which serves as an important role in revealing the mechanics of metal removal in a cutting process3,an adequate understanding of chip formation,especially serrated chips formation,is needed to improve and better optimize machining process operations.As analytical or experimental approaches have so far been unable to provide an adequate description of chip formation,researchers have turned to numerical approaches.In particular, finite element(FE)modeling has become an important technique in this context.8The reliability and accuracy of FE models heavily rely on the validity of the underlying constitutive models of working materials in functions of strain,strain rate,and temperature,which requires all relevant deformation variables during metal cutting be captured in an appropriate constitutive equation.9Therefore,it is crucial to acquire an accurate constitutive model to effectively describe the dynamic mechanical behavior in a manufacturing process.The Johnson-Cook(J-C)model is probably the one that is most widely employed in the finite element analysis of metal cutting processes.It correlates the material flow stress to the strain,strain rate,and effects of temperature.Unfortunately,the literature provides different parameters even for the same material,which are not reliable since they significantly affect simulation results(cutting forces,cutting temperature,chip morphology,etc.).10These discrepancies could be attributed,principally,to the different methods used for the determination of material constants.

The most common methodologies used to identify the J-C constitutive parameters in the literature can be divided into experimental and numerical approaches. Experimental approaches include static tests(tensile,compression)and dynamic tests (split-Hopkinson-pressure-bar technique(SHPB),Taylor test).Ozel and Karpat11identified constitutive material model parameters for high-strain rate metal cutting conditions using particle swarm optimization(PSO).The methodology was applied in predicting J-C constitutive model parameters,and flow stress data was obtained by using SHPB test data.Finally,identified results were compared with other solutions,which showed the advantage of the PSO algorithm.Peroni et al.12proposed an inverse method to identify the strain-rate and thermal sensitive material model of Glidcop materials.Flow stress data was obtained with a quasi-static test and an SHPB test.However,previous works10–13illustrate that flow stress data obtained from static tests cannot be used in metal cutting analyses due to the very low strain rates compared to those obtained in the case of machining operations.Although higher strain rates can be achieved during dynamic tests compared to static tests,they are still far from representing the real thermo-mechanical loading encountered.The levels of strain,strain rate,and temperature achieved with these experimental dynamic techniques are much lower than those encountered during a machining process:a maximum strain of about 50%and a strain rate of around 104s-1in dynamic tests,compared with strains in excess of 200%and strain rates of the order of 106s-1during a cutting process.14,15The modeled flow stress can correlate well with experimental results within the experimental ranges of strain,strain rate,and temperature.However,it may be considered invalid beyond the experimentally studied ranges,which is extrapolated according to the constitutive equations.

Numerical approaches,which identify constitutive data inversely using measured responses during orthogonal cutting tests as a reference,are very promising since they can model the material behavior under needed conditions.In this kind of approach,an inverse algorithm is used through adjusting material parameters until a good agreement between predicted and observed quantities(like cutting force or chip morphology)is reached.16Baker17proposed a new method to determine material parameters from machining simulations using inverse identification.This method relies on physical knowledge of a relation between observable quantities and parameters that can describe the material behavior.Ulutan and Ozel18proposed an inverse methodology using experimental force measurements to determine material model parameters.This methodology requires selecting meaningful values by experience,and it is restricted by the continuous chip geometry.Shatla et al.19proposed a ‘‘hybrid method” to determine the material constants of the JC constitutive equation.This method is focused on the minimization of the error between measured cutting forces and those predicted.Although a serrated chip was found for the cutting conditions,the agreement between the predicted and measured cutting forces was still good for the studied alloys.However,apart from being time consuming,these methods cannot give a unique solution.In addition,these estimated material constants were found to reflect only the experimental results in the range where they have been identified.Klocke et al.20proposed a reverse method to determine the constitutive models of Inconel 718 alloy and AISI 1045 based on FE simulations.In this methodology,a great quantity of FE simulations were performed,and two unknown constants of the J-C constitutive model were identified by comparing cutting forces and chip sizes.Shrot and Baker21used orthogonal cutting simulations to evaluate the continuous chip morphology,and obtained the constitutive parameters of the J-C model based on the Levenberg-Marquardt algorithm.In this program,the chip morphology was used as the criteria,and the number of unknown constants to be identified was limited to three.Agmell et al.22utilized a kalman filter to inversely identify the J-C constitutive model constants for AISI 1213.In their methodology,an enormous number of iterations were performed,and the coupling effects of the identified constants were assumed to be linear.

With the constant emerging of newly developed materials and the advancement of constitutive models,the mathematical equations of the constitutive model become more and more complicated,and the number of material constants involved gets even larger,which poses a much greater challenge to identify the constitutive parameters.Zerilli and Armstrong23developed a Z-A model with six constants involved,which was derived from the thermal activation theory of dislocations.Then the Z-A model was modified by incorporating a material failure function,where the number of parameters increased to ten.24Calamaz et al.25proposed a new material model(TANH)with nine constants based on a modified J-C constitutive model to introduce the strain softening phenomenon,while taking into account the material strain and strain rate hardening as well as the thermal softening phenomenon.Unfortunately,those inverse methods mentioned above lack the ability to identify the enormous amount of unknown constants from a cutting process,especially when the chip formation process is complicated.Furthermore,the standard inverse identification algorithms(like genetic algorithms or gradientbased methods)require a large number of iterations to match predictive and measured values of observables,which will be very time-consuming,and the identified results may not be unique.Consequently,it is urgent to develop an appropriate and low-cost strategy for this purpose.

To fill this gap,in this paper,a fast and reliable inverse methodology is proposed to identify constitutive parameters directly from machining experiments.Ti2AlNb intermetallic alloys are proposed as workpiece materials,whose dynamic mechanical behavior has not been researched,especially in high-strain-rate conditions.A reliable FE-machining model implemented with the TANH constitutive model is developed to establish a robust link between simulation observable quantities and material parameters.Achieving this would considerably reduce the efforts needed to run computationally expensive FE simulations iteratively.The cooperative particle swarm optimization algorithm(CPSO)is adopted as the inverse algorithm,which is especially suitable to identify enormous parameters among considerably large ranges.An objective function of the combination of predictive and experimental results(i.e.,cutting forces,chip geometry,and cutting temperature)is calculated to evaluate constitutive parameters in pending search ranges.Eventually,the identified parameters are verified by comparing the results obtained from FEM simulations with those from experiments under different cutting parameters.

2.Machining experiments

Fig.1 Orthogonal cutting experimental set-up.

Among the approximately 30 results variables obtained in a cutting process,cutting forces,chip geometries(the height of serrated chips)and cutting temperature are proposed for use,which can be easily identified and captured.Basic orthogonal cutting experiments were conducted on a turning lathe machine,and a piezoelectric dynamometer(Kistler 9527B)was fixed on the machined table to measure three-component cutting forces(Fx–the radial force,Fy–the axial force,and Fz–the cutting force).Ti2AlNb intermetallic alloys were proposed as the work material,which were well pretreated to a series of flat disks with a diameter of 190 mm and a thickness of 2 mm to allow an exact comparison with 2D FE-simulations,as shown in Fig.1.A microstructural observation by an optical microscope(Leica DM 6M)on the material reveals that uniformly distributed small lamellar O phase and equiaxed α2phase are within the B2matrix,as shown in Fig.2.The main properties of Ti2AlNb intermetallic alloys are shown in Table 1.Coated(TiAlN)carbide tools with a cutting edge length of 3 mm,a cutting edge angle of 90°,a rake angle γ0of 3°,and a clearance angle α0of 7°were used in the experiments.To remove the effect of tool wear,only a short cutting time of 1–2 s was used for each cutting parameter.Chips were collected for each experiment,mounted,and polished for further morphology measurements,as shown in Fig.3(a),where h1is the chip segment height and h2is the chip root height.It can be seen from Fig.3(b)that Fyis equal to zero,which illustrates that the cutting test is strictly orthogonal cutting.In addition,a tool-workpiece natural thermocouple was employed to measure the cutting temperature.The hot junction of the thermocouple was formed when the tool was cutting the workpiece material.The electromotive force signals between the hot and cold junctions of the thermocouple were recorded using an NI USB-6211 dynamic signal acquisition system.Then the cutting temperature can be calculated after the calibration of the electromotive force using a special calibration system,as illustrated by Stephenson.26The calibration curve of the Ti2AlNb alloy-tool thermocouple is shown in Fig.3(c).In order to make sure that the investigated parameters are unique and can be applied in a wide range of cutting conditions,the cutting speed v was varied from 20 m/min to 80 m/min,and the cutting depth apvaried from 0.06 mm to 0.15 mm,as shown in Table 2.θ is the average cutting temperature at the tool-workpiece interface,hcis the equivalent chip thickness,and G is the degree of serration,which are defined by hc=h2+(h1-h2)/2 and G=(h1-h2)/h1.The results show that with increasing the cutting speed,the serrated chips of Ti2AlNb alloys become more noticeable,and the cutting temperature increases rapidly,while the cutting forces keep almost constant.As the undeformed chip thickness increases,the cutting forces and cutting temperature increase accordingly,as well as the degree of serration.

Fig.2 Microstructures of Ti2AlNb intermetallic alloys.

Table 1 Main mechanical properties of Ti2AlNb.

Fig.3 Experimental observables.

Table 2 Cutting conditions and experimental results.

3.FE-model calibration

The proposed methodology of identifying the constitutive material behavior during machining processes is reverse,which requires the robust relations of simulation observables with varied constitutive model parameters.Therefore,an adiabatic two-dimensional finite element model of a cutting process is proposed.This model is based on the commercial ABAQUS/Explicit,which is suitable for analysis of dynamic and highly non-linear processes involving large material deformation.The machining tool is modeled as a rigid body with 3000 elements,and the workpiece as an isotropic body with 25000 elements.The initial temperatures of the workpiece and the tool are both set at 20 °C.The cutting tool rake angle is set at 3°,and the tool clearance angle is at 7°.Given that the cutting time is so short and the thermal conductivity of the workpiece material is low,only heat conduction is considered,and all the parts faces are assumed to be adiabatic.The boundary conditions of the model are shown in Fig.4.The general thermal and mechanical properties are presented in details in Table 327.

In finite element analysis,the behavior of the workpiece material requires an accurate and reliable material flow stress model.Due to the fact that the hyperbolic TANH model has been highly recommended by many researchers investigating the generation of serrated chips25,28,it has therefore been adopted here.The TANH constitutive model is given as follows:

Fig.4 Boundary conditions of finite element model for orthogonal cutting.

Table 3 General thermal and mechanical properties of the workpiece and the cutting tool.

The contact and friction behavior between the workpiece and the cutting tool represents one of the most important and complex aspects of machining processes,and has a great effect on the cutting forces and chip morphology.In this study,the friction at the tool-chip interface is modeled by a Coulomb limited Tresca law25which is given as follows:

where μ is the friction coefficient at the tool-chip interface,σnis the normal pressure,¯m is the fraction coefficient,τ is the shear stress of the workpiece,and σ0is the initial yield stress.

In addition,a chip formation criterion is introduced into FE models,which is based on the value of the equivalent plastic strain at element integration points.When the equivalent plastic strain reaches the strain at failureand the damage parameter exceeds 1,material failure takes place.If material failure takes place at all the integration points,the stiffness of the element is set to zero and remains zero for the rest of the calculation.The damage parameters used in this study are presented in Table 4.The strain at failure is given by the following equation:

where the strain at failure,is dependent on a nondimensional plastic strain rate,a dimensionless pressure-deviatoric stress ratio,p/q(where p is the pressure stress and q is the Mises stress),and a non-dimensional temperature,The strain at failure is defined by giving the failure parameters dd1-dd5.

4.Inverse algorithm for identification

In this paper,an evolutionary computational algorithm–CPSO is proposed to identify the unknown constants.CPSO,firstly developed by Van der Bergh and Engelbrecht29,is a stochastic,population-based optimization technique,which has been applied to solve many problems successfully.30In the CPSO algorithm,each particle in a population has a position and a velocity,which enables it to fly through the problem space and evolve over generations to find optima instead of dying and mutation mechanisms as genetic algorithms.Thus,this optimization algorithm is especially suitable to identify the parameters of the constitutive equation by considering a large number of independent material parameters.

In this study,48 particles are dedicated to cooperatively search for one ideal set of TANH parameters.The flow chart of the CPSO algorithm is given in Fig.5.The iterative approach of CPSO can be described as a minimization optimization process as follows:

①Initial positions and velocities of the particles are generated with random solutions.The objective function value is calculated for the current position of each particle,as defined by Eq.(4).The current position of each particle is set as the personal best position(Pl).Pl=[Bl,Cl,nl,ml,al,bl,cl,dl].Plwith the best value is set as the group best position(Q),and this value is stored.Q=[B,C,n,m,a,b,c,d].

②Modification of the position of a particle is evaluated by using its previous position information and its current velocity.The positions and velocities updating principles are expressed by Eq.(5)and Eq.(6),respectively.Each particle knows the distance between its current position and its best position(personal best)so far and the best position achieved in the group(Q)among all personal bests.The velocity is updated according to the previous velocity of the particle and the velocity of the particle towards Pland Q.This concept is similar to the human decision process where a person makes his/her decision using his/her own experiences and other people’s experiences.Additionally,the velocity updated in CPSO is stochastic due to the random numbers generated,which may cause an uncontrolled increase in the velocity and therefore instability in the search algorithm.Thus,maximum and minimum allowable velocities are selected and implemented in the algorithm.These velocities are selected depending on theparameters of the problem and limited to the dynamic range of the maximum position variable in each dimension.

Table 4 Failure parameters of the chip formation criterion.

Fig.5 Flow chart of the CPSO algorithm.

③The objective function value is calculated for the new position of each particle.If a better position is achieved by a particle,the Plvalue is replaced by the current value.As in Step 1,a Q value is selected among Plvalues.If the new value R(Q)is better than the previous value,the Q value is replaced by the current value and stored.

④Steps 2 and 3 are repeated until the iteration number reaches a predetermined iteration number or the best objective function value is achieved.

where l is the number of swarm,and k is the current iteration step.Fz0,Fx0,hc0,G0,and θ0are the experimental cutting results.Fz(l,k),Fx(l,k),hc(l,k),G(l,k),and θ(l,k)are the functions of simulated observables with independent parameters,which are established in Section 5.When R(Q)is converged to less than 1×10-5,then it assumes that this current position is the best optima.

Furthermore,in order to decrease the effect of the velocity towards the end of the search algorithm and confine the search in a small area to find optima accurately,the inertia weight w is calculated according to the following equation11:

where wmaxis the initial weight,which is set to be 0.9,wminis the final weight(0.2),and kmaxis the maximum iteration number.

5.Results and discussion

5.1.Link between simulation observables and constitutive constants

For the TANH constitutive model,constant A represents the yield stress,which can be determined from quasi static tensile/upsetting tests,while the other eight material constants B,C,n,m,a,b,c,and d are affected by the strain,strain rate,and temperature.Thus,these eight material constants need to be identified by an inverse method from a cutting process.Orthogonal design tests of machining simulation(L64(88))are conducted within an internal of±80%,which have eight levels for each constant,as shown in Table 5.Table 6 shows the simulation results and test arrangements,where the detailed data is given in Appendix A.The initial constitutive parameters A=1087.6 MPa,B=1557.7 MPa,C=0.00285,n=0.82,and m=1.51 are taken from the work of Wang and He31,32,which are derived by the SHPB test method under strain rate˙ε＜103and strain ε＜0.3,while the modified constants a=1.6,b=0.4,c=6,and d=1 are derived from the work of Calamaz.25Fig.6 shows an example of simulation results under v=60 m/min and ap=0.1 mm with the reference TANH model parameters.The average cutting temperature is calculated by taking from a series of contact points at the toolworkpiece interface,as shown in Fig.6(a).It is shown that the chip morphology using this constitutive model is serrated chips,which correlates well with the experimental ones.However,considering the specific values,the simulation observables(Fz,Fx,hc,G,θ)have large errors compared with those of experiments,where the errors can be calculated by Error=(Simulation-Experiment)/Experiment,respectively.The differences related to these comparisons with experiments are 30.4%,6.7%,6.3%,18.3%,and 8.7%,respectively,which induce a very big challenge to the inverse method.Fig.7 shows the correlation curves of the simulation observables with varied material constants.It can be seen that these eight parameters have different influences on the simulation observables.For instance,parameters B,C,m,and d have great positive effects on the cutting force,while parameter a has a negative effect on it.Meanwhile,parameters B,m,a,and n have great effects on the chip morphology.Thus, five empirical equations are drawn to describe the correlations between simulation observables and material constants based on the different impact weights on simulation results,as expressedin Eqs.(8)–(12).Since there may exist widely different parameters sets which can give similar observable quantities,FE-machining simulations under different cutting conditions are also performed to evaluate the identified parameters set.

Table 5 Orthogonal test factor levels configuration.

Table 6 Design of orthogonal tests for machining simulations with the TANH model(v=60 m/min and a p=0.1 mm).

Fig.6 FEM simulation results under v=60 m/min and a p=0.1 mm.

5.2.Inverse results analysis

The cooperative particle swarm optimization algorithms incorporated with simulation results are implemented into Matlab programs,and the iteration operation process is shown in Fig.8.Detailed operation results with the best optima as iteration goes are shown in Table 7.It is shown that the convergence rate of the object function value is very fast from the initial best object function value(0.8402)to 9.7×10-6within only 80 iterations.It is worthy to be noted that when the iteration step comes to 20,the object function value is 0.004,which indicates a high efficiency of the proposed inverse methodology.In order to evaluate the reliability of the inverse algorithm and prevent the optima solutions from being trapped into the local optimum,the inverse method is performed several times.Results show that although the initial positions of particles may not be the same as last one,their final Q optima still converges to B=1185.6 MPa,C=0.1,n=0.187,m=2.0,a=0.92,b=0.01,c=0.1,and d=1.5.At this moment,the observables(Fz,Fx,hc,G,θ)predicted by CPSO are 301 N,75.4 N,0.106 mm,0.406,and 786°C,respectively,which match well with the results obtained from experiments.

Fig.7 Effects of constitutive parameters on simulation observables under v=60 m/min and a p=0.1 mm.

5.3.Validation of inverse results for machining simulation

Fig.8 Object function value R(Q)variation with iteration steps.

The credibility of the presented inverse methodology is assessed through an evaluation of simulation results at various cutting conditions,incorporated with the optimum set of constitutive data,as shown in Table 8.It can be seen that the chip morphologies obtained from simulation results are very consistent with those from experiments,and the observables can be predicted with high accuracy.However,among the four observables,the worst coincident factor is the equivalent chip thickness(hc)with the largest error of 12.9%under v=40 m/min and ap=0.06 mm.It is assumed to be caused by the adopted chip formation criterion in FE-machining models.That is,when the equivalent plastic strain reaches the strain at failure,material failure at the integration point occurs,and the element is removed from the mesh.Although this chip formation criterion is based on physical mechanical theory and can lead to formation of chips,it can also result in a less material volume of chips than that of undeformed cutting materials,which leaves a thinner chip thickness.This phenomenon is especially obvious when the cutting depth is small.The solution can be further improved by developing a more accurate and reliable chip separation law.In addition,it is of crucial importance to ensure that the adopted links can accurately express the relations between the FE simulation results and the input variables,i.e.,TANH material parameters,especially the coupling effect of input parameters on the simulation results.These errors can be minimized largely by introducing a higher order in terms of input parameters on the computational performance.

5.4.Application of inverse results for machining simulation

Fig.9 shows the flow stress curves based on the initial constitutive parameters and the final optimal parameters.It can be seen that as the strain rate increases,both the initial constitutive model and the identified constitutive model increase greatly.Meanwhile,the latter is more sensitive to the strain rate,which has a higher flow stress value than that of the latter.The higher flow stress induces a greater cutting force.This is the reason why the cutting forces of the identified models can be greater than those of the initial models.Fig.9(b)shows the flow stress curves of the initial parameters and the identified ones under different temperatures.It can be observed that the thermal softening effect of the initial constitutive model is more enhanced than that of the identified one at low strains,which exhibits a lower flow stress value than that of the latter.However,at high strains,the flow stress of the identified modeldecreases at a higher rate than that of the initial one,which indicates that the thermo-plastic instability phenomenon is easier to occur for the identified model,and serrated chips get more noticeable.It is more obvious under a higher strain rate.Thus,in general,the hardening effect of this identified constitutive data is stronger than that of the original constitutive data under a low-strain condition to induce higher cutting forces,while the softening effect of the identified model is more enhanced at a high-strain state to lead to more noticeable serrated chips.

Table 7 Iteration process and results.

Table 8 Validation of experimental and simulation results.

Fig.9 Flow stress curves using the initial models set and the identified models set.

Consequently,considering the wide range of the applied unknown parameters number,the proposed methodology of identifying constitutive equations shows excellent results with respect to the predicted cutting forces and chip morphologies.Moreover,regardless of the material model adopted for simulations of cutting processes,it can provide a reliable and efficient framework for determination of flow stress data within not only the extreme ranges of strains,strain rates,and temperatures,but also more complicated mechanical phenomena,such as the strain softening effect or the thermal activation dislocation mechanism in metal cutting.The key feature of the current inverse approach is to adopt an independent optimization routine,which provides an opportunity to evaluate the influences of input parameters on simulation observables based on the reliability of the established links between observables and constitutive parameters without the need to run computationally expensive FE simulations iteratively.

6.Conclusions

1.Orthogonal cutting experiments are conducted under various cutting parameters.The results show that with increasing the cutting speed,serrated chips of Ti2AlNb alloys become more noticeable,and the cutting temperature increases rapidly,while the cutting forces keep almost constant.As the undeformed chip thickness increases,the cutting forces and cutting temperature increase accordingly,as well as the degree of serration.

2.FE-machining models,implemented with the modified J-C constitutive equation(TANH),are developed to establish the robust links between constitutive material parameters and simulation observables(Fz,Fx,hc,G,θ).The equations of constitutive parameters and the simulation observables relations are obtained,which can considerably reduce the efforts needed to run computationally expensive FE simulations iteratively.The results show that the cutting forces are more sensitive to the input parameters B and C,and the chip morphologies are more sensitive to n,b,a,and m,while the cutting temperature is more sensitive to B,C,n,a,and b.

3.A cooperative particle swarm optimization algorithm is if rstly proposed and implemented into the Matlab programs to inversely identify the optimal solutions.The inverse identification process shows that the convergence rate of the object function value is very fast from the initial best object function value(0.8402)to 9.7×10-6within only 80 iterations,which suggests that the cooperative particle swarm optimization algorithm has a vast advantage in identifying the constants within large ranges.The inverse results are validated under various cutting conditions,which show that the simulation results predicted by the identified model have high consistency with experimental ones,which illustrates the reliability of the proposed inverse methodology.

Considering the wide range of the applied unknown parameters number,the proposed methodology of identifying constitutive equations is appropriate to investigate the flow stress behavior in a machining process,and it is not limited to TANH constitutive equations and Ti2AlNb intermetallic alloys.

Acknowledgement

The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China(No.51475233).

Appendix A

Table A1 Detailed data of cutting simulation tests in Table 6.

Table A1 (continued)