key: cord-0266651-kf6mqv4e
authors: Bergs, Thomas; Hardt, Marvin; Schraknepper, Daniel
title: Determination of Johnson-Cook material model parameters for AISI 1045 from orthogonal cutting tests using the Downhill-Simplex algorithm
date: 2020-12-31
journal: Procedia Manufacturing
DOI: 10.1016/j.promfg.2020.05.081
sha: 7e4e50075aa4160452338c754cd344775dc1e43f
doc_id: 266651
cord_uid: kf6mqv4e

Abstract Despite the increasing digitalization of manufacturing processes in the context of Industry 4.0, the process design and development of machining processes poses major challenges for today’s manufacturing technology. Compared to the conventional process design, which is influenced by an empirical "trial-and-error" principle, the simulative process design offers the possibility of reducing development time and costs while at the same time improving the process understanding. A possible simulation technique to achieve these goals is the Finite Element Method (FEM). The FEM enables the calculation of the thermo-mechanical load spectrum underlying the machining process. Therefore, different input models are required. One of the most critical input models is the material model, which describes the constitutive material behavior. To determine the material model parameters, either (conventional) material tests, which require an extrapolation into the regime of metal cutting, or inverse techniques are used, where the process itself is used as a material test. Using the inverse technique, the model parameters are modified iteratively until a predefined agreement between simulations and experiments is achieved. The evaluation of the agreement bases on integral process variables, such as the cutting force, and their simulative counterparts. However, the procedure of the inverse determination requires high computational efforts and is not robust. This paper presents a novel approach to enhance the robustness of the inverse material model parameter determination from the cutting process. Orthogonal cutting tests on AISI 1045 steel have been conducted on a broaching machine tool over a range of different cutting speeds and undeformed chip thicknesses to set an experimental database. Thereby, the workpiece material was investigated in the two different heat treatments: normalized and coarse-grain annealed. The machining experiments showed differences in terms of the integral process results when comparing the two heat treatments. These results motivated for the development of a methodology capable to determine material model parameters robust and inversely from the machining process, which can be used with lower computational effort. To simulate the machining process, a Coupled-Eulerian-Lagrangian (CEL) model of the orthogonal cutting process has been set up. The material model parameters have been inversely determined using the Downhill-Simplex-Algorithm, which has been modified for this case. By using the Downhill-Simplex-Algorithm, it was possible to determine material model parameters within 17 iterations and achieving an average deviation between the experiment and the simulations below 10 %. Thereby, different process observables such as temperature, forces, and chip form have been used for the evaluation. Through this method, it is possible to determine material model parameters, which enable a good match between experiments and simulations with a low computational effort.

In the field of machining, the relevance of the fourth industrial revolution is especially reflected by an increasing demand of the virtualization of the process design [1] . In the state of the art, machining processes are empirically designed by a trial-and-error approach [2] . However, the empirical process design is limited in its capabilities, since it is of descriptive nature and not predictive. Additionally, this method of process design is expensive and time consuming [3] .

To overcome these limitations, simulation techniques are used that exhibit the capability to reduce the time to market of new processes and products [4] . The methods to model the metal cutting process can be divided into the following types: analytical, numerical, artificial intelligence (AI), and hybrid modeling [5] . An example for numerical methods is the Finite

In the field of machining, the relevance of the fourth industrial revolution is especially reflected by an increasing demand of the virtualization of the process design [1] . In the state of the art, machining processes are empirically designed by a trial-and-error approach [2] . However, the empirical process design is limited in its capabilities, since it is of descriptive nature and not predictive. Additionally, this method of process design is expensive and time consuming [3] .

To overcome these limitations, simulation techniques are used that exhibit the capability to reduce the time to market of new processes and products [4] . The methods to model the metal cutting process can be divided into the following types: analytical, numerical, artificial intelligence (AI), and hybrid modeling [5] . An example for numerical methods is the Finite

In the field of machining, the relevance of the fourth industrial revolution is especially reflected by an increasing demand of the virtualization of the process design [1] . In the state of the art, machining processes are empirically designed by a trial-and-error approach [2] . However, the empirical process design is limited in its capabilities, since it is of descriptive nature and not predictive. Additionally, this method of process design is expensive and time consuming [3] .

To overcome these limitations, simulation techniques are used that exhibit the capability to reduce the time to market of new processes and products [4] . The methods to model the metal cutting process can be divided into the following types: analytical, numerical, artificial intelligence (AI), and hybrid modeling [5] . An example for numerical methods is the Finite

In the field of machining, the relevance of the fourth industrial revolution is especially reflected by an increasing demand of the virtualization of the process design [1] . In the state of the art, machining processes are empirically designed by a trial-and-error approach [2] . However, the empirical process design is limited in its capabilities, since it is of descriptive nature and not predictive. Additionally, this method of process design is expensive and time consuming [3] .

To overcome these limitations, simulation techniques are used that exhibit the capability to reduce the time to market of new processes and products [4] . The methods to model the metal cutting process can be divided into the following types: analytical, numerical, artificial intelligence (AI), and hybrid modeling [5] . An example for numerical methods is the Finite 48th SME North American Manufacturing Research Conference, NAMRC 48 (Cancelled due to Element Analysis (FEA), which has found wide applications in modeling the machining process, especially in the scientific community [5] . Ever since the first application of Finite Element Method (FEM) in the field of engineering by Zienkiewicz [6] and especially in the field of machining by Klamecki [7] in the 1970s, the use of FEM techniques for modeling manufacturing processes increased significantly.

The major advantage of the FEM when modeling the machining process is the possibility to calculate process quantities, such as stresses, strains, and strain rates, which cannot be measured during the machining process [8] . However, these process quantities play a major role to understand the mechanisms within the process and are therefore necessary to enhance the process understanding.

When modeling the machining process by means of FEM, diverse input models are necessary, such as a friction model and a material model, which are essential for the success and reliability of the predicted results [9] . Among these models, the material model has a major impact on the results [10] [11] [12] [13] . The material models used for metal cutting simulations can be divided into empirical/phenomenological, semi-empirical, and physical-based material models. Thereby, empirical material models describe the material behavior as a function of the strain, strain rate, and temperature, whereas physical-based material model take fundamental microstructural properties such as the dislocation density into account [14] . Mostly, empirical material models are used in machining simulations, whereas physical-based material models are just rarely utilized due to enormous difficulties of modeling the basic material deformation mechanisms under loads of the metal cutting process. [15] However, as important as the selected material model are the underlying material model parameters. Commonly, these parameters are determined by quasi-static or dynamic material tests. An example for a dynamic material test, which has been used for determining material model parameters to describe high-strain and strain rate flow curves, is the Split-Hopkinson-Pressure-Bar (SHPB) test [16; 17] . When using the SHPB-test, strains up to 0.5, strain rates up to 5·10 5 s -1 , and temperatures up to 1,000 °C are achievable [18] . However, these conditions are far away from those encountered in the machining process. In the process, strains up to 2, strain rates up to 10 6 s -1 , and temperatures between 500 and 1,400 °C can occur [2; 19; 5] . Due to these differences in the occurring loads, extrapolation of the determined material behavior into the regime of metal cutting becomes necessary, which can lead to large deviations between the predicted and the actual material behavior [20] .

To circumvent the extrapolation of the determined material behavior in the regime of metal cutting alternative approaches are necessary. Within the last decade, inverse techniques have been used, where the process to be modeled is used as a material test itself. Thereby, the material model parameters are modified within a simulation until the predicted simulation results match the experimental results [21] . The conformity of simulation and experiments is evaluated based on integral process results, such as the cutting force, chip thickness, chip form or cutting temperature.

However, the procedure of the inverse determination is not robust and requires a large number of iterations and, therefore, high computational efforts [19] . To overcome these issues when inversely identifying material model parameters, different optimization strategies and algorithms have been used. In the field of sheet metal forming, Chaparro et al. investigated the inverse parameter identification of the Barlat material model using a genetic algorithm, a gradient-based algorithm, and a combination of both [22] . The results revealed that both algorithms are able to fit the numerical with the experimental data. In the field of machining, Özel and Karpat, used the evolutionary computational algorithm of cooperative Particle Swarm Optimization (cPSO) to determine the JC-parameter [23] . However, the underlying experimental data were obtained from SHPB-tests and from orthogonal cutting experiments in conjunction with a modified Oxley model. Therefore, the drawbacks of extrapolation and of the Oxley model are underlying this approach.

Franchi et al. developed an inverse optimization procedure to determine the JC-parameters of AISI 316 stainless steel and SAF 2507 super-duplex stainless steel as well as the Coulomb friction coefficient [24] . Therefore, a sequential approach, starting with an initial set of machining simulation based on a design of computer experiments (DOCE) and analysis of the numerical results in terms of cutting forces and temperatures was used. Based on these results, a regression model was developed serving as a surrogate model. Subsequent, a Multi-Island Genetic optimization algorithm was used to identify the best collection of JC-and friction coefficients by minimizing an objective function. However, the approach revealed large deviations between experimental and numerical results of up to 75 %.

Bosetti et al. inversely determined JC-material model parameters and the Tresca friction coefficient for AISI 304 stainless steel using the Downhill-Simplex-Algorithm (DSA) in combination with a genetic algorithm (GA) [25] . Since their approach focused on just one cutting condition, it remains questionable how well the determined parameters describe the material behavior for other cutting conditions. The underlying problem of the non-uniqueness of material model parameters has been widely reported in the literature [19; 26-28] .

Within this paper, a new methodology is used to determine the material model parameters from orthogonal cutting experiments using an optimization algorithm. To set an experimental database, orthogonal cutting experiments have been conducted on a broaching machine tool. As workpiece material AISI 1045 has been chosen in two different states of heat treatment. The material has been investigated in the normalized and in the coarse grain annealed state. The differences in the material behavior are initially analyzed by comparing the results of quasi-static tensile and shearing tests as well as of Charpy-impact tests, revealing differences in the two annealing states. Thereafter, a novel methodology of material parameter determination from the machining process is presented and applied to determine the Johnson-Cook material model parameters of AISI 1045 in the normalized state. To determine the material model parameters, the Downhill-Simplex algorithm, which has been modified to be applicable to the inverse problem of material model parameter determination, is used.

The paper is organized as follows: in the following chapter, the characterization of the workpiece material, including the results of quasi-static tests and impact tests are presented, followed by the experimental set-up of the orthogonal cutting experiments. In Chapter 3, the results of the orthogonal cutting tests are outlined and analyzed. The description of the orthogonal cutting model in the machining simulations is presented thereafter. In Chapter 6 the material model parameters are determined for AISI 1045 in the normalized state using an optimization algorithm, followed by the validation of the determined model parameters. Finally, a summary and conclusion of the results will be given in the last chapters.

This chapter is divided into three subchapters, presenting firstly the workpiece material AISI 1045 that is used in the experiments of this work, followed by a characterization of the material by means of conventional material tests. In the third subchapter, the experimental set-up of the orthogonal cutting experiments is outlined.

As workpiece material, the low-carbon steel AISI 1045 was investigated. The material has been focus of several researches so far, especially in the field of machining due to its wide application in the automotive industry, where it is used for components of medium loads such as crankshafts [29] . The chemical composition of the workpiece material has been determined by spark spectroscopy. The results, which have been averaged from four measurements, are summarized in Table 1 . The chemical compositions shows slight deviations from the nominal chemical composition as specified by the manufacturer, especially in terms of the carbon content. To investigate the influence of the material's grain size on the material behavior, the AISI 1045 has been annealed in two different states: normalized (N) and coarse-grain annealed (CG). The heat treatment conditions are summarized in Table 2 . After the heat treatment, the microsections showed a homogeneous microstructure without a line-structure from previous manufacturing processes. The microstructure of the normalized material consisted of globular perlite and ferrite, whereas the microstructure of the coarse grain annealed material consisted of globular pearlite and globular/lamellar ferrite, see Fig. 1 . The phase fraction of pearlite and ferrite as well as the average grain size of the two phases were determined using the software ImageJ. The results are summarized in Table 3 . Further, Table 3 contains the results of hardness measurements, which show a distinct deviation between the two different heat treatments. The given hardness values are averaged values, based on 16 measurements. Average ferrite grain size d / µm 7 11 Average pearlite grain size d / µm 12 37 Hardness HV / HV0.03 223±30 347±40

The differences in the hardness measurements are attributed to the differences in the phase fractions of the two investigated heat treatment conditions. The higher phase fraction of pearlite for the coarse grain material causing a higher hardness of the material, since pearlite exhibits a higher hardness than ferrite.

Besides the metallographic alterations, the differences of the two heat-treatments in terms of the material behavior under quasi-static and impact conditions have been investigated. The material behavior under quasi-static conditions has been determined under tensile and shear loading and the impact behavior by using the Charpy-impact test.

The quasi-static tests have been conducted on a Zwick Z100 universal testing machine with a maximal force of Fmax = 100 kN. The shear and tensile tests have been conducted with a strain rate of ̇= 0.001 −1 , resulting in a test speed of vtensile = 4.5 mm/min and vshear = 0.3 mm/min. Both, tensile and

Coarse Grain Annealed 50 µm 50 µm shear tests have been conducted three times for each state of annealing.

The tensile tests have been conducted according to DIN EN ISO 6892-1 using the A50 geometry. In Fig. 2 , the results of the tests are shown, revealing slight differences between the test repetitions. In comparison to the coarse grain annealed samples, the normalized samples show a slightly lower ultimate tensile strength of Rm = 700 MPa, whereas the average ultimate tensile strength of the coarse grain annealed samples is Rm = 724 MPa. On the other side, the yield strength of the normalized sample is , = 486 and of the coarse grain annealed samples , = 424

. However, the uniform elongation of the normalized samples is with Ag = 13 % about 33 % higher than the one of the coarse-grain annealed samples. The differences in the ultimate tensile strength are attributed to the phase differences of the two different states of annealing. The higher pearlite fraction of the coarse grain annealed samples results in an increased ultimate strength compared to the normalized samples. The higher uniform elongation of the normalized samples is attributed to the smaller grain size of the normalized material.

The shear tests have been conducted by using a modified flat tensile specimen, which was notched by two hooks, Fig. 3 . Therefore, shear conditions were enabled. The results of the shear tests show comparable differences, with slightly higher stresses for the coarse grain annealed sample and higher strains for the normalized samples, Fig. 3 .

Additional to the quasi-static tests, impact tests have been conducted according to DIN 50115 using Charpy-V-notched samples. The tests have been conducted on a Zwick/Roell system with a maximum impact energy of 50 J and have been repeated two times for each annealing condition. The conducted impact tests revealed differences between the investigated samples, with higher impact energies for the normalized material. The average necessary impact energy of the normalized samples was 24.5 J, whereas the one of the coarse grain annealed samples was 15.7 J. Except for one CGsample, all samples failed due to fracture. 

The orthogonal cutting experiments have been conducted on a tests bench built on a vertical broaching machine of type Forst RASX 8 x 2200 x 600 M/CNC, Fig. 4 . The broaching machine has a stroke length of 2,200 mm and a maximum cutting speed of vc = 150 m/min. For cutting speeds up to vc = 30 m/min the maximum broaching force is Fmax = 80 kN and for the high cutting speeds Fmax = 20 kN. In comparison to conventional broaching, the workpiece has been clamped into a customized fixture in the tool holder, where normally the broaching tool would be clamped into.

As cutting tool, a grooving insert tool of type CoroCut from Sandvik Coromant has been used. The grooving inserts were made out of the cemented carbide grade H13A, with an average grain size of 1-2 µm and a Co-content of 6 %. The rake angle of the tool was γ = 6° and the flank angle α = 3°. To circumvent the occurrence of built-up edge formation, the tools have been coated with a TiAlN-coating of 4 µm thickness. The cutting edge rounding was measured in average to rβ = 14 µm with a standard deviation of 2.1 µm for all tested tools

The cutting tool was clamped into a tool holder on a dynamometer type Z21289 from Kistler with a measuring range from -80 to +80 kN in the cutting force direction. The sampling rate was 50 kHZ. During the experiments both, cutting force Fc and cutting normal force FcN were measured. A high-speed camera of type Phantom v7.3 captured the chip formation process with a frame rate of 6,700 s -1 and a resolution of 800 x 600 pixels. To increase the intensity of light and to enhance the capture of the chip formation, a LED-light has been used. The cutting temperature has been measured by using a two-color pyrometer. Since the workpiece temperatures behind the cutting zone were too low to be captured by the twocolor pyrometer, the pyrometer has been used to measure the tool-side temperature of chip. Further, an infrared camera captured the workpiece temperature field. However, the results of these measurements will not be part of this paper and will be presented in the future. The orthogonal cutting experiments have been conducted by varying the cutting speed vc and the undeformed chip thickness h. The experimental design is summarized in Table 4 . The orthogonal cutting tests were repeated twice in order to enhance the statistical security. 

The cutting force Fc and cutting normal force FcN have been analyzed in the steady state of the measured signal. Therefore, the signal from 40 to 90 % of the total cutting time has been evaluated, Fig. 5 . The results of the cutting force and cutting normal force measurements for the investigated undeformed chip thicknesses and cutting speeds are shown in Fig. 5 . An increase of the cutting force and cutting normal force with increasing undeformed chip thickness can be seen for all investigated conditions and states of annealing of the material. The increase of the cutting force and cutting normal force with increasing undeformed chip thickness can be explained by the higher mechanical load which is involved in the chip formation process and is in accordance with observations from the literature. When increasing the cutting speed vc, a slight decrease of the cutting force and a more distinct decrease of the cutting normal force can be observed. The decrease of the cutting forces can be explained by the thermomechanical workpiece behavior. An increase of the cutting speed results in an increase of both, the mechanical and thermal load. The decrease of the force can therefore be attributed to a more dominating influence of the thermal softening, compared to the strain and strain rate hardening. It is further assumed that frictional alterations cause a decrease of the cutting normal force FcN, which is sensitive to the frictional conditions [30] .

Comparing the two different states of annealing reveals higher cutting and cutting normal forces of the normalized samples. This observation is in accordance with the results of the quasi-static tests and is attributed to the higher yield strength of the normalized material. The chip thicknesses have been measured at three different spots for all chips, using the recordings of the high-speed camera. The results of these measurements are shown in Fig. 6 , revealing an increase of the chip thickness with increasing the undeformed chip thickness. A large alteration of the chip thickness with increasing cutting speed cannot be observed. The comparison between the two states of annealing reveals small differences, with higher chip thicknesses for the normalized material. These differences are, however, within the measurement scattering. When measuring the temperature/temperature field in the cutting process, different measurement positions are possible. In this study, a two-color pyrometer was used to measure the temperature of the chip shortly after its formation. The displacement of the measuring position with different undeformed chip thicknesses showed no alterations.

The results of the temperature measurements using the twocolor pyrometer are summarized in Fig. 7 . When increasing the undeformed chip thickness, an increase of the chip temperature can be observed. However, an increase of the chip temperature for higher cutting speeds is not as distinct, as it is the case for higher undeformed chip thicknesses. Albeit, the shown temperatures do not necessarily represent the temperature close to the cutting zone. Due to higher tool-chip contact lengths or different chip curvatures, the measuring position can be further from the cutting zone, Fig. 7 .

In contrast to the force measurements, no pronounced differences between the two heat treatments can be identified for the chip temperature measurements. The chip temperature measurements show a large scatter. The deviations have to be taken into account when determining the material model parameters, since the simulations are not expected to be more accurate than the experiments.

The experimental results of the process observables of the two investigated states of heat treatment for AISI 1045 showed smaller deviations than expected from the quasi-static material tests. It seems doubtful that the simulations are capable to depict these differences in the process observables, which were in the experiments in average lower than 10 %. Therefore, just the experimental results of the normalized material will be used for the inverse determination of the material model parameters inversely by means of FEM-simulations. 

This chapter outlines the FEM-model used for the orthogonal cutting simulations. Therefore, the used models to describe the material behavior and the frictional behavior are described, followed by the Coupled-Eulerian-Lagrangian (CEL) model.

To model the material behavior under metal cutting conditions, the JC-model has widely been used, Equation (1). In the JC-model, the effects of strain , strain rate ̇, and temperature on the flow stress are modeled uncoupled by three separate terms [31] . The first bracket of Equation (1) expresses the strain hardening, which is in accordance with the Ludwik-equation [32] . The second bracket models the effect of strain-rate hardening and is formulated in a logarithmic form. The effect of thermal softening is expressed based on a power function [29] . In the JC-model, , , , , and are material constants. ̇0 is the reference strain rate with ̇0 = 0.1 −1 , 0 the reference temperature, and the melting temperature. [33] = ( + ⋅ ) (1 + ln (̇̇0)) ( 

Besides the material model, the friction model has a major influence on the simulated results [34] . In several studies, it has been shown that using a simple Coulomb friction model is not sufficient to describe the frictional behavior between the tool and the workpiece [35] . To overcome this drawback, several researches proposed different friction models, e.g. Özel [36] , Filice et al. [37] or Puls et al. [35] .

Puls et al. developed a friction test that is capable to reproduce the conditions encountered in the machining process in terms of relative velocities, temperatures, and normal pressures [35] . Therefore, the authors used an orthogonal highspeed deformation process on a broaching machine, where an indexable insert was rotated, resulting in an extreme negative rake angle, which suppressed the chip formation. Based on their findings, a temperature dependent friction model was developed, Equation (2).

In this study, the friction model according to Puls et al. is used, as well as the determined friction model parameters, which have been determined for AISI 1045 in a normalized state [35] . The parameters used for the friction model are summarized in Table 5 . The friction model was implemented into the simulation program ABAQUS/Explicit in a tabular form, whereby the friction coefficient was given for temperature steps of 10 °C until the melting temperature where it reaches zero.

When modeling the machining process, two different formulations are used: the Eulerian and the Lagrangian formulation [5; 38] . In the Eulerian approach, the mesh is fixed in space and material flows through element faces allowing large strains without the occurrence of mesh distortion [39] . However, for simulations of the machining process, the Eulerian formulation can only be used for the steady state, wherefore the knowledge of the final chip geometry is required [5] . In the Lagrangian formulation, the nodes of the mesh are attached to the material and follow the material deformation [40] .

To overcome the individual drawbacks of the Eulerian and Lagrangian formulations, see e.g. [5] , two other formulations have been developed and applied in modeling the machining process: the Arbitrary Lagrangian Eulerian (ALE) [41] and the Coupled Eulerian Lagrangian (CEL) [42] approach. In the ALE-formulation, the material flows through the meshanalogous to the Eulerian formulationand the element nodes are additional able to move free within the area.

The CEL-formulation has firstly been introduced to machining in the last five years [42; 43] . In the CELformulation, the workpiece material is modeled by the Eulerian formulation, allowing the material to flow freely through the fixed mesh. The tool on the other hand, is modeled by the Lagrangian formulation. The set-up of the CEL-model underlying this paper is shown in Fig. 8 . An inflow of the material in the Euler domain is used as boundary condition to model the cutting speed, Fig. 8 . The material leaves the Euler domain in form of a chip or flows out of the domain on the right hand side in the set-up. Within the Euler domain, an additional area is modeled to allow the chip to be formed. The Lagrangian tool is fixed in space and is assumed to be rigid.

The tool was modeled with the element type C3D4T, to allow the calculation of the temperature within the tool. The mesh size varied along the sides of the tool, ranging from 5 µm to 50 µm. Therefore, an accurate calculation of the temperature within the tool can be achieved. The Euler domain on the other side was modeled with EC3D8RT elements with a smallest mesh size of 5 µm within the zone of chip formation. The parameters to model the thermal and mechanical parameters of the tool material were taken from the literature [44; 45] .

To model the influence of the coating, a 4 µm thick layer of TiAlN-coating was modeled on the tool, represented by the green area in Fig. 8 . The data to model the thermal and the mechanical behavior were taken from the literature [46; 47] .

For all conducted simulations, a constant cutting length of lc = 3.3 mm was used to enable to reach the steady state conditions. To decrease the computational time, mass scaling was used by a factor of 1,000. Therefore, the computational time was decreased to approximately 2 h for the simulations of Table 6 . 

The method to determine the JC-material model underlying this paper bases on the Downhill-Simplex algorithm (DSA) (also called Nelder-Mead algorithm) [48] . In a previous study, the authors investigated the algorithm to re-identify the JCparameters [49] . In this study, the algorithm is used to identify the parameters from experimental results.

Generally, the DSA is a method for a multi-dimensional problem that can be employed to minimize the error between predictions and measurements [50] . When determining material models inversely, this capability can be used to minimize the error between the experimental results and the numerical results. Thereby, the model parameters are iterated until a pre-defined error or a number of iterations is reached. An advantage of the DSA, in comparison to other optimization algorithms such as the Levenberg-Marquardt algorithm which has been employed in the material model parameter determination [27; 51-53] , is that the DSA is a derivate-free algorithm.

When using the DSA, an initial simplex is needed. A simplex is a regular polytope that is defined by n + 1 vertices in a n-dimensional optimization problem [48; 54] . For a 3Dspace, the simplex is a tetrahedron and for the 2D-space a triangle. For the 2D-case, the procedure of the DSA is illustrated in Fig. 9 . The three points 1 , and +1 define the initial simplex. The worst vertex ( +1 ), in terms of the error value, is reflected around the centroid of the hyperplane that is formed by the two remaining vertices [48] . This operator is called reflection. The other operators of the DSA are expansion, internal contraction, and external contraction. In this study, the parameters of the four described operators were set to = 1 (reflection), = 0.5 (expansion), = 0.5 (internal contraction) and = 0.5 (external contraction).

To apply the Downill-Simplex algorithm to the problem of inverse parameter determination, a function has to be defined, which can be optimized by the algorithm. This function has to evaluate the deviation of the numerical from the experimental results. Therefore, an error function has been defined that evaluates the normalized deviation of the cutting Force Fc, the cutting normal Force FcN, chip thickness h', and chip temperature TC. The error function is given in Equation (3). The different integral process results are weighted by individual weighting factors. These have been chosen based on the scatter of the experimental results. A high scatter resulted in a low weighing factor, whereas a higher weighting factor has been used for low scatter. The weighting factors were set to = 0.40, = 0.25, = 0.15, and ℎ ′ = 0.20.

The lower weighting factor of the cutting normal force FcN compared to the weighting factor of the cutting force Fc is due to the high sensitivity of the cutting normal force to the frictional conditions. By utilizing a temperature-dependent friction model, an accurate modeling of the frictional conditions has been aimed. However, determining the frictional conditions is not the scope of this paper. 

This chapter outlines the procedure to determine the material model parameters of the JC-model for the material AISI 1045 in the normalized state. In order to reduce the computational effort of the procedure, the JC-parameter has been determined from the quasi-static tests. Therefore, the parameter is assumed to be 486 . The Downhill-Simplex algorithm is applied to determine four material model parameters. Hence, an initial simplex of five vertices is necessary. The vertices are randomly selected with the requirement to be within the defined domain of the parameters. The domain of the parameters is determined based on typical material model parameters from the literature for AISI 1045 plus an additional offset. The domain of the material model parameters was set to: ∈ [350 , 700 ] , ∈ [0.005, 0.15], ∈ [0.1, 0.9], and ∈ [0.1, 0.85] . The algorithm has been modified so that when a parameter is calculated to be outside of the parameter domain, it is projected onto the edge of the domain.

To determine the material model parameters, three cutting conditions were used covering the lower domain of the investigated cutting conditions. The undeformed chip thickness ℎ = 0.01 was not considered, since the measurements of the chip thickness were in the lower range of the measurable. As upper cutting condition, the undeformed chip thickness 

Modified simplex ℎ = 0.2 was chosen. Thus, the range of finishing and roughing conditions was covered. Further, the application of the determined material model parameters to cutting conditions outside of the calibration domain can be evaluated.

The development of the error function over the number of iterations is shown in Fig. 10 , where S1 to S5 represent the initial simplex. In Fig. 10 , a decreasing trend of the value of the error function can be observed. The algorithm finished after 13 iterations, after reaching a local minimum. However, an average error of 15.4 % over the three cutting conditions used for the parameter determination remained. The comparison of the experimental results and the simulated counterparts shows good agreement in terms of cutting force Fc, chip thickness h', and chip temperature TC for the investigated cutting conditions. The simulated cutting force deviated by less than 21 % from their experimental counterparts, the chip thickness by less than 13 % in average, and the chip temperature by less than 5 %. However, the simulated cutting normal force FcN deviated by large extends to the experiments by up to 264 % for the highest cutting condition.

The large deviations in terms of the cutting normal force FcN are attributed to the used friction model, which has a significantly influence on the cutting normal force. Even though that the cutting normal force was just taken into account by 10 % in the error function, the deviations cause a high value of the error function.

An improvement of this value is expected, when neglecting the cutting normal force FcN in the determination of the material model parameters. That is why a second approach has been followed, where the cutting normal force is not taken into account. In this approach, the weighting factors were chosen to = 0.50 , = 0 , = 0.15 , and ℎ ′ = 0.35 . The weighting factor ℎ′ has been increased as well in order to maintain the sum of 1 of the weighting factos.

For this approach, the development of the error function over the number of iterations is shown in Fig. 11 . In comparison to the first approach, a more distinct asymptotical development of the error function with increasing number of iterations can be seen. The algorithm finished after undercutting an average error value of lower than 10 %. The predefined criteria was reached after 17 iterations. For the determined material model parameters, the remaining average error value after 17 iterations was 8.1 %. For this parameter set, the cutting force Fc deviated by less than 10 % from the experimental results, the chip thickness by less than 8 % respective by 20 % for the high undeformed chip thickness of ℎ = 0.2 and the chip temperature by less than 11 %. It is remarkable that the deviation of the cutting normal force decreased for the second approach, where the cutting normal force was not evaluated in the algorithm. However, for the three cutting conditions used for the parameter calibration, the experimental cutting normal force was still 14 %, 72 %, and 148 % higher than the simulated one. A comparison between the experimental and the simulated results is shown in Fig. 12 and Fig. 13 . S3 S4 S5 I1 I2 I3 I4 I5 I6 I7 I8 I9 I10 I11 I12 S3  S4  S5  I1  I2  I3  I4  I5  I6  I7  I8  I9  I10  I11  I12  I13  I14  I15  I16 

Three cutting conditions have been used for the material model parameter determination. To validate the determined material model parameters, additional simulations for further cutting conditions have been conducted. Thereby, simulations for cutting conditions in the regime used for the determination as well as cutting conditions outside of the regime have been conducted. For the simulations outside of the regime used for the determination, the material behavior has to be extrapolated. The results of the simulations, as well as their experimental counterpart are shown in Fig. 14 .

As it can be seen in the Fig. 14 (a) , the simulated cutting force Fc within the regime of parameter determination are close to their experimental counterparts. For the lower undeformed chip thickness ℎ = 0.01 , which has not been considered for the material parameter determination, the cutting forces are also close to their experimental counterparts. In contrast, for higher undeformed chip thicknesses of ℎ = 0.3 and ℎ = 0.4 the simulated cutting force are overestimated. The overestimation of the cutting forces for higher undeformed chip thicknesses can be attributed to the parameter m of the thermal softening term. It is expected, that the thermal softening parameter is accurately enough for the domain of parameter calibration, but not for an extrapolation to higher cutting conditions.

For the simulated chip thicknesses and chip temperatures, comparable observations as for the simulated cutting forces can be made. Within the regime of parameter determination, a good agreement between experiments and simulations can be observed. However, for the higher feed rates, larger deviation between experiments and simulations occur.

The simulation of the cutting process by means of FEM is characterized by high computational times, especially in comparison to other simulation methods. These high computational times are one of the reasons why machining simulations using the FEM is not widely used in industry. For the inverse material parameter determination, lower computational times have to be aimed. For the determination of the material parameters of the second approach presented here, the total computational time was 4 days and 17 hours. By parallelization on different CPU, the simulation time was reduced to 1 day and 18 hours. 

Conventional material tests and orthogonal cutting experiments have been conducted on AISI 1045 in two different states of annealing. The results emphasized differences in the material behavior between the two states of annealing. These differences were more distinct for the material behavior under quasi-static conditions than for the cutting process. For the cutting process, the identified differences between the two states of annealing were smaller than the defined criterion for evaluating the agreement between the experiment and simulation. Consequently, only the material model parameters of one state of heat treatment were inversely calibrated within this work. If the agreement between experiment and simulation will be further enhanced, the differences in the material behavior depending on the state of heat treatment have to be taken into account. However, it is questionable how good an agreement between simulation and experiment can be at best for a wide range of process parameters when using the JC material model. This will be further investigated in the future. 

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Within this paper a novel approach, which is capable to determine the material model parameters inversely from the machining process within a short time has been presented. For the inverse parameter determination, multiple cutting conditions have been used for the calibration. Additionally, further cutting conditions outside the domain of calibration have been used for validation. The results demonstrated a close agreement between the experiments and the simulations within the domain of parameter determination. For cutting conditions outside the domain, large deviations between simulations and experiments can be observed. Therefore, the domain of parameter calibration has to be taken into account when using material model parameters to model the material behavior of the cutting process. The application of the model to a domain outside of the domain of calibration is expected to result in larger deviations due to extrapolating.

The deviations of the simulations from the experiments can be attributed, at least to some extent, to the used material and friction model. Especially the large differences of the cutting normal forces are expected to be due to the friction model. Further, the choice of the weighting factors underlying the evaluation have to be addressed critical. In order to investigate the influence of the weighting factors on the procedure of the inverse parameter determination, a sensitivity analysis will be presented in a future publication.

In the future further analysis of both parts, the experimental and the numerical, will be conducted. For the experimental part, the influence of the heat treatment as well as of alloying elements on the material behavior will be investigated. For the numerical part, the robustness of the used algorithm, the influence of the underlying algorithm parameters and the influence of the error function will be investigated. Additionally, when inversely determining the material model parameters from cutting experiments the achievable agreement between experiments and simulations will be investigated. An enhanced agreement is expected when taking more process observables, such as the temperature filed or the chip curl radius into account.

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The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for the funding of the depicted research within the project 365204822 "Development and verification of a constitutive approach for the determination of high-speed flow curves from the cutting process".