Modeling of Critical Cutting Speed of White Layer Formation in Hard-Cutting Process

： White layer exists on the machined surface of the hard-cutting and affects the surface quality and mechanical properties of a workpiece. Accurate predicting the critical cutting speed of white layer formation is of great significance for controlling the surface quality and selecting appropriate cutting parameters. In this work, an austenite transformation driving force calculation model of the white layer formation was established based on phase transformation thermodynamics theory, in which the influence of cutting temperature, stress and strain on the austenite transformation driving force in the hard-cutting process was taken into account. Second, a finite element (FE) model of the hard-cutting process was built by using hardened AISI52100 steel as cutting material. Then, a prediction model of critical cutting speed of the white layer formation was developed in combination with the austenite transformation driving force model and the hard-cutting FE model. Finally, the critical cutting speeds of the white layer formation at different chip thicknesses, tool rake angles and different levels of flank wear were simulated by using the critical cutting speed prediction model.


Introduction
Hard-cutting, which has advantages of high quality, high efficiency and environmental protection, is being widely used in finishing hardened steel. However, workpiece is subjected to extremely high temperature and severe plastic deformation as the result of high hardness, which inevitably leads to the dramatic changes in the machined surface. Scholars has observed that white layer is formed on the machined surface during hard-cutting process [1] and differ from the bulk material in microstructure [2], surface quality [3] and mechanical property [4].
The white layer has complex effects on the performance of the workpiece [5], it is desirable to be produced in some cases [6], yet not in others [7,8]. Therefore, modeling a critical cutting speed of white layer formation in the hard-cutting process is of vital importance for controlling the formation of the white layer and selecting reasonable cutting parameters.
In recent years, the properties and formation mechanism of the white layer have been explored in detail. The crystalline grain of the white layer is refined seriously, the hardness as well as the retained austenite content are higher than those of bulk material [9]. Chou and Evans [10] observed the white layer morphology in the hard turning of AISI52100 steel. They considered that the formation of the white layer is dominated by a rapid heat-cooling process, and cutting heat causes austenite and martensite transformations. Du et al. [11] analyzed the white layer of Ni based powder superalloy induced by cutting process, and found that phase transformation white layer can be observed when the cutting temperature is lower than the phase transformation temperature. They indicated that plastic deformation promotes the formation of the white layer. Zhang et al. [3,12] studied the formation mechanism of the white layer of AISI52100. The experimental results demonstrated that the white layer is formed by phase transformation, and plastic deformation provides the phase transformation driving force, which accelerates the formation of the white layer.
Experimental research has indicated that the white layer is induced by phase transformation in some cutting conditions [12,13]. It is of great significance to predict the white layer based on its properties and formation mechanism. In recent years, scholars have carried out in-depth exploration on the prediction of the white layer, in which the prediction model of white layer thickness is the most studied. Chou et al. [10] developed an empirical model based on the moving heat source. The model used the heat generated by plastic deformation and friction as moving heat source to calculate the temperature of the machined surface. If the machined surface temperature exceeds phase transformation temperature, the white layer is considered to be formed. Umbrello et al. [14] established a FE prediction model of the white layer thickness in dry hard-cutting process based on a hardness criterion. When the hardness of the machined surface exceeds matrix hardness, the white layer is determined to be produced.
Stress and strain have influence on the phase transformation temperature of the white layer [15], and thus, some scholars believed that the calculation model of the white layer thickness needs to be built on the basis of considering the effects of stress and strain. The influence of stress on austenite transformation temperature were taken into account by Ramesh et al., and the white layer thickness was predicted by using FE method [16]. Kong et al. [17] considered the effects of alloy elements, stress and strain on the phase transformation temperature of the white layer and established a critical austenite transformation temperature model of the white layer based on the phase transformation free energy theory. Then, the model was combined with a hard-cutting FE model to predict the thickness of phase transformation white layer at different cutting parameters. Zeng et al. [18] established a prediction model for white layer thickness based on phase transformation mechanism as well, the effects of stress, elastic and plastic strain on phase transformation temperature were considered.
Although some investigations into the modeling of the white layer in hard-cutting process were reported, certain problems have not been resolved. First, the research on the theoretical modeling of the white layer mainly focuses on the prediction of the white layer thickness. Nevertheless, the prediction of critical cutting speed of the white layer formation has not been studied in detail, yet establishing a model of critical cutting speed of the white layer formation has a guiding role in selecting reasonable cutting parameters. Second, driving force of austenite phase transformation plays significant roles in the white layer formation in the hard-cutting process. The kinetics of austenite formation involved in previous studies mainly focused on the heat treatment processes [19][20][21].
Nevertheless, plastic deformation was not included in these kinetics models, which has deeply influence on the formation of white layer. Therefore, stress and strain should be taken into account when calculating the austenite phase transformation driving force. Next, stress and strain cannot be measured in the process of cutting experiments, however, hard-cutting FE model can simulate these parameters accurately. Accordingly, combining the austenite phase transformation driving force model with the hard-cutting FE model to predict the critical cutting speed is worthy of study.
Accordingly, the following studies were carried out. A driving force model of the white layer austenite transformation was established based on the free energy change principle, and the influence of cutting temperature, stress and strain on the driving force were considered in the model. Then, a hard-cutting FE model was developed to simulate the cutting temperature, stress and strain distribution of the hard-cutting workpieces. Next, a critical cutting speed model of the white layer formation was established by combining the driving force model with the hard-cutting FE model. The critical cutting speed of the white layer formation calculated by the theoretical model was compared with the cutting experimental results. At last, the influences of cutting parameters, rake angle and different levels of flank wear on the critical cutting speed of the white layer formation were predicted and discussed.

Calculation model of austenite phase transformation driving force in hard-cutting process
Research by Zhang et al. [3] indicated that the retained austenite volume of the white layer is higher than that of the bulk material, demonstrating that austenite transformation occurs in the hard-cutting process. In the hard-cutting process, phase transformation white layer is induced by solid phase transformation. Free-energy change G  in metal solid transformation process is determined by phase transformation driving force and resistance. The solid phase transformation cannot be carried out unless G  <0, i.e. the phase transformation driving force is greater than the phase transformation resistance. High cutting temperature and plastic deformation provide the driving force for the white layer formation [22].

Driving force of austenite transformation provided by cutting temperature
Kooiker et al. [23] established a martensite to austenite reversion model and demonstrated that temperature is one of the main factors affecting the austenite transformation process. In the white layer austenitizing process, martensite M is parent phase, and austenite  is new phase. Fig. 1 shows the tendency of martensite and austenite free energy with temperature. The martensite cannot be transformed to the austenite unless molar free energy of austenite is lower than that of the martensite: where,  The free energy of a substance is given by [24]: where, H is molar enthalpy ( J/mol ), S is molar entropy ( J/(mol K)  ), T is temperature ( K ).
The free energy, enthalpy and entropy of a system change with temperature. In the constant-pressure process, molar enthalpy change H  due to temperature is written as [25]: where,   P CTis molar heat capacity at constant pressure ( J/(mol K)  ).
Molar entropy change S  due to temperature is [25]: Through regression analysis of experimental data, the relationship between the   P CT and temperature is described by a polynomial relation [25]: where, a , b , c are constants, the values of a , b , c of martensite and austenite are shown in Tab where, After substituting Eqs.8-11 into Eq.12, can be given as: 32 45.93 9.245 10 9.12 ln 4883.38 So far, the calculation model of austenite transformation driving force provided by cutting temperature was =0, the free energy of martensite is equal to that of austenite. On being less than 0, the molar free energy difference between austenite and martensite begins to provide driving force for M   transformation.

Driving force of austenite transformation provided by stress and strain
In the hard-cutting process, the white layer formation is affected by both plastic deformation and cutting temperature. Ramesh et al [16] investigated the influence of stress and strain on the austenite equilibrium transformation temperature cm A , and found that high stress and strain cause a decrease in cm A , which demonstrated that austenite transformation driving force can be provided by the plastic deformation in the hard-cutting process.
According to the thermodynamic principles of phase transformation, the free energy change due to stress P at constant temperature T : where, T G  is free energy change induced by stress ( J/mol ), V is molar volume of a substance ( 3 m /mol ).
As the molar volume of martensite differs from that of austenite, the T G  cannot be calculated by Eq. 14 directly. However, the free energy is state variable, T G  is only related to the initial and final state of the system, but not to the process. Therefore, T G  caused by pressure can be expressed by: is molar volume increment from martensite to austenite, P is compressive stress on the machined surface caused by the cutting process ( Pa ), 0 P is the stress on the surface before cutting process, which is atmospheric pressure. The stress produced by the hard-cutting process is much larger than atmospheric pressure, and thus, it is assumed 0 P = 0Pa.
Because the strain energy S W can be obtained by the hard-cutting FE model, its value is directly extracted from the post-processing results of the FE model instead of establishing its calculation model. Therefore, the driving force of austenite transformation induced by cutting temperature, stress and strain is described as: Where, the unit of S W is J/ mol .

FE modeling of hard-cutting process
According to the established model, the driving force of austenite transformation of the white layer is affected by cutting temperature, stress and strain. The stress P and strain energy S W cannot be measured directly during the cutting process, however, a hard-cutting FE model can predict the thermodynamic behavior in the machining process accurately. Therefore, an orthogonal cutting FE model was established by using ABAQUS software. In the orthogonal cutting process, the material is in the state of plane strain, and thus, a two-dimensional rather than three-dimensional orthogonal cutting FE model was established, as shown in Fig. 2.  The critical cutting speed model cannot fully consider the factors which affect the austenite transformation in the white layer owing to the complicacy of this process. Therefore, the prediction model was appropriately simplified and the following basic assumptions were made: ① The workpiece is free of impurities after heat treatment and the influence of cementite on the austenite transformation driving force is ignored.
② When the austenite transformation driving force provided by the hard-cutting process reaches the critical driving force, it is considered that austenite transformation can occur in the machined surface.

Extraction of cutting temperature, stress and strain energy
This model aims to predict the critical cutting speed of the white layer formation, therefore, only the data of the top surface of the workpiece were extracted to calculate the driving force and the subsurface was not considered.
The data extraction position is the contact point between the tool tip and the machined surface, as shown in Fig. 4.

Calculation of critical cutting speed of the white layer formation
After the data of T , P and S W extracting from the FE simulation results,

Data extraction point
Khodabakhshi et al. [27] measured the change of free energy in the austenite transformation process of carbon steel. The result showed that the free energy required for austenite transformation of carbon steel is 1156 J/mol , that is to say, the critical austenite transformation driving force is 1156 J/mol . The austenite transformation driving force calculated at different cutting speeds ( 0  = -10°, c a = 0.1mm, VB = 0 mm) was compared with the critical austenite transformation driving force, as shown in Fig. 5. At the cutting speeds of 28 m/min and 44 m/min, the austenite transformation driving force provided by the hard-cutting process is lower and higher than the critical austenite transformation driving force, respectively, which indicates that the white layer cannot and can be formed in the machined surface at these two cutting speeds. At the critical cutting speed, the austenite transformation driving force is equal to the critical austenite transformation driving force, and the critical cutting speed of white layer formation was determined by linear fitting method, which is 38m/min.

Model validation
To verify the accuracy of the simulation model, the predicted critical cutting speed were compared with the experimental results. The cutting experiments were performed in a MULTUS B400-W machining center, and the set-up of the hard-cutting experiments is shown in Fig. 6. The hard-cutting experiments were carried out at the cutting speeds of 28m/min, 34m/min and 44m/min ( 0  = -10°, c a = 0.1mm, VB = 0mm). The material and cutting tool used in the experiments are consistent with those in the FE model. The machined surfaces were observed by Scanning Electron Microscope (SEM), as shown in Fig. 7. Figs. 7 (a) and (b) indicate that only plastic deformation layer caused by the hard-cutting process is observed in the machined surface at the cutting speeds of 28m/min, 34m/min, and no white layer is produced. However, white layer with the thickness of less than 1μm is observed in the machined surface at the cutting speed of 44m/min, as shown in Fig. 7 (c). The  critical cutting speed of the white layer formation. As a result, the influences of cutting conditions on the critical cutting speed of the white layer formation were predicted and discussed.

Influence of chip thickness on the critical cutting speed of the white layer formation
The austenite phase transformation driving forces

Influence of rake angle on the critical cutting speed of the white layer formation
The tendency of with cutting speed at different rake angles is displayed in Fig. 9(a) (VB = 0 mm, ac = 0.1 mm). When the tool angle changes from negative rake angle to positive rake angle, the driving force decreases significantly. The tool rake angle has obvious influence on the cutting temperature, stress and strain. When 0  changes to positive rake angle, the extrusion and friction between tool rake face and chip are obviously reduced, leading to the decrease in cutting temperature, stress and strain, and thus the decrease in M G    . Fig. 9 (b) shows the change rule of critical cutting speed of the white layer formation with the rake angle. When 0  changes from -10 ° to 10 °, the critical cutting speed increases from 38 m/min to 59.9 m/min, the positive rake angle makes the critical cutting speed of white layer formation increase rapidly.

Conclusion
The models of the austenite transformation driving force and the critical cutting speed of the white layer formation were established. The conclusions can be derived as follows: (1) The austenite transformation driving force of the white layer in the hard-cutting process was derived mathematically, in which the effects of cutting temperature, stress and strain on the phase transformation driving force were taken into account. It indicates that both the cutting heat and the plastic deformation can provide driving force for austenite transformation.