Modeling and experimental verification of the dead metal zone for cutting force prediction in micro milling

In micro-cutting process, ploughing phenomenon occurring due to the dead metal zone (DMZ) leads to substantial ploughing force resulting in an obvious contribution to the total cutting force. To improve accuracy of the cutting force predicted, this paper aimed to explore the DMZ geometry related to the cutting depth and tool edge radius and thereof effect on cutting force. The prominent contribution of this research is to establish a new DMZ model by employing the slip-line field theory of the plastic formation. Based on the proposed model, DMZ are divided into shearing-dominated, mixed shearing and ploughing, and pure ploughing according to the minimum uncut chip thickness (MUCT). It is firstly proposed that the inner vertex of DMZ is the separation point of shearing effect and ploughing effect during metal cutting. The shape of the DMZ is theoretically calculated by an analytical way and verified by the simulation software. Finally, a cutting force model including shear force and ploughing force is established and verified by a series of experiments. The predicted cutting forces show remarkable agreement with those measured. The result proves that the separation point model is correct and can effectively demonstrate the ploughing force to accurately predict cutting force.


Introduction
With the progress of modern manufacturing technology, the demand for microcomponents in aviation, medical, electronics and other fields is gradually increasing.
The flourish of micro-cutting technologies has greatly met the requirements for small parts in these fields. Among available micro-machining technologies, micro-milling possessing some advantages of high precision, high efficiency and low cost has been widely used in contemporary micro-machining [1]. There is an arc at the tool edge as shown Fig. 1 (a) [2], the tool cannot be treated as completely sharp, which led to the simultaneous presence of shearing and ploughing during micro-machining process.
Some metal is detained nearby the tool edge named as dead metal zone (DMZ) which leads to ploughing and substantially affects the machining performance [3]. Ozturk and Altan [4] found a triangle-shaped zone existing in the front of the rounded tool edge in experiments ( as shown in Fig. 1 (b)). In this study, it was also found that the material in this zone almost remained static through finite element simulation (as shown in Fig.   1 (c)). Chips are incompletely sheared and partially ploughed due to the effect of DMZ, which leads to material around the tool edge excessive plastic deformed. Moreover, the shearing and ploughing phenomena can be divided into three cases according to the value of cutting depth: shearing-dominated, mixed shearing and ploughing, and pure ploughing [4]. Firstly, the shearing effect is dominant when the cutting depth is large enough, which leads to produce complete chips and DMZ. Secondly, for case of the cutting depth beyond a given height related to the tool edge radius, the shearing and the ploughing effect occur simultaneously, which leads to the partly chip formation and a 3 variable DMZ generation. Thirdly, the cutting depth is less than a critical value named the minimum uncut chip thickness (MUCT), which leads to only ploughing occurring without chips formation. The size of DMZ can affect the cutting force, which is crucial for descripting the micro-cutting processes. However, the available cutting force models for micro-machining process believed that the shearing force and ploughing force are generated from uncut chip thickness and the extrusion /friction between the clearance face and surface generated [5], respectively. Actually, ploughing force is also related to the uncut chip thickness in practical micro-cutting process [6]. To improve accuracy of the cutting force predicted, it is important for investigating the size of DMZ and thereof on cutting force. Altan [4]. (c) DMZ simulated by the finite element with the diagrams of material flow velocity vector [3].
The slip line field theory has been widely employed in the prediction of cutting force by many researchers. Merchant [7] initially assumed the shear zone as a thin plane for establishing an orthogonal cutting model. Afterwards, Li and Schafer [8] assumed the workpiece as ideal rigid plasticity to establish a triangular slip line field model. To accurately predict the cutting force, Childs [9] introduced the elastic contact of the toolchip to propose a new slip line field model. Furthermore, he compared the law of friction/stress models in the primary shear zone with those in secondary shear zone, and analyzed the temperature, strain and strain rate changes in the slip line field [10]. Oxley [11] presented a "parallel-side shear zone" model. Jin [12] considered the effect of rounded tool edge to propose a slip line field model for the micro-cutting process. 4 Above-mentioned cutting force models were established by employing slip line field theory and considered the material shear flow stress affected by the temperature, strain and strain rate. However, the DMZ can lead to the substantial ploughing force, which is critical to accurately predict cutting force in micro cutting process, there has been limited research on the effect of DMZ on cutting force.
Bissacco et al. [13] considered the influence of the rounded tool edge to develop a new cutting force model for micro end milling, which was based on an experimental investigation in orthogonal cutting. Yun et al. [14] presented a ploughing detection algorithm, and investigated the differences between ploughing and normal cutting by using the wavelet transformation of the cutting force signal in micro-milling processes.
Jin [15] introduced the tool edge radius and chip size to present the prediction cutting force for orthogonal micro-cutting process, which also included in the tool run-out, dynamometer dynamics and trajectory of tool. Wan [16] proposed a new method of analyzing and predicting the position of the three vertices of the DMZ, theoretically determined all the separation points. Hu [17] proposed a modified analytical slip line field model to describe the cutting process with negative rake angle by using a chamfered edge tools, starting from the assumption of the material with rigid plastic deformation. Although the above-mentioned models have been studied on the ploughing effect or the dead metal zone, quantitative analysis and calculation have not been carried out by combining the slip line field with DMZ.

Development of the DMZ model
As described in introduction, the shearing and ploughing phenomena can be divided into three cases according to the value of cutting depth. The DMZ will be proposed as three cases according to the value of cutting depth in this section.

Shearing-dominated DMZ model
When the cutting depth is large enough, the cutting force is dominated by the shearing effect. In this case, the chip formation is relatively complete. To accurately predict the size of the DMZ in the cutting process, the first priority is to establish a complete slip line field model. In orthogonal micro-cutting metal process, the deformation region of material is composed by three areas: namely the primary and secondary shear zone, and tertiary zone. Fang [18] believed that the radius of tool edge can affect the main shear zone thickness and thereof on the slip line field shape, and initially proposed the slip line field shape. Jin [12] proposed an improved slip line field model considering the effect of temperature, strain and strain rate on the adhesion and sliding area, which assumed the shear stress along rake face and the tool-chip friction constant. In this section, the DMZ is embedded into the slip line field model proposed for calculating the size of shearing-dominated DMZ. The slip line field considering the radius of tool edges is presented in Fig. 2. According to the Ref. [12], the shear zones are formed by and slip-lines perpendicular to each other. It is defined that counter-clockwise and clockwise couple are exerted the materials unit bounded by the slip-lines by the shear stress along the α and sliplines, respectively. Assuming the stress deformation and cutting conditions stable, it can be observed in Fig. 2 that the shear band is composed by four areas: 1) A polygonal area [GJBTNEQ] surrounded by slip lines is defined as the primary shear zone. Armarego and Brown [19] proposed that triangular QEG was the preflow effect in the shearing process. The slip line QG is the stress-free boundary. All slip lines in the triangle QEG intersect QG at a 45°. Green [20] proposed the tendency of GJNE area to be convex due to the rewinding effect of chips during metal cutting. The area JBTN is concave due to the friction between the chips and the tool rake face. The JBTN and GJNE areas are theoretically composed of radial lines and arcs, respectively.
2) A curved triangle BDJ is defined as the secondary shear zone. According to the Ref. [12], the direction of each slip line in this region is constant with rake angle 7 assuming the material already strain-hardened. The contact area of tool-chip can be divided into adhesion and sliding area. The chips in the sliding area slide with a constant coulomb friction coefficient, while, in the adhesion area, the ratio of friction stress to shear flow stress is constant. Both friction coefficient and adhesive shear stress are affected not only by tool and workpiece materials but also the surface roughness and temperature of rake face.
3) The polygon SAUTP is defined as the tertiary shear zone, which is mainly the ploughing effect resulting from the existence of rounded tool edge. Point P is the separation point formed during the cutting process. The material below the slip line PS will be squeezed and flow to the gap surface of the tool resulting in forming the so-called ploughing phenomenon.
4) The DMZ denoted by the BPS is located at the junction of the three main cutting deformation zones. The material above point B is mainly affected by the cutting force and slides upward to form chips, and the material below point S is mainly affected by the ploughing force to form a plough plane. In the BPS like-triangular area, the metal stagnation area called the DMZ is formed due to the simultaneous effect of adhesion of the tool, the shearing force, and the ploughing force.
The coordinates of point A are ( , − ). the shear stress not directly measured during the micro-cutting process [21]. In this paper, employing the ultimate strain theory [22] and the calculation method of friction angle [23], a method for obtaining ω is proposed. Referring to algorithm presented by Dewhurst [24] , the ω angle in the DMZ has the following formula: where τxy is the shear stress in the τ direction as shown in Fig. 5(a). k refers to the flow stress corresponding to the actual stress required for plastic flow at a certain instant in the deformation process. Due to the work-hardening effect during the single stretching process, the plastic flow stress is increasing with increment deformation. Flow stress is a function related to speed and temperature deformation, and deformation degree. For the convenience of calculation, the flow stress is set as a constant with approximate equality to the yield limit σc.
where , , and are the x and y direction normal stress, the average stress and shear yield strength, respectively.
μ is the coefficient of friction at point S, then the following equation can be derived: Where β is the friction angle of point S. Combining Eq. (6) with Eq. (7), it can be obtained that: Employing the yield criterion of von Mises, the relationship between μ and can be finally deduced [16], as following Eq. (9).

sin 2 cos 2
     (9) The ω can be calculated by Eq.(10): 11 Some algorithms for the friction angle of some materials are proposed by Kaymakci and Altintas [23] through experimental analysis, as shown in Table 1. Where h, n and v are uncut chip thickness in mm, normal rake angle in deg, and cutting velocity in m/min, respectively.  Malekian [25]. Point S and point B can be seen as two stagnation point in chips formation process, and Eq. (11) can be derived from the infinite strain criterion [16].
it can be obtained that: 12 ' '' 180 or 0 or 180 or '' ' 180 or 0 or 180 Based on the principle of maximum shear stress, Eq. (12) can be written as follows: where β is the friction angle, αe is equivalent rake angle, which can be calculated from Eq. (14) and related to point S.
It can be derived from Eq. (6)  As shown in Fig. 4, it can be concluded that BP is along the slip line in the triangle BPS, and SP is along the α slip line. According to the slip line field theory [12], the α slip lines are perpendicular to the β slip lines, therefore, ∠BPS approximately satisfies: 13 90 BPS    (16) Assume the coordinates of point P as ( , ), it can be calculated by mathematical geometry from Eq. (17) and Eq. (18) in the following: Combining the coordinates of points B, P and S, the size and shape of the triangle of the DMZ can be finally obtained.

Model expression
As shown in Fig. 7, when the cutting depth satisfied ℎ 1 < ℎ < ℎ 2 , where the h1 and h2 are calculated from section 2.1. For this case, discontinue chips can be generated during the cutting process resulting in the incomplete slip line field, which lacks the second deformation zone, so the slip line field model used in section 2.1 will no longer be suitable for this situation. Fang [26] used a negative rake angle chamfering tool to model the DMZ in the slip line field during cutting progress. On this basis, considering the DMZ, Hu et al. [17] improved it and obtained a modified slip line field model, which was verified by the finite element method. As shown in Fig. 8, by considering DMZ and material pre-shear zone, this paper improved the above model and further presented a slip line field suitable for cutting depth satisfies the situation of ℎ 1 < ℎ < ℎ 2 . 1) The first deformation zone is framed by DEFGHP. The equilateral right triangle area DEF represents the pre-cut zone, which is generated from the chips extruding and deforming before the workpiece material enters the main shear surface (DP).
2) The main slip line field is the sector area DBP. Through this area, the shearing speed in the first deformation zone gradually become the chip flowing speed.
3) The right triangle area PHS is the material area of the extruded workpiece, whose boundary is assumed to be an absolute straight line. There is no vertical flow of workpiece material at the DMZ-workpiece interface (PS), which is means that there is no material exchange between this zone and DMZ; 4) The triangle-like area PHS is the material retention area named DMZ, which is considered as a rigid body stably existing under the chamfered edge during cutting process. The point P is a stagnation point of the shearing effect and ploughing effect.
Due to the smaller cutting depth, the formation of the DMZ is incomplete.
According to the theory of maximum shear strain [25], the height of the upper vertex B of the DMZ is approximately the same as the depth of cut. 16

Model calculation
To ensure the continuity of the speed in the first deformation zone, THOMSEN [27] proposed that the rising height of the vertex D should be consistent with the rising height of the DMZ from the workpiece contact surface, which means that the vertical distance of DE from point D to the unprocessed surface should be equal to that of SP from point S to point P. The following relations can be obtained from Fig. 8: where is the angle between EG and the horizontal direction. The inclination degree of the pre-shear zone's upper surface DE is represented by the bow angle ρ. is the angle of the DB slip line in horizontal direction.
From geometric knowledge, the following formulas can be obtained:       (24) where ω at point S can be obtained from from Eq.(10) and the theoretical rake angle α' can be derived from Eq. (25) Fig. 9 shows the cutting process with the cutting depth equal to the MUCT resulting in no chip formation, where the height of MUCT is set as hlim, while hp and hs are defined as the plough thickness and the shear thickness, respectively. The lower apex of the DMZ is the separation point S, which is the same as the models established above. The DMZ is appearing when the material exceeds the point S.

Model calculation
According to the slip line model established in section 2.2, the triangle PSH shown in the Fig. 9 can be derived. hlim can be calculated through solving the triangle size, as following. In our previous research [2], the cutting force model in micro milling has been partially studied, in which, the axial cutting depth is divided into units of equal thickness (dz).
At any cutter rotation angle θm, when instantaneous uncut chip thickness h(θm) is less than hlim, there is no shearing effect during the cutting process, only the ploughing force occurs. For a cutting unit dz, the cutting force generating from the k-th cutter edge are decomposed into the tangential force (dFtk) and radial force (dFrk), which can be given in Eq.(37): where ktp and krp are the ploughing force coefficients in tangential direction and radial direction, respectively.
When the instantaneous uncut chip thickness is higher than hlim, both shearing force and ploughing force are simultaneously existing during the cutting process. where kts and krs are the shearing force coefficients in tangential direction and radial direction, respectively. The unit dz of the cutting depth can be written as: tan    m Rd dz (39) where R and ς are the tool radius and helix angle.
By conversing the tangential and radial force components into the where K is the number of teeth.
According to the Ref. [2], the cutting force coefficient can be fitted formula as a nonlinear function of the depth of cutting, as follows:  Table 2.

Theory and simulation model analysis
To assess the applicability of the model established in section 2, multiple sets of FEM simulations were carried out to simulate the size of DMZ by using Deform 2D/3D software. The tool is defined as pure rigid material participating in heat transfer. In the meshing process, Arbitrary Lagrange Eulerian (ALE) adaptive method is used to avoid too much deformation of materials around the cutter. Both the room temperature and the initial temperature condition for tools and workpieces are set as 20° C. In this study, Al6061 is selected as the workpiece material, and the width and cutting speed are kept at 2.5 mm and 104.67 m/min, respectively. The workpiece only moves in the horizontal direction and is fixed in the vertical direction. In order to ensure the simulation accuracy and efficiency, the mesh refinement is performed on the tool edge as well as the workpiece.
Subsequently, FEM simulation is mainly performed in two situations: the one is set the cutting depth constant to simulate the shape of the DMZ by the simulation step changing. The other way is using the constant simulation step to simulate the shape of the DMZ with the cutting depth changing. The shape can be judged by measuring the height of the three vertices of the triangle in the DMZ as shown in the Fig.11, where (a) and (b) are simulation diagram and theoretical diagram, respectively. The geometric dimensions of the tool are shown in Table 3. 23 Table 3 Geometric dimensions of the tool.  Table 4.   while the cutting depth is set as 16 μm. It is worth emphasizing that when the step size is higher than 200 steps, the size and shape of the dead model zone are approximately no longer changed. It can be seen that the DMZ is immediately generated once the tool edge touching the workpiece, which can be explained that the cutting depth meets the conditions for complete chip formation. In the first, due to only the upper half of the 25 tool edge touching the workpiece, the DMZ is located at a higher position, and the shape is not regular. When the rounded tool edge fully engages into the workpiece, the complete DMZ is formed. With the cutting process going on, the shape of the DMZ remains basically stable. The height of the three vertices of the DMZ obtained by simulation is compared with the theoretical position obtained by the previous calculation shown in Fig.12. As the shape of the DMZ becomes basically stable, the fitting error between the simulated position and the theoretical position is within 10%, which proves the theoretical modeling of the DMZ's correctness. rapidly, and basically remain stable when it reaches to a certain height. The separation point P of material ploughing and shearing, also has the same characteristics. When the 26 shape of the DMZ becomes basically stable, it can be seen that the fitting error between the simulated position and the theoretical position is within 10 %, which proves the theoretical modeling of the DMZ's correctness.
These simulation results indicate that the geometry of the DMZ is substantially affected by the radius of tool edge, which means that the radius of tool edge is directly engaged in the formation of DMZ and effects thereof on the shape of DMZ. However, once the cutting depth ensures that the chips can reach the height of h2, it will not affect the size of DMZ.

Experimental verification
To verify the correctness of the model presented in Section 2, a train of micro-milling experiments were carried out on the 3-axis micromachining center developed in-house with 0.5 μm resolution in each direction as shown in Fig.14. In this experiment, the test uses Al6061 as the material of the workpiece with the size of 60 mm × 10 mm × 3 mm.
The cutter with two flutes by superfine tungsten steel with TiAIN coated is used. The helix angle and tool radius are 35° and 0.8 mm, respectively. Kistler 9317B dynamometer is used to measure the cutting force, whose sensitivity is 25.65 pC/N and 25.37 pC/N in the X and Y directions, respectively. To remove high-frequency noise in the experiments, the cutting force signals are smoothed by a low-pass filter, whose cutoff frequency is set as 800 Hz. The experimental conditions and parameters are summarized in Table 5. 27 Fig. 14 the 3-axis micromachining center  The material selected for this experiment is Al6061, whose physical properties are shown in Table 6. The workpiece is treated as elastic-plastic behavior represented by where ,̇ and ̇0 are equivalent plastic strain, strain rate of equivalent plastic and reference plastic, respectively. T, Tmelt and Troom are the temperature of work, the material melting and the room, respectively. According the Ref. [29], n, m, A1, B1 and C1 are the material constants in constitutive relationship of the Johnson-Cook, which are list in Table 7.  Fig. 15 presents the differences between the cutting force predicted and that of experimentally measured in the X and Y directions for micro milling process. Among these comparisons, Fig.15 (a) presents the results at 0.8 μm feed per tooth, which is approach to or less than the value of MUCT lead to the pure ploughing phenomenon 30 occurring during the micro-milling process, therefore the ploughing force is relatively stable. The experimental results and the filtered simulation results have an error within 10%, which proves the correctness of the plough force modeling. Fig.15 (b) and Fig.15 (c) show the experiments where the feed per tooth is 2 μm and 4 μm, respectively, which can present the phenomenon of mixed ploughing and shearing in micro-milling process. In the cutting process corresponding to Fig.Fig. 15 (b), the feed per tooth is higher than the value of MUCT, which lead to the dominant share of ploughing force. However, due to the addition of shear force, it is also quite different from the pure ploughing force in Fig. 15 (a). Among the cutting forces in Fig. 15 (c), the feed per tooth is approach to that of macro milling resulting in the dominant shearing force, and the experimental results fit well with the filtered simulation results due to the effect of shearing force on cutting force almost ignored. It is worth noted that a minor fluctuation is appearing in the measured cutting force, this phenomenon is due to the slight vibration occurring during the micro-milling process. However, the peak value and wave of the measured force can not be substantially affected by these shapes of fluctuations, and the collected force still matches the predicted cutting force well in terms of shape, peak value, and global change trend. It is worth noting that there usually exists the phenomenon of the tool run-out and the eccentricity problems during the micro-milling process. This leads to the observation in the image: although the experimental data matches the predicted cutting force well in terms of shape, peak value and global change trend, there will still be some certain 31 errors. These issues will continue to be studied in subsequent research.

Conclusions
In the metal cutting process, the cutting force mainly includes shearing force and ploughing force, which should be accurately determined by the separation point in the process of modeling the cutting force. In this study, based on the plastic forming theory, the DMZ modeling is added to the slip line field model. The shear force starts to increase and keeps growing as the cutting depth rises while the 32 ploughing force also continues to increase until a complete DMZ is formed. Then the ploughing force will remain stable and stop increasing.
A train of experimental milling were performed to verify the proposed model with different cutting conditions. The measured cutting forces show a great agreement with that of predicted, which proves that the separation point proposed is effective to distinguish the ploughing phenomenon during metal cutting.

Authors' contributions
Bowen Song and Xiubing Jing contributed to the conceptualization, writing and editing.
Jian Xu and Huaizhong Li contributed to the methodology. Fujun Wang provided resources.

Compliance with ethical standards
Conflict of interest The authors declare that they have no conflicts of interest.
Ethical approval Not applicable.

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