1 Introduction

Micro-cutting has extensive applications in the manufacturing of 3D miniature components with desired durability, strength, and surface quality using metals or composites [1,2,3]. In micro-cutting, the uncut chip thickness is comparable in size with the cutting tool edge radius. This gives rise to a negative rake angle, which significantly affects the cutting force, materials, the deformation, and the tool wear.

Tremendous efforts have been devoted in analyzing micro-cutting process theoretically, numerically or experimentally. Kim et al. [4] suggested an orthogonal cutting model considering the cutting tool edge radius and the elastic recovery and investigated the plowing along the rounded edge and the sliding on the clearance face. Liu et al. [5] developed a finite element model based on strain gradient plasticity to examine the size effect, which was attributed to the tool edge radius and length scale of the material. Lai et al. [6] also performed FE simulations for micro-scale orthogonal machining considering the material strain gradient strengthening behavior, the tool edge radius and the fracture of the material. Subbiah et al. [7] observed material separation through a series of experiments and found that ductile fracture occurred due to finite inserts radii. Afazov et al. [8] presented an approach for predicting micro-milling cutting forces using FEM considering the effects of the cutting tool edge radius and the run-out. Vipindas et al. [9] investigated the effect of the cutting tool edge radius on the cutting force, the coefficient of friction, the surface roughness, and chip formation during micro end milling. Wu et al. [10] developed a comparison method to calculate the plowing force in micro-cutting by considering the cutting tool edge radius. Jun et al. [11] proposed a mechanistic model of micro end milling forces, with consideration of the effects of plowing, elastic recovery, effective rake angle, and flank face rubbing. An et al. [12] proposed a new deburring method in the study of ultraprecision micro-cutting. Due to the complexity of the micro-cutting process, considerable works were focusing on the FEM simulation to study the stress field, the large plastic deformation and the cutting force, etc. [8, 13,14,15,16,17,18,19].

A number of studies have been reported on the analytical and mechanistic models of micro-cutting process, based on slip-line field theory considering the effects of the cutting tool edge radius. Annoni et al. [20] used a slip-line field model to predict the cutting and the feed force based on the Waldorf model [21] under micro-scale cutting conditions considering effective rake angle. Jin et al. [22] established a slip-line field model to analyze the micro-cutting process with round edge. Fang [23, 24] proposed a slip-line model for the machining with a rounded-edge tool which consists of 27 slip-line sub-regions, each sub-region having its own physical meaning. It is noted that none of the above analytic models considered the material separation in cutting process.

In actual cutting process the work-piece material is removed by way of separation to form new surfaces. The creation of the new surfaces means that a portion of the work done by the cutting force in the cutting process is converted to surface energy. The specific surface work required is on the order of J/m2 for the formation of a new surface as calculated by the surface chemical free energy, which can be neglected in the calculation of the power consumption [25]. Based on this view, the traditional cutting force and shear angle analysis model only considers the work consumed by the plastic deformation and friction. However, the work required to produce a new surface in the analysis of the fracture behavior of the material is much higher, and the work per unit area required for the crack propagation during the metal ductile fracture can reach several hundred kJ/m2 [26]. Solid surface energy involves in the elastic and plastic deformation, cannot follow the definition for the free surface of the liquid. Thus, the surface chemical free energy is not suitable as a parameter characterizing the cutting process [27]. On the basis of the above considerations, the Ernst–Merchant cutting model has been modified applying the ductile fracture mechanics [28], and the interaction between cutting tool edge radius and material separation in micro-cutting process has been investigated using this methodology [29]. However, only the situation with a stable built-up edge was considered in [29], while in actual cutting process stable built-up edge may not generate under certain cutting conditions.

In this work, we proposed an analytical model of micro-cutting considering the factors of plastic flow, friction and ductile fracture, which can be used to estimate the shear angle and the material separation point position simultaneously. It should be pointed out that in the model no stable built-up edge is included.

2 Analytical micro-cutting model

In the process of the metal micro-cutting, the work-piece material experiences severe plastic deformation, and the ductile fracture occurs concurrently at the chip root, as shown in Fig. 1.

Fig. 1
figure 1

SEM images of the chip root illustrating: a severe plastic deformation and b ductile fracture

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Figure 1 a shows that severe plastic deformation occurs before and after the grain is cut off. Also, it can be seen that the chip root contour is basically the same as the tool edge, which implies that built-up edge is not obvious in front of the cutting tool edge. In Fig. 1b, the phenomenon of material tearing can be observed. This work focuses on the case that no stable built-up edge generates or the size of it can be negligible, therefore, the micro-cutting process can be modeled as Fig. 2, incorporating the effects of both the radius of tool edge and the material separation.

Fig. 2
figure 2

Analysis model for micro-cutting process

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In Fig. 2, A is the point on the edge radius at which the material separates. During the cutting process, the material above the point A forms the chip; whereas, that below is plowed under the edge to form the machined surface. New surfaces will be created due to the separation of the material. Therefore, the total work provided by the horizontal cutting force FC is consumed by the following: (1) the plastic deformation of the work-piece material; (2) the friction between the work-piece and the tool surface; and (3) creation of the new surfaces [27].

In Fig. 2, ϕ denotes the shear angle, and the angle θm corresponds to the location of the point where the material separation occurs and ρ is the edge radius of the cutting tool. h is the uncut chip thickness which is the vertical distance between the bottom of cutting tool and the free surface of the work-piece, α is the rake angel, αe is the effective rake angle which will be defined later. The vertical distance between point A on the edge radius and the bottom of the cutting tool can be expressed as:

$$ {h}_{\mathrm{m}\mathrm{in}}=\rho \left(1-\cos {\theta}_{\mathrm{m}}\right) $$
(1)

It is defined as minimum chip thickness in some literatures [30,31,32,33,34]. Therefore, the minimum chip thickness can be analyzed through determination of θm.

The slip-line field and its hodograph are shown in Fig. 3a and b, respectively, which are widely used in the analyses of cutting process [21]. Figure 3c shows the forces acting on the cutting tool with simplifying the curvilinear cutting edges as line segments.

Fig. 3
figure 3

a The slip-line field around the cutting tool edge [20]. b The hodograph of the slip-line field. c The normal and tangential forces acting on the cutting tool

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Based on the slip-line field model, the cutting power related to the shearing, friction, and fracture can be derived as follows:

$$ {F}_CV=\frac{\tau_y tVa\cos {\alpha}_e}{\sin \phi \cos \left(\phi +{\alpha}_e\right)}+\frac{F_rV\sin \phi }{\cos \left(\phi +{\alpha}_e\right)}+\frac{F_pV\sin \left({\zeta}_1-\lambda \right)}{\sin \left({\zeta}_1\right)}+ RaV $$
(2)

In the equation, the first term on the right-hand side is the plastic work, the second and third term are the work related to the friction, and the last one is the work required to generate the new surfaces [27]. V is the cutting speed, t is the depth of cut, τy is the shear yield strength, a is the width of cut, R represents the fracture toughness of the work-piece material and αe is the effective rake angle calculated as [11]:

$$ {\alpha}_e=\left\{\begin{array}{l}\alpha \frac{h+\rho \left({\alpha}_1\cos \alpha -\sin \alpha -1\right)}{h+\rho \left({\alpha}_2\cos \alpha -\sin \alpha -1\right)}\kern1em h\ge \rho \left(1+\sin \alpha \right)\\ {}\frac{1}{2}\left({\theta}_t+{\theta}_c-\pi \right)\kern6.5em h<\rho \left(1+\sin \alpha \right)\end{array}\right. $$
(3)

where

$$ {\displaystyle \begin{array}{l}{\alpha}_1=\frac{\alpha }{2}-\frac{\pi^2}{8\alpha }-\frac{\theta_c^2}{2\alpha }+\frac{{\pi \theta}_c}{2\alpha}\\ {}{\alpha}_2=\alpha +\frac{\pi }{2}-{\theta}_c\end{array}} $$
(4)
$$ {\displaystyle \begin{array}{l}{\theta}_t={\cos}^{-1}\left(1-\frac{h}{\rho}\right)\\ {}{\theta}_c={\cos}^{-1}\left(1-\frac{h_m}{\rho}\right)\end{array}} $$
(5)

Horizontal cutting force and surface tangential forces have the following relationship:

$$ {F}_C={F}_r\frac{\cos \left({\beta}_r+{\alpha}_e\right)}{\sin {\beta}_r}+{F}_p\frac{\sin \left(\lambda +{\beta}_p\right)}{\sin {\beta}_p} $$
(6)

In Eq. (6), βr and βp are the friction angles at tool–chip and tool–work interfaces, respectively. ζ1 is the slip-line angle related to the friction factor (m) at the tool–work-piece interface (AD) and can be determined according to slip-line field theory as

$$ {\zeta}_1=0.5{\cos}^{-1}(m) $$
(7)

In slip-line filed theory the frictional stress is assumed constant and proportional to the material shear flow stress k:

$$ \tau =m\cdot k $$
(8)

k is related to the uniaxial yield stress Y through the von Mises yield criterion

$$ k=\frac{Y}{\sqrt{3}} $$
(9)

Fr is the friction force between the tool rake face and the work-piece. The friction angle βp is determined as

$$ {\displaystyle \begin{array}{l}{\beta}_{\mathrm{p}}={\tan}^{-1}\left(\frac{\cos \left(2{\zeta}_1\right)}{1+\pi /2-2\xi +2{\zeta}_1-2\lambda +\sin \left(2{\zeta}_1\right)}\right)\\ {}\xi ={\sin}^{-1}\left(\frac{\sin \left(\lambda \right)}{\sqrt{2}\sin \left({\zeta}_1\right)}\right)\end{array}} $$
(10)

The plowing force is:

$$ {F}_p=m{\tau}_y{l}_{AD}a $$
(11)

Substituting Eqs. (6) and (11) into Eq. (2), one can obtain:

$$ {\displaystyle \begin{array}{l}\frac{F_C}{\mathrm{a}}=\frac{\tau_yt\cos {\alpha}_e}{Q\sin \varphi \cos \left(\varphi +{\alpha}_e\right)}+\frac{m{\tau}_y{l}_{AD}}{Q}\\ {}\kern2em \times \left[\frac{\sin \left({\varsigma}_1-\lambda \right)}{\sin \left({\varsigma}_1\right)}-\frac{\sin \left(\lambda +{\beta}_p\right)\sin {\beta}_r\sin \varphi }{\sin {\beta}_p\cos \left({\beta}_r+{\alpha}_e\right)\cos \left(\varphi +{\alpha}_e\right)}\right]+\frac{R}{Q}\end{array}} $$
(12)

where

$$ \lambda ={\theta}_m/2\kern0.5em ,\kern0.75em {l}_{AD}=2\rho \sin \frac{\theta_m}{2}\kern1em ,\kern0.5em t=h-{h}_{\mathrm{min}} $$
(13)
$$ Q=1-\frac{\sin {\beta}_r\sin \phi }{\cos \left({\beta}_r+{\alpha}_e\right)\cos \left(\phi +{\alpha}_e\right)} $$
(14)

According to the minimum energy principle, ϕ and θm should be such that the total power is the smallest, i.e., the first-order derivative of the total power with respect to ϕ or θm is zero (∂FC/∂φ = 0, ∂FC/∂θm = 0 ). Hence, two nonlinear equations can be obtained. By solving these two equations the effects of the cutting and material parameters to ϕ and θm can be analyzed.

3 Experimental validation and discussion

Experiments were carried out with oxygen-free copper (OFC) using uncoated tungsten carbide (WC) micro-cutting tool on the self-developed set-up for orthogonal cutting, as shown in Fig. 4a, and the cutting forces were measured with a table dynamometer (Kisler9256C).

Fig. 4
figure 4

a The set-up for orthogonal cutting. b SEM picture of the tool

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The edge radius of the tool was observed to be approximately 2.7 μm, shown as Fig. 4b. The rake angle of the cutting tool is 0° and the clearance angle is 7°. Quick-stopping was performed to freeze the chip, and the shear angles were measured through the SEM images of the chip root using the microscope with super depth of field, shown in Fig. 5.

Fig. 5
figure 5

SEM images of chip root and measurements of shear angles

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Table 1 shows the variations of the shear angle and the cutting force with the uncut chip thickness h.

Table 1 Measured values of shear angle and cutting force
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The MATLAB program was developed to conduct the numerical computations based on the second-order Newton-Raphson method. In the analyses, the cutting parameters employed to solve ϕ and the θm are given, shown in Table 2. The specific calculating procedure is depicted in the flowchart demonstrated in Fig. 6 and the numerical results are shown in Fig. 7.

Table 2 Cutting conditions (m = 0.85 [29])
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Fig. 6
figure 6

Flowchart of the calculating procedure

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Fig. 7
figure 7

a The comparison of cutting force (Fc) between calculating results and experimental measurements. b The computational and experimental results of the shear angle (Φ). c The variation of angle θm with uncut chip thickness. d The variation of minimum chip thickness (hmin) with uncut chip thickness. e The variation of fracture toughness with uncut chip thickness. f The variation of plowing force with uncut chip thickness

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From Fig. 7a, it can be seen that the calculated cutting forces based on the proposed model are almost the same as the experimental results. Figure 7b illustrates the variation of shear angle versus the uncut chip thickness, and it demonstrates that there is only a little difference between the experimental measurements and the calculated results. Angle θm decreases with the uncut chip thickness increasing shown as Fig. 7c. However, the minimum chip thickness increases as shown in Fig. 7d. Figure 7e indicates that the fracture toughness varies within 1.1 to 1.4 kJ/m2 corresponding to the variation of the uncut chip thickness within 5 to 25 μm. The fracture toughness value is also relatively low since oxygen-free copper is a soft material. The plowing forces were computed through Eq. (11); the results are shown in Fig. 7f, which indicates that the plowing force increases with the uncut chip thickness.

Karpat [29] investigated the effect of edge radius on material separation for two cutting conditions using the experimental data given by Woon et al. [36], considering a stagnant metal zone in front of the cutting edge. The research work is of great significance to analyzing the process of micro-cutting with stable built-up edge (BUE). Nevertheless, under certain cutting conditions BUE can be avoided. In our series experiments obvious stable BUE were not observed. In order to compare our model with that proposed by Karpat [27], the calculations of shear angle and fracture toughness were performed using the same cutting parameters as listed in Table 3, and the calculating results are provided in Table 4.

Table 3 Cutting conditions [29, 36] (m = 0.85)
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Table 4 Calculating results
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In Table 4, the data of the shear angle and the fracture toughness in parentheses were the results in [29]. The shear angles obtained by our model are very close to those of Karpat’s. However, the fracture toughness in our calculation is greater. It should be noticed that in Karpat’s model the rake angle near the cutting tool tip is positive due to the BUE, but in our model it is always negative (see Fig. 3a), which leads to higher fracture toughness. This is also the major difference between the two models.

The effects of fracture toughness, shear yield strength, and friction angle to shear angle, material separation position, and cutting force in the micro-cutting process are investigated numerically as follows.

In the analyses, the uncut chip thickness is taken as 0.016 mm. The cutting force, shear angle, and the angle related to material separation point were calculated with the changes of fracture toughness, shear yield strength and friction angle, respectively. The results are drawn in Fig. 8.

Fig. 8
figure 8

a The effect of the fracture toughness to the cutting force (Fc). b The effect of the fracture toughness to the shear angle (Φ) and angle θm. c The relationship between the shear yield stress to the cutting force (Fc). d The influence of the shear yield stress to the shear angle (Φ) and angle θm. e The variation of the cutting force (Fc) with the friction angle. f The effect of the friction angle to the hear angle (Φ) and angle θm

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Figure 8a, c, and e indicate that the cutting forces increase with all the three parameters. From Fig. 8b, it can be seen that with the increase of the fracture toughness, the shear angle ϕ decreases and the angle θm related to the material separation point increases, which imply that the chip formation becomes more difficult. When changing the shear yield stress from 500 MPa to 600 MPa with other parameters fixed, the shear angle ϕ increases and the angle θm decreases slightly (see Fig. 8d). The variation of θm is within 23.31°~23.13°, while ϕ varies from 22.67° to 23.58°.

In Fig. 8f, it can be seen that the friction angle influences the micro-cutting process greatly. The cutting force has a nonlinear relationship with the friction angle. Meanwhile, the changes of ϕ and θm are much larger than those in the cases of varying the fracture toughness and the shear yield stress. In practical cutting process cutting, fluid is often used to reduce the friction to lower the cutting force.

4 Conclusions

In this work, a micro-cutting model is developed based on the slip-line field theory considering the effects of cutting tool edge radius and material separation due to ductile fracture. The model proposed in this study focuses on the micro-cutting process with no stable built-up edge. This is its major difference from the model developed by Karpat [29], which deals with the micro-cutting process considering BUE in front of the cutting tool edge.

Based on the experimental data, the validation of the model was examined. The numerical results of the cutting forces and the shear angles agree with the experimental measurements very well, which indicates the model developed in this work is reasonable. Additionally, the minimum chip thickness can be determined using the model. The numerical investigations show that as the uncut chip thickness increases the shear angle increases, while the angle corresponding to the material separation point decreases. The minimum chip thickness increases slightly with the increases of uncut chip thickness. The plowing increases with uncut chip thickness.

Employing this model, the numerical investigations of the effects, the fracture toughness, the shear yield stress, and the friction to the micro-cutting process were performed. The results may be very helpful in selecting machining conditions and designing micro-cutting tools.