Analytical modeling of tool failure boundary map in milling titanium alloy

Abstract

As a typical aerospace difficult-to-machine material, tool failure in milling titanium alloy Ti6Al4V will reduce the stability of the milling process and affect the surface quality of the workpiece. Aiming at the fact that cemented carbide tools are prone to wear failure and breakage failure in milling titanium alloy, a safe tool failure boundary map is provided to ensure that the tools will not occur failure with the cutting parameters selected in the safe area during the prediction time. Based on the processing characteristics of Ti6Al4V, the failure boundary map mainly considers three forms of tool failure: flank wear, rake wear, and cutting edge breakage. By revealing the three failure mechanisms, the failure analytical model is established and the failure boundary map is obtained. Compared with the experimental results, it has good consistency, and the research results can provide a reference for the field of titanium alloy cutting process.

Introduction

Titanium alloy Ti6Al4V, as a typical difficult-to-machine material in aerospace, has poor thermal conductivity, low plasticity, high chemical activity, and small elastic modulus, which leads to small cutting deformation, high cutting temperature, large cutting force per unit area, and serious tool adhesion during cutting process, resulting in serious tool wear and poor surface quality,^1,2 affecting the development of titanium alloy cutting field. The tool failure boundary map provides a safe operating area for cutting tools by adding different types of failure boundaries, so as to ensure that the combination of cutting parameters in the safe area does not fail before the scheduled replacement time of the tool.

Many scholars at home and abroad have done a lot of research on tool failure mechanism and failure boundary map. Sun et al. ³ studied that under the condition of high-speed cutting, serious flank wear leads to tool failure. Gu ⁴ studied the failure mechanism of tool rake face by cutting railway steel with cemented carbide tools, and concluded that the cyclic load and thermo-mechanical coupling effect under the action of the chip mainly led to the failure of tool rake face. Cui et al. ⁵ established the tool material damage model, proposed the critical condition of tool damage, and revealed the tool failure mechanism under thermo-mechanical load. Cheng et al. ⁶ analyzed the failure mechanism of cemented carbide tool under cyclic impact load and the crack propagation law of tool material by milling 508 steel. The tool impact damage model was established, and the impact fracture limit condition of tool material was determined. Finally, the tool fatigue life model was established, which provided a theoretical basis for the study of tool failure mechanism. Okoshi and Sakai ⁷ proposed a three-dimensional wear map of the flank wear rate. The coordinate axes are the cutting speed, mechanical load, and flank wear volume, which can clearly express the change of flank wear rate with speed and load. Oosthuizen ⁸ proposed that the tool failure occurs under mechanical load and thermal load through the experiment on milling titanium alloy with PCD tool. A two-dimensional tool failure boundary map was constructed, which X direction is cutting speed and Y direction is the maximum undeformed chip thickness. The mechanism of each failure model was analyzed, and the failure boundary curve under each failure model was obtained. Xu ⁹ constructed a two-dimensional tool failure boundary map based on thermal-mechanical load through the experimental study of turning superalloy Inconel625 with cemented carbide tools and defined a safe cutting parameter area. Failure models and main wear mechanisms of different cutting parameters were plotted on the two-dimensional failure boundary map. Element diffusive behavior of coated cemented carbide tools and superalloy Inconel625 was analyzed statically, and the diffusive reference line was marked in the two-dimensional tool failure boundary map. Fan ¹⁰ established the tool wear diagram of high efficiency cutting titanium alloy related to cutting speed and feed per tooth, and carried out experimental verification. It provides an effective way for the reasonable selection of cutting parameters, and provides a simple and fast processing efficiency and cutting parameters for the rapid response of production and processing requirements.

However, at present, the failure boundary map established by scholars are obtained by experimental methods, which lacks the mechanism analysis and analytical model. In this paper, the milling of titanium alloy materials by cemented carbide tools is taken as the research object. By analyzing the mechanism under different failure modes, the failure models are constructed. According to the failure criteria, the tool failure boundary map under different cutting parameters is obtained, which provides a reference for the field of titanium alloy cutting.

Modeling of stress and temperature field of tool-chip interface and tool-workpiece interface

The stress field model of tool-chip and tool-workpiece is constructed by using the analytical method, and the stress distribution and wear location are obtained. Based on the contact relationship between tool and workpiece material, the stress field and the temperature field model are established. The stress and temperature field of tool-chip interface and tool-workpiece interface is given in this section.

Modeling of stress field of tool-chip interface and tool-workpiece interface

The normal contact stress distribution at the tool-chip interface is shown in Figure 1. The normal contact stress is the maximum at the nose of the tool and decreases along the tool rake face. It can be expressed as follows ¹¹

$${\sigma }_{\mathrm{rake}}\left(x\right)={\sigma }_{\mathrm{tip}-\mathrm{rake}}{(1-x/l_c)}^{\xi }$$

(1)

$${\sigma }_{\mathrm{tip}-\mathrm{rake}}=4\frac{1+b}{2+b}\frac{{\left(\cos {\beta }_n\right)}^2}{\sin \left[2({\phi }_n+{\beta }_n-{\alpha }_n)\right]}{\tau }_h$$

(2)

Figure 1.

Stress and temperature field on tool-chip interface.

Where σ_tip-rake is the maximum normal contact stress, ¹² l_c is the contact tool-chip length, ξ is the stress exponent, φ_n is the normal shear angle, α_n is the normal rake angle, β_n is the normal friction angle, b is the correlation index.

In the milling process, the undeformed chip thickness is uneven, so the chip thickness is changed. The chip thickness is introduced to improve the tool-chip contact length model, as shown in Figure 2. It can be express as ¹³

$$l_{\mathrm{c}}=\frac{h_{\max }({\xi }_1+2)}{2}\frac{\sin ({\phi }_n+{\beta }_n-{\alpha }_n)}{\cos {\phi }_n·\cos {\beta }_n·\sin {\eta }_c}$$

(3)

$$h_{\max }=R-\sqrt{R^2+4{f_z}^2-4f_z\sqrt{2R·a_e-{a_e}^2}}$$

(4)

Figure 2.

Schematic diagram of undeformed chip thickness.

Where ξ₁ is the relevant exponent, η_c is the chip flow angle, R is the tool radius, f_z is the feed per tooth, a_e is the cutting width.

The effective cutting time t for each tooth is the function of the cutting entry angle θ_in and the cutting exit angle θ_out ¹⁴

$$\{\begin{array}{c}{\theta }_{\mathrm{in}}=\arcsin \left(\frac{R-a_e}{R}\right)\hfill \\ {\theta }_{\mathrm{out}}=\frac{\pi }{2}+\arcsin \left(\frac{f_z}{2R}\right)\hfill \\ t=\frac{{\theta }_{\mathrm{out}}-{\theta }_{\mathrm{in}}}{2\pi }{t}_{\mathrm{all}}\hfill \end{array}$$

(5)

Where t_all is the total cutting time.

As shown in Figure 3, the normal stress field model after tool wear on the tool-workpiece interface can be expressed as ¹⁵

$${\sigma }_{\mathrm{flank}}(x\prime )=\{\begin{array}{c}{\sigma }_{\mathrm{tip}-\mathrm{flank}}\hfill \\ \multicolumn{1}{c}{{\sigma }_{\mathrm{tip}-\mathrm{flank}}{\left(\frac{x\prime }{V{B}_{\mathrm{CR}}}\right)}^2}\end{array}\begin{array}{c}\begin{array}{c}x\prime \in (V{B}_{\mathrm{CR}},\mathrm{VB})\hfill \end{array}\hfill \\ \multicolumn{1}{c}{x\prime \in (0,V{B}_{\mathrm{CR}})}\end{array}$$

(6)

Figure 3.

Stress field of tool-workpiece interface.

Where VB_CR is the critical point of the plastic flow region and the elastic contact region, where also represent the width of the elastic contact region, and σ_tip-flank is the normal stress at the tool-workpiece interface.

Modeling of temperature field of tool-chip interface and tool-workpiece interface

As shown in Figure 2, the maximum temperature at the tool-chip interface is not on the tool nose, but there is a certain distance from the tool nose. ¹⁶ Therefore, this paper adopts the temperature model established by Moufki et al. ¹⁷ The temperature of any point at the tool-chip interface is

$$\begin{array}{c}\multicolumn{1}{c}{T_{\mathrm{rake}}\left(x\right)=\frac{2}{15{l_c}^2}\frac{{\mu }_r{\sigma }_{\mathrm{tip}-\mathrm{rake}}\sqrt{v_{c}x}}{\sqrt{\pi \rho \mathrm{ck}}}}\\ (8x^2+15{l_c}^2-20l_{c}x)+T_0\hfill \end{array}$$

(7)

Similarly, the average temperature model at the tool-workpiece interface is

$$T_{\mathrm{flank}}\left(\mathrm{VB}\right)=\frac{7}{15}\frac{{\mu }_c{\sigma }_{\mathrm{tip}-\mathrm{flank}}\sqrt{V·\mathrm{VB}}}{\sqrt{\pi \rho \mathrm{ck}}}+T_0$$

(8)

Where μ_r is the tool-chip friction coefficient, μ_c is the tool-workpiece friction coefficient, the density of titanium alloy ρ = 4.45 × 10³ kg/m³, the thermal conductivity k = 5.44 W/m·K, the specific heat capacity c = 6.78 × 10⁻⁴ KJ/Kg·K, v_c is the chip velocity.

The relationship between chip velocity and cutting speed V can be expressed as ¹⁸

$$v_c=V\frac{\cos {\lambda }_s\sin {\phi }_n}{\cos {\eta }_c\cos ({\phi }_n-{\alpha }_n)}$$

(9)

Where λ_s is the cutting edge inclination angle.

Modeling of tool wear

Based on the stress and temperature field model of tool-chip interface and tool-workpiece interface, the prediction model of crater wear depth and flank wear width considering the abrasive wear, adhesive wear, and diffusive wear is constructed in this section.

Modeling of abrasive wear rate

In this paper, the abrasive wear rate model of tool-chip interface proposed by Rabinowicz et al. ¹⁸ is adopted, and its expression is

$$\begin{array}{c}{\stackrel{·}{V}}_{\mathrm{rake}-\mathrm{abrasion}-\mathrm{wear}}=\frac{\mathrm{d}{V}_{\mathrm{rake}-\mathrm{abrasion}-\mathrm{wear}}}{\mathrm{d}t}\hfill \\ =\frac{Q}{\pi P\tan \theta }{v}_c=M·v_c\hfill \end{array}$$

(10)

Similarly, the abrasive wear rate model of tool-workpiece interface is

$${\stackrel{·}{V}}_{\mathrm{flank}-\mathrm{abrasion}-\mathrm{wear}}=M·V$$

(11)

Where Q is the contact load of tool-chip interface, P is the material hardness of the tool rake face, θ is the average roughness angle, M = 2.37 × 10⁻¹¹. ¹⁹

Modeling of adhesive wear rate

In this paper, the improved adhesive wear rate model of tool-chip interface proposed by Usui et al. ²⁰ is adopted, and its expression is

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\mathrm{rake}-\mathrm{adhension}-\mathrm{wear}}=\frac{\mathrm{d}{V}_{\mathrm{rake}-\mathrm{adhension}-\mathrm{wear}}}{\mathrm{d}t}}\\ =N_1·{\sigma }_{\mathrm{rake}}·v_c·{\mathrm{e}}^{-\frac{N_2}{273+T_{\mathrm{rake}}(x)}}\hfill \end{array}$$

(12)

Similarly, the adhesive wear rate model of tool-workpiece interface is

$${\stackrel{·}{V}}_{\mathrm{flank}-\mathrm{adhension}-\mathrm{wear}}=N_1·{\sigma }_{\mathrm{flank}}·V·{\mathrm{e}}^{-\frac{N_2}{273+T_{\mathrm{flank}}(\mathrm{VB})}}$$

(13)

Where N_1 = 0.0004, N_2 = 7000. ¹⁹

Modeling of diffusive wear rate

As shown in Figure 4, In the process of cutting titanium alloy with cemented carbide tools, the main reason why the tool occurred diffusive wear is that the Co element on the tool surface flow to the chip under high temperature. ²⁰ The diffusive wear rate model of the tool-chip interface can be obtained by

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{V}}_{\mathrm{rake}-\mathrm{diffusion}-\mathrm{wear}}=\frac{\mathrm{d}{V}_{\mathrm{rake}-\mathrm{diffusion}-\mathrm{wear}}}{\mathrm{d}t}=\frac{J_{y=0}}{{\rho }_t}}\\ =\frac{2C_0}{{\rho }_t}\sqrt{\frac{v_c{D}_0}{\pi x}}{e}^{-\frac{Q}{2R_1(273+T_{\mathrm{rake}}\left(x\right))}}\hfill \end{array}$$

(14)

Similarly, the diffusive wear rate model of the tool-workpiece interface can be obtained by

$${\stackrel{·}{V}}_{\mathrm{flank}-\mathrm{diffusion}-\mathrm{wear}}=\frac{2C_0}{{\rho }_t}\sqrt{\frac{V{D}_0}{\pi ·\mathrm{VB}}}{e}^{-\frac{Q}{2R_1(273+T_{\mathrm{flank}}\left(\mathrm{VB}\right))}}$$

(15)

Where the concentration of diffusion substances C₀ = 0.0253 mol/mm³, the density of tool material ρ_t = 14.9 × 10³ kg/m³, the equation coefficient D₀ = 1.9 mm²/s, the activation energy Q = 114.4 kJ/mol, the gas constant R₁ = 8.315×10⁻³ kJ/mol/K.

Figure 4.

Schematic diagram of tool diffusive wear. ²¹ .

Modeling of crater wear rate on the tool rake face

In the existing research, ²² the crater wear model is established and the depth of crater wear is obtained by finite element simulation method. However, in this paper, the crater wear model is established by analytical method. The tool wear is the result of multiple wear mechanisms. ²³ Only considering one wear mechanism cannot reflect the real situation of tool wear. Therefore, abrasive wear, adhesive wear, and diffusive wear models is considered. Then, the crater wear rate model on the tool rake face is

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{d}{V}_{\mathrm{rake}}}{\mathrm{d}t}={\stackrel{·}{V}}_{\mathrm{abrasion}-\mathrm{wear}}+{\stackrel{·}{V}}_{\mathrm{adhension}-\mathrm{wear}}+{\stackrel{·}{V}}_{\mathrm{diffusion}-\mathrm{wear}}}\\ =M{v}_c+N_1·{\sigma }_n·v_c·{\mathrm{e}}^{-\frac{N_2}{273+T_{\mathrm{rake}}\left(x\right)}}\hfill \\ +\frac{2C_0}{{\rho }_t}\sqrt{\frac{v_c{D}_0}{\pi x}}{e}^{-\frac{Q}{2R_1(273+T_{\mathrm{rake}}\left(x\right))}}\hfill \end{array}$$

(16)

As shown in Figure 5, the crater wear volume ΔV_rake can be expressed as

$$\Delta V_{\mathrm{rake}}=\mathrm{KT}(x,t)\mathrm{dx}=({\int }_0^t\left(\frac{d{V}_{\mathrm{rake}}}{\mathrm{dt}}\right)\mathrm{dt}\prime )\mathrm{d}x$$

(17)

Figure 5.

Schematic diagram of crater wear curve.

Where KT(x, t) is the crater wear depth at x position from the nose of the rake face at time t.

Therefore, the crater wear depth KT(x, t) at different positions of the rake face related to the chip contact interface temperature and cutting time is obtained, and the crater wear model of the rake face is

$$\mathrm{KT}(x,t)=\left(\begin{array}{c}M{v}_c+N_1·{\sigma }_{\mathrm{rake}}\left(x\right)·v_c·{\mathrm{e}}^{-\frac{N_2}{273+T_{\mathrm{rake}}\left(x\right)}}\hfill \\ \multicolumn{1}{c}{+\frac{2C_0}{{\rho }_t}\sqrt{\frac{v_c{D}_0}{\pi x}}{e}^{-\frac{Q}{2R_1(273+T_{\mathrm{rake}}\left(x\right))}}}\end{array}\right)·t$$

(18)

Modeling of flank wear width

Similarly, considering abrasive wear, adhesive wear, and diffusive wear, the tool flank wear model is

$$\begin{array}{c}{V}_{\mathrm{flank}-\mathrm{wear}-\mathrm{mechanism}}\hfill \\ =\left(\begin{array}{c}{\stackrel{·}{V}}_{\mathrm{flank}-\mathrm{abrasion}-\mathrm{wear}}+{\stackrel{·}{V}}_{\mathrm{flank}-\mathrm{adhension}-\mathrm{wear}}\hfill \\ \multicolumn{1}{c}{+{\stackrel{·}{V}}_{\mathrm{flank}-\mathrm{diffusion}-\mathrm{wear}}}\end{array}\right)w\Delta t\hfill \\ =\left(\begin{array}{c}M·V+N_1·{\sigma }_{\mathrm{flank}}·V·{\mathrm{e}}^{-\frac{N_2}{273+T_{\mathrm{flank}}\left(\mathrm{VB}\right)}}\hfill \\ \multicolumn{1}{c}{+\frac{2C_0}{{\rho }_t}\sqrt{\frac{V{D}_0}{\pi ·\mathrm{VB}}}{\mathrm{e}}^{-\frac{Q}{2R_1(273+T_{\mathrm{flank}}\left(\mathrm{VB}\right))}}}\end{array}\right)w\Delta t\hfill \end{array}$$

(19)

After the milling tool is worn, the flank wear volume is shown in Figure 6. Then, the wear volume based on geometry is ²¹

$$\begin{array}{c}\multicolumn{1}{c}{V_{\mathrm{flank}-\mathrm{wear}-\mathrm{geometry}}=S_{\mathrm{ABCDEF}}·w\prime }\\ =(S_{\mathrm{AEF}}+S_{\mathrm{ABDE}}+S_{\mathrm{BCD}})·w\prime \hfill \\ =\left[\begin{array}{c}\frac{1}{2}{\left|\mathrm{AE}\right|}^2\tan {\gamma }_n+\left(\mathrm{VB}\right)\left|\mathrm{AE}\right|\hfill \\ \multicolumn{1}{c}{+\frac{1}{2}\cot \alpha {\left|\mathrm{AE}\right|}^2}\end{array}\right]·w\prime \hfill \end{array}$$

(20)

Figure 6.

Schematic diagram of tool flank wear volume based on geometry.

Where w is the cutting width of oblique angle, w′ is the width of chip.

In order to simplify equation (20), the higher-order term ${\left|\mathrm{AE}\right|}^2$ is ignored here and further calculated as follows

$$\begin{array}{c}\multicolumn{1}{c}{V_{\mathrm{flank}-\mathrm{wear}-\mathrm{geometry}}=\left(\mathrm{VB}\right)\left|\mathrm{AE}\right|\frac{w}{\cos {\lambda }_s}}\\ =\left(\mathrm{VB}\right)\frac{w}{\cos {\lambda }_s}·\frac{\left|\mathrm{AE}\right|·(\cos \alpha -\tan {\gamma }_n)}{\cot \alpha -\tan {\gamma }_n}\hfill \\ =\left(\mathrm{VB}\right)\frac{w}{\cos {\lambda }_s}·\frac{\Delta \mathrm{VB}}{\cot \alpha -\tan {\gamma }_n}\hfill \end{array}$$

(21)

The tool flank wear volume based on the mechanism is equivalent to that based on the geometry

$$\begin{array}{c}{V}_{\mathrm{flank}-\mathrm{wear}-\mathrm{geometry}}=\mathrm{VB}·\frac{w}{\cos {\lambda }_s}·\frac{\Delta \mathrm{VB}}{\cot \alpha -\tan {\gamma }_n}\hfill \\ =\left(\begin{array}{c}M·V+N_1·{\sigma }_{\mathrm{flank}}·V·{\mathrm{e}}^{-\frac{N_2}{273+T_{\mathrm{flank}}\left(\mathrm{VB}\right)}}\hfill \\ \multicolumn{1}{c}{+\frac{2C_0}{{\rho }_t}\sqrt{\frac{V{D}_0}{\pi ·\mathrm{VB}}}{\mathrm{e}}^{-\frac{Q}{2R_1(273+T_{\mathrm{flank}}\left(\mathrm{VB}\right))}}}\end{array}\right)·w\Delta t\hfill \\ =V_{\mathrm{flank}-\mathrm{wear}-\mathrm{mechanism}}\hfill \end{array}$$

(22)

Therefore, the wear rate that the tool flank wear width change with time can be obtained:

$$\begin{array}{c}\frac{\Delta \mathrm{VB}}{\Delta t}=\left\{\begin{array}{c}M·V+N_1·{\sigma }_{\mathrm{flank}}·V·{\mathrm{e}}^{-\frac{N_2}{273+T_{\mathrm{flank}}\left(\mathrm{VB}\right)}}\hfill \\ \multicolumn{1}{c}{+\frac{2C_0}{{\rho }_t}\sqrt{\frac{V{D}_0}{\pi ·\mathrm{VB}}}{\mathrm{e}}^{-\frac{Q}{2R_1(273+T_{\mathrm{flank}}\left(\mathrm{VB}\right))}}}\end{array}\right\}·\hfill \\ \frac{\cos {\lambda }_s(\cot \alpha -\tan {\gamma }_n)}{\mathrm{VB}}\hfill \end{array}$$

(23)

Modeling of tool cutting edge breakage

Establishment of equivalent uniaxial compressive stress model on cutting edge

Using the concept of damage equivalent stress, the complex triaxial stress state can be simplified to the uniaxial stress state, the uniaxial stress model is ⁵

$${\sigma }_c={\left\{\begin{array}{c}(1+b_v){〈\sigma 〉}^{+}:{〈\sigma 〉}^{+}-b_v{〈\mathrm{tr}\sigma 〉}^2\hfill \\ \multicolumn{1}{c}{+\frac{1-D}{1-\mathrm{hD}}[(1+b_v){〈\sigma 〉}^{-}:{〈\sigma 〉}^{-}-b_v{〈-\mathrm{tr}\sigma 〉}^2]}\end{array}\right\}}^{1/2}$$

(24)

Where σ_c is the damage equivalent stress, σ is the stress under triaxial state, b_v is the Poisson’s ratio of tool material, D is the damage value of tool material. Constant h = 0.2.

It is assumed that there are N initial micro-cracks in the tool material element. The total strain of the selected tool material element can be expressed as ⁵

$$\begin{array}{c}\multicolumn{1}{c}{\left(\begin{array}{c}{\epsilon }_1\hfill \\ \multicolumn{1}{c}{{\epsilon }_2}\end{array}\right)=\left(\begin{array}{c}{{\epsilon }_1}^e+\Delta {\epsilon }_1\hfill \\ \multicolumn{1}{c}{{{\epsilon }_2}^e+\Delta {\epsilon }_2}\end{array}\right)}\\ =\frac{(k+1)(b_v+1)}{4E}\left[\begin{array}{cc}1\hfill & \frac{k-3}{k+1}\hfill \\ \multicolumn{1}{c}{\frac{k-3}{k+1}}& 1\hfill \end{array}\right]\left(\begin{array}{c}{\sigma }_c\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)\hfill \\ +N\left[\begin{array}{cc}{S}_{11}\hfill & S_{12}\hfill \\ \multicolumn{1}{c}{S_{21}}& S_{22}\hfill \end{array}\right]\left(\begin{array}{c}{\sigma }_c\hfill \\ \multicolumn{1}{c}{0}\end{array}\right)\hfill \end{array}$$

(25)

Therefore, let ε₁ = ε, and the above equation is simplified to obtain

$${\sigma }_c=\frac{E}{1+\mathrm{EN}{S}_{11}}\epsilon$$

(26)

Where S₁₁, S₁₂, S₂₁, and S₂₂ are matrix elements.

Calculation of damage elastic modulus E_D

Under uniaxial compressive stress, the strain of the damaged material is determined by the elastic modulus E_D. It can be expressed as ⁵

$$E_D=\frac{{\sigma }_c}{\epsilon }=E(1-D)=\frac{E}{1+\mathrm{EN}{S}_{11}}$$

(27)

Where E is the elastic modulus.

The work done by uniaxial compressive stress W₁ can be expressed as ⁵

$$W_1=4\mathrm{ab}({\sigma }_1{\epsilon }_1+{\sigma }_2{\epsilon }_2)=4\mathrm{ab}{S}_{11}{{\sigma }_c}^2$$

(28)

Where a is the length of the selected element, which is the same order of magnitude the crack length b.

Further, the stress intensity factor of the crack array can be calculated by the following formula

$$K=F\sin \theta {[w\sin \frac{\pi (l_N+l_1)}{w}]}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$$

(29)

$$l_n=l_{n-1}+\stackrel{•}{l_n}\Delta t$$

(30)

Where l₁ is the length of the first impact crack, l₁ = 0.27c, 2c is the initial crack length, 2c = d. d is the particle diameter of the tool material. l_n is the length of the n time impact crack, and 2w is the initial crack spacing. ²⁴

The relationship between the crack propagation rate dl/dn and the stress intensity factor ΔK can be described by the Paris formula. The crack propagation rate of cemented carbide can be expressed as ²⁵

$$\stackrel{•}{l_n}=\frac{d{l}_n}{\mathrm{dn}}=8.3112\times {10}^{-16}(\Delta K{)}^{10.71}$$

(31)

Where n is the number of stress cycles, which also is the number of cutting impact. dl_n/dn is the crack propagation rate of the tool material under the nth impact.

$$W_e=2{\int }_0^{l_n}\frac{(k+1)(b_v+1)}{4E}{K}^2\mathrm{dl}$$

(32)

$$W_f=2c{\tau }_f\chi$$

(33)

Where W_e is the elastic strain energy caused by crack expansion, W_f is the friction energy released by the initial micro-crack sliding, τ_f is the normal component of stress caused by uniaxial compressive stress, χ is the sliding distance of the initial micro-crack surface, ⁵ and k is the correlation coefficient.

$$\chi =\frac{(k+1)(b_v+1)}{4E}c{\tau }_f{[w\sin \frac{\pi (l_n+l_1)}{w}]}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}[2\pi (l_n+l_1)]$$

(34)

$${\tau }_f=\frac{1}{2}{b}_v[{\sigma }_c-{\sigma }_c\cos 2\theta ]$$

(35)

$\stackrel{•}{l_0}$ and l₀ are the initial damage value of the tool material and the initial crack propagation rate respectively. After the tool material bears nth impact, the damage value of the tool material is D_n and the crack propagation length is l_n. The damage value D_n and crack opening propagation length l_n are determined as the initial condition when the tool material is subjected to (n+1)th impact.

$$2W_e+W_f=W_1$$

(36)

Substitute equations (32) and (33) into equation (36) to get S₁₁, the damage value, and the expression after the damage are obtained.

In the calculation of tool impact damage, the impact force F of milling cutter as intermittent cutting leads to tool tooth damage:

$$F=f_z{a}_p{V}^2\rho$$

(37)

Where f_z is the feed per tooth, a_p is the cutting depth, ρ is the material density.

Establishment of tool failure boundary map

By analyzing the mechanism under different failure modes, the failure model is constructed. According to the failure criteria given under different failure modes, the failure boundary map under different cutting parameters is obtained in this section.

Establishment of failure criterion for flank wear

When the tool mills titanium alloy workpiece with high cutting speed (90–120 m/min), the tool flank wear will be the more serious, which makes the tool fail firstly. ³ In this paper, the criterion of flank wear failure is that the average wear volume of flank face VB_ave is 0.3 mm, and the flank failure boundary is established.

Establishment of failure criterion for rake wear

When the tool mills titanium alloy workpiece with medium cutting speed (70–90 m/min), obvious crater wear occurs. Under this cutting speed, this wear form is more serious than other forms, which makes the tool fail firstly. This paper defines that the failure criterion of the crater wear on the rake face is that KT_max is 0.1 mm, and the failure boundary for the rake face is established.

Establishment of failure criterion for impact fracture after tool breakage

When the tool mills titanium alloy workpiece with low cutting speed (30–70 m/min), tool breakage is very likely to occur. It is assumed that there is no energy loss during the impact process and the kinetic energy during the impact process is all converted into the energy used for the fracture of the tool tooth, resulting in tool damage. Then, considering the impact fracture of the tool, the critical condition of the impact fracture of the cemented carbide tool can be expressed as ²⁶

$$F\ge \tau A=G\gamma A$$

(38)

Where G is the shear modulus of cemented carbide tool, γ is the shear strain of cemented carbide, and A is the cross-sectional area at fracture.

Further, according to the shear Hooke’s law

$$G=\frac{E_D}{2(1+b_v)}$$

(39)

Where E_D is elastic modulus and b_v is Poisson’s ratio.

Considering the cut-in impact, with the continuous cutting, the tool material continues to damage, the tool impact fracture conditions can be obtained:

$${\xi }_m=\frac{F}{m_A·a_p·\mathrm{VB}}\ge \left[\tau \right]=\frac{E_D\gamma }{2(1+b_v)}$$

(40)

Where ξ is the impact stress, m_A is correlation coefficient which obtained by experimental fracture area.

Establishment of tool failure boundary map

At present, the normal wear life of uncoated cemented carbide tool should be at least 30 min, so the tool reaches a failure criterion of certain failure forms within 30 min or produces tool breakage failure within the prediction time, which is considered that the tool fail in advance. As shown in Figure 7, a two-dimensional failure boundary map along the X direction as cutting speed and the Y direction as cutting width is established. According to three different failure criteria, three failure boundary curves are obtained respectively. Taking the intersection point of three failure boundaries as the center, the reasonable interval value is carried out, and the tool failure boundary map under different cutting parameters is established.

Figure 7.

The tool failure boundary map.

Experimental validation

Experimental design

The experimental tools used in this study are uncoated cemented carbide end mills, the tool material is YG6 cemented carbide. The tool geometric parameters are shown in Table 1.

Table 1.

Tool geometric parameters.

Parameter	Number
Diameter (mm)	10
Rake angle (°)	9
Flank angle (°)	8
Helix angle (°)	35
Tooth number	4

The experimental workpiece material is titanium alloy Ti6Al4V. VDL-1000E three-axis NC machine tool developed by Dalian Machine Tool Group is used for side milling in the experiment. Tool wear forms and the tool breakage are observed and measured by the ultra-depth three-dimensional microscope VHX-1000. The experimental site is shown in Figure 8.

Figure 8.

The experimental site.

Experimental result analysis

Comparison of flank wear width between analytic model results and experimental results

The model of tool flank wear is verified by experiments. The selected cutting parameters are shown in Table 2.

Table 2.

Cutting parameters.

NO	V (m/min)	ae (mm)	ap (mm)	fz (mm/r)
1	47	2	3	0.08
2	47	0.8	3	0.08
3	37	0.8	3	0.08
4	31	0.8	3	0.08
5	47	0.3	3	0.08

The tool flank wear volume VB is measured by the ultra-depth three-dimensional microscope VHX-1000. MATLAB software is used to calculate the tool flank wear model, and then the VB prediction results are compared with the experimental results, as shown in Figure 9.

Figure 9.

(a) Comparison of different cutting width between experimental and analytical results and (b) Comparison of different cutting speeds between experimental and analytical results.

The tool flank wear in the cutting process is described by theoretical modeling and experimental method. The comparison of flank wear width between analytical results and experimental results can be seen from Figure 9, and the trend is consistent with the prediction model. The maximum error is within 21%, which verifies the accuracy of the prediction model. It can be seen from Figure 9(a) that when the tool milling Ti6Al4V is between 30 and 55 m/min, the wear rate increases significantly with the increase of cutting speed. At the same time, it can be seen from Figure 9(b) that the wear rate increases significantly with the increase of cutting width. The error between the prediction model and the experimental model may be caused by the following reasons: the error in the prediction results of tool flank temperature field leads to the decrease of the accuracy of the prediction results of tool flank wear; milling vibration factor is not considered in the prediction model of flank wear; The accuracy of the established normal stress model is poor; In the side milling of Ti6Al4V, there is also oxidation wear in the side milling process, which is not considered in the wear model.

Comparison of the analytical results and experimental results of the model of tool crater wear depth

The experimental verification and analysis of the model established by the tool crater wear are carried out. The selected cutting parameters are shown in Table 3.

Table 3.

Cutting parameters.

NO	V (m/min)	ae (mm)	ap (mm)	fz (mm/r)
1	47	0.4	3	0.08
2	47	0.6	3	0.08
3	47	0.8	3	0.08

The wire cutting technology is used to cut and sample the vertical rake face, and the cross-section is observed by the ultra-depth three-dimensional microscope. Finally, in order to better obtain the crater wear profile curve, the image processing technology is used to scan the end mill cross-section to obtain the data points, as shown in Figure 10.

Figure 10.

Scanning figure of crater wear curve.

Since the tool has a variety of edge wear phenomena such as flank wear and micro-collapse edge wear near the cutting edge in the cutting process, the original cutting edge cannot be observed in the experimental observation process. The established crater wear model cannot consider the edge wear, so the influence of edge wear on the cutting edge is ignored. Finally, the data on the crater wear profile curve extracted from the experimental image is compared with analytical data. Since the cutting width has a great influence on the crater wear, this paper analyzes the crater wear depth under different cutting width.

Experimental results are basically consistent with the numerical analytical results in the variation trend. Since the crater wear cannot accurately describe the wear near the cutting edge, there is a large error at the first point. As shown in Figure 11, under the first cutting parameter, experimental value of the wear depth KT_max is 13.47 μm, analytical data is 16.02 μm, and the error is 18.93%. The maximum error of the crater wear profile is 4.36 μm. Under the second cutting parameter, experimental data is 14.46 μm, analytical data is 16.71 μm, and the error is 15.56%. The maximum error of the crater wear profile curve is 2.74 μm. Under the third cutting parameter, experimental data is 18.24 μm, analytical data is 20.21 μm, and the error is 10.78%. The maximum error of the crater wear profile curve is 3.94 μm. According to the experimental data of three groups, the wear depth of crater depth increases with the increase of cutting width.

Figure 11.

Comparison of analytical data and experimental data under different cutting parameters: (a) a_e = 0.4 mm, t = 1100 s, (b) a_e = 0.6 mm, t = 1100 s, and (c) a_e = 0.8 mm, t = 1100 s.

Comparison between analytical results and experimental results of tool breakage model

The model of cutting edge breakage is tested and analyzed. The selected cutting parameters are shown in Table 4. When tool breakage reaches 100 μm, it is considered that the tool fails. The failure mechanism of tool breakage is different from the tool wear, which is an abnormal failure phenomenon. In order to ensure the accuracy of the model, the parameters outside the cutting edge failure boundary are verified by 12 cutting edges of 3 end mills.

Table 4.

Cutting parameters.

NO	Tool number	V (m/min)	ae (mm)	ap (mm)	fz (mm/r)
1	1,2,3	31	2	3	0.08
2	4,5,6	47	1.8	3	0.08
3	7,8,9	56	1.4	3	0.08
4	10	31	0.8	3	0.08
5	11	47	0.8	3	0.08

Cutting parameters can be selected to ensure the reliability of the tool under large cutting parameters. According to the tool breakage failure criterion, the failure curve is drawn and the safety area is verified by different parameter points in Table 4, as shown in Figure 12. In Figure 12, the blue curve is the cutting edge breakage boundary obtained by the analytical model, and the three points are the parameters verified by the experiment.

Figure 12.

Comparison of analytical data and experimental data.

Experimental results are observed by the ultra-depth three-dimensional microscope. The measurement of tool breakage is shown in Figure 13.

Figure 13.

Measurement of tool breakage.

The experimental results show that the No. 4 and 5 experimental parameters are not damaged in the safe cutting area within the predicted time. The No.1, 2, and 3 experimental parameters are obviously damaged outside the safe cutting area, which proves that the tool occurs breakage failure. The experimental results are shown in Table 5.

Table 5.

Experimental results.

NO	Number of cutting edges	Average breakage (μm)	Time (s)
1	12	127	570
2	12	141	250
3	12	130	300
4	4	No obvious
5	4	No obvious

Experimental verification of the tool failure boundary map

In order to verify the accuracy of the tool failure boundary map, several parameters are added, in which the cutting depth and feed per tooth are consistent with the previous. The cutting parameters are shown in Table 6.

Table 6.

Cutting parameters.

NO	V (m/min)	ae (mm)	Average wear volume (μm)	Time (s)	Status
a	80	1	108	1800	Failure
b	70	0.8	181	1800	Safe
c	80	0.6	172	1800	Safe
d	95	0.6	326	1800	Safe
e	95	0.4	182	1800	Failure
f	110	0.5	309	1800	Safe

Combining three failure models, through the above 13 experimental cutting parameters, the failure boundary map is shown in Figure 14.

Figure 14.

The tool failure boundary map.

Conclusion

In view of the wear and breakage failure of cemented carbide tool in milling titanium alloy, a tool failure boundary map of safe operation area is provided to ensure that the tools will not occur failure with the cutting parameters selected in the safe area during the prediction time. Based on the cutting characteristics of titanium alloy, the failure boundary map mainly considers three kinds of tool failure forms including flank wear, crater wear, and tool cutting edge breakage, and it also reveals the three kinds of failure mechanism of tools. Then, the analytical models are established and compared with the experimental results, there is in good agreement, which provides a reference for the field of titanium alloy cutting and got the following conclusions:

1. Based on the analytical method, the temperature field model of the flank face is established, which including the influence of the flank wear. Considering abrasive wear, adhesive wear, and diffusive wear, the flank wear model of the end mills is established based on geometry. The model can describe the change rate of flank wear and obtain the flank wear. The experimental results are in good agreement with analytical results. The maximum error is within 21%, which verifies the accuracy of the prediction model and provides scientific and reasonable theoretical support for flank wear state recognition, tool life prediction, and tool geometry parameter optimization.

2. Based on the analytical method, the stress field, and temperature field models of the rake face are established, and the stress distribution and temperature distribution of the chip in the rake face during sliding are obtained. Then, a prediction model of tool crater wear depth considering abrasive wear, adhesive wear, and diffusive wear is established, and the prediction curve of crater wear is obtained. The results show that analytic model is basically consistent with the experimental crater wear profile curve in the variation trend. The maximum prediction error of KT_max of the crater wear depth is 18.93%, and the maximum error of the crater wear profile curve is 2.55 μm, which verifies that the established crater wear model has high prediction accuracy.

3. According to the cyclic load characteristics of the tool and the continuum damage mechanics, the damage value of the tool material is solved. Based on the damage value of the tool material, the cutting edge impact fracture boundary condition of the tool including the cutting parameters is established, and the safe area which can prevent tool from failing is divided to ensure that the tool does not fail with the selected cutting parameters in the safe area are during the predicted time, which also provides technical support for efficient milling of titanium alloys.

According to three different failure criteria, three failure boundary curves are obtained respectively and the tool failure boundary map under different cutting parameters is established. The experiment is finished to verify that the tool failure boundary map have high prediction accuracy