An investigation on cutting mechanism and thermomechanical behaviors at tool-chip interface in ultrasonic vibration–assisted cutting of Ti6Al4V alloy

Abstract

The thermomechanical behavior in tool-chip interface (TCI) is a key factor affecting the cutting performance. However, few investigations have been reported on the thermomechanical behaviors in tool-chip interface in ultrasonic vibration cutting (UVC). In this study, the thermomechanical behaviors in TCI are studied considering the ultrasonic vibration–induced alternation of contact stress and transient characteristics such as shear angle, tool-chip contact length, and flowing stress. The transient model for cutting force incorporating the effect of acoustic softening and transient characteristics is proposed. The predicted cutting forces and seizure area length are consistent with the experimental data. The mechanisms for the improvement of cutting performance in UVC are varied with cutting speed. Those findings suggest that the improvement of cutting performance is highly dependent on cutting speed, feed rate, as well as the coupling effect between ultrasonic vibration amplitude and feed rate. The ultrasonic vibration is also identified to improve the cutting performance prominently through promoting the crack initiation and propagation, as well as the generation of cyclic varied chip thickness. This is attributed to the changed cutting process introduced by ultrasonic vibration.

Appendices

Appendix 1. Modeling of acoustic softening effect on strength

On account of the researches on supersonic stress [11], the ultrasonic stress is formulated as

$$\sigma { = }\rho cU$$

(34)

where $\rho$, U, c are the density, speed of a specific material point, sound velocity of workpiece, respectively.

Supersonic intensity is given as

$$I = \frac{{\sigma^{2} }}{\rho c}{ = }\rho cU^{2}$$

(35)

The supersonic angular frequency is formulated as

$$\omega { = 2}\pi f$$

(36)

Then, the particle velocity is derived as

$$U = \omega \xi = 2\pi f\xi$$

(37)

where $f$ is the frequency of supersonic vibration, $\xi$ is the deformational distance.

The stress under supersonic vibration is obtained as

$$\sigma = \sigma_{{_{n} }} (1 - DI)^{e}$$

(38)

in which $\sigma_{{_{n} }}$ is the material stress.

At last, the supersonic stress is formulated as

$$\sigma = \sigma_{{_{n} }} (1 - {4}\pi^{{2}} f^{2} \xi^{2} D\rho c)^{e}$$

(39)

in which the constant D and e is the acoustic coefficient which are used to balance the influence of supersonic vibration on stress. The constant e is set as 1. Then

$$\sigma = \sigma_{{_{n} }} (1 - {4}\pi^{{2}} f^{2} \xi^{2} D\rho c)$$

(40)

The constant D is obtained with input of the stress with and without supersonic vibration.

The shear stress in acoustic vibration is given as

$$\tau = \frac{\sigma }{\sqrt 3 } = \tau_{n} (1 - {4}\pi^{{2}} f^{2} \xi^{2} D\rho c)$$

(41)

Appendix 2. Modeling of acoustic softening coefficient based on metal cutting

2.1 Stress without ultrasonic vibration

The strain and strain rate of shear band in conventional metal machining are given as

$$\varepsilon { = }\frac{\cos \alpha }{{\sin \phi \cos \left( {\phi - \alpha } \right)}}$$

(42)

$$\dot{\varepsilon }{ = }\frac{\nu \cos \alpha }{{\Delta d\cos (\phi - \alpha )}}$$

(43)

in which $v$ and $\Delta d$ are the cutting velocity and the thickness of shear zone.

According to Experimental researches [57, 70], the model of shear angle is proposed as

$$\phi { = }\frac{{5}}{{8}}\alpha { + }\frac{{1}}{{2}}\cos^{ - 1} \left[ {\exp \left( { - 52.5 \times 10^{ - 3} \left( {\frac{{\tau_{s} \sqrt {3} }}{{100c_{2} \rho }}} \right)^{{{0}{\text{.8}}}} \left( {\frac{{vh \times 10^{ - 3} }}{60\omega }} \right)^{{{0}{\text{.4}}}} } \right)} \right]$$

(44)

where $c_{2}$, $\omega$ are the specific heat, thermal conductivity, respectively.

The empirical formulation of shear zone is given by

$$\Delta d{ = }\frac{{h_{z} }}{5.9\sin \phi }$$

(45)

where $h_{z}$ is the feed rate.

According to the model of temperature in shear band [71], averaged temperature for Ti6Al4V is developed as follows.

$$T_{v} = T_{r} + \frac{{\varpi \tau_{s} \dot{\varepsilon }}}{{\rho c_{2} v\sin \phi }}$$

(46)

where $\varpi$, $\tau_{s}$ are the Taylor-Quinney coefficient, the shear flow stress, respectively.

The shear flowing stress is derived as

$$\tau_{s} { = }\frac{{1}}{{\sqrt {3} }}\left( {A + B(\frac{\varepsilon }{\sqrt 3 })^{n} } \right)\left\{ {1 + m\ln \frac{{\dot{\varepsilon }}}{{\dot{\varepsilon }_{0} }}} \right\}\left\{ {1 - \left( {\frac{{T_{v} - T_{r} }}{{T_{m} - T_{r} }}} \right)^{c} } \right\}$$

(47)

where $\dot{\varepsilon }_{{0}}$ is reference strain rate, $T_{v}$, $T_{m}$, $T_{r}$ are temperature of shear band, melting and room temperature of workpiece. A, B, C, m and n are Johnson–Cook material constants. The thermal attributes and material constant of workpiece are presented in Table 5 and 6.

Table 5 Thermal attributes of workpiece

Table 6 Johnson–Cook parameters of workpiece

The cutting speed, feed rate and cutting depth directional cutting forces $F_{tc}$, $F_{fc}$, $F_{{{\text{r}}c}}$ are measured in conventional oblique cutting experiments, then the resultant cutting force $F_{c}$ is determined.

$$F_{c} = \sqrt {F_{tc}^{2} + F_{fc}^{2} + F_{rc}^{2} }$$

(48)

According to the maximum stress principle, the shear stress and its relationship between shear angle is developed.

$$F_{s} = F_{c} \cos 45^{ \circ }$$

(49)

$$F_{s} = \tau_{s} (\frac{b}{\cos i})(\frac{h}{\sin \phi })$$

(50)

Based on the Eqs. (42)–(50), the material flowing stress $\tau_{s}$ without ultrasonic vibration is developed.

2.2 Stress under ultrasonic vibration in metal cutting

The strain and strain rate of shear band in UVC are given as

$$\varepsilon_{v} { = }\frac{{\cos \alpha_{v} }}{{\sin \phi_{v} \cos \left( {\phi_{v} - \alpha_{v} } \right)}}$$

(51)

$$\dot{\varepsilon }_{v} { = }\frac{{\nu_{v} \cos \alpha }}{{\Delta d_{v} \cos (\phi_{v} - \alpha_{v} )}}$$

(52)

where $v_{v}$ is the cutting velocity under ultrasonic vibration, $\Delta d_{v}$ is the thickness of shear zone in UVC.

The empirical formulations for shear zone and the analytical shear angle are presented as

$$\Delta d_{v} { = }\frac{{h_{z,v} }}{{5.9\sin \phi_{v} }}$$

(53)

$$\phi_{v} { = }\frac{{5}}{{8}}\alpha { + }\frac{{1}}{{2}}\cos^{ - 1} \left[ {\exp \left( { - 52.5 \times 10^{ - 3} \left( {\frac{{\tau_{s,v} \sqrt {3} }}{{100c_{2} \rho }}} \right)^{{{0}{\text{.8}}}} \left( {\frac{{v_{v} h_{v} \times 10^{ - 3} }}{60\omega }} \right)^{{{0}{\text{.4}}}} } \right)} \right]$$

(54)

The average temperature of shear band is given as follows.

$$T_{v,v} = T_{r} + \frac{{\varpi \tau_{s,v} \dot{\varepsilon }_{v} }}{{\rho c_{2} v_{v} \sin \phi_{v} }}$$

(55)

With consideration of ultrasonic influence on strain, strain rate, shear angle, thermal softening effect on stress and temperature, as well as those coupled effect, the plane shear flowing stress in ultrasonic vibration is derived as

$$\tau_{s,1} { = }\frac{{1}}{{\sqrt {3} }}\left( {A + B(\frac{{\varepsilon_{v} }}{\sqrt 3 })^{n} } \right)\left\{ {1 + m\ln \frac{{\dot{\varepsilon }_{v} }}{{\dot{\varepsilon }_{0} }}} \right\}\left\{ {1 - \left( {\frac{{T_{v,v} - T_{r} }}{{T_{m} - T_{r} }}} \right)^{c} } \right\}$$

(56)

When the acoustic softening effect on stress is taken into account, based on Eq. (8) in Appendix 1, the stress in supersonic vibration is derived as follows.

$$\tau_{s,v} = \tau_{s,1} (1 - {4}\pi^{{2}} f^{2} \xi^{2} D\rho c)$$

(57)

The average directional components of cutting forces $F_{tc,v}$, $F_{fc,v}$, $F_{{{\text{r}}c,v}}$ in the cutting speed, feed rate and cutting depth are measured in conventional oblique cutting experiments, then the resultant cutting force $F_{c}$ is determined.

$$F_{c,v} = \sqrt {F_{tc,v}^{2} + F_{fc,v}^{2} + F_{rc,v}^{2} }$$

(58)

According to the maximum stress principle, the shear stress and its relationship between shear angle is developed.

$$F_{s,v} = F_{c,v} \cos 45^{ \circ }$$

(59)

$$F_{s,v} = \tau_{s,v} (\frac{b}{\cos i})(\frac{{h_{v} }}{{\sin \phi_{v} }})$$

(60)

Based on the Eqs. (54) and (60), the average flowing stress $\tau_{s,v}$ with ultrasonic vibration is developed.

According to Eqs. (56)–(57), the acoustic softening coefficient is calculated.

Appendix 3

The coordinate system for measurement cutting forces is on account of conventional cutting, the forces in ultrasonic vibration should be normalized to this coordinate system. Figure 22 presents the relationship between coordinates in synthetic cutting velocity and conventional cutting velocity. The conversion model between two coordinate systems can be set up by the relationship between the synthetic and the traditional cutting velocity. The relationship of feed and tangential cutting force in two coordinate systems is presented as Fig. 23.

Fig. 22

The relationship of coordinate between synthetic and traditional cutting speed

Fig. 23

The cutting force relationship in two coordinate systems

The resultant force is developed as.

$$F_{c,v} { = }\sqrt {F_{t,t}^{2} + F_{f,t}^{2} }$$

(61)

$\theta_{1}$ is the angle between the resultant force in coordinates system are based on synthetic and traditional cutting velocity. The tangential and feed cutting force in coordinates of conventional cutting speed is equivalent of the measured force in orthogonal cutting. Those cutting forces are given by.

$$F_{tc} = F_{c,v} \cos (\beta_{v} - \alpha_{v} + \theta_{1} )$$

(62)

$$F_{fc} = F_{c,v} \sin (\beta_{v} - \alpha_{v} + \theta_{1} )$$

(63)

$$F_{c} { = }\sqrt {F_{t,c}^{2} + F_{f,c}^{2} }$$

(64)

Based on the model of oblique cutting in Ref. [72, 73], and the derived tool angle transformational model based on back rake-side rake [61], the x, y, z directional theoretical cutting force is deduced as

$$F_{z} = F_{t} = F_{c} (\cos \theta_{i} \cos (\theta_{n} + \theta_{1} )\cos i + \sin \theta_{i} \sin i)$$

(65)

$$F_{x} = F_{f} = F_{c} \cos \theta_{i} \sin (\theta_{n} + \theta_{1} )$$

(66)

$$F_{y} = F_{rc} = F_{c} (\sin \theta_{i} \cos i - \cos \theta_{i} \cos (\theta_{n} + \theta_{1} )\sin i)$$

(67)