**1. Introduction**

The demand for medical implants such as bone implantations and bone replacements is increasing due to joint diseases and the aging population. Titanium alloy has the characteristics of being non-magnetic, corrosion resistant, and possessing high strength and high toughness. In particular, metastable β titanium alloy is a new type of medical titanium alloy. Its elastic modulus is similar to that of human bones. It can effectively avoid the problem of stress shielding. In addition, the material does not contain cytotoxic elements. Therefore, it is favored in the medical field [1,2].

To improve the biocompatibility and wear resistance of medical titanium implants, micro texture on the surface is usually processed in order to realize surface modification [3,4]. At present, laser engraving, acid etching, alkali etching, and other methods are used to process the surface micro texture of titanium implants, but it is difficult to accurately control the geometry and surface morphology of the micro texture [5–7]. Moreover, micro textured surfaces processed by laser are prone to slag, irregular shape, and complex residual stress [8–10]. Therefore, UEVC can be used to solve the problems of difficult machining and unstable surface texture processing quality in titanium implants. It also overcomes the problems of high cutting temperature, heavy tool wear, and poor machining quality in the traditional cutting process [11–14].

UEVC was first proposed by the Japanese scholars Shamoto Eiji and Moriwaki Toshimichi [15]. By applying ultrasonic excitation in two or more directions of the tool, the

**Citation:** Wang, Z.; Pan, Y.; Zhang, Y.; Men, X.; Fu, X.; Ren, S. Study on the Material Removal Mechanism of Ultrasonic Elliptical Vibration Cutting of Medical β Titanium Alloy. *Micromachines* **2022**, *13*, 819. https:// doi.org/10.3390/mi13060819

Academic Editors: Xiuqing Hao, Duanzhi Duan and Youqiang Xing

Received: 28 April 2022 Accepted: 20 May 2022 Published: 25 May 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

tool tip can cut the workpiece along an elliptical path. In this paper, ultrasonic excitation is applied in the cutting direction and cutting depth direction of the tool in order to process the pit texture on the surface of the workpiece. Lotfi et al. [16] used UEVC to texture the surface of titanium alloy. It was found that micro texture was formed on the surface of the material under the impact of certain frequencies. Friction and wear tests were carried out. The results showed that the surface micro texture could effectively improve the friction and wear properties of titanium alloy. Zhang et al. [17] processed sinusoidal, sawtooth, oblique wave and other different groove nanostructures on hardened steel by means of amplitude control in ultrasonic elliptical vibration cutting. The restrictions of vibration conditions and tool geometry on machining shape were studied, and a compensation method for amplitude control command was proposed. Yang et al. [18] prepared ordered micro/nano grating structures on the surface of aluminum alloy using the ultrasonic elliptical vibration cutting process. The influence mechanism of micro texture on surface color and optical reflectivity was studied theoretically and experimentally.

The above research verified the machinability of UEVC in the processing of material surface micro texture. However, the high-quality processing of micro texture still needs to be deeply studied with respect to its mechanism, exploring the stress state, removal mode, chip shape and so on. Ma et al. [19] studied the effect of diamond tools on the critical cutting depth of brittle materials under the condition of ultrasonic vibration by performing groove cutting tests on brittle materials. It was found that under the condition of ultrasonic vibration, diamond tools can increase the critical cutting depth for the plastic cutting of brittle materials. Liu et al. [20] conducted molecular dynamics simulation using the improved model to explore the material removal mechanism of monocrystalline silicon under EVC. The results showed that the main material removal mechanism shifts from extrusion to shear in one vibration cycle. In addition, based on stress analysis, it was found that the formation mechanism of subsurface damage in the extrusion and shear stages is different. Huang et al. [21] developed a ductile zone machining model for UEVC of brittle materials based on the plastic zone machining model with the aim of achieving the maximum cutting depth, so as to maximize the machining efficiency while ensuring the machining surface quality. Liu et al. [22] studied the effect of amplitude on machined surface integrity in high-speed ultrasonic elliptical vibration milling of titanium alloy. It was found that the surface roughness increased with increasing vibration amplitude, and the surface residual compressive stress increased with increasing vibration amplitude. Gao et al. [23] used ultrasonic elliptical vibration milling to effectively improve the quality of the machined surface. The research found that, compared with ordinary milling, highspeed ultrasonic vibration milling demonstrated a stable improvement in the tool yield and surface roughness of the machined surface. The above research on the cutting mechanism of UEVC materials mostly focused on a single material removal method. Moreover, there is still a lack of theoretical research on micro texture processing of β titanium implants, which is a difficult-to-machine material.

This paper focuses on the technical problems of β titanium implant processing. The UEVC processing method is adopted. With the help of finite element simulation, the evolution laws of chip morphology, residual stress, and maximum principal stress in the machining process under different cutting trajectories are explored. The influence mechanism of the material removal process is revealed, and a complete model of the material removal process of UEVC is established. This provides theoretical guidance for the processing of the micro texture on the surface of β titanium implants.

### **2. UEVC Theoretical Model**

### *2.1. Kinematic Model*

UEVC is realized by applying periodic ultrasonic excitation with the same frequency and different amplitude to the cutting direction and cutting depth direction of the tool. Finally, the machining of micro texture on the workpiece surface is realized.

As shown in Figure 1, ultrasonic excitation is applied in the X direction (cutting speed direction) and the Y direction (cutting depth direction) to establish a UEVC process under ideal conditions. direction) and the Y direction (cutting depth direction) to establish a UEVC process under ideal conditions.

Finally, the machining of micro texture on the workpiece surface is realized.

UEVC is realized by applying periodic ultrasonic excitation with the same frequency and different amplitude to the cutting direction and cutting depth direction of the tool.

As shown in Figure 1, ultrasonic excitation is applied in the X direction (cutting speed

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**Figure 1.** Schematic diagram of UEVC. **Figure 1.** Schematic diagram of UEVC.

**2. UEVC Theoretical Model**

*2.1. Kinematic Model*

The tool tip trajectory equation is as follows: The tool tip trajectory equation is as follows:

$$x(t) = A\sin(2\pi ft + \varphi) \tag{1}$$

$$y(t) = B\cos(2\pi ft)\tag{2}$$

With the cutting speed, the trajectory equation of the tool tip relative to the workpiece is as follows: With the cutting speed, the trajectory equation of the tool tip relative to the workpiece is as follows:

$$x(t) = A\sin(2\pi ft + \varphi) + vt \tag{3}$$

$$y(t) = B\cos(2\pi ft)\tag{4}$$

where *A* and *B* are the amplitudes in X and Y directions, respectively. *f* is the ultrasonic vibration frequency, which is 20 kHz in this paper. *φ* is the phase difference of two-way sinusoidal excitation. *v* is the cutting speed. where *A* and *B* are the amplitudes in X and Y directions, respectively. *f* is the ultrasonic vibration frequency, which is 20 kHz in this paper. *ϕ* is the phase difference of two-way sinusoidal excitation. *v* is the cutting speed.

The speed of the tool tip relative to the workpiece can be derived from Equations (3) The speed of the tool tip relative to the workpiece can be derived from Equations (3) and (4):

$$v\_x(t) = 2\pi f A \cos(2\pi ft + \varphi) + v \tag{5}$$

$$v\_y(t) = -2\pi fB\sin(2\pi ft)\tag{6}$$

Different cutting paths can be obtained by adjusting the parameters (*A*, *B*, *f*, *φ* and *v*) according to the above formula. According to Formulas (5) and (6), if () ≥ 0 at any time *t*, the tool and workpiece will not be separated, which is called non-separated ultrasonic vibration cutting. If *t* makes () < 0, there will be a separation stage between the tool and the workpiece, which is called separated ultrasonic vibration cutting [24]. Aiming at the high-quality processing of β titanium implant surface micro texture, this paper only Different cutting paths can be obtained by adjusting the parameters (*A*, *B*, *f*, *ϕ* and *v*) according to the above formula. According to Formulas (5) and (6), if *vx*(*t*) ≥ 0 at any time *t*, the tool and workpiece will not be separated, which is called non-separated ultrasonic vibration cutting. If *t* makes *vx*(*t*) < 0, there will be a separation stage between the tool and the workpiece, which is called separated ultrasonic vibration cutting [24]. Aiming at the high-quality processing of β titanium implant surface micro texture, this paper only explores the process of separated UEVC.

### *2.2. Cutting Path Planning*

explores the process of separated UEVC.

and (4):

*2.2. Cutting Path Planning* Compared with traditional cutting, UEVC has more flexible trajectory control and more prominent advantages in the processing of material surface micro texture. Different cutting paths can be generated by adjusting the control parameters of cutting path (vibration frequency, phase difference, amplitude and cutting speed). Finally, micro textures Compared with traditional cutting, UEVC has more flexible trajectory control and more prominent advantages in the processing of material surface micro texture. Different cutting paths can be generated by adjusting the control parameters of cutting path (vibration frequency, phase difference, amplitude and cutting speed). Finally, micro textures with different shapes are processed on the material surface. The single-period trajectory can be obtained by adjusting the phase difference and amplitude using MATLAB software, as shown in Figure 2.

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**Figure 2.** Single-cycle cutting trajectory. **Figure 2.** Single-cycle cutting trajectory. speed to obtain the multi-cycle cutting trajectory under different parameters, as shown in

ware, as shown in Figure 2.

ware, as shown in Figure 2.

The single-cycle trajectory parameters are adjusted in combination with the cutting speed to obtain the multi-cycle cutting trajectory under different parameters, as shown in Figure 3. In the figure, *A* and *B* in Equations (5) and (6) are taken as 0.005. The single-cycle trajectory parameters are adjusted in combination with the cutting speed to obtain the multi-cycle cutting trajectory under different parameters, as shown in Figure 3. In the figure, *A* and *B* in Equations (5) and (6) are taken as 0.005. Figure 3. In the figure, *A* and *B* in Equations (5) and (6) are taken as 0.005.

with different shapes are processed on the material surface. The single-period trajectory can be obtained by adjusting the phase difference and amplitude using MATLAB soft-

with different shapes are processed on the material surface. The single-period trajectory can be obtained by adjusting the phase difference and amplitude using MATLAB soft-

**Figure 3.** Variation of cutting trajectory under different parameters. **Figure 3.** Variation of cutting trajectory under different parameters. **Figure 3.** Variation of cutting trajectory under different parameters.
