1 Introduction

In modern 3C industry (Computer, Communication and Consumer Electronics), the metal shell provides a better experience in both touch and visual than plastic one. Many companies have launched the products with metal shell since 2013, e.g., Apple, Samsung, HTC, Huawei, Lenovo, OPPO, ** grinding wheel topography on the workpiece via a geometric–kinematic model in a single-pass surface grinding process. In their model, table speed, wheel speed, wheel topography and original workpiece surface texture were concerned.

From the literature survey, many factors related to surface quality of the machined parts have been addressed, e.g., geometrical factor, micro-texture, and cutting parameters. However, there are few reports concerning the grinding texture on cutting tool flank face, in particular in the chamfer machining of hole. Therefore, this paper is focused on the relationship between the machined surface and grinding texture on flank face in the chamfer milling of stainless steel.

3 Manufacturing of Cutting Tool

Manufacturing of chamfer cutting tool starts with the establishment of the mathematical models including rake face, 1st flank face and 2nd flank face. Following that, the simulation of the models is carried out for the safety reason, and then machining process is driven by the simulated models.

3.1 Mathematical Model of Cutting Tool

There are two methods applied to establishment of cutting tool models, analytic geometry and differential geometry [30]. Analytic geometry is employed to establish the mathematical models of the cutting tool. Coordinate system xyz of cutting tool model is built firstly, where z axis is the central axis of cutting tool, and the tool tip is the origin of the system xyz (Figure 1). The coordinate system x1y1z1 can be obtained by the translation tx of xyz along x axis. The system x2y2z2 can be generated by rotation of x1y1z1 around y1 through an angle of α. By rotating x2y2z2 around x2 axis through an angle of β, the system x3y3z3 can be obtained. The system x4y4z4 is generated after moving x3y3z3 along y3 axis by ty. Finally, the system x5y5z5 is obtained by rotating x4y4z4 around x4 axis through an angle of ŋ. The transformation matrix M between xyz and x5y5z5 is calculated by Eq. (1):

$$\begin{aligned} M = \,& M_{xt} \times M_{y\eta } \times M_{xa} \times M_{yt} \times M_{x\beta } \\ = & \left[ {\begin{array}{*{20}c} {\cos (\alpha )} & {\sin (\alpha )\sin (\beta + \eta )} & { - \sin (\alpha )\cos (\beta + \eta )} & 0 \\ 0 & {\cos (\beta + \eta )} & {\sin (\beta + \eta )} & 0 \\ {\sin (\alpha )} & { - \cos (\alpha )\sin (\beta + \eta )} & { - \cos (\alpha )\cos (\beta + \eta )} & 0 \\ {t_{x} \cos (\alpha )} & {t_{x} \sin (\alpha )\sin (\beta + \eta ) + t_{y} \cos (\eta )} &{ - t_{x} \sin (\alpha )\cos (\beta + \eta ) + t_{y} \sin (\eta )} & 1 \\ \end{array} } \right], \\ \end{aligned}$$
(1)

where Mxt is a transformation matrix from xyz to x1y1z1, Mxŋ a transformation matrix from x1y1z1 to x2y2z2, Mxα a transformation matrix between x2y2z2 and x3y3z3, Myt a transformation matrix between x3y3z3 and x4y4z4, and Myt a transformation matrix from x4y4z4 to x5y5z5.

Figure 1
figure 1

Transformation of coordinate systems for cutting edges

Full size image

For the chamfer tool, a straight line is utilised as the cutting edge, the model of which is established easily in the system xyz, as shown in Eq. (2):

$$\begin{aligned} & \left\{ \begin{array}{l} x = x_{{o_{1} }} + t \times \cos \alpha , \hfill \\ y = 0, \hfill \\ z = t \times \sin \alpha , \hfill \\ \end{array} \right. \hfill \\ & \quad a \le t \le b, \hfill \\ \end{aligned}$$
(2)

where t is the parameter of the model, and a and b are the constraints of t.

The 2nd flank face is a three-dimensional surface which is constituted by cutting edge and the grinding path of the 2nd flank face. A straight line in x5y5z5 (Eq. 3) is selected as the grinding path of the 2nd flank face. Therefore, the path in xyz is calculated by substituting Eq. (3) into Eq. (1):

$$\left\{ \begin{aligned} x_{5} = t_{1} , \hfill \\ y_{5} = 0, \hfill \\ z_{5} = 0, \hfill \\ \end{aligned} \right. \quad c \le t_{1} \le d,$$
(3)

where t1 is the parameter of the model, and c and d are the constraints of t1.

3.2 Numerical Simulation on Cutting Tool Models

In order to validate the mathematical models (Section 3.1) of chamfer cutting tool, a numerical simulation is carried out in MATLAB, as shown in Figure 2, where there are the face and line, e.g., cutting edge, and 1st flank face between cutting edge and cutting path of the 2nd flank face. Here, the geometries of the chamfer cutting tool are determined by tool shape parameters (R, a, b) and cutting edge cross-section parameters (γ, β).

Figure 2
figure 2

Numerical simulation of chamfering tool

Full size image

3.3 Grinding Processes of Cutting Tool

The geometric parameters of chamfer cutting tool, as shown in Table 1, consist of diameter, edge length, tool length, tooth number, rake angle, the 1st flank angle, and the 2nd flank angle. The grinding processes is planned according to the features of cutting tool, as shown in Figure 3(a), consisting of four steps, i.e., rake angle grinding, edge side grinding, 2nd flank angle grinding, and 1st flank angle grinding. Rake angle grinding and edge side grinding are machined by a straight diamond grinding wheel (grit size: D10, concentration: C100, wheel grinding feed: 0.015 mm), whereas the 2nd flank angle grinding and 1st flank angle grinding are machined by a dish cup diamond grinding wheel (grit size: D64, concentration: C100, wheel grinding feed: 0.035 mm) (Figure 3b). Also, to generate five kinds of grinding textures on 1st flank face, five grinding strategies are developed (Figure 3c).

Table 1 Geometric parameters of chamfer tool
Full size table
Figure 3
figure 3

Simulation of grinding process

Full size image

The grinding data are generated based on the mathematical models of the tool, and they are imported into NUMROTO software embedded in the grinding machine. After setting up grinding parameters, the simulations are carried out on the four steps accordingly to validate the grinding data, as shown in Figure 3(a), which demonstrates the correctness of the data. Consequently, the cutting tools are machined in a grinding machine, as shown in Figure 4, where Figure 4(a) shows the carbide blank, Figure 4(b) reveals the grinding process in the grinding machine, and Figure 4(c) illustrates the chamfer cutting tool including 1st flank angle, 2nd flank angle, rake face and cutting edge, etc.

Figure 4
figure 4

Grinding process of chamfer cutting tool

Full size image

Figure 5 depicts the grinded cutting tools with different grinding textures on 1st flank face, where five 1st flank faces, on which there are five angles between cutting edge and the texture line, i.e., 0°, 15°, 30°, 45° and 60°, are displayed in Figure 5(a)–(e), respectively.

Figure 5
figure 5

The 1st flank faces

Full size image

4 Experimental Setups

In the test, the chamfers are machined on the drilled holes. A 304 stainless steels used as a mobile case are selected as the test workpiece. Figure 6 depicts the experimental setups. A 3-axis milling machine, VDL-1000E, is selected to carry out the test. Kistler 9257B is used to measure the cutting force combining charge amplifier and data acquisition. As the machining parameters, cutting speed is 14.32 m/min, feed rate is 0.01 mm/rev, and ap = 0.16 mm, which are employed in real industry.

Figure 6
figure 6

Experiment equipment

Full size image

KEYENCE VHX-1000 digital microscopeis utilised to observe the machined chamfered surface. A white light interferometer, Talysurf CCI PM, is selected to measure the surface roughness of machined chamfer.

5 Results and Discussions

The influences of grinding texture on cutting force, surface form and roughness are analysed accordingly in this section.

5.1 Cutting Force

Figure 7 shows the influences of grinding texture on the 3-direction cutting forces of x, y and z. With the grinding texture angle from 0° to 45°, cutting forces of x, y and z are decreased from x = 205.1 N, y = 207.5 N and z = 94 N to x = 119.6 N, y = 132.4 N and z = 67.7 N, respectively. Then the forces are increased to x = 147.1 N, y = 146.5 N, z = 72.6 N after the texture angle over 45°. Moreover, the cutting force values of x and y directions are higher than the cutting force values of z direction that changes with in a small range. The cutting forces of x and y directions are similar in trends and values. The lowest cutting force value is generated when the grinding texture equals 45°, and the highest cutting force is at the 0° grinding texture. According to the results, it is obvious that the cutting forces are influenced by the texture direction.

Figure 7
figure 7

Influence of grinding texture on cutting force

Full size image

5.2 Surface Forms and Roughness

Figure 8 shows the impact of the grinding texture angles on the machined surface forms, where the burn and the coarse texture are the main defect patterns. Generally, the surface quality is improved as the grinding texture angle is increased from 0° to 45°. However, the surface quality of machined products is deteriorated gradually when grinding texture is between 45° to 60°. In Figure 8(a), the coarse textures appear from bottom to top of the chamfer, and the burn is also generated. The burn is the major defect when the grinding texture angle is 15°, and comparing with 0° texture angle, the coarse texture defect is improved (Figure 8b). The burn spots are fewer at 30° grinding texture angle than at 15° angle (Figure 8c), and there is no scratch defect. There is no burn spot on the chamfer surface at 45° grinding texture angle (Figure 8d); however, there is many small coarse texture. When grinding texture angle is 60°, the burns are generated into an area, and they are more than 45° grinding texture angle (Figure 8e). Therefore, the surface quality is highly influenced by grinding texture, and it is the highest at 45° grinding texture.

Figure 8
figure 8

Surface forms of machined chamfers

Full size image

A 3D roughness criterion, arithmetical mean height Sa, is utilised to describe the roughness, and it expresses surface quality with higher-accuracy than the 2D roughness criteria. Figure 9 depicts the surface roughness curve and 3D forms of machined chamfer, and four samples are selected to measure Sa to raise the measurement accuracy. Sa is decreased with the texture angle raising from 0° to 45°, however, it is increased after 45° texture angle. The lowest Sa is generated at 45° texture angle.

Figure 9
figure 9

Surface roughness curve and 3D forms of chamfer surface

Full size image

The friction type between the flank face of the tools and the workpiece surface is mixed friction. The one part of the friction surface is separated by oil film, and the other part of the friction surface is contacted. The texture directions determine the frictional behaviour and lubricant retention of flank, which effects the cutting force, surface generation and roughness.

6 Conclusions

  1. (1)

    A set of cutting tools with five angles of texture angle, i.e., 0°, 15°, 30°, 45° and 60°, are modelled, simulated, machined and tested accordingly to evaluate the grinding texture on flank face. Cutting force, surface quality and surface roughness are observed.

  2. (2)

    The cutting force is decreased with texture angle increasing from 0° to 45°, and then is increased. The lowest cutting force is achieved at 45° grinding texture angle.

  3. (3)

    The burn and the coarse texture are the main defect patterns in chamfer milling of stainless steels. The best surface quality is achieved at 45° grinding texture.

  4. (4)

    Surface roughness Sa is decreased with texture angle increasing from 0° to 45°, and then is increased. It is obvious that the best state of surface roughness is archived by the tool with 45° grinding texture.

  5. (5)

    Combining the results of cutting force, surface quality, and surface roughness, it is obvious that the best machining state is obtained at 45° texture angle, i.e. the lowest cutting force, the best surface form, and the lowest surface roughness.