**2. Modeling**

#### *2.1. Modeling of the Indenter Structure*

The indenter tip shape greatly a ffects nanoscratching results, and there is no ideal Berkovich indenter due to the limitations of processing conditions. In addition, the indenter continuously wears as it is working; therefore, the indenter tip shape is di fferent for every test. The geometric shape and dimension parameters of a Berkovich indenter are shown in Figure 1. Assuming that a Berkovich indenter is a combination of a sphere and a triangular pyramid [25], the tip can be divided into three parts: the sphere (from Section 1 to the vertex of the nose in Figure 1a), the transition (from Section 1 to Section 2), and the pyramid (above Section 2). The following equations can be obtained according to the geometric relations:

$$d^\* = \frac{R}{\sin \alpha} - R \tag{2}$$

$$d\_2 = R(1 - \sin \alpha) \tag{3}$$

$$r\_0^2 = R^2 - \left(R - d\_1\right)^2\tag{4}$$

$$\tan \theta = \frac{r\_0}{d^\* + d\_1} \tag{5}$$

$$r = \sqrt{R^2 - \left(R + d\ast\right)^2 \sin^2 \theta} \tag{6}$$

$$l = 2\sqrt{r^2 - \left(r - \frac{d - d\_1}{\cos\theta}\right)^2} \tag{7}$$

where *R* is the indenter nose radius, *d*\* is the distance from the nose vertex to the top of the ideal indenter, α is the angle between the edge line and the centerline, θ is the angle between the edge plane and the centerline, *d*1 is the distance from Section 1 to the nose vertex, *d*2 is the distance from Section 2 to the nose vertex, *r*0 is the radius of Section 1, *r* is the circular arc radius of the intersection between the sphere and the triangular pyramid, and *l* is the length of the intersection between the section which is normally aligned to the centerline and the edge plane in the transition part.

**Figure 1.** Dimension parameters and geometric shape of the Berkovich indenter: (**a**) model diagram, (**b**) top view, (**c**) side view, and (**d**) 3-D solid model.

The normal projected area of the different indenter heights can be calculated using Equation (8).

$$A\_{\mathcal{P}} = \begin{pmatrix} \pi \left[ R^2 - \left( R - d \right)^2 \right] & (d \le d\_1) \\\\ \pi \tan^2 \alpha \left( d + d^\* \right)^2 - \frac{\arcsin \frac{l}{2 \tan \alpha (d + d^\*)}}{60} \pi \tan^2 \alpha \left( d + d^\* \right)^2 + \frac{3l}{2} \tan \alpha \left( d + d^\* \right) & (d\_1 < d < d\_2) \\\\ \frac{3 \sqrt{3}}{4} \tan^2 \alpha \left( d + \frac{R}{\sin \alpha} - R \right)^2. & (d \ge d\_2) \end{pmatrix} \tag{8}$$

#### *2.2. Modeling of the Critical Depth of Cut*

Based on the traditional cutting force model, indentation model, and scratch pattern, a new method considering the elastic recovery was designed and two assumptions were proposed: (1) As a rigid body, the indenter does not deform; however, it wears during the process of scratching. (2) The motion of the indenter is quasi-static. According to the characteristics of deformation, the process of scratching can be divided into three stages: the elasticity leading stage, the ductility leading stage, and the brittleness leading stage [26].

In the elasticity leading stage, the force applied on the indenter consists of an elastic restoring force, an adhesive force, and a frictional force. The elastic restoring force is a reactive force applied on the indenter caused by the elastic deformation of the part. The adhesive force between the two solids

(i.e., indenter and part) is complex [27]; therefore, a simplified approach that combines the adhesive and frictional forces is used in this study. All forces are decomposed into normal and tangential forces:

$$\begin{aligned} F\_{\rm ev} &= K\_1 A\_1 \\ F\_{\rm et} &= \mu K\_1 A\_1 \end{aligned} \tag{9}$$

where *K*1 is the average contact pressure between the indenter and workpiece, *A*1 is the projected area of the contact surface between the indenter and part (*A*1 = 3 √3(*d* + *<sup>d</sup>*∗)<sup>2</sup> tan<sup>2</sup> θ), and μ is the frictional and adhesive coefficient.

The study by Son et al. [28] showed that the minimum cutting depth producing chips can be expressed as

$$d\_m = R\left(1 - \cos\left(\frac{\pi}{4} - \frac{\beta}{2}\right)\right) \tag{10}$$

where β is the friction angle.

In the ductility leading stage, the force applied on the indenter consists of an elastic restoring force, frictional and adhesive forces, and cutting deformation force. The force analysis is illustrated in Figure 2. The frictional and adhesive forces consist of two parts: one caused by chip formation and one caused by elastic recovery. The cutting deformation force, which is a reaction force applied on the indenter caused by the deformation of the part, can be separated into a chip formation force and a plowing force. However, the plowing force can be ignored in this model since it is much weaker than the chip formation force [29]. All forces are decomposed into normal and tangential forces:

$$\begin{aligned} F\_{dn} &= F\_{dn1} - F\_{dn2} + F\_{dn3} \\ F\_{dt} &= F\_{dt1} + F\_{dt2} + F\_{dt3} \end{aligned} \tag{11}$$

where *Fdn*1 is the normal force component caused by the elastic restoring force, *Fdn*2 is the normal force component caused by the frictional and adhesive forces, *Fdn*3 is the normal force component caused by the chip formation force, *Fdt*1 is the tangential force component caused by the elastic restoring force, *Fdt*2 is the tangential force component caused by the frictional and adhesive force, and *Fdt*3 is the tangential force component caused by the chip formation force.

**Figure 2.** Force analysis model in the ductility leading stage.

The normal force components can be calculated using Equation (12).

$$\begin{aligned} F\_{dn1} &= K\_1 S\_1 \\ F\_{dn2} &= \mu (K\_1 + K\_2) S\_2 \sin \theta \\ F\_{dn3} &= K\_2 S\_2 \sin \theta \end{aligned} \tag{12}$$

where *K*2 is the cutting deformation contact stress, *S*1 is the projected area given by the shaded area in Figure 2, and *S*2 is the contact area between the region of ductile deformation, which is the area from the unmachined surface to the machined surface.

The areas *S*1 and *S*2, respectively, are

$$\begin{array}{l} S\_1 = \sqrt{3}[(d\_\mathsf{e} + d^\*)\tan\alpha + (d + d^\*)\tan\theta](d + d^\*)\tan\theta\\ S\_2 = \frac{(d - d\_\mathsf{e})}{2\cos\theta}[\sqrt{3}(d + d^\*)\tan\theta + \sqrt{3}(d\_\mathsf{e} + d^\*)\tan\theta] \end{array} \tag{13}$$

where *d*e is the part elastic recovery depth and is equal to the height difference between the scratching and residual depths. Note that the elastic recovery depth is not constant and increases linearly as the scratching depth increases [20].

The tangential force components are

$$\begin{aligned} F\_{dt1} &= K\_1 S\_2 \cos \theta \\ F\_{dt2} &= \mu (K\_1 + K\_2) S\_2 \cos \theta + \mu K\_1 S\_1 \\ F\_{dt3} &= \frac{F\_{dt3}}{\tan \theta} \end{aligned} \tag{14}$$

The normal and tangential forces, respectively, are

$$\begin{aligned} F\_{dn} &= K\_1 S\_1 - \mu (K\_1 + K\_2) S\_2 \sin \theta + K\_2 S\_2 \sin \theta \\ F\_{dt} &= K\_1 S\_2 \cos \theta + \mu (K\_1 + K\_2) S\_2 \cos \theta + \mu K\_1 S\_1 + K\_2 S\_2 \cos \theta \end{aligned} \tag{15}$$

In the brittleness leading stage, the average contact pressure and the cutting deformation contact stress show a zigzag change due to crack propagation and pop-in debris.

It is difficult to control the depth of processing, but controlling the cutting force, especially the normal force, is relatively easy, no matter whether ultra-precision grinding or single point diamond cutting is used. Therefore, the cutting force model of this section can be used to control the cutting depth through the cutting force.

The dislocation will appear when the part undergoes extrusion deformation [30]. The appearance of a cleavage crack will occur when one side's tensile stress reaches the limit under the action of the applied force. The theoretical cleavage strength can be expressed by [31]

$$
\sigma\_{\mathbb{C}} = \frac{1}{2} \sqrt{\frac{E\gamma}{a}} \tag{16}
$$

where *E* is elastic modulus, γ is surface energy per unit area, and *a* is the interplanar spacing. In the scratching process, the maximum stress in the part's machined surface is located at the tip of the indenter. If the maximum stress is less than the cleavage strength, no cracks will occur on the surface or subsurface of the part. The maximum stress is [32]

$$P\_0 = \frac{3}{2} K\_1 \tag{17}$$

The parameter *K*1 is obtained from Equation (15).

$$K\_{1} = \frac{F\_{\text{dn}}(\mu S\_{2} \cos \theta + S\_{2} \cos \theta) - F\_{\text{dt}}(S\_{2} \sin \theta - \mu S\_{2} \sin \theta)}{(S\_{1} - \mu S\_{2} \sin \theta)(\mu S\_{2} \cos \theta + S\_{2} \cos \theta) - (S\_{2} \cos \theta + \mu S\_{2} \cos \theta + \mu S\_{1})(S\_{2} \sin \theta - \mu S\_{2} \sin \theta)} \tag{18}$$

The critical condition of brittle materials during the scratching process is

$$
\sigma\_{\rm c} = \frac{3}{2} \mathcal{K}\_{\rm 1c} \tag{19}
$$

where *K*1*c* is the critical average contact pressure.
