*3.3. Chip Thickness*

Figure 5 assembles the chip shapes obtained by the cutting process. For *α* = 45◦ and 0◦ (and also for 15◦; see [21,22]), the chip had a roughly rectangular cross-section with a constant thickness *d*ch. This feature was easily explained by the action of the single dislocation system forming the chip. Even for *α* = −22.5◦, a small chip immediately adjacent to the rake face could be discerned, which was formed by the same dislocation system. In addition, for this rake angle, the other plastic processes discussed above contributed to forming a very broad chip extending a considerable distance in front of the tool. For *α* = −45◦, no chip was formed.

**Figure 5.** Side views of the chips formed during cutting.

We determined the chip thickness *d*ch as the extension of the chip perpendicular to the rake face (see Figure 1) and measured it at the base of the chip close to the PSZ; the values are assembled in Table 1. For the complex chip form of *α* = −22.5◦, we only considered the part of the chip adjacent to the rake face. The decline of *d*ch with decreasing *α* illustrated in Figure 6 could be rationalized as follows.

**Table 1.** Force and chip characteristics for various rake angles *α*: cutting force, *Fc*; thrust force, *Ft*; force components perpendicular, *N*, and parallel, *F*, to the rake surface of the tool; friction angle, *β*; and chip thickness, *d*ch. Force values are averaged over the last 30 Å of cutting. For the negative rake angles, data were also obtained for a cutting length of 200 Å see Appendix B; in the rows marked by (\*), averages over the last 100 Å of cutting are presented.


From mass conservation, the chip thickness, *d*ch, is related to the cutting depth, *d*, by:

$$d\_{\rm ch} = d \, \frac{\sin \psi}{\sin \phi'}\tag{4}$$

This relation may simply be obtained by noting that the length of the PSZ, *b*, obeys both *d* = *b* sin *φ* and *d*ch = *b* sin *ψ*; see Figure 1. Using Equation (1), Equation (4) thus predicts that *d*ch = 0 for *α* = −35◦ and *d*ch = *d* for *α* = 2*φ* − 90◦ = 19.5◦, since then, *ψ* = *φ*; see Figure 1.

Figure 6 compares our simulation results with the simple theory of Equation (4). Since only the shear angle enters this theory, the fair agreement obtained shows that indeed, the single shear plane model can describe the cutting process well.

**Figure 6.** Chip thickness, *d*ch, as a function of the rake angle *α*. Simulation results (squares) are compared to the theoretical prediction, Equation (4). For the negative rake angles, data were also obtained for a cutting length of 200 Å (circles); see Appendix B.

## *3.4. Forces*

The force on the tool was readily measured in an MD simulation. Figure 7 shows the evolution of the force with cutting length; the components of the force in the cutting direction, *Fc*, and normal to the original surface, *Ft*, which are denoted as cutting and thrust forces, respectively, are displayed. After the tool contacted the substrate (at Cutting Length 0), the forces started increasing. The increase was, however, not monotonous, since the generation of dislocations led to force drops. Towards the end of the cut, a sort of steady-state appeared to develop—with the possible exception of the thrust force at rake angle *α* = −45◦. We assemble the averages of the forces over the last 30 Å of the cut in Table 1.

**Figure 7.** (**a**) Cutting force, *Fc*, and (**b**) thrust force, *Ft*, as a function of the cutting length *L*.

The thrust forces increased with decreasing *α* showing that a stronger perpendicular force was needed to keep a wider tool at its prescribed depth. With decreasing *α*, also the cutting force increased; this increase illustrated the fact that larger forces were needed to shear the material and form the chip. Here, the case of *α* = −45◦ was exceptional, as the cutting force was quite small, even below the values of *α* = 0◦. This could be explained by the fact that in this case, no chip was formed (see Section 3.3), and we hence, did not have a case of real "cutting" here.

## *3.5. Force Angle*

From the forces determined in the simulation, we could calculate the friction angle; see Table 1. It allowed us to calculate the force angle; we display it in Figure 8. It showed that in the regime of chip formation, *α* ≥ 0, the force angle was constant, *χ* = 15.3◦ ± 0.7◦, and only increased for negative *α*, where no chips were formed. This feature—constant *χ* for constant *φ*–is important as the relation between these two angles is an important ingredient of available cutting theories.

**Figure 8.** Force angle, *χ*, as a function of the rake angle *α*. For the negative rake angles, data were also obtained for a cutting length of 200 Å (circles); see Appendix B.

Mechanical theories of cutting result in a linear relationship between shear angle *φ* and force angle *χ* [6],

$$
\Phi = \mathfrak{c}\_1 - \mathfrak{c}\_2 \chi. \tag{5}
$$

From theoretical arguments, the law, Equation (5), is derived with *c*<sup>1</sup> = 45◦ and *c*<sup>2</sup> = 1 or 0.5, depending on whether it is assumed that shear should occur in the direction of maximum shear stress or the power needed for cutting is minimized, respectively [2,3,5–7,50,51]. In particular, the so-called Merchant's law [2] with *c*<sup>1</sup> = 45◦ and *c*<sup>2</sup> = 0.5 forms the basis of textbook treatments of cutting [51]. On the other hand, from experimental data compiled for mild steel by Pugh [52], coefficients of *c*<sup>1</sup> = 32◦ and *c*<sup>2</sup> = 0.44 can be read off (Figure 2) [6]. A shear angle of *φ* = 54.7◦ would hence correspond to a force angle of *χ* = −52◦; this corresponds to a force pointing out of the surface, which is totally unphysical. On the other hand, our measured force angle of *χ* = 15.3◦ corresponded to *φ* = 25.3◦, which was in direct contrast to the shear observed in the simulation.

Our results also lay strongly outside the simple theoretical dependencies *c*<sup>1</sup> = 45◦ and *c*<sup>2</sup> = 1 or 0.5, which would predict a force angle of −9.7◦ (*c*<sup>2</sup> = 1) and −19.4◦ *c*<sup>2</sup> = 0.5. Again, the large shear angle would require a negative force angle.

We note that MD simulation of cutting an isotropic material (metallic glass) corroborated Merchant's law [2]), i.e., Equation (5) with *c*<sup>1</sup> = 45◦ and *c*<sup>2</sup> = 0.5 [53].

We concluded that our results of single crystal cutting showed a fixed force angle for a fixed shear angle, in agreement with available cutting theories. However, the value of *χ* was incompatible with these theories.
