Rake angle effect in cutting-based single atomic layer removal

When minimum cutting depth is down to a single atomic layer, two portions of the cutting tool, namely, cutting edge and lowest atoms of the cutting tool, are involved in the cutting-induced material removal. Correspondingly, there are different critical rake angles for those two portions of the tool, different from the nominal rake angle in conventional cutting and edge radius-induced effective rake angle in nanocutting. Both they should be considered in atomic and close-to-atomic cutting to obtain the defect-free processed surface with an ideal crystalline structure. Molecular dynamics modelling is carried out to investigate the critical rake angles to enable single atomic layer removal on monocrystalline Cu (1 1 1) surface. The analysis results clearly indicate that the critical rake angles of nanometric cutting edge and the lowest tool atoms for single atomic layer removal are among the range of (–70°, –65°) and (–17°, –14°), respectively. To achieve single atomic layer removal, the tool edge radius is suggested to be not greater than 2 nm. The research findings would provide theoretical guidelines to the cutting tool design for the application of mechanical cutting of high-performance atomic scale devices.


Introduction
Atomic and close-to-atomic scale manufacturing (ACSM) has become one of the most promising and extremely-significant fields of manufacturing science and technology for the manufacturing of atomic devices [1][2][3].Mechanical cutting is one of promising subtractive processing methods to enable ACSM by material removal at atomic and close-to-atomic scale (ACS).Design and manufacture of specific cutting tools for ACS cutting are of great significance to realize such an objective [4].Rake angle is one of important geometrical parameters for cutting tools, which has a profound influence on material removal behaviour.A large number of studies have been conducted to clearly elucidate the influence of tool rake angle on conventional machining and micro/nano cutting process [5][6][7][8][9].
As cutting depth changes from conventional scale to the nanoscale, the role of rake angle, has been greatly emphasized.In conventional machining, the tool rake face directly contacts with workpiece materials, enabling chip formation by material shearing [10][11][12].When cutting depth decreases to the nanoscale, since cutting depth is comparable or lower than the cutting-edge radius, there is a negative effective rake angle formed within the contact zone between cutting tool and workpiece materials.It has played the dominated role in extrusiondriven material removal [13][14][15][16].
When cutting depth is further decreased to ACS, evidently lower than edge radius, it can be expected that like in nanocutting, the effective rake angle has a significant influence on material removal behaviour in ACS cutting.However, it is still not well understood, hindering the design of next-generation cutting tool for enabling ACSM.
Molecular dynamics (MD) simulation has been extensively applied to investigate the mechanical machining mechanism of various engineering materials, such as metals, semiconductor materials and ceramics materials [17][18][19].For instance, based on MD simulation, He et al. studied the nanometric cutting mechanism of lutetium oxide single crystal [20].Wu et al. investigates the amorphization and dislocation evolution mechanisms of single crystalline 6H-SiC [21].The research findings have significantly advanced the fundamental understanding of material deformation and removal mechanism involved in mechanical machining.
In this study, serials of MD analysis are performed to investigate the effect of tool rake angle on surface generation during the cutting process of single crystal copper towards single atomic layer removal.The effective rake angles for enabling single atomic layer removal are determined.The research findings indicate that two effective rake angles exist to affect the material removal during ACS cutting process.

Analysis on the effective rake angles in ACS cutting
It should be highlighted that micro cutting typically refers to the mechanical cutting process with a nominal cutting depth of 1−999 μm, while nano-cutting usually adopts a nominal cutting depth of 1−100 nm.As for ACS cutting, its generally-used nominal cutting depth is smaller than 1 nm.
As illustrated in Fig. 1, the cutting edge could be regarded as the envelop curve of the outermost atoms.In ACS cutting process, the real material removal process is dependent on not only nanometric cutting tool edge but also the atoms located at the lowest point of the cutting tool.Therefore, cutting-based single atomic layer removal could be regarded as the result of two portions of cutting tools, namely, nanometric cutting edge and lowest tool atoms.To conveniently describe the cutting mechanism, the nanometric cutting edge is defined to nanotool, while lowest atoms are defined to atom-tool.

Calculation of the effective rake angle of nanometric tool ()
As the cutting depth used in ACS cutting is comparable or lower than the size of workpiece atoms, it is usually much smaller than the edge radius of the cutting tool.Thus, the effective rake angle is always negative.In nanocutting, as cutting depth is significantly larger than the atomic radius r ( ) w of workpiece material, the material atoms could be treated as points for easy understanding in Fig. 2a.The influence of atomic radius (r w ) could be ignored.As given in Fig. 2a, the effective rake angle ( E ) in nanometric cutting, can be derived by: ) arcsin(1 ) Where R t is tool edge radius and a is cutting depth.However, different from nanocutting, in ACS cutting, the effect of the atomic radius of workpiece and cutting tool material could no longer be ignored, in Fig. 2b, the workpiece atoms are treated as spheres with a radius of r W .The E for the nanometric cutting tool could be calculated as: In micro/nano cutting, since both cutting depth (a) and edge radius (R ) t is far larger than the workpiece atomic radius (r w ), so the effective rake angle could be derived in Eq. ( 1).In ACS cutting process, differently, both a and R t is comparable to r w .For specific workpiece material, as r w is one constant value, the effective rake angle is mainly dependent on the cutting depth (a) and cutting-edge radius (R t ).Moreover, from Eq. ( 2), it can be determined that the E of nanometric cutting edge would be easily affected by process parameters (cutting depth, etc.), tool geometrical parameters and workpiece material properties.

Calculation of the effective rake angle of atomic tool ( e )
As shown in Fig. 1, the atoms located at the lowest position of cutting tool play the role of the atomic cutting tool in ACS cutting.As cutting depth changes, there are three typical cases of contact states between the atomic cutting tool and workpiece atom within the targeted first atomic layer, as shown in Fig. 3. Correspondingly, there are different effective rake angles ( e ) of atomic tool at different cutting depths, as illustrated in Fig. 4. Here, the e is defined as the included angle between the common tangent of workpiece and tool atoms and vertical line.Depending on cutting depth, the e is significantly changed as follows: (1) < 0 e when cutting depth(a) is lower than + r r w t (see Fig. 3a).(2) = 0 e when cutting depth(a) is equalling to + r r w t (see Fig. 3b).
For each case, the e could be derived from Fig. 4, and it can be expressed as the follows: As per Eq. ( 3), it can be found that the e of the atom-tool is dependent on the cutting depth (a), and atomic radiuses (r r and w t ) of workpiece and cutting tool.For a specific diamond-copper cutting system, as the atomic radiuses of workpiece and tool material are constant value, e is only dependent on cutting depth.By changing cutting depth, the influence of the e of the atomic tool could be studied in the cutting-based single atomic layer removal process.

Cutting process simplification
In conventional machining, the rake angle (γ) is the angle between the rake face of cutting tool and vertical direction, it is an important geometry parameter of cutting tool in real mechanical machining, as illustrated in Fig. 5.In the ACS cutting, since cutting depth is much smaller than cutting edge radius, the cutting-edge radius effect should be considered.In the cutting process, there would be an effective rake angle (γ E ) formed between the cutting edge radius and workpiece materials.Besides, due to the interactions between the lowest atoms of cutting tool and the workpiece atoms, another effective rake angle (γ e ) is existed, as shown in Fig. 3 and 4.
The influence of tool edge radius and cutting depth on ACS cutting process could be studied from the perspective of effective rake angle.For this reason, in the present study, the rounded edge cutting tool is simplified into a sharp cutting tool with specific rake angle as illustrated in Fig. 5.As there are two effective rake angles ( e and E ) in the ACS cutting process, the influence of E on cutting mechanism is first studied.On this basis, the influence of the e is investigated by changing the cutting depth.

Simulation model and protocol
Molecular dynamics analysis is employed to analyse the critical rake angles in cutting-based single atomic layer removal.The cutting model mainly consists of a monocrystalline copper workpiece and a diamond cutting tool.The model parameters used are summarized in Table 1.In our recent study, it has been found that the defect-free processed surface could be preferably achieved on the (1 1 1) plane of monocrystalline copper [22].Therefore, in this study, those analyses are conducted on (1 1 1) surface, as shown in Fig. 6.The effect of cutting speed on the surface generation in cutting-based single atomic layer removal has been studied [22].And it is found that the cutting should be preferably conducted at a lower cutting speed (smaller than 42.5 m/ s) in order to obtain a defect-free processed surface on (1 1 1) plane of monocrystalline copper [22].Considering the high-computational cost in MD simulation, a cutting speed of 10 m/s is adopted, and it has been used in other MD-based studies on nanocutting [23].As for other simulation parameters that are not justified, they are usually-adopted in MD simulation, such as time step of 1 s.
The x and y-dimension of the copper workpiece are along [1 -1 0] and [1 1 2] and the cutting is carried out along the [-1 1 0] direction.In the workpiece, there are three kinds of copper atoms, including thermostatic layer, Newtonian layer and boundary layer.During the cutting process, the boundary layer is kept fixed to eliminate the possible cutting-induced motion of the workpiece.The atoms in the thermostatic layer are kept at a temperature of 293 K to simulate the heat dissipation in the mechanical cutting process.The Newtonian atoms, which obey newton's law during the cutting process, are used for the investigation of the underlying cutting mechanism.In the y-direction of the cutting model, the periodical boundary condition is employed, while the fixed boundary condition is adopted in the x-and z-directions.All molecular  dynamics simulations are performed using LAMMPS [24], and OVITO [25] are used to visualize the MD simulation results.
In molecular dynamics modelling, it is critically significant to reasonably select the potential function to accurately describe the interatomic interactions.For copper-diamond cutting system used in the present work, there are interatomic interactions among workpiece copper atoms (Cu-Cu), those among diamond carbon atoms in the cutting tool (C-C) and the interactions between workpiece copper and tool carbon atoms (Cu-C).To accurately describe the effect of tool effective rake angles on cutting-based single atomic layer removal, the Embedded atom method (EAM) potential [26] and Morse function are employed to describe the Cu-Cu interactions and Cu-C interactions [27].For the C-C interaction, as its strength is far larger than those of Cu-C and Cu-Cu, it is neglected in this study.

Results and discussion
As shown in Section 2, there are two effective rake angles of the same cutting tool during ACS cutting process.As per MD simulation results, both them would have a significant effect on material removal behaviour, significantly affecting surface generation in the cuttingbased single atomic layer removal.To decouple their effects, first, the sharp diamond tools with different negative rake angles are adopted process.The simulation results are further studied from various aspects, including workpiece surface topography, displacement vector and cutting forces, etc.

Surface generation
The aim of ACS cutting is to obtain the defect-free proceed surface by cutting-based single atomic layer removal.Thus, the surface morphologies of the processed surfaces by mechanical cutting with the use of the cutting tools with various rake angles are first analysed.

Effects of nano-tool rake angle on surface generation
As shown in Fig. 3c, the e of atom-tool is 0°, when cutting depth (a) is + r r w t .To eliminate the possible influence of negative e or positive e on surface generation, the cutting depth of about ( + r r w t ) is used.For a copper-diamond cutting system, r w (copper) and r t (carbon) are 1.28 Å and 0.7 Å. Fig. 7 shows the surface morphologies of Cu (1 1 1) surfaces at different γ E .At the cutting velocity of 10 m/s and cutting depth of 2 Å, single atomic layer removal can be achieved, when the γ E is (-65°, 0°).Though there are only several surface defects located at the cuttingin and/or -exit edge of the workpiece, the processed surfaces exhibit an ideal crystalline structure.Differently, when the γ E is comparable to or smaller than -70°, there Fig. 5. Simplification of tool edge radius in atomic or close-to-atomic cutting.

Table 1
Model parameters in MD simulations.are more surface defects generated on the processed surface.Moreover, the number of surface defects tends to increase with the decrease of γ E .
In addition, the surface composition of the newly-formed Cu (1 1 1) surfaces is studied.The workpiece atoms are coloured based on the number of atomic layers.As shown in Fig. 8, two kinds of surface composition could be observed.For the γ E of (-30°, 0°), the newlyprocessed Cu (1 1 1) surface only consists of skyblue atoms, namely, those within second layer, indicating that first atomic layer has been fully removed from the workpiece surface.
As γ E decreases to (-30°, -65°), an extremely small number of atoms within the first atomic layer is pressed into the processed surface.Meanwhile, combing Figs.7 and 8, it can be determined that single atomic layer removal could be also realized, but the processed surface consists of the atoms within two atomic layers.
When tool rake angle is below about -65°, a large number of first layer atoms are pressed into the second atomic layer, while several surface defects are generated.As illustrated, the newly formed surface consists of atoms from three atomic layers, including first (blue), second (skyblue), and third (yellow) atomic layers.Thus, such case of atomic layer removal is regarded as multi-atomic layer removal.
There is one critical γ E to enable single atomic layer removal, which is about (-70°, -65°).Moreover, as γ E changes, there are three types of atomic layer removal regimes, including layer-by-layer removal, single atomic layer removal, and multi-atomic layer removal [28].

Effects of atom-tool rake angle on surface generation
Fig. 9 shows the surface morphologies of the processed surface with different combinations of γ E and cutting depths (a).As shown, at one specific γ E , as cutting depth changes, the surface morphologies have also been significantly changed.It should be due to the influence of the e of atom-tool.At the γ E of -10°, -35°, and -60°, when cutting depth is lower than 1.1 Å, namely, °26. 4 e , there is no material removal.When cutting depth is larger than 1.1 Å and smaller than 1.6 Å, plastic material removal occurs, but the defect-free surface could not be obtained.Only when the cutting depth is larger than about 1.6 Å, the processed surface exhibits ideal crystalline structure.According to Eq. (3), the e for the plastic material removal on Cu (1 1 1) surface should be about (-26.4°,9.6°), while e for stable and continuous material removal is about (-11°, 9.6°).
As for those surfaces obtained at the γ E of -85°, no defect-free surface is obtained, despite of different cutting depths.
Therefore, to enable cutting-based single atomic layer removal, both effective rake angles of nano-tool and atom-tool should be synergistically ensured.

Chip formation
Fig. 10 shows the atomic displacement distribution along the cutting direction of workpiece material at the cutting depth of 2 Å.At the rake angle of larger than about -70°, those atoms with the largest atomic displacement are closely contacted with the rake face.Based on Lai's justification method for chip formation in MD simulations [13], it can be determined that the critical E of nano-tool for chip formation is about (-65°, -70°).When E is larger than -65°, there is chip formation during cutting process.When E is lower than about -70°, there is no chip formation.The piled material in front of the rake plane will be extruded to generate the new processed surface.As for the critical rake angle of (-65°, -70°), it should be the transformation phase between chip formation and no-chip formation.

Subsurface deformation mechanism
In nano-cutting process, there is elastic and plastic deformation on the processed workpiece surface [15], inducing the formation of subsurface defects.Differently, in ACS cutting process, only elastic deformation occurs on the processed surface with no subsurface defects remained after cutting [29].Fig. 11 illustrates the defect structures in the workpiece subsurface.Here the workpiece atoms are coloured based on the centro-symmetry parameter (CSP).To clearly show subsurface defects, the atoms with the CSP of smaller than 3 are omitted, which represent the atoms with perfect FCC structure.
It can be seen in Fig. 11 that as the increase, there are more defect atoms (CSP > 3) formed within the contact zone between the cutting tool and workpiece materials, indicating the increasingly evident cutting-induced deformation on the workpiece surface.
Nevertheless, at the end of the cutting, there is no defect remained in workpiece subsurface (see Fig. 12) when the is larger than about -80°.It indicates that there is only elastic deformation on the processed surface.After the tool passes over the workpiece surface, the deformed part would recover completely.Differently, when is decreased to about -85°, there are subsurface defects formed.It indicates that the occurrence of elastic and plastic deformation on the processed workpiece surface.The plastically deformed part would finally lead to lasting deformation, thereby leading to the formation of workpiece subsurface defects.
Therefore, a reasonable rake angle of the cutting tool should be F n and F t tends to increase with the E , which further induces the change of atomic displacement behaviour during the cutting process, as illustrated in Fig. 14.
In the cutting-based single atomic layer removal, F n and F t could provide the necessary compressive stress and shear stress to enable the material deformation and removal along with normal and tangential directions on the processed surface.As tool rake angle changes, there are several typical kinds of cases discussed as follows.
Case 1. (F n < F t ): When E is larger than about -28°, both F t and F n is lower, but F (about 20 nN) is larger than F n (about 5 nN), indicating that the F t plays a more important role in the material removal process.The targeted first atomic layer will undergo an evident dislocationdominated material removal process.
As given in Fig. 14a, there is one slip plane generated between the topmost two atomic layers.The materials below the slip plane approximately remains immobilized while those above this plane are slipped along the cutting direction.With the advance of the cutting tool, the slipped materials are gradually formed into chips and removed.As for those materials below slip plane, they approximately remain immobilized.It is supposed to be one cross-section of a shearstress driven edge dislocation.At such case, the F t would provide the necessary shear stress to enable dislocation motion, as illustrated in Fig. 14.But the F n is too small to provide the compressive stress to initialize the plastic deformation on the processed surface.Consequently, only slight elastic deformation occurs.

Case 2. (F n ≈ F t ): When
E is (-38°, -28°), F n is regarded to be approximately equalling to F t .At this case, the material removal process is approximately the same with that in Case 1.The only difference is that a part of first layer atoms has been pressed into the processed surface, which should be ascribed to the increase of the F n .

Case 3. (F n > F t ):
For the E of (-65°, -38°), though the F n (about 30 nN) is larger than F t (about 20 nN), both F t and F n would have a great influence on the material deformation and removal process.
As illustrated in Fig. 14b, the material removal process is approximately consistent with that at Case 2. However, due to the change of E , the cutting tool could impose a larger normal F n on the workpiece surface, which provides a higher compressive stress to the occurrence of the workpiece surface deformation.Consequently, But an extremely small number of first layer atoms is pressed into the processed surface, see Fig. 8d-f, further leading to the slip of second atomic layer along cutting direction.

Case 4. (F n > > F t ): When
E is smaller than about -65°, F n is evidently larger than F t , thus, the F n has become the dominated driving forces for material deformation and removal.It will provide a higher compressive stress to extrude large number of first layer atoms into the second atomic layer, as shown in Fig. 8h-i, which induces the continual slip of second atomic layer (see Fig. 14c and d).When E is about -85°, the third atomic layer also slips along cutting direction.Meanwhile, F n is also large enough to induce plastic deformation on the processed surface, leading to the generation of subsurface defects after cutting, as analysed in Section 4.3.

Effects of atom-tool rake angle on cutting forces
As analysed in Section 2.2, the is mainly dependent on the cutting depth(a).Fig. 15 shows the dependence of F t and F n on the cutting depth at various of -10°, -35°, -60°and -85°, which corresponds to four typical cases in Section 4.4.1.
There is one critical cutting depth (a c ) for F t in each case, due to the workpiece atomic sizing effect [22].For instance, for the rake angle of -10°, when cutting depth is larger than about 1−1.1 Å(a c ), namely, of (-29.7°,-26.4°),F t is evidently increased from about 0 nN to about 23 nN.It indicates that the shear stress has exceeded the critical resolved shear stress for dislocation slip along cutting direction, when cutting depth is larger than such critical value.Correspondingly, there is a critical e for the occurrence of material removal.Differently, F n has exhibited yield-like changes, when are -10°, -35°, and -60°, respectively.When the cutting depth is lower than above-mentioned a c , F n tends to linearly grow, as shown in Fig. 15a-c, indicating that elastic deformation occurs on the workpiece surface along the normal direction.However, when cutting depth is larger than such critical value, due to the influence of tool rake angles, F n exhibits evidently different changes, thereby induces different cases of atomic displacement behaviors, as illustrated in Fig. 14.
Differently, when the rake angle is about -85°, as given in Fig. 15d, F n is finally converged to about 145 nN, far larger than F t of 40 nN.As a result, the elastic-plastic deformation occurs on the processed surface.As per the above analysis, with an increase in cutting depth, there are two critical cutting depths, which affect the cutting forces during the cutting process and further influence the cutting-induced material deformation and removal.Correspondingly, according to Eq. ( 3), the critical e could be calculated, as in Table 2. Clearly, the γ E has a little influence on the critical γ e .
Overall, both the effective rake angles of the cutting tools would influences the cutting forces, thereby affecting the material removal in the cutting process.

Stress distribution
To indicate the distribution of compressive and/or tension stress of workpiece during ACS cutting using a sharp cutting tool, Fig. 16 shows the first principle stress distribution of workpiece.Clearly, the processed surface exhibits tension stress, due to the extrusion and friction of cutting tool.There is also one compressive stress concentred zone in each case.
Fig. 17 gives the hydrostatic pressure distribution of workpiece at the E of -10°, -35°, -60°and -85°, respectively.When E is -10°, the compressive stress is mainly concentred in front of rake face, which is mainly located in the undeformed material zone.However, as the E changes, both location and area of the concentred zone of compressive stress are gradually changed.As depicted above, during the cutting process, the compressive stress would dominate the elastic and/or deformation of the processed surface.Therefore, as the E changes from -10°to -85°, there is more-and-more evident elastic deformation on the processed surface.Once the compressive stress is large enough, elastic and plastic deformation would be initialized on the processed workpiece surface and subsurface defects are formed, as analysed in Section 4.3.
In ACS cutting, the shear stress dominates the chip formation by dislocation motion.Fig. 18 gives the maximum shear stress distribution of workpiece.At the of -10°, there is one evident shear stress concentrated zone before cutting edge.When the is decreased to -85°, the area of such a shear stress concentrated zone is evidently decreased.
Overall, as the rake angle of cutting tool changes, the stress state of the workpiece during ACS cutting has been significantly changed.There are shear stress, compressive stress and tension stress-dominated zones, respectively.At a lower rake angle, like -10°, the larger shear stress would dominate chip formation by dislocation motion, while the lower compressive stress enables elastic deformation on the processed surface, as illustrated in Fig. 19a.After cutting, the elastically deformed part springs back, and the processed surface exhibits tension stress.
With a decrease in , the area of shear stress zone is decreased, while that of the compressive stress zone is continually increased, as displayed in Fig. 19b.When the is decreased to lower than one threshold, the compressive stress would enable elastic-plastic deformation while the shear stress is very small to only enable the elastic-plastic deformation instead of chip formation, see Fig. 19c.

Fabrication of multi-atomic layer groove by single atomic layer removal
In order to determine whether the controlled single atomic layer removal can be achieved by diamond cutting, four-time consecutive cutting on Cu (1 1 1) surface are conducted to fabricate one rectangular groove with the thickness of 4 atomic layers.From the above analysis, E is larger than about -65°, single atomic layer removal could be achieved.Here, the E is -10°and the cutting depth is 2 Å.As shown in Fig. 20, after four cutting processes, one groove with the targeted thickness of 4 atomic layers is successfully fabricated.
The newly-generated Cu (1 1 1) surface has ideal crystallographic structures after each cutting.It clearly indicates that controllable single atomic layer removal can be achieved on the Cu(1 1 1) surface at reasonable tool rake angle and process parameters.

Single atomic layer removal by a rounded cutting edge
Above analysis shows that there is a critical E of the cutting tool for enabling controlled single atomic layer removal.The E for cuttingbased single atomic layer should be (-70°, -65°).According to Eq. ( 2), it can be determined that only when the cutting-edge radius is smaller than about 2 nm, the E is larger than -65°.When the tool edge radius is larger than 3.5 nm, the E has decreased to be lower than about -70°.Hence, the critical tool edge radius is about 2-3.5 nm.In order to achieve a stable single atomic layer removal on Cu (1 1 1) surface, the tool edge radius of smaller than 3.5 nm should be used.To verify it, MD   simulations are conducted to study the single atomic layer removal using the tool edge radius of 2 nm.Fig. 19 shows the simulation results of ACS cutting with the tool edge radius of 2 nm and the cutting depth of 2 Å.It can be seen that there is chip formation over the entire cutting process Fig. 21.
With the advance the cutting tool, the first atomic layer atoms (blue) has been removed from the workpiece surface.Moreover, as per the surface topography and surface composition of the processed as shown in Fig. 22, it could be determined that single atomic layer removal has been achieved, thereby theoretically verifying the predicted tool edge radius based on effective rake angle calculation.

Conclusions
In this study, the effect of rake angle on atomic and close-to-atomic scale cutting mechanism in cutting-based single atomic layer removal is

Fig. 1 .
Fig. 1.Schematic illustration of the cutting-based single atomic layer removal process.

Fig. 7 .Fig. 8 .
Fig. 7. Effect of rake angle on surface morphologies of Cu(1 1 1) surfaces.Atoms are coloured according to their z-direction heights.The blue circles are applied to highlight the surface defects.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9 .Fig. 10 .Fig. 11 .
Fig. 9. Influence of the cutting depths on surface morphologies at different γ E .Atoms are coloured as per their z-direction heights.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12 .Fig. 13 .
Fig. 12. Subsurface defects of the workpiece at the cutting distance of 8 nm.Atoms are coloured as per their CSPs.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 15 .
Fig. 15.influences of cutting depth on cutting forces at different rake angles.

Fig. 16 .
Fig. 16.Influence of on first principle stress of the workpiece.

Fig. 17 .
Fig. 17.Influence of on hydrostatic pressure distribution of the workpiece.

Fig. 18 .
Fig. 18.Influence of on maximum shear stress of the workpiece.

Fig. 19 .
Fig. 19.Schematic illustrations for workpiece stress distribution in ACS cutting using a sharp cutting tool.

Fig. 20 .
Fig. 20.Surface morphologies of the processed surface after four cutting.

Fig. 21 .
Fig. 21.Simulation results of single atomic layer cutting at cutting edge radius of 2 nm.

Fig. 22 .
Fig. 22. Surface topography of the processed surface using a rounded edge tool with an edge radius of 2 nm.

Table 2
Critical cutting depths and e .
e Cutting depth (a)