Numerical simulation of materials-oriented ultra-precision diamond cutting: review and outlook

Ultra-precision diamond cutting is a promising machining technique for realizing ultra-smooth surface of different kinds of materials. While fundamental understanding of the impact of workpiece material properties on cutting mechanisms is crucial for promoting the capability of the machining technique, numerical simulation methods at different length and time scales act as important supplements to experimental investigations. In this work, we present a compact review on recent advancements in the numerical simulations of material-oriented diamond cutting, in which representative machining phenomena are systematically summarized and discussed by multiscale simulations such as molecular dynamics simulation and finite element simulation: the anisotropy cutting behavior of polycrystalline material, the thermo-mechanical coupling tool-chip friction states, the synergetic cutting responses of individual phase in composite materials, and the impact of various external energetic fields on cutting processes. In particular, the novel physics-based numerical models, which involve the high precision constitutive law associated with heterogeneous deformation behavior, the thermo-mechanical coupling algorithm associated with tool-chip friction, the configurations of individual phases in line with real microstructural characteristics of composite materials, and the integration of external energetic fields into cutting models, are highlighted. Finally, insights into the future development of advanced numerical simulation techniques for diamond cutting of advanced structured materials are also provided. The aspects reported in this review present guidelines for the numerical simulations of ultra-precision mechanical machining responses for a variety of materials.

Ultra-precision diamond cutting is a promising machining technique for realizing ultra-smooth surface of different kinds of materials. While fundamental understanding of the impact of workpiece material properties on cutting mechanisms is crucial for promoting the capability of the machining technique, numerical simulation methods at different length and time scales act as important supplements to experimental investigations. In this work, we present a compact review on recent advancements in the numerical simulations of material-oriented diamond cutting, in which representative machining phenomena are systematically summarized and discussed by multiscale simulations such as molecular dynamics simulation and finite element simulation: the anisotropy cutting behavior of polycrystalline material, the thermo-mechanical coupling tool-chip friction states, the synergetic cutting responses of individual phase in composite materials, and the impact of various external energetic fields on cutting processes. In particular, the novel physics-based numerical models, which involve the high precision constitutive law associated with heterogeneous deformation behavior, the thermo-mechanical coupling algorithm associated with tool-chip friction, the configurations of individual phases in line with real microstructural characteristics of composite materials, and the integration of external energetic fields into cutting models, are highlighted. Finally, insights into the future development of advanced numerical simulation techniques for diamond cutting of advanced * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

Introduction
Ultra-smooth surface with nanoscale surface roughness, ultrahigh surface integrity and eliminated subsurface damage has been crucially desired for promoting functionalities of components and parts for decades [1][2][3][4][5]. In particular with the increasing demands of applications in extreme conditions of high temperature, high pressure and strong corrosive [6,7], the preparation of ultra-smooth surface of emerging advanced structured materials is also greatly needed. Among wide ranges of advanced manufacturing techniques, ultra-precision single point diamond turning (SPDT) utilizing synthetical or natural single-crystal diamond (SCD) tools with nanoscale cutting edge radius has become a popular one, due to its achievable accurate form accuracy, moderate machining efficiency, high machined surface finish, high degree of machining freedom and low subsurface damage, etc [8][9][10][11][12]. For instance, ultra-smooth surface with nanoscale or even sub-nanoscale surface flatness can be readily prepared by ultra-precision diamond cutting, for its achievable nanometer machining accuracy [13,14].
In the ultra-precision SPDT process, the combination of linear slideways with sub-nanometer programming resolution and SCD tool with sharpness of tens of nanometers jointly leads to deterministic removal of workpiece material with thickness down to nanometer range. While internal microstructures of workpiece materials have feature sizes that are comparable with both the cutting edge radius of SCD tool and the depth of cut (DOC), the microstructure-dominated properties of specimen have a large influence on the diamond cutting process. Furthermore, the size effect associated with the ultra-small material removal thickness magnifies the impact of elastic-plastic deformation of workpiece materials in diamond cutting process [15,16]. Thus, workpiece material and diamond tool are highly coupled in the diamond cutting process, for which revealing the dependence of machining response on the properties of workpiece material is essentially needed.
Specifically, the dependence of diamond cutting on the properties of workpiece material can be embodied in below aspects. Firstly, the machined surface integrity of polycrystalline materials in diamond cutting process exhibits strong anisotropic characteristics, given the comparable internal grain size with DOC. For instance, the formation of surface steps at grain boundary (GB) with a height of few nanometers, caused by the anisotropic deformations in neighboring grain interiors, is commonly found in the diamond cutting of polycrystalline metals such as gold [17], copper [18][19][20][21][22][23], aluminum alloy [24] and ZnSe [25]. While the surface integrity is deteriorated by the formation of GB surface steps, discovering the formation mechanisms and suppressing strategies of GB surface steps are greatly needed.
Secondly, while the tool-chip friction states strongly affect both the machined surface morphology and chip formation in diamond cutting, the exchange of cutting heat between chip and tool, which largely determines the sticking and slip friction states, is closely related to the thermomechanical properties of the workpiece material, given the ultra-small material removal thickness [26]. While the friction states closely dominate the cutting stability, it is essentially required to reveal the role of thermo-mechanical coupling played in determining the tool-chip friction states.
Thirdly, while the superior combined properties of composites originate from the complementation of properties of individual phases, their machinability is severely restricted by the low synergetic deformation between constitutive phases with dramatically different mechanical properties. For instance, cavities on machined surface of SiC particle-reinforced Al matrix composites (SiCp/Al) caused by particle-matrix interface debonding, or roughening on machined surface of carbon fiber-reinforced plastic (CFRP) caused by fiber fracture, are inevitably formed in their cutting processes, which dramatically lower the machined surface integrity [27][28][29][30][31]. Thus, how to modulate the differences in machining response between constitutive phases is critical for facilitating the machinability of composites.
Finally, various external energetic fields-assisted diamond cutting processes, such as ultrasonic vibration-assisted [32][33][34], thermal-assisted [35][36][37][38][39][40][41][42] and ion implantationassisted [43][44][45][46][47], have been increasingly employed to promote the ductile machinability of difficult-to-machine materials. While the effectiveness in applying the energetic field assistance for facilitating the critical DOC for brittle-to-ductile (BTD) transition has been well demonstrated, fundamental understanding of the interaction between workpiece material and energetic field is critical for tailoring the field configurations to optimize the energetic field-assisted cutting performance. Therefore, a thorough understanding of the cuttinginduced evolutions of microstructures and properties of workpiece materials, as well as their correlations with cutting responses, are crucial for enhancing the capability of ultraprecision diamond cutting.
Besides experimental research, various numerical simulation approaches, such as finite element (FE) simulation at the microscopic scale and molecular dynamics (MD) simulation at the nanoscale, have become more popular for its capability to provide dynamic insights into on-going diamond cutting processes of a varieties of materials, such as material deformation, chip formation, cutting force evolution and surface formation, etc [48][49][50]. Despite of the wide applications of different simulation methods utilized in the exploration of diamond cutting process, there are still issues or challenges that are needed to be addressed for better comparison of predicted results with experimental data. Firstly, the accurate constitutive law for describing internal microstructure-governed heterogeneous deformation of workpiece material and corresponding damage law for realizing chip formation are greatly desired for addressing the anisotropic cutting response of polycrystalline materials, which provide the base for the understanding and suppressing of machining anisotropy-induced deterioration of surface finish. Secondly, the configurations of individual phases in the physics-based simulation models of composite materials should be in line with their real microstructural characteristics, such as polygon particle shapes and random particle sizes in SiCp/Al, and unidirectional fiber arrangements in CFRP, which are crucial for comprehensively revealing the particle-tool/fiber-tool interactions in diamond cutting processes of composites. Thirdly, the comprehensive algorithm describing both the thermal and mechanical behaviors of tool-chip contact is critical for providing high predication accuracies of material flow, chip formation, residual stress, machined surface quality and cutting force. Finally, the integration of ultrasonic vibration, laser or ion fields into cutting models that mostly mimics experimental configuration without loss of physics is also important for revealing the tool-field-material interactions, as well as their impacts on the cutting processes.
In the present work, we present a compact review on the recent advances in advanced numerical simulations of diamond cutting of a variety of materials, which differs in properties, microstructures and constituents. The framework of this review is illustrated in figure 1. The review is constructed as follows. In section 2 for fundamental understanding of numerical simulation methods of diamond cutting process is presented. In section 3 for traditional diamond cutting without external energetic field-assistance, different modes operated in diamond cutting of metallic, hard brittle materials and composite materials, such as anisotropic dislocation slip, toolworkpiece interaction, phase transformation and cracking, are reviewed. In section 4 for external energetic field-assisted diamond cutting, the interaction of vibration, heat and ion fields with workpiece material and tools are presented. Finally, in sections 5 and 6 we briefly summarize the paper and present the outlooks, respectively.

Numerical simulation methods for diamond cutting process
At present, MD simulations at the nanoscale and FE simulations at the microscale are becoming more popular for studying the diamond cutting mechanisms of different kinds of materials. Figure 2 shows the schematic illustration of MD model and FE model of diamond cutting.
In MD simulations of diamond cutting, the nanoscale structural evolution of materials can be identified by deterministically integrating motions of individual atoms following the Newtonian second law of motion. Specifically, the motion states of all atoms can be accurately captured based on the chosen empirical potential describing the interatomic interactions in the simulated material system, thus providing dynamic atomic details of the ongoing cutting process. In the MD model of diamond cutting, the workpiece material is usually given as three zones, namely thermostat zone, boundary zone and Newtonian zone, respectively. In addition, other settings in MD simulations, such as model size, ensemble, boundary condition, atom type, lattice structure, etc, also play an important role in revealing the diamond cutting mechanisms of materials. MD simulations of diamond cutting are mainly realized by MD modeling, simulation and visualization software. With the development of computational technology, there are many commercial or open source MD simulation software (LAMMPS, IMD, etc) and visualization software (Ovito, VMD, Atomeye, etc) available. In 1980, MD simulation was utilized for the first time to investigate the cutting mechanism of monocrystalline copper [61]. Recently, the diamond cutting mechanisms of different materials including metallic materials [49,50], hard brittle materials [40][41][42]48], and composite materials [62][63][64] have been extensively investigated through MD simulations. However, MD simulations suffer greatly from the small length scale and time scale that are significantly lower than the natural processes being studied, thus qualitatively predicted results by MD simulations are mainly expected.
To bridge the gap between experiment and numerical simulation with respect to size effects in diamond cutting, quantitatively comparison of FE simulation results with experimental data is expectable. Both the length scale and time scale of FE simulations fulfill the requirements by corresponding experimental investigations. In FE simulations of diamond cutting, the specimen is given as finite number of meshes. Therefore, the mesh shape, mesh type, mesh division method and mesh density directly affect the computational accuracy and efficiency of the FE simulations. In particular, the material constitutive equation with the consideration of strain rate, yield strength and temperature, can quantitatively characterize the relationship between the strain and stress of the material, thus determining the reliability of FE simulation results. In the past decades, FE simulation methods are becoming more popular for studying the residual stress, cutting force, chip formation, machined surface quality and temperature distribution in diamond cutting of materials [22,26]. At present, Abaqus, Advantedge, Deform and Ansys are commonly used commercial FE simulation software for diamond cutting process. Furthermore, the scale bridging between MD simulation and FE simulation can be realized by utilizing the output values of material properties especially the interfaces by MD simulations as the input parameters in relevant FE simulations [65].   [35], Copyright (2022), with permission from Elsevier. Reprinted from [51], Copyright (2022), with permission from Elsevier. Reprinted from [52], Copyright (2020), with permission from Elsevier. Reprinted from [53], Copyright (2018), with permission from Elsevier. Reprinted from [54], Copyright (2009), with permission from Elsevier. Reprinted from [55], Copyright (2003), with permission from Elsevier. Reprinted from [56], Copyright (2015), with permission from Elsevier. Reprinted from [57], Copyright (2020), with permission from Elsevier. Reprinted from [58], Copyright (2016), with permission from Elsevier. Reprinted from [59], Copyright (2009), with permission from Elsevier. used engineering materials for their excellent mechanical properties and desirable thermal properties, diamond cutting technology is becoming more and more important for preparing ultra-smooth of non-ferrous metals. Since the surface formation and chip formation in metal cutting process at high strain rates and temperature are accompanied by severe plastic deformation of workpiece materials under high temperature and strain rates, as well as large strains, the physics-based plasticity model is critical for the accurately prediction of diamond cutting of non-ferrous metals. Specifically, both the deformation behavior and properties of workpiece material under diamond cutting with micro/nanoscale material removal thickness exhibit strong anisotropic characteristics, i.e. elastic modulus and dislocation density are different in different crystallographic orientations with different atomic packing spacings. Thus, the capability of accurately addressing the crystallographic anisotropy is central for the physics-based simulation model for diamond cutting.

Traditional diamond cutting
The governing mechanisms of plasticity of metals under diamond cutting, such as successive events of dislocation nucleation and subsequent slip, can be well described by MD simulations, in which the atomic arrangements in different crystallographic orientations can be directly constructed. Thus, MD simulation is capable of revealing the anisotropic cutting mechanisms of polycrystalline metals. Previous MD simulations of diamond cutting of polycrystalline metals demonstrated that the heterogeneous deformation behavior between adjacent grains leads to a strong anisotropic machining response near the GBs [66,67]. In particular, the anisotropic plastic deformation behavior between different grains leads to the formation of the pile-up in GB, i.e. GB steps, which plays an important role in determining the final machined surface morphology [68], as presented in figure 3(a).
In addition to MD simulations, FE simulations at the microscopic scale are also becoming more popular for investigating the cutting processes of a varieties of metal materials. However, the isotropic constitutive model is often used in previous FE simulations of diamond cutting without the consideration of anisotropic deformation behavior, thus amplifying the error in the quantitative comparison between simulation and experimental results. In diamond cutting process, the anisotropic deformation behavior between different grains strongly affects the microscopic deformation behavior of polycrystalline materials. For instance, the formation of GB steps accommodated by the heterogeneous deformation behavior between adjacent grains strongly affects the machined surface integrity in diamond cutting. Thus, the high precision constitutive law that can accurately describe the crystal orientation-dependent anisotropic deformation behavior of materials is greatly desired. Crystal plasticity FE (CPFE) simulations, in which the crystal plasticity model as the constitutive law is implemented through user subroutines in ABAQUS FE code, have been widely utilized to study the grain level anisotropic response of metallic materials under diamond cutting. The mechanisms of chip formation and surface formation in diamond cutting of single crystal metal have been widely investigated through CPFE simulations based on the activated dislocation slip system characteristics and the shear strain criterion [69,70]. However, previous CPFE simulations for diamond cutting of metallic materials mainly dealt with single crystal materials, and there is lack of CPFE simulations of diamond cutting of its polycrystal counterpart.
Recently, CPFE model based on the user subroutine (UMAT or VUMAT) has been employed to elucidate the anisotropic cutting mechanisms of polycrystalline copper in diamond cutting [71,72]. As shown in figures 3(b) and (c), the machining surface quality, chip profile and cutting force exhibit significant anisotropic characteristics due to the occurrence of GB migration and anisotropic dislocation activities [22]. While the single crystal copper possesses an uniform surface integrity accompanied with regular chip lamellar in a long range of scales, the polycrystalline copper exhibits a strong anisotropy between different crystal grains accompanied by the formation of complex chip lamellar [23].
With the fundamental understanding of anisotropic cutting behavior of polycrystalline metals, the suppressing strategy of anisotropic deformation through the optimization of both intrinsic parameters of workpiece microstructures and extrinsic cutting parameters can be revealed. Specifically, surface steps with a height of several nanometers may appear at the GBs of machined surface due to the anisotropic elastoplastic deformation of grains with different orientations, thus how to eliminate the anisotropic deformation of workpiece material is critical. It was found that a large GB orientation angle and rake angle or a small tool cutting edge radius is beneficial to suppress the GB surface step formation of polycrystalline gold [17] and copper [22] by diamond cutting. In addition, the high cutting speed result in low surface roughness of polycrystalline copper within a certain range of DOC due to the low material recovery associated with the short toolworkpiece contact time [73]. As shown in figure 3(d), within a certain misorientation angle, the GB step height increases with the increase of misorientation angle [74].

Tool-chip friction state and tool wear.
In addition to the crystallographic orientation-governed cutting anisotropy, the tool-chip friction state in diamond cutting of metallic materials can also be discovered by numerical simulations. The tool-chip friction state associated with thermomechanical coupling behavior strongly influences the cutting status with ultra-fine material removal volume. Specifically, the thermo-mechanical coupling tool-chip friction behavior strongly dependents on temperature fluctuation and cutting force in diamond cutting, and which is usually described by sticking-sliding friction model. The tool-chip friction state in diamond cutting of metallic materials can be effectively  [68], Copyright (2021), with permission from Elsevier); (b) Cross-sectional morphology of microstructural evolution of polycrystalline copper after diamond cutting characterized by transmission electron microscope (TEM) and CPFE simulation (Reprinted from [22], Copyright (2019), with permission from Elsevier); (c) Chip morphology of polycrystalline and single crystal copper after diamond cutting (Reprinted by permission from Springer Nature Customer Service Centre GmbH: Springer Nature [23], Copyright (2020)); (d) GB steps formation of polycrystalline copper with different misorientations after diamond cutting (Reprinted from [74], Copyright (2017), with permission from Elsevier). captured by the sticking-sliding friction criterion embedded in the FE model [75][76][77]. However, different friction states including sticking and sliding can strongly affect the variation of the simulated machining force and the chip morphology. Therefore, the friction model that can simultaneously consider the coexistence of sliding zone and sticking zone in cutting process of metals has been proposed, which demonstrates the high accuracy of predicted tool-chip contact length and machining force variation from experimental results [78]. Although the tool-chip friction leads to the generation of high temperature in the machining zone and excessive deformation of the workpiece material, most of the previous FE simulations based on sticking-sliding friction model have not consider the variation of the mechanical properties of the material with the cutting temperature. Thus, an updated sticking-sliding friction model incorporated with thermos-mechanical coupling algorithm is critical for comprehensively describing the toolchip friction in diamond cutting. Recently, the friction model has been improved with the comprehensive consideration of friction stress and corresponding thermo-mechanical coupling response to distinguish different friction states in terms of the relative velocity field [26], as shown in figure 4(a).
While diamond tool is a critical component influencing the cutting process, simulations on the cutting-induced structural change of diamond tool are also performed. One aspect is addressing the diamond tool wear mechanisms. Mechanical wear and chemical wear are the two primary wear mechanisms of diamond tool. Furthermore, chemical wear such as oxidation reaction and graphitization is more significant than mechanical wear especially for ferrous metals, because of the ultrahigh hardness of diamond material. As shown in figures 4(b) and (c), the initial graphitization of diamond and subsequent chemical reaction between carbon atoms and iron atoms dominate the wear mechanisms of pure iron under diamond cutting. It is found that the graphitization occurs simultaneously through an intermediate active state of atom groups rather than atom by atom. Furthermore, the cutting-induced graphitization of diamond tool is strongly affected by crystal orientation. Specifically, the diamond (100) surface has the strongest resistance to graphitization, followed by the diamond (111) surface, and the least resistant is the diamond (011) surface [79]. Similar chemical wear mechanisms have also been observed in diamond cutting of aluminum alloys [80]. The atomic vibration of ferrous metal is accompanied by a cutting-induced increase of the atomic distance. Subsequently, carbon atoms tend to diffuse into the lattice of ferrous metal due to the smaller carbon atom diameter compared to iron atoms, resulting into the bond formation between iron atom and carbon atom [81], as shown in figure 4(b).
The other aspect is suppressing tool wear by introducing micro-nano textures on diamond cutting tool. The presence of surface micro-textures on diamond tool can produce predominant functionalities compared with untextured tools [82][83][84][85][86]. In the field of diamond cutting, the retention and accessibility of lubricant can be improved by the reasonable micro-texture design on diamond tool surface, thus reducing the machining force, the cutting temperature and the tool-chip interface friction [87]. Although the service life of ceramic and cemented carbide tools can be prolonged by introducing micro-nano textures on tool surface, there are few related studies on SCD tool, mainly due to that the micro-nano textures on the surface of SCD tool are hardly to be prepared by traditional processing methods. Recently, femtosecond pulsed laser micromachining with low thermal effect, high resolution and high precision has been utilized to prepare the surface micro-nano textures on SCD surface [88][89][90][91], as shown in figure 4(d). As shown in figure 4(e), subsequent experimental results of diamond cutting of oxygen-free copper demonstrate that the anti-friction performance of diamond tool is greatly improved by introducing micro-textures on the diamond tool surface [91].
The advantages of diamond cutting utilizing textured diamond tools are analyzed from the nanoscale by MD simulations. MD models of diamond tool with different textures, such as arc-shaped texture tool and V-shaped texture tool, are established. It is found that as compared to untextured diamond tool, the diamond cutting by textured diamond tool causes lower machining force and cutting temperature, less compressive stress and hydrostatic stress [92]. Finally, by establishing a series of three-dimensional MD models of diamond cutting with textured single crystal diamond tool, the influences of texture shape, texture factor, texture width, texture depth and texture direction on material removal behavior are studied from the atomic level [93].

Ductile machinability of hard brittle materials
While hard brittle materials with hard and brittle characteristics have promising applications in high-performance components for their extraordinary chemical, physical and mechanical properties, their machinability under diamond cutting have also been discovered by numerical simulations for decades. In this section, numeric simulations of diamond cutting of different types of hard brittle materials, including the silicon (Si), germanium (Ge), gallium nitride (GaN), silicon carbide (SiC), gallium arsenide (GaAs), glass and tungsten carbide, are briefly reviewed.
Realizing the ductile-mode cutting is essentially needed to achieve ultra-smooth surface of hard brittle materials, which requires the fundamental understanding of the ductile and brittle deformation mechanisms. Previous studies have demonstrated that the dislocation slip and phase transformation are the two main modes of ductile deformation of hard brittle materials under diamond cutting [40][41][42]. In particular for the high pressure phase transformation, including the amorphization and the structural transformation from the original phase to another crystal phase, acts as one of the dominant mechanisms for the ductile deformation of hard brittle materials in diamond cutting. For instance, the phase transformation from the original crystal phase to the amorphous phase for Si, Ge, GaAs, SiC, GaN, tungsten carbide is the fundamental mechanism leading to their ductile deformation [94][95][96][97][98][99][100][101][102][103][104][105][106][107][108]. In addition, the phase transformation from diamond cubic structure to other structure phase is observed in diamond cutting of single crystal Ge and Si [98,108]. Unlike crystalline materials, the plastic deformation of silica glass residing in non-crystalline structure is dominated by rearrangement of the atomic structure and surface densification in diamond cutting processes [109,110].
The phase transformation involved in the diamond cutting of 6H-SiC is captured through MD simulations based on the coordination number distribution related to structure characteristics, as shown in figure 5(a). The coordination number for per atom can be identified by the post-processing visualization software OVITO [97]. Correspondingly, the amorphous structure is also found in the subsurface layer adjacent to the machined surface in diamond cutting experiment of 6H-SiC, as shown in figure 5(b). The brittle deformation behavior of hard brittle materials is dominated by crack initiation and propagation. The crack propagation along the cleavage planes and GBs dominates the brittle failure of hard brittle materials, respectively [41,111,112]. Figures 6(a) and (b) shows the MD simulation and experimental results of subsurface damage of single crystal 6H-SiC in diamond cutting at a DOC of 20 nm, respectively. The cleavage process occurs in the subsurface layer of the machined region, as shown in figure 6(b). Correspondingly, MD simulation results also capture the formation of microcracks and subsequent crack propagation, as shown in figure 6(a). A similar phenomenon is also observed in diamond cutting of hard brittle GaAs, in which cracks tend to form along the cleavage plane between the amorphous layer and the crystal [111].
In addition, rational selection of machining conditions can also achieve ductile mode cutting of hard brittle materials. One aspect is the material parameters associated with the microstructure evolution. Previous MD simulations for diamond cutting of hard brittle materials demonstrate that the machined surface integrity strongly depends on the microstructural characteristics of material, including cutting orientation, machined surface orientation and grain size, etc [113][114][115][116]. For instance, the ductile machinability of single crystal 6H-SiC under different cutting orientations at a DOC of 2 nm is investigated. Although for all cutting orientations the material operates in ductile-mode cutting with continuous chip formation and without crack initiation and propagation, the subsurface damage layer (SDL) depth, i.e. the distance between the lowest permanently displaced atoms and the material surface, varies for different cutting orientations [115], as shown in figure 7(a). Goel et al [117] found that single crystal silicon has a pronounced amorphization tendency than that of polycrystalline silicon, which signifies that GBs eases the material removal process, as shown in figure 7(b).
In addition to microstructural parameters of material, processing parameters, including cutting speed, DOC, tool feed rate, tool inclination angle, tool sharpness, tool nose radius, tool edge radius and tool rake angle also have a strong impact on the machined surface quality of hard brittle materials [118][119][120][121][122][123][124]. In particular for tool rake angle, previous studies demonstrate that the negative rake angle is beneficial for realizing ductile-mode cutting and inhibiting the formation of cracks in diamond cutting of hard brittle materials [120,123,125]. Specifically, as the tool rake angle decreases, the stress state of workpiece material can be transited from shearing to extrusion without removing any material in the end.

Synergetic deformation behavior of composite materials
While composite materials composed of no less than two distinct phases have desirable combined properties of constitutive phases, the dramatically different properties between matrix phase and dispersed phase results into the low machinability of either particulate composites or fibrous composites. Therefore, revealing the synergetic deformation mechanisms as well as their parameter dependence between individual phases under cutting action is crucial for promoting the machinability of composite materials by diamond cutting.
The physical model of composites that mostly address experimentally observed microstructural characteristics is prerequisite for achieving high prediction accuracy of composites cutting process by numeric simulations. Unlike the singlephase crystalline substance, the modeling of composite materials is more complicated due to the synergetic deformation behavior among individual phases. In particular, the presence of interface between particle/fiber and matrix strongly affects the machinability of composite materials. Single-phase hard brittle material is generally modeled only by one constitutive law describing the material deformation behavior and one damage law enabling chip formation. However, the material deformation behavior of different phases in composite material is described by different constitutive laws. In particular for the interface between adjacent phases, the mechanical properties of the matrix-particle/fiber interface have a strong impact on the machinability of composite materials. Specifically, different cutting parameters lead to different stress concentration characteristics developed near the matrix-particle/fiber interface, resulting in the stiffness degradation of the interface and subsequent interface debonding. However, in the most of previous FE simulations the particle/fiber-matrix interface is regarded as an ideal interface without the consideration of its mechanical deformation behavior, resulting in the description deviations of debonding behavior of matrix-particle/fiber interface. Furthermore, the particle shape is usually set as a regular polygon with uniform particle distribution in the most of previous FE simulations, which is inconsistent with the random particle shape and distribution characteristics observed in real composite materials.
As typical representative composite materials, the machinability of SiCp/Al and CFRP under diamond cutting has been widely studied by FE simulations [126][127][128][129]. More recently, in diamond cutting of SiCp/Al, different types of constitutive laws are introduced to capture the initiation of particle cracking, the post-failure behavior of the particle and the plastic deformation behavior of Al matrix [128]. In particular, cohesive zone model (CZM) governed by the traction-separation law with zero thickness cohesive elements is introduced to precisely describe the particle-matrix debonding behavior caused by excessive tensile and shear stress in diamond cutting. The high stress concentration from cutting edge extrusion can lead to particle breakage beneath machined surface. Although the breakage of the hard particles on the cutting path can lead to the formation of cavities, the subsequent plastic deformation of the matrix can cover the previously formed cavities, thereby leading to the realization of high machined surface quality, as shown in figure 8(a). Xu et al [129] found that the friction coefficient is inversely related to the normal force in diamond cutting of unidirectional CFRP. Furthermore, the macro-fracture and micro-fracture of unidirectional CFRP occurred at the SDL after the fiber is strongly deflected and the tool-fiber contact point, respectively [127].
The synergetic deformation behavior between individual phases of composites under cutting can be tailored by rational parameter selection. One aspect is the influence of particle size and distribution of particle reinforced Al matrix composites, as well as the fiber orientation of CFRP. For instance, when the tool acts on the top and middle of the particle, the particle usually breaks in the direction of stress concentration. When the tool acts on the bottom of the particle, it is usually pulled out from the matrix material [130]. Another aspect is the influence of processing parameters on the machined surface quality of composites. It is found that reducing the DOC or increasing the cutting speed can improve the machined surface quality of composites [131]. Furthermore, the tool rake angle has an important influence on the chip morphology of SiCp/Al in diamond cutting. The tool rake angle is inversely proportional to the jaggedness of the serrated chip, and the negative rake angle promotes chip breakage, as presented by FE simulations shown in figure 8(b). In addition, the use of a large number of cutting steps accompanied by a small DOC for material removal is beneficial for achieving the ultra-high surface quality of composites. Compared with the single-pass strategy, the multi-pass strategy is also beneficial for improving the machined surface quality of unidirectional CFRP and SiCp/Al in orthogonal cutting [132][133][134]. Specifically, under the condition of the same total DOC, the use of multi-pass cutting strategies can reduce the surface roughness and subsurface damage, as shown in figure 8(c).

Field-assisted diamond cutting
While the machinability of hard brittle materials is strongly relied on the operating ductile-mode cutting, increasing the critical DOC for the BTD transition is critical for promoting the machining capability of SPDT on hard brittle materials. Furthermore, the rapid severe wear of diamond tool also significantly lowers the processing accuracy of ferrous and hard brittle materials. In addition to conventional diamond cutting, recently field-assisted diamond cutting processes, such as ultrasonic vibration-assisted, thermal-assisted and ion implantation-assisted diamond cutting, have been successfully applied to increase the critical DOC and suppress tool wear in diamond cutting of difficult-to-machine materials.

Vibration-assisted diamond cutting
In the vibration-assisted diamond cutting process, onedimensional or two-dimensional ultrasonic frequency oscillations with micrometer magnitudes are superimposed to diamond tool, which transmits the continuous cutting to intermittent cutting that facilitates the separation of chip from rake face of cutting tool, as shown in figure 9. Consequently, both the cutting force, cutting heat and tool wear are lowered significantly in vibration-assisted diamond cutting from ordinary diamond cutting [136][137][138][139].
One aspect is the vibration-assisted diamond cutting of ferrous metals. In ultrasonic elliptical vibration assisted cutting (UEVAC) of materials, the maximum shear angle in UEVAC is normally larger than that in ordinary cutting (OC) [140]. Correspondingly, figure 10(a) illustrates the shear angle in UEVAC and OC of magnesium alloy from FE simulations. The applying of ultrasonic vibration causes intermittent contact between workpiece and tool, resulting in reduced time for interatomic chemical reactions between workpiece and tool, which in turn promotes the flow of cutting fluid and the reduction of friction [141]. Furthermore, the wear extent of diamond tool in UEVAC is much smaller than that in OC when the cutting distance is the same [142,143]. It should be noted that the investigation of wear mechanisms of diamond tool by FE simulation has been a long-standing challenge, since diamond tool is usually treated as rigid body without the consideration of its chemical wear in FE simulation. In addition to experimental methods, tool wear is usually investigated by means of MD simulations, in which atomic level chemical wear including oxidation reaction and graphitization can be captured, as discussed in section 3.1.2.
In particular for hard brittle materials, the intermittent contact between workpiece and tool suppresses the cracking and decreases cutting force, thus promoting the ductile machinability accompanied with increased critical DOC. The contact interaction between workpiece and diamond tool, as well as the thermo-pyrochemical reaction and corresponding tool wear, can be significantly decreased by applying UEVAC, thus promoting the machinability of tungsten carbide [144]. The small vibration frequency or high amplitude increases the material removal rate and reduces the stress concentration and subsurface damage thickness of single crystal silicon [145]. In one cutting cycle, the main material removal mechanism transits from extrusion deformation dominated by shear stress to shear deformation dominated by tensile stress [146]. Compared with OC, the maximum shear stress and hydrostatic stress are smaller in UEVAC [147], as shown in figure 10(b). Furthermore, the vibration amplitude ratio of the cutting direction to the DOC direction of 3.5 is suggested as a suitable selection for realizing the ultra-smooth surface of single crystal silicon [147].
Zhu et al [148] found that compared with OC, the phase transformation layer associated with subsurface damage becomes thinner by applying UEVAC than that by applying OC, thereby achieving better machined surface quality of single crystal silicon. The main material removal mechanisms of hard brittle 3C-SiC dominated by amorphization and crack propagation are significantly suppressed by applying UEVAC [149]. The influence of tool trajectory and the position of Si particles on the machining response of RB-SiC is also studied by FE simulation [150]. By applying UEVAC, the critical BTD transition depth for RB-SiC is increased by ten times than that of OC. In addition, the high vibration frequency, as well as the large amplitude ratio of the cutting direction to the DOC direction, are conducive to realizing the ductile-mode cutting of RB-SiC.

Thermal-assisted diamond cutting
Laser-assisted cutting (LAC) and heat treatment-assisted cutting are classified as thermally enhanced machining [38]. In LAC, an intensive localized heating is applied to the workpiece surface prior to cutting, the resulting decrease in the material strength caused by high temperature softening thus promote the machinability of the materials. The LAC can be divided into two types, relies in the configuration of laser source with respect to diamond tool. Specifically, one type is in-process heating LAC (In-LAC), in which the laser beam directly passes through the transparent diamond tool to irradiate the cutting edge; the other type is pre-heated LAC (Pre-LAC), in which the laser heat source irradiates the material surface in front of the tool rake face, as shown in figure 11 [151]. The Pre-LAC system is easily integrated and adjusted. Therefore, the material can be plastically removed more easily due to the thermal softening effect that occurs in the shear zone of the material. However, material removal efficiency is low in Pre-LAC due to the time delay between laser heating and chip formation. The high temperature on material surface induced by high laser power may introduce local yield, thermal cracks and large heat affected zone. Thus, Pre-LAC is not suitable for ultra-precision cutting of hard brittle materials accompanied by ultra-high hardness. Furthermore, most cutting tools used in Pre-LAC are cemented carbide or ceramic materials rather than SCD.
During In-LAC, the material below the cutting edge is heated by a focused laser beam with a spot diameter of micrometers through the transparent diamond tool, thus minimizing the heat affected zone and energy consumption with ideal machining accuracy [151,152]. Previous experimental studies have demonstrated that the ultra-smooth surface of hard brittle materials such as germanium, binderless tungsten carbide and silicon can be achieved with a significant increase in their critical DOCs by using In-LAC technique [153][154][155][156]. Recently, the cutting mechanisms of hard brittle materials in laser-assisted diamond cutting have been widely studied through simulation methods. In laser-assisted diamond cutting, different laser powers correspond to different heating temperatures, which can be derived based on the heat conduction theory [108]. Specifically, the heating temperature increases with increasing laser power. Therefore, in the MD simulation of laser-assisted diamond cutting, the whole workpiece is heated to a specific temperature that is dependent on the selected laser power.
The strength and hardness of material generally decreases with increasing temperature in laser-assisted diamond cutting [156]. Thus, the coupling of laser heat source with cutting model considering the temperature-dependent material properties is crucial for the accurately prediction of the machining response of materials in laser-assisted diamond cutting process. In laser-assisted diamond cutting, the thermal softening effect leads to the improved ductile deformation behavior of material accompanied by the suppressed amorphization and cracking events related to subsurface damage, as well as the enhanced dislocation mobility and formation of stacking fault related to plastic deformation, which in turn result into a reduction in the material hardness and cutting force [108,157]. Furthermore, the recrystallization from the amorphous phase to the metastable crystalline phase occurs due to the heat annealing effect induced by laser irradiation, and the dislocation activity is significantly increased by applying In-LAC. Thus, the ductile machinability of hard brittle materials is significantly promoted with the increased critical DOC under In-LAC.
While the thermomechanical properties of workpiece materials are a key factor in their diamond cutting, heat treatment before cutting is expected to improve the machinability of workpiece materials, especially for hard brittle materials. The hardness and strength of material can be reduced by thermal softening effect in heating pretreatment, which exhibits a cutting mechanism similar to that of laser-assisted diamond cutting. In heat treatment-assisted diamond cutting, the material surface is heated to a predetermined temperature prior to cutting, which can be realized by a resistor heating module or LAC apparatus. So, the fundamental mechanisms behind the heat treatment-assisted diamond cutting are similar to the aforementioned laser-assisted diamond cutting.
The lubrication and recrystallization process of the amorphous layer are the two important mechanisms for inhibiting the subsurface damage at heating temperature in nano-cutting of silicon. As shown in figure 12(a), the SDL thickness in low temperature is larger than that in high temperature. For single crystal 3C-SiC, the specific cutting energy of different orientated surfaces at 2000 K is 33%-43% lower than that at 300 K [42]. Dislocation nucleation and stacking fault formation are more pronounced in high temperature than that in low temperature, as shown in figure 12(b). In addition, the atomic tool wear can be significantly suppressed during high temperature cutting [158]. Recently, Zhao et al studied the heat treatmentassisted diamond cutting mechanisms of polycrystalline 3C-SiC by utilizing multiscale modeling and simulation, in which the temperature-dependent properties of GB are derived from MD simulation results of CZM-based tensile/shear loading, and further are used as the input parameters of FE model [41]. In particular, the FE simulation results demonstrate that with the increment of pretreatment temperature, the material removal mode transits from brittle-mode cutting related with inter-granular fracture to ductile-mode cutting related with intra-granular fracture, as shown in figure 12(c).

Ion implantation-assisted diamond cutting
Recently, ion implantation-assisted diamond cutting has become a popular machining technique due to its capability to achieve ultra-smooth machined surface of hard brittle materials [159]. Figure 13 shows the illustration of ion implantation-assisted diamond cutting of hard brittle materials. Ions are accelerated by electrostatic accelerator and then implanted into the workpiece material by ion implanter. The modified layer dominated by amorphization in ion implantation process greatly lower the brittleness and hardness of the material. The ductile machinability of hard and brittle materials is significantly improved due to the collision of high energy ions, which has been experimentally confirmed [44][45][46].
Recently, the damage evolution and material removal mechanisms in ion implantation-assisted cutting have been   widely studied by MD simulation. Amorphous holes appear accompanied with localized stress during the ion implantation process. Figure 14(a) shows the dynamic evolution of lattice damage and hydrostatic stress of single crystal silicon during ion implantation process. Lattice damage absorbs the shear strain energy in nano-cutting process, and the easy plastic yield results in pronounced shear bands generated in the machining process [161][162][163]. Therefore, the material ductility is enhanced and the critical DOC associated with the BTD transition is increased. The reduction in machining force indicates a reduction in machining energy and tool wear.
The subsurface damage, shear stress and hydrostatic stress of 6H-SiC in nano-cutting can be effectively reduced by applying ion implantation, which is conducive to activating dislocation movement and promoting ductile-mode cutting [47], as shown in figure 14(b). Material modification accompanied by the formation of amorphous layer occurs during ion implantation process, which facilitates the shear slip within the material in subsequent cutting. In addition, the temperature rising rate, minimum DOC and material removal rate can also be increased by ion implantation [164]. Furthermore, the increase of ion implantation dose is proposed to effectively form a uniform modified thick layer with promoted machinability and suppressed tool wear [165]. Even in brittle-mode cutting, crack propagation is blocked at the interface between crystalline layer and amorphous layer, and there is no subsurface damage extending into the crystalline layer of single crystal 6H-SiC.

Conclusion
In the present work, we present a brief review on the application of numerical simulations in addressing the impact of properties and microstructures of workpiece materials on the diamond cutting mechanisms of different types of workpiece materials, such as metallic materials, hard brittle materials and composite materials. In addition, the effect of applying external energy fields to the diamond cutting of difficult-tomachine materials is also discussed. The conclusions are summarized as follows: (a) The anisotropic deformation behavior among single crystal grains in diamond cutting of polycrystalline materials can be well described at the microscopic scale by CPFE simulation, which provides bases for the fundamental understanding of formation mechanisms as well as suppressing strategies of GB surface steps on machined surface. (b) The variation of tool-chip friction state with cutting temperature can be effectively captured by the thermomechanical coupling sticking-sliding friction criterion embedded in the FE model. In addition, the diamond tool wear can be suppressed by introducing textures on cutting tool. (c) The fundamental understanding of phase transformation and cracking events through simulations is crucial for revealing the BTD transition mechanisms of hard brittle materials, thus enabling the rational selection of optimized parameters for enhanced ductile machinability. (d) The physics-based numerical model is critical for providing predicted results that are in line with experimental data for composites materials. The real microstructural characteristics of reinforced phase as well as the proper treatment of reinforced phase-matrix interfaces are essentially needed to accurately represent the tool-phases interactions in numerical simulations of diamond cutting of composites. (e) The configuration of external fields (vibration field, thermal field and ion implantation field) and their interactions with workpiece materials without loss of physics is critical for revealing the mechanisms of field-assisted diamond cutting of difficult-to-machine materials with enhanced machinability by numerical simulations.

Future research perspectives
Future research on the numerical simulations of materialsoriented diamond cutting could be further recommended from below aspects, mainly aiming for increasing the prediction accuracy of simulation results for advanced structured materials compared to experimental data.
(a) Development of empirical potentials capable of accurately describing the deformation behavior of newly emerged materials.
The development of novel advanced structured materials with superior properties also brings challenge for their numerical simulation. In particular for MD simulations, the accuracy of MD simulation results strongly depends on the used interatomic empirical potential, thus novel empirical potential should be developed. Furthermore, although there are many different types of potentials for the same material, there are certain deviations in the description of material deformation behavior by different potentials. For instance, the melting temperatures of single crystal Si and Ge are generally overestimated by the commonly used Tersoff potential [61,62], which may lead to bias in addressing the thermodynamic issues in MD simulations of diamond cutting. Therefore, the development of more reliable potentials for specific material in diamond cutting process is one of the main future goals.

(b) Development of high precision physics-based FE model
Developing high precision physics-based FE model especially for composite materials can be pushed through the detailed microstructures characterization of composite materials by the computerized tomography scan technique, which combines a series of x-ray images extracted from different angles around material bulk, and can be further utilized as the input parameters into the FE model base on the representative volume element algorithm. In such a way, the highly accurate physics-based simulation model of composite materials consistent with experimentally observed microstructural characteristics, such as the polygon particle shape, random particle size and unidirectional fiber arrangement, can be constructed with high prediction accuracy of numerical simulation.
(c) Development of coupled FE model of CPFE and CZM Developing coupled FE model consists of CPFE model and CZM is crucial to describe both the dislocation slip-dominated inhomogeneous plastic deformation and brittle cracking for hard brittle materials machining. Specifically, CPFE model can describe the anisotropic ductile deformation behavior of workpiece material associated with dislocation slip with DOC down to nanometer range, while CZM can describe the cracking response including different types of crack forms such as median crack, radial crack as well as half-penny crack, with DOC above the critical value for BTD.
(d) Development of thermo-mechanical coupling algorithm for interface Developing thermo-mechanical coupling algorithm particularly for interface governed by the traction-separation law, such as the capability of describing degradation of interface properties with increasing temperature, is crucial for predicting the machining response of materials in thermal-assisted cutting. For instance, the temperature-dependent properties of GB of polycrystalline materials can be derived from MD simulation results of CZM-based tensile/shear loading, and further used as the input parameters of subsequent FE model of thermal-assisted diamond cutting.