Evaluation of surface and subsurface damages for diamond turning of ZnSe crystal.

With single-point diamond turning (SPDT), a series of samples are processed under different cutting parameters. The brittle-ductile transition depth of ZnSe crystal is obtained, and the damages of the samples are measured from surface and subsurface damage depths as well as damage density. The effects of cutting parameters on the damages are investigated quantitatively. The results show the cutting depth has a minor while the feed has a major effects on the damages. Also, the interaction effect between feed and cutting depth is very small for surface damage depth or damage density, while it is large for subsurface damage depth. Based on the indentation mechanics and the kinetic characteristics of SPDT, a model is proposed to evaluate the surface and subsurface damage depths of ZnSe crystal by cutting parameters. The model has an average relative error less than 15.0%, which could be further used to obtain the depth and the removal characteristics of cracks in shoulder region.


Introduction
Optical components from ZnSe crystal are utilized primarily in near infrared (NIR) and infrared (IR) imaging system, including IR lasers, night vision systems, radiometers, and IR pyrometers [1,2]. Single-point diamond turning (SPDT) technique is often used to prototype these optical components, which should have submicrometer form accuracy and nanometers surface finish. However, due to the high brittleness and low hardness of ZnSe crystal, the unwanted surface and subsurface damages, such as fractures, pits and micro cracks, may remain in the final machined surface [3]. These damages will increase the light absorption and scattering, thereby introducing a detrimental effect on the performance of the optical system [4]. Especially in the field of high-power lasers, these defects would cause localized heating and even an explosion [5]. Therefore, it is essential to investigate the evaluation and mitigation methods of the surface and subsurface damages in ZnSe crystal in SPDT process.
To evaluate the damage in the brittle material processed by SPDT, many methods have been proposed. Nakamura et al. [6] used the scanning force microscope and scanning laser microscope to evaluate the surface and subsurface cracks in nano-scale machined silicon. Pizani et al. [7] utilized the Raman scattering to characterize the subsurface damage in silicon processed by SPDT. Yan et al. [8] made use of the cross-sectional transmission electron microscopy to observe the subsurface damage of single crystalline silicon in microcutting. Lai et al. [9] used the Raman spectroscopy to investigate the subsurface deformation of monocrystalline germanium after SPDT. To reveal the damage mechanism of Si by single point diamond grinding, Zhang et al. [10] utilized the scanning electron microscope, X-ray photoelectron spectroscopy, Raman spectroscopy, energy dispersive spectroscopy and X-ray diffraction (XRD) to examine the machined Si surface. They also characterized the surface damage (hard particle dislodgement, micro-cracks, and grinding grooves) of the machined TN85 cermet by a scanning electron microscope in backscattering mode [11]. Guo et al. [12] observed the subsurface damage of ground TiC-based cermet hemisphere couples by focused ion beam. These measurement methods could characterize the subsurface damage qualitatively, but the experimental equipment and instruments are generally expensive. In addition to direct measurement, many theoretical models were established to evaluate the damage. Blackley et al. [13] proposed a phenomenological model of brittle-ductile transition to quantitatively determine median crack depth. Zong et al. [14] and Huang et al. [15] used the Vickers indentation model to calculate the equivalent length of median crack in diamond turning process. But these models could be only used to calculate the median crack depth at the brittle-ductile transition position. Yu et al. [16] presented a damaged region analysis method to determine the subsurface damage depth in fast tool servo diamond turning process, but the method was preferable to the micro-structured surface with sinusoidal wave along radial direction. Therefore, effective method for evaluating the surface and subsurface damages in the brittle material processed by SPDT is still worthy of study.
The surface and subsurface damages could be mitigated by ductile mode cutting and partial ductile mode cutting. In ductile mode cutting, the material is machined by plastic flow instead of brittle fracture deriving a damage-free surface [17]. It is accepted that all brittle materials would experience a transition from ductile mode to brittle mode with the increase of undeformed chip thickness in SPDT process. When the undeformed chip thickness is lower than the brittle-ductile transition depth of the material, the ductile mode cutting will be achieved. Tie et al. [18] used the spiral turning method to achieve the super smooth surface of KDP crystal by ductile mode cutting. The surface roughness could reach about 1.5 nm. Mao et al. [19] used diamond turning with a slow tool servo to machine the spherical concave microlens array on single-crystal silicon (001) by ductile mode cutting. The microlens array had a form error of ∼300 nm peak-valley. Although the ductile mode cutting has been successfully utilized to machine various brittle materials, its processing efficiency is very poor, it is not widely adopted in the SPDT process. Therefore, the partial ductile mode cutting, with higher production efficiency than ductile mode cutting, was proposed recently. In partial ductile mode cutting, the brittle fractures are generated but they do not penetrate into the final cut surface. Zong et al. [14] demonstrated that the partial ductile mode cutting could be achieved when the median crack at the brittle-ductile transition position of shoulder region did not penetrate into the cut surface. They carried out the partial ductile mode cutting experiments on ZnS crystal with the oblique diamond cutting. Wang et al. [20] believed that all cracks (i.e., median and lateral cracks) in shoulder region could not penetrate into the cut surface in partial ductile mode cutting. They processed the KDP crystal by partial ductile mode cutting. When the crack depth was smaller than or equal to the distance between the crack origin and the cut surface, the crack did not penetrate into the cut surface [21]. Therefore, for the partial ductile mode cutting, it is essential to predict the depths of all median and lateral cracks in shoulder region during the SPDT process, which has never been studied before.
In the machining process of brittle material, brittle-ductile transition depth is a very important parameter which determines the cutting mode directly. In SPDT, brittle-ductile transition depth has the same meaning as critical undeformed chip thickness. There are many factors influencing the brittle-ductile transition depth. Blake et al. and Nakasuji et al. [22,23] found that, for germanium and silicon, the tool rake angle and clearance angle had great effects on the brittle-ductile transition depth but cutting speed showed negligible effect. Meanwhile, the brittle-ductile transition depth was influenced slightly by nose radius for germanium while it was sensitive to the nose radius for silicon. What's more, the brittle-ductile transition depth varied with cutting direction for anisotropic materials [24,25]. For example, An et al. [26] found that the brittle-ductile transition depth of KDP crystal on (001) plane varied between 160 nm and 220 nm. Although a great deal of research on brittle-ductile transition depth has been conducted, the brittle-ductile transition depth of ZnSe crystal has never been studied.
In this paper, using XRD and plunge cutting experiments, the structural property and brittleductile transition depth of ZnSe crystal are analyzed, respectively. With the SPDT, a series of surfaces are processed under different cutting parameters designed by Latin hypercube design, and their surface and subsurface damage depths, as well as damage density, are measured. The response surface model and Pareto charts are constructed to analyze the effect of cutting parameters on surface and subsurface damage depths as well as damage density. Based on the indentation mechanics and the kinetic characteristics of SPDT, a model is established to evaluate the surface and subsurface damage depths of cutting samples. Furthermore, the depth and removal characteristics of cracks in shoulder region are discussed. This study would be valuable for surface and subsurface damage evaluations, as well as the design and manufacturing of ZnSe-based optical components. Figure 1(a) shows the experimental setup of SPDT process. The indentation mechanics could be applied to simulate ultra-precision cutting, grinding, or polishing, and all machining tools involved can be categorized as sharp indenter because its edge radius or grit size is extremely small [17]. Figure 1(b) shows the configuration of median and lateral cracks in a typical single-edge orthogonal process. It can be observed that a deformed zone is generated below the workpiece surface, and the lateral and median cracks are initiated from the bottom of the deformed zone. The lateral crack approximately spreads parallel to the workpiece surface [27]. Figure 1(c) shows the cutting geometry viewed in the plane normal to the cutting direction. The origin of coordinate system xO 0 y in Fig. 1(c) is the center of the trajectory of the 0th cut. The x coordinate axis is along the feed direction, and the y coordinate axis is perpendicular to the feed direction. It can be found that each cut is operated on the deformed surface generated by the last cut except for the first one. Every point between adjacent two cuts has different undeformed chip thickness h i s, which increase from zero to a large value. For example, the h i between the 0th cut and the 1st cut varies from zero at point J to a maximum value at point E. The maximum undeformed chip thickness h m , is the length of segment EF, which is expressed as

Theoretical model
where f, R and a p are the feed, tool nose radius, and cutting depth, respectively [28]. Figure 1(c) also indicates that the critical undeformed chip thickness h c exists at the brittleductile transition position. When h i < h c , the materials would be removed in ductile mode; when h i ≥ h c , the crack propagation would start to prevail. In this paper, the h c of ZnSe crystal is obtained by plunge cutting experiments. The depth C li and width C wi of lateral crack, and the depth of median crack C mi at each h i are expressed as where k 1 and k 2 are dimensionless constants which depend on the indenter system. For Vicker's indenter, k 1 = 0.226 and k 2 ≈ 7 [29,30]; α is the half apex angle of tool; E, H v , K c and ν are the elastic modulus, harness, fracture toughness and Poisson's ratio of the workpiece, respectively; F i is the normal load. In SPDT process, the angle α can be expressed as where β and γ e are the clearance angle and the effective rake angle of cutting tool, respectively [31]. The effective rake angle γ e is given by where r and γ are the edge radius and the nominal rake angle of cutting tool, respectively [32]. The normal cutting force at each h i in brittle material removal mode can be written as [33]: Figure 1(c) shows the transition angles ϕ i and θ i at each h i . The relationship between them is when θ i ∈ [asin(0.5f /R), asin(f /R)], the origin point D 1i of the crack caused by the 1st cut occurs on the left of line O 1 Q and ϕ i is a negative value; when θ i ∈ [asin(f /R), acos(1−a p /R)], the crack origin point D 1i occurs on the right of line O 1 Q and ϕ i is a positive value. As shown in Fig. 1(c), assuming point L ni (n = 1, 2, 3. . . ) is the intersection point of the nth cut-induced median and lateral cracks at each h i , the x and y coordinates of point L ni in coordinate system xO 0 y are respectively , the x and y coordinates of point M ni at each h i are respectively Assuming point N ni is the endpoint of the nth cut-induced median crack at each h i , the x and y coordinates of point N ni are respectively The median crack produces subsurface damage and the lateral crack determines the removal of material in the form of hemi-spherical packets from the bulk material, i.e., surface damage. The penetration depths of lateral and median cracks into the cut surface are the surface and subsurface damage depths, respectively. Assuming SD ni and SSD ni represent the nth cut-induced surface and subsurface damage depths at each h i respectively, SD ni and SSD ni can be expressed as Because the distribution of cracks induced by different cuts are the same, SD 1i = SD ni and SSD 1i = SSD ni . Assuming SD nm and SSD nm represent the maximum SD ni and SSD ni within the range of h i from h c to h m respectively, and assuming SD m and SSD m represent the surface and subsurface damage depths of the final cut surface respectively, then SD m = SD nm = SD 1m and SSD m = SSD nm = SSD 1m . To obtain θ i and then evaluate the surface and subsurface damages, an auxiliary line O 1 H perpendicular to edge O 0 C 1i of triangle ∆O 0 O 1 C 1i is added, as shown in Fig. 1 Using Eqs. (1)- (19), parameters θ i and ϕ i , SD ni and SSD ni , SD m (or SD nm ) and SSD m (or SSD nm ) can be calculated by the material properties of workpiece, tool geometries and cutting parameters. In other words, the depths of all cracks in shoulder region, and the surface and subsurface damage depths of the final cut surface could be evaluated based on Eqs. (1)- (19). The calculation procedure is shown in Fig. 2 and described as follows: (1) according to the cutting parameters and tool geometries, h m is calculated by Eq. (1) firstly; (2) dividing the range of 0 to h m into enough equal intervals, each corresponding h i could be obtained; (3) the transition angles θ i and ϕ i at each h i could be calculated by Eqs. (19) and (8) respectively, and the half apex angle α could be calculated by Eqs. (5) and (6); (4) according to the material properties of workpiece, the normal load F i , C li , C wi and C mi at each h i could be obtained by Eq. (7) and Eqs.
(2)-(4), respectively; (5) the x and y coordinates of points L ni (x Lni , y Lni ), M ni (x Mni , y Mni ) and N ni (x Nni , y Nni ) could be obtained by Eqs. (9)-(16); (6) finally, the SD ni and SSD ni at each h i could be calculated by Eq. (17) and Eq. (18) respectively, then the SD m (or SD nm ) and SSD m (or SSD nm ) could be obtained.

Response surface model
Pareto charts are usually utilized to investigate the effects of input parameters (sample points in the design space) on the response, which could be obtained based on response surface model. Using the quadratic model, the response surface model could be established by where A is the response; u is the amount of input parameters; x 1 , x 2 , x 3 , . . . x u are the input parameters; is the fitting error; a 0 is constant term; a i and a ii are the linear and quadratic effects of x i , respectively; a ij is the linear-by-linear interaction between x i and x j . The coefficients (i.e., a i , a ii , a ij ) would be fitted out using the least square method based on several groups of known input parameters and responses. Then, contribution rate (CR) of each input parameter on the response could be obtained by where uu = (u + 1)(u + 2)/2−1. In this paper, the input parameters are selected by optimal and random Latin hypercube designs, two typical designs of experiment [34,35].

Cutting experiments and measurement
Before cutting experiments, XRD analysis was carried out to obtain the structural property of ZnSe crystal (Φ25 ×1 mm) without preexisting defects. Table 1 shows the mean material properties of polycrystalline ZnSe crystal. Then a series of cutting experiments were carried out using the four-axis SPDT machine Moore Nanotech 350FG. The cutting experiments were divided into two parts. In the first part, the brittle-ductile transition depth of ZnSe crystal was obtained by plunge cutting experiments. The length of each groove is larger than 1000 µm, and the cutting depth of each groove increases continuously from 0 µm to ∼2 µm. Five cutting tools (No. 23031, 50925, 24555, 22192 and 24557), five cutting speeds (50 mm/min, 100 mm/min, 200 mm/min, 400 mm/min, and 800 mm/min), and five cutting directions (0°, 45°, 90°, 135°, and 180°) were considered to investigate the brittle-ductile transition depth. Table 2 shows the geometric parameters of five cutting tools. In the second part, two groups of surfaces were cut under different cutting parameters (cutting speed: 800 mm/min; feed f : 0.5 to 5.0 µm/rev; and cutting depth a p : 0.5 to 5.0 µm). Each group has sixteen small surfaces, where eight surfaces were cut by tool 22192 and the other eight by tool 24557, respectively. The cutting parameters for the first group and the second group were determined by optimal and random Latin hypercube designs respectively with u = 2 and w = 8, as shown in Table 3. Figure 3 shows the schematic diagram of cutting experiments and approximate measuring positions (the left edge, the center and the right edge) for XRD analysis. After cutting experiments, the ZYGO optical profiler was used to observe and measure the morphologies of each groove and each small surface. Five random measurements of SD m were carried out for each small surface, and their average value is regarded as the final SD m . The SSD m was measured by chemical etching. Each small surface was etched several times for 10s each using HNO 3 solutions (6 mol/L) [36]. Ten random measurements of surface roughness (p -v value) were carried out for each small surface after each etching, and the maximum surface roughness during the etching was regarded as the SSD m [37]. Each groove and each small surface were cleaned with alcohol and acetone before each measurement.

Structural property of ZnSe
For crystal material, the brittle-ductile transition depth greatly depends on the crystal orientation and cutting direction. This may result in that the brittle fractures occur in some orientations while the ductile surface is generated in other orientations. Polycrystalline materials are usually treated as a series of single crystals with random crystal orientations. Figure 4 shows the XRD spectra of ZnSe crystal under different measuring positions, indicating the crystal is cubic sphalerite structure with three strong peaks of (111), (220) and (311). The grain size perpendicular to (hkl) crystal plane can be calculated according to the Scherrer equation. The grain size of ZnSe crystal is about 288.2 nm by averaging the grain sizes under the three strong peaks and different measuring positions. The range of cutting depth is usually 0.5 to 5.0 µm, larger than the grain size. Therefore, the main brittle removal of finish cutting is the grain removal, and the crack would be generated in both radial and horizon directions [39].

Brittle-ductile transition depth of ZnSe
Plunge cutting experiment is the most direct method to measure the brittle-ductile transition depth by observing the transition of the cutting mode from ductile to brittle [40]. Figure 5 shows the surface morphology and profile of the groove cut by tool 22192 with cutting speed 100 mm/min and cutting direction 135°. It can be found from Fig. 5(a) that, with the increase of cutting depth along the cutting direction, the groove gradually broadens and becomes deeper, and the density of micro cracks gradually increases. The length and the maximum width of the groove are about 1200 µm and 2 µm, respectively. It can be found from Fig. 5(b) that there is a significant difference between the ductile and brittle regions, i.e., a clear brittle-ductile transition boundary can be identified on the groove surface. The surface in ductile region is very smooth while the surface in brittle region is very coarse. It is interesting from Fig. 5(a) that the ductile region does not transit into an entirely brittle region while it goes into the partially brittle region. This is because the undeformed chip area is a part of the circle with the radius equal to the tool nose radius. In this area, the maximum undeformed chip thickness occurs in the middle of the groove, and the undeformed chip thickness decreases toward the both sides of the groove. It may finally reduce to zero near both edges of the groove. Therefore, when the first time the undeformed chip thickness approaches the brittle-ductile transition depth during the plunge cutting process, the brittle fracture would take place in the middle of the groove while the ductile surface would still exist in the area towards the edges of the groove. In addition, there is a critical time (T c ) when the brittle fracture starts to occur, and there is a time (T b ) when the cutting tool starts to detach from workpiece, as shown in Figs. 5(a) and 5(b). When the cutting speed is 100 mm/min, T c = 0.04s and T b = 0.58s. Figure 5(b) also shows the groove profile and initial surface profile. The groove profile is located in the middle of the groove, and the initial surface profile is parallel to the groove profile. Each groove profile could be divided into three parts, i.e., the uncut profile, the smooth profile corresponding to ductile region, and the rugged profile corresponding to brittle region. Catastrophic shifts usually start to appear in the rugged profile and the uncut profile of the groove almost coincides with the initial surface profile. As shown in Fig. 5(b), the brittle-ductile transition depth h c can be measured as the distance between the profiles of initial surface and the groove at the point where the smooth groove profile starts to change to the rugged profile. Figure 5(c) shows the brittle-ductile transition depths under different cutting tools, which are obtained by averaging the brittle-ductile transition depths under different cutting directions. Incertitude is given at one standard deviation. Because the intrinsic heat generated by cutting process under the cutting speed of 50 mm/min to 800 mm/min is insufficient to cause any noticeable material softening, the effect of cutting speed on brittle-ductile transition depth can be ignored. Figure 5(c) shows that the brittle-ductile transition depths under tools 23031 and 24557 are slightly larger than those under tools 50925 and 24555, respectively. This indicates that the higher the nose radius, the larger the brittle-ductile transition depth. Figure 5(c) also shows that the brittle-ductile transition depths under tools 24555 and 24557 are larger than those under other three tools, indicating that the negative rake angle is desirable. The negative rake angle may have an effect of achieving hydrostatic stress field for avoiding brittle fracture. Moreover, Fig. 5(c) shows the measured minimum h c under different cutting tools.

Effect of cutting parameters on surface and subsurface damages
Figure 6(a) shows the morphologies of surface 4 cut by tool 22192 before and after etching for 10s. No fracture can be observed before etching, while the subsurface damage is exposed after etching. This indicates that it is possible that subsurface damage occurs without any indication of surface damage, which may be due to the high-pressure phase phenomena. Figures 6(b) and 6(c) show the morphologies of surface 6 cut by tools 22192 and 24557, respectively. It can be seen that the surface damages are distributed randomly, which may be attributed to the structural property of ZnSe crystal. Meanwhile, less surface damage is produced under tool 22192 compared with that under tool 24557. The damage density (DD) and SD m are utilized to characterize the surface damage. The damage density is defined as the ratio of the area of damages to that of the image. The image with damages is processed by the procedures such as gray scale processing, binary operation, edge extraction, regional division, etc. Then the damage density can be calculated by counting the fraction of dark pixels in the image. Figure 6(d) shows the edge extraction of damage based on Fig. 6(c). The SD m is measured by ZYGO optical profiler directly, as shown in Fig. 6(e). The SSD m is obtained by chemical etching. Figure 6(f) displays the surface roughness (SR) evolutions of the first group of surfaces during the chemical etching. The SR increases firstly and then reaches a plateau with the maximum SR equal to the SSD m [37]. Table 3 shows the measured SD m , damage density and SSD m for the first group of surfaces. Measured SD m ≈ 0 or DD ≈ 0 means there is no damage. Measured SSD m ≈ 0 means there is no obvious change of SR with the etching time. For example, although Fig. 6(a) shows the exposed subsurface damage after etching, the measured SSD m ≈ 0 because of no obvious change of SR with the etching time, as shown in Fig. 6(f). In other words, SSD m is very small at this point. The Pareto charts are constructed to investigate the effect of cutting parameters on SD m , DD and SSD m , as shown in Fig. 6(g). All coefficients in Eq. (20) are given in Table 4. It can be seen that the contribution rate (CR) of feed f is much larger than that of cutting depth a p for SD m or DD or SSD m . This indicates that the cutting depth has a minor effect on the surface and subsurface damages. It can be also seen that the interaction effect between feed f and cutting depth a p on SD m or DD is very weak, but it increases obviously on SSD m . This indicates that the interaction effect should be emphasized for subsurface damage.

Comparison of calculated and measured surface and subsurface damage depths
The surface damage depth SD m and subsurface damage SSD m can be calculated by the material properties of workpiece, tool geometries, cutting parameters using Eqs. (1)- (19). For ZnSe crystal with anisotropic properties along different crystal orientations and cutting directions, its mean material properties are used. Figures 7(a) and 7(c) show the calculated and measured SD m s and SSD m s for the first group of surfaces. It can be found that the calculated and measured SD m s and SSD m s have the same change trend with the surface number (i.e., cutting parameters). However, the calculated SD m is much smaller than the measured one, and the calculated SSD m is much larger than the measured one. Some calculated SD m s or SSD m s are zero, which means that the cutting may be ductile mode cutting or partial ductile mode cutting. The indentation model could be used to accurately determine the crack sizes with error below 5% [31]. However, why is there a large difference between the calculated and measured values? The main reason is that the constants k 1 = 0.226 and k 2 = 7 in Eqs.
(2)-(4) are more applicable to typical Vicker's indenter (like a pyramid with effective angle 136°), which has a certain difference with the cutting tool [41]. Therefore, the values of k 1 and k 2 should be adjusted to match the SPDT process. The measured SD m s and SSD m s for the first group of surfaces are utilized to fit out the constants k 1 and k 2 using the least square method. Figure 7(a) shows the calculated SD m s after adjustment of k 1 = 0.378 for tool 22192 and k 1 = 0.327 for tool 24557. Defining the relative error between the experimental and calculated values is the maximum relative errors between the experimental and calculated values are 5.9% and 12.2% for tools 22192 and 24557, respectively. .4%, respectively. The average relative errors are smaller than 15%, indicating that the calculation after adjustment could be used to effectively evaluate the SD m and SSD m . Because the optimal Latin hypercube design has a great space filling capacity, and the random Latin hypercube design has a certain random characteristic, the above evaluation would have strong universality, especially for cutting parameters f = 0.5-5.0 µm/rev and a p = 0.5-5.0 µm. Tools 22192 and 24557 are the representatives of those tools with zero and negative rake angles, respectively. Therefore, the above method would also provide a reference for other cutting tools. The cutting edge is considered to be perfect without defects in above evaluation, but actually the tool wear usually exists. Moreover, the surface of ZnSe crystal behaviors anisotropy. These are two reasons for the difference between calculated and measured values. Therefore, the future research should consider the tool wear and crystal orientation.

Depth prediction of cracks in shoulder region
With the adjustment of k 1 and k 2 , the SD m (or SD 1m ) and SSD m (or SSD 1m ) could be evaluated using Eqs. It can be found that no surface damage occurs when feed f ≤ 1.5 µm/rev, while no subsurface damage occurs when feed f ≤ 1.0 µm/rev. This indicates that subsurface damage can occur without any indication of surface damage, which is consistent with our experimental results. It can be also found that the SD m or SSD m generally increases obviously with the feed f under certain cutting depth a p . As feed f increases, the brittle-ductile transition position moves down toward the tool center line and the damage replicates into the final cut surface more easily. Under certain feed f, the SSD m and SD m fluctuate with the cutting depth a p , indicating that the cutting depth has a minor effect on the surface and subsurface damages. More specially, the SD m decreases slightly at first and then nearly remains unchanged with the cutting depth a p under any feed f. Meanwhile, as the cutting depth a p increases, the SSD m decreases slightly at first and then nearly remains unchanged when 1.5 µm/rev ≤ f ≤ 2.5 µm/rev, while the SSD m increases slightly at first and then nearly keeps unchanged when 3.0 µm/rev ≤ f ≤ 5.0 µm/rev. This indicates that the effect of cutting depth a p on SD m is almost irrelevant to the feed f, but the effect on SSD m varies with the feed f. This is why the interaction effect between feed and cutting depth is weak for SD m but large for SSD m . Similarly, the SD 1i and SSD 1i , i.e., the depths of all cracks in shoulder region, can be predicted using Eqs. (1)- (19). Figures 8(c) and 8(d) show the variations of SD 1i and SSD 1i with ductile mode cutting h i under different cutting parameters using the geometries of tool 22192. When SD 1i = 0 or SSD 1i = 0, the crack does not occur, and the cutting is ductile mode cutting. When SD 1i < 0 or SSD 1i < 0, the crack is generated but it does not approach the cut surface, and the cutting is partial ductile mode cutting; when SD 1i > 0 or SSD 1i > 0, the crack penetrates into the cut surface. Figure 8(c) shows that with the increase of undeformed chip thickness h i , the SD 1i equals to zero firstly, then jumps and increases gradually when f = 5 µm/rev and a p = 0.5 µm; the SD 1i equals to zero firstly, then jumps and increases gradually, and finally decreases when f = 5 µm/rev and a p = 5 or 3 µm, or f = 3 µm/rev and a p = 5 or 3 or 0.5 µm; and the SD 1i equals to zero firstly, then plunges and decreases gradually when f = 0.5 µm/rev. Figure 8(d) shows that the SSD 1i has the same change trend with the SD 1i . This indicates that it is possible that the damage exists underneath the cut surface if we just make the damage depth at the brittle-ductile transition position is not larger than zero. This seems contradictory with the research by Zong et al. [14], who demonstrated that the partial ductile mode cutting could be achieved as long as the median crack at the brittle-ductile transition position did not penetrate into the cut surface. Wang et al. [20] admitted that all cracks in shoulder region could not penetrate into the cut surface in partial ductile mode cutting, but they found that the depths of all cracks would be no more than zero in the actual processing of KDP crystal. They proposed a loose condition for achieving partial ductile mode cutting, which can be expressed as where f c is the critical feed.

Removal characteristics of cracks in shoulder region
In partial ductile mode cutting, two conditions (1) the depths of all cracks generated in shoulder region are not larger than zero and (2) all cracks generated in the previous cut could be removed by the following cut, should be fulfilled simultaneously. Because the distributions of cracks induced by different cuts are the same, condition (1) could be described as SD 1m ≤ 0 and SSD 1m ≤ 0 As shown in Fig. 1(c), assuming that the depths of lateral crack M L1i M R1i and median crack D 1i N 1i within the range of h i from h c to h m , are no more than zero (marked in red), to remove all cracks M L1i M R1i and D 1i N 1i by the following cut, point M 1i and N 1i should not be outside the circle formed by the following cut. Therefore, condition (2) could be described as where x M1i , y M1i , x N1i and y N1i could be calculated by Eqs. (11)- (16). Experiments have indicated that the cutting may be partial ductile mode cutting when feed f = 0.5 µm/rev under tool 22192 or 24557. Dividing the range of angle θ i , i.e., [asin(0.5f /R), acos(1−a p /R)], into 999 equal intervals, the number of cracks can be counted by evaluating whether a lateral or median crack at each angle θ i is generated. Figure 9(a) shows the total number of cracks (lateral and median cracks) induced by the first cut, and the number of cracks removed by the following cut, vary with the cutting depths when f = 0.5 µm/rev under tool 24557. It can be found that the total number of cracks is equal to zero at first and then increases with cutting depth a p . The cracks are not removed within the first six cuts. The number of cracks removed by the 9th or 10th cut increases at first and then decreases with the cutting depth a p . Figure 9(b) shows the trajectories of points M 1i and N 1i (see Fig. 1), and the 7th, 8th, 9th, and 10th cuts when feed f = 0.5 µm/rev, a p = 3.7 µm under tool 24557. The trajectories of points M 1i and N 1i represent the damage region within the deformed surface, and the actual microfracture-induced damages are shown in the inset of Fig. 9(b). It can be found that the damage region is between the trajectories of the 7th and the 10th cuts. It can be also found that the trajectories of points M 1i and N 1i near the uncut surface firstly intersect with the 8th cut, and those away from the uncut surface finally intersect with the 10th cut. This indicates that the 1st cut-induced cracks are not removed until the 8th cut, which means that more following cuts may be needed to remove the cracks induced by the previous cut. This also indicates that the cracks near the uncut surface are removed first, and those near the cut surface are then removed by the following cut. Furthermore, the trajectories of point M 1i , point N 1i , and each cut are discussed under different cutting depths when f = 0.5 µm/rev. The same removal characteristics of the cracks in shoulder region could be obtained.

Conclusions
This paper investigates the influence factors of brittle-ductile transition depth of ZnSe crystal, and the relationship between the cutting parameters and the damages in ZnSe crystal. This paper also proposes the evaluation model of surface and subsurface damage depths, and analyzes the depth and removal properties of cracks in shoulder region. The results could provide some useful guidance for the design and manufacturing of ZnSe-based optical devices. The conclusions are summarized as follows: (1) The tool with higher negative rake angle or larger nose radius could introduce greater brittle-ductile transition depth.
(2) The cutting depth has a minor effect on the surface and subsurface damages, but the feed has a great effect. The interaction effect between feed and cutting depth is very small for surface damage, but it increases obviously for subsurface damage. It is also possible that subsurface damage occurs without any surface damage.
(3) By adjusting constants k 1 and k 2 by Latin hypercube design, the proposed model could be utilized to efficiently evaluate the surface and surface damage depths of ZnSe crystal. The model could be also used to predict the depths of all cracks in shoulder region.