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Abstract
Based on the fracture mechanics and grinding kinematics, a theoretical model is developed to determine various subsurface damage (SSD) parameters and roughness Rz of the ground brittle material with consideration of the material removal mode and spring back. Based on the image processing, a digital method is proposed to extract various SSD parameters from the cross-section micrograph of the ground sample. To verify the model and method, many fused silica samples are ground under different processing parameters, and their SSD depth and roughness Rz are measured. The research results show the average SSD depth (SSDa) can be expressed as SSDa = χ1Rz4/3 + χ2Rz (χ1 and χ2 are coefficients). The SSDa is closer to half of the maximum SSD depth (SSDm) as the wheel speed decreases or the grinding depth, feed speed, or abrasive diameter increases. The SSD length or density basically increases linearly with the increase of the SSDm. The digital method is reliable with a largest relative error of 6.65% in SSD depth, extraction speed of about 1.63s per micrograph, and good robustness to the micrograph size and small-scale residue interference. The research will contribute to the evaluation of SSDs and the optimization of the grinding process of fused silica.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Due to the good mechanical, thermal, and optical properties of fused silica, such as high hardness, high-temperature resistance, and superior light transmission rate in both visible and near IR spectrums, the optical components processed from fused silica are widely used in the field of optical communication, laser technologies, and optical instruments, etc. However, subsurface damages (SSDs) are inevitably generated during the grinding of fused silica, which is usually the first step in processing optical components [1]. For the ground fused silica, the main form of the SSDs is micro-cracks rather than stacking faults or dislocations [2]. The SSDs could weaken the mechanical strength, enhance the local temperature due to the light absorption by the impurities in the micro-cracks, and improve the laser damage susceptibility in the field intensification, which could induce macroscopic damages to optical components [3,4]. To reduce the SSDs, some post-processes, lapping and polishing, are usually used which are notoriously inefficient and costly, yet, the time fully depends on the SSDs left by the grinding [5]. Consequently, it is important to comprehensively characterize and accurately predict the grinding-induced SSDs.
Regarding the characterization of the SSDs, the maximum SSD depth and SSD depth distribution have been investigated, where the latter is usually characterized by the cumulative crack obscuration at various depths below the ground surface [6]. The maximum SSD depth is most focused on those optical components in ultra-precision optical systems, such as high-energy laser elements and EUV lithography substrates [7]. It can be used to determine the maximum material removal for ultra-smooth and near defect-free machining. Considering that some residual SSDs have a small influence on the service performance of optical components in general optical applications, and some residual SSDs can even improve the processing efficiency of post-processes, the SSD depth distribution is more concerned. Nevertheless, only the characterization parameter of the cumulative crack obscuration is not enough due to the complex distribution of the grinding-induced SSDs. More SSD parameters, such as average SSD depth, SSD length and density, are also required to comprehensively characterize the SSDs. Although the SSDs can be detected by magnetorheological finishing wedge [6], cross-section microscopy [8], chemical etching [9], confocal microcopy [10], how to extract more SSD parameters accurately and rapidly still needs to be investigated since the morphology of subsurface cracks is usually irregular.
Regarding the prediction of the SSDs, many empirical and theoretical models of SSD depth have been established. The former mainly depends on the fitting of experimental data. For example, Blaineau et al. [4] and Pashmforoush et al. [11] developed relationship models between SSD depth and processing parameters based on regressive analysis. The latter is usually based on the formation mechanism of subsurface cracks and the removal process of brittle materials. For example, to predict the SSD depth of optical glass, Dong et al. [12] correlated the indentation load with the processing parameters for cup grinding and Wang et al. [13] correlated the indentation load with the cutting force in rotary ultrasonic face milling based on the indentation fracture mechanics. Li et al. [14], Esmaeilzare et al. [15], and Solhtalab et al. [16] reported that the SSD depth could be predicted by the roughness in the grinding or lapping of brittle optical materials. Combining the theory of scratch fracture mechanics and grinding kinematics, Yao et al. [17], Li et al. [18] and Xiao et al. [19] developed the relationship models among SSD depth, roughness, and grinding parameters for BK7 glass and fused silica. Due to the limitation of the current characterization parameters of the SSDs, the above empirical and theoretical models always stay in the prediction of the maximum SSD depth. So far, other SSD parameters have been less involved in the prediction.
In this paper, based on the kinematics of horizontal surface grinding and the fracture mechanics of brittle materials scratched by a sharp indenter, a theoretical model for determining various SSD parameters and roughness Rz is developed. Based on the image processing technique, a digital method is proposed to accurately and rapidly extract various SSD parameters from the cross-section micrographs of the ground samples. Forty-five fused silica samples are ground under different processing parameters, and their SSD depth and roughness Rz are measured for validating the model and method. This study is of great significance for the comprehensive evaluation of SSDs and can provide guideline for the optimization of the grinding process of fused silica.
2. Theoretical model of SSD parameters
2.1 Instantaneous SSD depth and peak-valley value
Figure 1 shows the lateral-median crack system induced by a sharp indenter scratching for the horizontal surface grinding of brittle materials. There is a critical undeformed chip thickness hc. When it is larger than the instantaneous penetration depth hi (hc > hi) of the abrasive grit, an elastoplastic zone is formed beneath the abrasive grit; when hc ≤ hi, the lateral crack and median crack are generated below the elastoplastic zone, which are respectively deflected toward and almost perpendicular to the ground surface. The value of hc can be calculated by [20]:
Fig. 1. The lateral-median crack system induced by a sharp indenter scratching for the horizontal surface grinding of brittle materials.
The instantaneous penetration depth hi ranges from zero to the maximum undeformed chip thickness hm, which can be expressed as [21]:
Combining Eq. (1) with Eq. (2), the critical brittle fracture angle φc can be expressed as:
The instantaneous rotation angle φi ranges from zero to the maximum rotation angle φm, which can be expressed as [21]:
The maximum undeformed chip thickness hm can be given by [22]:
Based on Eq. (2) and Eq. (6), it can be obtained:
whereAs shown in Fig. 2, the brittle material is usually removed in (i) ductile mode (hc > hi) where the stress within the shear band exceeds the limit value so that the material is removed by plastic flows like a metal, (ii) brittle mode (hc ≤ hi) where the lateral crack remains in the workpiece while the material is still removed by plastic flows, and (iii) brittle mode (hc ≤ hi) where the lateral crack propagates and extends to the workpiece surface so that the material above the lateral crack is completely removed by brittle fragments. The main difference between mode (i) and mode (ii) is that there are no cracks in mode (i) whereas lateral and median cracks appear in mode (ii). In mode (ii), since the lateral crack does not extend to the workpiece surface, only the material in the penetration region of abrasive grit is removed by plastic flows.
Those median cracks penetrating into the workpiece generate the SSDs [24]. Combining Fig. 1 with Fig. 2, the instantaneous SSD depth SSDi is determined by the instantaneous median crack depth Cmi, rotation angle φi, and penetration depth hi, which can be expressed as:
The value of Cmi can be calculated by [25]:
whereThe value of Cli can be calculated by [25]:
whereAs shown in Fig. 3, the realistic ground surface profile varies in the range from hicosφi to Clicosφi. The instantaneous peak-valley value PVi of the profile is calculated by:
In precision or ultra-precision grinding, the cutting depth ap (micron scale) is much smaller than the wheel diameter de (millimeter scale). According to Eq. (5), the maximum rotation angle φm is very small, then cosφi ≈ 1. According to Eq. (10) and Eq. (15), the values of SSDi and PVi can be respectively calculated by:
In the actual grinding, modes (ii) and (iii) occur simultaneously in the brittle mode. In the actual measurement, it is difficult to distinguish the difference in SSDi between modes (ii) and (iii), and only one measuring basis, the grit tip trajectory, is used. In this situation, the values of SSDi in modes (ii) and (iii) are the same, i.e., Eq. (16) can be changed to
2.2 Determination of SSD parameters and Rz
According to Eq. (18), the value of SSDi increases monotonously with the value of φi in the brittle mode. The maximum SSD depth SSDm appears at φm, which can be calculated by:
According to the definition of roughness Rz in the standard ISO 4287:1998, Rz is the maximum height of the assessed profiles, it can be obtained:
Combining Eq. (19) with Eq. (20), the relationship between SSDm and Rz can be expressed as:
In this paper, the average SSD depth SSDa is defined as the average of all instantaneous SSD depth SSDi, which can be approximately calculated as:
Since φc<<φm and φc ≈ 0, according to Eq. (20) and Eq. (22), the value of SSDa can be approximately calculated by:
Combining Eq. (19) with Eq. (23), it can be obtained:
Equation (24) indicates that the SSDa is less than half of the SSDm. As shown in Fig. 4, if Δ is very small, the SSDa approximates half of the SSDm, and Eq. (18) can also be changed to:
The instantaneous SSD length SSDli = SSDi/cosφi. Since cosφi ≈ 1, the total SSD length SSDl can be approximately equal to the total SSD depth, which can be calculated by:
Equation (21) and Eq. (23) indicate that there is a relationship model between SSDm or SSDa and Rz, that is, SSDm or SSDa = χ1Rz4/3 + χ2Rz (χ1 and χ2 are coefficients). Since it is difficult to completely capture all the subsurface cracks in the range of 0 to φm in the actual measurement. Equation (27) is only used to reveal that there may be a linear relationship between SSDl and SSDm.
3. Digital extraction of SSD parameters
3.1 Recognition of subsurface crack tip
Since the morphology of subsurface cracks is usually irregular, it is difficult to manually measure or calculate the SSD length and density. Meanwhile, many SSD depths need to be captured to calculate the average SSD depth, which is a tedious task. To address these problems, a digital method is proposed to extract various SSD parameters. The premise of extracting the SSD depth, length, and density is to accurately locate the subsurface crack tip from the cross-section micrograph of the ground sample, as shown in Fig. 5(a). A black gap and some residues (pits, scratches and impurities, etc.) are remained in the cross-section micrograph, which interfere with the recognition of subsurface cracks. The epoxy resin is used to fix the ground sample, and the black gap between the epoxy resin and fused silica is generated due to the fall of the epoxy resin during the process of sample preparation [25]. To remove the residues, the cross-section micrograph is processed using the morphological erosion-reconstruction method, which can be expressed as:
Fig. 5. Recognition of subsurface crack tip: (a) original image; (b) gray scale image; (c) binary image; (d) erosion-reconstruction image; (e) profile detection image; (f) lowermost pixel extraction image.
3.2 Recognition of ground surface
The ground surface is the measuring basis, which lies in the middle of the gap here. To retain only the gap and remove all subsurface cracks and residues, many times of erosion operations are conducted on the erosion-reconstruction image, i.e., Fig. 5(d), as shown in Figs. 6(a)–6(e). The matrix B1 in the erosion process is changed to
Fig. 6. Recognition of ground surface: (a) erosion once; (b) erosion twice; (c) erosion three times; (d) erosion four times; (e) erosion five times; (f) profile detection image; (g) intermediate pixel extraction image; (h) the fitted ground surface profile; (i) original image and the fitted ground surface profile in red.
As the erosion time increases, the gap becomes progressively thinner and the residues as well as subsurface cracks eventually disappear. During the erosion process, the image starts from several connected domains to a single connected domain. The first single-connected-domain image, i.e., Fig. 6(c), is conducted by the Canny edge detection algorithm to extract the profile of the shrunken gap, as shown in Fig. 6(f). The ground surface profile is extracted by calculating the intermediate pixels between the lowermost and uppermost pixels of the detected profiles in each pixel column, as shown in Fig. 6(g). The profile can be approximated as a linear function y = ax + b, where x and y are the pixel coordinates. For Fig. 6(h), a = −0.006, b = 400.679. Figure 6(i) shows the original image and the fitted ground surface profile.
3.3 Calculation of SSD parameters
The length c (unit: µm) of each pixel can be calculated directly from the provided micrograph. As shown in Fig. 5(a), the upper-left corner of the image shows the size of 1024 × 1024 (unit: pixel) and 256 × 256 (unit: µm), then the length c = 0.25 µm. After identifying the subsurface crack tip, ground surface and scale bar, the SSD depth can be calculated as:
Fig. 7. (a) The fitted ground surface profile and the linear connection of the lowermost pixel with its top five peaks (marked by ‘+’); (b) The profile of subsurface crack.
4. Experimental
Forty-five polished fused silica samples with the same roughness Ra of 2 nm and dimensions of 34 × 34 × 1 mm3 were prepared. The material properties of fused silica were shown in Table 1. Grinding experiments were conducted by a CNC Machining Center (S6150A, LOKSHUN). Three resin bond diamond wheels were used with average abrasive diameters (dg) of 120 µm, 100 µm and 80 µm, respectively. Their mesh sizes were 120, 140 and 170, respectively. The diamond-wheels had the same diameter (dw = 125 mm) and width (10 mm). The fraction and volume fraction of the abrasive grit were χ = 0.5 and υ = 25%, respectively. The dressing of the diamond wheels was performed by a dressing wheel before each experiment to preclude the dulling of abrasive grits. The dressing parameters were shown in Table 2. Grinding depth (ap = 2–36 µm), wheel speed (vs = 2–36 m/s), and feed speed (vw = 200–3600 mm/min) were determined by the Optimum Latin hypercube design, as shown in Table 3. The Latin hypercube design is a typical experimental design method with a great space filling capacity. A cross-section for each sample was cut and polished to a mirror, and then etched with HF/HNO3 (80/20%v.) for about 10 s to expose the SSDs. The epoxy resin (DFR103-A/B) was used to wrap and fix the samples during the polishing. An aspheric surface optics measurement instrument (PGI 3D, Taylor Hobson) and a laser scanning confocal microscopy (OLS4000, Olympus Corporation) were used to measure the surface roughness Rz and surface/subsurface morphology of ground samples. All samples were cleaned by alcohol and deionized water, and then dried. The experimental setup and the related measurements are shown in Fig. 8.
Table 2. Dressing parameters
Table 3. Processing parameters for grinding experiments and the values of roughness (Rz), the maximum and average SSD depths (SSDm, SSDa), SSD length (SSDl), and SSD density (SSDd) for all ground samples.
5. Result and discussion
5.1 Surface and subsurface morphologies of ground samples
Figures 9(a)–9(c) show the surface morphologies for the randomly selected regions from ground samples 33, 11, and 36. It can be observed that most of the fractures intersect with each other, thus forming the craters. Due to the random distribution of the abrasive grits on the grinding wheel, the crater size varies in different regions. Five surface morphologies and five surface profiles are obtained for each sample. The average surface roughness Rz is calculated as shown in Table 3. Figures 10(a)–10(h) show the cross-section micrographs for the randomly selected ground samples. Subsurface cracks can be clearly observed, introducing the crater-like surface morphology and complex SSD distribution. Some polishing-induced residual pits, scratches, and impurities can also be found. Figures 10(a)–10(h) also show the top five SSD depths and their average value measured manually with the SSD numbered from deep to shallow. In addition to SSD depth, SSD length and density are extracted to characterize SSDs comprehensively. If five cross-section micrographs are selected for each ground sample and five SSD depths need to be measured for each micrograph, 1125 SSD depths, 225 SSDms, and 225 SSDas will be calculated for 45 ground samples. This is a heavy and tiring task. Moreover, it is difficult to measure the SSD length or density by hand due to the irregular shape of subsurface cracks. Zhao et al. [27,28] have proposed a SSD extraction method for the ground optical glass based on the image processing technique. However, it can not recognize the ground surface involving epoxy resin, gap, and residues like Fig. 5(a). Meanwhile, it can not extract SSD length or density other than the maximum SSD depth.
Fig. 9. Surface morphologies for the randomly selected regions from ground samples (a) 33, (b) 11, and (c) 36. The sample numbers are randomly selected.
Fig. 10. Cross-section micrographs including the top five SSD depths and their average for the randomly selected ground samples (a) 33, (b) 11, (c) 36, (d) 27, (e) 19, (f) 25, (g) 2, and (h) 43. The sample numbers are randomly selected. SSDm and SSDa denote the maximum and average SSD depths, respectively.
5.2 Verification of digital extraction of SSD parameters
Since the intersection of subsurface cracks near the ground surface causes a significant uncertainty in the number count of cracks, only the top ten SSD depths are digitally extracted based on Figs. 10(a)–10(h), and the top five SSD depths are compared with the measured values to verify the extraction method, as shown in Fig. 11. It can be observed that, among forty SSD depths, eight SSDms, and eight SSDas (for all eight cross-section micrographs in different scales), the largest relative errors are 6.65%, 6.61%, and 4.82%, respectively, and the smallest relative errors are 0.02%, 0.02%, and 1.39%, respectively, indicating the proposed method is accurate to a large extent. The total time for extracting all SSD parameters is 13.01 seconds when a laptop (Processor: Intel Core i7-8550U 1.8 GHz, Installed memory: 16.0 GB) is used, showing an extraction speed of ∼1.63 seconds per micrograph. The method is also validated with other cross-section micrographs of different sizes shown in Ref. [15,27], as shown in Figs. 12(a)–12(c). It can be found that, among three cross-section micrographs, subsurface crack tips can be located accurately. This indicates that the method can successfully recognize the SSDs from the images of different sizes, which may attract metrology instrument companies and optics production department. Table 3 shows the extracted values of SSDm, SSDa (the average of the top ten SSD depths), SSDl, and SSDd for all ground samples.
Fig. 11. The top ten extracted SSD depths and the top five measured SSD depths corresponding to Figs. 10(a)–10(h).
Fig. 12. The original, the fitted ground surface profile and the linear connection of the lowermost pixel with its several top peaks (marked by ‘+’) corresponding to the cross-section micrographs shown in Ref. [15,27].
The digital method can accurately and quickly extract more SSD parameters, but it does not work well for some cross-section micrographs involving large-scale pits, scratches, and impurities. It can be observed from Fig. 13(a) that the subsurface crack and the ground surface can be recognized successfully, while the lowermost pixel is extracted for the scratch edge but not for the crack. It can be found from Fig. 13(b) that the ground surface can be extracted, while those subsurface cracks covered by impurities cannot be completely detected, and the pit tip is mistakenly considered as the crack tip. These failures are mainly caused by the inappropriate operations in the sample preparation (polishing, cleaning, etc.) rather than the method itself.
Fig. 13. Recognition of subsurface crack and ground surface for the samples with large-scale (a) scratches and (b) pits/impurities.
5.3 Verification of theoretical model of SSD parameters
Based on the indentation fracture mechanics of brittle materials, Li et al. [14] proposed a model SSDm = χRz4/3 (χ is a coefficient) by investigating the indentation-induced lateral-median crack system. According to the values of SSDm and Rz shown in Table 3, the model can be expressed as SSDm = 4.35×Rz4/3. Based on the kinematics relationship between abrasive grit and workpiece in horizontal surface grinding and the fracture mechanics of brittle materials scratched by a sharp indenter, another model is developed as SSDm = χ1Rz4/3 + χ2Rz (χ1 and χ2 are coefficients) (Eq. (2)1), which can be expressed as SSDm = 5.00×Rz4/3−1.14×Rz according to the actual values, as shown in Fig. 14(a). Similarly, the values of SSDa with Rz can be fitted as SSDa = 2.08×Rz4/3 or SSDa = 1.37×Rz4/3 + 1.25×Rz, proving the reasonability and validity of Eq. (2)3, as shown in Fig. 14(b). Comparing Fig. 14(a) and Fig. 14(b), the SSDa is slightly less than half of the SSDm, i.e., SSDa < 0.5×SSDm. It should be noted that the SSDa is the average of the top ten SSD depths, which has a certain difference with the definition of SSDa where all subsurface cracks are involved. Due to the intersection of subsurface cracks near the ground surface, it is difficult to determine the number of all cracks and the depth of each crack. The top ten SSD depths are representative since the rest SSD depths are very small. According to the values of SSDl, SSDd and SSDm shown in Table 3, the variations of SSDl and SSDd with SSDm are shown in Fig. 14(c) and Fig. 14(d), respectively. It can be observed that either SSDl or SSDd increases almost linearly with SSDm (SSDl = 6.60 + 2.83×SSDm, SSDd = 0.444 + 0.018×SSDm) regardless of the existence of scatter points.
Fig. 14. (a) SSDm expressed as a function of Rz; (b) SSDa expressed as a function of Rz; (c) Variation of SSDl with SSDa; (d) Variation of SSDd with SSDa.
Equation (24) indicates that the SSDa approximates half of the SSDm when Δ is very small, which is evaluated by the ratio δ = Δ/(0.5×SSDm)×100%. According to the material parameters and processing parameters, Δ can be determined by Eq. (25). Since grinding will produce a large impact at the moment of contact between the abrasive grit and workpiece. It is unreasonable to use the static fracture toughness to study the dynamic crack initiation under the impact load. Clifton et al. [29] studied the dynamic fracture law using the plate impact test, revealing that the dynamic fracture toughness of brittle material is about 30% of the static value. Mahmoud et al. [30] found that the minimum and maximum semi-apex angles α of abrasive grit are 39° and 71°, respectively after 462 measurements over 150 abrasive grits. Taking α = 39°, α = 71° and 0.3×Kc, the value of δ is calculated for different ground samples, as shown in Fig. 15(a). The average ratio δ = 17.6% for α = 39° and δ = 15.4% for α = 71°, indicating it is reasonable that the SSDa approximates half of the SSDm. The response surface model and Pareto chart are prepared to investigate the processing parameters on the ratio δ. It can be seen from Fig. 15(b) that the interaction effect between any two processing parameters is much smaller than the main effect. The contribution rate of wheel speed vs denotes an increase effort, while grinding depth ap, feed speed vw, or abrasive diameter dg has a decrease effort. This indicates that the ratio δ increases with wheel speed while decreases with grinding depth, feed speed, or abrasive diameter. Therefore, the SSDa is more closed to half of the SSDm at a smaller wheel speed or a larger grinding depth, feed speed, or abrasive diameter.
Fig. 15. (a) Variation of SSDi with rotation angle φi; (b) Contribution rate of processing parameters on ratio δ.
6. Conclusions
This paper proposes a theoretical model to determine various SSD parameters as well as Rz and a digital method to extract various SSD parameters for the ground brittle materials. The grinding experiments of fused silica are conducted to validate the theoretical model and extraction method. Combining the theoretical model, digital extraction and experimental results, the relationship among various SSD parameters is revealed. The main findings include:
- (1) The maximum and average SSD depths (SSDm and SSDa, respectively) can be expressed as SSDm or SSDa = χ1Rz4/3 + χ2Rz (χ1 and χ2 are coefficients). The SSDa is less than half of the SSDm. As the wheel speed decreases or the grinding depth, feed speed, or abrasive diameter increases, the SSDa gradually approaches the half of SSDm. Moreover, the SSD length or density basically increases linearly with the increase of the SSDm.
- (2) The digital method can extract the maximum/average SSD depth, SSD length, and SSD density from the cross-section micrograph of the ground sample. The maximum relative error between the extracted and measured SSD depths is 6.65%. The extraction speed is fast, approximately 1.63s per micrograph. The method exhibits good robustness to the micrograph size and small-scale residue interference.
Funding
National Natural Science Foundation of China (12104074, 62173098); China Postdoctoral Science Foundation (2020M683233); Chongqing Special Postdoctoral Science Foundation (XmT20200021, XmT20200043).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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