Theoretical model and digital extraction of subsurface damage in ground fused silica

: 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 (SSD a ) can be expressed as SSD a = χ 1 Rz 4 / 3 + χ 2 Rz ( χ 1 and χ 2 are coefficients). The SSD a is closer to half of the maximum SSD depth (SSD m ) 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 SSD m . 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.


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 postprocesses, 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.

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 h c .When it is larger than the instantaneous penetration depth h i (h c > h i ) of the abrasive grit, an elastoplastic zone is formed beneath the abrasive grit; when h c ≤ h i , 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 h c can be calculated by [20]: where E, K c , and H s are the elastic modulus, fracture toughness, and scratch hardness of the workpiece material, respectively.The instantaneous penetration depth h i ranges from zero to the maximum undeformed chip thickness h m , which can be expressed as [21]: where φ i is the instantaneous rotation angle of the abrasive grit; d e is the effect wheel diameter; a p is the cutting depth.The value of d e can be calculated by: where d w , v w and v s are the wheel diameter, feed speed, and wheel speed, respectively.Combining Eq. (1) with Eq. ( 2), the critical brittle fracture angle φ c can be expressed as: The instantaneous rotation angle φ i ranges from zero to the maximum rotation angle φ m , which can be expressed as [21]: The maximum undeformed chip thickness h m can be given by [22]: where α is the semi-apex angle of the abrasive grit; C is the grit surface density which can be calculated by [23]: where χ is the fraction of the abrasive grit; d g is the equivalent diameter of the abrasive grit; υ is the volume fraction of the abrasive grit.Based on Eq. ( 2) and Eq. ( 6), it can be obtained: where As shown in Fig. 2, the brittle material is usually removed in (i) ductile mode (h c > h i ) 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 (h c ≤ h i ) where the lateral crack remains in the workpiece while the material is still removed by plastic flows, and (iii) brittle mode (h c ≤ h i ) 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 SSD i is determined by the instantaneous median crack depth C mi , rotation angle φ i , and penetration depth h i , which can be expressed as: The value of C mi can be calculated by [25]: where where κ ≈ 2.231; α 0 ≈ 0.096; α 1 = 2/3 and α 2 = 2 for pyramidal indenter [26]; µ is the spring back factor of the material, which is defined as the ratio of residual depth to scratch depth.
The value of C li can be calculated by [25]: where As shown in Fig. 3, the realistic ground surface profile varies in the range from h i cosφ i to C li cosφ i .The instantaneous peak-valley value PV i of the profile is calculated by: Fig. 3. Realistic ground surface profile for determining peak-valley value PV i .
In precision or ultra-precision grinding, the cutting depth a p (micron scale) is much smaller than the wheel diameter d e (millimeter scale).According to Eq. ( 5), the maximum rotation angle φ m is very small, then cosφ i ≈ 1.According to Eq. ( 10) and Eq. ( 15), the values of SSD i and PV i 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 SSD i between modes (ii) and (iii), and only one measuring basis, the grit tip trajectory, is used.In this situation, the values of SSD i in modes (ii) and (iii) are the same, i.e., Eq. ( 16) can be changed to

Determination of SSD parameters and Rz
According to Eq. ( 18), the value of SSD i increases monotonously with the value of φ i in the brittle mode.The maximum SSD depth SSD m appears at φ m , 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 SSD m and Rz can be expressed as: In this paper, the average SSD depth SSD a is defined as the average of all instantaneous SSD depth SSD i , which can be approximately calculated as: Since φ c <<φ m and φ c ≈ 0, according to Eq. ( 20) and Eq. ( 22), the value of SSD a can be approximately calculated by: Combining Eq. ( 19) with Eq. ( 23), it can be obtained: where ∆ can be expressed as: Equation (24) indicates that the SSD a is less than half of the SSD m .As shown in Fig. 4, if ∆ is very small, the SSD a approximates half of the SSD m , and Eq. ( 18) can also be changed to: The instantaneous SSD length SSD li = SSD i /cosφ i .Since cosφ i ≈ 1, the total SSD length SSD l can be approximately equal to the total SSD depth, which can be calculated by: where k is the number of the instantaneous SSD depth SSD i in the range of 0 to φ m .Equation (21) and Eq. ( 23) indicate that there is a relationship model between SSD m or SSD a and Rz, that is, SSD m or SSD a = χ 1 Rz 4/3 + χ 2 Rz ( χ 1 and χ 2 are coefficients).Since it is difficult to completely capture all the subsurface cracks in the range of 0 to φ m in the actual measurement.Equation ( 27) is only used to reveal that there may be a linear relationship between SSD l and SSD m .

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: where E is the erosion image; I e is the matrix of the input image; B 1 is the shrinkage matrix, a 30 × 1 identity matrix here; j is a certain element in matrix I e ; the operator ⊗ denotes the erosion operation.In morphological reconstruction, the dilation operation is performed repeatedly on the erosion image until the pixel values of the image do no longer change, which can be expressed as: where P q is the matrix of reconstruction image after the qth iteration (q = 1, 2, 3. . .); B 2 is the expansion matrix, B 2 = [0 1 0; 1 1 1; 0 1 0]; B2 is the reflection of B 2 ; and operator ⊕ denotes the dilation operation.Taking Fig. 5(a) as an example, to improve the calculation speed, the micrograph is first converted into a gray scale image and then a binary image as the input image, as shown in Fig. 5

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.
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.

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: where SSD pixel is the SSD depth expressed in the unit of pixel; (x, y) is the pixel coordinate of the detected crack tip.The crack tip is extracted from the lowermost pixel (Fig. 5(f)) by the function findpeaks, returning the indices where the peaks occur.Figure 7(a) shows the fitted ground surface profile and the linear connection of the lowermost pixel with its five peaks (marked by '+').The five crack tips correspond to the top five SSD depths.The profile of subsurface crack can be separated from profiles (Fig. 5(e)) by removing all pixels above the fitted ground surface profile, as shown in Fig. 7(b).The skeleton algorithm is used to refine the subsurface crack profile to single pixel width.In this paper, the SSD density SSD d is defined as the ratio of the total SSD length SSD l within a distance to the distance.For a cross-section micrograph, the values of SSD l and SSD d can be approximately calculated by: where L c and L g are the number of white pixels for the subsurface crack profile and the fitted ground surface profile, respectively.Thus, a digital method of extracting various SSD parameters from the cross-section micrograph of the ground sample is proposed.

Experimental
Forty-five polished fused silica samples with the same roughness Ra of 2 nm and dimensions of 34 × 34 × 1 mm 3 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 (d g ) 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 (d w = 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 (a p = 2-36 µm), wheel speed (v s = 2-36 m/s), and feed speed (v w = 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/HNO 3 (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.

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 SSD m s, and 225 SSD a s 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.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 SSD m , SSD a (the average of the top ten SSD depths), SSD l , and SSD d for all ground samples.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.

Verification of theoretical model of SSD parameters
Based on the indentation fracture mechanics of brittle materials, Li et al. [14] proposed a model SSD m = χRz 4/3 ( χ is a coefficient) by investigating the indentation-induced lateral-median crack system.According to the values of SSD m and Rz shown in Table 3, the model can be expressed as  SSD m = 4.35×Rz 4/3 .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 SSD m = χ 1 Rz 4/3 + χ 2 Rz ( χ 1 and χ 2 are coefficients) (Eq.( 2)1), which can be expressed as SSD m = 5.00×Rz 4/3− 1.14×Rz according to the actual values, as shown in Fig. 14(a).Similarly, the values of SSD a with Rz can be fitted as SSD a = 2.08×Rz 4/3 or SSD a = 1.37×Rz 4/3 + 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 SSD a is slightly less than half of the SSD m , i.e., SSD a < 0.5×SSD m .It should be noted that the SSD a is the average of the top ten SSD depths, which has a certain difference with the definition of SSD a 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 SSD l , SSD d and SSD m shown in Table 3, the variations of SSD l and SSD d with SSD m are shown in Fig. 14(c) and Fig. 14(d  Equation (24) indicates that the SSD a approximates half of the SSD m when ∆ is very small, which is evaluated by the ratio δ = ∆/(0.5×SSDm )×100%.According to the material parameters and processing parameters, ∆ 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°a nd 71°, respectively after 462 measurements over 150 abrasive grits.Taking α = 39°, α = 71°a nd 0.3×K c , 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 SSD a approximates half of the SSD m .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 v s denotes an increase effort, while grinding depth a p , feed speed v w , or abrasive diameter d g 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 SSD a is more closed to half of the SSD m at a smaller wheel speed or a larger grinding depth, feed speed, or abrasive diameter.

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 (SSD m and SSD a , respectively) can be expressed as SSD m or SSD a = χ 1 Rz 4/3 + χ 2 Rz ( χ 1 and χ 2 are coefficients).The SSD a is less than half of the SSD m .As the wheel speed decreases or the grinding depth, feed speed, or abrasive diameter increases, the SSD a gradually approaches the half of SSD m .Moreover, the SSD length or density basically increases linearly with the increase of the SSD m .
(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.

Fig. 1 .
Fig. 1.The lateral-median crack system induced by a sharp indenter scratching for the horizontal surface grinding of brittle materials.

Fig. 4 .
Fig. 4. Instantaneous SSD depth SSD i as a function of rotation angle φ i .
(b) and Fig.5(c), respectively.Figure5(d)shows the erosion-reconstruction image, indicating that almost all residues have been removed.The Canny edge detection algorithm is employed to extract the profile of subsurface crack, as shown in Fig.5(e).The crack tip is obtained by extracting the lowermost pixels of the detected profile in each pixel column, as shown in Fig.5(f).
5(d), as shown in Figs.6(a)-6(e).The matrix B 1 in the erosion process is changed to

Fig. 6 .
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.

Fig. 7 .
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.

Fig. 12 .
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].

Fig. 13 .
Fig. 13.Recognition of subsurface crack and ground surface for the samples with large-scale (a) scratches and (b) pits/impurities.
), respectively.It can be observed that either SSD l or SSD d increases almost linearly with SSD m (SSD l = 6.60 + 2.83×SSD m , SSD d = 0.444 + 0.018×SSD m ) regardless of the existence of scatter points.

Fig. 14 .
Fig. 14.(a) SSD m expressed as a function of Rz; (b) SSD a expressed as a function of Rz; (c) Variation of SSD l with SSD a ; (d) Variation of SSD d with SSD a .

Fig. 15 .
Fig. 15.(a) Variation of SSD i with rotation angle φ i ; (b) Contribution rate of processing parameters on ratio δ.

Table 3 . Processing parameters for grinding experiments and the values of roughness (Rz), the maximum and average SSD depths (SSD m , SSD a ), SSD length (SSD l ), and SSD density (SSD d ) for all ground samples.
p (µm) v s (m/s) v w (mm/min) d g (µm)