Evaluation of Fracture Strength of Ceramics Containing Small Surface Defects Introduced by Focused Ion Beam

The aim of this study was to clarify the effects of micro surface defects introduced by the focused ion beam (FIB) technique on the fracture strength of ceramics. Three-point bending tests on alumina-silicon carbide (Al2O3/SiC) ceramic composites containing crack-like surface defects introduced by FIB were carried out. A surface defect with area in the range 19 to 35 µm was introduced at the center of each specimen. Test results showed that the fracture strengths of the FIB-defect specimens depended on area. The test results were evaluated using the evaluation equation of fracture strength based on the process zone size failure criterion and the area parameter model. The experimental results indicate that FIB-induced defects can be used as small initial cracks for the fracture strength evaluation of ceramics. Moreover, the proposed equation was useful for the fracture strength evaluation of ceramics containing micro surface defects introduced by FIB.


Introduction
Ceramics have excellent mechanical, tribological, and thermal properties in comparison with metallic materials. However, there is a risk that failure initiates from defects due to the low fracture toughness of ceramics. Therefore, it is very important to quantitatively evaluate the influence of defects on fracture strength for maintenance management of ceramic components.
It is reported that the fracture toughness for small flaws is lower than that for large cracks. Thus, conventional linear elastic fracture mechanics (LEFM) is not applicable for predicting the fracture strength of ceramics containing small flaws [1]. Many authors have proposed various models for the relationship between defect size and fracture strength [2][3][4][5][6]. Ando et al. proposed the process zone size failure criterion and reported that the fracture strength of ceramics containing small surface cracks could be evaluated by this criterion [7]. This model can predict the fracture strength of a material having a semicircular crack introduced perpendicular to the tensile stress. Murakami and Endo proposed the √ area parameter model to predict the fatigue limit of metals having a surface crack with arbitrary shape [8]. Here, the √ area is the square root of the area of a surface defect projected onto the direction of the maximum tensile stress. The √ area parameter model is widely applied to evaluate the fatigue limit of metals [9,10]. In a previous study, we proposed a new evaluation equation for the fracture strength of ceramics combining the process zone size failure criterion and the √ area parameter model [11]. This equation is useful for the evaluation of fracture strength of ceramics having a surface crack with arbitrary shape and orientation introduced by the indentation method.
However, it is known that tensile residual stress is generated around the indentation when a pre-crack is introduced on the surface of a ceramic material by the indentation method [12]. Therefore, elimination of the tensile residual stress by surface polishing or annealing in inert gas is necessary [13]. If the tensile residual stress is not eliminated, the influence of residual stress on the fracture strength should be considered to predict the fracture strength with high accuracy [13]. The dimensions of a defect at the origin of a fracture in an actual ceramic material are equivalent to the grain size, which is less than 20 µm in a typical ceramic. However, the dimensions of defects introduced in most of the previous studies were larger than 20 µm because it is difficult to introduce small surface cracks by the indentation method [14][15][16].
In this study, we introduced crack-like surface defects on the surface of ceramics by the focused ion beam (FIB) technique. The FIB technique is a high-precision working method that uses the effect of sputtering by ion beam scanning on the surface of a material. Sakamoto et al. reported that the fatigue limits of FIB-notched specimens were equivalent to the fatigue limits of annealed fatigue-cracked specimens of carbon steel [17]. Rokio and Solin evaluated small crack initiation and propagation from small FIB-defect in high strength steel and reported that a FIB notch to initiate small cracks is a useful way to test small crack growth in high strength steels [18]. However, to our knowledge, the influence of FIB-introduced surface defects on the fracture strength of ceramics has not yet been studied. In this paper, the effects of microscopic surface defects introduced by FIB on the fracture strength of an Al 2 O 3 /SiC composite are clarified. Regarding the relationship between fracture strength and defect size, the consistencies of the test results and the values predicted by the proposed evaluation equation were verified.

Evaluation Equation of Fracture Strength
In this section, an outline of the proposed equation for fracture strength evaluation is described [11]. The equation is based on the process zone size failure criterion [7] and the √ area parameter model [8,19].
Ando et al. proposed the following equation to predict the fracture strength (σ C ) of ceramics with a small surface crack focusing on the process zone size at the crack tip and [7] where K IC (MPa·m 1/2 ) is the fracture toughness, and σ 0 (MPa) is the fracture strength of a smooth specimen assuming a defect is sufficiently small, and σ C (MPa) is the fracture strength of a specimen containing a surface crack. The equivalent crack length, a e (µm) is defined as On the other hand, Murakami showed that the maximum stress intensity factor (K max ) at the edge of a surface crack with arbitrary shape under tensile stress (σ ∞ ) could be approximated within 10% error by [19] where √ area is the square root of the area of a surface defect projected onto the direction of the maximum tensile stress. When σ C is applied to a surface crack, the K max value reaches K IC . Equation (3) leads to The relationship between fracture strength σ C and the crack size √ area is expressed as follows based on Equations (1), (2) and (4) This is the equation to evaluate the fracture strength of a ceramic material containing a surface defect with arbitrary shape. Figure 1 shows the relationship between σ C and √ area of alumina reinforced by silicon carbide (Al 2 O 3 /SiC) [11]. The dotted line indicates the σ C predicted based on conventional LEFM. The predicted values of σ C based on LEFM are overestimate for √ area less than 20 µm. Thus, conventional LEFM is not applicable to ceramics with small cracks. On the other hand, the solid line indicates the σ C predicted based on Equation (5). Although the σ C predicted based on Equation (5) were slightly smaller than those of the experimental results, they agreed relatively well with the experimental results in the range of 5-20 µm. In this study, the relationship between √ area of an FIB defect and fracture strength was evaluated using K IC and σ 0 as the basic mechanical properties of the sample material.
This is the equation to evaluate the fracture strength of a ceramic material containing a surface defect with arbitrary shape. Figure 1 shows the relationship between σC and √ of alumina reinforced by silicon carbide (Al2O3/SiC) [11]. The dotted line indicates the σC predicted based on conventional LEFM. The predicted values of σC based on LEFM are overestimate for √ less than 20 µm. Thus, conventional LEFM is not applicable to ceramics with small cracks. On the other hand, the solid line indicates the σC predicted based on Equation (5). Although the σC predicted based on Equation (5) were slightly smaller than those of the experimental results, they agreed relatively well with the experimental results in the range of 5-20 µm. In this study, the relationship between √ of an FIB defect and fracture strength was evaluated using KIC and σ0 as the basic mechanical properties of the sample material.

Test Specimen
Alumina reinforced by silicon carbide (Al2O3/SiC) was selected as a test material in this study. The Al2O3 powder (AKP-50, Sumitomo Chemical Industry Co. Ltd., Tokyo, Japan) used in this study had a mean particle size of 0.2 µm. The SiC powder (ultrafine grade, Ibiden Co. Ltd., Gifu, Japan) used had a mean particle size of 0.27 µm. The samples were prepared using a mixture of Al2O3 with 15 vol % SiC powder. To this mixture, isopropanol was added, and the mixture was blended thoroughly for 24 h by ball milling. After drying and comminution, the mixture was hot-pressed at 1973 K and 35 MPa for 2 h in an N2 atmosphere [20]. The hot-pressed plates were cut into bending test specimens measuring 3 × 4 × 20 mm. The surface of each specimen was polished to a mirror-like finish.

Specimen Preparations
To heal surface cracks induced during machining, all the bending test specimens were heattreated at 1573 K for 1 h in air, which are the optimum healing conditions for surface cracks in Al2O3/SiC [20]. These specimens are called "smooth" specimens. The σ0 value was evaluated by the strength distribution of the smooth specimens.
An FIB defect was introduced at the center of each smooth specimen by using a multi beam SEM-FIB system JIB-4501 (by JEOL Ltd., Tokyo, Japan). These specimens are called the "FIB-defect"

Test Specimen
Alumina reinforced by silicon carbide (Al 2 O 3 /SiC) was selected as a test material in this study. The Al 2 O 3 powder (AKP-50, Sumitomo Chemical Industry Co. Ltd., Tokyo, Japan) used in this study had a mean particle size of 0.2 µm. The SiC powder (ultrafine grade, Ibiden Co. Ltd., Gifu, Japan) used had a mean particle size of 0.27 µm. The samples were prepared using a mixture of Al 2 O 3 with 15 vol % SiC powder. To this mixture, isopropanol was added, and the mixture was blended thoroughly for 24 h by ball milling. After drying and comminution, the mixture was hot-pressed at 1973 K and 35 MPa for 2 h in an N 2 atmosphere [20]. The hot-pressed plates were cut into bending test specimens measuring 3 × 4 × 20 mm. The surface of each specimen was polished to a mirror-like finish.

Specimen Preparations
To heal surface cracks induced during machining, all the bending test specimens were heat-treated at 1573 K for 1 h in air, which are the optimum healing conditions for surface cracks in Al 2 O 3 /SiC [20].
These specimens are called "smooth" specimens. The σ 0 value was evaluated by the strength distribution of the smooth specimens.
An FIB defect was introduced at the center of each smooth specimen by using a multi beam SEM-FIB system JIB-4501 (by JEOL Ltd., Tokyo, Japan). These specimens are called the "FIB-defect" specimens. Figure 2 shows the detail of the FIB technique process. A test specimen was fixed on a jig with carbon tape. The FIB defect was processed while monitoring the processing state using the SEM. Figure 3a schematically shows the shape of an FIB defect. The surface lengths (2c) ranged from 20 to 60 µm and the depths (d) ranged from 19 to 24 µm. The conditions of the FIB technique are presented in Table 1. The area of a defect was evaluated based on the fracture surface observation.
For comparison, "pre-crack + polished" specimens were prepared in the following procedure. First, a Knoop indentation was made at the center of each smooth specimen. Figure 3b shows the shape of a Knoop pre-crack. The indentation loads ranged from 9.8 to 49 N. Next, these specimens were polished using a 1-µm diamond slurry to eliminate the tensile residual stress around the indentations. The amount of material removed from the surface was 4 to 6 times the indentation depth. The ASTM standard C1421 suggests removing 4.5 to 5 times the indentation depth [21]. Thus, the removed depth in this study almost met the standard. specimens. Figure 2 shows the detail of the FIB technique process. A test specimen was fixed on a jig with carbon tape. The FIB defect was processed while monitoring the processing state using the SEM. Figure 3a schematically shows the shape of an FIB defect. The surface lengths (2c) ranged from 20 to 60 µm and the depths (d) ranged from 19 to 24 µm. The conditions of the FIB technique are presented in Table 1. The area of a defect was evaluated based on the fracture surface observation. For comparison, "pre-crack + polished" specimens were prepared in the following procedure. First, a Knoop indentation was made at the center of each smooth specimen. Figure 3b shows the shape of a Knoop pre-crack. The indentation loads ranged from 9.8 to 49 N. Next, these specimens were polished using a 1-µm diamond slurry to eliminate the tensile residual stress around the indentations. The amount of material removed from the surface was 4 to 6 times the indentation depth. The ASTM standard C1421 suggests removing 4.5 to 5 times the indentation depth [21]. Thus, the removed depth in this study almost met the standard.    specimens. Figure 2 shows the detail of the FIB technique process. A test specimen was fixed on a jig with carbon tape. The FIB defect was processed while monitoring the processing state using the SEM. Figure 3a schematically shows the shape of an FIB defect. The surface lengths (2c) ranged from 20 to 60 µm and the depths (d) ranged from 19 to 24 µm. The conditions of the FIB technique are presented in Table 1. The area of a defect was evaluated based on the fracture surface observation. For comparison, "pre-crack + polished" specimens were prepared in the following procedure. First, a Knoop indentation was made at the center of each smooth specimen. Figure 3b shows the shape of a Knoop pre-crack. The indentation loads ranged from 9.8 to 49 N. Next, these specimens were polished using a 1-µm diamond slurry to eliminate the tensile residual stress around the indentations. The amount of material removed from the surface was 4 to 6 times the indentation depth. The ASTM standard C1421 suggests removing 4.5 to 5 times the indentation depth [21]. Thus, the removed depth in this study almost met the standard.

Measurement of Fracture Toughness
The fracture toughness K IC value was evaluated by the indentation fracture (IF) method. Vickers indentations were made on the polished surface of a bending test specimen by the application of an indentation load of 49 N for 15 s. Then, K IC (Pa·m 1/2 ) was calculated using the following equation in accordance with the Japan Industrial Standards [22] where E (Pa) is the Young's modulus, P (N) is the indentation load, a (m) is half the diagonal length of the indentation, and c (m) is half the surface crack length. The units of calculated values of K IC were converted into "MPa·m 1/2 ". Crack length was measured using an optical microscope with total magnifications of 580× and higher to avoid the measurement error of K IC values [23].

Measurement of Fracture Strength
The σ C values of each specimen were measured by a three-point bending test, as shown in Figure 4, at room temperature in air. The span length was 16 mm, and the crosshead speed was 0.5 mm/min. After the bending test, the fracture surfaces were observed by laser microscopy, and the size of the defects at the fracture origins was measured.

Measurement of Fracture Toughness
The fracture toughness KIC value was evaluated by the indentation fracture (IF) method. Vickers indentations were made on the polished surface of a bending test specimen by the application of an indentation load of 49 N for 15 s. Then, KIC (Pa·m 1/2 ) was calculated using the following equation in accordance with the Japan Industrial Standards [22] = 0.026 , where E (Pa) is the Young's modulus, P (N) is the indentation load, a (m) is half the diagonal length of the indentation, and c (m) is half the surface crack length. The units of calculated values of KIC were converted into "MPa·m 1/2 ". Crack length was measured using an optical microscope with total magnifications of 580× and higher to avoid the measurement error of KIC values [23].

Measurement of Fracture Strength
The σC values of each specimen were measured by a three-point bending test, as shown in Figure  4, at room temperature in air. The span length was 16 mm, and the crosshead speed was 0.5 mm/min. After the bending test, the fracture surfaces were observed by laser microscopy, and the size of the defects at the fracture origins was measured.

Mechanical Properties of the Sample Material
In this section, the calculated results of KIC and σ0 are presented. An average and a standard deviation of the KIC value of four samples measured by the IF method were 3.50 ± 0.05 MPa·m 1/2 . The scatter of the KIC value is quite small.
The σ0 value of the sample material was estimated using the strength distribution of the smooth specimens. The two-parameter Weibull function is expressed by where F(σC) is the cumulative failure probability based on the median rank method, α is the scale parameter, and β is the shape parameter. Figure 5 shows the two-parameter Weibull distribution of the smooth specimens. The test results were plotted as the relationship between ln (σC) and ln{ln(1/(1 − F)} led by Equation (7). The second horizontal and vertical axis show σC and F values, respectively. Two parameters were determined using the maximum likelihood method based on JIS R1625 [24]. The α value was 8.81, and the β value was 871 MPa. In Figure 5, a thick solid line and thin solid lines show the approximation line and the 90% confidence interval, respectively. The 90% confidence interval values of α and β are (αL, αU) = (6.72, 11.6), (βL, βU) = (838, 910). In the Weibull distribution,

Mechanical Properties of the Sample Material
In this section, the calculated results of K IC and σ 0 are presented. An average and a standard deviation of the K IC value of four samples measured by the IF method were 3.50 ± 0.05 MPa·m 1/2 . The scatter of the K IC value is quite small.
The σ 0 value of the sample material was estimated using the strength distribution of the smooth specimens. The two-parameter Weibull function is expressed by where F(σ C ) is the cumulative failure probability based on the median rank method, α is the scale parameter, and β is the shape parameter. Figure 5 shows the two-parameter Weibull distribution of the smooth specimens. The test results were plotted as the relationship between ln (σ C ) and ln{ln(1/(1 − F)} led by Equation (7). The second horizontal and vertical axis show σ C and F values, respectively. Two parameters were determined using the maximum likelihood method based on JIS R1625 [24]. The α value was 8.81, and the β value was 871 MPa. In Figure 5, a thick solid line and thin solid lines show the approximation line and the 90% confidence interval, respectively. The 90% confidence interval values of α and β are (α L , α U ) = (6.72, 11.6), (β L , β U ) = (838, 910). In the Weibull distribution, the F(σ C ) value increased as the defect dimensions in the specimens decreased. Because the size of defects in a smooth the specimen with F(σ C ) = 99% considered to be sufficiently small, the σ C value at F(σ C ) = 99% is regarded as σ 0 . This evaluation method of σ 0 showed a good estimation results of σ C as shown in Figure 1 [11]. Substituting the values of α and β of the smooth specimen and F(σ C ) = 99% in Equation (7), σ 0 = 1036 MPa was calculated.
Materials 2018, 11, x FOR PEER REVIEW 6 of 9 the F(σC) value increased as the defect dimensions in the specimens decreased. Because the size of defects in a smooth the specimen with F(σC) = 99% considered to be sufficiently small, the σC value at F(σC) = 99% is regarded as σ0. This evaluation method of σ0 showed a good estimation results of σC as shown in Figure 1 [11]. Substituting the values of α and β of the smooth specimen and F(σC) = 99% in Equation (7), σ0 = 1036 MPa was calculated.

Fracture Strength of FIB-Defect Specimens
All specimens containing FIB-induced surface defects fractured at the FIB-defect area as a result of the three-point bending tests. Figure 6a,b show laser microscope images of fractured surfaces of a FIB-defect specimen and a pre-crack + polished specimen, respectively. The surface length 2c, the depth d, and the √ of the FIB-defects and pre-cracks after polishing were measured by these observations. Table 2 shows the results of the bending tests for the FIB-defect specimens and the precrack + polished specimens. The sizes of the FIB defects were in the following range: √ = 19-35 µm. The σC value of the FIB-defect specimens decreased with increasing √ . Similar to the precrack + polished specimens, σC showed strong √ -dependency in the FIB-defect specimens.

Fracture Strength of FIB-Defect Specimens
All specimens containing FIB-induced surface defects fractured at the FIB-defect area as a result of the three-point bending tests. Figure 6a,b show laser microscope images of fractured surfaces of a FIB-defect specimen and a pre-crack + polished specimen, respectively. The surface length 2c, the depth d, and the √ area of the FIB-defects and pre-cracks after polishing were measured by these observations. Table 2 shows the results of the bending tests for the FIB-defect specimens and the pre-crack + polished specimens. The sizes of the FIB defects were in the following range: √ area = 19-35 µm. The σ C value of the FIB-defect specimens decreased with increasing √ area. Similar to the pre-crack + polished specimens, σ C showed strong √ area-dependency in the FIB-defect specimens. the F(σC) value increased as the defect dimensions in the specimens decreased. Because the size of defects in a smooth the specimen with F(σC) = 99% considered to be sufficiently small, the σC value at F(σC) = 99% is regarded as σ0. This evaluation method of σ0 showed a good estimation results of σC as shown in Figure 1 [11]. Substituting the values of α and β of the smooth specimen and F(σC) = 99% in Equation (7), σ0 = 1036 MPa was calculated.

Fracture Strength of FIB-Defect Specimens
All specimens containing FIB-induced surface defects fractured at the FIB-defect area as a result of the three-point bending tests. Figure 6a,b show laser microscope images of fractured surfaces of a FIB-defect specimen and a pre-crack + polished specimen, respectively. The surface length 2c, the depth d, and the √ of the FIB-defects and pre-cracks after polishing were measured by these observations. Table 2 shows the results of the bending tests for the FIB-defect specimens and the precrack + polished specimens. The sizes of the FIB defects were in the following range: √ = 19-35 µm. The σC value of the FIB-defect specimens decreased with increasing √ . Similar to the precrack + polished specimens, σC showed strong √ -dependency in the FIB-defect specimens.

Comparison Between Predicted Fracture Strength and Test Results
The σ C of the sample material was predicted substituting K IC = 3.50 MPa·m 1/2 and σ 0 = 1036 MPa into Equation (5). Figure 7 shows the predicted σ C values and the bending test results. For the pre-crack + polished specimens, the predicted σ C values matched the experimental results. Thus, the usefulness of this equation was confirmed. The errors between the experimental and predicted results were generally within ±15%. It is thought that the prediction errors are mainly caused by the approximation errors of K max by Equation (3). For the FIB-defect specimens, the experimental σ C are between the predicted σ C based on LEFM and Equation (5). The predicted values of σ C based on LEFM and Equation (5) give the upper and lower limit, respectively. Similar tendency that Equation (5) gives lower σ C was observed in the results shown in Figure 1. The results indicate that a FIB-defect can be used as a small initial crack for fracture strength evaluation of ceramics. It was found that the evaluation equation was useful for evaluating the fracture strengths of ceramics containing micro surface defects introduced by FIB.

Comparison Between Predicted Fracture Strength and Test Results
The σC of the sample material was predicted substituting KIC = 3.50 MPa·m 1/2 and σ0 = 1036 MPa into Equation (5). Figure 7 shows the predicted σC values and the bending test results. For the precrack + polished specimens, the predicted σC values matched the experimental results. Thus, the usefulness of this equation was confirmed. The errors between the experimental and predicted results were generally within ±15%. It is thought that the prediction errors are mainly caused by the approximation errors of Kmax by Equation (3). For the FIB-defect specimens, the experimental σC are between the predicted σC based on LEFM and Equation (5). The predicted values of σC based on LEFM and Equation (5) give the upper and lower limit, respectively. Similar tendency that Equation (5) gives lower σC was observed in the results shown in Figure 1. The results indicate that a FIB-defect can be used as a small initial crack for fracture strength evaluation of ceramics. It was found that the evaluation equation was useful for evaluating the fracture strengths of ceramics containing micro surface defects introduced by FIB.

Conclusions
This study investigated the fracture strengths of Al2O3/SiC samples in which micro surface defects had been introduced by the FIB technique. The test results were evaluated using the proposed evaluation equation of fracture strength. The results are summarized as follows: (1) The micro surface defects in the size range of √ = 19-35 µm were introduced on the surface of Al2O3/SiC by the FIB technique.
(2) The fracture strengths of the FIB-defect specimens showed √ dependency. This result is similar to that obtained for the pre-crack + polished specimens.
(3) FIB-defect can be used as a small initial crack for fracture strength evaluation of ceramics. (4) The fracture strengths of the FIB-defect specimens predicted by the evaluation equation matched

Conclusions
This study investigated the fracture strengths of Al 2 O 3 /SiC samples in which micro surface defects had been introduced by the FIB technique. The test results were evaluated using the proposed evaluation equation of fracture strength. The results are summarized as follows: (1) The micro surface defects in the size range of √ area = 19-35 µm were introduced on the surface of Al 2 O 3 /SiC by the FIB technique.
(2) The fracture strengths of the FIB-defect specimens showed √ area dependency. This result is similar to that obtained for the pre-crack + polished specimens.
(3) FIB-defect can be used as a small initial crack for fracture strength evaluation of ceramics. (4) The fracture strengths of the FIB-defect specimens predicted by the evaluation equation matched the test results. It was confirmed that the proposed equation for fracture strength evaluation was useful for ceramics containing micro surface defects introduced by FIB.