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

materials

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

Nanako Sato 1 and Koji Takahashi 2,* ID 1 Graduate School of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama 240-8501, Japan; [email protected] 2 Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama 240-8501, Japan * Correspondence: [email protected]; Tel.: +81-45-339-4017

Received: 9 February 2018; Accepted: 19 March 2018; Published: 20 March 2018

Abstract: 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.

Keywords:√Al2O3/SiC; focused ion beam; surface defect; fracture strength; process zone size failure criterion; area parameter model

1. 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 [2–6]. 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.

Materials 2018, 11, 457; doi:10.3390/ma11030457 www.mdpi.com/journal/materials Materials 2018, 11, 457 2 of 9

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 [14–16]. 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 Al2O3/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.

2. 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 (σC) of ceramics with a small surface crack focusing on the process zone size at the crack tip and [7]

2 π KIC πσC = ae sec − 1 , (1) 8 σ0 2σ0

1/2 where KIC (MPa·m ) is the fracture toughness, and σ0 (MPa) is the fracture strength of a smooth specimen assuming a defect is sufﬁciently small, and σC (MPa) is the fracture strength of a specimen containing a surface crack. The equivalent crack length, ae (µm) is deﬁned as

2 1 KIC ae = , (2) π σc

On the other hand, Murakami showed that the maximum stress intensity factor (Kmax) at the edge of a surface crack with arbitrary shape under tensile stress (σ∞) could be approximated within 10% error by [19] q √ ∼ Kmax = 0.650σ∞ π area, (3) √ where area is the square root of the area of a surface defect projected onto the direction of the maximum tensile stress. When σC is applied to a surface crack, the Kmax value reaches KIC. Equation (3) leads to q √ KIC = 0.650σC π area . (4) Materials 2018, 11, 457 3 of 9

√ MaterialsThe 2018 relationship, 11, x FOR PEER between REVIEW fracture strength σC and the crack size area is expressed as follows3 of 9 based on Equations (1), (2) and (4) √ 3.382 √ 2 = cos− 3.38σ0 area ∙2σ0 . (5) σ = cos 1 √ · . (5) C 2 3.382 √ π·KIC + 3.38σ0 area π This is the equation to evaluate the fracture strength of a ceramic material containing a surface 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 defect with arbitrary shape. Figure1 shows the relationship between σC and area of alumina reinforced by silicon carbide (Al2O3/SiC) [11]. The dotted line indicates the σC predicted based on 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 conventional LEFM. The predicted values of σ based on LEFM are overestimate for area less than 20 µm. Thus, conventional LEFM is not applicableC to ceramics with small cracks. On the other hand, 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 the solid line indicates the σ predicted based on Equation (5). Although the σ predicted based on Equation (5) were slightly smallerC than those of the experimental results, they agreedC relatively well 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√ √ with the experimental results in the range of 5–20 µm. In this study, the relationship between area of of an FIB defect and fracture strength was evaluated using KIC and σ0 as the basic mechanical an FIB defect and fracture strength was evaluated using K and σ as the basic mechanical properties properties of the sample material. IC 0 of the sample material.

√ FigureFigure 1.1. RelationshipRelationship betweenbetween fracturefracture strengthstrength andand √areaof of Al Al2O2O3/SiC3/SiC [[11].11].

3.3. ExperimentalExperimental ProceduresProcedures

3.1.3.1. TestTest SpecimenSpecimen

AluminaAlumina reinforcedreinforced byby siliconsilicon carbidecarbide (Al(Al22OO33/SiC) was was selected selected as as a a test test material material in in this study. TheThe AlAl22OO33 powder (AKP-50, Sumitomo Chemical Industry Co. Ltd., Tokyo, Japan)Japan) usedused inin thisthis studystudy hadhad aa mean mean particle particle size size of of 0.2 0.2µm. µm. The The SiC SiC powder powder (ultraﬁne (ultrafine grade, grade, Ibiden Ibiden Co. Ltd.,Co. Ltd., Gifu, Gifu, Japan) Japan) used hadused a meanhad a particlemean particle size of 0.27size µofm. 0.27 The µm. samples The samples were prepared were prepared using a mixture using a ofmixture Al2O3 withof Al152O vol3 with % SiC15 vol powder. % SiC To thispowder. mixture, To isopropanolthis mixture, was isopro added,panol and was the mixtureadded, wasand blended the mixture thoroughly was blended for 24 h bythoroughly ball milling. for After24 h by drying ball milling. and comminution, After drying the and mixture comminution, was hot-pressed the mixture at 1973 was K and hot-pressed 35 MPa for at 21973 h in K an and N2 atmosphere35 MPa for 2 [20 h ].in The an hot-pressedN2 atmosphere plates [20]. were The cut hot-pressed into bending plates test were specimens cut into measuring bending 3test× 4specimens× 20 mm. measuring The surface 3 of× 4 each × 20 specimenmm. The wassurfac polishede of each to specimen a mirror-like was ﬁnish.polished to a mirror-like finish. 3.2. Specimen Preparations 3.2. SpecimenTo heal surface Preparations cracks induced during machining, all the bending test specimens were heat-treated at 1573To K heal for 1surface h in air, cracks which induced are the optimum during machining, healing conditions all the forbending surface test cracks specimens in Al2O were3/SiC heat- [20]. treated 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”

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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” specimens. Figure2 shows the detail of the FIB technique process. A test specimen was ﬁxed on a jig with carbonMaterials tape. 2018 The, 11, x FIB FOR defectPEER REVIEW was processed while monitoring the processing state using4 of 9 the SEM. Materials 2018, 11, x FOR PEER REVIEW 4 of 9 Figure3a schematicallyspecimens. Figure shows 2 shows the theshape detail of of the an FIB FIB tec defect.hnique process. The surface A test specimen lengths was (2 fixedc) ranged on a jig from 20 to 60 µm andwithspecimens. the carbon depths Figure tape. (d )The 2 ranged shows FIB defect the from detail was 19 processedof to the 24 FIBµ m.whiletechnique The monitoring conditions process. the A testprocessing of specimen the FIB state techniquewas using fixed the on SEM. area jig presented in Table1.Figure Thewith carbon area 3a schematically of tape. a defect The FIB shows was defect evaluatedthe was shape processed of basedan FIB while defect. on monitoring the The fracture surface the processing lengths surface (2 c observation.state) ranged using from the SEM.20 to Figure 3a schematically shows the shape of an FIB defect. The surface lengths (2c) ranged from 20 to For comparison,60 µm and the depths “pre-crack (d) ranged + polished” from 19 to 24 specimens µm. The conditions were preparedof the FIB technique in the followingare presented procedure. in60 Tableµm and 1. Thethe depthsarea of (ad )defe rangedct was from evaluated 19 to 24 based µm. The on conditionsthe fracture of surface the FIB observation. technique are presented First, a Knoopin TableFor indentation 1.comparison, The area wasof “pre-crack a defe madect was at+ polished” evaluated the center specimenbased of each on sthe were smooth fracture prepared specimen.surface in theobservation. following Figure 3 procedure.b shows the shape of a KnoopFirst, pre-crack.For a Knoop comparison, indentation The indentation“pre-crack was made + polished” loads at the rangedcenterspecimen of fromeachs were smooth 9.8prepared to specimen. 49 in N. the Next, followingFigure these 3b procedure.shows specimens the were polished usingshapeFirst, a of Knoop 1-a µKnoopm indentation diamond pre-crack. slurrywas The made indentation to eliminateat the centerloadsthe rangedof each tensile fromsmooth residual 9.8 specimen.to 49 N. stress Next, Figure around these 3b specimensshows the indentations.the shape of a Knoop pre-crack. The indentation loads ranged from 9.8 to 49 N. Next, these specimens The amountwere of polished material using removed a 1-µm from diamond the surfaceslurry to waseliminate 4 to 6the times tensile the residual indentation stress around depth. the The ASTM indentations.were polished The using amount a 1-µm of materialdiamond removed slurry to from eliminate the surface the tensile was 4 residualto 6 times stress the indentationaround the standard C1421depth.indentations. The suggests ASTM The standard removingamount ofC1421 material 4.5 suggests to5 removed times removing the from indentation 4.5 the to surface 5 times was depththe indentation4 to [ 216 times]. Thus, depth the indentation the[21]. removed Thus, depth in this studythedepth. almostremoved The ASTM metdepth thestandard in this standard. study C1421 almost suggests met removingthe standard. 4.5 to 5 times the indentation depth [21]. Thus, the removed depth in this study almost met the standard.

Figure 2. PhotographsFigure 2. Photographs of the of FIB the FIB process: process: (a (a)) ﬁxing fixing method method of test of testspecimen; specimen; (b) inside (b )of inside device. of device. Figure 2. Photographs of the FIB process: (a) fixing method of test specimen; (b) inside of device.

Figure 3. Schematic illustrations of surface defects: (a) FIB-defect specimen; (b) pre-crack + polished Figure 3. specimen.FigureSchematic 3. Schematic illustrations illustrations of surface surface defects: defects: (a) FIB-defect (a) FIB-defect specimen; (b specimen;) pre-crack + polished (b) pre-crack + polished specimen.specimen. Table 1. FIB processing conditions. Table 1. FIB processing conditions. TableIon Source 1. FIB processing Ga Liquid conditions. Metal Ions AcceleratingIon Source voltage Ga Liquid 30 kVMetal Ions AcceleratingProve current voltage 3000 30 kV pA IonProve SourceDose current Ga190–280 Liquid 3000 nC/µm pA Metal 2 Ions Dose 190–280 nC/µm2 Accelerating voltage 30 kV

Prove current 3000 pA µ 2 Dose 190–280 nC/ m

Materials 2018, 11, 457 5 of 9 Materials 2018, 11, x FOR PEER REVIEW 5 of 9

3.3.3.3. Measurement ofof FractureFracture ToughnessToughness

The fracture toughness KIC value was evaluated by the indentation fracture (IF) method. Vickers The fracture toughness KIC value was evaluated by the indentation fracture (IF) method. Vickers indentationsindentations werewere mademade onon thethe polishedpolished surfacesurface ofof aa bendingbending testtest specimenspecimen byby thethe applicationapplication ofof anan indentation load of 49 N for 15 s. Then, KIC (Pa·m1/21/2) was calculated using the following equation in indentation load of 49 N for 15 s. Then, KIC (Pa·m ) was calculated using the following equation in accordanceaccordance withwith thethe JapanJapan IndustrialIndustrial StandardsStandards [[22]22] 1 1 = 0.026E 2 P 2 a , (6) K = 0.026 , (6) IC 3 c 2 where E (Pa) is the Young’s modulus, P (N) is the indentation load, a (m) is half the diagonal length 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 of the indentation, and c (m) is half the surface crack length. The units of calculated values of K converted into “MPa·m1/2”. Crack length was measured using an optical microscope with totalIC were converted into “MPa·m1/2”. Crack length was measured using an optical microscope with total magnifications of 580× and higher to avoid the measurement error of KIC values [23]. magniﬁcations of 580× and higher to avoid the measurement error of KIC values [23].

3.4.3.4. Measurement ofof FractureFracture StrengthStrength

The σC values of each specimen were measured by a three-point bending test, as shown in Figure The σC values of each specimen were measured by a three-point bending test, as shown in Figure4, at4, roomat room temperature temperature in in air. air. The The span span length length was was 16 16 mm, mm, and and the the crosshead crosshead speed speed was was 0.5 0.5 mm/min.mm/min. AfterAfter thethe bendingbending test,test, thethe fracturefracture surfacessurfaces werewere observedobserved by laser microscopy, andand thethe sizesize ofof thethe defectsdefects atat thethe fracturefracture originsorigins waswas measured.measured.

FigureFigure 4.4. SchematicSchematic diagram diagram ofof the the three-point three-point loadin loadingg system system and and size size and and dimensions dimensions of test of testspecimen. specimen.

4.4. ResultsResults and EvaluationEvaluation ofof FractureFracture StrengthStrength

4.1. Mechanical Properties of the Sample Material 4.1. Mechanical Properties of the Sample Material

In this section, the calculated results of KIC and σ0 are presented. An average and a standard In this section, the calculated results of KIC and σ0 are presented. An average and a standard 1/2 deviation of the KIC value of four samples measured by the IF method were 3.50 ± 0.05 MPa·m . deviation of the KIC value of four samples measured by the IF method were 3.50 ± 0.05 MPa·m1/2. The The scatter of the KIC value is quite small. scatter of the KIC value is quite small. The σ0 value of the sample material was estimated using the strength distribution of the smooth The σ0 value of the sample material was estimated using the strength distribution of the smooth specimens.specimens. TheThe two-parametertwo-parameter WeibullWeibull functionfunction isis expressedexpressed byby α σC F(σC)= =1−exp−1 − exp − , ,(7) (7) β where F(σ ) is the cumulative failure probability based on the median rank method, α is the scale where F(σCC) is the cumulative failure probability based on the median rank method, α is the scale parameter,parameter, andand β isis the the shape shape parameter. parameter. Figure Figure 55 showsshows the the two-parameter two-parameter Weibull Weibull distribution distribution of of the smooth specimens. The test results were plotted as the relationship between ln (σ ) and the smooth specimens. The test results were plotted as the relationship between ln (σC) and ln{ln(1/(1C ln{ln(1/(1 − F)} led by Equation (7). The second horizontal and vertical axis show σ and F values, − F)} led by Equation (7). The second horizontal and vertical axis show σC and F values,C respectively. respectively.Two parameters Two were parameters determined were determinedusing the maximum using the likelihood maximum method likelihood based method on JIS based R1625 on [24]. JIS R1625The α value [24]. Thewasα 8.81,value and was the 8.81, β value and was the 871β value MPa. was In Figure 871 MPa. 5, a thick In Figure solid5 ,line a thick and solidthin solid line andlines thinshow solid the linesapproximation show the approximation line and the 90% line andconfiden the 90%ce interval, conﬁdence respectively. interval, respectively. The 90% confidence The 90% interval values of α and β are (αL, αU) = (6.72, 11.6), (βL, βU) = (838, 910). In the Weibull distribution,

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Materials 2018, 11, x FOR PEER REVIEW 6 of 9 Materials 2018, 11, x FOR PEER REVIEW 6 of 9 conﬁdence interval values of α and β are (αL, αU) = (6.72, 11.6), (βL, βU) = (838, 910). In the Weibull distribution,the F(σC) value the increasedF(σ ) value as increasedthe defect as dimensions the defect dimensionsin the specimens in the decreased. specimens decreased.Because the Because size of the F(σC) value increasedC as the defect dimensions in the specimens decreased. Because the size of thedefects size in of a defectssmooth in the a specimen smooth the with specimen F(σC) = 99% with consideredF(σ ) = 99% to be considered sufficiently to besmall, sufﬁciently the σC value small, at defects in a smooth the specimen with F(σC) = 99% consideredC to be sufficiently small, the σC value at theF(σCσ) = value99% is at regardedF(σ ) = 99% as σ is0. This regarded evaluation as σ . Thismethod evaluation of σ0 showed method a good of σ showedestimation a good results estimation of σC as F(σC) C= 99% is regardedC as σ0. This evaluation0 method of σ0 showed a good0 estimation results of σC as resultsshown ofin σFigureas shown 1 [11]. in Substituting Figure1[ 11 the]. Substituting values of α theand values β of the of smoothα and β specimenof the smooth and F specimen(σC) = 99% and in shown in FigureC 1 [11]. Substituting the values of α and β of the smooth specimen and F(σC) = 99% in FEquation(σ ) = 99% (7), in σ0 Equation = 1036 MPa (7), wasσ = calculated. 1036 MPa was calculated. EquationC (7), σ0 = 1036 MPa was0 calculated.

Figure 5. Two-parameter Weibull distribution of fracture strength of the smooth specimens. FigureFigure 5. 5. Two-parameterTwo-parameter Weibull Weibull distribution of fracturefracture strength of the smooth specimens.

4.2. Fracture Strength of FIB-Defect Specimens 4.2. Fracture Fracture Strength of FIB-Defect Specimens All specimens containing FIB-induced surface defects fractured at the FIB-defect area as a result 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 of the three-point bending tests. Figure 66a,ba,b showshow laserlaser microscopemicroscope imagesimages ofof fracturedfractured surfacessurfaces ofof aa FIB-defect specimen and a pre-crack + polished specimen, respectively. The surface length 2c, the FIB-defect specimenspecimen andand a a pre-crack pre-crack + polished+ polished specimen, specimen, respectively. respectively. The surfaceThe surface length length 2c, the 2 depthc, the depth d, and√ the √ of the FIB-defects and pre-cracks after polishing were measured by these depthd, and thed, andarea theof √ the FIB-defects of the FIB-defects and pre-cracks and pre-cracks after polishing after were polishing measured were by measured these observations. by these observations. Table 2 shows the results of the bending tests for the FIB-defect specimens and the pre- observations.Table2 shows theTable results 2 shows of the the bending results testsof the for bend theing FIB-defect tests for specimens the FIB-defect√ and the specimens pre-crack and + polished the pre- crack + polished specimens. The sizes of the FIB defects were in the following range:µ √ = 19–35 crackspecimens. + polished The sizesspecimens. of the FIBThe defects sizes of were the FIB in the defects following were√ range:in the followingarea = 19–35 range:m √. The σ =C 19–35value µm. The σC value of the FIB-defect specimens decreased with increasing √. Similar to the pre- µm.of the The FIB-defect σC value specimensof the FIB-defect decreased√ specimens with increasing decreased witharea .increasing Similar to √ the pre-crack. Similar +to polishedthe pre- crack + polished specimens, σC showed strong √-dependency in the FIB-defect specimens. crackspecimens, + polishedσC showed specimens, strong σC showedarea-dependency strong √ in the-dependency FIB-defect specimens.in the FIB-defect specimens.

√ Figure 6. LaserLaser microscope microscope images images of of fracture fracture surfaces: surfaces: (a)( FIB-defecta) FIB-defect specimen, specimen, √area = 19.1= 19.1 µm,µ m,σC Figure 6. Laser microscope images of fracture surfaces:√ (a) FIB-defect specimen, √ = 19.1 µm, σC = 687 MPa; (b) pre-crack + polished specimen, √ = 64.6 µm, σC = 400 MPa. =σ C687= 687MPa; MPa; (b) pre-crack (b) pre-crack + polished + polished specimen, specimen, √area = 64.6= 64.6 µm,µ m,σC σ= C400= 400MPa. MPa.

Table 2. Bending test results. Table 2. Bending test results. Surface Fracture Surface Depth √ Fracture Type of Defect Length Depth √ Strength Type of Defect Length d (µm) (µm) Strength 2c (µm) d (µm) (µm) σC (Mpa) 2c (µm) σC (Mpa) FIB-defect test 20.8 19.4 19.1 687 FIB-defect test 20.8 19.4 19.1 687 specimen 20.0 23.2 20.4 715 specimen 20.0 23.2 20.4 715

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Table 2. Bending test results.

Surface Length 2c √ Fracture Strength Type of Defect Depth d (µm) area (µm) (µm) σC (Mpa) Materials 2018, 11, x FOR PEER REVIEW 7 of 9 20.8 19.4 19.1 687 20.0 23.2 20.4 715 33.4 21.1 25.3 597 33.4 21.1 25.3 597 FIB-defect test 32.332.3 22.5 26.2 26.2 557 557 specimen 32.832.8 23.8 27.4 27.4 592 592 46.246.2 24.1 32.6 32.6 530 530 60.160.1 21.3 35.2 35.2 508 508 Pre-crackPre-crack + + 121121 44.0 64.6 64.6 400 400 polishedpolished test 124124 47.5 68.0 68.0 322 322 specimentest specimen 228228 106 137 137 253 253

4.3.4.3. ComparisonComparison BetweenBetween PredictedPredicted FractureFracture StrengthStrength andand TestTest Results Results

The σC of the sample material was predicted substituting KIC = 3.50 MPa·m1/21/2 and σ0 = 1036 MPa The σC of the sample material was predicted substituting KIC = 3.50 MPa·m and σ0 = 1036 MPa into Equation (5). Figure 7 shows the predicted σC values and the bending test results. For the pre- into Equation (5). Figure7 shows the predicted σC values and the bending test results. crack + polished specimens, the predicted σC values matched the experimental results. Thus, the For the pre-crack + polished specimens, the predicted σC values matched the experimental results. Thus,usefulness the usefulness of this equation of this was equation confirmed. was The conﬁrmed. errors between The errors the experimental between the and experimental predicted results and predictedwere generally results werewithin generally ±15%. withinIt is thought±15%. Itthat is thoughtthe prediction that the predictionerrors are errors mainly are caused mainly causedby the approximation errors of Kmax by Equation (3). For the FIB-defect specimens, the experimental σC are by the approximation errors of Kmax by Equation (3). For the FIB-defect specimens, the experimental between the predicted σC based on LEFM and Equation (5). The predicted values of σC based on LEFM σC are between the predicted σC based on LEFM and Equation (5). The predicted values of σC based onand LEFM Equation and Equation (5) give (5)the give upper the and upper lower and lowerlimit, limit,respectively. respectively. Similar Similar tendency tendency that that Equation Equation (5) gives lower σC was observed in the results shown in Figure 1. The results indicate that a FIB-defect (5) gives lower σC was observed in the results shown in Figure1. The results indicate that a FIB-defect cancan bebe usedused asas aa smallsmall initialinitial crackcrack forfor fracturefracture strengthstrength evaluationevaluation ofof ceramics.ceramics. ItIt waswas foundfound thatthat thethe evaluationevaluation equationequation waswas usefuluseful forfor evaluatingevaluating thethe fracturefracture strengthsstrengths ofof ceramicsceramics containingcontaining micromicro surfacesurface defectsdefects introducedintroduced byby FIB.FIB.

FigureFigure 7.7. ComparisonComparison betweenbetween thethe predictedpredicted fracturefracture strengthstrength andand thethe bendingbendingtest testresults. results.

5.5. ConclusionsConclusions

ThisThis studystudy investigatedinvestigated thethe fracturefracture strengthsstrengths ofof Al Al2O2O33/SiC/SiC samples in whichwhich micromicro surfacesurface defectsdefects hadhad beenbeen introducedintroduced byby thethe FIBFIB technique.technique. TheThe testtest resultsresults werewere evaluatedevaluated usingusing thethe proposedproposed evaluationevaluation equationequation ofof fracturefracture strength.strength. TheThe resultsresults areare summarizedsummarized as as follows: 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 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.

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√ (1) The micro surface defects in the size range of area = 19–35 µm were introduced on the surface of Al O /SiC by the FIB technique. 2 3 √ (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.

Acknowledgments: This study was supported in part by a grant for Scientific Research (B) (16H04231) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. Author Contributions: N. Sato and K. Takahashi conceived and designed the experiments; N. Sato performed the experiments and evaluated the data; N. Sato and K. Takahashi wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Usami, S.; Kimoto, H.; Takahashi, I.; Shida, S. Strength of ceramic materials containing small flaws. Eng. Fract. Mech. 1986, 23, 745–761. [CrossRef] 2. Hoshide, T. Grain fracture model and its application to strength evaluation in engineering ceramics. Eng. Fract. Mech. 1993, 44, 403–408. 3. Li, C.W.; Lui, S.C.; Goldacker, J. Relation between strength, microstructure, and grain—Bridging characteristics in in situ reinforced silicon nitride. J. Am. Ceram. Soc. 1995, 78, 449–459. [CrossRef] 4. Taylor, D. Predicting the fracture strength of ceramic materials using the theory of critical distances. Eng. Fract. Mech. 2004, 71, 2407–2416. [CrossRef] 5. Munz, D. What can we learn from R-curve measurements? J. Am. Ceram. Soc. 2007, 90, 1–15. [CrossRef] 6. Askes, H.; Susmel, L. Understanding cracked materials: Is linear elastic fracture mechanics obsolete? Fatigue Fract. Eng. Mater. Struct. 2015, 38, 154–160. [CrossRef] 7. Ando, K.; Kim, B.A.; Iwasa, M.; Ogura, N. Process zone size failure criterion and probabilistic fracture assessment curves for ceramics. Fatigue Fract. Eng. Mater. Struct. 1992, 15, 139–149. [CrossRef] 8. Murakami, Y.; Endo, M. Quantitative evaluation of fatigue strength of metals containing various small defects or cracks. Eng. Fract. Mech. 1983, 17, 1–15. 9. Saberifar, S.; Mashreghi, A.R.; Mosalaeepur, M.; Ghasemi, S.S. The interaction between non-metallic inclusions and surface roughness in fatigue failure and their influence on fatigue strength. Mater. Des. 2012, 35, 720–724. [CrossRef] 10. Beretta, S.; Romano, S. A comparison of fatigue strength sensitivity to defects for materials manufactured by AM or traditional processes. Int. J. Fatigue 2017, 94, 178–191. [CrossRef] 11. Sato, N.; Takahashi, K. Prediction of fracture strength for ceramics containing a surface defect with arbitrary shape. Trans. Jpn. Soc. Spring Eng. 2018, 63. in press. 12. Marshall, D.B.; Lawn, B.R. Residual stress effects in sharp contact cracking: Indentation fracture mechanics. J. Mater. Sci. 1979, 14, 2001–2012. [CrossRef] 13. Chou, I.A.; Chan, H.M.; Harmer, M.P. Effect of Annealing Environment on the Crack Healing and Mechanical Behavior of Silicon Carbide-Reinforced Alumina Nanocomposites. J. Am. Ceram. Soc. 1998, 81, 1203–1208. [CrossRef] 14. Li, Z.; Ghosh, A.; Kobayashi, A.; Bradt, R.C. Indentation fracture toughness of sintered silicon carbide in the Palmquist crack regime. J. Am. Ceram. Soc. 1989, 72, 904–911. [CrossRef] 15. Takahashi, K.; Iwanaka, K.; Osada, T.; Koike, H. Increase in strength of partially stabilized zirconia after shot peening. J. Mater. Eng. Perform. 2015, 24, 3573–3578. [CrossRef] 16. Corić,D.;´ Renjo, M.M.; Curković,L.´ Vickers indentation fracture toughness of Y-TZP dental ceramics. Int. J. Refract. Met. Hard Mater. 2017, 64, 14–19. [CrossRef] Materials 2018, 11, 457 9 of 9

17. Sakamoto, J.; Takahashi, Y.; Aono, Y.; Noguchi, H. Method for Assessing Applicability of an Artiﬁcial Flaw as a Small initial crack for fatigue limit evaluation and its application to a drill Hole and an FIB processed sharp notch in annealed 0.45% carbon steel. J. Test. Eval. 2013, 41, 194–199. [CrossRef] 18. Roiko, A.; Solin, J. Measurement of small cracks initiating from inclusions, Focused Ion Beam notches and drilled holes. Int. J. Fatigue 2014, 62, 154–158. [CrossRef] 19. Murakami, Y. Analysis of stress intensity factors of modes I, II and III for inclined surface cracks of arbitrary shape. Eng. Fract. Mech. 1985, 22, 101–114. [CrossRef] 20. Osada, T.; Nakao, W.; Takahashi, K.; Ando, K.; Saito, S. Kinetics of Self-crack-healing of alumina/Silicon carbide composite including oxygen partial pressure effect. J. Am. Ceram. Soc. 2009, 92, 864–869. [CrossRef] 21. ASTM International. Standard Test Methods for Determination of Fracture Toughness of Advanced Ceramics at Ambient Temperature; ASTM C1421-10; ASTM International: West Conshohocken, PA, USA, 2010. 22. Japanese Standards Association. Testing Methods for Fracture Toughness of Fine Ceramics at Room Temperature; JIS R1607; Japanese Standards Association: Tokyo, Japan, 2015. 23. Miyazaki, H.; Yoshizawa, Y. Correlation of the indentation fracture resistance measured using high-resolution optics and the fracture toughness obtained by the single edge-notched beam (SEPB) method for typical structural ceramics with various microstructures. Ceram. Int. 2016, 42, 7873–7876. [CrossRef] 24. Japanese Standards Association. Weibull Statics of Strength Data for Fine Ceramics; JIS R1607; Japanese Standards Association: Tokyo, Japan, 2010.

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