PHILOSOPHICAL MAGAZINE. 2003. VOL. 83. No. 6.745-76 Taylor Francis Combined mode-I and mode-II fracture of ceramics with crack-face grain and or whisker bridging TAKASHI AKATSUT, KENJI SAITO, YASUHIRO TANABE and EIICHI YASUDA Materials and Structures Laboratory, Tokyo Institute of Technology Nagatsuta 4259, Midori, Yokohama 226-8503, Japan [Received I May 2002 and accepted in revised form 17 September 2002 ABSTRACT Crack-face grain and or whisker bridging in ceramics was investigated unde combined mode-I and mode-ll loading. A novel technique for analysing the stress shielding at the crack tip caused by the bridging was proposed, in which the critical values of the local mode-I and mode-II stress intensity factors were numerically derived from an azimuthal angle at the onset of noncoplanar crack xtension using the mixed-mode failure criteria. The wedging effect, which induced local mode-I crack opening at the tip, was identified under the combined-mode loading on polycrystalline alumina as well as an alumina atrix composite reinforced with silicon carbide whiskers. The ef accelerated with the increase in the mode-II comps, effective for nd loading and the decrease in the bridging zone length. stress shielding due to the whisker bridging was not only effective for mode-I but also for mode-lI crack opening. §1. INTRODUCTION The low fracture toughness of ceramics is undesirable for expanding their application for structural components. One of the most effective ways to toughen ceramics is to reinforce them with fibres or whiskers. It is well known that the high fracture toughness of ceramic composites reinforced with fibres or whiskers is ypically achieved by stress shielding due to crack-face bridging(Evans 1990, Becher 1991, Akatsu et al. 1999), which decreases the stress concentration at the crack tip. The stress shielding toughening in the composites has been certified and discussed mostly under nominally applied pure mode-I loading. It is, however, very important to examine the shielding under combined mode-L, mode-II and/or mode- III loading, because cracks in the composites may be oriented at an arbitrary angle to the far-field loading direction or subjected to multiaxial stresses Failure criteria under mixed-mode loading have been proposed including: (i the maximum hoop stress(MHS) criterion(Erdogan and Sih 1963) (i the maximum strain-energy release rate (MEr) criterion(Nuismer 1975 (ii) the minimum strain-energy density(MED) criterion(Sih 1974)and (iv) the Singh-Shetty (1989)empirical criterion T Author for correspondence. Email: takashi akatsu@msl titech ac jp Philosopical Magazine IssN 1478-6435 print/ISSN 1478-6443 online 2003 Taylor Francis Ltd ttp//www.tandf.co.uk/journals DOl:10.1080/014186102100046303
PHILOSOPHICAL MAGAZINE, 2003, VOL. 83, NO. 6, 745–764 Combined mode-I and mode-II fracture of ceramics with crack-face grain and/or whisker bridging Takashi Akatsuy, Kenji Saito, Yasuhiro Tanabe and Eiichi Yasuda Materials and Structures Laboratory, Tokyo Institute of Technology, Nagatsuta 4259, Midori, Yokohama 226-8503, Japan [Received 1 May 2002 and accepted in revised form 17 September 2002] Abstract Crack-face grain and/or whisker bridging in ceramics was investigated under combined mode-I and mode-II loading. A novel technique for analysing the stress shielding at the crack tip caused by the bridging was proposed, in which the critical values of the local mode-I and mode-II stress intensity factors were numerically derived from an azimuthal angle at the onset of noncoplanar crack extension using the mixed-mode failure criteria. The wedging effect, which induced local mode-I crack opening at the tip, was identified under the combined-mode loading on polycrystalline alumina as well as an alumina matrix composite reinforced with silicon carbide whiskers. The effect was accelerated with the increase in the mode-II component of nominally applied loading and the decrease in the bridging zone length. It was also found that the stress shielding due to the whisker bridging was not only effective for mode-I but also for mode-II crack opening. } 1. Introduction The low fracture toughness of ceramics is undesirable for expanding their application for structural components. One of the most effective ways to toughen ceramics is to reinforce them with fibres or whiskers. It is well known that the high fracture toughness of ceramic composites reinforced with fibres or whiskers is typically achieved by stress shielding due to crack-face bridging (Evans 1990, Becher 1991, Akatsu et al. 1999), which decreases the stress concentration at the crack tip. The stress shielding toughening in the composites has been certified and discussed mostly under nominally applied pure mode-I loading. It is, however, very important to examine the shielding under combined mode-I, mode-II and/or modeIII loading, because cracks in the composites may be oriented at an arbitrary angle to the far-field loading direction or subjected to multiaxial stresses. Failure criteria under mixed-mode loading have been proposed including: (i) the maximum hoop stress (MHS) criterion (Erdogan and Sih 1963), (ii) the maximum strain-energy release rate (MER) criterion (Nuismer 1975), (iii) the minimum strain-energy density (MED) criterion (Sih 1974) and (iv) the Singh–Shetty (1989) empirical criterion. Philosophical Magazine ISSN 1478–6435 print/ISSN 1478–6443 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/0141861021000046303 { Author for correspondence. Email: takashi_akatsu@msl.titech.ac.jp
t Akatsu et al The main concern in previous studies dealing with mixed-mode fracture has been to predict reliably the azimuthal angle 0c and the critical load Pc for noncoplanar crack propagation under an untried combination of nominal mode-I, mode-II and or mode-III loading through the criteria. However, the discrepancy between the prediction and observation has been recognized frequently. Even an empirical para meter without any physical insight is utilized to eliminate the discrepancy(Singh and Shetty 1989). It is important not only to predict Bc and Pc precisely but also to elucidate the discrepancy. The discrepancy observed in alumina (Li and Sakai 1996)and graphite (Li et al. 1999) materials consisting of rather coarse grains has attributed to the wedging effect. The effect, which induces local mode-I crack open- ing at the tip under combined-mode loading, is caused by sliding between deflected crack faces. This was confirmed through the model calculation of a two-dimensional eriodic zigzag-crack (Tong et al. 1995a, b, Carlson and Beevers 1985) under the MHS criterion. The wedging effect seems to be plausible, but the model used for the estimation of the effect is too simple to describe real crack propagation in ceramic polycrystals and composites, because the crack deflection is actually demonstrated in three dimensions with large irregularity. We need to develop a further and realistic consideration of the stress shielding due to crack-face interlocking and or bridging under combined-mode loading. In this paper, a novel technique for analysing the stress shielding under com- bined-mode loading is introduced, in which the critical values of the local mode-I and mode-II stress intensity factors at a crack tip are numerically and individually derived from ]c. According to the technique, the stress disturbance for each mode can be estimated without using any models of the crack-face interaction. Three kinds of ceramic are adopted to make a mutual comparison of the stress shielding under mixed-mode loading; float glass for no bridging, polycrystalline alumina for relatively weak crack-face grain interlocking, and an alumina matrix composite reinforced with silicon carbide whiskers for strong crack-face bridging. The stress shielding under mixed-mode loading is also examined as a function of the bridging zone length. Crack-face bridging under mixed-mode loading is discussed in detail $2. ANALYSIS METHODOLOGY 2.1. Fracture criteria under combined-mode crack opening cture criteria under combined-mode crack opening are generally grouped into three categories as follows The MHS criterion (Erdogan and Sih 1963). The mode-I stress intensity factor KI of a crack subjected to nominally applied mixed-mode loading is given as a function of the parametric angle 0 around a crack tip(figure 1)as follows Ki(e)=cos K,()cos(/->Kn(0)sin e where Ki(0)and Kn(0)are the stress intensity factors defined in the direction of the crack-face for mode-I and mode- lI crack openings respectively. Under MHS, the crack begins to propagate when the following condition is satis- fied
The main concern in previous studies dealing with mixed-mode fracture has been to predict reliably the azimuthal angle c and the critical load Pc for noncoplanar crack propagation under an untried combination of nominal mode-I, mode-II and/ or mode-III loading through the criteria. However, the discrepancy between the prediction and observation has been recognized frequently. Even an empirical parameter without any physical insight is utilized to eliminate the discrepancy (Singh and Shetty 1989). It is important not only to predict c and Pc precisely but also to elucidate the discrepancy. The discrepancy observed in alumina (Li and Sakai 1996) and graphite (Li et al. 1999) materials consisting of rather coarse grains has attributed to the wedging effect. The effect, which induces local mode-I crack opening at the tip under combined-mode loading, is caused by sliding between deflected crack faces. This was confirmed through the model calculation of a two-dimensional periodic zigzag-crack (Tong et al. 1995a, b, Carlson and Beevers 1985) under the MHS criterion. The wedging effect seems to be plausible, but the model used for the estimation of the effect is too simple to describe real crack propagation in ceramic polycrystals and composites, because the crack deflection is actually demonstrated in three dimensions with large irregularity. We need to develop a further and realistic consideration of the stress shielding due to crack-face interlocking and/or bridging under combined-mode loading. In this paper, a novel technique for analysing the stress shielding under combined-mode loading is introduced, in which the critical values of the local mode-I and mode-II stress intensity factors at a crack tip are numerically and individually derived from c. According to the technique, the stress disturbance for each mode can be estimated without using any models of the crack-face interaction. Three kinds of ceramic are adopted to make a mutual comparison of the stress shielding under mixed-mode loading; float glass for no bridging, polycrystalline alumina for relatively weak crack-face grain interlocking, and an alumina matrix composite reinforced with silicon carbide whiskers for strong crack-face bridging. The stress shielding under mixed-mode loading is also examined as a function of the bridging zone length. Crack-face bridging under mixed-mode loading is discussed in detail. } 2. Analysis methodology 2.1. Fracture criteria under combined-mode crack opening Fracture criteria under combined-mode crack opening are generally grouped into three categories as follows. (i) The MHS criterion (Erdogan and Sih 1963). The mode-I stress intensity factor KI of a crack subjected to nominally applied mixed-mode loading is given as a function of the parametric angle around a crack tip (figure 1) as follows: KIð Þ¼ cos 2 KIð Þ0 cos2 2 3 2 KIIð Þ0 sin ; ð1Þ where KIð0Þ and KIIð0Þ are the stress intensity factors defined in the direction of the crack-face for mode-I and mode-II crack openings respectively. Under MHS, the crack begins to propagate when the following condition is satis- fied: 746 T. Akatsu et al.
Combined mode-l and mode-ll fracture of ceramics 747 K lla KIrin(e 0)食 Crack sKip(e KIln(o KIri(o) Klip(8) KI Figure 1. Schematic illustration of local mode-I crack opening and mode-lI crack opening at the tip. The local stress intensity factors are characterized by the parametric {K1()}=Klc where the value defined at the onset of crack extension is denoted by the ubscript c, ee is a crack deflection angle with respect to the crack face, at which Ki reaches the maximum, given as =2m1A0)(x0/y 1∫1{K(o)}e1[{K(0)} (3) and Kle is the critical value of KI for crack propagation, which coincides with the local fracture toughness at the tip (ii) The MER criterion (Nuismer 1975). The strain energy release rate g of a crack undergoing nominal mixed-mode loading is given as a function of 0, as ()=h2(){K1(0)}2+2h2(0)K1(0)Kn(0)+h2(6){Kn(O)}2](4) whe for plane stress E
KI c f g ð Þ c ¼ KIc ð2Þ where the value defined at the onset of crack extension is denoted by the subscript c, c is a crack deflection angle with respect to the crack face, at which KI reaches the maximum, given as c ¼ 2 tan1 1 4 f g KIð Þ0 c f g KIIð Þ0 c þ 1 4 fKIð Þg 0 c f g KIIð Þ0 c 2 þ 8 1=2 ð3Þ and KIc is the critical value of KI for crack propagation, which coincides with the local fracture toughness at the tip. (ii) The MER criterion (Nuismer 1975). The strain energy release rate g of a crack undergoing nominal mixed-mode loading is given as a function of , as follows: gð Þ¼ 1 E 0 ½h1ð Þ f g KIð Þ0 2 þ 2h12ð Þ KIð Þ0 KIIð Þþ 0 h2ð Þ f g KIIð Þ0 2 ð4Þ where E0 ¼ E for plane stress; E 1 2 for plane strain; 8 < : ð5Þ Combined mode-I and mode-II fracture of ceramics 747 Figure 1. Schematic illustration of local mode-I crack opening and mode-II crack opening at the tip. The local stress intensity factors are characterized by the parametric angle .
T. Akatsu et al h1(6) (1+cos0)(1-0032+0.041) h2(6) sin(1-0.0032+0027) h()=5s2(5)(5-3s804+si0018+09 E is Youngs modulus and v is Poissons ratio Under the MEr criterion, the crack starts to extend when the following is satisfied ig(ec)s=ge where 8 at which g reaches the maximum is derived from the following C2(6){K1(0) with C1()=20(5)n6(1-008+003) (10) C2(0)=c(2/(3cos-1)+0242-0s and ge is the critical value of g correlated with Kle through ge=KI/E' (ii) The MED criterion (Sih 1974). The strain-energy density factor S is defined as lollows S=a(){K1(0)}2+2a2(0)k(O)Kn(0)+a2(6){Kn(0) (0)=1(1+cos O)(-cos 8 6(2 cos 0 (12) ()={(+1)(1-cos6)+(1+cosb(3cos6-1)} u is the rigidity given by and K is given as follows
h1ð Þ¼ 1 2 cos2 2 ð Þ 1 þ cos 1 0:0032 þ 0:0414 ; h12ð Þ¼ cos2 2 sin 1 0:0032 þ 0:0274 ð6Þ h2ð Þ¼ 1 2 cos2 2 ð Þ 5 3 cos 1 þ sin2 ½ Þ 0:168 þ 0:02 cosð3 ¼ 2 p ; ð7Þ E is Young’s modulus and is Poisson’s ratio. Under the MER criterion, the crack starts to extend when the following is satisfied: g c f g ð Þ c ¼ gc; ð8Þ where c at which g reaches the maximum is derived from the following relationship: C1 c ð Þ C2 c ð Þ ¼ f g KIIð Þ0 c f g KIð Þ0 c ; ð9Þ with C1ð Þ¼ 1 2 cos 2 sin 1 0:048 2 þ 0:033 4 ; C2ð Þ¼ 1 2 cos 2 ð Þþ 3 cos 1 0:2422 0:085 4 ð10Þ and gc is the critical value of g correlated with KIc through gc ¼ K2 Ic=E 0 . (iii) The MED criterion (Sih 1974). The strain-energy density factor S is defined as follows: S ¼ a1ð Þ f g KIð Þ0 2 þ 2a12ð Þ KIð Þ0 KIIð Þþ 0 a2ð Þ f g KIIð Þ0 2 ; ð11Þ where a1ð Þ¼ 1 16 ð Þ 1 þ cos ð Þ cos ; a12ð Þ¼ 1 16 sin ð2 cos þ 1Þ; ð12Þ a2ð Þ¼ 1 16 f g ð Þ þ 1 ð Þþ 1 cos ð Þ 1 þ cos ð Þ 3 cos 1 ; is the rigidity given by ¼ E 2 1ð Þ þ ð13Þ and is given as follows: 748 T. Akatsu et al.
Combined mode-/and mode-ll fracture of ceramics 749 3-v for plane stress, (14) 3-4v for plane strain the MEd criterion, a crack suffering nominal mixed-mode loading to propagate at an angle Be where S reaches a minimum, when the Ing criterion is met: Sc where S is the critical value of S given by S=[(1-2v)/4u]kie for plane strain, and Smin is the minimum of s determined under the following con- where ee is driven from the following formula [Kn(o))) sin 0c(-6cos 8e+K-1) {K1(0)}e {k(0)2-2sm2 +sin6(2cos6-K+1)=0 Further modification of the criteria is often carried out to minimize the discre- pancy between the predicted and the observed values of (Ki(O))e and (Kn(0) through the Singh-Shetty formula (iv) The Singh-Shetty(1989)empirical criterion. In each theory described above a series of a combination of (K(O))c and (Kn(O)c under an arbitrary mix ture of mode-I and mode-II loading are represented as an envelope in a KI versus Ku diagram (figure 2). Singh and Shetty (1989) introduced the following formula to draw the envelope in the diagram iK1(O)e_((Kn(0)))2 where C is the parametric constant utilized to fit the prediction to the obser 2. 2. Determination of stress shielding at a crack tip 6 The stress shielding due to crack-face interlocking and/or bridging decreases the ess concentration at the tip. The local stress intensity factors at the tip in the ck-face direction, Kltip() for mode-I crack opening and Klltip(O) for mode-I crack opening, are simply given at the onset of crack propagation as follows {K1m(O0)}=(K1)-kB [Klltip(O)Jc=(Kla)e-Klb where Kla and Kla are the nominal stress intensity factors for mode-I crack and mode-II crack opening respectively. Klb and Klb are the shielded stress intensity
¼ 3 1 þ for plane stress; 3 4 for plane strain: 8 >: ð14Þ Under the MED criterion, a crack suffering nominal mixed-mode loading begins to propagate at an angle c where S reaches a minimum, when the following criterion is met: Smin ¼ Sc; ð15Þ where Sc is the critical value of S given by Sc ¼ ½ð1 2Þ=4 K2 Ic for plane strain, and Smin is the minimum of S determined under the following condition: oS oat ¼c ¼ 0; ð16Þ where c is driven from the following formula: f g KIIð Þ0 c f g KIð Þ0 c 2 sin cð Þ 6 cos c þ 1 þ 2 f g KIIð Þ0 c f g KIð Þ0 c 2 cos2 c 2 si n2 c ð Þ 1 cos c þ sin c ð Þ¼ 2 cos c þ 1 0: ð17Þ Further modification of the criteria is often carried out to minimize the discrepancy between the predicted and the observed values of {KI(0)}c and {KIIð0Þgc through the Singh–Shetty formula. (iv) The Singh–Shetty (1989) empirical criterion. In each theory described above, a series of a combination of {KI(0)}c and {KII(0)}c under an arbitrary mixture of mode-I and mode-II loading are represented as an envelope in a KI versus KII diagram (figure 2). Singh and Shetty (1989) introduced the following formula to draw the envelope in the diagram: f g KIð Þ0 c KIc þ f g KIIð Þ0 c CKIc 2 ¼ 1; ð18Þ where C is the parametric constant utilized to fit the prediction to the observation. 2.2. Determination of stress shielding at a crack tip The stress shielding due to crack-face interlocking and/or bridging decreases the stress concentration at the tip. The local stress intensity factors at the tip in the crack-face direction, KItip(0) for mode-I crack opening and KIItip(0) for mode-II crack opening, are simply given at the onset of crack propagation as follows: KItipð Þ0 c ¼ KIa ð ÞcKIb; ð19Þ fKIItipð Þg 0 c ¼ KIIa ð ÞcKIIb; ð20Þ where KIa and KIIa are the nominal stress intensity factors for mode-I crack opening and mode-II crack opening respectively. KIb and KIIb are the shielded stress intensity Combined mode-I and mode-II fracture of ceramics 749
750 t Akatsu et al KK ne(oO) (Km(), Kk une(o). determined by ee observed KI (k),(Km) Y drawn by the fracture criterion Figure 2. Schematic illustration of the failure envelope in the KrKn diagram. The shielded stress intensity factors, Kb values, are derived from the comparison between the local stress intensity factor Ktin and nominal stress intensity factor Ka value factors at the onset of crack propagation for mode-I crack opening and mode-lI crack opening respectively. In the mixed-mode failure criterion described above, an eally straight crack without any crack-face interaction was presumed. Thus, KI(0) and Kn(e) in each criterion are definitely substituted for Kuip()and Kllp(e)respec tively. We then focus on the azimuthal angle 8 of the noncoplanar crack propaga tion, which must be determined on the basis of the local stress field in the vicinity of the tip regardless of any crack-face interaction. Reconsidering the criteria, it is found that the crack deflection angle 8c is commonly defined as a function of the ratio of (KItip(O))e to( Klltip(O))e. In other words, (KItip(0))/(Klltip(O))e can be numerically derived from experimental ec by the use of equations(3), (9)and(17). Both Kltip(O) and Kltip(O)are then evaluated by the combination of (Klip(0)/(Klltip(o))e and each criterion; equations(1)and (2)for the MHS criterion, equations(4)and( 8)for the mer criterion, and equations (11)and(15)for the MEd criterion On the other hand,(Kla) and(Klla)e are determined according to the conventional fracture mechanics explained in the following section. Finally, stress shielding at a crack tip under mixed-mode loading is individually estimated for mode-I and mode-II crack openings through the numerical evaluation of Klb and Kllb values using equa- tions(19) and(20)respectively. The procedure to derive the kb values described in his section is schematically illustrated in figure 2. S3. EXPERIMENTAL DETAILS As mentioned in $l, three kinds of ceramic have been selected to make a mutual comparison of the stress shielding under combined-mode loading. The float glass was supplied by Ashahi Glass Co, Ltd, Japan. The polycrystalline alumina was made through hot-pressing fine alumina powders of high purity(TM-100: Taimei Chemicals Co, Ltd, Japan). The alumina powders mixed with silicon carbide whiskers(TWS400, Tokai Carbon Co, Ltd, Japan)were hot pressed to fabricate the alumina matrix composite with a whisker volume fraction of 20%. The hot pressing was carried out at a maximum temperature of 1500.C for the monolithic
factors at the onset of crack propagation for mode-I crack opening and mode-II crack opening respectively. In the mixed-mode failure criterion described above, an ideally straight crack without any crack-face interaction was presumed. Thus, KIðÞ and KIIðÞ in each criterion are definitely substituted for KItipðÞ and KIItipðÞ respectively. We then focus on the azimuthal angle c of the noncoplanar crack propagation, which must be determined on the basis of the local stress field in the vicinity of the tip regardless of any crack-face interaction. Reconsidering the criteria, it is found that the crack deflection angle c is commonly defined as a function of the ratio of {KItip(0)}c to {KIItip(0)}c. In other words, fKItipð0Þg=fKIItipð Þg 0 c can be numerically derived from experimental c by the use of equations (3), (9) and (17). Both KItip(0) and KIItip(0) are then evaluated by the combination of fKItipð0Þg=fKIItipð Þg 0 c and each criterion; equations (1) and (2) for the MHS criterion, equations (4) and (8) for the MER criterion, and equations (11) and (15) for the MED criterion. On the other hand, (KIaÞc and (KIIaÞc are determined according to the conventional fracture mechanics explained in the following section. Finally, stress shielding at a crack tip under mixed-mode loading is individually estimated for mode-I and mode-II crack openings through the numerical evaluation of KIb and KIIb values using equations (19) and (20) respectively. The procedure to derive the Kb values described in this section is schematically illustrated in figure 2. } 3. Experimental details As mentioned in } 1, three kinds of ceramic have been selected to make a mutual comparison of the stress shielding under combined-mode loading. The float glass was supplied by Ashahi Glass Co., Ltd, Japan. The polycrystalline alumina was made through hot-pressing fine alumina powders of high purity (TM-100; Taimei Chemicals Co., Ltd, Japan). The alumina powders mixed with silicon carbide whiskers (TWS400, Tokai Carbon Co., Ltd, Japan) were hot pressed to fabricate the alumina matrix composite with a whisker volume fraction of 20%. The hot pressing was carried out at a maximum temperature of 15008C for the monolithic 750 T. Akatsu et al. Figure 2. Schematic illustration of the failure envelope in the KINKII diagram. The shielded stress intensity factors, Kb values, are derived from the comparison between the local stress intensity factor Ktip and nominal stress intensity factor Ka values
Combined mode-I and mode-lI fracture of ceramics alumina and 1750.C for the composite under a uniaxial pressure of 33 MPa in a flowing argon gas atmosphere for I h. The hot-pressed billets were cut and polished with diamond tools to create a beam of size 3 mm x 4 mm x 40 mm. The procedures for the powder mixing, the hot pressing and the machining have been described in detail elsewhere (Yasuda et al. 1991) Beams were utilized to measure Pe and ]c under combined mode-I and mode-II loading in two distinct ways: one was the controlled surface flaw(CSF)method and the other was the asymmetrical four-point bend test of a single-edge pre-cracked beam(SEPB)and a single-edge notched beam(SENB). In the CSF method a surface crack with a half-penny shape was first introduced in the centre of the beam by indentation with a sharp diamond stylus of Knoop geometry. Then, the region residually stressed around the indentation was fully removed by surface grinding and polishing(Akatsu et al. 1996). The surface crack whose geometry changed from a half-penny to a half-ellipsoid because of the grinding propagated through the symmetrical four-point bend test with an inner span of 10 mm and an outer spa of 30mm, using a testing machine (AG-100kNG, Shimadzu Co, Ltd, Japan)at a cross-head speed of 0.5 mm min. The rather fast speed of the cross-head movement selected to obviate subcritical crack growth below the critical load of cata- strophic fracture. The advantage of the CSF method is easy adjustment of crack length as well as the combination of mode-I and mode-II loadings by changing the maximum indentation load and the parametric angle a between the longer diagonal line of the indentation and the longitudinal direction of the beam respectively(figure 3) On the contrary to its advantage, one of the disadvantages of the CsF method is the complicated analysis required to determine the critical stress intensity factor under the combined mode of loading because the combination of local mode-I mode-II and mode-Ill stresses varies along the contour of the half-ellipsoid surface crack. The stress intensity factors for each mode crack opening at an arbitrary point 2 Crack front Half-circle a Crack Surface on the tensile side of a bend beam Figure 3. Schematic illustration of a half-ellipsoidal surface crack of a beam for the CsF method
alumina and 17508C for the composite under a uniaxial pressure of 33 MPa in a flowing argon gas atmosphere for 1 h. The hot-pressed billets were cut and polished with diamond tools to create a beam of size 3 mm 4 mm 40 mm. The procedures for the powder mixing, the hot pressing and the machining have been described in detail elsewhere (Yasuda et al. 1991). Beams were utilized to measure Pc and c under combined mode-I and mode-II loading in two distinct ways: one was the controlled surface flaw (CSF) method and the other was the asymmetrical four-point bend test of a single-edge pre-cracked beam (SEPB) and a single-edge notched beam (SENB). In the CSF method, a surface crack with a half-penny shape was first introduced in the centre of the beam by indentation with a sharp diamond stylus of Knoop geometry. Then, the region residually stressed around the indentation was fully removed by surface grinding and polishing (Akatsu et al. 1996). The surface crack whose geometry changed from a half-penny to a half-ellipsoid because of the grinding propagated through the symmetrical four-point bend test with an inner span of 10 mm and an outer span of 30 mm, using a testing machine (AG-100kNG, Shimadzu Co., Ltd, Japan) at a cross-head speed of 0.5 mm min1 . The rather fast speed of the cross-head movement was selected to obviate subcritical crack growth below the critical load of catastrophic fracture. The advantage of the CSF method is easy adjustment of crack length as well as the combination of mode-I and mode-II loadings by changing the maximum indentation load and the parametric angle between the longer diagonal line of the indentation and the longitudinal direction of the beam respectively (figure 3). On the contrary to its advantage, one of the disadvantages of the CSF method is the complicated analysis required to determine the critical stress intensity factor under the combined mode of loading, because the combination of local mode-I, mode-II and mode-III stresses varies along the contour of the half-ellipsoid surface crack. The stress intensity factors for each mode crack opening at an arbitrary point Combined mode-I and mode-II fracture of ceramics 751 Figure 3. Schematic illustration of a half-ellipsoidal surface crack of a beam for the CSF method
t Akatsu et al A with a parametric angle o(see figure 3)are given as follows(Kassir and Shih 1966) [Kltip(O)) oe(rd)- sit o(-k2 cos2o (Klltip(0))=0e(rd)k2sin a cos a cOS (1 (21) Kutip(0))=0e(nd) /(1-v) sin a cos a sD(1-k2cos2o)-/ where ae is the maximum tensile stress applied on the surface of the beam for the four-point bend test given by 3Pe(l1-l2) (22) w and t are the width and thickness respectively of the beam. I, and l2 are the outer and inner spans respectively of the bend test. In this study, 11= 30 mm and 12= 10 mm were selected. Both k and k are given as follows k2=1-k=1 where d and c are the depth and half-length respectively of the surface crack(see figure 3. E(k)is the second sort of perfect ellipse integration given as follows E(k)=(1-k2sin2o) /2do B is given as follows: (k2-)E(k)+k2K(k) where K()is the first sort of perfect ellipse integration given by (I-k2sin The complicated combined-mode stress intensity of the surface crack was examined n detail using the MHS criterion. Numerical consideration of the combination of equations(1)and(21)revealed that the maximum of Klip (0c) of the half-ellipsoidal surface crack appears either on the surface (o=0) or at the bottom (o=T/2 depending on the parametric angle a as well as the ratio of d to c, for example a shallow crack with a large a starts to expand from the bottom(o= /2)on the one hand, and a rather deep crack with a small a begins to propagate from the surface (o=0)on the other hand. The maximum of Kltip(e)must be determined under combined mode-I and mode-III crack opening in the former, and combined mode-I and mode-II crack opening in the latter. The bold curve in figure 4 shows the boundary between the former and the latter, which is numerically given through the combination of equations(1)and(21), striking a balance between o=0 and o=T/2. The areas above and below the line correspond to the former and the latter respectively. In this study, (Kla)e and(Klla) were measured for a half-ellipsoidal
A with a parametric angle (see figure 3) are given as follows (Kassir and Shih 1966): fKItipð Þg 0 c ¼ cðpdÞ 1=2 sin2 E kð Þ ð1 k2 cos2 Þ 1=4 ; fKIItipð Þg 0 c ¼ cðpdÞ 1=2 k2 sin cos k0 cos B ð1 k2 cos2 Þ 1=4 ; ð21Þ KIIItipð Þ0 c ¼ cðpdÞ 1=2 ð Þ 1 k2 sin cos sin B 1 k2 cos2 1=4 ; where c is the maximum tensile stress applied on the surface of the beam for the four-point bend test given by c ¼ 3Pc l1 l2 ð Þ 2wt2 : ð22Þ w and t are the width and thickness respectively of the beam. l1 and l2 are the outer and inner spans respectively of the bend test. In this study, l1 ¼ 30 mm and l2 ¼ 10 mm were selected. Both k and k0 are given as follows: k2 ¼ 1 k02 ¼ 1 d c 2 ; ð23Þ where d and c are the depth and half-length respectively of the surface crack (see figure 3). EðkÞ is the second sort of perfect ellipse integration given as follows: E kð Þ¼ ðp=2 0 ð1 k2 sin2 Þ 1=2 d : ð24Þ B is given as follows: B ¼ k2 E kð Þþ k02 K kð Þ; ð25Þ where KðkÞ is the first sort of perfect ellipse integration given by K kð Þ¼ ðp=2 0 d ð1 k2 sin2 Þ 1=2 : ð26Þ The complicated combined-mode stress intensity of the surface crack was examined in detail using the MHS criterion. Numerical consideration of the combination of equations (1) and (21) revealed that the maximum of KItipðcÞ of the half-ellipsoidal surface crack appears either on the surface ð ¼ 0Þ or at the bottom ð ¼ p=2Þ depending on the parametric angle as well as the ratio of d to c; for example a shallow crack with a large starts to expand from the bottom ( ¼ p=2Þ on the one hand, and a rather deep crack with a small begins to propagate from the surface ( ¼ 0) on the other hand. The maximum of KItipðcÞ must be determined under combined mode-I and mode-III crack opening in the former, and combined mode-I and mode-II crack opening in the latter. The bold curve in figure 4 shows the boundary between the former and the latter, which is numerically given through the combination of equations (1) and (21), striking a balance between ¼ 0 and ¼ p=2. The areas above and below the line correspond to the former and the latter respectively. In this study, ðKIaÞc and ðKIIaÞc were measured for a half-ellipsoidal 752 T. Akatsu et al.
Combined mode-/and mode-ll fracture of ceramics 753 90r Combined 8 70FK/Ku criterion Half-ellipsoida at the bottom surface crack 60 50 Combined KKn criterion on the surface 40 30 0.20.40.60.81 Ratio of the depth to half-length of a surface crack, d/c Figure 4. Boundary between mode-I and mode-II and between mode-I and mode-II criteri for a half-ellipsoidal surface crack. The crack propagates from the surface for the mode-I and mode-II criterion. and from the bottom for the mode -I and mode -II criterion surface crack with the combination of a and d/c in the area below the bold curve in figure 4 through the following equations: Gc(兀t (Kla) (1-k2) (27) (Kmlale= ge(xd)k- sina B(-k) It should be noted that 2c is the representative crack size under combined mode-I and mode-lI opening Another disadvantage of the CSF method is the experimental difficulty of com bined-mode fracture with the large mode-II component of the nominally applied loading. In order to overcome the difficulty in the Csf method the asymmetric four point bend test of a SEPB and a SenB as shown in figure 5 was carried out A crack was prepared at the centre of the bend beam using the bridge indentation technique for the SEPB Japanese Industrial Standard (JIS)R1607, 1995). For comparison, a SENB was adopted to confirm crack propagation without any crack-face interaction under the mixed mode loading, in which a very sharp notch with a radius of curva- ture of about 10 um(Nishida et al. 1995)was introduced at the centre. The bend test T In the case of a pure mode-I crack opening in the CsF method(o=r/2), Kltip reaches the maximum at the bottom(o= r/2) of the half-ellipsoidal surface crack. The critical value (Kla)=ae(nd)/E(k). The depth d should represent crack size under pure mode-I opele f the nominal mode-I stress intensity factor was evaluated through the following equati
surface crack with the combination of and d/c in the area below the bold curve in figure 4 through the following equationsy: KIa ð Þc ¼ cðpdÞ 1=2 sin2 E kð Þ 1 k2 1=4 ; KIIa ð Þc ¼ cðpdÞ 1=2 k2 sin cos k0 B 1 k2 1=4 : ð27Þ It should be noted that 2c is the representative crack size under combined mode-I and mode-II opening. Another disadvantage of the CSF method is the experimental difficulty of combined-mode fracture with the large mode-II component of the nominally applied loading. In order to overcome the difficulty in the CSF method, the asymmetric fourpoint bend test of a SEPB and a SENB as shown in figure 5 was carried out. A crack was prepared at the centre of the bend beam using the bridge indentation technique for the SEPB (Japanese Industrial Standard (JIS) R1607, 1995). For comparison, a SENB was adopted to confirm crack propagation without any crack-face interaction under the mixed mode loading, in which a very sharp notch with a radius of curvature of about 10 mm (Nishida et al. 1995) was introduced at the centre. The bend test Combined mode-I and mode-II fracture of ceramics 753 { In the case of a pure mode-I crack opening in the CSF method ð ¼ p=2Þ, KItip reaches the maximum at the bottom ð ¼ p=2Þ of the half-ellipsoidal surface crack. The critical value of the nominal mode-I stress intensity factor was evaluated through the following equation: ðKIaÞc ¼ cðpdÞ 1=2 =EðkÞ: The depth d should represent crack size under pure mode-I opening. Figure 4. Boundary between mode-I and mode-II and between mode-I and mode-II criteria for a half-ellipsoidal surface crack. The crack propagates from the surface for the mode-I and mode-II criterion, and from the bottom for the mode-I and mode-II criterion
Crack or Notch Figure 5. Schematic illustration of the asymmetric four-point bending of a SEPB or a SenB he combination of Kla and Kla values is adjusted by changing the offset was carried out with the same machine at the same cross-head speed for the same eason as those in the CSF method (Kla)e and(KIla) values for both the SEPB and the senB specimens tested through the asymmetrical bending are determined as follows(Hua et al. 1982) r3/2 Fr Q r1/ sss PeSo Pe h1=15-9544()+456.9(7)-8237(7)+5776/) hn=0316+1708-2371 where So, S, and S, are the distance from the centre of the beam to the crack, to the inner support and to the outer support respectively(see figure 5). The combination of Kla and Klla values was adjusted by changing So. In this study, S,= 5mm and S2=15mm were selected. a is the length of the pre-crack or the notch, which was prepared to be about half the beam thickness and was measured through optical microscopic observation.(Kla) under nominally applied pure mode-I loading was measured through the three-point bend test of the SEPB and sEnb S using the following equation
was carried out with the same machine at the same cross-head speed for the same reason as those in the CSF method. ðKIaÞc and ðKIIaÞc values for both the SEPB and the SENB specimens tested through the asymmetrical bending are determined as follows (Hua et al. 1982): KIa ð Þc ¼ M wt3=2 FI; KIIa ð Þc ¼ Q wt1=2 FII; ð28Þ M ¼ S2 S1 S1 þ S2 PcS0; Q ¼ S2 S1 S1 þ S2 Pc; ð29Þ FI ¼ 11:5 95:44 a t þ 456:9 a t 2 823:79 a t 3 þ575:76 a t 4 ; FII ¼ 0:280 1:605 a t þ 17:608 a t 2 23:77 a t 3 þ14:825 a t 4 ; ð30Þ where S0, S1 and S2 are the distance from the centre of the beam to the crack, to the inner support and to the outer support respectively (see figure 5). The combination of KIa and KIIa values was adjusted by changing S0. In this study, S1 ¼ 5 mm and S2 ¼ 15 mm were selected. a is the length of the pre-crack or the notch, which was prepared to be about half the beam thickness and was measured through optical microscopic observation. ðKIaÞc under nominally applied pure mode-I loading was measured through the three-point bend test of the SEPB and SENB specimens using the following equation: 754 T. Akatsu et al. Figure 5. Schematic illustration of the asymmetric four-point bending of a SEPB or a SENB specimen. The combination of KIa and KIIa values is adjusted by changing the offset distance S0