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