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