Modelling of combined reinforcement of ceramic composites 523 The ZrO2 in the vicinity of the crack tip will take place t-m transformation and a transformation zone is formed(Fig. 3). It is assumed that the transformation zone is comprised of a continuum of dilation particles near crack tip. The hydrostatic stress,Em(=3Ei, i= l, 2, 3), applied on each dilation particle is given b 2m =20+)Pcos(0/2) where Kp is the elastic intensity factor determined for the actual geometry of a cracked specimen at a Fig. 3. Schematic of combine toughening of whisker and given load P, r and 0 are the position co-ordinates of an expansion center relative to the crack tip, v is Poissons ratio If Em reaches a critical value, ac, t-m transfor- crack. We still assume that the whole crack is a mation will occur including a dilation of about 4% mode I crack, although the crack is deflected in the and a shear distortion of about 7%.6 A uniform vicinity of the crack tip. As a is the angle between dilation transformation of a ZrOz particle will the crack faces, the crack opening displacement, produce a reduction in near crack-tip intensity 2u,(x), at point x is relative to the applied intensity, and the stress intensity factor induced is 2uy=a(a-x)(ao≤x≤a) (11) Ak=(Gcos(3/2)1/(2 m) "x p2 dA (9) then the crack tip opening displacement CTOD is where G is the shear modulus and da is the volume dilation strain of each Zro, particle When u(x)>uy(2uyc is the critical opening dis- A shear strain, &r, and dilation, e, take place placement of the crack face), the whisker at point x multaneously during a Zro 2 particle transform- will break and the crack closure stress o, (x)Vw=0o ing, and the toughening contribution of the shear The stress intensity factor K, due to the external strain is of the same order of magnitude as that of oad p is the volume dilation. 2 To simplify the calculation of the contribution from transformation toughen 2Bn VaF(a/w) ing, we presume a shear strain factor, f, which reflects the effect of shear strain. So the transfor. mation effect from both shear strain and dilation where F(a/w is a weight function. 5 strain can be obtained by As a crack propagates, the fracture resistance K, △K=V=mG(f+ cos(30/2)/(2x)r32 Kr=Km+△Kx+△Kb+△Kd rare (10) where Km is the fracture toughness of the matrix, and△Ky,△ Kh and△ Kd are the contributions of transformation, whisker bridging and crack deflec where A denotes the region of the transformation tion, respectively. Equation (14)includes three zone, (1+f is assumed to represent shear strain toughening mechanisms, transformation, whisker and dilation and V_m is the volume fraction of the bridging and crack deflection ZrO which takes place t-m transformed in the total zro 3 EXPERIMENTAL PROCEDURE 2.1.2 Fracture resistance equation For the notch specimen, the initial notch length, ao, Al2O3 powder with a average size of about 0.1 um is much longer than the crack growing length, is used as the basic material ZrO2 powder stabilised a(= a-ao), until the instable propagation of the by 2mol%Y203 and Sic whisker was prepared asThe ZrO2 in the vicinity of the crack tip will take place t±m transformation and a transformation zone is formed (Fig. 3). It is assumed that the transformation zone is comprised of a continuum of dilation particles near crack tip. The hydrostatic stress, m ˆ 1 3 ii; i ˆ 1; 2; 3 ÿ ), applied on each dilation particle is given by m ˆ 2 1… † ‡  3 Kp  2r p cos…† … =2 8† where Kp is the elastic intensity factor determined for the actual geometry of a cracked specimen at a given load P, r and  are the position co-ordinates of an expansion center relative to the crack tip,  is Poisson's ratio. If m reaches a critical value, c, t±m transfor￾mation will occur including a dilation of about 4% and a shear distortion of about 7%.6 A uniform dilation transformation of a ZrO2 particle will produce a reduction in near crack-tip intensity relative to the applied intensity, and the stress intensity factor induced is
k ˆ ‰ Š G cos… † 3=2 = … † 2 1=2 r 3=2 h idA …9† where G is the shear modulus and dA is the volume dilation strain of each ZrO2 particle. A shear strain, "r , and dilation, "T, take place simultaneously during a ZrO2 particle transform￾ing, and the toughening contribution of the shear strain is of the same order of magnitude as that of the volume dilation.12 To simplify the calculation of the contribution from transformation toughen￾ing, we presume a shear strain factor, f, which re¯ects the e€ect of shear strain. So the transfor￾mation e€ect from both shear strain and dilation strain can be obtained by
Ks ˆ VtÿmG  f ‡ 1  "T … …‰ Š cos… † 3=2 =  2 1=2 r 3=2 " # rdrd …10† where A denotes the region of the transformation zone, (1+f) is assumed to represent shear strain and dilation, and Vtÿm is the volume fraction of the ZrO2 which takes place t±m transformed in the total ZrO2. 2.1.2 Fracture resistance equation For the notch specimen, the initial notch length, a0, is much longer than the crack growing length,
a …ˆ a ÿ a0), until the instable propagation of the crack. We still assume that the whole crack is a mode I crack, although the crack is de¯ected in the vicinity of the crack tip. As  is the angle between the crack faces, the crack opening displacement, 2uy… † x , at point x is 2uy ˆ … † a ÿ x … †… a0  x  a 11† then the crack tip opening displacement CTOD is CTOD ˆ …† … a ÿ a0 12† When u x… † uyc (2uyc is the critical opening dis￾placement of the crack face), the whisker at point x will break and the crack closure stress y… † x Vw ˆ 0o. The stress intensity factor Kp due to the external load P is Kp ˆ 3PL 2BW2  a p F a…† … =W 13† where F a… † =W is a weight function.15 As a crack propagates, the fracture resistance Kr is Kr ˆ Km ‡
Ks ‡
Kb ‡
Kd …14† where Km is the fracture toughness of the matrix, and
Ks,
Kb and
Kd are the contributions of transformation, whisker bridging and crack de¯ec￾tion, respectively. Equation (14) includes three toughening mechanisms, transformation, whisker bridging and crack de¯ection. 3 EXPERIMENTAL PROCEDURE Al2O3 powder with a average size of about 0.1m is used as the basic material. ZrO2 powder stabilised by 2mol%Y2O3 and SiC whisker was prepared as Fig. 3. Schematic of combine toughening of whisker and transformation. Modelling of combined reinforcement of ceramic composites 523