November 2007 R Curves and Crack-Stability Map 3555 IL. Theoretical Analysis Y(,=)=0.2139+ 1.0770 (13) The applied crack-driving force for a surface crack, expressed in rms of the stress intensity due to far-field loading, is given by the following equation 1.0770 Ka(c)=OvC (6 (14) where Y, a stress-intensity coefficient, is a function of the surface crack shape, relative crack depth, and the mode of loading, ar Newman9.10 have calculated values of y for surface cracks of The solid line in Fig. I plots the solutions of Eq.(12),i.e semi-elliptical shape( semi-major axis, c, and semi-minor axis, a) for different crack penetrations(a/w, where w is the beam thick and a specific value of the parameter, (/w)=0.0568, pertinent ness)in bending by finite-element analysis(FEA). Further, ob- to the Ce-TZP/AL,O, ceramic discussed in the following section servations of crack fronts on fracture surfaces indicate that the he resulting plot maps two regimes: a regime of stable crack aspect ratio of the crack decreases linearly with stable growth rowth and a regime of unstable crack growth with two impor through the beam thickness tant implications. For a toughness ratio, Ko/Koo>0.197, crack growth is always unstable for any initial crack length. For ratios <0. 197, crack growth is unstable for initial crack lengths c< n, it is stable for ci< c< c, and it is again unstable for c>c. The dotted-dashed vertical line in Fig. I corre- Raju and Newman's FEA results, used in conjunction with sponds to the transformation-toughened Ce-TZP/AlO Eq. (7), lead to the following equation for y in terms of a single ceramic with Ko=8.90 MPa. m, Ko=1. 11 MPa - m2.203 variable ca Ko/Koo=0.125. For these toughness parameters, the crack-sta- bility map defines(ci/2)=0.184 and(c5/)=1.103 Y(c)=0.2139+1.0770 In the limiting case, w>>7, the stress-intensity coefficient, Y, (8) is independent of the crack length and Eq(12)reduces to the Tollowing fo The substitution of Eqs.(6)and(5)into Eq (4) leads to the foll exp Y(c)ovc=Koc -(Koo -Ko)exp The dashed line in F Similarly, the slopes of the crack-driving force and the crack a plot of Eq(15)and rep- esents the crack-stabilit growth resistance functions can b ted to obtain the fol crack-growth regime is or the limiting case relative to the more generalized presented by Eq (12). In the following section, the implications of crack stability on fracture 「Y()dY (K-K0) are Ce-TZP/ALO Equation(9)divided by Eq(10)obtains the following equa Ill. Surface Cracks and R Curves for Ce-TZP/AlO3 tion for c the critical crack size for transition from sta ble to unstable growth, or vice versa Ramachandran et al. measured R curves for Ce-TZP/AlO3 by measuring the lengths of surface cracks initiated at pores or dy (11) r Ce. TZPlAlo Inst:hetRick G:newth Equation (I1)can be transformed into the following conve- nient form Instable Crack Grwth 05 In Eq(12), the stress-intensity coefficient and its deriva ms of the normalized parameters, (c/) and (/w) resistanceII. Theoretical Analysis The applied crack-driving force for a surface crack, expressed in terms of the stress intensity due to far-field loading, is given by the following equation: KaðcÞ ¼ Ys ffiffi c p (6) where Y, a stress-intensity coefficient, is a function of the surface crack shape, relative crack depth, and the mode of loading, and s is the maximum tensile stress in bending of a beam. Raju and Newman9,10 have calculated values of Y for surface cracks of semi-elliptical shape (semi-major axis, c, and semi-minor axis, a) for different crack penetrations (a/w, where w is the beam thick￾ness) in bending by finite-element analysis (FEA). Further, ob￾servations of crack fronts on fracture surfaces indicate that the aspect ratio of the crack decreases linearly with stable growth through the beam thickness11: a c ¼ 1 a w (7) Raju and Newman’s FEA results, used in conjunction with Eq. (7), lead to the following equation for Y in terms of a single variable c: YðcÞ ¼ 0:2139 þ 1:0770 w w þ c   (8) The substitution of Eqs. (6) and (5) into Eq. (4) leads to the following equation: Yðc
Þs ffiffiffiffi c
p ¼ K1 ðK1 K0Þ exp c
l   (9) Similarly, the slopes of the crack-driving force and the crack￾growth resistance functions can be equated to obtain the fol￾lowing equation: s ffiffiffiffi c
p Yðc
Þ 2c
þ dY dc ðc
Þ   ¼ ðK1 K0Þ l exp c
l   (10) Equation (9) divided by Eq. (10) obtains the following equa￾tion for c
, the critical crack size for transition from stable to unstable growth, or vice versa: Yðc
Þ Yðc
Þ 2c
þ dY dc ðc
Þ   ¼ K1l ðK1 K0Þ exp c
l   1 (11) Equation (11) can be transformed into the following conve￾nient form: K0 K1 ¼ 1 exp c
l   2c
l Y c
l ; l w   Y c
l ; l w   þ 2c
l l dY dc c
l ; l w   8 >>< >>: 9 >>= >>; þ 1 0 BB@ 1 CCA 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 (12) In Eq. (12), the stress-intensity coefficient and its derivative have been written in terms of the normalized parameters, (c
/l) and (l/w): Y c
l ; l w   ¼ 0:2139 þ 1:0770 1 þ c
l l w   (13) dY dc c
l ; l w   ¼ 1:0770 w 1 þ c
l l w  2 (14) The solid line in Fig. 1 plots the solutions of Eq. (12), i.e., values of (c
/l) for different values of the parameter, K0/KN, and a specific value of the parameter, (l/w) 5 0.0568, pertinent to the Ce-TZP/Al2O3 ceramic discussed in the following section. The resulting plot maps two regimes: a regime of stable crack growth and a regime of unstable crack growth with two impor￾tant implications. For a toughness ratio, K0/KN40.197, crack growth is always unstable for any initial crack length. For ratios o0.197, crack growth is unstable for initial crack lengths, c < c
1, it is stable for c
1 < c < c
2, and it is again unstable for c > c
2. The dotted–dashed vertical line in Fig. 1 corre￾sponds to the transformation-toughened Ce-TZP/Al2O3 ceramic with KN 5 8.90 MPa m1/2, K0 5 1.11 MPa m1/2, and K0/KN 5 0.125. For these toughness parameters, the crack-sta￾bility map defines ðc
1=lÞ ¼ 0:184 and ðc
2=lÞ ¼ 1:103. In the limiting case, w  l, the stress-intensity coefficient, Y, is independent of the crack length and Eq. (12) reduces to the following form: K0 K1 ¼ 1 exp c
l   2 c
l þ 1
2 6 6 4 3 7 7 5 (15) The dashed line in Fig. 1 shows a plot of Eq. (15) and rep￾resents the crack-stability map for the limiting case. The stable crack-growth regime is compressed for the limiting case relative to the more generalized solution represented by Eq. (12). In the following section, the implications of crack stability on fracture strength are examined for the transformation-toughened Ce-TZP/Al2O3. III. Surface Cracks and R Curves for Ce-TZP/Al2O3 Ramachandran et al. 5 measured R curves for Ce-TZP/Al2O3 by measuring the lengths of surface cracks initiated at pores or Fig. 1. Crack-stability map showing the ranges of crack length for stable and unstable growth in a ceramic exhibiting rising crack-growth resistance. November 2007 R Curves and Crack-Stability Map 3555