3322 Journal of the American Ceramic Society--Zok Vol. 89. No. I Sidebar C. Mechanics of Notch Sensitivity The mechanics underpinning notched fracture of CFCCs is well established. It is based on continuum descriptions of inelastic deformation, through either cohesive zone models (for Class I and III)or plasticity-type descriptions coupled with an appropriate criterion for fracture initiation( Class II). For the former, two types of cohesive zones are required(Fig. 19). One is aligned parallel to the loading direction and characterized by a shear traction law, t(os ) the other is perpendicular, with a tensile traction law, o(8, ) In an infinite isotropic notched body, the effective fracture energy I is dictated by the interplay between the two bands. That is, when the strength to of the shear zone is small relative to that of the tensile band, o, extensive shear deformation occurs, causing reduction in the peak stress within the tensile band. Otherwise, the tensile band ruptures before shear bands have an opportunity to develop and the fracture energy is the same as that of the tensile band, To. Pertinent numerical results showing the effects of the strength ratio t/o, are plotted on Fig. C1. Beneficial effects of the shear bands ar btained when to/o<0. 2 When the notch length is finite, the condition for composite fracture is again dictated by the interplay between the two bands terestingly, when plotted against the normalized notch length, ao/ach, with ach Er/o, results for the normalized strength fall essentially onto a single, monotonically decreasing band, only weakly dependent on t/o,(Fig. C2(b). Evidently,the beneficial effects of the shear band are embodied in the fracture energy r(to/oo. The trend is described by the empirical formula in Eq. (19). This formula not only fits the numerical results but also yields the correct values in the limits of very short and very long notches. That is, ON/oo+I as aolach-0(the notch insensitive domain)and oN/oo-(ao/ ach)-/ (the Griffith stress) when ao/ach≥1 Modifications to the analysis must be made to account for finite specimen width and elastic anisotropy 6 This is accomplished by re-defining the characteristic length scale ach such that the predicted strength in the Griffith limit is correct. The g(p, n)o F(ao/w)(I-o/w) where W is specimen width, F(ao W is the usual geometric factor, and g(p, 2)characterizes the elastic anisotropy. Composites with balanced symmetric fiber lay-ups exhibit cubic in-plane symmetry, whereupon 2= E1/E2= 1, V12=V21 and g(p,) g(p,λ) Traction Laws (a) (1+0.1(1-p)-0016(1-p2+0.004-p)10 rith (E1E2)1/2 ∠ The subscripts I and 2 refer to the principal material axes, H is the shear modulus. and v is poisson's ratio Shear Band 团 (b) Griffith stress Tensile Band 显 0.125 δt Notch length, ao/ach Fig Cl. Notch tip damage processes in porous matrix continuous finite-notched body) and(b) tensile strength(edge-notched semi-infinite fiber ceramic composites. body).(Adapted from Suo et al. and He et al.)Sidebar C. Mechanics of Notch Sensitivity The mechanics underpinning notched fracture of CFCCs is well established. It is based on continuum descriptions of inelastic deformation, through either cohesive zone models (for Class I and III) or plasticity-type descriptions coupled with an appropriate criterion for fracture initiation (Class II). For the former, two types of cohesive zones are required (Fig. 19). One is aligned parallel to the loading direction and characterized by a shear traction law, t(ds); the other is perpendicular, with a tensile traction law, s(dt). In an infinite isotropic notched body, the effective fracture energy G is dictated by the interplay between the two bands. That is, when the strength to of the shear zone is small relative to that of the tensile band, so, extensive shear deformation occurs, causing reduction in the peak stress within the tensile band. Otherwise, the tensile band ruptures before shear bands have an opportunity to develop and the fracture energy is the same as that of the tensile band, Go. Pertinent numerical results showing the effects of the strength ratio to/so are plotted on Fig. C1.58 Beneficial effects of the shear bands are obtained when to/sor0.2. When the notch length is finite, the condition for composite fracture is again dictated by the interplay between the two bands. Interestingly, when plotted against the normalized notch length, ao/ach, with ach  EG=s2 o, results for the normalized strength fall essentially onto a single, monotonically decreasing band, only weakly dependent on to/so (Fig. C2(b)).58 Evidently, the beneficial effects of the shear band are embodied in the fracture energy G(to/so). The trend is described by the empirical formula in Eq. (19). This formula not only fits the numerical results but also yields the correct values in the limits of very short and very long notches. That is, sN/so-1 as ao/ach-0 (the notch insensitive domain) and sN=so ! ðpao= achÞ 1=2 (the Griffith stress) when ao/ach  1. Modifications to the analysis must be made to account for finite specimen width and elastic anisotropy.56 This is accomplished by re-defining the characteristic length scale ach such that the predicted strength in the Griffith limit is correct. The result is ach  EG gðr; lÞ½ soF að Þ o=W ð Þ 1 ao=W 2 where W is specimen width, F(ao/W) is the usual geometric factor, and g(q, k) characterizes the elastic anisotropy.59,60 Composites with balanced symmetric fiber lay-ups exhibit cubic in-plane symmetry, whereupon l  E1=E2 ¼ 1, n12 ¼ n21 and g(q, k) is given by: gð Þ¼ r; l 1 þ r 2
1=2 1 þ 0:1ð1 rÞ 0:016ð1 rÞ 2 þ 0:002ð1 rÞ 3  with r  ð Þ E1E2 1=2 2m12 ð Þ n12n21 1=2 The subscripts 1 and 2 refer to the principal material axes, l is the shear modulus, and n is Poisson’s ratio. Fig. C1. Notch tip damage processes in porous matrix continuous- fiber ceramic composites. Fig. C2. Effects of strength ratio to/so on (a) composite toughness (in- finite-notched body) and (b) tensile strength (edge-notched semi-infinite body). (Adapted from Suo et al. 57 and He et al. 58). 3322 Journal of the American Ceramic Society—Zok Vol. 89, No. 11