D. Leguillon et al. Journal of the European Ceramic Society 26(2006)343-349 0.3 cause crack deflection,for which the assumption seems to be 0.2 Acknowledgement v=0.16 This work was supported by the French Ministry 0.1 Research in the ACI program "Surfaces and Interfa 2001-2003 02 03 References nctiong(Eq (12))(solid lines)vs the Youngs moduli ra- 1. Chartier, T, Merle, D and Besson, J. L, Laminar ceramic composites wo different Poissons ratio compared to the toughness ratio J. Eur Ceram Soc. 1995. 15. 101-10 Gp/Ga(dashed line) at the porous/dense interface 2. Clegg, w.J. Blanks, K.S., Davis, J. B. and Lanckmans, F, Porous es as crack deflecting interlayers a minor role. It is assumed that both dense and porous ceram- Eng. Mater,1997,132-136,1866-1869 ics have the same Poissons ratio. Two values are compared Blanks, K. S, Kristofferson, A, Carlstrom, E. and Clegg, W. J in Fig. 12: v=0.16(B4C)(note that v=0. 17 for SiC)and Crack deflection in ceramic laminates using porous interlayers. J. EE. Ceram.Soc.,1998,18,1945-1951 a realistic value met in many materials v=0. 3. Clearly the 4. Davis. J. B. Kristoffersson. A. Carlstrom. E. and Clegg. W. J. Fab- deviation in Youngs moduli ratio causing deflection is weak (6%)(Fig 12). Using the porosity function H(Eq (13),it leads to about l% deviation in the pore volume fraction V. It llaires monolithiques et composites en Carbure de silicin. Ph. D. thesis no. 282TD, Ecole des mines, St- is obviously a negligible effect. 6. Reynaud, properties and mechanical behaviour of Sic dense-porous laminates 9. Conclusion J.Eur. Ceram.Soe,2005,25,589597 7. Tariolle, S, Reynaud, C, Thevenot, F, Chartier, T. and Besson, J The first conclusion to draw is that deflection is very L, Preparation and mechanical properties of SiC-SiC and B:C-B laminates. J. Solid State Chem. 2004. 177. 487-492. difficult to promote by porous layers obtained by the ad 8. Tariolle, S, Carbure de bore monolithique et preux er composites dition of spherical pore forming particles. This prediction correlates well with experimental results: Reynaud and co- 328TD. Ecole des Mines. St-Etienne. France. 2004 orkers'-did not observe any extensive deflection in Sic 9. Ma, J, Wang, H, Weng, L. and Tan, G. E. B, Effec below v=42%(i. e. VN40%)and Tariolle et al. 7, found a interlayers on crack deflection in ceramic laminates. J. ignificant deflection for a rather high value V=52%(1.e Soc.,2004,24,825-831 M. Y and Va46%)in B4 C. It is in a good agreement with the present between dissimilar elastic materials. Int. J. Solids struct. 153-167 model, while the one based on the second He and Hutchin- 11 He. M. Y and Hutchinson, J. w Kinking of a crack out of an son approach, I as proposed by Clegg et al. 2-4 tends to sig- nificantly underestimate the experimental values. The other 12 terface. J. App..,1989,111,270-278 herti Tazi, O, Comportement a la rupture d'un assemblage de well-known He and Hutchinson approach neglects the lam materiaux fragiles, Ph D. thesis, University P. and M. Curie, Paris inated micro-structure of the material and gives an erroneous france. 2005 low value of the porosity causing crack deflection 13. Leguillon, D, Strength or toughness? A criterion for crack onset at notch. Eur J. Mech. A/Solids. 2002. 21.61-72 Finally, emphasis must be put on Figs. 7 and 8 that are 14. Leguillon, D, Sanchez-Palencia, E, Computation of Singular Solu- in a way intrinsic, porosity does not occur explicitly. They tions in Elliptic Problems and Elasticity. Masson, Paris, John Wiley play the role of 'master curves' and can be used whatever New York, 1987. the dependence of the elastic and fracture properties on the 15. Fujita, H, Jefferson, G. McMeeking, R.M. and Zok,F.W,Mul- porosity. The only assumption is that the Youngs modulus lite/alumina mixtures for use as porous m oxide fiber com. and the toughness of the porous material follow the same rule posites. J. Am. Ceram. Soc., 2004, 87(2), This may be wrong for small porosity values. But the present 16. Martin, E. and Leguillon, D, Energetic conditions for interfacial fa ure in the vicinity of a matrix crack in brittle matrix composites. Int. analysis deals only with large values of the porosity that can J. Solids struct.,2004,41,6937-6948D. Leguillon et al. / Journal of the European Ceramic Society 26 (2006) 343–349 349 Fig. 12. The function g (Eq. (12)) (solid lines) vs. the Young’s moduli ra￾tio Ep/Ed for two different Poisson’s ratio compared to the toughness ratio Gc p/Gc d (dashed line) at the porous/dense interface. a minor role. It is assumed that both dense and porous ceram￾ics have the same Poisson’s ratio. Two values are compared in Fig. 12: ν = 0.16 (B4C) (note that ν = 0.17 for SiC) and a realistic value met in many materials ν = 0.3. Clearly the deviation in Young’s moduli ratio causing deflection is weak (<6%) (Fig. 12). Using the porosity function H (Eq. (13)), it leads to about 1% deviation in the pore volume fraction V. It is obviously a negligible effect. 9. Conclusion The first conclusion to draw is that deflection is very difficult to promote by porous layers obtained by the ad￾dition of spherical pore forming particles. This prediction correlates well with experimental results: Reynaud and co￾workers5–7 did not observe any extensive deflection in SiC below V˜ = 42% (i.e. V ≈ 40%) and Tariolle et al.7,8 found a significant deflection for a rather high value V˜ = 52% (i.e. V ≈ 46%) in B4C. It is in a good agreement with the present model, while the one based on the second He and Hutchin￾son approach,11 as proposed by Clegg et al.,2–4 tends to sig￾nificantly underestimate the experimental values. The other well-known He and Hutchinson approach10 neglects the lam￾inated micro-structure of the material and gives an erroneous low value of the porosity causing crack deflection. Finally, emphasis must be put on Figs. 7 and 8 that are in a way intrinsic, porosity does not occur explicitly. They play the role of ‘master curves’ and can be used whatever the dependence of the elastic and fracture properties on the porosity. The only assumption is that the Young’s modulus and the toughness of the porous material follow the same rule. This may be wrong for small porosity values. But the present analysis deals only with large values of the porosity that can cause crack deflection, for which the assumption seems to be valid. Acknowledgement This work was supported by the French Ministry of Research in the ACI program “Surfaces and Interfaces 2001–2003”. References 1. Chartier, T., Merle, D. and Besson, J. L., Laminar ceramic composites. J. Eur. Ceram. Soc., 1995, 15, 101–107. 2. Clegg, W. J., Blanks, K. S., Davis, J. B. and Lanckmans, F., Porous interfaces as crack deflecting interlayers in ceramic laminates. Key Eng. Mater., 1997, 132–136, 1866–1869. 3. Blanks, K. S., Kristofferson, A., Carlstrom, E. and Clegg, W. J., ¨ Crack deflection in ceramic laminates using porous interlayers. J. Eur. Ceram. Soc., 1998, 18, 1945–1951. 4. Davis, J. B., Kristoffersson, A., Carlstrom, E. and Clegg, W. J., Fab- ¨ rication and crack deflection in ceramic laminates with porous inter￾layers. J. Am. Ceram. Soc., 2000, 83(10), 2369–2374. 5. Reynaud, C., C´eramiques lamellaires monolithiques et composites en Carbure de Silicium. Ph.D. thesis no. 282TD, Ecole des Mines, St￾Etienne, France, 2002. 6. Reynaud, C., Thevenot, F., Chartier, T. and Besson, J. L., Mechanical ´ properties and mechanical behaviour of SiC dense-porous laminates. J. Eur. Ceram. Soc., 2005, 25, 589–597. 7. Tariolle, S., Reynaud, C., Thevenot, F., Chartier, T. and Besson, J. ´ L., Preparation and mechanical properties of SiC–SiC and B4C–B4C laminates. J. Solid State Chem., 2004, 177, 487–492. 8. Tariolle, S., Carbure de Bore monolithique et poreux et composites lamellaires, ´elaboration, propri´et´es, renforcement. Ph.D. thesis no. 328TD, Ecole des Mines, St-Etienne, France, 2004. 9. Ma, J., Wang, H., Weng, L. and Tan, G. E. B., Effect of porous interlayers on crack deflection in ceramic laminates. J. Eur. Ceram. Soc., 2004, 24, 825–831. 10. He, M. Y. and Hutchinson, J. W., Crack deflection at an interface between dissimilar elastic materials. Int. J. Solids Struct., 1989, 25(9), 153–167. 11. He, M. Y. and Hutchinson, J. W., Kinking of a crack out of an interface. J. Appl. Mech., 1989, 111, 270–278. 12. Cherti Tazi, O., Comportement a la rupture d’un assemblage de ` materiaux fragiles, Ph.D. thesis, University P. and M. Curie, Paris, ´ France, 2005. 13. Leguillon, D., Strength or toughness? A criterion for crack onset at a notch. Eur. J. Mech. A/Solids, 2002, 21, 61–72. 14. Leguillon, D., Sanchez-Palencia, E., Computation of Singular Solu￾tions in Elliptic Problems and Elasticity. Masson, Paris, John Wiley, New York, 1987. 15. Fujita, H., Jefferson, G., McMeeking, R. M. and Zok, F. W., Mul￾lite/alumina mixtures for use as porous matrices in oxide fiber com￾posites. J. Am. Ceram. Soc., 2004, 87(2), 261–267. 16. Martin, E. and Leguillon, D., Energetic conditions for interfacial fail￾ure in the vicinity of a matrix crack in brittle matrix composites. Int. J. Solids Struct., 2004, 41, 6937–6948