J. Pascual et al. Journal of the European Ceramic Society 28(2008)1551-1556 1555 and all together prevent the recognition of a threshold by simple grains, which occurred at the as-sintered surfaces. Since analysis of strength data in the discussed case racture originated due to the same flaw population in both An interesting consequence of this behaviour is that the fit batches, the fracture statistics of both batches is identical. ting procedure for three-parameter Weibull distributions gives if the total stress(applied stress plus residual stress)at the unstable results in many cases. For example the omission of a position of the flaws is correctly taken into account single data point may have significant influence on the deter-. The layered architecture causes compressive residual stresses mined threshold stress. In our analysis the threshold stress has of approximately sa yer, which cause an increase in strength the outer alumina lay not been fitted to the data, it has been set equal to the residual 158 MPa. This also causes a lower bound stress in the outer layer of the laminate. Only if the threshold of the strength(threshold stress stress is well defined, a stable fitting of the other two parameters is possible on the basis of a small number of tests Ack Finally, it should be realised that this simple analysis is only valid if the stress field can be considered to be almost constant The authors thank g. de portu from the istituto di scienza e over the extension of the crack, ie. if the crack size is small Tecnologia dei Materiali Ceramici of the Consiglio Nazionale cases a more complicated e first compressed layer. In other delle Ricerche, Faenza, Italy (ISTEC-CNR)for providing the compared to the thickness lysis based on the analysis of the specimens. Special thanks go to Prof. R. Morrell for revising stress intensity factors of cracks would be necessary. 6. 404 The the manuscript critical( Griffith) crack size can be determined from strength data of via ac=(l/t)(Kc/Yor)". The fracture toughness in the references first alumina layer is Kc= 3.8 MPa,m. The geometry factor of a surface crack is approximately Y=l. with these assump 1. Danzer, R, Mechanical performance and lifetime prediction. In Concise tions the critical crack sizes for batches a and range from cyclopedia of Advanced Ceramic Materials, about 15-35 um. This analysis fits to the fractographic evidence, Press, Oxford UK, 1991. Pp. 286-299 ig.2. 3. DanzerR. Mechanical failure of advanced ceramics: the value of fractog. 4. Conclusions phy. Key Eng. Mater, 2002. 223. 1-18 General conclusions related to the behaviour and fracture Printing Office. Washington, 2007 statistics of laminates strengthened by compressive stresses and 5. Wachtman, J.B. Mechanical Properties of Ceramics.Wiley-Interscience more special conclusions on the behaviour of the investigated 6. Danzer, R, Lube, T. Supancic, P and Damani, R, Fracture of ceramics. laminate can be drawn from this work. In general it holds that: Adv Eng Mat, submitted for publication 7. Weibull w.. A statistical distribution function ech.,1951,18,293-298 layered architecture of the specimens if the outer layer of the & Weibull, W, A Statistical Theory of the strength of Materials. General- specimen has compressive residual stresses 9. Jayatilaka, A.d. Sand Trustrum, K, Statistical approach to brittle fracture The residual compressive stress causes a lower bound(thresh- J Mater. Sci. 1977 26-1430. old) for the strength, therefore a high amplitude of the 10. Danzer, R, A general strength distribution function for brittle materials. J. compressive stress is beneficial Eur ceran.Soc.,1992,10,461-472 The threshold is masked by the scatter of the strength data. 11. ENV 843-5., Advanced Tecnical Ceramics, Monolithic Ceramics, Mechan- Therefore it can hardly be recognised in a conventional ical Properties at Room Temperature. Part 5. 12. Laraia, V.J. and Heuer, A H, Novel composite microstructure and mechani- Weibull diagram. cal behavior of mollusk shell. J. Am. Ceram Soc., 1989, 72(11),2177-2179 For this type of laminate the appropriate fracture statistical 13. Currey, J D, Mechanical properties of mother of pearl in tension. Proc. R approach is the three-parameter Weibull method. The two- oC.lond.,1977,B196,443-463 parameter Weibull approach is not appropriate and should 14. Chan, H. M, Layered ceramics: processing and mechanical behaviour not be used, since its application can cause inappropriate 15. Clegg, W J,Kendall,K. Alford, N M Button, T Wand Birchall, J D outer layer material (were fracture initiates)the inappro- 16. Clegg, W. J, The fabrication and failure of laminar ceramic composites. priate"two-parameter Weibull modulus"of the laminate is Acta. Metall. Mater,1992,40(11),3085-3093. increased. It even depends on the number of tests undertaken 17. Virkar. A. V. Huang, J. L and Cutler, R. A, Strengthening of oxide ceram- For extrapolations to very high reliabilities the inappropriate ics by transformation-induced stresses. J. Am. Ceram. Soc., 1987, 70(3), two-parameter Weibull distribution gives conservative results.18.Rao,M.P,Sanchez-Herencia,AJ,Beltz,GE,McMeeking,R.Mand Lange, F. F, Laminar ceramics that exhibit a threshold strength. Science, In the case of the investigated alumina-alumina/zirconia lam- 1999,286.102-105 19. Gee, L.A., Dobedoe, R. S, Vann, R, Lewis, M. H. Blugan G. and Kubler Ceran,2005,104(3).103-109. Fracture always initiated (in the alumina specimens as well 20. de Portu, G, Micele, L, Guicciardi, S, Fujimura, S, Pezzotti, Gand as in the laminate specimens) at abnormally large alumina kiguch. Y, effect of residual stress on the fracture behaviour of notchedJ. Pascual et al. / Journal of the European Ceramic Society 28 (2008) 1551–1556 1555 and all together prevent the recognition of a threshold by simple analysis of strength data in the discussed case. An interesting consequence of this behaviour is that the fit￾ting procedure for three-parameter Weibull distributions gives unstable results in many cases. For example the omission of a single data point may have significant influence on the deter￾mined threshold stress. In our analysis the threshold stress has not been fitted to the data, it has been set equal to the residual stress in the outer layer of the laminate. Only if the threshold stress is well defined, a stable fitting of the other two parameters is possible on the basis of a small number of tests. Finally, it should be realised that this simple analysis is only valid if the stress field can be considered to be almost constant over the extension of the crack, i.e. if the crack size is small compared to the thickness of the first compressed layer. In other cases a more complicated analysis based on the analysis of the stress intensity factors of cracks would be necessary.6,40,41 The critical (Griffith) crack size can be determined from strength data2–6 σf via ac = (1/π)(Kc/Yσf) 2. The fracture toughness in the first alumina layer is Kc = 3.8 MPa√m. 33 The geometry factor of a surface crack is approximately Y = 1. With these assump￾tions the critical crack sizes for batches A and L range from about 15–35m. This analysis fits to the fractographic evidence, Fig. 2. 4. Conclusions General conclusions related to the behaviour and fracture statistics of laminates strengthened by compressive stresses and more special conclusions on the behaviour of the investigated laminate can be drawn from this work. In general it holds that: • The strength of ceramics can significantly be increased by a layered architecture of the specimens if the outer layer of the specimen has compressive residual stresses. • The residual compressive stress causes a lower bound (thresh￾old) for the strength, therefore a high amplitude of the compressive stress is beneficial. • The threshold is masked by the scatter of the strength data. Therefore it can hardly be recognised in a conventional Weibull diagram. • For this type of laminate the appropriate fracture statistical approach is the three-parameter Weibull method. The two￾parameter Weibull approach is not appropriate and should not be used, since its application can cause inappropriate extrapolations. Compared to the Weibull modulus of the outer layer material (were fracture initiates) the inappro￾priate “two-parameter Weibull modulus” of the laminate is increased. It even depends on the number of tests undertaken. For extrapolations to very high reliabilities the inappropriate two-parameter Weibull distribution gives conservative results. In the case of the investigated alumina–alumina/zirconia lam￾inate: • Fracture always initiated (in the alumina specimens as well as in the laminate specimens) at abnormally large alumina grains, which occurred at the as-sintered surfaces. Since fracture originated due to the same flaw population in both batches, the fracture statistics of both batches is identical, if the total stress (applied stress plus residual stress) at the position of the flaws is correctly taken into account. • The layered architecture causes compressive residual stresses in the outer alumina layer, which cause an increase in strength of approximately 158 MPa. This also causes a lower bound of the strength (threshold stress). Acknowledgement The authors thank G. de Portu from the Istituto di Scienza e Tecnologia dei Materiali Ceramici of the Consiglio Nazionale delle Ricerche, Faenza, Italy (ISTEC-CNR) for providing the specimens. Special thanks go to Prof. R. Morrell for revising the manuscript. References 1. Danzer, R., Mechanical performance and lifetime prediction. In Concise Encyclopedia of Advanced Ceramic Materials, ed. R. J. Brook. Pergamon Press, Oxford UK, 1991. pp. 286–299. 2. Munz, D. and Fett, T., Ceramics. Springer, Berlin, Heidelberg, 1999, pp. 298. 3. Danzer, R., Mechanical failure of advanced ceramics: the value of fractog￾raphy. Key Eng. Mater., 2002, 223, 1–18. 4. Quinn, G. D., Fractography of Glasses and Ceramics. US Government Printing Office, Washington, 2007. 5. Wachtman, J. B., Mechanical Properties of Ceramics. Wiley-Interscience, New York, Chichester, 1996, pp. 448. 6. Danzer, R., Lube, T., Supancic, P. and Damani, R., Fracture of ceramics. Adv. Eng. Mat., submitted for publication. 7. Weibull, W., A statistical distribution function of wide applicability. J. Appl. Mech., 1951, 18, 293–298. 8. Weibull, W., A Statistical Theory of the Strength of Materials. General￾stabens Litografiska Anstalts Forlag, Stockholm, 1939, p. 45. ¨ 9. Jayatilaka, A. d. S. and Trustrum, K., Statistical approach to brittle fracture. J. Mater. Sci., 1977, 12, 1426–1430. 10. Danzer, R., A general strength distribution function for brittle materials. J. Eur. Ceram. Soc., 1992, 10, 461–472. 11. ENV 843-5, Advanced Technical Ceramics, Monolithic Ceramics, Mechan￾ical Properties at Room Temperature. Part 5: Statistical Evaluation, 1997. 12. Laraia, V. J. and Heuer, A. H., Novel composite microstructure and mechani￾cal behavior of mollusk shell. J. Am. Ceram. Soc., 1989, 72(11), 2177–2179. 13. Currey, J. D., Mechanical properties of mother of pearl in tension. Proc. R. Soc. Lond., 1977, B196, 443–463. 14. Chan, H. M., Layered ceramics: processing and mechanical behaviour. Annu. Rev. Mater. Sci., 1997, 27, 249–282. 15. Clegg, W. J., Kendall, K., Alford, N. M., Button, T. W. and Birchall, J. D., A simple way to make tough ceramics. Nature, 1990, 347, 455–457. 16. Clegg, W. J., The fabrication and failure of laminar ceramic composites. Acta. Metall. Mater., 1992, 40(11), 3085–3093. 17. Virkar, A. V., Huang, J. L. and Cutler, R. A., Strengthening of oxide ceram￾ics by transformation-induced stresses. J. Am. Ceram. Soc., 1987, 70(3), 164–170. 18. Rao, M. P., Sanchez-Herencia, A. J., Beltz, G. E., McMeeking, R. M. and ´ Lange, F. F., Laminar ceramics that exhibit a threshold strength. Science, 1999, 286, 102–105. 19. Gee, I. A., Dobedoe, R. S., Vann, R., Lewis, M. H., Blugan, G. and Kubler, ¨ J., Enhanced fracture toughness by ceramic laminate design. Adv. Appl. Ceram., 2005, 104(3), 103–109. 20. de Portu, G., Micele, L., Guicciardi, S., Fujimura, S., Pezzotti, G. and Sekiguchi, Y., Effect of residual stress on the fracture behaviour of notched