S. Bueno et al. /Journal of the European Ceramic Society 28 (2008)1961-1971 provided for the fine-grained materials with homogeneous where E=E/(1-v)is the generalized Youngs modulus for microstructures. All materials presented different levels of plane strain(E is the Youngs modulus and v is the Poisson's microcracks in the"as sintered"state. A latter work on ratio) microcrack-free and fine-grained alumina+ 10 vol %o aluminium The activation of toughening mechanisms during the frac- titanate fabricated from alumina and aluminium titanate mix- ture of ceramic materials gives rise to inelastic strain processes tures showed that second phase grains as well as matrix grains that produce additional release of the elastic energy accumulated could act as bridges in the wake of the propagating crack. 8 in the material at the moment of fracture initiation and/or con- This material presented increased thermal shock resistance than tributing to the retardation of crack growth.The inelastic strain a monophase alumina of similar grain size while aining levels achieved in ceramic materials can be enough to restrict the strength direct utilization of linear elastic fracture toughness parameters The initial objective of this work was to investigate the since they become dependent on testing and specimen geometry possibilities of crack bridging in fine-grained, homogeneous for non-linear materials. 18-20 and microcrack -free alumina-10 vol %o aluminium titanate com The rising R-curve behaviour, increasing Kic or GIc posites for flaw tolerance. Reaction sintering of alumina and with crack extension(Aa), has traditionally been the most titania was used as processing route. I 5 The microstruc- utilized approach to analyze deviations from the linear tures of the reaction sintered materials were different than behaviour induced by toughening in dense and fine-grained nat of the previously studied material, with a bimodal dis- ceramics. 9, 21-22 In equilibrium conditions, the applied stress tribution of aluminium titanate grains with nanoparticles intensity factor, KI, is balanced by the crack growth resistance, located at the alumina grain boundaries. The characteriza- K, and maximum values of this latter, Ko, are reached when tion of the fracture process in the composites and monophase the process zone is completely developed. alumina materials, combining different fracture parameters In order to build the R curve of the materials, crack growth gether with fractographic observations, has allowed determin- resistance and crack length values during crack extension are ing the extreme effect of the grain boundary characteristics needed. The"in situ" measurement of crack length can be a prob- in the fracture process. The major toughening mechanism lem especially for materials such as alumina-aluminiumtitanate identified in the composite studied here has been microcrack- composites, constituted by phases with large differences in hard- ness and in which residual stresses are present. The low quality of polished surfaces of relatively large specimens(e.g: bending 2. Quantification of fracture toughness bars with lateral face dimension 50mm x 6 mm)of such mate- rials makes the identification and monitoring of the propagating In general, the linear elastic fracture behaviour of ceramic crack enormously difficult Alternatively, the R curves can be determined by the indirect critical stress intensity factor in mode I, KiC, and critical strain method that defines an equivalent crack length as a function energy release rate, GIC. For three-point-bend beams, the val- of the elastic compliance of the specimen, C 23-25 For par- ues of Kic can be determined from the notch depths and the allelepiped bars with straight through notches tested in three maximum loads reached in the tests according to the general points bending, the expression provided by Guinea et al. can stress intensity formulation, valid for any notch depth, a, in line be utilized(eq (4)) elastic materials(Eq(1)6 (CEB) 3PL 2BW3/2 X Y(a) [CEB+qI(CEB)2+q2(CEB)+g3 where P is the maximum load, L is the span, B and w are the where E, a and B have the same meaning as before(eq.(1) width and the thickness of the bars, a is the normalized notch and qi(i=1, 2, 3) are parameters that depend on the L/W ratio length(a=alW) and Y(a)is a shape function depending on the (2.5 <(L/W)< 16) span to thickness ratio(L/W, Eq(2)) In a lesser extent, the non-linear fracture toughness paran J-integral and work of fracture, ywoF, have been used br (199+0.83a-0.31a2+0.14a3+4W/L) and co-workers. 26 Bradt and co-workers 27 and Sakai et al. 18 Y(a)=x(-0.09-0.42a+0.82a2-0.31a2) (1-a)32×(1+3a) (2) to characterize ceramic materials with coarser microstructures and higher levels of non-linearity such as refractories and fiber From Kic and Young s modulus, Gic can be calculated according The J-integral is an energy term that generalizes the energy to the analysis of Irwin that relates the stress-derived fracture release rate, G, to include non-linear elastic and inelastic toughness(Kic)and the energy-derived fracture toughness( Gic) behaviours and that describes the total energy of the crack-tip for plane strain conditions(Eq ( 3): stress-strain field. The critical value, JIC, constitutes a fracture criterion for materials where the toughening occurs along lim- ited crack propagation such as those that present small bridging1962 S. Bueno et al. / Journal of the European Ceramic Society 28 (2008) 1961–1971 provided for the fine-grained materials with homogeneous microstructures. All materials presented different levels of microcracks in the “as sintered” state. A latter work on microcrack-free and fine-grained alumina + 10 vol.% aluminium titanate fabricated from alumina and aluminium titanate mix￾tures showed that second phase grains as well as matrix grains could act as bridges in the wake of the propagating crack.8 This material presented increased thermal shock resistance than a monophase alumina of similar grain size while maintaining strength. The initial objective of this work was to investigate the possibilities of crack bridging in fine-grained, homogeneous and microcrack-free alumina–10 vol.% aluminium titanate com￾posites for flaw tolerance. Reaction sintering of alumina and titania was used as processing route. 15 The microstruc￾tures of the reaction sintered materials were different than that of the previously studied material, with a bimodal dis￾tribution of aluminium titanate grains with nanoparticles located at the alumina grain boundaries. The characteriza￾tion of the fracture process in the composites and monophase alumina materials, combining different fracture parameters together with fractographic observations, has allowed determin￾ing the extreme effect of the grain boundary characteristics in the fracture process. The major toughening mechanism identified in the composite studied here has been microcrack￾ing. 2. Quantification of fracture toughness In general, the linear elastic fracture behaviour of ceramic materials is quantified by the following toughness parameters: critical stress intensity factor in mode I, KIC, and critical strain energy release rate, GIC. For three-point-bend beams, the val￾ues of KIC can be determined from the notch depths and the maximum loads reached in the tests according to the general stress intensity formulation, valid for any notch depth, a, in linear elastic materials (Eq. (1)) 16: KI = 3PL 2BW3/2 × Y(α) (1) where P is the maximum load, L is the span, B and W are the width and the thickness of the bars, α is the normalized notch length (α = a/W) and Y(α) is a shape function depending on the span to thickness ratio (L/W, Eq. (2)). Y(α)= √α(1.99 + 0.83α − 0.31α2 + 0.14α3 + 4(W/L) ×(−0.09 − 0.42α + 0.82α2 − 0.31α3)) (1 − α) 3/2 × (1 + 3α) (2) FromKIC and Young’s modulus,GIC can be calculated according to the analysis of Irwin that relates the stress-derived fracture toughness (KIC) and the energy-derived fracture toughness (GIC) for plane strain conditions (Eq. (3)): GIC = K2 IC E (3) where E = E/(1 − ν2) is the generalized Young’s modulus for plane strain (E is the Young’s modulus and ν is the Poisson’s ratio). The activation of toughening mechanisms during the frac￾ture of ceramic materials gives rise to inelastic strain processes that produce additional release of the elastic energy accumulated in the material at the moment of fracture initiation and/or con￾tributing to the retardation of crack growth.17 The inelastic strain levels achieved in ceramic materials can be enough to restrict the direct utilization of linear elastic fracture toughness parameters since they become dependent on testing and specimen geometry for non-linear materials.18–20 The rising R-curve behaviour, increasing KIC or GIC with crack extension (a), has traditionally been the most utilized approach to analyze deviations from the linear behaviour induced by toughening in dense and fine-grained ceramics.19,21–22 In equilibrium conditions, the applied stress intensity factor, KI, is balanced by the crack growth resistance, KR, and maximum values of this latter, K∞, are reached when the process zone is completely developed. In order to build the R curve of the materials, crack growth resistance and crack length values during crack extension are needed. The “in situ” measurement of crack length can be a prob￾lem especially for materials such as alumina–aluminium titanate composites, constituted by phases with large differences in hard￾ness and in which residual stresses are present. The low quality of polished surfaces of relatively large specimens (e.g.: bending bars with lateral face dimension 50 mm × 6 mm) of such mate￾rials makes the identification and monitoring of the propagating crack enormously difficult. Alternatively, the R curves can be determined by the indirect method that defines an equivalent crack length as a function of the elastic compliance of the specimen, C. 23–25 For par￾allelepiped bars with straight through notches tested in three points bending, the expression provided by Guinea et al.16 can be utilized (Eq. (4)): α = (CE B) 1/2 [CE B + q1(CE B) 1/2 + q2(CE B) 1/3 + q3] 1/2 (4) where E , α and B have the same meaning as before (Eq. (1)) and qi (i = 1, 2, 3) are parameters that depend on the L/W ratio (2.5 ≤ (L/W) ≤ 16). In a lesser extent, the non-linear fracture toughness parameter J-integral and work of fracture, γWOF, have been used by Li and co-workers,26 Bradt and co-workers27 and Sakai et al.18 to characterize ceramic materials with coarser microstructures and higher levels of non-linearity such as refractories and fiber reinforced ceramic matrix composites. The J-integral is an energy term that generalizes the energy release rate, G, to include non-linear elastic and inelastic behaviours and that describes the total energy of the crack-tip stress–strain field.28 The critical value, JIC, constitutes a fracture criterion for materials where the toughening occurs along lim￾ited crack propagation such as those that present small bridging zones.29