V.M. Solaro, M. Bertoldi Acta Materialia 54(2006)4929-4937 laminates can lead to defects in the form of tunnel cracks [13] and fracture in the ceramic laminate[10]. The use of near-surface stresses that hinder the growth of surface cracks has been extensively exploited in glasses 4, 15]. Surface flaws represent typical defects in ceramics and glasses: in fact, once the process has been optimised reduce or eliminate volume defects [16, 17]. surface flav which are normally generated during surface finishing and in service, become strength limiting. Furthermore, surface defects are important when bodies are subjected to bend ing, as is often the case in ceramic components. Recently, Sglavo et al. have proposed that creation of a residual a maximum compression at a certain Fig. I. T-curve that allows the stable growth phenomenon in the interval lepth from the surface can arrest surface cracks [18]. This co, Cr). Straight ines represent various Keat values and are used to pproach has been applied to silicate glasses by producing evaluate the stable growth interval and final strength,oF the residual stress field through a double ion-exchange process [19, 20]. the interval co, ce] and subjected to Kext= T(c1), will prop- Residual stresses in ceramic materials can arise from dif- agate instantaneously up to a length c2 within the interval ferences in the thermal expansion coefficient of the consti- [co, CF] and then grow stably up to cp for higher Kext values tuting grains or phases, uneven sintering rates or phase The presence of residual stresses inside the material can transformations associated with specific volume change. be responsible for a T-curve like that shown in Fig. 1. If the As described below, the residual stress profile developed simple model represented in Fig. 2 is considered, residual after sintering can be controlled by changing the thickness stresses, ores are responsible for the stress intensity factor and the stacking order of the layers. In this approach a pre- [21,22 dictable strength can be obtained by changing the multi- layer"architecture". In the present work this approach Kres o(x)(2,)d applied to the design and the production of alumina/ ZirconIa compo aminates where h is a weight function and w is the width of the body. In the present analysis discontinuous stepwise stress pro file is adopted and perfect adhesion between different lam 2. Theory inae is hypothesised. In addition, it is assumed that each layer is characterised by a constant fracture toughness The first step in the approach mentioned above is to value. Ki develop a procedure for designing ceramic components Under the influence of the external load(Kext, crack ith high mechanical reliability. These components ar propagation occurs when the sum(Kres t Kext) equals the characterised by narrow scatter in fracture strength, which fracture toughness, Kc, of the material at the crack tip is obtained by stable growth of defects before final failure. Including the residual stresses as a part of the crack resis- In this way an invariant final strength, which is indepen- tance of the material, the"apparent"fracture toughness dent of the initial flaw size can be attained can be defined Stable crack propagation is possible when fracture toughness, T, is a growing function of crack length, c, stee. T=ki-Kres (3) per than the applied stress intensity factor, Kext, which is It is clear from Eqs.(2)and (3)that for compressive generally defined as Kext=yac., where y is the shape fac- residual stresses(negative) there is a beneficial effect on tor and o the applied stress. Analytically, stable growth T. In addition, given a proper residual stress profile, it occurs when the following condition is satisfied [1, 213 T(c), a It has been demonstrated elsewhere that the range of stable crack growth is finite [18]. This is shown schemati 白 cally in Fig. I where the interval [co, cF]represents the range where cracks can grow in a stable fashion: all defect included in such interval propagate to the same maximum △ores value before final failure, thus leading to a unique strength value(F in Fig. 1). More precisely, if kinetic effects are eglected the stable crack growth interval can be extended down to co: in fact, a flaw with generic size, CI, enclosed in Fig. 2. Crack model considered in the present worklaminates can lead to defects in the form of tunnel cracks [13] and fracture in the ceramic laminate [10]. The use of near-surface stresses that hinder the growth of surface cracks has been extensively exploited in glasses [14,15]. Surface flaws represent typical defects in ceramics and glasses: in fact, once the process has been optimised to reduce or eliminate volume defects [16,17], surface flaws, which are normally generated during surface finishing and in service, become strength limiting. Furthermore, surface defects are important when bodies are subjected to bend￾ing, as is often the case in ceramic components. Recently, Sglavo et al. have proposed that creation of a residual stress profile with a maximum compression at a certain depth from the surface can arrest surface cracks [18]. This approach has been applied to silicate glasses by producing the residual stress field through a double ion-exchange process [19,20]. Residual stresses in ceramic materials can arise from dif￾ferences in the thermal expansion coefficient of the consti￾tuting grains or phases, uneven sintering rates or phase transformations associated with specific volume change. As described below, the residual stress profile developed after sintering can be controlled by changing the thickness and the stacking order of the layers. In this approach a pre￾dictable strength can be obtained by changing the multi￾layer ‘‘architecture’’. In the present work this approach is applied to the design and the production of alumina/ mullite/zirconia composite laminates. 2. Theory The first step in the approach mentioned above is to develop a procedure for designing ceramic components with high mechanical reliability. These components are characterised by narrow scatter in fracture strength, which is obtained by stable growth of defects before final failure. In this way an invariant final strength, which is indepen￾dent of the initial flaw size, can be attained. Stable crack propagation is possible when fracture toughness, T, is a growing function of crack length, c, stee￾per than the applied stress intensity factor, Kext, which is generally defined as Kext = wrc 0.5, where w is the shape fac￾tor and r the applied stress. Analytically, stable growth occurs when the following condition is satisfied [1,21]: Kext ¼ T ðcÞ; dKext dc 6 dT ðcÞ dc : ( ð1Þ It has been demonstrated elsewhere that the range of stable crack growth is finite [18]. This is shown schemati￾cally in Fig. 1 where the interval [c0, cF] represents the range where cracks can grow in a stable fashion: all defects included in such interval propagate to the same maximum value before final failure, thus leading to a unique strength value (rF in Fig. 1). More precisely, if kinetic effects are neglected, the stable crack growth interval can be extended down to c 0; in fact, a flaw with generic size, c1, enclosed in the interval ½c 0; cF and subjected to Kext = T(c1), will prop￾agate instantaneously up to a length c2 within the interval [c0, cF] and then grow stably up to cF for higher Kext values. The presence of residual stresses inside the material can be responsible for a T-curve like that shown in Fig. 1. If the simple model represented in Fig. 2 is considered, residual stresses, rres, are responsible for the stress intensity factor [21,22]: Kres ¼ Z c 0 rresðxÞh x c ; c w dx; ð2Þ where h is a weight function and w is the width of the body. In the present analysis discontinuous stepwise stress pro- file is adopted and perfect adhesion between different lam￾inae is hypothesised. In addition, it is assumed that each layer is characterised by a constant fracture toughness value, Ki C. Under the influence of the external load (Kext), crack propagation occurs when the sum (Kres + Kext) equals the fracture toughness, Ki C, of the material at the crack tip. Including the residual stresses as a part of the crack resis￾tance of the material, the ‘‘apparent’’ fracture toughness can be defined as T i ¼ Ki C  Kres: ð3Þ It is clear from Eqs. (2) and (3) that for compressive residual stresses (negative) there is a beneficial effect on T. In addition, given a proper residual stress profile, it K ψ σ c cF c0 T F c0 * c1 c2 Fig. 1. T-curve that allows the stable growth phenomenon in the interval (c0, cF). Straight lines represent various Kext values and are used to evaluate the stable growth interval and final strength, rF. c res(x) res,i x σ σ Δσ i xi-1 layer i res,i Fig. 2. Crack model considered in the present work. 4930 V.M. Sglavo, M. Bertoldi / Acta Materialia 54 (2006) 4929–4937