V.M. Sglavo, M. Bertoldi/Composites: Part B 37 (2006)481-489 development of the residual stresses in ceramic multilayer is the effect of an external load, flaw with generic size (c1) opportunely controlled, materials characterized by high enclosed in the interval CA, cBI and subjected to Kext=Tc1), racture resistance and limited strength scatter can be designed will propagate instantaneously up to a length within the interval and produced. By varying the nature, thickness and stacking [CA, CB] and then grow stably up to cB for higher Kext values order of the laminae, the residual stress profile developed after The arguments proposed so far are absolutely general sintering can be tailored to promote the growth of surface regardless the reasons for the non-constant fracture toughness. cracks in a stable manner before final failure. In this way, The presence of residual stresses inside the material can be strength predictable and variable as needed can be obtained by responsible for a T-curve like that shown in Fig. 1. If the simple changing the multilayer architecture. Such approach is model represented in Fig. 2 is considered, which corresponds to presented in the present work and reduced to practice for the a surface crack in a ceramic laminate, residual stresses, Ores, are production of alumina/zirconia and alumina/mullite composite correlated to the stress intensity factor [17, 18 2. Theory The aim of the present work is to set up a design procedure where h is a weight function and w is the finite width of the body useful to produce ceramic components with high mechanical In the present analysis discontinuous stepwise stress profile reliability, i.e. characterized by limited strength scatter and, is considered according to the laminated structure subjected to possibly, high fracture resistance. In order to reach such bending loads of relevance here(Fig. 2). Perfect adhesion target, the idea is advanced that stable growth of defects could between different laminae is also hypothesized. In addition, it ccur before final failure. In this way, regardless the initial is assumed that each layer is characterized by a constant law size, an invariant final strength can be attained. For fracture toughness value, Kd example, stable crack propagation is possible when fracture Under the influence of the external load(Kext, crack toughness, T, is a growing function of crack length, c, steeper propagation occurs when the sum (Kres+Kext)equals the than the applied stress intensity factor, Kext, which is generally fracture toughness, Kc, of the material at the crack tip. If the defined as Kext=yoc 0.5, where y is the shape factor and a the residual stresses are supposedly considered as a material applied stress. Analytically, stable growth occurs when the property, theapparent fracture toughness can be defined as following condition is satisfied [1, 17 T=Kc-K dExt dr(c) Next=r(c)dcs It is clear from Eqs. (2)and (3)that for compressive residual stresses(negative) there is a beneficial effect on T. In addition, It has been demonstrated elsewhere that the stability range, if given a proper residual stress profile, it should be possible to any,is finite [12]. This is shown schematically in Fig. I, where obtain T being a steep growing function of c. Moreover,as the interval [CA, cB] represents the range, where cracks can grow surface flaws have been considered, T-curve is unique for any in a stable fashion under the effect of an external load. As a defect and, therefore, it can be considered as fixed with respect direct consequence all the defects included in such interval to the surface of the body. Consequently, crack length(c)and same maximum value before final failure upo depth from the surface (x) can be regarded as identical ading, thus leading to a unique strength value. To be more quantities in the subsequent analysis precise, if kinetic effects are limited or neglected, the stable In order to understand, the effect of residual stress intensity crack growth interval can be extended down to CA; in fact, under and location on the apparent fracture toughness, it is useful to Acres 个不不 r2 layer i y,√ x Fig. 1. T-curve that allows the stable growth B).Straight lines are used to evaluate the stable growth interval and final strength, aF. Fig. 2. Crack model considered in the present work.development of the residual stresses in ceramic multilayer is opportunely controlled, materials characterized by high fracture resistance and limited strength scatter can be designed and produced. By varying the nature, thickness and stacking order of the laminae, the residual stress profile developed after sintering can be tailored to promote the growth of surface cracks in a stable manner before final failure. In this way, strength predictable and variable as needed can be obtained by changing the multilayer ‘architecture’. Such approach is presented in the present work and reduced to practice for the production of alumina/zirconia and alumina/mullite composite laminates. 2. Theory The aim of the present work is to set up a design procedure useful to produce ceramic components with high mechanical reliability, i.e. characterized by limited strength scatter and, possibly, high fracture resistance. In order to reach such target, the idea is advanced that stable growth of defects could occur before final failure. In this way, regardless the initial flaw size, an invariant final strength can be attained. For example, stable crack propagation is possible when fracture toughness, T, is a growing function of crack length, c, steeper than the applied stress intensity factor, Kext, which is generally defined as KextZjsc0.5, where j is the shape factor and s the applied stress. Analytically, stable growth occurs when the following condition is satisfied [1,17]: Kext ZTðcÞ dKext dc % dTðcÞ dc (1) It has been demonstrated elsewhere that the stability range, if any, is finite [12]. This is shown schematically in Fig. 1, where the interval [cA, cB] represents the range, where cracks can grow in a stable fashion under the effect of an external load. As a direct consequence all the defects included in such interval propagate to the same maximum value before final failure upon loading, thus leading to a unique strength value. To be more precise, if kinetic effects are limited or neglected, the stable crack growth interval can be extended down to c
A; in fact, under the effect of an external load, flaw with generic size (c1), enclosed in the interval c
A; cB
and subjected to KextZT(c1), will propagate instantaneously up to a length within the interval [cA, cB] and then grow stably up to cB for higher Kext values. The arguments proposed so far are absolutely general, regardless the reasons for the non-constant fracture toughness. 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, which corresponds to a surface crack in a ceramic laminate, residual stresses, sres, are correlated to the stress intensity factor [17,18] Kres Z ðc 0 sresðxÞh x c ; c w dx (2) where h is a weight function and w is the finite width of the body. In the present analysis discontinuous stepwise stress profile is considered according to the laminated structure subjected to bending loads of relevance here (Fig. 2). Perfect adhesion between different laminae is also hypothesized. In addition, it is assumed that each layer is characterized by a constant fracture toughness value, Ki C. Under the influence of the external load (Kext), crack propagation occurs when the sum (KresCKext) equals the fracture toughness, Ki C, of the material at the crack tip. If the residual stresses are supposedly considered as a material property, the ‘apparent’ fracture toughness can be defined as: Ti ZKi CKKres (3) 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 should be possible to obtain T being a steep growing function of c. Moreover, as surface flaws have been considered, T-curve is unique for any defect and, therefore, it can be considered as fixed with respect to the surface of the body. Consequently, crack length (c) and depth from the surface (x) can be regarded as identical quantities in the subsequent analysis. In order to understand, the effect of residual stress intensity and location on the apparent fracture toughness, it is useful to Fig. 1. T-curve that allows the stable growth phenomenon in the interval (cA, cB). Straight lines are used to evaluate the stable growth interval and final strength, sF. Fig. 2. Crack model considered in the present work. 482 V.M. Sglavo, M. Bertoldi / Composites: Part B 37 (2006) 481–489