V.M. Solaro, M. Bertoldi Acta Materialia 54(2006)4929-4937 layers; this was observed in previous work [23] when the layers maintain with respect to the layer previously encoun- layer thickness was greater than a critical value, te=kc/ tered by the propagating crack. The equations contain 2n 0.34(1+vo, oc being the compressive stress and v Pois- parameters(x;, Aoi). Two conditions have to be satisfied son's ratio. Layer thickness and compressive stress are (forces equilibrium and equivalence between the sum of therefore mutually dependent and it is not possible to single-layer thickness and the total laminate thickness), lesign the desired mechanical behaviour by using only a which leaves 2n-2 degrees of freedom for defining the de square-wave stress(single layer) profile. Fortunately, these sired T-curve problems can be overcome by considering a multilayered It is important to note that in Eq (9)the elastic modulus structure of the different layers is assumed to be the same. It has been Before considering a complex multilayer profile, it is demonstrated elsewhere that the error in estimating T is useful to analyse another simple case. Consider two stress less than 10% if the Youngs modulus variation amongst profiles obtained by the combination of simple square- the layers is less than 33%[23, 24]. wave profiles of diffe nd identical exte sion(Fig. 5). This situation corresponds to laminates with 3. Design of the laminate wo layers of difTerent composition and identical thickness. The actual order of the two layers is the only difference Eq (9) suggests guidelines for the properties of the stress between the two examined profiles. It is clear from profile which would promote stable growth of surface ig. 5 that the order of the compressive layers is impor- cracks. The T-curve should be a monotonically increasing tant both for the final strength and the stability interval. function of c, which requires a continuous increase of the Such a consideration is general and the final conclusion compressive stresses from the surface towards the internal can be drawn that the compression intensity in successive layers. A stress-free or slightly tensile stressed layer is pre- layers must grow continuously to obtain a properly ferred on the surface since this allows the lower boundary designed T-curve. of the stable growth interval to move towards the surfac At this point, the principle of superposition can be used which envelopes the smaller flaws within such an interval calculate the T-curve for a general multi-step profile. We The risk of edge cracking and delamination phenomena onsider n layers, with n steps of amplitude Ao;(Fig. 2), is reduced by using multi-step profiles, which reduces the equal to the stress increase of layer j with respect to the pre- thickness of the most stressed layer vious one. A general equation, which defines the apparent The residual stress profile that develops within a ceramic fracture toughness for layer i in the interval [xi-l, xi] laminate is related to the composition/microstructure and (Fig. 2), can be obtained thickness of the layers and to their stacking order. Accord ing to the theory of composite plies [25], in order to main- tain flatness during in-plane loading, as in the case of biaxial residual stresses developed upon processing, lami- xi_I<x<x (9) nate structures must possess a number of symmetri properties. If each layer is isotropic, as is the case for where i indicates the layer rank and x is the starting depth ramiCs w tions the sum being calculated for a diferent number structure, and i出 is s ymm etrical, then th terms for each i. This represents a mathematical transla- pic, its response to loading is similar to that of a homoge- tion of the"memory"effect of stress history that deeper neous plate [25]. Regardless of the physical source of residual stresses their presence in a co-sintered multilayer is related to the constraining effect. As perfectly adherent layers, every layer must deform similarly and at the same rate as the others. The difference between free deformation or free deforma- tion rate of one layer with respect to the average value of V4, V5 5 V stresses Such stresses can be either viscous or elastic in nat ure and can be relaxed or maintained within the material depending on temperature, cooling rate and material prop- erties. With the exception of the edges, if thickness is much smaller than the other dimensions, each layer can be con- sidered to be in a biaxial stress state At this point the fundamental task in properly designing a symmetric multilayer is the estimation of the biaxial Fig. 5. Residual stress and corresponding T-curve for two simple square- residual stresses In the common case of stresses developed wave profiles placed in different order. from differences in thermal expansion coefficients only, thelayers; this was observed in previous work [23] when the layer thickness was greater than a critical value, tc ¼ K2 C= ½0:34ð1 þ mÞr2 c , rc being the compressive stress and m Pois￾son’s ratio. Layer thickness and compressive stress are therefore mutually dependent and it is not possible to design the desired mechanical behaviour by using only a square-wave stress (single layer) profile. Fortunately, these problems can be overcome by considering a multilayered structure. Before considering a complex multilayer profile, it is useful to analyse another simple case. Consider two stress profiles obtained by the combination of simple square￾wave profiles of different amplitude and identical exten￾sion (Fig. 5). This situation corresponds to laminates with two layers of different composition and identical thickness. The actual order of the two layers is the only difference between the two examined profiles. It is clear from Fig. 5 that the order of the compressive layers is impor￾tant both for the final strength and the stability interval. Such a consideration is general and the final conclusion can be drawn that the compression intensity in successive layers must grow continuously to obtain a properly designed T-curve. At this point, the principle of superposition can be used to calculate the T-curve for a general multi-step profile. We consider n layers, with n steps of amplitude Dri (Fig. 2), equal to the stress increase of layer j with respect to the pre￾vious one. A general equation, which defines the apparent fracture toughness for layer i in the interval [xi1,xi] (Fig. 2), can be obtained: T ¼ Ki C Xi j¼1 2Y c p 0:5 Drres;j p 2  arcsin xj1 c h i xi1 < x < xi; ð9Þ where i indicates the layer rank and xj is the starting depth of layer j. Eq. (9) represents a short notation of n different equations, the sum being calculated for a different number of terms for each i. This represents a mathematical transla￾tion of the ‘‘memory’’ effect of stress history that deeper layers maintain with respect to the layer previously encoun￾tered by the propagating crack. The equations contain 2n parameters (xi,Dri). Two conditions have to be satisfied (forces equilibrium and equivalence between the sum of single-layer thickness and the total laminate thickness), which leaves 2n  2 degrees of freedom for defining the de￾sired T-curve. It is important to note that in Eq. (9) the elastic modulus of the different layers is assumed to be the same. It has been demonstrated elsewhere that the error in estimating T is less than 10% if the Young’s modulus variation amongst the layers is less than 33% [23,24]. 3. Design of the laminate Eq. (9) suggests guidelines for the properties of the stress profile which would promote stable growth of surface cracks. The T-curve should be a monotonically increasing function of c, which requires a continuous increase of the compressive stresses from the surface towards the internal layers. A stress-free or slightly tensile stressed layer is pre￾ferred on the surface since this allows the lower boundary of the stable growth interval to move towards the surface, which envelopes the smaller flaws within such an interval. The risk of edge cracking and delamination phenomena is reduced by using multi-step profiles, which reduces the thickness of the most stressed layer. The residual stress profile that develops within a ceramic laminate is related to the composition/microstructure and thickness of the layers and to their stacking order. Accord￾ing to the theory of composite plies [25], in order to main￾tain flatness during in-plane loading, as in the case of biaxial residual stresses developed upon processing, lami￾nate structures must possess a number of symmetrical properties. If each layer is isotropic, as is the case for ceramics with fine and randomly oriented crystalline micro￾structure, and the stacking order is symmetrical, then the laminate remains flat upon sintering and, being orthotro￾pic, its response to loading is similar to that of a homoge￾neous plate [25]. Regardless of the physical source of residual stresses, their presence in a co-sintered multilayer is related to the constraining effect. As perfectly adherent layers, every layer must deform similarly and at the same rate as the others. The difference between free deformation or free deforma￾tion rate of one layer with respect to the average value of the whole laminate accounts for the creation of residual stresses. Such stresses can be either viscous or elastic in nat￾ure and can be relaxed or maintained within the material depending on temperature, cooling rate and material prop￾erties. With the exception of the edges, if thickness is much smaller than the other dimensions, each layer can be con￾sidered to be in a biaxial stress state. At this point the fundamental task in properly designing a symmetric multilayer is the estimation of the biaxial residual stresses. In the common case of stresses developed from differences in thermal expansion coefficients only, the x x1 x2 x3 R 2σ R x1 x2 x3 x1 x, c K ψ x2 x3 x x1 res x2 x3 K ψ –σ res –σ R 2σ R x, c T T σ σ Fig. 5. Residual stress and corresponding T-curve for two simple square￾wave profiles placed in different order. 4932 V.M. Sglavo, M. Bertoldi / Acta Materialia 54 (2006) 4929–4937