V.M. Sglavo, M. Bertoldi/Composites: Part B 37(2006)481-489 thickness and the total laminate thickness), 2n-2 are the symmetrical, the laminate remains flat upon sintering and remaining degrees of freedom suitable to define the desired being orthotropic, its response to loading is similar to that of a T-curve homogeneous plate [22] It is important to point out that in the calculations carried out Regardless the physical source of residual stresses, their to obtain Eq. (9) the approximation is made that the elastic presence in co-sintered multilayer is related to constraining modulus of the different layers is constant. It has been effect. Under the condition of perfectly adherent layers, every demonstrated elsewhere that the approximation in T estimate lamina must deform similarly and at the same rate of the others does not exceeds 10%o if the Youngs modulus variation is less The difference between free deformation or free deformation than33%[20.21] rate of the single lamina 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 nature and can 3. Laminates design be relaxed or maintained within the material depending on temperature, cooling rate and material properties. With the Eq (9)suggests some considerations about the conditions that exception of the edges, if thickness is much smaller than the a proper stress profiles should possess to promote the stable other dimensions, each layer can be considered to be in a growth of surface cracks. The stable propagation of surface biaxial stress state. defects is possible only when the T-curve is a monotonic increasing function of c and this requires a continuous increase of symmetric multilayer is the estimate of the biaxial residual the compressive stresses from the surface towards intemal layers. stresses. In the common case of stresses developed from A stress-free or slightly tensile stressed layer is also preferred on differences in thermal expansion coefficients only, the the surface, since this allows to move the lower boundary of the following conditions (related to forces equilibrium, compat stable growth interval towards the surface. It is important to point ibility and constitutive model) must be satisfied out that, according to Eq.(9), the effect of the surface layer is transferred to all internal laminae. The surface tensile layer has in 10ms/=0;=e1+a△T=e0=Ee1(10) act a reducing effect on the T-curve for any crack length and for this reason its depth and intensity must be limited, the maximum a'; being the thermal expansion coefficient, E= E /(l-vi) (vi=Poisson's ratio, Ei=Young modulus), e; the elastic strain, stress being, otherwise, too low. In addition, by using multi-step e: the deformation. The system defined by Eq(10)represent a profiles it is possible to reduce the thickness of the most stressed layer with the introduction of intermediate layers before and set of 3n+I equations and 3n+I unknowns(oi, Ei, ein e). The solution of such linear system allows to calculate the residual beyondit. The risk of edge cracking and delamination phenomena stress in the generic layer i(among n layers)as are reduced accordingly The residual stress profile that develops within a ceramic Ores i =E(d-a)AT laminate is related either to the composition/microstructure and thickness of the laminae and to their stacking order. according where T=TSF-TRT (TSE=stress free temperature, TRT room temperature) and a is the average thermal expansion to the theory of composite plies [22 in order to maintain coefficient of the whole laminate defined as flatness during in-plane loading, as in the case of biaxial esIdual stresses developed upon processing, laminate structu=∑E1/∑E isotropic, like in ceramic laminae with fine and randomly t; being the layer thickness. In this specific case, the residual oriented crystalline microstructure, and the stacking order is stresses are, therefore, generated upon cooling after sintering. It AZ=1 AM-1 has been shown in previous works that TsF represents the AM041乒m temperature below which the material can be considered to behave as a perfectly elastic body and visco-elastic relaxation phenomena do not occur [23]. It must be pointed out that the reported analysis, corresponding to the development of stresses AZ20,35m from differences in thermal expansion coefficients only, can be AMld, 4 m easily generalized when other differential strain developers are active like those associated to martensitic phase transformations In this case, the compatibility equation in Eq (10)becomes AM0. 540 E1=er+a△T+er=e where Er represents the strain associated to phase transformation Eq(9)represents the fundamental tool for the design of a ceramic laminate with pre-defined mechanical properties Fig.6. Architecture of the AZ-1 and AM-1 laminates Layers thickness and Different ceramic layers can be stacked together in order to composition are reported(dimensions are not in scale) develop after sintering a specific residual stress profile that canthickness and the total laminate thickness), 2nK2 are the remaining degrees of freedom suitable to define the desired T-curve. It is important to point out that in the calculations carried out to obtain Eq. (9) the approximation is made that the elastic modulus of the different layers is constant. It has been demonstrated elsewhere that the approximation in T estimate does not exceeds 10% if the Young’s modulus variation is less than 33% [20,21]. 3. Laminates design Eq. (9) suggests some considerations about the conditions that a proper stress profiles should possess to promote the stable growth of surface cracks. The stable propagation of surface defects is possible only when the T-curve is a monotonic increasing function of c and this requires a continuous increase of the compressive stresses from the surface towards internal layers. A stress-free or slightly tensile stressed layer is also preferred on the surface, since this allows to move the lower boundary of the stable growth interval towards the surface. It is important to point out that, according to Eq. (9), the effect of the surface layer is transferred to all internal laminae. The surface tensile layer has in fact a reducing effect on the T-curve for any crack length and for this reason its depth and intensity must be limited, the maximum stress being, otherwise, too low. In addition, by using multi-step profiles it is possible to reduce the thickness of the most stressed layer with the introduction of intermediate layers before and beyondit. The risk of edge cracking and delamination phenomena are reduced accordingly. The residual stress profile that develops within a ceramic laminate is related either to the composition/microstructure and thickness of the laminae and to their stacking order. According to the theory of composite plies [22], in order to maintain flatness during in-plane loading, as in the case of biaxial residual stresses developed upon processing, laminate structure has to satisfy some symmetry conditions. If each layer is isotropic, like in ceramic laminae with fine and randomly oriented crystalline microstructure, and the stacking order is symmetrical, the laminate remains flat upon sintering and, being orthotropic, its response to loading is similar to that of a homogeneous plate [22]. Regardless the physical source of residual stresses, their presence in co-sintered multilayer is related to constraining effect. Under the condition of perfectly adherent layers, every lamina must deform similarly and at the same rate of the others. The difference between free deformation or free deformation rate of the single lamina 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 nature and can be relaxed or maintained within the material depending on temperature, cooling rate and material properties. With the exception of the edges, if thickness is much smaller than the other dimensions, each layer can be considered to be in a biaxial stress state. At this point the fundamental task to properly design a symmetric multilayer is the estimate of the biaxial residual stresses. In the common case of stresses developed from differences in thermal expansion coefficients only, the following conditions (related to forces equilibrium, compat￾ibility and constitutive model) must be satisfied Xn iZ1 sres;iti Z0 3i Zei CaiDT Z 3 si ZE
i ei (10) ai being the thermal expansion coefficient, E
i ZEi=ð1KniÞ (niZPoisson’s ratio, EiZYoung modulus), ei the elastic strain, 3i the deformation. The system defined by Eq. (10) represent a set of 3nC1 equations and 3nC1 unknowns (si, 3i, ei,3). The solution of such linear system allows to calculate the residual stress in the generic layer i (among n layers) as sres;i ZE
i ða
KaiÞDT (11) where DTZTSFKTRT (TSFZstress free temperature, TRTZ room temperature) and a
is the average thermal expansion coefficient of the whole laminate defined as a
ZXn 1 E
i tiai Xn 1 E
i ti . (12) ti being the layer thickness. In this specific case, the residual stresses are, therefore, generated upon cooling after sintering. It has been shown in previous works that TSF represents the temperature below which the material can be considered to behave as a perfectly elastic body and visco–elastic relaxation phenomena do not occur [23]. It must be pointed out that the reported analysis, corresponding to the development of stresses from differences in thermal expansion coefficients only, can be easily generalized when other differential strain developers are active like those associated to martensitic phase transformations. In this case, the compatibility equation in Eq. (10) becomes 3i Zei CaiDT C3T Z3 (13) where 3T represents the strain associated to phase transformation [8,24]. Eq. (9) represents the fundamental tool for the design of a ceramic laminate with pre-defined mechanical properties. Different ceramic layers can be stacked together in order to develop after sintering a specific residual stress profile that can Fig. 6. Architecture of the AZ-1 and AM-1 laminates. Layers thickness and composition are reported (dimensions are not in scale). V.M. Sglavo, M. Bertoldi / Composites: Part B 37 (2006) 481–489 485