Availableonlineatwww.sciencedirect.com Part B: engineering ELSEVIER Composites: Part B 37(2006)481-489 Design and production of ceramic laminates with high mechanical reliability Vincenzo M. Sglavo*Massimo bertoldi I Dipartimento di Ingegneria dei Materiali e Tecnologie Industriali, Universita di Trento, Via Mesiano, 77, 38050 Trento, Italy Available online 20 March 2006 Abstract a procedure for designing innovative ceramic laminates characterized by high mechanical reliability is proposed in this work. A fracture mechanics approach has been considered to define the stacking sequence, thickness and composition of the different laminae on the basis of the requested strength and of the defect size distribution. Once the different laminae are stacked together a residual stress profile is generated upon cooling after sintering because of the differential thermal expansion coefficient. Such residual stress profile is conceived in order to allow stable growth of surface defects upon bending and guarantee limited strength scatter. As an example, the proposed approach is used to design and produce ceramic laminates in the alumina-zirconia and alumina-mullite system. Mechanical performances of the produced materials are discussed in terms of the generated residual stress profile and compared to parent monolithic ceramics. C 2006 Elsevier ltd. all rights reserved Keywords: A. Ceramic-matrix composites; B. Residual/internal stress; B. Strength; B. Fracture toughness; Ceramic laminates 1. Introduction been improved by introducing low-energy paths for growin crack in laminated structures [2-6] or by introducing Ceramics are commonly considered as brittle materials. In compressive residual stresses [7, 8]. Laminates presenting spite of this, their very attractive physical and chemical threshold strength have been also successfully produced by properties make such materials suitable for different appli- alternating thin compressive layers and thicker tensile layers cations. The limited fracture toughness associated to the [9]. Unfortunately, the most important limitations of such presence of flaws generated either upon processing and in laminates is that they can be used only with specific service is responsible for their limited mechanical reliability. orientations with respect to the applied load and, for example, The resulting strength scatter is usually too large to allow safe they are not easily suitable to produce real components such design,unless statistical approaches identifying acceptable plates, shells or tubes as usually required in typical minimum failure risk are used. In addition, fracture usually applications occurs in a catastrophic manner in absence of any warning of The idea that surface stresses can hinder the growth of the incipient rupture [1] surface cracks has been extensively exploited in the past Many efforts have been made in the past to increase the especially on glasses [10-11]. Sglavo and Green have recently mechanical reliability of glasses and ceramics. Higher fracture proposed that the creation of a residual stress profile in glass toughness have been attained through the exploitation of the with a maximum compression at a certa pth from the reinforcing action of grain anisotropy or second phases or the surface can arrest surface cracks and result in higher failure promotion of crack shielding effects associated to phase- stress and limited strength variability [12-14 One can point transformation or micro-cracking [1]. In such cases, a precise out that surface flaws represent the most typical defects in microstructure control is always required and this is achievable ceramic and glasses: in fact, once the processing procedures are conditions. As an alternativefracture behavior of ceramics has generate volume defects [15, 16], surface flaws are normally only surface defects become critical when a body is subjected rresponding author Tel. +39 461 882468: fax: +39 40 to bending and not to tension, as it is usually the case in ceramic E-mail address: vincenzo. sglavo@ unitn. it(V M. Sglavo) components Now at Eurocoating Spa, Via Al Dos de la Roda 60, 38057 Pergine Residual stresses in ceramic materials can arise either from differences in the thermal expansion coefficient of the 1359-8368/S- see front matter 2006 Elsevier Ltd. All rights reserved. constituting grains or phases, uneven sintering rates or doi: 10. 1016/ martensitic phase transformations. As described below, if the
Design and production of ceramic laminates with high mechanical reliability Vincenzo M. Sglavo *, Massimo Bertoldi 1 Dipartimento di Ingegneria dei Materiali e Tecnologie Industriali, Universita` di Trento, Via Mesiano, 77, 38050 Trento, Italy Available online 20 March 2006 Abstract A procedure for designing innovative ceramic laminates characterized by high mechanical reliability is proposed in this work. A fracture mechanics approach has been considered to define the stacking sequence, thickness and composition of the different laminae on the basis of the requested strength and of the defect size distribution. Once the different laminae are stacked together a residual stress profile is generated upon cooling after sintering because of the differential thermal expansion coefficient. Such residual stress profile is conceived in order to allow stable growth of surface defects upon bending and guarantee limited strength scatter. As an example, the proposed approach is used to design and produce ceramic laminates in the alumina–zirconia and alumina–mullite system. Mechanical performances of the produced materials are discussed in terms of the generated residual stress profile and compared to parent monolithic ceramics. q 2006 Elsevier Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites; B. Residual/internal stress; B. Strength; B. Fracture toughness; Ceramic laminates 1. Introduction Ceramics are commonly considered as brittle materials. In spite of this, their very attractive physical and chemical properties make such materials suitable for different applications. The limited fracture toughness associated to the presence of flaws generated either upon processing and in service is responsible for their limited mechanical reliability. The resulting strength scatter is usually too large to allow safe design, unless statistical approaches identifying acceptable minimum failure risk are used. In addition, fracture usually occurs in a catastrophic manner in absence of any warning of the incipient rupture [1]. Many efforts have been made in the past to increase the mechanical reliability of glasses and ceramics. Higher fracture toughness have been attained through the exploitation of the reinforcing action of grain anisotropy or second phases or the promotion of crack shielding effects associated to phasetransformation or micro-cracking [1]. In such cases, a precise microstructure control is always required and this is achievable only with a careful control of starting material and process conditions. As an alternative, fracture behavior of ceramics has been improved by introducing low-energy paths for growing crack in laminated structures [2–6] or by introducing compressive residual stresses [7,8]. Laminates presenting threshold strength have been also successfully produced by alternating thin compressive layers and thicker tensile layers [9]. Unfortunately, the most important limitations of such laminates is that they can be used only with specific orientations with respect to the applied load and, for example, they are not easily suitable to produce real components such as plates, shells or tubes as usually required in typical applications. The idea that surface stresses can hinder the growth of surface cracks has been extensively exploited in the past especially on glasses [10–11]. Sglavo and Green have recently proposed that the creation of a residual stress profile in glass with a maximum compression at a certain depth from the surface can arrest surface cracks and result in higher failure stress and limited strength variability [12–14]. One can point out that surface flaws represent the most typical defects in ceramic and glasses: in fact, once the processing procedures are optimized to reduce or remove heterogeneities that can generate volume defects [15,16], surface flaws are normally created during surface finishing or upon service. In addition, only surface defects become critical when a body is subjected to bending and not to tension, as it is usually the case in ceramic components. Residual stresses in ceramic materials can arise either from differences in the thermal expansion coefficient of the constituting grains or phases, uneven sintering rates or martensitic phase transformations. As described below, if the Composites: Part B 37 (2006) 481–489 www.elsevier.com/locate/compositesb 1359-8368/$ - see front matter q 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2006.02.001 * Corresponding author Tel.: C39 461 882468; fax: C39 461 881945. E-mail address: vincenzo.sglavo@unitn.it (V.M. Sglavo). 1 Now at Eurocoating Spa, Via Al Dos de la Roda 60, 38057 Pergine Valsugana, Italy
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
V.M. Sglavo, M. Bertoldi/Composites: Part B 37(2006)481-489 483 analyze some special cases. First of all, if the reference model As shown in Fig. 3(b) a stability range exists between x (Fig. 2) is thought to correspond to an edge crack in a semi- and the tangent point between Kext and T. One can observe nfinite body, Eq (2)can be simplified as that for increasing xI the strength decreases and the stability interval width increases. Since, both high strength and large yos ores()T2-2)05 stable growth interval are desirable, an intermediate value of (πc)5 (4) x, has to be considered in the perspective laminate design On the other side an increase of or is useful to increase both the stable growth range and the maximum stress. In where Y=1.1215. One could point out that such simplification addition, if Kc increases, the maximum stress is higher but is not rigorous as Y maintains a slight dependence on x/e [18]. the stability range decreases though one must considered Nevertheless, this allows to perform the calculations in closed that Kc is a parameter that depends on the material form without loosing of generality selection and it is not usually modified in the desi One very simple situation corresponds to the step profil procedure shown in Fig. 3(a) defined as: A more realistic residual stress shape is the square-wave rs=00<x<x1 Ures =-or XI<x<too Ores =00<x<x Ores =or xI<x<x2 In this case, T can be analytically calculated as rs=0x2<x<+∞ 0<x<x1 In this case, the T-curve can be calculated both analytically nd by using the principle of superposition [1, 17]. The square- wave profile can be considered in fact as the sum of two simple T=Kc+2r n[2 arcsin x,<x<+o step profiles with stresses of identical amplitude but opposite sign placed at different depths (xI and x2). The apparent (6) fracture toughness become T=Kc-2y 一 arcsin xI<x<x T=Ko-2y TtR( )-=( x<x<+∞ yx,√C(b) Fig. 3. Step residual stress profile(a) and comesponding apparent fracture toughness(b). The effects of intensity (left)and location(right) of the residual stress are
analyze some special cases. First of all, if the reference model (Fig. 2) is thought to correspond to an edge crack in a semiinfinite body, Eq. (2) can be simplified as Kres Z Y ðpcÞ 0:5 ðc 0 sresðxÞ 2c c2Kx2 0:5 dx (4) where Yz1.1215. One could point out that such simplification is not rigorous as Y maintains a slight dependence on x/c [18]. Nevertheless, this allows to perform the calculations in closed form without loosing of generality. One very simple situation corresponds to the step profile shown in Fig. 3(a) defined as: sres Z 0 0!x!x1 sres ZKsR x1!x!CN ( (5) In this case, T can be analytically calculated as: T ZKC 0!x!x1 T ZKC C2Y c p 0 @ 1 A 0:5 sR p 2 Karcsin x0 c 0 @ 1 A 0 @ 1 A x1!x!CN 8 >>>>< >>>>: (6) As shown in Fig. 3(b) a stability range exists between x1 and the tangent point between Kext and T. One can observe that for increasing x1 the strength decreases and the stability interval width increases. Since, both high strength and large stable growth interval are desirable, an intermediate value of x1 has to be considered in the perspective laminate design. On the other side, an increase of sR is useful to increase both the stable growth range and the maximum stress. In addition, if KC increases, the maximum stress is higher but the stability range decreases though one must considered that KC is a parameter that depends on the material selection and it is not usually modified in the design procedure. A more realistic residual stress shape is the square-wave profile (Fig. 4) defined as: sres Z0 0!x!x1 sres ZKsR x1!x!x2 sres Z0 x2!x!CN 8 >< >: (7) In this case, the T-curve can be calculated both analytically and by using the principle of superposition [1,17]. The squarewave profile can be considered in fact as the sum of two simple step profiles with stresses of identical amplitude but opposite sign placed at different depths (x1 and x2). The apparent fracture toughness becomes: T Z KC 0!x!x1 T Z KCK2Y c p 0 @ 1 A 0:5 sR p 2 Karcsin x1 c 0 @ 1 A 2 4 3 5 x1!x!x2 T Z KCK2Y c p 0 @ 1 A 0:5 sR arcsin x2 c 0 @ 1 AKarcsin x1 c 0 @ 1 A 2 4 3 5 x2!x!CN 8 >>>>>>>>>>>< >>>>>>>>>>>: (8) Fig. 3. Step residual stress profile (a) and corresponding apparent fracture toughness (b). The effects of intensity (left) and location (right) of the residual stress are shown. V.M. Sglavo, M. Bertoldi / Composites: Part B 37 (2006) 481–489 483
V.M. Sglavo, M. Bertoldi/Composites: Part B 37 (2006)481-489 therefore, mutually dependent also for this reason and it is not possible to design the desired mechanical behavior by using a quare-wave stress(single layer)profile, only. Fortunately, almost all these problems can be overcome by considering a multilayered structure Before moving towards a more complex profile, it is useful to analyze another simple case. Consider two stress profiles obtained by the combination of simple square-wave profiles of different (double, for simplicity) amplitude and identical extension(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 important either for the final strength and the stability interval. Such consideration is general and the final conclusion can be drawn that the compression intensity in successive layers must be Fig. 4. Square-wave stress profile and relative apparent fracture toughness continuously growing to obtain a properly designed T-curve At this point, the principle of superposition can be used to This special case is useful to discuss an important point. calculate the T-curve for a general multi-step profile. The Depending on the width(x2-x1)of the compressive layer, the proposed approach can be extended in fact to n layers provided tangent point can fall beyond the position x2. In this case, the that n step profiles with amplitude Ao, (Fig. 2), equal to the stress stable growth range is automatically defined by the interval [ri, increase of layer with respect to the previous one, are considered x2] and the maximum stress is lower than the tangent stress. A general equation, which defines the apparent fracture toughness Strength and instability point become, therefore, mutually for layer i in the interval [i-I,xi ( Fig. 2), can be obtained independent within a certain degree. certain depth from the surface is, therefore, suitable to generate T=kc- arcsin a stable growth range for surface defects. Unfortunately, this simple solution is not actually practicable because the forces x;<x<x equilibrium in the component is not satisfied. In addition, in order to achieve elevated strength, the required compressive where i indicates the layer rank and x; is the starting depth of stress is usually very high and localized intense interlaminar layer. Eq (9)represents a short notation of n different equations, shear stresses can be generated; these can be then responsible the sum being calculated for different number of terms for each i for delamination between layers. Edge cracking can also arise This represents a mathematical translation of the ' memory at the interface between highly compressed laminae. This effect of stress history that deeper layers maintain with respect to phenomenon was analyzed in a previous work [19] to occur the layer previously encountered by the propagating crack. when the layer thickness is larger than a critical value, Regardless the layer order, since 2n parameters(r, Ad )are now c=K2/0.341+v)02], oe being the compressive stress and v available and two conditions have to be satisfied(forces the Poisson's ratio Layer thickness and compressive stress are, equilibrium and equivalence between the sum of single layer v√v Fig. 5. Residual stress and corresponding T-curve for two simple square-wave profiles placed in different order
This special case is useful to discuss an important point. Depending on the width (x2Kx1) of the compressive layer, the tangent point can fall beyond the position x2. In this case, the stable growth range is automatically defined by the interval [x1, x2] and the maximum stress is lower than the tangent stress. Strength and instability point become, therefore, mutually independent within a certain degree. A single compressive layer of proper thickness placed at a certain depth from the surface is, therefore, suitable to generate a stable growth range for surface defects. Unfortunately, this simple solution is not actually practicable because the forces equilibrium in the component is not satisfied. In addition, in order to achieve elevated strength, the required compressive stress is usually very high and localized intense interlaminar shear stresses can be generated; these can be then responsible for delamination between layers. Edge cracking can also arise at the interface between highly compressed laminae. This phenomenon was analyzed in a previous work [19] to occur when the layer thickness is larger than a critical value, tcZK2 C=½0:34ð1CnÞs2 c , sc being the compressive stress and n the Poisson’s ratio. Layer thickness and compressive stress are, therefore, mutually dependent also for this reason and it is not possible to design the desired mechanical behavior by using a square-wave stress (single layer) profile, only. Fortunately, almost all these problems can be overcome by considering a multilayered structure. Before moving towards a more complex profile, it is useful to analyze another simple case. Consider two stress profiles obtained by the combination of simple square-wave profiles of different (double, for simplicity) amplitude and identical extension (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 important either for the final strength and the stability interval. Such consideration is general and the final conclusion can be drawn that the compression intensity in successive layers must be continuously growing 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. The proposed approach can be extended in fact to n layers provided that n step profiles with amplitude Dsj (Fig. 2), equal to the stress increase oflayerj with respectto the previous one, are considered. A general equation, which definesthe apparent fracturetoughness for layer i in the interval [xiK1, xi] (Fig. 2), can be obtained T Z Ki CKX i jZ1 2Y c p 0:5 Dsres;j p 2 Karcsin xjK1 c h i xiK1!x!xi (9) 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 different number of terms for each i. This represents a mathematical translation of the ‘memory’ effect of stress history that deeper layers maintain with respect to the layer previously encountered by the propagating crack. Regardless the layer order, since 2n parameters (xi, Dsi) are now available and two conditions have to be satisfied (forces equilibrium and equivalence between the sum of single layer Fig. 4. Square-wave stress profile and relative apparent fracture toughness. Fig. 5. Residual stress and corresponding T-curve for two simple square-wave profiles placed in different order. 484 V.M. Sglavo, M. Bertoldi / Composites: Part B 37 (2006) 481–489
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 can
thickness 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, compatibility 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 ðaKaiÞDT (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