Journal of the American Ceramic Society-Sglavo et al. No.10 and can be relaxed or maintained within the material depending TableL. Materials Properties Used to Estimate the Stress ture, cooling rate, and material properties. With the exception of the edges, if thickness is much smaller than the Distribution and the Apparent Fracture Toughnes other dimensions. each lamina can be considered to be in a bi- Material E(GPa) Kc(MPam0)x(10-6°- axial stress state One fundamental task in the opportune designing of a sym- AMO 4(14)0 3.6(0.2) 7.75 metric multilayer is the estimate of the biaxial residual stresses. AM10378-3680.2340.2333.3(0.2) In the common case where stresses develop from differences in AM20 0.2380.2373.1(0.3) 7.30 thermal expansion coefficients only, the residual stress in the AM30 0.242-0.24126(0.2) 7.12 generic layer i(among n layers) can be written as: AM40 0.2460.2442.4(0.2) 88 Numbers between parenthesis correspond to the standard deviation. Elastic =E1(-m)△T (7) modulus and Poissons ratio values correspond to calculated Voigt-Reuss bounds for AMI0-AM40 composites vi(vi= Poissons ratic in order to promote the stable growth of surface defects as deep TSF-TRT (TSF =stress free temperature, TRT as≈l80um ture), and a is the average thermal expansion On the basis of the aforementioned analysis, once the Young whole laminate. defined as: ion coefficient, and fracture toughness for each layer are established, the residual stress distribution and the corresponding apparent fracture toughness curve for each =∑E1/∑Et laminate can be estimated In this study room temperature equal to 25.C and stress-free temperature equal to 1200C were es- with t; being the layer thickness. In this specific case the residual tablished as indicated in previous works. 2.23 The properties of stresses are therefore generated upon cooling after sintering. It the materials required for the calculation are summarized in Table I. Young modulus and Poissons ratio values for AMz has been shown in previous works that the TSF represents the composites shown in Table I temperature below which the material can be considered to be- bounds according to previous results, Young modulus and have as a perfectly elastic body and visco-elastic relaxation phe Poissons ratio equal to 229 GPa and 0.27 live nomena do not occu considered for pure mullite. Poisson's ratio equal to 0. 23 was Equation( 6) represents the fundamental tool for the design of a ceramic laminate with pre-defined mechanical properties. Dif- ssumed for pure alumina. The difference between the bounds s lower than 7% and 1.3% for elastic modulus and Poisson ferent ceramic layers can be stacked together in order to develop a specific residual stress profile after sintering that can be eval ratio, respectively, for mullite content below 40%0. Therefore, the average of the values reported in Table I has been used for uated by Eq. (7). Once the correlated kc curve is calculated by Eq(6), strength and fracture behavior are directly defined. By the evaluation of Eqs (7)and( 8)thus accepting an error equ hanging the stacking order and composition of the laminae it is well as the thermal expansion coefficient and fracture toughness stress. In addition, if the shape of kc is also tailored, reported below. The stress distribution was calculated by Eq. (7) defects can be forced to grow in a stable way before reaching the and the corresponding T-curve estimated according to Eq(6) ritical condition, thus obtaining single-value strength. As an example, a ceramic laminate composed of different The residual stress profile and the kc, curve for the engineered composite are shown in Fig 4. In the same graphs the applied ayers belonging to the alumina/mullite system has been de- stress intensity factor corresponding to the maximum stress igned and produced in the research work as documented in this paper. The architecture of the laminate is reported in Fig. 3 (strength) equal to N 400 MPa and the cracks depth interval Here the different alumina/ mullite composites are labeled are also shown. Since Kci was calculated step by step(Eq.(6)). alumina monolithic(AM)z where z corresponds to the volume the corresponding diagram is discontinuous at the boundary percent content of mullite. The composition and thickness of the between layers, this refecting the discontinuities in the ores d layers and the composite architecture are selected to produce a agram(Fig 4(a). One can easily suppose that the real kc, trend ceramic laminate with a" constant"strength of A 400 MPa are is continuous and that the discontinuities in Fig. 4 correspond to nathematical artifacts only. It is for this reason that the max- shown in Fig. 3. The apparent fracture toughness curve and imum sustainable stress was only calculated approximately correlated residual stress profile were correspondingly tailored equaling about 400 MP AMO, 41um AM20, 44um Ill. Experimental Procedure AM40, 43um Ceramic laminates corresponding to the material designed in the AM20, 44um previous section were produced and characterized. As previous- y pointed out, the thermal expansion coefficient as required for AM10,42um the development of the residual stress profile was tailored by considering composites in the alumina/mullite system for the production of the single lamina The ceramic powders used in the present work are reported AMO, 540 um in Table Il. a-alumina(ALCOA, Leetsdale, PA, A-16SG Dso=0.4 um) was considered as the fundamental starting ma- terial. High purity and fine mullite(KCM Corp, Nagoya, Japan, KM101, Dso=0.77 um) powder was chosen as the sec- symmetry axis Green laminae were produced by tape casting water-based Fig 3. Structure of the alumina/mullite multilayer designed and pro- vanC R. T. Vander Norwalk, CT)as dispersant and duced in the pi t work. The actual layers thickness and composition acrylic emulsions(B-1235, DURAMAX, Rohm Haas are reported(dimensions are not in scale Philadelphia, PA)as A lower-Tg acrylic emulsion(B-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 lamina can be considered to be in a bi￾axial stress state. One fundamental task in the opportune designing of a sym￾metric multilayer is the estimate of the biaxial residual stresses. In the common case where stresses develop from differences in thermal expansion coefficients only, the residual stress in the generic layer i (among n layers) can be written as: si ¼ E
i ða
aiÞDT (7) where ai is the thermal expansion coefficient, E
i ¼ Ei=ð1 niÞ(ni 5 Poisson’s ratio, Ei 5 Young modulus), DT 5 TSF–TRT (TSF 5 stress free temperature, TRT 5 room tempera￾ture), and a
is the average thermal expansion coefficient of the whole laminate, defined as: a
¼ Xn 1 E
i ti ai= Xn 1 E
i ti (8) with 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 the TSF represents the temperature below which the material can be considered to be￾have as a perfectly elastic body and visco-elastic relaxation phe￾nomena do not occur.22 Equation (6) represents the fundamental tool for the design of a ceramic laminate with pre-defined mechanical properties. Dif￾ferent ceramic layers can be stacked together in order to develop a specific residual stress profile after sintering that can be eval￾uated by Eq. (7). Once the correlated K
C curve is calculated by Eq. (6), strength and fracture behavior are directly defined. By changing the stacking order and composition of the laminae it is therefore possible to produce a material with predefined failure stress. In addition, if the shape of K
C is also tailored, surface defects can be forced to grow in a stable way before reaching the critical condition, thus obtaining single-value strength. As an example, a ceramic laminate composed of different layers belonging to the alumina/mullite system has been de￾signed and produced in the research work as documented in this paper. The architecture of the laminate is reported in Fig. 3. Here the different alumina/mullite composites are labeled as alumina monolithic (AM)z where z corresponds to the volume percent content of mullite. The composition and thickness of the layers and the composite architecture are selected to produce a ceramic laminate with a ‘‘constant’’ strength of 400 MPa are shown in Fig. 3. The apparent fracture toughness curve and correlated residual stress profile were correspondingly tailored in order to promote the stable growth of surface defects as deep as 180 mm. On the basis of the aforementioned analysis, once the Young modulus, thermal expansion coefficient, and fracture toughness for each layer are established, the residual stress distribution and the corresponding apparent fracture toughness curve for each laminate can be estimated. In this study room temperature equal to 251C and stress-free temperature equal to 12001C were es￾tablished as indicated in previous works.22,23 The properties of the materials required for the calculation are summarized in Table I. Young modulus and Poisson’s ratio values for AMz composites shown in Table I correspond to Voigt–Reuss bounds;17 according to previous results,24 Young modulus and Poisson’s ratio equal to 229 GPa and 0.27, respectively, were considered for pure mullite. Poisson’s ratio equal to 0.23 was assumed for pure alumina.17 The difference between the bounds is lower than 7% and 1.3% for elastic modulus and Poisson’s ratio, respectively, for mullite content below 40%. Therefore, the average of the values reported in Table I has been used for the evaluation of Eqs (7) and (8) thus accepting an error equal to 4% at the highest. The elastic modulus for pure alumina as well as the thermal expansion coefficient and fracture toughness for AMz composites were measured on monolithic samples as reported below. The stress distribution was calculated by Eq. (7) and the corresponding T-curve estimated according to Eq. (6). The residual stress profile and the K
C;i curve for the engineered composite are shown in Fig. 4. In the same graphs the applied stress intensity factor corresponding to the maximum stress (strength) equal to 400 MPa and the cracks depth interval are also shown. Since K
C;i was calculated step by step (Eq. (6)), the corresponding diagram is discontinuous at the boundary between layers, this reflecting the discontinuities in the sres di￾agram (Fig. 4(a)). One can easily suppose that the real K
C;i trend is continuous and that the discontinuities in Fig. 4 correspond to mathematical artifacts only. It is for this reason that the max￾imum sustainable stress was only calculated approximately, equaling about 400 MPa. III. Experimental Procedure Ceramic laminates corresponding to the material designed in the previous section were produced and characterized. As previous￾ly pointed out, the thermal expansion coefficient as required for the development of the residual stress profile was tailored by considering composites in the alumina/mullite system for the production of the single laminae. The ceramic powders used in the present work are reported in Table II. a-alumina (ALCOA, Leetsdale, PA, A-16SG, D50 5 0.4 mm) was considered as the fundamental starting ma￾terial. High purity and fine mullite (KCM Corp., Nagoya, Japan, KM101, D50 5 0.77 mm) powder was chosen as the sec￾ond phase. Green laminae were produced by tape casting water-based slurries. Suspensions were prepared by using NH4-PMA (Dar￾van Cs R. T. Vanderbilt Inc., Norwalk, CT) as dispersant and acrylic emulsions (B-1235, DURAMAXs , Rohm & Haas, Philadelphia, PA) as binder. A lower-Tg acrylic emulsion (B￾AM0, 41µm AM20, 44µm AM30, 48µm AM40, 43µm AM20, 44µm AM10, 42µm AM0, 540 µm symmetry axis Fig. 3. Structure of the alumina/mullite multilayer designed and pro￾duced in the present work. The actual layers thickness and composition are reported (dimensions are not in scale). Table I. Materials Properties Used to Estimate the Stress Distribution and the Apparent Fracture Toughness Material E (GPa) n KC (MPa  m0.5) a (106 1C1 ) AM0 394 (14) 0.23 3.6 (0.2) 7.75 AM10 378–368 0.234–0.233 3.3 (0.2) 7.63 AM20 361–344 0.238–0.237 3.1 (0.3) 7.30 AM30 345–324 0.242–0.241 2.6 (0.2) 7.12 AM40 328–306 0.246–0.244 2.4 (0.2) 6.88 Numbers between parenthesis correspond to the standard deviation. Elastic modulus and Poisson’s ratio values correspond to calculated Voigt–Reuss bounds for AM10–AM40 composites. 2828 Journal of the American Ceramic Society—Sglavo et al. Vol. 88, No. 10