V M. Sglavo, M. Bertoldi Acta Materialia 54(2006 )4929-4937 following conditions (related to forces equilibrium, com- 4z30, 35 Hm patibility and constitutive law) must be satisfied 6.88 Grsst=0 7.75 1=e1+x△T=E, (10) a; being the thermal expansion coefficient, E=E/(l-vi) (Vi is Poissons ratio, Ei is Youngs modulus), e i the elastic strain and &i the deformation. The system defined by AZ40,520m Eq(10) represents a set of 3n t l equations and 3n+ 1 ppm/°C ns(o, ei, Ei, e). The solution of this linea ar system allows the residual stress in the ith layer to be calculated a ds=E(-x)△T where AT= TSF- TRT(TSE is stress-free temperature, TRT is room temperature)and a is the average thermal expan- Fig. 6. Architecture of the AMZ laminate. Layer thickness, composition sion coefficient of the whole laminate. defined as and thermal expansion coefficient( Fig. 7)are reported(dimensions are not ∑E Erti (12) order to promote the stable growth of surface defects as where f; is the layer thickness. In this instance the residual stresses are therefore generated upon cooling after sinter On the basis of the aforementioned analysis, once the Youngs modulus, Poissons ratio, thermal expansion coef- ng. It has been shown in previous studies that SE repre- ficient and fracture toughness for each layer are determined sents the temperature below which the material can b considered to behave elastically without viscoelastic relax the residual stress distribution and the corresponding appar- ent fracture toughness curve for each laminate can be esti mated. In this study, room temperature is 25C and the q. (9)represents the fundamental tool for the stress-free condition is assumed at 1200 C as discussed in designing a ceramic laminate with predefined mechanical earlier works [26, 27]. The properties of the materials properties. Different ceramic layers can be stacked together in order to develop a specific residual strese required for the calculation are summarised in Table I and in Fig. 7. Youngs modulus and Poissons ratio values for profile from sintering, by using Eq (11). Since the stress AM and AZ composites shown in Table I correspond to level in Eq (11)does not depend on stacking order, the Voigt-Reuss bounds [21]; according to previous results sequence of laminae can still be changed provided the [281. Young's modulus and Poisson's ratio equal to symmetry condition is maintained. Once the stress pro- 229 GPa and 0.27, respectively, were considered for pure file is defined, the apparent fracture toughness can be mullite. The elastic modulus for pure alumina and zirconia estimated from Eq.(9). By changing the stacking order and composition of the layers, i.e. the laminate architec- was measured on monolithic samples as reported elsewhere ture, it is therefore possible to produce a material with a unique and predefined failure stress Table 1 ceramic laminates composed of layers belonging to the Materials properties used to estimate stress distribution and apparent alumina/zirconia and alumina/mullite systems were designed in the work reported here. The thermal expansion Material E(GPa) stress profile was tailored by changing the composition of At0 394(14) coefficient required for the development of the residual AN 36(0.2) 0.23 0.2340.233 the single laminae. The architecture of the engineered lam- AM20 3.1(0.3) 0.2380.237 0.242-0.241 inate is reported in Fig. 6. The notation"AZw/y"or AM40 0.246-0.244 AMw/y stands for alumina(A), zirconia(Z) or mullite AZl0 3.5(0.3) (M), while w corresponds to the volume percent content A220 0.242-0.240 AZ30 of zirconia or mullite and y to the layer thickness in micro- AZ40 337-308 3.9(0.3) 0.2480.245 metres. The composition and thickness of the layers and 318-287 4.5(0.3) 0.2540.251 their stacking order were selected to produce ceramic lam Numbers in parentheses correspond to the standard deviation. Elastic inates with a"constant"strength, aF, approximately equal modulus and Poisson,'s ratio values correspond to calculated Voigt-Reuss to 700 MPa. The apparent fracture toughness curve and bounds for AM10-AM40 composite corresponding residual stress profile were also tailored in Ref.[27]following conditions (related to forces equilibrium, com￾patibility and constitutive law) must be satisfied: Pn i¼1 rres;iti ¼ 0; ei ¼ ei þ aiDT ¼ e; ri ¼ E i ei; 8 >>>< >>>: ð10Þ ai being the thermal expansion coefficient, E i ¼ Ei=ð1  miÞ (mi is Poisson’s ratio, Ei is Young’s modulus), ei the elastic strain and ei the deformation. The system defined by Eq. (10) represents a set of 3n + 1 equations and 3n + 1 unknowns (ri, ei,ei,e). The solution of this linear system allows the residual stress in the ith layer to be calculated as rres;i ¼ E i ða  aiÞDT ; ð11Þ where DT = TSF  TRT (TSF is stress-free temperature, TRT is room temperature) and a is the average thermal expan￾sion coefficient of the whole laminate, defined as a ¼ Xn 1 E i tiai ,Xn 1 E i ti; ð12Þ where ti is the layer thickness. In this instance the residual stresses are therefore generated upon cooling after sinter￾ing. It has been shown in previous studies that TSF repre￾sents the temperature below which the material can be considered to behave elastically without viscoelastic relax￾ation [26]. Eq. (9) represents the fundamental tool for the designing a ceramic laminate with predefined mechanical properties. Different ceramic layers can be stacked together in order to develop a specific residual stress profile from sintering, by using Eq. (11). Since the stress level in Eq. (11) does not depend on stacking order, the sequence of laminae can still be changed provided the symmetry condition is maintained. Once the stress pro- file is defined, the apparent fracture toughness can be estimated from Eq. (9). By changing the stacking order and composition of the layers, i.e. the laminate architec￾ture, it is therefore possible to produce a material with a unique and predefined failure stress. Ceramic laminates composed of layers belonging to the alumina/zirconia and alumina/mullite systems were designed in the work reported here. The thermal expansion coefficient required for the development of the residual stress profile was tailored by changing the composition of the single laminae. The architecture of the engineered lam￾inate is reported in Fig. 6. The notation ‘‘AZw/y’’ or ‘‘AMw/y stands for alumina (A), zirconia (Z) or mullite (M), while w corresponds to the volume percent content of zirconia or mullite and y to the layer thickness in micro￾metres. The composition and thickness of the layers and their stacking order were selected to produce ceramic lam￾inates with a ‘‘constant’’ strength, rF, approximately equal to 700 MPa. The apparent fracture toughness curve and corresponding residual stress profile were also tailored in order to promote the stable growth of surface defects as deep as 150 lm. On the basis of the aforementioned analysis, once the Young’s modulus, Poisson’s ratio, thermal expansion coef- ficient and fracture toughness for each layer are determined, the residual stress distribution and the corresponding appar￾ent fracture toughness curve for each laminate can be esti￾mated. In this study, room temperature is 25 C and the stress-free condition is assumed at 1200 C, as discussed in earlier works [26,27]. The properties of the materials required for the calculation are summarised in Table 1 and in Fig. 7. Young’s modulus and Poisson’s ratio values for AM and AZ composites shown in Table 1 correspond to Voigt–Reuss bounds [21]; according to previous results [28], Young’s modulus and Poisson’s ratio equal to 229 GPa and 0.27, respectively, were considered for pure mullite. The elastic modulus for pure alumina and zirconia was measured on monolithic samples as reported elsewhere AZ30, 35 µm AZ0, 40 µm AM40, 90 µm AZ0, 40 µm AZ40, 520 µm symmetry axis 8.37 7.75 6.88 7.75 8.68 (ppm / ˚C) α Fig. 6. Architecture of the AMZ laminate. Layer thickness, composition and thermal expansion coefficient (Fig. 7) are reported (dimensions are not to scale). Table 1 Materials properties used to estimate stress distribution and apparent fracture toughness Material E (GPa) KC (MPa m1/2) m AM0/AZ0 394 (14) 3.6 (0.2) 0.23a AM10 378–368 3.3 (0.2) 0.234–0.233 AM20 361–344 3.1 (0.3) 0.238–0.237 AM30 345–324 2.6 (0.2) 0.242–0.241 AM40 328–306 2.4 (0.2) 0.246–0.244 AZ10 375–360 3.5 (0.3) 0.236–0.235 AZ20 356–332 3.6 (0.2) 0.242–0.240 AZ30 337–308 3.9 (0.3) 0.248–0.245 AZ40 318–287 4.5 (0.3) 0.254–0.251 AZ100 204 (8) – 0.29a Numbers in parentheses correspond to the standard deviation. Elastic modulus and Poisson’s ratio values correspond to calculated Voigt–Reuss bounds for AM10–AM40 composites. a Ref. [27]. V.M. Sglavo, M. Bertoldi / Acta Materialia 54 (2006) 4929–4937 4933