T Lube et al. Joumal of the European Ceramic Sociery 27(2007)1449-1453 In fracture mechanics, both residual and applied stresses are Table I usually included in the crack driving force. However, it is useful Properties of the material layers to consider the residual stresses as part of the crack resistance E( atech(10-6K-) Ko(MPam Thus, in laminates with compressive stress at the surface, the (0-1150°C) higher resistance to failure results from a reduction of the crack 8 driving force rather than from an increase in the intrinsic material AZ (60A40Z) 305 0.2579.24 resistance to crack extension The term Kres(a)can-as an approximation- be assessed by means of the weight function approach, that allows us to calcu- layers, A=tAz/ta is the layer thickness ratio, and e= EA/EAZ late the stress intensity factor K(a), for an edge crack of length a The indexes make reference to the materials. Since the reason for an arbitrary stress distribution acting normal to the fracture of the residual stresses is a deformation constraint( due to the path. The weight function procedure developed by Bueckner thermal mismatch), the stress profile has to be proportional to simplifies the determination of K(a) since most of the numeri- the Youngs modulus in the corresponding layer. For the assess- cal methods require separate calculations for each given stress ment of the residual stress profile the different elastic moduli distribution and each crack length. This method is of particular were considered interest when the material is submitted to a"complicated"stress In this paper, symmetrical N-layer laminates with compres- profile such as creep, residual stresses in tempered glasses, or sive stresses at the surface were studied(N being an odd number esidual stresses in multilayers. Applying this concept to our to fulfil the condition of symmetry). All the layers made of the residual stress profile O res results same material(A or AZ, respectively) have the same thick ness, so the laminate is well defined by the thicknesses tA h(r, a)]res(x)dx (3) and [AZ, or the total thickness W and A. Through the paper, W will be considered constant and equals to W=1.5 mm, accord- where h(r, a) is the suitable weight function, a the crack length, ing to a possible design condition. The corresponding effective and x is the distance from the surface R-curves are calculated according to the procedure explained Previous works by Fett et al. 0, I validate the applicabili above. The influence of the residual stress field, defined by of this methodology to inhomogeneous materials. The weight geometrical and material properties, on the apparent R-curve function presented in Eq (4)was developed using the boundary is examined in detail. The results are expressed for the lami- collocation method. 12 It models materials with an homogeneous nated system Al2O3-xAl203/(1-x)ZrO2, but the conclusions Young's modulus It will be used as a first approximation For can be extended for any ceramic multilayer system with ideally inhomogeneous materials a suitable weight function will depend strong interfaces on E(). The consequences of using this simplified weight func- tion for a laminate will be discussed later 2. Experimental procedure The weight function is This study entailed the use of a high-purity (99.7%)alumina (x, a)-=Vrdvi-cal -(a/wo)i powder(Alcoa A16-SG, Alcoa Aluminium Co, New York, NY with an average particle size of 0.3 um, and a zirconia powder (TZ3Y-S, Tosoh Corp, Tokyo, Japan)doped with 3 mol% Y203 (-)2+4m(-2)"(G with an average particle size of 0.3 um. On the basis of previous experiments, > the different powders were mixed with organic (4) binders, dispersants, plasticizers, and solvents to obtain suitable where the coefficients A, can be found in 8, 12 w is the total slips for tape casting. Slurry compositions were the same for thickness. It is worth of note that is not dependent on Youngs both Al2 O3 and AlO3 /ZrO2 composite powders. Sheets of pure modulus exclusively in the case of a homogeneous material. alumina(A)and of the composite containing 60 vol %o alumina In order to calculate the residual stresses in a laminate the and 40 vol% zirconia(Az)were produced following approximation was used. Far away from the free Table I presents the material properties for the different surface, 4 the residual stress, OR, in each layer is uniform and layers, where E and v were measured by impulse excitation biaxial. For the different layers a or aZ technique, a by means of a dilatometer between 20 and 1200C ( single-edge-V-notch beam in four point bending test). 16. 7 re and the intrinsic toughness Ko following the VAMAS procedure 1+(N+1)/(N-1)(e/入) tA (N +l) 3. Results and discussion As shown by previous authors, the apparent R-curve in where E=E/(I-v), v being the Poisson's ratio, Aa= multilayers presents an oscillating behaviour.4(see Fig. 1) (aAz -aA)the difference of the thermal expansion coefficients The toughness increases in of AZ and A, respectively, Tsf the temperature below which the increasing crack length and reaches a local maximum at the inter residual stresses arise, To the room temperature, N the number of face. It decreases in the tensile layers reaching a local minimum1450 T. Lube et al. / Journal of the European Ceramic Society 27 (2007) 1449–1453 In fracture mechanics, both residual and applied stresses are usually included in the crack driving force. However, it is useful to consider the residual stresses as part of the crack resistance. Thus, in laminates with compressive stress at the surface, the higher resistance to failure results from a reduction of the crack driving force rather than from an increase in the intrinsic material resistance to crack extension. The term Kres(a) can – as an approximation – be assessed by means of the weight function approach,6 that allows us to calcu￾late the stress intensity factor K(a), for an edge crack of length a for an arbitrary stress distribution acting normal to the fracture path. The weight function procedure developed by Bueckner7 simplifies the determination of K(a) since most of the numeri￾cal methods require separate calculations for each given stress distribution and each crack length. This method is of particular interest when the material is submitted to a “complicated” stress profile such as creep,8 residual stresses in tempered glasses,9 or residual stresses in multilayers.2 Applying this concept to our residual stress profile σres results: Kres(a) = a 0 h(x, a)σres(x) dx, (3) where h(x, a) is the suitable weight function, a the crack length, and x is the distance from the surface. Previous works by Fett et al.10,11 validate the applicability of this methodology to inhomogeneous materials. The weight function presented in Eq. (4) was developed using the boundary collocation method.12 It models materials with an homogeneous Young’s modulus. It will be used as a first approximation. For inhomogeneous materials a suitable weight function will depend on E(x). The consequences of using this simplified weight func￾tion for a laminate will be discussed later. The weight function is h(x, a) = 2 πa 1 √1 − (x/a)(1 − (a/W))1.5 ×  1 − a W 1.5 +Aνμ 1 − x a ν+1 a W μ , (4) where the coefficients Aνμ can be found in.8,12 W is the total thickness. It is worth of note that is not dependent on Young’s modulus exclusively in the case of a homogeneous material. In order to calculate the residual stresses in a laminate the following approximation was used.13 Far away from the free surface,14 the residual stress, σR, in each layer is uniform and biaxial. For the different layers A or AZ: σres,A = −E A Tsf T0 α dT 1 + ((N + 1)/(N − 1)(e/λ)) and σres,AZ = −σR,A tA tAZ (N + 1) (N − 1), (5) where E = E/(1 − ν), ν being the Poisson’s ratio, α = (AZ − αA) the difference of the thermal expansion coefficients of AZ and A, respectively, Tsf the temperature below which the residual stresses arise, T0 the room temperature, N the number of Table 1 Properties of the material layers E (GPa) ν αtech (10−6 K−1) (0–1150 ◦C) K0 (MPa m1/2) A 391 0.241 8.64 3.8 AZ (60A40Z) 305 0.257 9.24 4.28 layers, λ = tAZ/tA is the layer thickness ratio, and e = E A/E AZ. The indexes make reference to the materials. Since the reason of the residual stresses is a deformation constraint (due to the thermal mismatch), the stress profile has to be proportional to the Young’s modulus in the corresponding layer. For the assess￾ment of the residual stress profile the different elastic moduli were considered. In this paper, symmetrical N-layer laminates with compres￾sive stresses at the surface were studied (N being an odd number to fulfil the condition of symmetry). All the layers made of the same material (A or AZ, respectively) have the same thick￾ness, so the laminate is well defined by the thicknesses tA and tAZ, or the total thickness W and λ. Through the paper, W will be considered constant and equals to W = 1.5 mm, accord￾ing to a possible design condition. The corresponding effective R-curves are calculated according to the procedure explained above. The influence of the residual stress field, defined by geometrical and material properties, on the apparent R-curve is examined in detail. The results are expressed for the lami￾nated system Al2O3 − xAl2O3/(1 − x)ZrO2, but the conclusions can be extended for any ceramic multilayer system with ideally strong interfaces. 2. Experimental procedure This study entailed the use of a high-purity (99.7%) alumina powder (Alcoa A16-SG, Alcoa Aluminium Co., New York, NY) with an average particle size of 0.3 m, and a zirconia powder (TZ3Y-S, Tosoh Corp., Tokyo, Japan) doped with 3 mol% Y2O3 with an average particle size of 0.3m. On the basis of previous experiments,15 the different powders were mixed with organic binders, dispersants, plasticizers, and solvents to obtain suitable slips for tape casting. Slurry compositions were the same for both Al2O3 and Al2O3/ZrO2 composite powders. Sheets of pure alumina (A) and of the composite containing 60 vol% alumina and 40 vol% zirconia (AZ) were produced. Table 1 presents the material properties for the different layers, where E and ν were measured by impulse excitation technique, α by means of a dilatometer between 20 and 1200 ◦C and the intrinsic toughness K0 following the VAMAS procedure (single-edge-V-notch beam in four point bending test).16,17 3. Results and discussion As shown by previous authors, the apparent R-curve in multilayers presents an oscillating behaviour3,4 (see Fig. 1). The toughness increases in the layers under compression with increasing crack length and reaches a local maximum at the inter￾face. It decreases in the tensile layers reaching a local minimum