C R Chen et al. Acta Materialia 55(2007)409-421 tip at interface 2 which is far below the intrinsic toughness. Appendix 1. Analytical estimate of the thermal residual When the crack tip is in layer 3, Jfr increases again with the stresses increase of a In Fig. 9 the experimental values of the fracture tough Since the layers in our composite lie parallel to the y-z less are also plotted. These values were evaluated from the plane, very low stresses in the x-direction o can be measured fracture loads of the specimens, using the above- expected therefore, we can assume ox=0 in the following mentioned procedure to determine Far. These values fit context. In the case of plane strain, the conditions &==0 well to the numerically predicted values and ox=0 yield the stresses in the z-direction G2=Vy-Ex△T 6. Conclusion where a denotes the substitute value of the CtE(see Sec o In this paper, a procedure has been presented to predict tion 4.2). The elastic strain components ef and ef follow as Gy+wa△T =-x△T experiments to measure the intrinsic fracture toughness of the single phases and to determine the material data The elastic strain energy density can then be calculated (Youngs modulus, Poissons ratio, CTE)which are neces- sary for the stress and strain analysis and the calculation of =5(arey +o26g +E(x△T)2)(A13 the residual stress state. The numerical modeling includes a conventional finite element stress analysis and the calcula- Eq.(Al. 3) is valid for any material point under thermal ion of the crack driving force based on the concept of con- and/or mechanical loading if plane strain conditions pre- figurational forces. The procedure yields the fracture vail and the stresses in the x-direction are zero. The strain toughness of the composite as a function of the crack energy density o consists of two parts, one depending on length. The spatial variation of both the elastic modulus the stress in the y-direction y and the other due to the ther and the thermal residual stresses influence the fracture mal expansion or contraction. Note that the stresses in the y-direction o, depend on both the external loading and the In the investigated bending bar, consisting of layers thermal expansion/contraction made of alumina and an alumina-zirconia composite If no external loading applies and only the eigenstrain the variations of the residual stresses produce much lar- aAT is active, apart from the end regions, the strain state ger shielding and anti-shielding effects than the variation Ey is independent of the position vector, say ey. AT=c with c of the elastic modulus. In particular, the compressive being a constant. The stress o, Ar follows as esidual stress state in the surface layer is of high rele vance for inhibiting the stable crack growth of surface yAT= Ec Er'AT (A1.4) cracks, since the fracture toughness of the composite is several times higher than the intrinsic fracture toughness. The constant c is found by utilizing the global equilibrium The numerical predictions fit well to the experimentally condition, Jo y Rdx=0. This integral can easily be evalu- measured fracture toughness data. The described proce- ated, since E, aAT, v are constant in the individual layers dure can be used in future to improve the design of mul- (1+v)(EAAA + EAZAZlAZ tilayer composites with arbitrary shape and loading △T (A15) conditions (EALA EAzIaz) The parameters (A and (Az denote the total thickness of all Acknowledgements A-and AZ-layers, respectively. Poissons ratio v is hereby considered to be the same for both materials which is a The work was partly supported by the European Com- good approximation for most structural ceramics. In our munity,s Human Potential Programme under contract case v= VA=VAZ A 0.25 HPRN-CT-2002-2003. J. Pascual acknowledges the finan The evaluation yields for the data of our laminate com- cial support provided through the European Commu- posite v AT=-144 and +166 MPa for the A-and az- nity's Human Potential Programme under contract material, respectively, which is identical to the numerical HPRN-CT-2002-00203.R. Danzer and J. Pascual express results of Fig. 2 for plane strain conditions their thanks to francis chalvet and goffredo de portu from ISTEC Faenza for providing the specimens. C Appendix 2. Analytical estimate of the crack driving force Chen, F.D. Fischer, and O. Kolednik gratefully acknowl- edge the financial support by the Osterreichische A2. 1. Expressions for Jfar and Cinh Forschungsforderungsgesellschaft mbh. the Province of Styria, the Steirische Wirtschaftsforderungsgesellschaft To evaluate the crack driving force according to Eq. (3) mbH and the Municipality of Leoben under the frame both an estimate of the far-field J-integral Jfar and the of the Austrian Kplus Programme material inhomogeneity term Cinh are needed First we con-tip at interface 2 which is far below the intrinsic toughness. When the crack tip is in layer 3, Jfr increases again with the increase of a. In Fig. 9 the experimental values of the fracture tough￾ness are also plotted. These values were evaluated from the measured fracture loads of the specimens, using the above￾mentioned procedure to determine Jfar. These values fit well to the numerically predicted values. 6. Conclusions In this paper, a procedure has been presented to predict and optimize the fracture toughness of a multilayer ceramic composite made of different phases. The procedure requires experiments to measure the intrinsic fracture toughness of the single phases and to determine the material data (Young’s modulus, Poisson’s ratio, CTE) which are neces￾sary for the stress and strain analysis and the calculation of the residual stress state. The numerical modeling includes a conventional finite element stress analysis and the calcula￾tion of the crack driving force based on the concept of con- figurational forces. The procedure yields the fracture toughness of the composite as a function of the crack length. The spatial variation of both the elastic modulus and the thermal residual stresses influence the fracture behavior. In the investigated bending bar, consisting of layers made of alumina and an alumina–zirconia composite, the variations of the residual stresses produce much lar￾ger shielding and anti-shielding effects than the variation of the elastic modulus. In particular, the compressive residual stress state in the surface layer is of high rele￾vance for inhibiting the stable crack growth of surface cracks, since the fracture toughness of the composite is several times higher than the intrinsic fracture toughness. The numerical predictions fit well to the experimentally measured fracture toughness data. The described proce￾dure can be used in future to improve the design of mul￾tilayer composites with arbitrary shape and loading conditions. Acknowledgements The work was partly supported by the European Com￾munity’s Human Potential Programme under contract HPRN-CT-2002-2003. J. Pascual acknowledges the finan￾cial support provided through the European Commu￾nity’s Human Potential Programme under contract HPRN-CT-2002-00203. R. Danzer and J. Pascual express their thanks to Francis Chalvet and Goffredo de Portu from ISTEC Faenza for providing the specimens. C.R. Chen, F.D. Fischer, and O. Kolednik gratefully acknowl￾edge the financial support by the O¨ sterreichische Forschungsfo¨rderungsgesellschaft mbH, the Province of Styria, the Steirische Wirtschaftsfo¨rderungsgesellschaft mbH and the Municipality of Leoben under the frame of the Austrian Kplus Programme. Appendix 1. Analytical estimate of the thermal residual stresses Since the layers in our composite lie parallel to the y z plane, very low stresses in the x-direction rx can be expected; therefore, we can assume rx ” 0 in the following context. In the case of plane strain, the conditions ez ” 0 and rx ” 0 yield the stresses in the z-direction: rz ¼ mry Ea DT ðA1:1Þ where a* denotes the substitute value of the CTE (see Sec￾tion 4.2). The elastic strain components eel y and eel z follow as e el y ¼ ð1 m2Þ E ry þ ma DT ; e el z ¼ a DT ðA1:2Þ The elastic strain energy density / can then be calculated as / ¼ 1 2 ðry e el y þ rze el z Þ ¼ 1 2 1 m2 E r2 y þ Eða DT Þ 2  ðA1:3Þ Eq. (A1.3) is valid for any material point under thermal and/or mechanical loading if plane strain conditions pre￾vail and the stresses in the x-direction are zero. The strain energy density / consists of two parts, one depending on the stress in the y-direction ry and the other due to the ther￾mal expansion or contraction. Note that the stresses in the y-direction ry depend on both the external loading and the thermal expansion/contraction. If no external loading applies and only the eigenstrain a*DT is active, apart from the end regions, the strain state ey is independent of the position vector, say ey,DT = c with c being a constant. The stress ry,DT follows as ry;DT ¼ Ec 1 m2 EaDT 1 m ðA1:4Þ The constant c is found by utilizing the global equilibrium condition, R h 0 ry;DT dx ¼ 0. This integral can easily be evalu￾ated, since E, a*DT, m are constant in the individual layers: c ¼ ð1 þ mÞðEAa AtA þ EAZa AZtAZÞ ðEAtA þ EAZtAZÞ DT ðA1:5Þ The parameters tA and tAZ denote the total thickness of all A- and AZ-layers, respectively. Poisson’s ratio m is hereby considered to be the same for both materials, which is a good approximation for most structural ceramics. In our case m = mA = mAZ 0.25. The evaluation yields for the data of our laminate com￾posite ry,DT = 144 and +166 MPa for the A- and AZ￾material, respectively, which is identical to the numerical results of Fig. 2 for plane strain conditions. Appendix 2. Analytical estimate of the crack driving force A2.1. Expressions for Jfar and Cinh To evaluate the crack driving force according to Eq. (3), both an estimate of the far-field J-integral Jfar and the material inhomogeneity term Cinh are needed. First we con- 418 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421