October 2005 Tailored Residual Stresses in Ceramic laminates 2827 model depicted in Fig. I can be considered. The residual stress, Ores(x) which is a function of the distance from the surface, x ores (n only, is correlated with a stress intensity factor defined as 1)/v(x/c)ore(x)dx (1) crack length and y a function The generic system in Fig. I can be al ected to external loads, correlated with the stress intensity factor, Kext. Crack ture toughness, Kc, of the material around the crack tip. ll s propagation occurs when the sum(Kres+ Kext equals the esidual stresses are hypothesized as a material characteristic Kc( he apparent fracture toughness can be defined by combining Ko th the stress intensity factor correlated with K Thus, crack propagates when Kext =kc. It is clear that com- pressive (i.e, negative) residual stresses have a beneficial effect on crack propagation resistance. If the simple situation where the residual stress possesse ve ed in Fig. 2 is considere calculated For an edge crack in a semi-infinite body subjected to residual stresses, Eq. (I)can be rewritten as: X1 vccr Fig. 2. Plot of the step residual stress profile(a) and of the correspond- ng apparent fracture toughness(b). The dashed tangent straight line is drawn to calculate the failure stress for crack lengths lower than ccr. with Y a 1. 12. It is correct to point out here that this is not slight dependence on x/c. Nevertheless, such approximation sition; then, by using the principle of s nae of different compo- rigorously true for non-uniform loading, since Y maintains a a ceramic laminate constituted by lar corre- simplifies the calculations while avoiding the loss of generality sponding apparent fracture toughness can be evaluated The apparent fracture toughness correlated with the residual Referring to the generic step profile shown in Fig. 1, if each stress profile shown in Fig. 2(a) becomes step has an amplitude o,=of-1+Ao the apparent fracture toughness in layer i(xi_I <x< x) can be defined Kc=k 0<x<x Kc=kc+2) ()(-amcm() x1<x<+∞ kci=ko Assuming the presence of surface cracks smaller than a critical length, cer, the material strength, of, can be calculated by means of the graphical construction reported in Fig. 2(b)from the tan- In the calculations carried out to obtain Eq(the approxima- gent line correlated with the applied stress intensity factor de- tion is made that the elastic modulus of the different layers constant. Nevertheless. it has been demonstrated elsewhere that the approximation in Kci estimate does not exceed 10% if Kex=Yor√re the Young modulus variation is less than 33%.4One should note that the hypotheses made purport that perfect adhesion The addition or subtraction of step profiles such as in exists between layers and no delamination occurs; such hypoth ig 2(a)allows the reproduction of the stress generated within eses have been experimentally verified as reported later in the resent paper 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. i.e. the composite architecture. According to the theory of composite plies, in order to maintain flatness during in-plane loading, as in the case of biaxial residual stresses developed during produc- tion, laminate structure must conform to a number of symmetry conditions. If each layer is isotropic, like ceramic laminae with 4乙 layer i fine and randomly oriented crystalline microstructure, and the tacking order is symmetrical, the laminate remains fat upon ering and being orthotropic, its response to loading is sim- to that of a homogeneous plate Regardless of the physical source of residual stresses, their presence in a co-sintered multilayer is related to constraining effect. When the different layers perfectly adhere to each other every lamina must deform similarly and at the same rate as the others The difference between free deformation or free defor- Fig1. Schematic representation of the model used for the calculation mation rate of the single lamina with respect to the average val- of the apparent fracture toughness in a multilayered body subjected to ue of the whole laminate accounts for the creation of residual residual stresses, Ore(x), with a surface crack (length=c) tresses Such stresses can be either viscous or elastic in naturemodel depicted in Fig. 1 can be considered. The residual stress, sres(x), which is a function of the distance from the surface, x, only, is correlated with a stress intensity factor defined as: Kres ¼ 2 c p
0:5 Z c 0 cðx=cÞsresðxÞ dx (1) being c the crack length and c a function of x/c. 17 The generic system in Fig. 1 can be also subjected to external loads, correlated with the stress intensity factor, Kext. Crack propagation occurs when the sum (Kres1Kext) equals the frac￾ture toughness, KC, of the material around the crack tip. If the residual stresses are hypothesized as a material characteristic, the apparent fracture toughness can be defined by combining KC with the stress intensity factor correlated with sres: K
C ¼ KC Kres (2) Thus, crack propagates when Kext ¼ K
C. It is clear that com￾pressive (i.e., negative) residual stresses have a beneficial effect on crack propagation resistance. If the simple situation where the residual stress possesses a simple step-profile as reported in Fig. 2 is considered, K
C can be calculated. For an edge crack in a semi-infinite body subjected to residual stresses, Eq. (1) can be rewritten as: Kres ¼ Y ðpcÞ 0:5 Z c 0 sresðxÞ 2c ðc2 x2Þ 0:5 dx (3) with Y 1.12. It is correct to point out here that this is not rigorously true for non-uniform loading, since Y maintains a slight dependence on x/c. Nevertheless, such approximation simplifies the calculations while avoiding the loss of generality. The apparent fracture toughness correlated with the residual stress profile shown in Fig. 2(a) becomes: K
C ¼ KC 0 < x < x1 K
C ¼ KC þ 2Y c p
0:5 sR p 2 arcsin x1 c

x1 < x < þ1 ( (4) Assuming the presence of surface cracks smaller than a critical length, ccr, the material strength, sf, can be calculated by means of the graphical construction reported in Fig. 2(b) from the tan￾gent line correlated with the applied stress intensity factor de- fined as: Kext ¼ Ysf ffiffiffiffiffi pc p (5) The addition or subtraction of step profiles such as in Fig. 2(a) allows the reproduction of the stress generated within a ceramic laminate constituted by laminae of different compo￾sition; then, by using the principle of superposition, the corre￾sponding apparent fracture toughness can be evaluated. Referring to the generic step profile shown in Fig. 1, if each step has an amplitude sj 5 sj11Dsj, the apparent fracture toughness in layer i (xi1 oxoxi) can be defined as: K
C;i ¼KC;i Xi j¼1 2Y c p
0:5 Dsres;j p 2 arcsin xj1 c h i
  (6) In the calculations carried out to obtain Eq. (3) the approxima￾tion is made that the elastic modulus of the different layers is constant.18 Nevertheless, it has been demonstrated elsewhere that the approximation in K
Ci; estimate does not exceed 10% if the Young modulus variation is less than 33%.19,20 One should note that the hypotheses made purport that perfect adhesion exists between layers and no delamination occurs; such hypoth￾eses have been experimentally verified as reported later in the present paper. 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, i.e., the composite architecture. According to the theory of composite plies,21 in order to maintain flatness during in-plane loading, as in the case of biaxial residual stresses developed during produc￾tion, laminate structure must conform to a number of symmetry conditions. If each layer is isotropic, like 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 sim￾ilar to that of a homogeneous plate.21 Regardless of the physical source of residual stresses, their presence in a co-sintered multilayer is related to constraining effect. When the different layers perfectly adhere to each other, every lamina must deform similarly and at the same rate as the others. The difference between free deformation or free defor￾mation rate of the single lamina with respect to the average val￾ue of the whole laminate accounts for the creation of residual stresses. Such stresses can be either viscous or elastic in nature c xi −1 layer i σres(x) ∆σres,i xi σres,i Fig. 1. Schematic representation of the model used for the calculation of the apparent fracture toughness in a multilayered body subjected to residual stresses, sres(x), with a surface crack (length 5 c). x x1 σ res x c 1 KC * ( ) x ( ) x Yσ π f c ccr KC − σ KC * (a) (b) Fig. 2. Plot of the step residual stress profile (a) and of the correspond￾ing apparent fracture toughness (b). The dashed tangent straight line is drawn to calculate the failure stress for crack lengths lower than ccr. October 2005 Tailored Residual Stresses in Ceramic Laminates 2827