D. Leguillon et al. /Journal of the European Ceramic Sociery 26(2006)343-349 (index pen). Deflect promoted if the above inequality holds true for deflection but is wrong for penetration, it lead Ep/Ed fpen(Ep/Ed 0.6 where Gpen and Gder are the toughness of the next layer(pen- 04 etration mechanism)and of the interface(deflection mecha nism). It will be assumed in the following that the toughness of the interface is that of the porous material. Indeed, if the interface was stronger then the crack would grow within the porous medium at a short distance from the interface. Such a choice is also suggested in Fujita et al. I5 03 Clearly the dimensionless crack increment lengths ud Fig. 5. Youngs moduli ratio vs pore volume fraction V: SiC with Polyamide and Apen play a role in the above relation. Now we make the particles -7(diamonds), SiC with com starch particles, 7, (squares),BaC following reasonable additional assumption: if the crack pen with corn starch particles(triangles). Shear modulus ratio vs pore volume Apen=1. The deflected extension length remains to be deter mined. It could be done using a maximum stress criterion. 3 For simplicity, we assume he toughness parameters. They can be expressed in terms of the porosity as proposed in the next section hH11mt么u udef=Ppen=I (10) A simplification of the geometry of the since it refers to a characteristic length of the microstructure part of the material being replaced by a homogenized(aver (the layer thickness) that does not exist in their approach aged )one. Fig. 4 compares the function g defined in Eq(12) (Fig 2a and b). Nevertheless, complete computations relying when 3 and 4 layers are kept around the crack tip. Obviously, on a stress criterion have been carried out and it has been the simplified geometry retained in this model seems to be but that the final results are not deeply modified l orosity observed that the deflection length increases with the sufficient to our purpose Finally the criterion takes the simplified form: Ep 4. Elastic and fracture parameters of the porous with The two Figs. 5 and 6 show that a unique function depend E Adet(ep/Ed, D)-A(Ep/E ng on the volume fraction of pores V can be used to express (12) the elastic and fracture parameters of the porous material. In Ed/ Apen( Ep/Ed, 1)-A(Ep/ the first case(Eq (13)the parameters depend linearly on the It is clear from this expression that the crucial point is the volume fraction of pores V, whereas they depend linearly on knowledge of two relevant data: the ratios of the elastic and 02 g 4 layers 0.2 PreD Fig. 6. Toughness ratio vs. pore volume fraction V: Fig 4. The function g(Eq(12)(solid lines)vs the Young's moduli ratio particles -7(diamonds ) SiC with com starch particle quare), B4C Ep/Ed for two models of geometry(3 and 4 layers)at the porous/de with corn starch particles7, (circles), SiC with PTFE (triangles). H(V (Eq (13))dashed line, K(V(Eq (14) solid line.346 D. Leguillon et al. / Journal of the European Ceramic Society 26 (2006) 343–349 (index pen). Deflection is promoted if the above inequality holds true for deflection but is wrong for penetration, it leads to: fdef(Ep/Ed, µdef) fpen(Ep/Ed, µpen) ≥ Gc def Gc pen (9) where Gc pen and Gc def are the toughness of the next layer (pen￾etration mechanism) and of the interface (deflection mecha￾nism). It will be assumed in the following that the toughness of the interface is that of the porous material. Indeed, if the interface was stronger then the crack would grow within the porous medium at a short distance from the interface. Such a choice is also suggested in Fujita et al.15 Clearly the dimensionless crack increment lengths µdef and µpen play a role in the above relation. Now we make the following reasonable additional assumption: if the crack pen￾etrates the next layer then it breaks it completely: pen = e ⇒ µpen = 1. The deflected extension length remains to be deter￾mined. It could be done using a maximum stress criterion.13 For simplicity, we assume here: µdef = µpen = 1 (10) This choice is less arbitrary than that of He and Hutchinson11 since it refers to a characteristic length of the microstructure (the layer thickness) that does not exist in their approach (Fig. 2a and b). Nevertheless, complete computations relying on a stress criterion have been carried out and it has been observed that the deflection length increases with the porosity but that the final results are not deeply modified.12,16 Finally the criterion takes the simplified form: g Ep Ed  ≥ Gc def Gc pen (11) with g Ep Ed  = Adef(Ep/Ed, 1) − A(Ep/Ed, 0) Apen(Ep/Ed, 1) − A(Ep/Ed, 0) (12) It is clear from this expression that the crucial point is the knowledge of two relevant data: the ratios of the elastic and Fig. 4. The function g (Eq. (12)) (solid lines) vs. the Young’s moduli ratio Ep/Ed for two models of geometry (3 and 4 layers) at the porous/dense interface. Fig. 5. Young’s moduli ratio vs. pore volume fraction V: SiC with Polyamide particles5–7 (diamonds), SiC with corn starch particles3,7,8 (squares), B4C with corn starch particles7,8 (triangles). Shear modulus ratio vs. pore volume fraction V: B4C with corn starch particles8 (circles). The dashed line is the function H(V) (Eq. (13)), the solid line is the function K(V) (Eq. (14)). toughness parameters. They can be expressed in terms of the porosity as proposed in the next section. A simplification of the geometry of the inner domain (Figs. 2a and 3) has been used. Only three or four layers have been considered around the crack tip, the remaining part of the material being replaced by a homogenized (aver￾aged) one. Fig. 4 compares the function g defined in Eq. (12) when 3 and 4 layers are kept around the crack tip. Obviously, the simplified geometry retained in this model seems to be sufficient to our purpose. 4. Elastic and fracture parameters of the porous ceramic The two Figs. 5 and 6 show that a unique function depend￾ing on the volume fraction of pores V can be used to express the elastic and fracture parameters of the porous material. In the first case (Eq. (13)) the parameters depend linearly on the volume fraction of pores V, whereas they depend linearly on Fig. 6. Toughness ratio vs. pore volume fraction V: SiC with polyamide particles5–7 (diamonds), SiC with corn starch particles3,7,8 (squares), B4C with corn starch particles7,8 (circles), SiC with PTFE particles3 (triangles). H(V) (Eq. (13)) dashed line, K(V) (Eq. (14)) solid line.