D. Leguillon et al. /Journal of the European Ceramic Sociery 26(2006)343-349 0.3 、H 0.6 025 Fig9. The function h(Eq (16)(solid line)vs the volume fraction of pores Fig. Il. The function g(Eq (12))(solid line)vs the Young s moduli ratio and the He and Hutchinson approach(HH, dotted line)compared to the Ep/Ea for different thicknesses ratio ep/ea(0. 25, 0.5, 1, 2)compared to the toughness ratio H(V)=GS/Gs(Eq (13))(dashed line) toughness ratio Gp/Ga(dashed line)at the porous/dense interface the present analysis). Between the two values, the primary crack changes direction but kinks out immediately. In this (19) latter case, the work of fracture is not strongly increased, the The answers brought by the present analysis and the He and Hutchinson one 0 differ now significantly, the agreement 7. The influence of the porous layers thickness with the experiments will be discussed below The criterion can be also plotted versus the volume fraction Throughout this paper, the porous material is considered as of pore V. It is illustrated in the two following figures derived homogenous, this implies that the pores size is much smaller from the single Fig8. In the first one(Fig. 9), the elastic and than the layers thickness. This property must not be forgotten fracture parameters depend linearly on the volume fraction of especially in this section. The porous layers thickness can pores(Eq (16),whereas in the second (Fig. 10)they depend be diminished provided it does not interfere with the pores linearly on the surface fraction of pores(Eq. (17). Clearly diameter. Typically, at least one decade must separate these the predicted porosity that causes crack deflection(arrows in two characteristic lengths, the ratio between the pore diameter Figs. 9 and 10)is above 40% in both cases. If one does not and the layer thickness must not exceed 0.1 forget the residual porosity in the sintered parts(Eq. (15)) Numerical results show that thin porous layers tend to pro- these results are in a good agreement with the experiments mote crack deflection In Fig. 11(azoom of Fig 8 in the range of Reynaud and co-workers",and Tariolle",, while the He 0-0.3 for Ep/Ed), the function g (Eq (12)is plotted versus and Hutchinson approach 0 underestimates it. Blanks et al. the Young s moduli ratio Ep/Ed for different values of the found a wide range of values: between 34% and 44%. below thicknesses ratio pled, where ep and ed are respectively the 34%no deflection was observed. above 44% an extensive porous and dense layer thicknesses. Results are summarized deflection was obtained (this last value is in agreement with in Table 1. The model of porosity relies on function H defined in Eq (12). The porosity required to promote deflection de- creases with the porous layers thickness. Nevertheless, this effect remains small, the reduction is only about 10% when he relative porous layers thickness is divided by 8. 8. The influence of the poisson 's ratios The Poissons ratios of the components have been omitted in the above discussion. The next figure shows that they play 041 Table 1 Youngs moduli ratio Ep/Ed and pore volume fraction V promoting crack Fig 10. The function k(Eq (17)(solid line)vs the surface fraction of pores and the He and Hutchinson approach o(HH, dotted line)compared to the toughness ratio(Eq (14))K(V)= GS/GA(dashed line). 0.445 0.416348 D. Leguillon et al. / Journal of the European Ceramic Society 26 (2006) 343–349 Fig. 9. The function h (Eq. (16)) (solid line) vs. the volume fraction of pores and the He and Hutchinson approach10 (HH, dotted line) compared to the toughness ratio H(V) = Gc p/Gc d (Eq. (13)) (dashed line). is: Gc def Gc pen = Gc p Gc d (19) The answers brought by the present analysis and the He and Hutchinson one10 differ now significantly, the agreement with the experiments will be discussed below. The criterion can be also plotted versus the volume fraction of pore V. It is illustrated in the two following figures derived from the single Fig. 8. In the first one (Fig. 9), the elastic and fracture parameters depend linearly on the volume fraction of pores (Eq. (16)), whereas in the second (Fig. 10) they depend linearly on the surface fraction of pores (Eq. (17)). Clearly the predicted porosity that causes crack deflection (arrows in Figs. 9 and 10) is above 40% in both cases. If one does not forget the residual porosity in the sintered parts (Eq. (15)), these results are in a good agreement with the experiments of Reynaud and co-workers7,8 and Tariolle8,9 while the He and Hutchinson approach10 underestimates it. Blanks et al.3 found a wide range of values: between 34% and 44%. Below 34% no deflection was observed, above 44% an extensive deflection was obtained (this last value is in agreement with Fig. 10. The function k (Eq. (17)) (solid line) vs. the surface fraction of pores and the He and Hutchinson approach10 (HH, dotted line) compared to the toughness ratio (Eq. (14)) K(V) = Gc p/Gc d (dashed line). Fig. 11. The function g (Eq. (12)) (solid line) vs. the Young’s moduli ratio Ep/Ed for different thicknesses ratio ep/ed (0.25, 0.5, 1, 2) compared to the toughness ratio Gc p/Gc d (dashed line) at the porous/dense interface. the present analysis). Between the two values, the primary crack changes direction but kinks out immediately. In this latter case, the work of fracture is not strongly increased, the design goal is not attained. 7. The influence of the porous layers thickness Throughout this paper, the porous material is considered as homogenous, this implies that the pores size is much smaller than the layers thickness. This property must not be forgotten, especially in this section. The porous layers thickness can be diminished provided it does not interfere with the pores diameter. Typically, at least one decade must separate these two characteristic lengths, the ratio between the pore diameter and the layer thickness must not exceed 0.1. Numerical results show that thin porous layers tend to pro￾mote crack deflection. In Fig. 11 (a zoom of Fig. 8 in the range 0–0.3 for Ep/Ed), the function g (Eq. (12)) is plotted versus the Young’s moduli ratio Ep/Ed for different values of the thicknesses ratio ep/ed, where ep and ed are respectively the porous and dense layer thicknesses. Results are summarized in Table 1. The model of porosity relies on function H defined in Eq. (12). The porosity required to promote deflection de￾creases with the porous layers thickness. Nevertheless, this effect remains small, the reduction is only about 10% when the relative porous layers thickness is divided by 8. 8. The influence of the Poisson’s ratios The Poisson’s ratios of the components have been omitted in the above discussion. The next figure shows that they play Table 1 Young’s moduli ratio Ep/Ed and pore volume fraction V promoting crack deflection for various porous/dense layers thickness ratios ep/ed ep/ed 2 1 0.5 0.25 Ep/Ed 0.120 0.150 0.175 0.205 V 0.461 0.445 0.432 0.416