D. Leguillon et al / Journal of the European Ceramic Society 26(2006)343-349 the largest surface fraction of pores S in the second case(Eq (14) Ep= H(Ed; Gp=H(V)Gd with (13) Ep=k(Ed; Gp=k(vGa with 4S K(V)=1-= (14) where Gp and Ga are the toughness respectively of the porous and the dense ceramics Fig. 7. The function g(Eq (12))(solid line)vs the Youngs moduli ratie In both cases a cubic lattice of spherical voids is assumed Ep/Ed and the He and Hutchinson approach (HH, dotted line)compare colation condition of pores(V=T/6=0.52). Such a choice is consistent with the process used to create porosity by addition (see Section 3) of spherical pyrolysable particles of constant diameters. 2-8 Higher volume fraction of pores can be obtained using parti- GAf=G cles of different sizes It must be pointed out that, throughout this paper, the term The function g(Eq (12)(solid line)and the toughness ra- "dense"is related to the stiffer material. The sintering leads tio(dashed line)are plotted versus the Youngs moduli ratio to a residual close porosity in the bulk(2.5% in SiC and 6% in (Fig. 7). As expected, this figure shows clearly that no de B4C) independent of that obtained by addition of pore form- flection can occur at such an interface, since the inequality ing agents. This residual porosity is formed of pores much (Eq (II)) never holds true. The He and Hutchinson(HH) smaller than those resulting from the addition of particles. It approach 0 is also plotted(dotted line)and leads to the same can be ignored in the present analysis but has to be taken into conclusion account in the measures V=V-Vo (15) 6. Deflection at the porous/dense interface where V is the actual porosity and Vo the initial(residual) one if no pore forming agent is added The so-called next layer is now a dense one. The function As a consequence of Eqs. (13)and(14), the deflection g(Eq (12))and the toughness ratio are again plotted versus criterion(Eq. (11)) can be rewritten the Youngs moduli ratio( Fig 8)at a porous/dense interface It is only assumed that the two ratios(Youngs moduli and h(V)≥H(v) with h(V)=g (16 toughness)follow the same rule(whatever this rule, i.e. Eqs (16)and(17)or any other ). The toughness ratio in Eq (I1) k(V)≥K(V) with k(V)=g (17) 0.8 The two following Figs. 5 and 6 exhibit experimental mea- res of the Youngs moduli and the toughness ratios for the Gp/Gd,/ two types of laminates and different additional pore form ing agents: corn starch, polyamide and PTFE. Data are taken from Blanks et al. Reynaud and co-workers 5-7and Tariolle 0.4 et al. 7, 8 In both cases the better matching is obtained using the surface fraction of pores(Eq(17)(dashed lines) Ep/Ed 5. Deflection at the dense/porous interface Fig 8. The function g(Eq (12))(solid line)vs the In this case, the so-called next layer is a porous one Ep ea and the He and Hutchinson approach o(HH, dotted line)compare and the toughness ratio in Eq.(9)or Eq(11)equals 1 5 to the toughness ratio Gs/G&(dashed line)at the porous/dense interfaceD. Leguillon et al. / Journal of the European Ceramic Society 26 (2006) 343–349 347 the largest surface fraction of pores S in the second case (Eq. (14)): Ep = H(V)Ed; Gc p = H(V)Gc d with H(V) = 1 − 6V π (13) or Ep = K(V)Ed; Gc p = K(V)Gc d with K(V) = 1 − 4S π = 1 − 6V π 2/3 (14) where Gc p and Gc d are the toughness respectively of the porous and the dense ceramics. In both cases a cubic lattice of spherical voids is assumed and the parameters of the porous ceramic vanish at the per￾colation condition of pores (V = π/6 = 0.52). Such a choice is consistent with the process used to create porosity by addition of spherical pyrolysable particles of constant diameters.2–8 Higher volume fraction of pores can be obtained using parti￾cles of different sizes.9 It must be pointed out that, throughout this paper, the term “dense” is related to the stiffer material. The sintering leads to a residual close porosity in the bulk (2.5% in SiC and 6% in B4C) independent of that obtained by addition of pore form￾ing agents. This residual porosity is formed of pores much smaller than those resulting from the addition of particles. It can be ignored in the present analysis but has to be taken into account in the measures: V = V˜ − V0 (15) where V˜ is the actual porosity and V0 the initial (residual) one if no pore forming agent is added. As a consequence of Eqs. (13) and (14), the deflection criterion (Eq. (11)) can be rewritten: h(V) ≥ H(V) with h(V) = g  1 − 6V π  (16) or k(V) ≥ K(V) with k(V) = g 1 − 6V π 2/3  (17) The two following Figs. 5 and 6 exhibit experimental mea￾sures of the Young’s moduli and the toughness ratios for the two types of laminates and different additional pore form￾ing agents: corn starch, polyamide and PTFE. Data are taken from Blanks et al.,3 Reynaud and co-workers5–7 and Tariolle et al.7,8 In both cases the better matching is obtained using the surface fraction of pores (Eq. (17)) (dashed lines). 5. Deflection at the dense/porous interface In this case, the so-called next layer is a porous one and the toughness ratio in Eq. (9) or Eq. (11) equals 115 Fig. 7. The function g (Eq. (12)) (solid line) vs. the Young’s moduli ratio Ep/Ed and the He and Hutchinson approach10 (HH, dotted line) compared to the toughness ratio Gc p/Gc d (dashed line) at the dense/porous interface. (see Section 3): Gc def = Gc pen = Gc p (18) The function g (Eq. (12)) (solid line) and the toughness ra￾tio (dashed line) are plotted versus the Young’s moduli ratio (Fig. 7). As expected, this figure shows clearly that no de- flection can occur at such an interface, since the inequality (Eq. (11)) never holds true. The He and Hutchinson (HH) approach10 is also plotted (dotted line) and leads to the same conclusion. 6. Deflection at the porous/dense interface The so-called next layer is now a dense one. The function g (Eq. (12)) and the toughness ratio are again plotted versus the Young’s moduli ratio (Fig. 8) at a porous/dense interface. It is only assumed that the two ratios (Young’s moduli and toughness) follow the same rule (whatever this rule, i.e. Eqs. (16) and (17) or any other). The toughness ratio in Eq. (11) Fig. 8. The function g (Eq. (12)) (solid line) vs. the Young’s moduli ratio Ep/Ed and the He and Hutchinson approach10 (HH, dotted line) compared to the toughness ratio Gc p/Gc d (dashed line) at the porous/dense interface