guillon et al. Journal of the European Ceramic Sociery 26(2006)343-349 Fig. 1. Macrostructure of boron carbide composite with interlayers obtained by the use of corn starch(left), microstructure of porous boron carbide obtained with 20 vol. of corn starch(right) ticles are burned out during the debinding step, prior to sin- The present model relies on a two-scale analysis where ering Dense and porous layers have the same thickness af- the laminated structure is taken into account as explained in ter sintering(e a 100 um). Laminated specimens made of 20 Section 2. The criterion proposed in Section 3 derives from an layers have been tested under 3-point flexure loading. De- energy balance and can take into account a complementary tails on the fabrication of specimens and the measurement stress criterion. This makes it possible to avoid the above- of elastic and failure parameters can be found in Reynaud mentioned drawback, since virtual crack extension lengths and co-workers,6 and in Tariolle et al. 7,8 Fig. 1 shows mi- are known. The criterion is expressed in terms of two relevant shapes within the porous interlayer, and depicts the pore parameters: the Young s moduli and toughness ratios of the crographs of the layered B4C material dense and porous layers In Section 4, experiments show that The analyses of crack deflections by interfaces are gen- these ratios can be written quite simply as functions of the erally based on two models due to He and Hutchinson porosity. Sections 5 and 6 are devoted to the study of the crack Both are carried out in an unbounded domain made of two deflection at the dense/porous and porous/dense interfaces. A elastic materials. In the first one, the primary crack lies in short Section 7 is dedicated to the analysis of the role of the one material and impinges on the interface. Two virtual crack relative thickness of porous layers. Section 8 deals with the extensions are considered, either deflected along the interface minor influence of Poissons ratios (Fig. 2b)or straight in the adjacent layer. The energy release rates at the tip of these two extensions are compared. The 2. The asymptotics of the problem drawback of this approach is the arbitrary choice of the two increment lengths. In the second He and Hutchinson model, I The model is based on a two-scale analysis, the small pa- the primary crack is along the interface and the ability of the rameter being the layers thickness. At the macro scale the crack to kink out of the interface is studied. The two criteria laminated micro-structure is ignored, as a first approxima- involve the toughness of the materials and of the interface. tion, the whole laminate is treated as a homogeneous mate- Curiously, it is often this second paper that is used to inter- rial. It is homogenized using a rule of mixture for simplicity pret the experimental results of cracks deflection in ceramic since more sophisticated homogenization processes do not laminates, although the main assumption, a long primary in- bring significant differences in the final results. 12 There is a terface crack, is not fulfilled. Moreover, in any case, it is clear primary crack for which the tip undergoes the classical mode from Fig. 2b that the laminated environment of the crack tip I singular field. The antisymmetric mode II is inhibited due is ignored in these approaches to the symmetries Within this framework. the dis placement solution(so- called far field) prior to any crack growth can be written in ED °(x1,x2)={(0,0)+kr√F1(6)+ (1) Here xi and x2 stand for the Cartesian coordinates and rand 8 for the polar ones. The coefficient ky is the stress intensity factor and the angular shape function is denoted u(0 Considering now a small crack extension e, the perturbed solution is expressed as a correction brought to the initial term(Eq ( 1)) Fig 2 Schematic view of the inner(unbounded)stretched domain for a U((x1, x2)=U(1, x2)+small correction deflection at a porous/dense interface: (a)the present analysis(P: porous, D dense, H: homogenized); (b) He and Hutchinson approach The small correction is assumed to vanish as e-0344 D. Leguillon et al. / Journal of the European Ceramic Society 26 (2006) 343–349 Fig. 1. Macrostructure of boron carbide composite with interlayers obtained by the use of corn starch (left), microstructure of porous boron carbide obtained with 20 vol.% of corn starch (right). ticles are burned out during the debinding step, prior to sin￾tering. Dense and porous layers have the same thickness af￾ter sintering (e ≈ 100
m). Laminated specimens made of 20 layers have been tested under 3-point flexure loading. De￾tails on the fabrication of specimens and the measurement of elastic and failure parameters can be found in Reynaud and co-workers5,6 and in Tariolle et al.7,8 Fig. 1 shows mi￾crographs of the layered B4C materials and depicts the pore shapes within the porous interlayers.8 The analyses of crack deflections by interfaces are gen￾erally based on two models due to He and Hutchinson.10,11 Both are carried out in an unbounded domain made of two elastic materials. In the first one,10 the primary crack lies in one material and impinges on the interface. Two virtual crack extensions are considered, either deflected along the interface (Fig. 2b) or straight in the adjacent layer. The energy release rates at the tip of these two extensions are compared. The drawback of this approach is the arbitrary choice of the two increment lengths. In the second He and Hutchinson model,11 the primary crack is along the interface and the ability of the crack to kink out of the interface is studied. The two criteria involve the toughness of the materials and of the interface. Curiously, it is often this second paper that is used to inter￾pret the experimental results of cracks deflection in ceramic laminates, although the main assumption, a long primary in￾terface crack, is not fulfilled. Moreover, in any case, it is clear from Fig. 2b that the laminated environment of the crack tip is ignored in these approaches. Fig. 2. Schematic view of the inner (unbounded) stretched domain for a deflection at a porous/dense interface: (a) the present analysis (P: porous, D: dense, H: homogenized); (b) He and Hutchinson approach.10 The present model relies on a two-scale analysis where the laminated structure is taken into account as explained in Section 2. The criterion proposed in Section 3 derives from an energy balance and can take into account a complementary stress criterion. This makes it possible to avoid the above￾mentioned drawback, since virtual crack extension lengths are known. The criterion is expressed in terms of two relevant parameters: the Young’s moduli and toughness ratios of the dense and porous layers. In Section 4, experiments show that these ratios can be written quite simply as functions of the porosity. Sections 5 and 6 are devoted to the study of the crack deflection at the dense/porous and porous/dense interfaces. A short Section 7 is dedicated to the analysis of the role of the relative thickness of porous layers. Section 8 deals with the minor influence of Poisson’s ratios. 2. The asymptotics of the problem The model is based on a two-scale analysis, the small pa￾rameter being the layers thickness. At the macro scale the laminated micro-structure is ignored, as a first approxima￾tion, the whole laminate is treated as a homogeneous mate￾rial. It is homogenized using a rule of mixture for simplicity, since more sophisticated homogenization processes do not bring significant differences in the final results.12 There is a primary crack for which the tip undergoes the classical mode I singular field. The antisymmetric mode II is inhibited due to the symmetries. Within this framework, the displacement solution (so￾called far field) prior to any crack growth can be written in plane elasticity: U- 0(x1, x2) = U- 0(0, 0) + kI √r u- I (θ) +··· (1) Here x1 and x2 stand for the Cartesian coordinates and r and θ for the polar ones. The coefficient kI is the stress intensity factor and the angular shape function is denoted u- I (θ). Considering now a small crack extension , the perturbed solution is expressed as a correction brought to the initial term (Eq. (1)): U- (x1, x2) = U- 0(x1, x2) + small correction (2) The small correction is assumed to vanish as → 0.