D. Leguillon et al / Journal of the European Ceramic Society 26(2006)343-349 and porous layers and in the homogenized remaining part. The third equation expresses that the crack faces are free of traction. Finally, the last one is the matching condition with he mode I term involved in the far field(Eq (1)) Similarly, a crack extension e either in the next layer or along the interface(a deflection at the interface porous/dense is illustrated in Fig. 2a)leads to the following expansion =U0,0)+k√eW(y1,y2,)+ where u= e/e is the dimensionless crack extension length Here w must fulfil the same system of equations(Eq (4)), the traction free condition(Eq. (43)) being extended to the faces of the new extension of the crack 3. The deflection criterion Within this framework, the leading term of the change in Fig 3. The inner stretched domain and its artificial outer boundary r. The potential energy between the two states(prior to and follow- origin is located at the tip of the primary crack (i.e. before branching) ing a crack extension)is written: 13 The micro scale is obtained by stretching the domain 8W=ki/A/Ep E around the primary crack tip by 1/e, e being the layer thick- ness. Considering the limit e-0, the problem is now set- where d stands for the specimen depth(plane elasticity ). Ed tled in an unbounded domain, so-called inner domain. In or- and Ep are respectively the Youngs moduli of the dense and der to have tractable computations, this domain is artificially the porous ceramics. Poisson's ratios play a minor role as bounded at a large distance(roo>>1, where roo is the radius shown below in Fig 12. The function A is numerically derived of the artificial boundary and I the dimensionless stretched from the displacement field W using a contour integral: 13, thickness of the layers, say r=200 for instance, see Fig 3) from the primary crack tip. Moreover, only few dense(D v(W(y1,y,p),√pu(6) ig. 2a)and porous(P in Fig. 2a) layers, say 3 or 4, are kept being replaced by the homogenized material (H in Fig et with The validity of this simplification will be discussed in Section v(y, 5)=2/(o(U)xy-o(y)ny)ds Using the change of variable y =xi le(p=rle), Uo can be I is any contour in the inner domain surrounding and located far from the crack tip and its extension, n is its normal pointing U(1, x2)=y(ey, ey2) toward the crack tip. For practical reasons, the outer artificial boundary ro is selected( see Fig 3). The integral y in Ec =(0,0)+kr√ew(y1,y2,0)+ (3) (7)is contour independent for any and y fulfilling th equilibrium n(Eq. 41) where the 0 in W recalls that in a first step there is no crack a necessary condition for the crack growth is a conse extension. The function w is the solution to the follo quence of an energy balance 8W≥c→kf(p Vy·a=0 g=C: V,w g. n=along the crack faces E A(Ep/Ed, p)-A(Ep/Ed, O) The first equation is the balance of momentum(equilibrium). Here G is the toughness in the direction of fracture and ed The symbol nabla Vy holds for derivatives with respect to is the newly created crack surface. This expression(Eq (8)) yI and y2. The second equation is the constitutive law, C must be considered twice, once for a deflection(index def in is the elastic operator, it takes different values in the dense e following) and once for a penetration in the next layerD. Leguillon et al. / Journal of the European Ceramic Society 26 (2006) 343–349 345 Fig. 3. The inner stretched domain and its artificial outer boundary Γ ∞. The origin is located at the tip of the primary crack (i.e. before branching). The micro scale is obtained by stretching the domain around the primary crack tip by 1/e, e being the layer thick￾ness. Considering the limit e→0, the problem is now set￾tled in an unbounded domain, so-called inner domain. In or￾der to have tractable computations, this domain is artificially bounded at a large distance (r∞  1, where r∞ is the radius of the artificial boundary and 1 the dimensionless stretched thickness of the layers, say r∞ = 200 for instance, see Fig. 3) from the primary crack tip. Moreover, only few dense (D in Fig. 2a) and porous (P in Fig. 2a) layers, say 3 or 4, are kept in the vicinity of the primary crack tip, the remaining part being replaced by the homogenized material (H in Fig. 2a). The validity of this simplification will be discussed in Section 3. Using the change of variable yi = xi/e (ρ = r/e), U- 0 can be expanded as (near field): U- 0(x1, x2) = U- 0(ey1, ey2) = U- 0(0, 0) + kI √e W- (y1, y2, 0) +··· (3) where the 0 in W- recalls that in a first step there is no crack extension. The function W- is the solution to the following problem:    −∇y · σ = 0 σ = C : ∇yW- σ · n- = 0 along the crack faces W- behaves like √ρ u- I (θ) at infinity (4) The first equation is the balance of momentum (equilibrium). The symbol nabla
y holds for derivatives with respect to y1 and y2. The second equation is the constitutive law, C is the elastic operator, it takes different values in the dense and porous layers and in the homogenized remaining part. The third equation expresses that the crack faces are free of traction. Finally, the last one is the matching condition with the mode I term involved in the far field (Eq. (1)). Similarly, a crack extension either in the next layer or along the interface (a deflection at the interface porous/dense is illustrated in Fig. 2a) leads to the following expansion: U- (x1, x2) = U￾eµ(ey1, ey2) = U- 0(0, 0) + kI √e W- (y1, y2, µ) +··· (5) where µ = /e is the dimensionless crack extension length. Here W- must fulfil the same system of equations (Eq. (4)), the traction free condition (Eq. (43)) being extended to the faces of the new extension of the crack. 3. The deflection criterion Within this framework, the leading term of the change in potential energy between the two states (prior to and follow￾ing a crack extension) is written:13 W = k2 I  A Ep Ed , µ − A Ep Ed , 0  ed (6) where d stands for the specimen depth (plane elasticity). Ed and Ep are respectively the Young’s moduli of the dense and the porous ceramics. Poisson’s ratios play a minor role as shown below in Fig. 12. The functionAis numerically derived from the displacement field W- using a contour integral:13,14 A Ep Ed , µ = ψ(W- (y1, y2, µ), √ρ u- I (θ)) (7) with ψ(U- , V- ) = 1 2 Γ (σ(U- )n-V- − σ(V- )n-U- ) dS Γ is any contour in the inner domain surrounding and located far from the crack tip and its extension, n- is its normal pointing toward the crack tip. For practical reasons, the outer artificial boundary Γ ∞ is selected (see Fig. 3). The integral ψ in Eq. (7) is contour independent for any U- and V- fulfilling the equilibrium equation (Eq. 41). A necessary condition for the crack growth is a conse￾quence of an energy balance: W ≥ Gc d ⇒ k2 I f Ep Ed , µ ≥ Gc (8) with f Ep Ed , µ = A(Ep/Ed, µ) − A(Ep/Ed, 0) µ Here Gc is the toughness in the direction of fracture and d is the newly created crack surface. This expression (Eq. (8)) must be considered twice, once for a deflection (index def in the following) and once for a penetration in the next layer