2142 D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 Two intensity factors k Eq. (3)and K Eq. (5)are involved in the previous pressions. Their computation is based on contour integrals y(Leguillon and San- chez-Palencia, 1987) (r) y(yPu) y(ru, r), pupU) For any fields U and y satisfying the equilibrium equation in a wedge and stress free boundary conditions on the edges, y is a contour independent integral which is defined by y(U,D=o(Unv-o(nnu)ds C is any contour in n2o to compute k or Din to compute K surrounding the origin and starting and finishing at, the primary crack stress free edges. The unit normal n to C points towards the origin Indeed UD and y Eq.(7)are a priori unknown, and must be replaced by the corresponding finite element approximations. Note that in Eq. (7)V can be used in place of y since y(,,p! 4. Application to fracture mechanics 4.I. Differential and incremental approaches on The previous expansions Eqs. ( 1)(6)express the effect of a small perturbation a solution to a structural problem. Above, the perturbation is a narrow ligament remaining between a main crack tip and an interface, but it can be also a short crack increment with small dimensionless length n. Roughly speaking, this increment is a forward growth while the ligament is a"backward"one. Replacing formally e with n in the inner and outer expansions and substituting these relations in the potential energy expression allow, at the leading order, the change in potential energy between the unperturbed(before crack growth) and perturbed (after crack growth) states to be defined(Leguillon, 1989) H(0)-W(n)=k2Km2+ The energy release rate is the driving force associated with n. It is the derivative of w with respect to this variable G=limW(O-W( limk2Kn2 n-07 0 Obviously, this limit is meaningful for the classical crack tip singularity characterized by 2=1/2. It vanishes if 2>1/2(weak singularity, a>0) and it tends to infinity if n<1/2(strong singularity, a<o). These situations are met in case of a crack
2142 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 Two intensity factors k Eq. (3) and K Eq. (5) are involved in the previous expressions. Their computation is based on contour integrals C (Leguillon and Sanchez-Palencia, 1987) k5 C(U0 ,r−lu− ) C(rlu+ ,r−lu− ) , K5 C(Vˆ 1 ,rlu+ ) C(r−lu− ,rlu+ ) . (7) For any fields U and V satisfying the equilibrium equation in a wedge and stress free boundary conditions on the edges, C is a contour independent integral which is defined by C(U,V)5E C (s(U)nV2s(V)nU) ds. (8) C is any contour in V0 to compute k or Vin to compute K surrounding the origin and starting and finishing at, the primary crack stress free edges. The unit normal n to C points towards the origin. Indeed U0 and Vˆ 1 Eq. (7) are a priori unknown, and must be replaced by the corresponding finite element approximations. Note that in Eq. (7) V1 can be used in place of Vˆ 1 since C(rlu+ , rlu+ )=0. 4. Application to fracture mechanics 4.1. Differential and incremental approaches The previous expansions Eqs. (1)–(6) express the effect of a small perturbation on a solution to a structural problem. Above, the perturbation is a narrow ligament remaining between a main crack tip and an interface, but it can be also a short crack increment with small dimensionless length h. Roughly speaking, this increment is a forward growth while the ligament is a “backward” one. Replacing formally e with h in the inner and outer expansions and substituting these relations in the potential energy expression allow, at the leading order, the change in potential energy between the unperturbed (before crack growth) and perturbed (after crack growth) states to be defined (Leguillon, 1989) W(0)2W(h)5k 2 Kh2l 1 …. (9) The energy release rate is the driving force associated with h. It is the derivative of 2W with respect to this variable G5lim h→0 W(0)−W(h) h 5lim h→0 k 2 Kh2l−1 1…. (10) Obviously, this limit is meaningful for the classical crack tip singularity characterized by l=1/2. It vanishes if l.1/2 (weak singularity, a.0) and it tends to infinity if l,1/2 (strong singularity, a,0). These situations are met in case of a crack
D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 2143 impinging on an interface between two elastic materials, depending if the crack lies on the soft side (weak singularity) or on the stiff side(strong singularity). These two particular cases play an important role in fracture mechanics of heterogeneous materials(Leguillon and Sanchez-Palencia, 1992) If G=0 the critical value G(material toughness) is never reached and the griffith criterion cannot be fulfilled whatever the applied loads. This is a drawback of the classical differential theory which may be overcome with the help of the following incremental condition derived from expansion Eq (9) w(0)-W(n)=G n= (or-(n-k"Kn2-IeG (11) n Eq.( 1)is a necessary condition for fracture and provides an incremental criterion which coincides with the differential theory when it holds true(=1/2)and which can still bc used when this theory fails. Compared to the usual Griffith criterion, it contains the additional unknown parameter n. This incremental approach will be used in the next sections On the opposite, if G-o, the criterion is violated for any non-zero applied load as small as it can be. Eq. (10)implicitly assumes that the derivative -aw/an exists but this existence is clearly questionable. A modified criterion is examined by the authors(Leguillon et al., 1999)in this special case. It is based on the principle of maximum decrease of the total energy as suggested by Lawn(1993) 4.2. He and Hutchinson criterion for interface deflection He and Hutchinson(1989)have considered the problem of a crack impinging(n ligament) on an interface between two isotropic elastic materials: the crack lies in one material and can either penetrate the other one or branch along the interface For simplicity, we limit here the comparison to a crack normal to the interface and a penetration(dimensionless length np)or a double symmetric deflection along the interface(total dimensionless length 2nd)(Fig. 4). From Eq.(I1), deflection is pro- moted if Kp( np (12) where G@ and G2) are the respective toughness of the interface and of material 2 and where Kd and K, are the intensity factors Eq. (5)extracted from the term yo1V2)of an inner expansion considering a unit penetration or a unit deflection (F1g.5) In addition to the respective toughness of material 2 and of the interface, such a criterion Eq(12)requires the knowledge of the elementary increments in the two directions(or at least their ratio) except if 2 =1/2, but in that case the problem turns to be a classical crack branching one in a homogeneous material(Leguillon, 1993) with anisotropic toughness. The cracks' extension lengths should be related to the material and the interface microstructure. Moreover such increments' sizes should differ and often remain unknown
D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 2143 impinging on an interface between two elastic materials, depending if the crack lies on the soft side (weak singularity) or on the stiff side (strong singularity). These two particular cases play an important role in fracture mechanics of heterogeneous materials (Leguillon and Sanchez-Palencia, 1992). If G=0 the critical value Gc (material toughness) is never reached and the Griffith criterion cannot be fulfilled whatever the applied loads. This is a drawback of the classical differential theory which may be overcome with the help of the following incremental condition derived from expansion Eq. (9) W(0)2W(h)$Gch⇒ W(0)−W(h) h 5k 2 Kh2l−1 $Gc. (11) Eq. (11) is a necessary condition for fracture and provides an incremental criterion which coincides with the differential theory when it holds true (l=1/2) and which can still bc used when this theory fails. Compared to the usual Griffith criterion, it contains the additional unknown parameter h. This incremental approach will be used in the next sections. On the opposite, if G→`, the criterion is violated for any non-zero applied load as small as it can be. Eq. (10) implicitly assumes that the derivative 2∂W/∂h exists but this existence is clearly questionable. A modified criterion is examined by the authors (Leguillon et al., 1999) in this special case. It is based on the principle of maximum decrease of the total energy as suggested by Lawn (1993). 4.2. He and Hutchinson criterion for interface deflection He and Hutchinson (1989) have considered the problem of a crack impinging (no ligament) on an interface between two isotropic elastic materials: the crack lies in one material and can either penetrate the other one or branch along the interface. For simplicity, we limit here the comparison to a crack normal to the interface and a penetration (dimensionless length hp) or a double symmetric deflection along the interface (total dimensionless length 2hd) (Fig. 4). From Eq. (11), deflection is promoted if Kd Kp S 2hd hp D 2l−1 $ G(i) c G(2) c , (12) where G(i) c and G(2) c are the respective toughness of the interface and of material 2 and where Kd and Kp are the intensity factors Eq. (5) extracted from the term V1 (y1,y2) of an inner expansion considering a unit penetration or a unit deflection (Fig. 5). In addition to the respective toughness of material 2 and of the interface, such a criterion Eq. (12) requires the knowledge of the elementary increments in the two directions (or at least their ratio) except if l=1/2, but in that case the problem turns to be a classical crack branching one in a homogeneous material (Leguillon, 1993) with anisotropic toughness. The cracks’ extension lengths should be related to the material and the interface microstructure. Moreover such increments’ sizes should differ and often remain unknown
144 D. Leguillon er al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2167 material 2 interf crackI material 1 material 2 nterface crack material 1 Fig. 4. Penetration or deflection of a crack impinging on an interface. naterial 2 material 2 1 interface material 1 material 1 interface ck crac crack Fig. 5. Stretched inner domain with a unit length penetration or deflection To make possible the comparison between the two crack behaviours He and Hut- chinson add the assumption that the two perturbation lengths(penetration and deflection) are equal. The HH criterion reads Ga ga where Ga and Gp are the deflection and penetration energy release rates. He and Hutchinson calculate these quantities, using integral equations and Muskhelishvili method, at a same distance a from the impinging point, on the deflected and penetrat ing branches (i.e. n=np=a). This assumption is slightly different from our own
2144 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 Fig. 4. Penetration or deflection of a crack impinging on an interface. Fig. 5. Stretched inner domain with a unit length penetration or deflection. To make possible the comparison between the two crack behaviours He and Hutchinson add the assumption that the two perturbation lengths (penetration and deflection) are equal. The HH criterion reads Gd Gp $ G(i) c G(2) c , (13) where Gd and Gp are the deflection and penetration energy release rates. He and Hutchinson calculate these quantities, using integral equations and Muskhelishvili’s method, at a same distance a from the impinging point, on the deflected and penetrating branches (i.e. hd=hp=a). This assumption is slightly different from our own
D. Leguillon er al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 2145 (Leguillon et al., 1999)2na=np=a leading together with Eq.(12)to the so-called LS (Leguillon and Sanchez-Palencia, 1992)criterion Kd go Once the above equality assumptions have been made, criteria Eqs. (13)and(14) are independent of any length, otherwise they are not(see He et al., 1994, for HH and Eq(12) for LS). Although they look similar, these two criteria are slightly different. HH assume the penetration and deflection geometries and study the local fields at the tip of the new extensions. It is thus consistent to carry out the analysis at the same distance of the primary crack tip. On the contrary, in the present Ls approach, the question is to determine the energy balance which allows creation of crack extensions In this context, it is consistent to examine equal crack extensions It makes an important difference in case of symmetrical double deflection along the interface. In the HH case the total interface debonding length is 2a whereas it must be a in the ls one An attempt to introduce different crack increments is proposed by Ahn et al. (1998) which is shown to fit experimental data. However their approach is not an asymptotic one, based on a structural computation; it depends on the applied loads, on the geometry of the specimen and on the actual length of the increments, not only on their ratio as above Eq.(12)(i.e. even if the increments are taken as equal, their results depend on the common value a) t A comparison between He and Hutchinson's results and the present criterion LS (14)is shown in Fig. 6 for different values of the first Dundurs parameter a 2.2 Kd/Kp Gd/Gp He Hutchinson 0.6 0,4 -1-08-06-0,40,2 020,40,60,81 dundurs Fig. 6. Comparison between HH Eq (13)and LS Eq(14) criteria
D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 2145 (Leguillon et al., 1999) 2hd=hp=a leading together with Eq. (12) to the so-called LS (Leguillon and Sanchez-Palencia, 1992) criterion Kd Kp $ G(i) c G(2) c . (14) Once the above equality assumptions have been made, criteria Eqs. (13) and (14) are independent of any length, otherwise they are not (see He et al., 1994, for HH and Eq. (12) for LS). Although they look similar, these two criteria are slightly different. HH assume the penetration and deflection geometries and study the local fields at the tip of the new extensions. It is thus consistent to carry out the analysis at the same distance of the primary crack tip. On the contrary, in the present LS approach, the question is to determine the energy balance which allows creation of crack extensions. In this context, it is consistent to examine equal crack extensions. It makes an important difference in case of symmetrical double deflection along the interface. In the HH case the total interface debonding length is 2a whereas it must be a in the LS one. An attempt to introduce different crack increments is proposed by Ahn et al. (1998) which is shown to fit experimental data. However their approach is not an asymptotic one, based on a structural computation; it depends on the applied loads, on the geometry of the specimen and on the actual length of the increments, not only on their ratio as above Eq. (12) (i.e. even if the increments are taken as equal, their results depend on the common value a). A comparison between He and Hutchinson’s results and the present criterion LS Eq. (14) is shown in Fig. 6 for different values of the first Dundurs parameter a Fig. 6. Comparison between HH Eq. (13) and LS Eq. (14) criteria