PHYSICS OF SOLIDS Pergamon outa or tn48 (2000)2137-216] s or son www.elsevier.com/locate/jmps Interface debonding ahead of a primary crack D. Leguillon a, .C. Lacroix a. E. Martin b Laboratoire de Modelisation en Mecanique, CNRS UMR 7607, Universite Pierre et Marie Curie, Paris 6.8 rue du Capitaine Scott, 75015 Paris, France Laboratoire de genie Mecanique, UPRES 496, IUT A, Universite de bordeaux I, 33405 Talence, Received 2 July 1999; received in revised form 18 November 1999 Abstract The debonding of an interface between two elastic materials is assumed to occur ahead of a primary crack lying in material 1. Conditions for such a mechanism are derived from an asymptotic analysis and depend on the elastic mismatch between the two materials as well as on material I and interface toughnesses. The ligament width between the primary crack tip and the interface and the debond nucleation length can be determined if the interface strength is known. A similar mechanism involving a crack nucleation ahead of the primary crack in material 2 is also examined. Deflection prediction is derived from an energetic criterion which is in general more favourable to deflection than the He and Hutchinson one. The former decreases asymptotically towards the latter. Moreover, if the crack lies in the stiffest material esults show that these nucleation mechanisms are highly improbable. If material 2 and inter- face strengths are known, a stress criterion is naturally derived. The presented analysis is independent of the applied loads and of the geometry of the specimen. 2000 Elsevier Science Ltd. All rights reserved Keywords: A Crack branching and bifurcation; B Ceramic material; C. Asymptotic analysis 1. Introduction Fibre reinforced ceramic matrix composites and ceramic laminates are increasingly considered for various applications because their fabrication can be tailored to meet ecific properties like high strength at high temperatures(Moya, 1995; Evans author.Fax:+33-1-4427-5259 dol@ccr jussieu. fr (D. Leguillon) 0022-5096/00/S- see front matter 2000 Elsevier Science Ltd. All rights reserved PI:S0022-5096(99)00101-5
Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 www.elsevier.com/locate/jmps Interface debonding ahead of a primary crack D. Leguillon a,*, C. Lacroix a , E. Martin b a Laboratoire de Mode´lisation en Me´canique, CNRS UMR 7607, Universite´ Pierre et Marie Curie, Paris 6, 8 rue du Capitaine Scott, 75015 Paris, France b Laboratoire de Ge´nie Me´canique, UPRES 496, IUT A, Universite´ de Bordeaux I, 33405 Talence, France Received 2 July 1999; received in revised form 18 November 1999 Abstract The debonding of an interface between two elastic materials is assumed to occur ahead of a primary crack lying in material 1. Conditions for such a mechanism are derived from an asymptotic analysis and depend on the elastic mismatch between the two materials as well as on material 1 and interface toughnesses. The ligament width between the primary crack tip and the interface and the debond nucleation length can be determined if the interface strength is known. A similar mechanism involving a crack nucleation ahead of the primary crack in material 2 is also examined. Deflection prediction is derived from an energetic criterion which is in general more favourable to deflection than the He and Hutchinson one. The former decreases asymptotically towards the latter. Moreover, if the crack lies in the stiffest material, results show that these nucleation mechanisms are highly improbable. If material 2 and interface strengths are known, a stress criterion is naturally derived. The presented analysis is independent of the applied loads and of the geometry of the specimen. 2000 Elsevier Science Ltd. All rights reserved. Keywords: A. Crack branching and bifurcation; B. Ceramic material; C. Asymptotic analysis 1. Introduction Fibre reinforced ceramic matrix composites and ceramic laminates are increasingly considered for various applications because their fabrication can be tailored to meet specific properties like high strength at high temperatures (Moya, 1995; Evans, * Corresponding author. Fax: +33-1-4427-5259. E-mail address: dol@ccr.jussieu.fr (D. Leguillon). 0022-5096/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 00 22 -5096(99)00101-5
2138 D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 1997). The mechanical response of such materials can differ substantially from that of bulk materials due to the presence of interfaces. To enable a non brittle mode of failure, the interface strength between adjoining phases must be adjusted to decrease the stress concentration effect of any crack that occurs in a phase, reducing the likelihood that it will propagate into the next phase The capability of an interface to deflect a crack is usually analysed in terms of competition between deflection and penetration for a stationary crack terminating at the interface at a normal or an oblique angle. This approach provides a crack defec tion condition involving a strength ratio or a toughness ratio to be compared with a critical ratio depending on the elastic mismatch between the two phases(He and Hutchinson, 1989; Gupta et al., 1992). This criterion has been applied to describe the crack deflection observed in various combinations of brittle bimaterials(Kumar and Singh, 1998; Liu et al., 1998; Martin et al., 1998). In spite of the uncertainties related to the determination of the properties of micron sized materials, the criterion sometimes fails to predict interfacial deflection(Warrier et al., 1997; Kovar et al 1997)or interfacial penetration(Ahn et al., 1998). Among the various reason invoked to explain such discrepancies, a different mechanism of crack deflection can be postulated( Cook and Gordon, 1964): interfacial failure occurs ahead of the main crack and deflection results from linking between the interfacial crack and the pri- mary crack. This mechanism has been experimentally evidenced in some bimaterial systems(Theocaris and Milios, 1983; Warrier et al., 1997; Zhang and Lewandowski, 1997; Majumdar et al., 1998)or suggested as a result of a micromechanical analysis (Pagano, 1998). Further, Lee et al. (1996)and Clegg et al.(1997) considered crack deflection as an interaction between the primary crack and the growth of an interface defect. It must be pointed out that the modelling of this deflection mechanism has al ways been restricted to an interface between similar materials The purpose of this paper is to establish a crack deflection model taking into account the interface debonding ahead of a primary crack. The analysis is performed within the framework of two dimensional linear elasticity. Asymptotic expressions are used to derive the change in potential energy induced by crack nucleation growth. The competition between the growth of the main crack and the interface debonding(crack nucleation) is first investigated. The derived criterion is compared with the experimental results obtained by Lee et al. (1996)on cracked brittle lami- nates submitted to a four-point bending test Further, the competition between the interface debonding and the interface pen- etration ahead of the primary crack is examined. The approach leads to an energy multi-criterion involving the ligament width, the various crack extensions along and across the interface and the elastic mismatch of the bimaterial components 2. The notched bimaterial specimen The analysis is restricted to plane strain elasticity. We consider(Fig. 1)a bima- terial specimen( thickness e) made of two isotropic layers bonded together along an interface with perfect transmission conditions: displacements and tractions are
2138 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 1997). The mechanical response of such materials can differ substantially from that of bulk materials due to the presence of interfaces. To enable a non brittle mode of failure, the interface strength between adjoining phases must be adjusted to decrease the stress concentration effect of any crack that occurs in a phase, reducing the likelihood that it will propagate into the next phase. The capability of an interface to deflect a crack is usually analysed in terms of competition between deflection and penetration for a stationary crack terminating at the interface at a normal or an oblique angle. This approach provides a crack deflection condition involving a strength ratio or a toughness ratio to be compared with a critical ratio depending on the elastic mismatch between the two phases (He and Hutchinson, 1989; Gupta et al., 1992). This criterion has been applied to describe the crack deflection observed in various combinations of brittle bimaterials (Kumar and Singh, 1998; Liu et al., 1998; Martin et al., 1998). In spite of the uncertainties related to the determination of the properties of micron sized materials, the criterion sometimes fails to predict interfacial deflection (Warrier et al., 1997; Kovar et al., 1997) or interfacial penetration (Ahn et al., 1998). Among the various reasons invoked to explain such discrepancies, a different mechanism of crack deflection can be postulated (Cook and Gordon, 1964): interfacial failure occurs ahead of the main crack and deflection results from linking between the interfacial crack and the primary crack. This mechanism has been experimentally evidenced in some bimaterial systems (Theocaris and Milios, 1983; Warrier et al., 1997; Zhang and Lewandowski, 1997; Majumdar et al., 1998) or suggested as a result of a micromechanical analysis (Pagano, 1998). Further, Lee et al. (1996) and Clegg et al. (1997) considered crack deflection as an interaction between the primary crack and the growth of an interface defect. It must be pointed out that the modelling of this deflection mechanism has always been restricted to an interface between similar materials. The purpose of this paper is to establish a crack deflection model taking into account the interface debonding ahead of a primary crack. The analysis is performed within the framework of two dimensional linear elasticity. Asymptotic expressions are used to derive the change in potential energy induced by crack nucleation or growth. The competition between the growth of the main crack and the interface debonding (crack nucleation) is first investigated. The derived criterion is compared with the experimental results obtained by Lee et al. (1996) on cracked brittle laminates submitted to a four-point bending test. Further, the competition between the interface debonding and the interface penetration ahead of the primary crack is examined. The approach leads to an energy multi-criterion involving the ligament width, the various crack extensions along and across the interface and the elastic mismatch of the bimaterial components. 2. The notched bimaterial specimen The analysis is restricted to plane strain elasticity. We consider (Fig. 1) a bimaterial specimen (thickness e) made of two isotropic layers bonded together along an interface with perfect transmission conditions: displacements and tractions are
D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 2139 material 2 cracklE material 1 Fig 1. The notched bimaterial specimen under four-point bending continuous across the interface. The isotropy assumption is not essential and in-axis orthotropy can be considered as well. There is a transverse crack lying in one of the components, denoted material 1(Youngs modulus E1, Poissons ratio vi), and perpendicular to the interface. Material 2(E2, v2) is unbroken. The crack length is L and its tip is located at a distance from the interface (L+/=e/2). This distance is assumed to be small with respect to a characteristic dimension of the specimen(th thickness e for instance, k<<e) and to the crack length (<<L). We set ee, e is a small dimensionless parameter(the dimensionless ligament width). This specimen is submitted to a given loading geometry (a four-point bending for instance as illus- trated on Fig. 1) In the following the thickness e is set to unity. This is a way to define dimen sionless lengths; each physical length and displacement is divided by e. The corre- sponding domain is denoted n2 Numerical results are presented essentially for three different contrasts between the elastic properties of the components. A strong contrast characterized either by EE2=0. 1 or EE2=10, and the absence of contrast E/ E2=l(homogeneous specimen). In each case Poissons ratio is taken to be identical (V=V2=0. 3). Special situations are considered for comparison with the He and Hutchinson(HH)criterion to match with their data. In that comparison, Dundurs coefficients(a, B)are used to characterize the mismatch between two isotropic materials(Dundurs, 1969),a<0 corresponding to a material I stiffer than the other and reciprocally 3. Far and near fields- matched asymptotics Details of the matched asymptotics procedure are presented in Leguillon(1993): the main lines are recalled here. The displacement solution UE to the problem can be described with the help of two expansions U(x1x2)=U(x1-x2)+f(e)U(x1x2)+.where f(e)0 as e-0 The first term U(xI, x2) is the solution to the so-called"unperturbed problem""i
D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 2139 Fig. 1. The notched bimaterial specimen under four-point bending. continuous across the interface. The isotropy assumption is not essential and in-axis orthotropy can be considered as well. There is a transverse crack lying in one of the components, denoted material 1 (Young’s modulus E1, Poisson’s ratio n1), and perpendicular to the interface. Material 2 (E2, n2) is unbroken. The crack length is L and its tip is located at a distance l from the interface (L+l=e/2). This distance is assumed to be small with respect to a characteristic dimension of the specimen (the thickness e for instance, l,,e) and to the crack length (l,,L). We set l=ee, e is a small dimensionless parameter (the dimensionless ligament width). This specimen is submitted to a given loading geometry (a four-point bending for instance as illustrated on Fig. 1). In the following the thickness e is set to unity. This is a way to define dimensionless lengths; each physical length and displacement is divided by e. The corresponding domain is denoted Ve . Numerical results are presented essentially for three different contrasts between the elastic properties of the components. A strong contrast characterized either by E1/E2=0.1 or E1/E2=10, and the absence of contrast E1/E2=1 (homogeneous specimen). In each case Poisson’s ratio is taken to be identical (n1=n2=0.3). Special situations are considered for comparison with the He and Hutchinson (HH) criterion to match with their data. In that comparison, Dundurs coefficients (a, b) are used to characterize the mismatch between two isotropic materials (Dundurs, 1969), a,0 corresponding to a material 1 stiffer than the other and reciprocally. 3. Far and near fields — matched asymptotics Details of the matched asymptotics procedure are presented in Leguillon (1993); the main lines are recalled here. The displacement solution Ue to the problem can be described with the help of two expansions Ue (x1,x2)5U0 (x1,x2)1f1(e)U1 (x1,x2)1… where f1(e)→0 as e→0. (1) The first term U0 (x1, x2) is the solution to the so-called “unperturbed problem” in
2140 D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 material 2 material 1 Fig. 2. The unperturbed outer domain o2. which the ligament of length / is neglected, i. e the crack impinges on the interface formally corresponding to the geometry obtained with E=0 denoted Q2(Fig. 2) The second term is a small correction to the first approximation, U(xix is a solution to a problem based on the same unperturbed structure n. The exact formu- lation of this second problem is a consequence of the matching rules as explained below. From a numerical point of view, these terms are quite easy to compute, since there is no small ligament. Finite elements do not necessitate a particularly refined mesh in this area to account for the geometry Indeed, such a solution Eq. (1) is valid outside a vicinity of the ligament(which is absent in the unperturbed structure); it is a so-called"outer"expansion. Thus, the precise knowledge of the solution requires another description near the ligament. It is obtained, first by stretching the initial domain by l] and then considering the limit domain as 8-0. It is an unbounded one Q, with a semi-infinite crack leaving a unit ligament between its tip and the interface(Fig. 3) On this domain spanned by the space variables y=x E, the second expansion, the (x1x2)=(eyy2)=1y2)+Fi(e)(iy2)+… where F(e0 as a-0. To define correctly the different terms of this expansion conditions at infinity must be prescribed. They derive from the matching rules Expansion Eq. (1)holds true outside a vicinity of the ligament and the other one q.(2)inside. There must exist an intermediate region in which both are valid: in material 2 nterface material 1 crack Fig. 3. The stretched inner domain Qn with a unit ligament
2140 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 Fig. 2. The unperturbed outer domain V0 . which the ligament of length l is neglected, i.e. the crack impinges on the interface, formally corresponding to the geometry obtained with e=0 denoted V0 (Fig. 2). The second term is a small correction to the first approximation, U1 (x1,x2) is a solution to a problem based on the same unperturbed structure V0 . The exact formulation of this second problem is a consequence of the matching rules as explained below. From a numerical point of view, these terms are quite easy to compute, since there is no small ligament. Finite elements do not necessitate a particularly refined mesh in this area to account for the geometry. Indeed, such a solution Eq. (1) is valid outside a vicinity of the ligament (which is absent in the unperturbed structure); it is a so-called “outer” expansion. Thus, the precise knowledge of the solution requires another description near the ligament. It is obtained, first by stretching the initial domain by l/e and then considering the limit domain as e→0. It is an unbounded one Vin, with a semi-infinite crack leaving a unit ligament between its tip and the interface (Fig. 3). On this domain spanned by the space variables yi =xi /e, the second expansion, the “inner” one, writes Ue (x1,x2)5Ue (ey1,ey2)5V0 (y1,y2)1F1(e)V1 (y1,y2)1 … , (2) where F1(e)→0 as e→0. To define correctly the different terms of this expansion, conditions at infinity must be prescribed. They derive from the matching rules. Expansion Eq. (1) holds true outside a vicinity of the ligament and the other one Eq. (2) inside. There must exist an intermediate region in which both are valid: in Fig. 3. The stretched inner domain Vin with a unit ligament
D. Leguillon er al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 2141 other words the outer terms behaviour approaching the ligament area must match with the inner terms"behaviour going at infinity More precisely, U(x1-X2) has a singular behaviour approaching the crack tip LP(x1x2)=0,0)+kr丝()+ where r and 0 are the polar coordinates with origin at the crack tip. The exponent is positive, smaller than I and, depending on the relative stiffness of the substrates can be larger, equal to or smaller than 1/2(the classical crack tip singularity)(Zak and Williams, 1963). Two modes are usually involved in such a situation, but it is assumed here that the applied loads only generate a symmetrical one in Eq (3)(three or four-point bending tests on notched specimens for instance). Coefficient k is the intensity factor. We avoid the expression"stress"intensity factor which we reserve for the usual fracture mechanics crack tip singularity. The explanation for the index + will be found below This leads to Lv1y2)=((0,0)F1(e)=kex,y(y1y2)=p2t(0)+(0v1y2) where p=r/E. Here, PIGi,2)is the solution to a well-posed problem with decaying conditions at infinity and it is once more a singular behaviour which is involved p(13y2)Kp-y(6)asp→∞, where K denotes the corresponding intensity factor. It is independent of the applied loads, of the global geometry of the structure and of the actual length ee of the perturbation. It depends only on local material properties and geometry; more pre- cisely it depends on the shape of the perturbation not on its size. Function a singular behaviour at infinity whereas the other, with a positive exponent expresses a singular behaviour at the origin. This entails the complementary matching f(e)=kKe,(x1x2)=ry(6)+C(x1x2) where U(x1x2) is the solution to a well-posed problem in the unperturbed structure no. The outer expansion Eqs. (1),(3)and(6) provides a formulation for the remote fields whereas the inner expansion Eqs. (2),(4)and(5) plays the same role for the near field The different terms in the asymptotics as well as the singular modes and their dual counterparts can be evaluated as indicated in Leguillon and Sanchez-Palencia (1987). The far(outer) fields are obtained by classical finite element computations in the domain n2o. The computation difficulty of the inner expansion terms comes from the infinite domain ( n. It is necessary to bound it artificially at a large distance from the perturbation(ligament with a length unity) to perform computations. From a theoretical point of view, y Gi, y2) has a finite energy while Vvi J2)has not, but his difference disappears on a bounded domain and vv1y2) can be computed directly with prescribed displacements Put(@)along the artificially created bound- ary. Such a domain must be large but the problems under consideration are inde pendent of the applied loads and can be solved once and for all
D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 2141 other words the outer terms’ behaviour approaching the ligament area must match with the inner terms’ behaviour going at infinity. More precisely, U0 (x1,x2) has a singular behaviour approaching the crack tip U0 (x1,x2)5U0 (0,0)1k rlu+ (q)1 … , (3) where r and q are the polar coordinates with origin at the crack tip. The exponent l is positive, smaller than 1 and, depending on the relative stiffness of the substrates, can be larger, equal to or smaller than 1/2 (the classical crack tip singularity) (Zak and Williams, 1963). Two modes are usually involved in such a situation, but it is assumed here that the applied loads only generate a symmetrical one in Eq. (3) (three or four-point bending tests on notched specimens for instance). Coefficient k is the intensity factor. We avoid the expression “stress” intensity factor which we reserve for the usual fracture mechanics crack tip singularity. The explanation for the index + will be found below. This leads to V0 (y1,y2)5U0 (0,0),F1(e)5ke l ,V1 (y1,y2)5rlu+ (q)1Vˆ 1 (y1,y2), (4) where r=r/e. Here, Vˆ 1 (y1,y2) is the solution to a well-posed problem with decaying conditions at infinity and it is once more a singular behaviour which is involved Vˆ 1 (y1,y2)|K r−lu− (q) as r→`, (5) where K denotes the corresponding intensity factor. It is independent of the applied loads, of the global geometry of the structure and of the actual length ee of the perturbation. It depends only on local material properties and geometry; more precisely it depends on the shape of the perturbation not on its size. Function r−lu− (q) is the dual mode to rlu+ (q); the former, with a negative exponent expresses a singular behaviour at infinity whereas the other, with a positive exponent expresses a singular behaviour at the origin. This entails the complementary matching f1(e)5k K e2l ,U1 (x1,x2)5r−lu− (q)1Uˆ 1 (x1,x2), (6) where Uˆ 1 (x1,x2) is the solution to a well-posed problem in the unperturbed structure V0 . The outer expansion Eqs. (1), (3) and (6) provides a formulation for the remote fields whereas the inner expansion Eqs. (2), (4) and (5) plays the same role for the near fields. The different terms in the asymptotics as well as the singular modes and their dual counterparts can be evaluated as indicated in Leguillon and Sanchez-Palencia (1987). The far (outer) fields are obtained by classical finite element computations in the domain V0 . The computation difficulty of the inner expansion terms comes from the infinite domain Vin. It is necessary to bound it artificially at a large distance from the perturbation (ligament with a length unity) to perform computations. From a theoretical point of view, Vˆ 1 (y1, y2) has a finite energy while V1 (y1,y2) has not, but this difference disappears on a bounded domain and V1 (y1,y2) can be computed directly with prescribed displacements rlu+ (q) along the artificially created boundary. Such a domain must be large but the problems under consideration are independent of the applied loads and can be solved once and for all
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
2146 D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 while the second one vanishes B=0. The agreement between our calculations and He and Hutchinsons results is quite good. It is to be pointed out that HH criterion ha been first defined with the assumption B=0. It has been shown next by Martinez and Gupta(1994) that the influence of B on the criterion remains weak over the range B=-0.2, 0.2. The LS criterion, based on a numerical-analytical approach, can be carried out whatever B and extends without additional complexity to in-axis ortho- ropic substrates The right part a>0 of Fig. 6 corresponds to a weak singularity, when the main crack lies in the softest of the two components and the left one a<o corresponds to a strong singularity when the stiffest material is broken 5. Interface debonding ahead of the crack tip Herein, we do not propose an improved branching criterion but the analysis of the debonding mechanism ahead of the crack tip, disregarding at first any competition with the penetration event. This is highlighted by the forthcoming results involving only material 1 and interface toughnesses G@) and G, but not the second substrate The geometry includes a ligament ahead of the main crack and is first considered prior to any interface debonding. The aim of this section is to analyse the competition between the growth of this main crack(the shortening of the ligament)and a possible debonding of the interface ahead of the crack tip 5.1. The primary crack growth t We suppose that the main crack is growing within material I towards the interface Is consider an increment Sl<<I of the initial crack length L (remember that 7 is not the crack length but the ligament width). The corresponding dimensionless parameter is Se=Slle. The associated length in the stretched domain is SelE(Fig. 7) The energy release rate is minus the derivative of the potential energy W with material 2 interface stretched crack rack material 1 ig. 7. The main crack growth in the stretched domain Q2n
2146 D. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 while the second one vanishes b=0. The agreement between our calculations and He and Hutchinson’s results is quite good. It is to be pointed out that HH criterion has been first defined with the assumption b=0. It has been shown next by Martinez and Gupta (1994) that the influence of b on the criterion remains weak over the range b=20.2, 0.2. The LS criterion, based on a numerical-analytical approach, can be carried out whatever b and extends without additional complexity to in-axis orthotropic substrates. The right part a.0 of Fig. 6 corresponds to a weak singularity, when the main crack lies in the softest of the two components and the left one a,0 corresponds to a strong singularity when the stiffest material is broken. 5. Interface debonding ahead of the crack tip Herein, we do not propose an improved branching criterion but the analysis of the debonding mechanism ahead of the crack tip, disregarding at first any competition with the penetration event. This is highlighted by the forthcoming results involving only material 1 and interface toughnesses G(1) c and G(i) c , but not the second substrate toughness G(2) c . The geometry includes a ligament ahead of the main crack and is first considered prior to any interface debonding. The aim of this section is to analyse the competition between the growth of this main crack (the shortening of the ligament) and a possible debonding of the interface ahead of the crack tip. 5.1. The primary crack growth We suppose that the main crack is growing within material 1 towards the interface. Let us consider an increment dl,,l of the initial crack length L (remember that l is not the crack length but the ligament width). The corresponding dimensionless parameter is de=dl/e. The associated length in the stretched domain is de/e (Fig. 7). The energy release rate is minus the derivative of the potential energy W with Fig. 7. The main crack growth in the stretched domain Vin