Availableonlineatwww.sciencedirect.com SCIENCE DIRECT JOURNAL OF THE IECHANICS AND Journal of the Mechanics and Physics of Solids HYSICS OF SOLIDS 54(2006)266-287 www.elsevier.comlocate/jmps The roles of toughness and cohesive strength crack deflection at interfaces J.P. Parmigiana, I. M.D. Thouless, b, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA Department of Materials Science, Engineering, Unirersity of Michigan, Ann Arbor, MI 48109, US.A Received 23 February 2005: received in revised form I September 2005: accepted 6 September 2005 Abstract In order to design composites and laminated materials, it is necessary to understand the issues that govern crack deflection and crack penetration at interfaces. Historically, models of crack deflection have been developed using either a strength-based or an energy-based fracture criterion. However, in general, crack propagation depends on both strength and toughness. Therefore in this paper, crack deflection has been studied using a cohesive-zone model which incorporates both strength and toughness parameters simultaneously. Under appropriate limiting conditions, this model reproduces earlier results that were based on either strength or energy considerations alone. However, the general model reveals a number of interesting results. Of particular note is the apparent absence of any lower bound for the ratio of the substrate to interface toughness to guarantee crack penetration. It appears that, no matter how tough an interface is, crack deflection can always be induced if the strength of the interface is low enough compared to the strength of the substrate. This may be of significance for biological applications where brittle organic matrices can be bonded by relatively tough organic layers. Conversely, it appears that there is a lower bound for the ratio of the substrate strength to interfacial strength, below which penetration is guaranteed no matter how brittle the interface. Finally, it is noted that the effect of modulus mismatch on crack deflection is very sensitive to the mixed-mode failure criterion for the interface, particularly if the cracked layer is much stiffer than the substrate C)2005 Elsevier Ltd. All rights reserved Keywords: Crack deflection; Crack penetration; Interfacial fracture; Toughness; Cohesive strength Corresponding author. Tel. +1734 7635289: fax: +17346473170 E-mail address: thouless(@ umich.edu(M. D. Thouless). Current address: Department of Mechanical Engineering, Oregon State University, Corvallis, OR 97331 USA 022-5096/S.see front matter o 2005 Elsevier Ltd. All rights reserved doi:10.1016/ . mps.2005.09.002
Journal of the Mechanics and Physics of Solids 54 (2006) 266–287 The roles of toughness and cohesive strength on crack deflection at interfaces J.P. Parmigiania,1, M.D. Thoulessa,b, a Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA b Department of Materials Science, & Engineering, University of Michigan, Ann Arbor, MI 48109, USA Received 23 February 2005; received in revised form 1 September 2005; accepted 6 September 2005 Abstract In order to design composites and laminated materials, it is necessary to understand the issues that govern crack deflection and crack penetration at interfaces. Historically, models of crack deflection have been developed using either a strength-based or an energy-based fracture criterion. However, in general, crack propagation depends on both strength and toughness. Therefore, in this paper, crack deflection has been studied using a cohesive-zone model which incorporates both strength and toughness parameters simultaneously. Under appropriate limiting conditions, this model reproduces earlier results that were based on either strength or energy considerations alone. However, the general model reveals a number of interesting results. Of particular note is the apparent absence of any lower bound for the ratio of the substrate to interface toughness to guarantee crack penetration. It appears that, no matter how tough an interface is, crack deflection can always be induced if the strength of the interface is low enough compared to the strength of the substrate. This may be of significance for biological applications where brittle organic matrices can be bonded by relatively tough organic layers. Conversely, it appears that there is a lower bound for the ratio of the substrate strength to interfacial strength, below which penetration is guaranteed no matter how brittle the interface. Finally, it is noted that the effect of modulus mismatch on crack deflection is very sensitive to the mixed-mode failure criterion for the interface, particularly if the cracked layer is much stiffer than the substrate. r 2005 Elsevier Ltd. All rights reserved. Keywords: Crack deflection; Crack penetration; Interfacial fracture; Toughness; Cohesive strength ARTICLE IN PRESS www.elsevier.com/locate/jmps 0022-5096/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2005.09.002 Corresponding author. Tel.: +1 734 7635289; fax: +1 734 6473170. E-mail address: thouless@umich.edu (M.D. Thouless). 1 Current address: Department of Mechanical Engineering, Oregon State University, Corvallis, OR 97331, USA
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 1. Introduction I.I. Contribution of crack deflection to toughening Crack deflection and delamination at interfaces play a major role in the performance of many composite systems. Brittle materials such as ceramics, concrete or epoxies can be toughened by the addition of relatively brittle fibers, provided crack deflection occurs at the interfaces between the fibers and the matrix. If crack deflection does not occur. a crack propagating through the matrix will continue unimpeded when it encounters the fiber. This results in little or no toughening, as relatively little energy is dissipated by fracture of a brittle fiber. Conversely, if crack deflection does occur, then the crack is effectively blunted Furthermore, if the crack circumvents the fibers and continues to propagate without netrating them, the intact fibers left behind in the crack wake bridge the crack surfaces Fig. la). Significant contributions to toughening can then be provided by energy that is dissipated by friction at the debonded fiber-matrix interfaces( Campbell et al., 1990; Evans and Marshall, 1989: Aveston et al., 1971; Aveston and Kelly, 1973). Similar effects occur during the fracture of composites reinforced by whiskers or particles(Evans et al, 1989; Ruhle et al., 1987; Becher and Wei, 1984), or of polycrystalline materials(Khan et al 2000: Cook, 1990). Deflection along interfaces in these materials results in toughening by crack bridging, frictional pull-out, or crack deflection(Faber and Evans, 1983a, b) (Fig. 1b) Laminated composites provide another class of engineering materials for which crack deflection at interfaces plays a crucial role in their mechanical properties(Kovar et al 1997, 1998: Clegg, 1992; Chan, 1997; Korsunsky, 2001)(Fig. Ic). Deflection along multiple interlaminar interfaces results in dissipation of energy by the delamination process. In addition, cracks often need to be re-initiated in undamaged plies in such materials; this process also contributes to the strength of the composite. Deflection of cracks along the interfaces in single and multilayer coatings(Fig. Id)provides the same mechanism protecting substrates, which is of particular use in wear applications(Xia et aL., 2004: Luo etal.,2003) Many natural materials are composites, and rely upon crack deflection to provide exceptional levels of toughness(Nardone and Prewo, 1988: Folsom et al., 1992; He et al Fig. 1. Manifestations of crack deflection in composites and multi-layered materials. (a) Crack bridging in a fiber. reinforced composite.(b) Crack deflection in a whisker- or particle-reinforced composite(c) Delamination in a laminated composite.(d) Delamination in a multi-layered film on a substrate
1. Introduction 1.1. Contribution of crack deflection to toughening Crack deflection and delamination at interfaces play a major role in the performance of many composite systems. Brittle materials such as ceramics, concrete or epoxies can be toughened by the addition of relatively brittle fibers, provided crack deflection occurs at the interfaces between the fibers and the matrix. If crack deflection does not occur, a crack propagating through the matrix will continue unimpeded when it encounters the fiber. This results in little or no toughening, as relatively little energy is dissipated by fracture of a brittle fiber. Conversely, if crack deflection does occur, then the crack is effectively blunted. Furthermore, if the crack circumvents the fibers and continues to propagate without penetrating them, the intact fibers left behind in the crack wake bridge the crack surfaces (Fig. 1a). Significant contributions to toughening can then be provided by energy that is dissipated by friction at the debonded fiber-matrix interfaces (Campbell et al., 1990; Evans and Marshall, 1989; Aveston et al., 1971; Aveston and Kelly, 1973). Similar effects occur during the fracture of composites reinforced by whiskers or particles (Evans et al., 1989; Ruhle et al., 1987; Becher and Wei, 1984), or of polycrystalline materials (Khan et al., 2000; Cook, 1990). Deflection along interfaces in these materials results in toughening by crack bridging, frictional pull-out, or crack deflection (Faber and Evans, 1983a, b) (Fig. 1b). Laminated composites provide another class of engineering materials for which crack deflection at interfaces plays a crucial role in their mechanical properties (Kovar et al., 1997, 1998; Clegg, 1992; Chan, 1997; Korsunsky, 2001) (Fig. 1c). Deflection along multiple interlaminar interfaces results in dissipation of energy by the delamination process. In addition, cracks often need to be re-initiated in undamaged plies in such materials; this process also contributes to the strength of the composite. Deflection of cracks along the interfaces in single and multilayer coatings (Fig. 1d) provides the same mechanism for protecting substrates, which is of particular use in wear applications (Xia et al., 2004; Luo et al., 2003). Many natural materials are composites, and rely upon crack deflection to provide exceptional levels of toughness (Nardone and Prewo, 1988; Folsom et al., 1992; He et al., ARTICLE IN PRESS Fig. 1. Manifestations of crack deflection in composites and multi-layered materials. (a) Crack bridging in a fiberreinforced composite. (b) Crack deflection in a whisker- or particle-reinforced composite. (c) Delamination in a laminated composite. (d) Delamination in a multi-layered film on a substrate. J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 267
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 1993: Tu et al, 1996). For example, wood consists of aligned long, hollow cylindrical cells (Ashby and Jones, 1998; Wainwright et al., 1982), and common experience shows that attempts to fracture wood perpendicular to its grain are hindered by crack deflection along the grains. Shells of many animals provide examples of materials with exceptional toughness created by combining a hard, brittle, inorganic mineral with a compliant protein (Kessler et al., 1996). The protein exists at the interfaces between the mineral components, and provides a bonding agent that can delaminate and dissipate energy when an attempt is made to fracture the shell. It is clear that the balance between the organic interface and inorganic matrix is highly optimized for the evolutionary purposes of the shell 1. 2. Previous analyses of crack deflection The optimization of composites that exhibit crack deflection and interfacial delamina- tion requires an understanding of how the interfacial and bulk properties affect the mechanics of the problem. The role of crack deflection at interfaces was first recognized and analyzed about forty years ago by Cook and Gordon (1964). Their analysis used a trength-based fracture criterion. They considered a matrix crack perpendicular to a fiber having identical elastic properties as the matrix. The matrix crack was modeled as an ellipse with a very high aspect ratio, and the results of Inglis(1913)were used to investigate the stresses around the crack tip. Cook and Gordon(1964)noted that the maximum normal stress ahead of, and co-planar with, the crack is about five times greater than the maximum normal stress perpendicular to the crack tip. Based on this observation, they suggested that a fiber needs to be about five times stronger than the interface between it and the matrix to prevent fiber fracture, and to allow crack deflection to occur This concept was extended many years later by Gupta et al.(1992), who used earlier work(Zak and Williams, 1963: Williams, 1957, Swenson and Rau, 1970) on the stress field around a sharp crack at bimaterial interfaces, to look at the criterion for determining whether a crack at normal incidence to a bimaterial interface would deflect or not Comparisons between the maximum normal stress across the interface and the maximum normal stress ahead of the crack allowed predictions to be made about whether defection or penetration should occur. For example, in an elastically homogeneous system, the results indicated that crack deflection should occur if the material ahead of the crack is more than about three and a half times stronger than the interface. While giving a slightly different value for the ratio of the two strengths required for deflection, this result is consistent with the earlier work of Cook and Gordon(1964). Furthermore, this work showed that crack deflection along the interface becomes much less likely if the cracked matrix is stiffer than the second phase, with crack deflection becoming essentially impossible if there is a compliant second phase embedded in a rigid matrix. Conversely, the tendency for crack deflection increases slightly when the second phase is stiffer than the matrix These analyses follow an Inglis(1913) or strength-based approach to fracture. Both nalyses lead to design criteria for composites and laminates that are based on the ratio of the strengths of the interface and second phase. An alternative approach using interfacial fracture mechanics Rice(1988)follows that of Griffith(1920)and others(Irwin, 1957; Kies and Smith, 1955: Orowan, 1949), and is based on an energy criterion. Many authors have ed linear-elastic fracture mechanics to look at crack deflection from an en perspective(He and Hutchinson, 1989: Thouless et al., 1989: Martinez and gupta, I
1993; Tu et al., 1996). For example, wood consists of aligned long, hollow cylindrical cells (Ashby and Jones, 1998; Wainwright et al., 1982), and common experience shows that attempts to fracture wood perpendicular to its grain are hindered by crack deflection along the grains. Shells of many animals provide examples of materials with exceptional toughness created by combining a hard, brittle, inorganic mineral with a compliant protein (Kessler et al., 1996). The protein exists at the interfaces between the mineral components, and provides a bonding agent that can delaminate and dissipate energy when an attempt is made to fracture the shell. It is clear that the balance between the organic interface and inorganic matrix is highly optimized for the evolutionary purposes of the shell. 1.2. Previous analyses of crack deflection The optimization of composites that exhibit crack deflection and interfacial delamination requires an understanding of how the interfacial and bulk properties affect the mechanics of the problem. The role of crack deflection at interfaces was first recognized and analyzed about forty years ago by Cook and Gordon (1964). Their analysis used a strength-based fracture criterion. They considered a matrix crack perpendicular to a fiber having identical elastic properties as the matrix. The matrix crack was modeled as an ellipse with a very high aspect ratio, and the results of Inglis (1913) were used to investigate the stresses around the crack tip. Cook and Gordon (1964) noted that the maximum normal stress ahead of, and co-planar with, the crack is about five times greater than the maximum normal stress perpendicular to the crack tip. Based on this observation, they suggested that a fiber needs to be about five times stronger than the interface between it and the matrix to prevent fiber fracture, and to allow crack deflection to occur. This concept was extended many years later by Gupta et al. (1992), who used earlier work (Zak and Williams, 1963; Williams, 1957; Swenson and Rau, 1970) on the stress field around a sharp crack at bimaterial interfaces, to look at the criterion for determining whether a crack at normal incidence to a bimaterial interface would deflect or not. Comparisons between the maximum normal stress across the interface and the maximum normal stress ahead of the crack allowed predictions to be made about whether deflection or penetration should occur. For example, in an elastically homogeneous system, the results indicated that crack deflection should occur if the material ahead of the crack is more than about three and a half times stronger than the interface. While giving a slightly different value for the ratio of the two strengths required for deflection, this result is consistent with the earlier work of Cook and Gordon (1964). Furthermore, this work showed that crack deflection along the interface becomes much less likely if the cracked matrix is stiffer than the second phase, with crack deflection becoming essentially impossible if there is a compliant second phase embedded in a rigid matrix. Conversely, the tendency for crack deflection increases slightly when the second phase is stiffer than the matrix. These analyses follow an Inglis (1913) or strength-based approach to fracture. Both analyses lead to design criteria for composites and laminates that are based on the ratio of the strengths of the interface and second phase. An alternative approach using interfacial fracture mechanics Rice (1988) follows that of Griffith (1920) and others (Irwin, 1957; Kies and Smith, 1955; Orowan, 1949), and is based on an energy criterion. Many authors have used linear-elastic fracture mechanics to look at crack deflection from an energy perspective (He and Hutchinson, 1989; Thouless et al., 1989; Martinez and Gupta, 1993; ARTICLE IN PRESS 268 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 Penetration Fig. 2. Details of the crack deflection problem modeled by He and Hutchinson(1989). The macroscopic view shown in(a). A comparison is made between the conditions for(b)a small kink to extend across the interface, an (c)a small kink to extend al He et al, 1994; Lu and Erdogan, 1983: Tullock et al., 1994). These generally follow the approach of Cotterell and Rice(1980), where the energy-release rates of kinks at different angles ahead of a main crack are considered The ratio of the energy-release rates in different directions is taken to be proportional to the critical ratio of the toughnesses required to trigger fracture in the different directions deflection or penetration occurs when a crack impinges a bimaterial interface in a normal direction was examined by comparing the energy-release rate at the tip of a small kink extending across the interface, p, with the energy-release rate at the tip of a small kink deflected along the interface d(Fig. 2). The condition for crack deflection along the interface can be written as where Ti is the toughness of the interface under the appropriate mixed-mode conditions, and Is is the toughness of the material (substrate) ahead of the interface. One particularly well-known result is that when the elastic properties across the interface are identical, and the kinks are vanishingly small, crack deflection occurs if the toughness of the material on the other side of the interface is more than approximately four times the mixed-mode toughness of the interface(He and Hutchinson, 1989; Thouless et al., 1989) 1.3. Problem addressed in the present work In the analyses described above, two different fracture criteria were used: a stress- base criterion and an energy-based criterion. These lead to two different types of material parameters forming the basis for design of interfaces. A stress-based fracture criterion leads to the deflection-penetration criterion being expressed in terms of the relative strengths of the interface and second phase. An energy-based fracture criterion leads to the
He et al., 1994; Lu and Erdogan, 1983; Tullock et al., 1994). These generally follow the approach of Cotterell and Rice (1980), where the energy-release rates of kinks at different angles ahead of a main crack are considered. The ratio of the energy-release rates in different directions is taken to be proportional to the critical ratio of the toughnesses required to trigger fracture in the different directions. Of particular note is the work by He and Hutchinson (1989), with corrections (He et al., 1994; Martinez and Gupta, 1993). In their work, the problem of determining whether crack deflection or penetration occurs when a crack impinges a bimaterial interface in a normal direction was examined by comparing the energy-release rate at the tip of a small kink extending across the interface, Gp, with the energy-release rate at the tip of a small kink deflected along the interface, Gd (Fig. 2). The condition for crack deflection along the interface can be written as Gi Gs o Gd Gp , (1) where Gi is the toughness of the interface under the appropriate mixed-mode conditions, and Gs is the toughness of the material (substrate) ahead of the interface. One particularly well-known result is that when the elastic properties across the interface are identical, and the kinks are vanishingly small, crack deflection occurs if the toughness of the material on the other side of the interface is more than approximately four times the mixed-mode toughness of the interface (He and Hutchinson, 1989; Thouless et al., 1989). 1.3. Problem addressed in the present work In the analyses described above, two different fracture criteria were used: a stress-based criterion and an energy-based criterion. These lead to two different types of material parameters forming the basis for design of interfaces. A stress-based fracture criterion leads to the deflection–penetration criterion being expressed in terms of the relative strengths of the interface and second phase. An energy-based fracture criterion leads to the ARTICLE IN PRESS f s f s k Penetration f s k Deflection (c) (b) (a) Fig. 2. Details of the crack deflection problem modeled by He and Hutchinson (1989). The macroscopic view is shown in (a). A comparison is made between the conditions for (b) a small kink to extend across the interface, and (c) a small kink to extend along the interface. J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 269
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 leflection-penetration criterion being expressed in terms of the relative toughnesses of the interface and second phase. At the present time, there is no crack deflection analysis that bridges these two historically distinct views of fracture. It is this gap in the understanding of the mechanics of interfaces that motivated the present stud The cohesive-zone view provides a coherent analytical framework for fracture that naturally incorporates both strength and energy criteria. Cohesive-zone modeling has its origins in the early models of Dugdale(1960)and Barenblatt (1962) that considered the effects of finite stresses at a crack tip. A cohesive-zone model incorporates a region of material ahead of the crack (the"cohesive zone") having a characteristic traction separation law that describes the fracture process. In a typical traction-separation law, the tractions across the crack plane increase with displacement up to a maximum cohesive strength, and then decay to zero at a critical opening displacement. When the critical displacement is reached, the material in the cohesive zone is assumed to have failed, and the crack advances. This approach to modeling fracture became particularly useful with the advent of sophisticated computational techniques, since it allowed crack propagation to be predicted for different geometries(Hillerborg et al., 1976: Needleman, 1987, 1990; Tvergaard and Hutchinson, 1992; Ungsuwarungsri and Knauss, 1987). The fracture behavior in a single mode of deformation tends to be dominated by two characteristic quantities of the traction-separation law-a characteristic toughness(the area under the curve),I, and a characteristic strength(closely related to the cohesive strength for many traction-separation laws), a. Cohesive-zone models provide a particularly powerful approach for analyzing fracture since their predictions appear to be fairly insensitive to the details of the traction-separation law, being dependent only on these two characteristic The dependence of cohesive models on both strength and toughness parameters makes them a natural bridge between the two traditional views of fracture(Parmigiani and Thouless, 2006). By varying the parameters of a cohesive model it is possible to move from a regime in which fracture is controlled only by the toughness, through a regime in which both toughness and strength control fracture, to a regime in which only strength dominates fracture. The relative importance of these two parameters is indicated by comparing the fracture-length scale, Er/a(where E is the modulus of the material) to the appropriate characteristic length, L of the geometry(Suo et al., 1993). When the fracture- length scale is relatively small, i. e, the non-dimensional group Er/GL is very small, the toughness controls fracture; when the fracture-length scale is relatively large, the strength controls fracture. In the intermediate range, both parameters are important. Consideration of the fracture-length scale immediately highlights an inherent problem with energy-based analyses of crack deflection at interfaces. These models invoke a pre- existing kink along the interface. This kink has to be very small in comparison to any other characteristic dimension of the problem, so that asymptotic solutions for the crack-tip stress field can be used. However, the length of the kink then becomes the characteristic dimension that the fracture-length scale must be compared to, in order to determine whether fracture is controlled by energy or stress. Therefore, the kink has to be short compared to any other dimensions of the problem, for crack-tip asymptotic solutions to be valid; but, simultaneously, the kink has to be long compared to the fracture length scale, so 2The shape of the traction-separation curves can occasionally affect fracture. For example, there are laws in hich the characteristic strength is not related to the cohesive strength(Li et al., 2005a, b)
deflection–penetration criterion being expressed in terms of the relative toughnesses of the interface and second phase. At the present time, there is no crack deflection analysis that bridges these two historically distinct views of fracture. It is this gap in the understanding of the mechanics of interfaces that motivated the present study. The cohesive-zone view provides a coherent analytical framework for fracture that naturally incorporates both strength and energy criteria. Cohesive-zone modeling has its origins in the early models of Dugdale (1960) and Barenblatt (1962) that considered the effects of finite stresses at a crack tip. A cohesive-zone model incorporates a region of material ahead of the crack (the ‘‘cohesive zone’’) having a characteristic tractionseparation law that describes the fracture process. In a typical traction-separation law, the tractions across the crack plane increase with displacement up to a maximum cohesive strength, and then decay to zero at a critical opening displacement. When the critical displacement is reached, the material in the cohesive zone is assumed to have failed, and the crack advances. This approach to modeling fracture became particularly useful with the advent of sophisticated computational techniques, since it allowed crack propagation to be predicted for different geometries (Hillerborg et al., 1976; Needleman, 1987, 1990; Tvergaard and Hutchinson, 1992; Ungsuwarungsri and Knauss, 1987). The fracture behavior in a single mode of deformation tends to be dominated by two characteristic quantities of the traction-separation law—a characteristic toughness (the area under the curve), G, and a characteristic strength (closely related to the cohesive strength for many traction–separation laws), s^. Cohesive-zone models provide a particularly powerful approach for analyzing fracture since their predictions appear to be fairly insensitive to the details of the traction-separation law, being dependent only on these two characteristic parameters.2 The dependence of cohesive models on both strength and toughness parameters makes them a natural bridge between the two traditional views of fracture (Parmigiani and Thouless, 2006). By varying the parameters of a cohesive model it is possible to move from a regime in which fracture is controlled only by the toughness, through a regime in which both toughness and strength control fracture, to a regime in which only strength dominates fracture. The relative importance of these two parameters is indicated by comparing the fracture-length scale, EG=s^ 2 (where E is the modulus of the material) to the appropriate characteristic length, L of the geometry (Suo et al., 1993). When the fracture-length scale is relatively small, i.e., the non-dimensional group EG=s^ 2 L is very small, the toughness controls fracture; when the fracture-length scale is relatively large, the strength controls fracture. In the intermediate range, both parameters are important. Consideration of the fracture-length scale immediately highlights an inherent problem with energy-based analyses of crack deflection at interfaces. These models invoke a preexisting kink along the interface. This kink has to be very small in comparison to any other characteristic dimension of the problem, so that asymptotic solutions for the crack-tip stress field can be used. However, the length of the kink then becomes the characteristic dimension that the fracture-length scale must be compared to, in order to determine whether fracture is controlled by energy or stress. Therefore, the kink has to be short compared to any other dimensions of the problem, for crack-tip asymptotic solutions to be valid; but, simultaneously, the kink has to be long compared to the fracture length scale, so ARTICLE IN PRESS 2 The shape of the traction–separation curves can occasionally affect fracture. For example, there are laws in which the characteristic strength is not related to the cohesive strength (Li et al., 2005a, b). 270 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 that an energy criterion for fracture is appropriate. If it is assumed that an interface can support singular stresses, the kink can be taken to the limit of zero length with no conceptual difficulty. However, if an interface with a finite cohesire strength does not contain a physical kink of a finite length, both energy and stress are expected to play a role in initiating fracture along the interface In this paper, a cohesive-zone model is used to analyze the problem of crack deflection at interfaces. Of major concern are(i) an elucidation of the roles of the interfacial strength, the interfacial toughness, the substrate strength and the substrate toughness on crack deflection, and (i) an understanding of the conditions under which any of these parameters might dominate design considerations. These issues are addressed by using a cohesive-zone analysis to look at the general problem of crack deflection at different fracture-length scales, in the absence of any pre-existing kinks. The results of the calculations are presented in non-dimensional terms for a wide range of parameter space, so that the effects of different strength and toughness values on the transition are fully explored. The roles of mixed-mode failure criteria and modulus mismatch across the interface are also explored nally, in the appendix, cohesive-zone models are used to look at kinked cracks. The results of these calculations are used to make a connection with existing energy-based analyses of crack deflection, and to show that the numerical approach used in this paper can accurately capture the classical energy-based criteria for this phenomenon, provided the fracture-length scales are small enough, and that appropriate assumptions about the kinks are made 2. Numerical results 2. Cohesive- zone model A cohesive-zone model was used to analyze crack deflection at interfaces. This problem requires a mixed-mode implementation of the model. Often, mixed-mode effects are modeled by combining normal and shear displacements into a single parameter that is used in a traction-separation law to indicate overall load-carrying ability(Tvergaard and Hutchinson, 1993). However, an alternative approach is to use separate and independent laws for mode I and mode Il, each being functions of only the normal and shear displacements, respectively. The ability to specify the mode-I and mode-II strength and toughness values independently appears to be necessary to capture some experimental results (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006). Since the traction-separation laws are prescribed independently, they need to be coupled through a mixed-mode failure criterion Such a failure criterion relates the normal and shear placements at which the load-bearing capability of the cohesive-zone elements fail. In this work. a linear failure criterion of the form 1/1+m/u=1 was used, where gI is the mode-I energy-release rate, TI is the mode-I toughness, n is the mode-II energy release rate, and Tu is the mode-II toughness. In this formulation, the toughness is defined as the total area under the traction-separation law, and the energy release rate is defined as the area under the traction-separation law at any particular instant of interest (Yang and Thouless, 2001). While simple, this linear criterion allows a fairly rich range of mixed-mode behavior to be mimicked, from what we will call a
that an energy criterion for fracture is appropriate. If it is assumed that an interface can support singular stresses, the kink can be taken to the limit of zero length with no conceptual difficulty. However, if an interface with a finite cohesive strength does not contain a physical kink of a finite length, both energy and stress are expected to play a role in initiating fracture along the interface. In this paper, a cohesive-zone model is used to analyze the problem of crack deflection at interfaces. Of major concern are (i) an elucidation of the roles of the interfacial strength, the interfacial toughness, the substrate strength and the substrate toughness on crack deflection, and (ii) an understanding of the conditions under which any of these parameters might dominate design considerations. These issues are addressed by using a cohesive-zone analysis to look at the general problem of crack deflection at different fracture-length scales, in the absence of any pre-existing kinks. The results of the calculations are presented in non-dimensional terms for a wide range of parameter space, so that the effects of different strength and toughness values on the transition are fully explored. The roles of mixed-mode failure criteria and modulus mismatch across the interface are also explored. Finally, in the appendix, cohesive-zone models are used to look at kinked cracks. The results of these calculations are used to make a connection with existing energy-based analyses of crack deflection, and to show that the numerical approach used in this paper can accurately capture the classical energy-based criteria for this phenomenon, provided the fracture-length scales are small enough, and that appropriate assumptions about the kinks are made. 2. Numerical results 2.1. Cohesive-zone model A cohesive-zone model was used to analyze crack deflection at interfaces. This problem requires a mixed-mode implementation of the model. Often, mixed-mode effects are modeled by combining normal and shear displacements into a single parameter that is used in a traction-separation law to indicate overall load-carrying ability (Tvergaard and Hutchinson, 1993). However, an alternative approach is to use separate and independent laws for mode I and mode II, each being functions of only the normal and shear displacements, respectively. The ability to specify the mode-I and mode-II strength and toughness values independently appears to be necessary to capture some experimental results (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006). Since the traction–separation laws are prescribed independently, they need to be coupled through a mixed-mode failure criterion. Such a failure criterion relates the normal and shear displacements at which the load-bearing capability of the cohesive-zone elements fail. In this work, a linear failure criterion of the form GI=GI þ GII=GII ¼ 1 (2) was used, where GI is the mode-I energy-release rate, GI is the mode-I toughness, GII is the mode-II energy release rate, and GII is the mode-II toughness. In this formulation, the toughness is defined as the total area under the traction-separation law, and the energyrelease rate is defined as the area under the traction-separation law at any particular instant of interest (Yang and Thouless, 2001). While simple, this linear criterion allows for a fairly rich range of mixed-mode behavior to be mimicked, from what we will call a ARTICLE IN PRESS J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 271
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 ""Griffith criterion"for which there is a single value of the critical energy-release rate required for fracture (i.e, Tu=Ti, to one in which fracture occurs only in response to mode-I loading(n>i). The use of Eq (2)in cohesive-zone analyses has been shown to do an excellent job of describing experimental results (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006), and it can mimic mixed-mode fracture criteria for linear-elastic fracture mechanics(LEFM), if the phase angle is defined as y= arctan√乡n/, where y has its usual definition under LEFM conditions of y= arctan(Kn/Kn), and Kn and KI are the nominal mode-II and mode-I stress-intensity factors acting at a crack tip Hutchinson and Suo, 1992) The general forms of the mode-I and mode-lI traction-separation laws used in this study are shown in Fig 3. The mode-I cohesive strength is o, the mode-ll cohesive strength is t, the mode-I toughness is TI, and the mode-lI toughness is Ill. Generalized forms for the traction-separation laws have been used, as the precise shape does not generally have a Mode l Mode ll Fig 3. Schematic illustration of the(a)mode-I, and(b) mode-ll traction-separation laws used for the cohesive. zone model in this paper. Throughout this paper the values of 81/5 and 52/8 were kept at fixed values of 0.01 and 0. 75, respectivel
‘‘Griffith criterion’’ for which there is a single value of the critical energy-release rate required for fracture (i.e., GII ¼ GI), to one in which fracture occurs only in response to mode-I loading (GIIbGI). The use of Eq. (2) in cohesive-zone analyses has been shown to do an excellent job of describing experimental results (Yang and Thouless, 2001; Kafkalidis and Thouless, 2002; Li et al., 2006), and it can mimic mixed-mode fracture criteria for linear-elastic fracture mechanics (LEFM), if the phase angle is defined as c ¼ arctan ffiffiffiffiffiffiffiffiffiffiffiffiffiffi GII=GI p , (3) where c has its usual definition under LEFM conditions of c ¼ arctanðKII=KIÞ, and KII and KI are the nominal mode-II and mode-I stress-intensity factors acting at a crack tip (Hutchinson and Suo, 1992). The general forms of the mode-I and mode-II traction-separation laws used in this study are shown in Fig. 3. The mode-I cohesive strength is s^, the mode-II cohesive strength is t^, the mode-I toughness is GI, and the mode-II toughness is GII. Generalized forms for the traction–separation laws have been used, as the precise shape does not generally have a ARTICLE IN PRESS I = d 0 c n ˆ c 0 (a) 1 Mode I ˆ Γ ∫ 2 t c 0 (b) 1 Mode II 2 II = d 0 c Γ ∫ Fig. 3. Schematic illustration of the (a) mode-I, and (b) mode-II traction-separation laws used for the cohesivezone model in this paper. Throughout this paper the values of d1=dc and d2=dc were kept at fixed values of 0.01 and 0.75, respectively. 272 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 significant effect on fracture. The strength and toughness(area under the curve) are the two dominant parameters that control fracture, and cracks can propagate only if both the ode-I and energy criteria are met. As discussed above, the use of separate mode-I and II laws allows for a general investigation of fracture, encompassing problems in which shear fracture has physical significance and problems in which pure shear only results in slip, not fracture The cohesive-zone modeling was implemented within the commercial finite-element package ABAQUS(version 6.3-1), as described by Yang(2000 ). Three- and four-node linear, plane-strain elements were used for the continuum elements. The elements for the cohesive zone were defined using the ABAQUS UEL feature, the traction-separation laws of Fig 3, and the failure criterion of Eq(2). These were implemented in a FORTRAN subroutine. An example of the code used is given in Parmigiani(2005) While several different geometries could have been used to study the problem of crack deflection, the work in this paper focuses on a laminated system subject to a uniform tensile displacement, as shown in Fig. 4. A layer of thickness h, with an elastic modulus of Ef and a Poissons ratio of v/, is bonded to a substrate of thickness d. The layer of thickness h has a crack that extends from the free surface to the interface. and that is normal to the interface. The substrate has an elastic modulus of e and a poisson 's ratio of Vs. For all the calculations reported in this paper, the substrate is ten times thicker than the cracked layer, so that d= 10h. Plane-strain conditions are assumed, so that the two Dundurs parameters can be defined as(Dundurs, 1969) Ef-es +e 8=E(1-2)/(1-y)-E(1-2)/(1-) 2(Er+es) where,E=E/(1-12). If the substrate cracks, it will do so under pure mode-I conditions therefore, only the mode-I fracture properties of the substrate are required. The mode-I substrate toughness is designated as Is, and the mode-I strength is designated asas. The crack impinging on interface cohesive-zone eleme The laminated geometry used to study crack deflection in this paper. A layer h and with an modulus of Er and a Poissons ratio of v is bonded to a substrate of thickne d=10. The ubstrate has an elastic modulus of e and a poisson's ratio of v. There is a crack surface to the interface and is normal to the interface Sets of cohesive elements exist e crack in the ubstrate and along the interface. There is a plane of symmetry along the crack, and is loaded by a uniform displacement applied to the ends of the specimen
significant effect on fracture. The strength and toughness (area under the curve) are the two dominant parameters that control fracture, and cracks can propagate only if both the stress and energy criteria are met. As discussed above, the use of separate mode-I and mode-II laws allows for a general investigation of fracture, encompassing problems in which shear fracture has physical significance and problems in which pure shear only results in slip, not fracture. The cohesive-zone modeling was implemented within the commercial finite-element package ABAQUS (version 6.3-1), as described by Yang (2000). Three- and four-node, linear, plane-strain elements were used for the continuum elements. The elements for the cohesive zone were defined using the ABAQUS UEL feature, the traction-separation laws of Fig. 3, and the failure criterion of Eq. (2). These were implemented in a FORTRAN subroutine. An example of the code used is given in Parmigiani (2005). While several different geometries could have been used to study the problem of crack deflection, the work in this paper focuses on a laminated system subject to a uniform tensile displacement, as shown in Fig. 4. A layer of thickness h, with an elastic modulus of Ef and a Poisson’s ratio of nf , is bonded to a substrate of thickness d. The layer of thickness h has a crack that extends from the free surface to the interface, and that is normal to the interface. The substrate has an elastic modulus of Es and a Poisson’s ratio of ns. For all the calculations reported in this paper, the substrate is ten times thicker than the cracked layer, so that d ¼ 10h. Plane-strain conditions are assumed, so that the two Dundurs parameters can be defined as (Dundurs, 1969) a ¼ E¯ f E¯ s E¯ f þ E¯ s , (4) and b ¼ E¯ f ð1 2nsÞ=ð1 nsÞ E¯ sð1 2nf Þ=ð1 nf Þ 2ðE¯ f þ E¯ sÞ , (5) where, E¯ ¼ E=ð1 n2Þ. If the substrate cracks, it will do so under pure mode-I conditions; therefore, only the mode-I fracture properties of the substrate are required. The mode-I substrate toughness is designated as Gs, and the mode-I strength is designated as s^s. The ARTICLE IN PRESS d = 10 h h 2L = 220 h Ef ,νf Es,νs crack impinging on interface cohesive-zone elements Fig. 4. The laminated geometry used to study crack deflection in this paper. A layer of thickness h and with an elastic modulus of Ef and a Poisson’s ratio of nf is bonded to a substrate of thickness d, where d ¼ 10h. The substrate has an elastic modulus of Es and a Poisson’s ratio of ns. There is a crack that extends from the top surface to the interface, and is normal to the interface. Sets of cohesive elements exist ahead of the crack in the substrate and along the interface. There is a plane of symmetry along the crack, and the system is loaded by a uniform displacement applied to the ends of the specimen. J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 273
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 interface fails under mixed-mode conditions, and a mixed-mode analysis is required for crack propagation along the interface. The mode-I cohesive parameters of the interface are designated as TI and di, while the mode-II cohesive parameters of the interface are designated as Tui and ti 2. 2. Effects of fracture length-scales on crack deflection In this section, a general study of the effect of fracture parameters on crack deflection presented A macroscopic crack is assumed to impinge directly on an interface as shown in Fig. 4. There are no pre-existing kinks or flaws ahead of this main crack, and there is an axis of symmetry along the plane of the crack. Two rows of cohesive elements are placed at the tip of the crack; one row along the interface, and the other row in the substrate. A uniform tensile displacement is applied to the ends of the specimen. The required mesh density for the numerical calculations was determined by selecting several meshes, running the simulations, and analyzing the results to verify what mesh density gave consistent solutions that were within an acceptable range for the uncertainty of the results The transition between crack deflection and crack penetration for the geometry shown in Fig. 4 depends on the following material and geometrical parameters Ef,Es,vs Ts, Gi, ti, as, h, d Using the dundurs result for the effects of mismatched moduli across plane interfaces (Dundurs, 1969), these parameters can be re-expressed in the following non-dimensional groups for the plane-strain conditions considered in this paper: o,B, T:/Erh, E Ti/ah, 6s/G, T/Ti, Tui/Ti, t:/oi, d/h The additional non-dimensional parameters 81/8c and 82/8 which describe the shape of the traction-separation laws(Fig. 3)were kept at constant values of 0.01 and 0.75 throughout the paper. These parameters do not play a significant role in the fracture process and transition in failure mechanism The quantity Ti/Eh was fixed at 1.0 x 10- for all the results presented in this paper This is a physically reasonable value for the parameter, and numerical studies indicated that even fairly significant changes around this level had a negligible influence on the failure transition. The initial studies were conducted with a=0, and B=0(so that Ef=Es=E, and vf=vs=v), and with IIi= TIli=Ii and ti=dj. With these parameters fixed, a series of calculations was performed holding the non-dimensional interface fracture- length scale at a constant value of ET ah=0.01. After running umerical calculations with a given set of as/ai and Is/Ti, a note was made as to whether the crack first began to grow along the interface or through the substrate. In these calculations, the onset of crack growth along either plane was defined as occurring when the first element in a cohesive zone failed. 3 A systematic exploration of how the failure mechanism depended on the magnitude of as/Gi and Ts/ri allowed a failure-mechanism map to be plotted, as shown in Fig. 5. In this figure, which has axes of as/a; and Ts /Ti, constant values of the non-dimensional substrate SThe issue of whether a crack might kink off an interface, after first deflecting along it, was not explored here However, crack deflection off interfaces has been studied experimentally and modeled using a cohesive-zone approach in a paper by Li et al. (2005a)
interface fails under mixed-mode conditions, and a mixed-mode analysis is required for crack propagation along the interface. The mode-I cohesive parameters of the interface are designated as GIi and s^i, while the mode-II cohesive parameters of the interface are designated as GIIi and t^i. 2.2. Effects of fracture length-scales on crack deflection In this section, a general study of the effect of fracture parameters on crack deflection is presented. A macroscopic crack is assumed to impinge directly on an interface as shown in Fig. 4. There are no pre-existing kinks or flaws ahead of this main crack, and there is an axis of symmetry along the plane of the crack. Two rows of cohesive elements are placed at the tip of the crack; one row along the interface, and the other row in the substrate. A uniform tensile displacement is applied to the ends of the specimen. The required mesh density for the numerical calculations was determined by selecting several meshes, running the simulations, and analyzing the results to verify what mesh density gave consistent solutions that were within an acceptable range for the uncertainty of the results. The transition between crack deflection and crack penetration for the geometry shown in Fig. 4 depends on the following material and geometrical parameters: Ef ; Es; nf ; ns;GIi; GIIi;Gs; s^i; t^i; s^s; h; d. Using the Dundurs result for the effects of mismatched moduli across plane interfaces (Dundurs, 1969), these parameters can be re-expressed in the following non-dimensional groups for the plane-strain conditions considered in this paper: a; b;Gi=E¯ f h;E¯ f Gi=s^ 2 i h; s^s=s^i;Gs=GIi; GIIi=GIi; t^i=s^i; d=h. The additional non-dimensional parameters d1=dc and d2=dc which describe the shape of the traction-separation laws (Fig. 3) were kept at constant values of 0.01 and 0.75 throughout the paper. These parameters do not play a significant role in the fracture process and transition in failure mechanism. The quantity Gi=Eh¯ was fixed at 1:0 106 for all the results presented in this paper. This is a physically reasonable value for the parameter, and numerical studies indicated that even fairly significant changes around this level had a negligible influence on the failure transition. The initial studies were conducted with a ¼ 0, and b ¼ 0 (so that Ef ¼ Es ¼ E, and nf ¼ ns ¼ n), and with GIi ¼ GIIi ¼ Gi and t^i ¼ s^i. With these parameters fixed, a series of calculations was performed holding the non-dimensional interface fracture-length scale at a constant value of E¯ Gi=s^ 2 i h ¼ 0:01. After running numerical calculations with a given set of s^s=s^i and Gs=Gi, a note was made as to whether the crack first began to grow along the interface or through the substrate. In these calculations, the onset of crack growth along either plane was defined as occurring when the first element in a cohesive zone failed.3 A systematic exploration of how the failure mechanism depended on the magnitude of s^s=s^i and Gs=Gi allowed a failure-mechanism map to be plotted, as shown in Fig. 5. In this figure, which has axes of s^s=s^i and Gs=Gi, constant values of the non-dimensional substrate ARTICLE IN PRESS 3 The issue of whether a crack might kink off an interface, after first deflecting along it, was not explored here. However, crack deflection off interfaces has been studied experimentally and modeled using a cohesive-zone approach in a paper by Li et al. (2005a). 274 J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287
J.P. Parmigiani, M.D. Thouless/J. Mech. Phys. Solids 54(2006)266-287 r/o E998958最标号6°标工 Er/o defleraclong Er2h=000 Ratio of the substrate strength to interface strength, a /o Fig. 5. The results of a set of calculations for Er/a h=0.01, a=B=0, Ti/Eh= 1.0 x 10-, Tu=lmi=r ii, and d/h= 10. The plot shows the regimes in which crack penetration or crack deflection will occur in /ri and a./@i space. The error bars indicate the range of uncertainty of the transition. fracture-length scale follow a parabolic form. The error bars on this figure are associated with numerical considerations, and show the uncertainty with which the boundary of the transition could be determined. In this context, it should be noted that, occasionally, the demands of the geometry created numerical difficulties close to the transition. In the worst cases the numerical simulation failed before the onset of crack growth. When this happened, the transition was quantified by examining the values of p/Ts and sd/ri at the point of the numerical instability, where p is the energy-release rate for penetration of the crack into the substrate and d is the energy-release rate for deflection of the crack along the interface. These ratios measure the extent of the appropriate traction-separation law that has been traversed and indicate how close the elements of the cohesive zone are to failure As might be intuitively expected, the failure-mechanism map of Fig. 5 shows that crack deflection is promoted by high values of both as ai and Ts/Ti. Conversely, crack enetration is promoted by low values of these two ratios. At larger values of the non- dimensional substrate fracture-length scale, the failure mechanism is controlled by the strength ratio; at smaller values, it becomes more sensitive to the toughness ratio However, even in this latter range, there is no indication that only toughness controls the failure mechanism. Indeed, if there is a lower bound on the toughness ratio required to guarantee crack penetration, it is much lower than the range that could be explored by the present calculations. Conversely, there does appear to be a vertical asymptote representing a critical value of the strength ratio below which crack penetration is guaranteed, irrespective of the toughness ratio. This implies that crack penetration will always occur if the substrate
fracture-length scale follow a parabolic form. The error bars on this figure are associated with numerical considerations, and show the uncertainty with which the boundary of the transition could be determined. In this context, it should be noted that, occasionally, the demands of the geometry created numerical difficulties close to the transition. In the worst cases, the numerical simulation failed before the onset of crack growth. When this happened, the transition was quantified by examining the values of Gp=Gs and Gd =Gi at the point of the numerical instability, where Gp is the energy-release rate for penetration of the crack into the substrate and Gd is the energy-release rate for deflection of the crack along the interface. These ratios measure the extent of the appropriate traction-separation law that has been traversed, and indicate how close the elements of the cohesive zone are to failure As might be intuitively expected, the failure-mechanism map of Fig. 5 shows that crack deflection is promoted by high values of both s^s=s^i and Gs=Gi. Conversely, crack penetration is promoted by low values of these two ratios. At larger values of the nondimensional substrate fracture-length scale, the failure mechanism is controlled by the strength ratio; at smaller values, it becomes more sensitive to the toughness ratio. However, even in this latter range, there is no indication that only toughness controls the failure mechanism. Indeed, if there is a lower bound on the toughness ratio required to guarantee crack penetration, it is much lower than the range that could be explored by the present calculations. Conversely, there does appear to be a vertical asymptote representing a critical value of the strength ratio below which crack penetration is guaranteed, irrespective of the toughness ratio. This implies that crack penetration will always occur if the substrate ARTICLE IN PRESS 0 10 20 30 40 50 60 70 80 01234567 Ratio of the substrate strength to interface strength, σs / σi ^ ^ Ratio of the substrate toughness to interface toughness, Γs / Γi deflection along interface penetration into substrate E Γi /σi 2 h = 0.01 Γi /Eh = 1.0 x 10-6 ^ E Γs /σs 2 h = 0.1 E Γs /σs 2 h = 0.01 E Γs /σs 2 h = 0.001 ^ ^ ^ Fig. 5. The results of a set of calculations for E¯ Gi=s^ 2 i h ¼ 0:01, a ¼ b ¼ 0, Gi=Eh¯ ¼ 1:0 106 , GIi ¼ GIIi ¼ Gi, s^i ¼ t^i, and d=h ¼ 10. The plot shows the regimes in which crack penetration or crack deflection will occur in Gs=Gi and s^s=s^i space. The error bars indicate the range of uncertainty of the transition. J.P. Parmigiani, M.D. Thouless / J. Mech. Phys. Solids 54 (2006) 266–287 275
