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