3910 lee et al. CRACK DEFLECTION AND TOUGHENING M1 Beam 1 h1 Beam 2 Pw Wedge fhi M Primary Crack 网 h1 h2 Interface Debonding Fig. 6. (a)Suo and Hutchinson [28] representation of the specimen after the debonding of the int ahead of the primary crack and(b)axial forces and bending moments acting in the specimen under wedge can now be used to describe the crack driving forces 0.05 and 50 um. As it is the largest flaws which woren hat P,an In the same manner, equation(3)can be applied to half-length a was taken as 50 um. Figure 7 shows that the other loading states, such as four point bending such flaws can grow if the interfacial fracture energy, and tension[28]. In the case of four point bending P: R, is less than about 200- for the experimental is equal to P: in magnitude and P, is zero. In the case condition where the wedging displacement was of tension when the right end of the specimen is 80 Am. This corresponds to the interface having a constrained so as not to allow rotation, all the force fracture energy less than 54% of that of the matrix and moment components have non-zero values. (the measured fracture energy of bulk PMMa was The presence of the bridging ligament between the 372J m-2)and is in reasonable agreement with the primary crack and the interface debond produces a experimental value of 60% statically indeterminate structure. It is not straight- Figure 7 also shows the crack driving forces introduce an additional boundary evaluated using the J-integral as described in condition required to solve equations (4)-(7); Appendix A2. It can be seen that there is reasonably therefore, finite element analysis, described in the good agreement between the J-integral and the Appendix, is used to find the edge loading conditions prediction from equation (3)down to an interfacial for equation (3)and to evaluate the driving force for defect size of about 50 um at which the peak the growth of the interface defects. a value of P- for J-integral values is 191Jm-. However, at defect ach interfacial crack length and P, can be obtained sizes less than 50 um, the finite clement analysis from Fig. A2 in Appendix A2. Substituting the value predicts that the crack driving force should decrease of Ph and Pw enables values for P, P2, Mi and M2 to in an almost linear mann be obtained from equations (4H7), which can be Growth of the interfacial defects will occur substituted into equation(3)to give the crack driving provided that the driving force is greater than the force, 9,, of an interfacial crack of a given length. fracture energy. This implies that the interface must This is shown in Fig. 7 where it can be seen that the contain defects larger than a, in Fig. 8 for an interface rack driving force decreases as the length of the with fracture energy R, as shown schematically Fig. 8. If the fra gy of the interface
3910 LEE et al.: CRACK DEFLECTION AND TOUGHENING hi I_ a d M2 . Beam 3 L (a) Crack l hi ~ .... t_ _l.t rface_ _ l ® P2 Interface Debonding (b) P3 Fig. 6. (a) Suo and Hutchinson [28] representation of the specimen after the debonding of the interface ahead of the primary crack and (b) axial forces and bending moments acting in the specimen under wedge loading. Further, noting that P3 and M3 are zero, equation (3) can now be used to describe the crack driving forces for the growth of interracial defects in wedging. In the same manner, equation (3) can be applied to the other loading states, such as four point bending and tension [28]. In the case of four point bending P~ is equal to P: in magnitude and P3 is zero. In the case of tension, when the right end of the specimen is constrained so as not to allow rotation, all the force and moment components have non-zero values. The presence of the bridging ligament between the primary crack and the interface debond produces a statically indeterminate structure. It is not straightforward to introduce an additional boundary condition required to solve equations (4)-(7); therefore, finite element analysis, described in the Appendix, is used to find the edge loading conditions for equation (3) and to evaluate the driving force for the growth of the interface defects. A value of P, for each interfacial crack length and Pb can be obtained from Fig. A2 in Appendix A2. Substituting the value of Pb and P~ enables values for P~, P2, M~ and M2 to be obtained from equations (4)-(7), which can be substituted into equation (3) to give the crack driving force, if,, of an interfacial crack of a given length. This is shown in Fig. 7 where it can be seen that the crack driving force decreases as the length of the interfacial crack increases. As mentioned earlier, the interface contains flaws with a half-length between 0.05 and 50 pm. As it is the largest flaws which would be expected to cause interface failure, the flaw half-length a was taken as 50 pm. Figure 7 shows that such flaws can grow if the interracial fracture energy, R, is less than about 200 J m -~ for the experimental condition where the wedging displacement was 80 pm. This corresponds to the interface having a fracture energy less than 54% of that of the matrix (the measured fracture energy of bulk PMMA was 372 J m -2) and is in reasonable agreement with the experimental value of 60%. Figure 7 also shows the crack driving forces evaluated using the J-integral as described in Appendix A2. It can be seen that there is reasonably good agreement between the J-integral and the prediction from equation (3) down to an interfacial defect size of about 50pm at which the peak J-integral values is 191 J m -2. However, at defect sizes less than 50 pm, the finite element analysis predicts that the crack driving force should decrease in an almost linear manner. Growth of the interfacial defects will occur provided that the driving force is greater than the fracture energy. This implies that the interface must contain defects larger than ac in Fig. 8 for an interface with fracture energy R~ as shown schematically in Fig. 8. If the fracture energy of the interface
LEE et al. CRACK DEFLECTION AND TOUGHENING 3911 200 K-field E Solution Beam Bending Model 100 Bending Model FEM 0 5000 10000050100 Half Debond Length (um) Fig. 7.(a)Comparison of the crack driving forces obtained from global model Equation(3)] and FEM J-integral)and(b)the details of the short defect regime is higher than the peak, denoted as mat in Fig. 8, the primary crack and the interface defect or debond crack deflection will not occur. Then the critical crackcrack(see Section 5) deflection condition will be the ratio of this crack Figure 7 suggests that the crack driving force for resistance to the fracture energy of the matrix Rm, i.e. an interfacial defect can be explained in terms of two R/R, which, in this case, is 0.51. This value is only regimes, depending on defect or crack length. When valid for the specific condition where the wedge the crack length is long, it can be explained using the displacement is 80 um and it might be expected that compliance method Figure A2 shows the changes in increasing the wedge displacement will result in a compliance for wedge loading: the compliance varies higher ratio, as it will increase the crack driving force in a linear manner in the regime of interest. For a at the interface defect. However, an upper limit to fixed grip case, the crack driving force can be this ratio is obtained from the consideration of the expressed in terms of the compliance, C,as path stability of the interface debond crack (see Section 4.2)and the path stability of the T-shaped crack formed as a result of the interaction between 9 max An Equilibrium og Araa〓s ac= Critical Defect Size af= Final Crack Lengt Fig. 8. Schematic illustration of the critical defect size required for the interface debonding ahead of the he final debond length for a given interface fracture energy
LEE et al.: CRACK DEFLECTION AND TOUGHENING 3911 s- 200 200 l E150~ ~ / ~ieultidon E I~ ~ I/ Beam ilOOI .__ ~ Beam Bending Model ~J .~100~-/ / Model Bending -~ I/~--- FEM OI ~ I , , I , I , I, I , I , I , I , I 01 h I 0 5000 10000 0 50 100 Half Debond Length (llm) (a) (b) Fig. 7. (a) Comparison of the crack driving forces obtained from global model [equation (3)] and FEM (J-integral) and (b) the details of the short defect regime. is higher than the peak, denoted as fqm,x in Fig. 8, crack deflection will not occur. Then the critical crack deflection condition will be the ratio of this crack resistance to the fracture energy of the matrix R~, i.e. R/Rm, which, in this case, is 0.51. This value is only valid for the specific condition where the wedge displacement is 80/~m and it might be expected that increasing the wedge displacement will result in a higher ratio, as it will increase the crack driving force at the interface defect. However, an upper limit to this ratio is obtained from the consideration of the path stability of the interface debond crack (see Section 4.2) and the path stability of the T-shaped crack formed as a result of the interaction between the primary crack and the interface defect or debond crack (see Section 5). Figure 7 suggests that the crack driving force for an interfacial defect can be explained in terms of two regimes, depending on defect or crack length. When the crack length is long, it can be explained using the compliance method. Figure A2 shows the changes in compliance for wedge loading: the compliance varies in a linear manner in the regime of interest. For a fixed grip case, the crack driving force can be expressed in terms of the compliance, C, as 1 u~dC ~J'- 2 C: da" (8) | : / \ ae= Equilibrium ~/Area~ ~J Crack Length .............. ..... 'i ............ R, ac = Critical Defect Siz Crac. Half Debond Length Fig. 8. Schematic illustration of the critical defect size required for the interface debonding ahead of the primary crack and the final debond length for a given interface fracture energy