cfa mater..Vol.44.No.10.pp.3903922.199 Pergamon Copynght ['1996 Acta Metallurgie PIIS1359645496)00087 GROWTH OF INTERFACE DEFECTS AND ITS EFFECT ON CRACK DEFLECTION AND TOUGHENING CRITERIA W. LEE, S J. HOWARD and w.J. CLEGG Department of Materials Science, University of Cambridge. Pembroke Street Cambridge CB 3QZ England (Receied 29 June 1995: in revised form 31 January1996 Abstract-The role of interface defects in crack deflection at planar interfaces direct observation of interface and crack growth in laminates of poly( methyl ate)and a finite element analysis. It was observed that crack deflection occurred by the growth ce defects ahead of the growing primary crack. It was also observed that the growth of these def sure toughening through crack deflection, but that a further condition, that any interface efects must not kink out of the interface, must be also satisfied. A finite element an observed deflection mechanism suggests that initial crack deflection is possible when fracture energy is less than 64% of that of the bulk material, which is consistent with experimental lowever he analysis also predicts that this is somewhat dependent on layer thicknesses and gl ing states for relatively longer interface defects. Copyright C 1996 Acta Metallurgica ine 1 INTRODUCTION bulk material acting normal to the direction of crack extension. In this case it was predicted that the It is well known that the presence of weak interfaces interface must have a strength less than about 0.35 ransverse to a crack growing in a brittle material that of the matrix where there was no elast causes the crack to be deflected with a consequent mismatch to ensure crack deflection increase in the resistance to crack growth [1]. In Kendall 7] suggested that, as there was no driving certain cases the onset of deflection may also be force for the opening of an interface ahead of the required if other energy absorbing mechanisms, such as fibre pull-out are to operate[2, 3]. The conditions n crack, crack deflection could only occur by the for a crack to be deflected at an interface were first primary through-thickness crack changing its path once it had reached the interface Crack deflection investigated by Cook and Gordon [4]. They found was assumed to occur when the force required to that the stress component opening an elliptical shape crack.aw, has a very high value at the crack tip grow the interfacial crack was less than that required to grow the crack across the interface. An energy and decreases with distance from the crack tip based analysis predicted that crack deflection would whilst the stress component acting perpendicular to then occur if the fracture energy of the interface is less the interface. Or is zero at the crack tip but rises to than about 10-20% of the bulk the exact value a maximum at one crack tip radius from the crack tip depending on the relative thickness of the two layers and then decreases The ratio of the peak value of ox which was supported by model experiments.More to o, is 1/5 and from this it was inferred that an recently the problem has been analysed extensively by interface with a theoretical tensile strength of less He and Hutchinson (8)and Martinez and Gupta[9] than 1/5 that of the matrix will debond ahead of the for the case where a primary crack terminates at an interface between two semi-infinite planes. Crack However when a crack with a sharp tip is deflection was assumed to occur when ,R:i.e.when have the same value and decrease monotonically with distance from the crack tip at the same rate [5]. For his case, Gupta et al. [6 proposed that the deflection R (2) criterion would then b where sa is the strain energy release rate of the (I)deflected crack, s, is that of the penetrating crack, R, is the fracture energy of the interface and Rm is that where o is the interface strength, omf is the strength of the bulk material beneath the interface, since then of the bulk material, ax(0") is the stress acting the condition for propagation in the interface will be ormal to the interface and o (90 )is the stress in the met at a lower applied load than that for penetration
~ Pergamon PIi S 1359-6454(96)00068-7 Acta mater. Vol. 44. No. 10. pp. 3905-3922. 1996 Copyright ( 1996 Acta Metallurgica Inc. Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 1359-6454'96 $15.00 + 0.00 GROWTH OF INTERFACE DEFECTS AND ITS EFFECT ON CRACK DEFLECTION AND TOUGHENING CRITERIA W. LEE, S. J. HOWARD and W. J. CLEGG Department of Materials Science, University of Cambridge, Pembroke Street. Cambridge CB2 3QZ, England (Received 29 June 1995; in revised form 31 Januao' 1996) Abstract--The role of interface defects in crack deflection at planar interfaces has been studied by the direct observation of interface and crack growth in laminates of poly(methyl methacrylate) and a finite element analysis. It was observed that crack deflection occurred by the growth of interface defects ahead of the growing primary crack. It was also observed that the growth of these defects is not sufficient to ensure toughening through crack deflection, but that a further condition, that any growing interface defects must not kink out of the interface, must be also satisfied. A finite element analysis based on the observed deflection mechanism suggests that initial crack deflection is possible when the interfacial fracture energy is less than 64% of that of the bulk material, which is consistent with experimental values. However, the analysis also predicts that this is somewhat dependent on layer thicknesses and global loading states for relatively longer interface defects. Copyright © 1996 Acta Metallurgica In('. l. INTRODUCTION It is well known that the presence of weak interfaces transverse to a crack growing in a brittle material causes the crack to be deflected with a consequent increase in the resistance to crack growth [1]. In certain cases the onset of deflection may also be required if other energy absorbing mechanisms, such as fibre pull-out, are to operate [2, 3]. The conditions for a crack to be deflected at an interface were first investigated by Cook and Gordon [4]. They found that the stress component opening an elliptical shape crack, a,,, has a very high value at the crack tip and decreases with distance from the crack tip, whilst the stress component acting perpendicular to the interface, a ..... is zero at the crack tip but rises to a maximum at one crack tip radius from the crack tip and then decreases. The ratio of the peak value of a.,.~ to a, is 1/5 and from this it was inferred that an interface with a theoretical tensile strength of less than 1/5 that of the matrix will debond ahead of the main crack allowing crack deflection to occur. However when a crack with a sharp tip is considered, both tr~ and o,, in the plane of the crack have the same value and decrease monotonically with distance from the crack tip at the same rate [5]. For this case, Gupta et al. [6] proposed that the deflection criterion would then be a~ a~x(O °) R~: i.e. when Ri Cffd < (2) Rm ffp where f#a is the strain energy release rate of the deflected crack, (~ is that of the penetrating crack, R~ is the fracture energy of the interface and Rm is that of the bulk material beneath the interface, since then the condition for propagation in the interface will be met at a lower applied load than that for penetration 3905
Lee et al. CRACK DEFLECTION AND TOUGHENING across the interface. thus the critical condition is met 2. EXPERIMENTAL when u p=R/R. This approach gave a critical S,/s, ratio of 0.26 for a doubly defected crack [9] Specimens containing planar interfaces with a and 0.25 for a singly deflected crack [8]when there is range of diferent properties were prepared by no elastic modulus mismatch across the interface. pressing together sheets of poly(methyl methacry This result is in good agreement with early late), PMMA, at temperatures above the glass late nmental work [7] and has been confirmed by transition temperature. Sheets, approx, 150 mm later experimentation [10]. Gupta et al. [6] and square and 5 mm thick, were washed in soap an Martinez and Gupta [9 considered the effect of water and then briefly rinsed in acetone and then elastic orthotropy and He et al. [11] extended He water again before drying with purified compressed and Hutchinsons original analysis by considering air. The sheets were then pressed together in a the effect of residual stress. These analyses showed pre- heated press at temperatures between 110 and that critical crack deflection criterion can be 140C at a pressure of 1. 5 MPa for 30 min before changed due to elastic orthotropy and to the cooling down to room temperature under pressure presence of residual stresses. Numerical method gave The pressed blocks, with a final thickness of 8 mm, almost identical solutions to the analytical solutions were then cut into test specimens approx. 150 mm long and 20 mm wide. The cut edges were then Most of the theories essentially consider whether polished to allow observation of the crack as it he driving force for the crack to grow along the approached the interface. Starter notches, 1.75 mm interface will exceed the fracture energy of the deep and 0. 34 mm wide, were cut into the specimens interface, before the driving force for the primary using a diamond impregnated wire. crack exceeds the fracture energy of the matrix. Some preliminary tests were carried out in tension The conditions under which deflection will occur and in four point bending, but growth of the crack are therefore those in which the primary crack is from the notch occurred so rapidly that it was stationary and terminated at the interface. Such an difficult to observe the crack as it was deflected at the analysis will predict in which direction the crack will interface. Only when the notches were cut close to the grow as the applied force on the material is increased interface, approx. 500 um above the interface, was it from zero. However, this is not always the situation observed that the interface debonded before the main in most real systems, where the applied force is such crack grew from the notch. However, it was not clear that the primary crack is moving, often in an unstable whether the same(debonding of the interface before manner. Therefore, it is not clear that the the formation of a T-shaped crack)would occur if the consideration of the path of such a stationary crack crack had a sharp tip. Therefore it was necessary to at an interface provides a general criterion for crack observe what would happen when a sharp crack deflection grows toward the interface. Many of the loading In addition all the work described above considers geometries designed for slow crack growth require only homogeneous interfaces with low fracture special apparatus or complex specimen geometry and is not unusual for structural solids, analysis (19-22]. Instead a wedge loading method including ceramics, to contain defects especially shown in Fig. I was used. A wedge 0.5 mm thick, was at interphase boundaries [3, 13-17]. Considering driven to the root of the notch with the base of the this structural effect, recently Mammoli et al. [18] specimen completely supported. This caused a very und that the presence of defects in the interface small crack to be initiated at the notch-tip. the load can increase the critical R /Re ratio for crack normal to the interface was then slowly removed deflection allowing the specimen to bend and the crack to grow Despite this work it is still not clear how a crack By controlling the rate of this unloading, crack is defected at the interface and how other factors growth rates as low as 10 um s- could be obtained influence the crack deflection criteria. The first aim of crack and the interface to be observed easily Whi such as global loading states and layer thicknesses allowing the interaction between a growing prima this paper is therefore to observe how growing cracks this enabled slow crack growth to be observed, it was are deflected at interfaces and the structure of the not possible to measure the load applied by the interface. Subsequently, a finite element analysis, wedge. Instead this was estimated using finite based on the experimental observations of the crack element analysis, where the displacement of the notch deflection process and of the interface structure, faces was taken to be the difference between the is carried out to establish the properties that thickness of the wedge and the width of the notch, interfaces must have if deflection is to occur as a which is 80 um in this case. Figure A2 in the consequence of the interaction between a primary Appendix shows how the wedging load is predicted crack and an interface defect of various lengths. to vary with displacement for different interfacia Finally, the effects of the global stress state and the crack lengths geometry of the specimen on the criterion for crack Experiments on specimens of the same dimensions deflection and on the subsequent toughening containing no interface were carried out and showed behaviour are discussed that under the conditions used here, the crack would
3906 LEE et al.: CRACK DEFLECTION AND TOUGHENING across the interface. Thus the critical condition is met when fq~/f#p = R,/Rm. This approach gave a critical f#~/f#p ratio of 0.26 for a doubly deflected crack [9] and 0.25 for a singly deflected crack [8] when there is no elastic modulus mismatch across the interface. This result is in good agreement with early experimental work [7] and has been confirmed by later experimentation [10]. Gupta et al. [6] and Martinez and Gupta [9] considered the effect of elastic orthotropy and He et al. [11] extended He and Hutchinson's original analysis by considering the effect of residual stress. These analyses showed that critical crack deflection criterion can be changed due to elastic orthotropy and to the presence of residual stresses. Numerical method gave almost identical solutions to the analytical solutions [121. Most of the theories essentially consider whether the driving force for the crack to grow along the interface will exceed the fracture energy of the interface, before the driving force for the primary crack exceeds the fracture energy of the matrix. The conditions under which deflection will occur are therefore those in which the primary crack is stationary and terminated at the interface. Such an analysis will predict in which direction the crack will grow as the applied force on the material is increased from zero. However, this is not always the situation in most real systems, where the applied force is such that the primary crack is moving, often in an unstable manner. Therefore, it is not clear that the consideration of the path of such a stationary crack at an interface provides a general criterion for crack deflection. In addition all the work described above considers only homogeneous interfaces with low fracture energy. It is not unusual for structural solids, including ceramics, to contain defects especially at interphase boundaries [3, 13-17]. Considering this structural effect, recently Mammoli et al. [18] found that the presence of defects in the interface can increase the critical RJRm ratio for crack deflection. Despite this work it is still not clear how a crack is deflected at the interface and how other factors such as global loading states and layer thicknesses influence the crack deflection criteria. The first aim of this paper is therefore to observe how growing cracks are deflected at interfaces and the structure of the interface. Subsequently, a finite element analysis, based on the experimental observations of the crack deflection process and of the interface structure, is carried out to establish the properties that interfaces must have if deflection is to occur as a consequence of the interaction between a primary crack and an interface defect of various lengths. Finally, the effects of the global stress state and the geometry of the specimen on the criterion for crack deflection and on the subsequent toughening behaviour are discussed. 2. EXPERIMENTAL Specimens containing planar interfaces with a range of different properties were prepared by pressing together sheets of poly(methyl methacrylate), PMMA, at temperatures above the glass transition temperature. Sheets, approx. 150mm square and 5 mm thick, were washed in soap and water and then briefly rinsed in acetone and then water again before drying with purified compressed air. The sheets were then pressed together in a pre-heated press at temperatures between 110 and 140°C at a pressure of 1.5 MPa for 30 min before cooling down to room temperature under pressure. The pressed blocks, with a final thickness of 8 mm, were then cut into test specimens approx. 150 mm long and 20 mm wide. The cut edges were then polished to allow observation of the crack as it approached the interface. Starter notches, 1.75 mm deep and 0.34 mm wide, were cut into the specimens using a diamond impregnated wire. Som e preliminary tests were carried out in tension and in four point bending, but growth of the crack from the notch occurred so rapidly that it was difficult to observe the crack as it was deflected at the interface. Only when the notches were cut close to the interface, approx. 500/tm above the interface, was it observed that the interface debonded before the main crack grew from the notch. However, it was not clear whether the same (debonding of the interface before the formation of a T-shaped crack) would occur if the crack had a sharp tip. Therefore it was necessary to observe what would happen when a sharp crack grows toward the interface. Many of the loading geometries designed for slow crack growth require special apparatus or complex specimen geometry and analysis [19-22]. Instead a wedge loading method shown in Fig. 1 was used. A wedge, 0.5 mm thick, was driven to the root of the notch with the base of the specimen completely supported. This caused a very small crack to be initiated at the notch-tip. The load normal to the interface was then slowly removed, allowing the specimen to bend and the crack to grow. By controlling the rate of this unloading, crack growth rates as low as 10/am s -~ could be obtained, allowing the interaction between a growing primary crack and the interface to be observed easily. Whilst this enabled slow crack growth to be observed, it was not possible to measure the load applied by the wedge. Instead, this was estimated using finite element analysis, where the displacement of the notch faces was taken to be the difference between the thickness of the wedge and the width of the notch, which is 80#m in this case. Figure A2 in the Appendix shows how the wedging load is predicted to vary with displacement for different interfacial crack lengths. Experiments on specimens of the same dimensions containing no interface were carried out and showed that, under the conditions used here, the crack would
LEE et al. CRACK DEFLECTION AND TOUGHENING Interface at the notch root Knife Wedge Rigid si Rigid Supt Unloading by the Remova of the Rigid Support an Further crack Growth Fig. I. Schematic illustration of the wedge loading used to obtain slow crack growth grow across about 70% of the full thickness of the tures. In this case, no further growth of the primary mpie. Thus, there was sufficient driving force to crack was observed suggesting that the growth of the ensure that a crack would grow across any interface interface debond crack had absorbed strain energy i present in the mid-plane of a specimen. The preference to the primary crack terfacial fracture energies of the specimens, where In the specimens pressed at intermediate tempera- rack deflection occurred on wedge loading, were tures shorter interface debond cracks were observed ubsequently measured using the four point bend The primary crack then grew and eventually it was delamination test (23). In those specimens where the arrested by the short interface debond crack to form primary crack stopped before it reached the interface a T-shaped crack. In this case the defected crack debond crack, further loading was applied to typically kinked into the matrix when further loading completely break the cracked layer and to make was applied in bending as shown in Fig 3.Whilst the tarter notches for the four point bend delamination length of the interface debond crack became shorter test. The fracture energy of bulk PMMA was as the interface became tougher, half debond lengths measured as 372 J m: using notched specimens shorter than about 500 um were not observed tested in three-point bending [24]. The Young's The interface fracture energies of the specimens modulus was measured as 3.0 GPa using an tested in wedge loading were subsequently measured unnotched specimen in four point bending. The using the four point bend delamination test. By interfacial structure of the PMMA laminates was also relating the interface fracture energy measured in a observed using scanning electron microscopy four point bend test to the debon during wedge loading, the trend shown in Fig. 4 was 3. OBSERVATIONS OF CRACK DEFLECTION obtained. The interfacial fracture energies of the specimens having a half interface debond length less When wedge loading was applied to the specimens, than about 750 um could not be measured using the crack deflection was not always observed. In such four point bend test since the interface debond crack cases. the primary through-thickness crack grew kinked into the matrix after propagating a very short across the interfaces, which were pressed at 140 C In distance. The implications of this are discussed in pecimens pressed at lower temperatures, the growth Section 5. Figure 4 suggests that the initial deflection of the primary crack towards the interface caused the of the primary crack at the interface is still possible interface to delaminate ahead of the growing crack, when the fracture energy of the interface is as high as as shown in Fig. 2. The distance between the tip of about 60% of that of the matrix the growing crack and the interface was somewhat Scanning electron microscopy (SEM)of variable but was generally of the order of 50 interface showed that it was not completely un Once initiated the interfacial crack grew extremely but contained some unbonded regions with total rapidly in the specimens pressed at lower tempera- lengths varying between 0. 1 and 100 um. Figure 5
® Interface LEE et al.: F CRACK DEFLECTION AND TOUGHENING F Knife Wedge Notch )ot 3907 PMMA model composite Rigid Support Rigid Support Unloading by the Removal of the Rigid Support and Further Crack Growth Fig. 1. Schematic illustration of the wedge loading used to obtain slow crack growth. grow across about 70% of the full thickness of the sample. Thus, there was sufficient driving force to ensure that a crack would grow across any interface present in the mid-plane of a specimen. The interfacial fracture energies of the specimens, where crack deflection occurred on wedge loading, were subsequently measured using the four point bend delamination test [23]. In those specimens where the primary crack stopped before it reached the interface debond crack, further loading was applied to completely break the cracked layer and to make starter notches for the four point bend delamination test. The fracture energy of bulk PMMA was measured as 372Jm-: using notched specimens tested in three-point bending [24]. The Young's modulus was measured as 3.0GPa using an unnotched specimen in four point bending. The interfacial structure of the PMMA laminates was also observed using scanning electron microscopy. 3. OBSERVATIONS OF CRACK DEFLECTION When wedge loading was applied to the specimens, crack deflection was not always observed. In such cases, the primary through-thickness crack grew across the interfaces, which were pressed at 140°C. In specimens pressed at lower temperatures, the growth of the primary crack towards the interface caused the interface to delaminate ahead of the growing crack, as shown in Fig. 2. The distance between the tip of the growing crack and the interface was somewhat variable but was generally of the order of 50#m. Once initiated the interfacial crack grew extremely rapidly in the specimens pressed at lower temperatures. In this case, no further growth of the primary crack was observed suggesting that the growth of the interface debond crack had absorbed strain energy in preference to the primary crack. In the specimens pressed at intermediate temperatures, shorter interface debond cracks were observed. The primary crack then grew and eventually it was arrested by the short interface debond crack to form a T-shaped crack. In this case the deflected crack typically kinked into the matrix when further loading was applied in bending as shown in Fig. 3. Whilst the length of the interface debond crack became shorter as the interface became tougher, half debond lengths shorter than about 500 #m were not observed. The interface fracture energies of the specimens tested in wedge loading were subsequently measured using the four point bend delamination test. By relating the interface fracture energy measured in a four point bend test to the debond length measured during wedge loading, the trend shown in Fig. 4 was obtained. The interfacial fracture energies of the specimens having a half interface debond length less than about 750/tm could not be measured using the four point bend test since the interface debond crack kinked into the matrix after propagating a very short distance. The implications of this are discussed in Section 5. Figure 4 suggests that the initial deflection of the primary crack at the interface is still possible when the fracture energy of the interface is as high as about 60% of that of the matrix. Scanning electron microscopy (SEM) of the interface showed that it was not completely uniform but contained some unbonded regions with total lengths varying between 0.1 and 100#m. Figure 5
LEE er al.: CRACK DEFLECTION AND TOUGHENING Fig. 2. Photograph showing crack deflection which occurred by the debonding of the interface ahead of the growing primary crack shows a typical interface structure in the PMma the matrix fracture energy. The suggested mechanism laminates used in this work. by which crack deflection occurs is also supported by The experimental observations are supported by the recent experimental observation of a Zro: Al O the earlier observations of Phillipps et al. [25] laminate system[26] and SiC/ C laminate system [10] crack deflection in Sic C laminates occurred where the interface defects grew before the primary the interface fracture energy was measured at 56% of crack reached the interface Fig 3. Photograph showing the formation of a T-shaped crack and the subsequent kinking of the crack t of the interface at the initial stage of crack deflection
3908 LEE et al.: CRACK DEFLECTION AND TOUGHENING Fig. 2. Photograph showing crack deflection which occurred by the debonding of the interface ahead of the growing primary crack. shows a typical interface structure in the PMMA laminates used in this work. The experimental observations are supported by the earlier observations of Phillipps et al. [25] where crack deflection in SiC/C laminates occurred when the interface fracture energy was measured at 56% of the matrix fracture energy. The suggested mechanism by which crack deflection occurs is also supported by the recent experimental observation of a ZrO:/AbO~ laminate system [26] and SiC/C laminate system [10] where the interface defects grew before the primary crack reached the interface. Fig. 3. Photograph showing the formation of a T-shaped crack and the subsequent kinking of the crack out of the interface at the initial stage of crack deflection
LEE el al. CRACK DEFLECTION AND TOUGHENING by whether the observed interfacial defect can grow rather than whether a primary crack can change its path upon reaching a weak defect-free interface. To The shortest half- debond investigate this point further. consider the beam length observed (500 um shown in Fig. 6, which is loaded by a combination of axial forces and bending moments. Note that the axial force P, and bending moment M, acting in beam I are not zero as there is a small bridging ligament between the primary crack and the interface and that only half the whole geometry is considered due to symmetry. Using beam theory it can be shown that the driving force for the growth of the interfacia crack is [27] Half Debond Length(mm) °5++盒 Fig. 4. The relation between the half-debond length of the interface and the fracture energy of the interface(measured separately from the same specimen). Note that half-debond where b is the width of beam. E is Youngs dicates that crack deflection is not possible when the modulus, A is the cross-sectional area of the beam fracture energy of the interface is higher than about 60% of and I is the second moment of area of the beam the matrix for this loading arrangement and which is bh:12 specimen geometry. In the specific case considered in the experime hown in Fig. 6(b). from force and moment equilibrium conditions, the force and moment GROWTH OF AN INTERFACE DEFECT AHEAD OF components for equation (3)are THE PRIMARY CRACK PI=P.-Pn 4.1. The origin of the driring force for the growth of the interfacial defects ) The observed crack deflection process suggests that there is a physical mechanism which produces a driving force for interfacial crack growth before the M=Pmn-2)+几 rimary crack reaches the interface. It is suggested that, when defects exist in the interface. the condition 十 for the occurrence of crack deflection is determined E=5. 1 Fn K y Fig. 5. SEM photograph showing a typical interface structure including the defects
E 0.8 0.6 LEE et al.: CRACK DEFLECTION AND TOUGHENING The shortest half-debond length observed (500 gm) 0.2 0 , , J ~ , ~ I , ~ : T , ~ , I J , ~ I 0 2 4 6 8 10 Half Debond Length (mm) Fig. 4. The relation between the half-debond length of the interface and the fracture energy of the interface (measured separately from the same specimen). Note that half-debond lengths shorter than 500 pm were not observed, which indicates that crack deflection is not possible when the fracture energy of the interface is higher than about 60% of that of the matrix for this loading arrangement and specimen geometry. 3909 4. GROWTH OF AN INTERFACE DEFECT AHEAD OF THE PRIMARY CRACK 4.1. The origin of the driving .force for the growth of the interfacial defects The observed crack deflection process suggests that there is a physical mechanism which produces a driving force for interfacial crack growth before the primary crack reaches the interface. It is suggested that, when defects exist in the interface, the condition for the occurrence of crack deflection is determined by whether the observed interfacial defect can grow, rather than whether a primary crack can change its path upon reaching a weak defect-free interface. To investigate this point further, consider the beam shown in Fig. 6, which is loaded by a combination of axial forces and bending moments. Note that the axial force P~ and bending moment M~ acting in beam 1 are not zero as there is a small bridging ligament between the primary crack and the interface and that only half the whole geometry is considered due to symmetry. Using beam theory, it can be shown that the driving force for the growth of the interfacial defect or crack is [27] 1[ P~ M~ + P~ M~ P~ M~ 1 (qi= ~IA~ + E,L ~-A,.+ E,.I ,_ E3A~ (3) where b is the width of the beam, E is Young's modulus, A is the cross-sectional area of the beam and I is the second moment of area of the beam, which is bh~;12. In the specific case considered in the experiment shown in Fig. 6(b), from force and moment equilibrium conditions, the force and moment components for equation (3) are PI = P~- Pb (4) P-" = (P,, - Pb) (5) f'h = " M,. (7) Fig. 5. SEM photograph showing a typical interface structure including the defects
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
LEE et al. CRACK DEFLECTION AND TOUGHENING Thus, a linear compliance crack length relation through-thickness crack and the interface is equal to means that the crack driving force is inversely the half length of the flaw proportional to the square of the crack length e above analysis predicts that as an interfacial For short interface defects located near the tip of flaw grows, the crack driving force. s, increase the primary crack, it is assumed that the crack driving before decreasing. Thus once an interfacial defect force is mainly influenced by the stress field starts to grow, when ,=R; it could continue to do associated with the crack. A further so until the crack dr assumption can be made such that the existence of a to the fracture energy of the interface (Fig. 8) small defect ahead of the tip of the primary crack will However, not all of the available elastic energy will not alter the stress field of the primary crack. In this be used to drive the crack to the equilibrium position case, using Bueckner's principle [29]. the stress so that the remaining energy, described by the are intensity factor for the interfacial defect can be shown under the curve of and above the line R,, must be either dissipated through the loading mechanism or stored in the sample. Experiments on SiC/C K=1 2o(a dx laminates have shown that a substantial fraction of this excess elastic energy can be stored and then used drive interface cracks further than the equilibrium position [30] A first estimate at an upper bound to this amount ()the excess energy is used to drive interfacial crack growth Figure 9 shows the variation of the observed When the length of the defect is much smaller than and predicted defect crack lengths against interfacial the distance between itself and the tip of the primary toughness, Ri, when this effect is considered. It can crack, it can be assumed that the stress acting near the be seen that the debond length is unde defect does not change significantly and therefore the by the analysis suggested above, particularly for stress a,(x) is approximately constant along the line tough interfaces or short debond lengths. a possibl of the defect. In this case. the integrals in equation explanation for this is that there is additional excess (9), are an ra and o ia respectively. The contribution strain energy associated with the unstable propa of shear stress is negligible in this case since o, <a gation of the primary crack before the interfacial [5]. The stress intensity factor is then nyna. Thus defect grows. This additional energy may contribute proportional to the square root of the length of the to the growth of the interfacial defect, with varying defect and the crack driving force, g, will be efficiency with respect to initial flaw length, as the proportional to the length of the defect. Thus a linear interfacial defect extends. dependence of the crack driving force on the crack Similar analyses have been repeated for other length is expected for short defects [see Fig. 7(b)]. loading states such as bending and tension. Using the So far it has been assumed that the distance same specimen geometries and properties as in the between the tip of the growing primary crack and the wedge loading case, the variation of the crack driving interface was 50 um. Varying this showed that the forces with the interface defect half-length were peak in the crack driving force is reached when the predicted for three- point and four-point bending by ratio of the distance between the tip of the primary FEM and by equation (3), as shown in Fig 10. The crack and the interface to the half-length of the faw, loading was applied in such a manner that it produces dj a, is equal to unity. Further. the values of these the same K at the tip of the primary crack. It can be peak crack driving forces are not particularly seen that whilst the crack driving forces are dependent on the flaw length as shown in Table 1, substantially different for long cracks, the peak values explaining the relative independence of the observed for three-point and four-point bending are close to distance between the primary crack and the interface those obtained for wedge loading. The crack driving on the fracture energy of the interface. This means force for a very short defect(a 50 um)develops in that, if the interface has the highest possible a similar way as explained with equation (9). Thi toughness that will allow crack deflection, interfacial indicates that once the defect size is smaller than debonding will occur when the distance between the certain length scale. i. e. K-field, the driving force influenced by the global loading state, at least for the Table 1. variation of the peak crack driving force for a d-I for loading arrangements considered so far. This is also different loading true of the variation of the phase angle of loading Loading state Crack driving force(Jm-) quation(10)) as shown in Fig. Il. The phase angles ng, tension and three -and four-pe End-fixed tension 179 175 172 166 157 14 187 186 180 with half-lengths up to 25 um (d= 50 um and h=4 mm
3912 LEE et al.: CRACK DEFLECTION AND TOUGHENING Thus, a linear compliance/crack length relation means that the crack driving force is inversely proportional to the square of the crack length. -For short interface defects located near the tip of the primary crack, it is assumed that the crack driving force is mainly influenced by the stress field associated with the primary crack. A further assumption can be made such that the existence of a small defect ahead of the tip of the primary crack will not alter the stress field of the primary crack. In this case, using Bueckner's principle [29], the stress intensity factor for the interfacial defect can be shown to be [24] and K~ = 1 ~ 2a,,(x)a dx W/n'-~ J0 x//a 2 - x "~ KI, = x/rt~--~l f° ~j~72 2a,,(x)a dx. (9) When the length of the defect is much smaller than the distance between itself and the tip of the primary crack, it can be assumed that the stress acting near the defect does not change significantly and therefore the stress a.(x) is approximately constant along the line of the defect. In this case. the integrals in equation (9), are ~r,:,rra and a.zra respectively. The contribution of shear stress is negligible in this case since ~r v << a.. [5]. The stress intensity factor is then or. x/~. Thus proportional to the square root of the length of the defect and the crack driving force, ff~ will be proportional to the length of the defect. Thus a linear dependence of the crack driving force on the crack length is expected for short defects [see Fig. 7(b)]. So far it has been assumed that the distance between the tip of the growing primary crack and the interface was 50 ~m. Varying this showed that the peak in the crack driving force is reached when the ratio of the distance between the tip of the primary crack and the interface to the half-length of the flaw, d/a, is equal to unity. Further, the values of these peak crack driving forces are not particularly dependent on the flaw length as shown in Table 1, explaining the relative independence of the observed distance between the primary crack and the interface on the fracture energy of the interface. This means that, if the interface has the highest possible toughness that will allow crack deflection, interfacial debonding will occur when the distance between the Table 1. Variation of the peak crack driving force for a/d = 1 for different loading states and interracial flaw lengths Crack driving force (J m--') Loading state Flaw length (,urn) 1 5 10 25 50 100 Wedging 178 181 188 187 192 195 Four-point bending 182 185 186 187 186 180 End-fixed tension 179 175 172 166 157 143 through-thickness crack and the interface is equal to the half length of the flaw. The above analysis predicts that as an interfacial flaw grows, the crack driving force, f#, increases before decreasing. Thus once an interfacial defect starts to grow, when ~ = Rj it could continue to do so until the crack driving force again becomes equal to the fracture energy of the interface (Fig. 8). However, not all of the available elastic energy will be used to drive the crack to the equilibrium position. so that the remaining energy, described by the area under the curve of if, and above the line R, must be either dissipated through the loading mechanism or stored in the sample. Experiments on SiC/C laminates have shown that a substantial fraction of this excess elastic energy can be stored and then used to drive interface cracks further than the equilibrium position [30]. A first estimate at an upper bound to this amount of crack growth is given by the condition that all of the excess energy is used to drive interfacial crack growth. Figure 9 shows the variation of the observed and predicted defect crack lengths against interfacial toughness, R, when this effect is considered. It can be seen that the debond length is under-estimated by the analysis suggested above, particularly for tough interfaces or short debond lengths. A possible explanation for this is that there is additional excess strain energy associated with the unstable propagation of the primary crack before the interfacial defect grows. This additional energy may contribute to the growth of the interfacial defect, with varying efficiency with respect to initial flaw length, as the interfacial defect extends. Similar analyses have been repeated for other loading states such as bending and tension. Using the same specimen geometries and properties as in the wedge loading case, the variation of the crack driving forces with the interface defect half-length were predicted for three-point and four-point bending by FEM and by equation (3), as shown in Fig. 10. The loading was applied in such a manner that it produces the same K at the tip of the primary crack. It can be seen that whilst the crack driving forces are substantially different for long cracks, the peak values for three-point and four-point bending are close to those obtained for wedge loading. The crack driving force for a very short defect (a < 50 ~m) develops in a similar way as explained with equation (9). This indicates that once the defect size is smaller than a certain length scale, i.e. K-field, the driving force for the growth of these defects is not significantly influenced by the global loading state, at least for the loading arrangements considered so far. This is also true of the variation of the phase angle of loading [equation (10)] as shown in Fig. 11. The phase angles for wedge loading, tension and three- and four-point bending are almost identical for interfacial defects with half-lengths up to 25pm (d= 50pro and h = 4 ram)
LEE el aL. CRACK DEFLECTION AND TOUGHENING 250 Experimental Predicted(FEM) 0 100001500020000 Half Debond Length (um) Fig 9. Modified crack driving force curve in Fig. 7 after the consideration of kinetic effect on crack growth and its comparison with experimental results 4.2,T magnitude of the crack driving forces for the growth of short defects between the wedge loading and It was mentioned in Section 4.1 that any interface bending, the crack driving forces develop in a may deflect primary cracks once it contains defects somewhat different way when uniaxial tension is and there is enough driving force for ts growth applied. Whilst the peak in the crack driving force for However, this is true only when the interface debond d/a= I is not significantly sensitive to the actual size crack does not kink out of the interface and when the of the defects for wedge loading and bending, it primary crack stops with no further growth. It is shows a relatively strong dependence on defect size expected that these two effects will put an upper limit for tension as shown in Table 1. This suggests that the to the critical R Rm ratio which allows crack development of the crack driving force in bending deflection. The path stability problem is described and tension are somewhat different, with the loading here and the growth of the primary crack and in bending being more favourable for crack deflection subsequent interface crack growth are discussed in to occur by the mechanism observed 200 Four Point Bending (Beam Theory) Four Point Bending( FEM) Tension(Beam Theory 100 Three Point ending(FEM) E868巴 Tension(FEM) 00100001500020000 5000 1500020000 Half Debond Length (am) Half Debond Length (um) Fig. 10. Variation of crack driving forces under(a)three-point bending and four-point bending and(b)
LEE et al.: CRACK DEFLECTION AND TOUGHENING 3913 250 E Q) 2 o LL O~ .~LL 0 0 "E 200 150 1 O0 50 • ~ Experimental ~~=m BB ,Predicted (FEM) 0 L i , i I ~ , , , 1 i L , , I , , , , I 0 5000 10000 15000 20000 Half Debond Length (pm) Fig. 9. Modified crack driving force curve in Fig. 7 after the consideration of kinetic effect on crack growth and its comparison with experimental results. Whilst there is no significant difference in the magnitude of the crack driving forces for the growth of short defects between the wedge loading and bending, the crack driving forces develop in a somewhat different way when uniaxial tension is applied. Whilst the peak in the crack driving force for d/a = l is not significantly sensitive to the actual size of the defects for wedge loading and bending, it shows a relatively strong dependence on defect size for tension as shown in Table 1. This suggests that the development of the crack driving force in bending and tension are somewhat different, with the loading in bending being more favourable for crack deflection to occur by the mechanism observed. 4.2. The crack path of the growing interfacial defect It was mentioned in Section 4.1 that any interface may deflect primary cracks once it contains defects and there is enough driving force for its growth. However, this is true only when the interface debond crack does not kink out of the interface and when the primary crack stops with no further growth. It is expected that these two effects will put an upper limit to the critical R/Rm ratio which allows crack deflection. The path stability problem is described here and the growth of the primary crack and subsequent interface crack growth are discussed in Section 5. ,,-. 200 E • 150 o LL 100 > L- a -~ 50 0 200 ,~__ Four Point Bending (Beam Theory) E m 150 ---- Four Point Bending (FEN) \ \\ u_ Three Point .~ 100 a -~ 50 0 i i . , I , , , , I . . , , [~'7 . , I 5000 10000 15000 20000 Half Debond Length (pm) Beam Theory) Tension (FEM) 0 .... I .... I .... i , , , , I 0 5000 10000 15000 20000 Half Debond Length (llm) (a) (b) Fig. 10. Variation of crack driving forces under (a) three-point bending and four-point bending and (b) uniaxial tension
LEE ef al.: CRACK DEFLECTION AND TOUGHENING Tension Four point 0.82 10 094 S&H (Tensio Three Point 10 HH0.94 0.1 10 100 1000 10000 Half Debond Length(um) interface. Markers are predictions by Suo and Hutchinson [28]. The agreement is good for long ons for cracks not propagating in a right-hand y-axis. Pure opening loading is predicted self-similar fashion under mixed mode loading are ery short defects as assumed earlier in well understood [31, 32]. It is well established that a Section 4.1. The loading then deviates away from crack growing in a homogeneous material tends pure opening at short defect lengths. Initially, the towards a mode i path [33]. The mode mixity of a phase angle is negative. A negative y in this case crack is normally characterised by the phase angle, v, simply means that the sense of the shear stress intensity is reversed and that the interfacial crack will tend to kink upwards into the cracked layer. As the crack grows further, the phase angle At the peak phase angles, 15 in wedging and 28 in The conditions required for a crack propagating in an tension, the critical Ri /Ron ratios to prevent crack interface to kink out of the interface have been kinking are about 0.74-0.88. The phase angle then described by He et al. [34]. They have found that for falls slowly to zero. This final variation, whe a given combination of opening and shear loading at a> l mm(h=4 mm), is predicted accurately by the the tip of an interfacial crack, as characterised by the analysis of Suo and Hutchinson [28] when the phase angle t, there is a critical ratio of interfacial appropriate edge loading conditions, determined by and matrix toughnesses, R/Rm, which must not be FEM, are used. The variation observed in this global exceeded if the crack is to remain in the interface. regime seems reasonable, since, by inspection of As the interfacial defect grows, the phase angle of table 2, it can be seen that the fractional difference loading it experiences varies, as shown in Fig. 11. The in the bending moments, Mi and M2, falls as the phase angle was determined by FEM and is shown on interfacial debond length B, suggesting upo the left-hand y-axis. The critical R, /Rm ratios to inspection of equation(3) xial loading of the prevent crack kinking for the range of phase angles specimen, P, and P2, be observed, as predicted by He et al., are shown on the the imposition of equal and opposite bending Table 2. variation of the moment components M, and M] with half-debond lengths(hi h2- 4 mm. d=50;m) Half-debond length 50 um 500 um I mr
3914 LEE et al.: CRACK DEFLECTION AND TOUGHENING 30 0.72 20 0.82 • 10 0.94 -~ #. o 1.o -10 0.94 O. 1 1 10 1 O0 1000 10000 Half Debond Length (l.tm) Fig. I l. Variation of phase angles with the growth of the interface defect ahead of the primary crack under various global loading states and corresponding RURm ratios required to prevent crack kinking out of the interface. Markers are predictions by Suo and Hutchinson [28]. The agreement is good for long crack lengths. The reasons for cracks not propagating in a self-similar fashion under mixed mode loading are well understood [31, 32]. It is well established that a crack growing in a homogeneous material tends towards a mode I path [33]. The mode mixity of a crack is normally characterised by the phase angle, ~b, defined as , [Kn'~ i,D' = tan- t-~'l). (10) The conditions required for a crack propagating in an interface to kink out of the interface have been described by He et al. [34]. They have found that for a given combination of opening and shear loading at the tip of an interracial crack, as characterised by the phase angle @, there is a critical ratio of interfacial and matrix toughnesses, RURm, which must not be exceeded if the crack is to remain in the interface. As the interracial defect grows, the phase angle of loading it experiences varies, as shown in Fig. I I. The phase angle was determined by FEM and is shown on the left-hand y-axis. The critical RURm ratios to prevent crack kinking for the range of phase angles observed, as predicted by He et al., are shown on the right-hand y-axis. Pure opening loading is predicted for very short defects as assumed earlier in Section 4.1. The loading then deviates away from pure opening at short defect lengths. Initially, the phase angle is negative. A negative ~, in this case simply means that the sense of the shear stress intensity is reversed and that the interfacial crack will tend to kink upwards into the cracked layer. As the crack grows further, the phase angle increases, passing through zero, and rises to a peak. At the peak phase angles, 15 ° in wedging and 28 ° in tension, the critical RURm ratios to prevent crack kinking are about 0.74-0.88. The phase angle then falls slowly to zero. This final variation, where a > 1 mm (/7 = 4 mm), is predicted accurately by the analysis of Suo and Hutchinson [28] when the appropriate edge loading conditions, determined by FEM, are used. The variation observed in this global regime seems reasonable, since, by inspection of Table 2, it can be seen that the fractional difference in the bending moments, M~ and M2, falls as the interfacial debond length increases, suggesting upon inspection of equation (3) that the axial loading of the specimen, P, and P2, becomes less significant. Since the imposition of equal and opposite bending Table 2. Variation of the moment components M~ and M2 with half-debond lengths (h~ = h2 = 4 ram, d = 50/~m) Half-debond length 50 #~m 500 #~m I mm 2 mm 4 mm 8 mm Wedging M~ (Nm) 0.8205 0.6930 0.6440 0.5604 0.4389 0.3152 M: (Nm) 1.451 1.315 1.128 0.8648 0.5905 0.3753 Four-point M~ (Nm) 0.2971 0.2559 0.2907 0.3399 0.3749 0.3624 bending Mz (Nm) 1.172 1.685 1.574 1.393 1.150 0.8757 End-fixed Mj (Nm) 0.3082 0.2728 0.3139 0.3704 0.4163 0.4216 tension M: (Nm) 1.198 1.167 1.043 0.8955 0.7005 0.4878