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Debonding mechanisms in the presence of an interphase in composites
PII:
Acta mater. Vol. 46, No. 15, pp. 5237±5247, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00222-5 1359-6454/98 $19.00 + 0.00
DEBONDING MECHANISMS IN THE PRESENCE OF AN INTERPHASE IN COMPOSITES Y.-F. LIU{, Y. TANAKA and C. MASUDA National Research Institute for Metals, 1-2-1 Sengen, Tsukuba 305, Japan (Received 2 January 1998; accepted 19 June 1998) AbstractÐThe debonding mechanism of an interphase in layered and ®ber-reinforced composites was investigated with an emphasis on crack de¯ection and penetration. A single-edged notched model consisting of three phases was analyzed by the boundary element method. The dependence of elastic ®elds involved in the model on elastic constant combinations was revealed. Interfacial debonding other than the competition between crack de¯ection and penetration may be initiated due to the high tensile and shear stress concentration before a main crack propagates to meet the next interface. The ratio of energy release rates for in®nitesimal de¯ecting and penetrated cracks, de®ned as R Gd =Gp , was computed when a main crack reached an interphase interface. The results obtained modi®ed those for two-phase models. Discussion regarding fracture sequences observed in Ti-15-3 alloy reinforced with unidirectionally aligned SCS-6 ®ber was made. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Interphase layers with thicknesses ranging from sub-micron to millimeter order occur in ®lm±substrate structures, welded structures, laminates comprising of metal and/or ceramics, and ®berreinforced systems. Although there exist various fracture modes due to the presence of an interphase, debonding at the interface is an important fracture mechanism to achieve high toughness materials or to control the crack propagation path [1, 2]. Early eorts to investigate the debonding mechanism date back to Cook and Gordon [3]. They considered possible interfacial debonding occurrence before a main crack tip met the interface lying ahead of a main crack. Many other investigators focused on the situation where a main crack momentarily impinges on the interface and thus the crack de¯ection and penetration behavior is addressed [4±14]. Local crack tip stresses [4, 7, 13, 14] and dierences of fracture energy along de¯ected and penetrated cracks [8±12] were employed to determine the crack path. He and Hutchinson [9] were the ®rst to pay particular attention to the competition between crack penetration and de¯ection and they solved several important crack con®gurations which approached a bimaterial interface from dierent angles to the interface. He et al. further considered the in¯uence of thermal stresses on the competition between crack penetration and de¯ection [10]. Although work by He and Hutchinson treated the competition within in®nite media, Tullock et al. [12] accounted for ®nite-sized geometrical and boundary {To whom all correspondence should be addressed.
condition eects and calculated competition conditions using the boundary element method (BEM) in a bimaterial double-edged notch (DEN) specimen. In the manner of He and Hutchinson's solution procedures [9], the eect of anisotropy for dissimilar materials on the orthotropic level with respect to competition conditions was also studied [11]. However, dissimilar systems with an elastic constant jump at the interface were not considered in the Cook and Gordon mechanisms and quantitative results were not available in their treatment. Furthermore, when the interphase is taken into account, a three-phase material model becomes necessary where previous analytical results involving crack de¯ection and penetration mechanisms may not be applicable. Such a model related to the above fracture mechanisms was investigated in this study. The dependence of elastic ®elds on the elastic constants, the in¯uence of material properties and the thickness of the interphase on the mechanisms for crack de¯ection and penetration were analyzed. The results were applied to several material combinations found in ®ber-reinforced composites. 2. ANALYSIS
2.1. Problem statement The model in the present work was a singleedged notch (SEN)-like specimen composed of three-phase materials and subject to uniform tensile loading as shown in Fig. 1. The three materials, numbered 1, 2 and 3, respectively, were assumed to be isotropic and elastic with the elastic properties being given in Fig. 1. Of particular interest here is
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2.2. Dependence of elastic ®elds on the elastic constants It is known that stress ®elds for a bimaterial system depend on the material property mismatch in the form of Dundurs' parameters, a and b [15, 16]. For the three-phase model shown in Fig. 1, two sets of Dundurs' parameters, namely, a1 and b1 for Materials 1 and 2, and a2 and b2 for Materials 2 and 3 are de®ned as [16] mi 1 ÿ i1 ÿ mi1 1 ÿ i mi 1 ÿ i1 mi1 1 ÿ i
1
mi 1 ÿ 2i1 ÿ mi1 1 ÿ 2i mi 1 ÿ i1 mi1 1 ÿ i
2
ai
bi
Fig. 1. A three-phase material SEN-like model analyzed in this study. Also shown are material and geometric de®nitions.
the case in which the main crack propagates into the next material perpendicular to the dissimilar material interface. The thickness values of Materials 1, 2 and 3 were initially set at w1=100, w2=1 and w3=100 without a speci®c unit for an elastic problem, while w2 was changed later to obtain three ratios of w2/w3=0.01, 0.1 and 0.2 to simulate dierent physical backgrounds. Such a model corresponds to a precracked layered specimen and may be considered a two-dimensional approximation to the crack frontal situation present in many ®ber-reinforced composite systems, with the crack front just in contact with the cylindrical ®ber. This specimen is experimentally feasible. Three cases were considered: (a) the relative tendency to de¯ect or penetrate when a main crack reached the interface between Materials 1 and 2, (b) the interface between Materials 1 and 2 survived and the main crack penetrated into Material 2, and (c) competition between de¯ection and penetration when the crack reached the interface between Materials 2 and 3. For Case (b), the main crack tip experiences only Mode I loading, so the main crack will tend to propagate straight ahead when no local microstructural changes such as crystal anisotropy or porosity in the interphase (Material 2) are present. For Cases (a) and (c), the relative tendency for the main crack to de¯ect or penetrate was addressed by the energy criterion [9]. All the problems were analyzed under elastostatic plane strain conditions for the stated geometric ratios and various elastic mismatches.
where ni and mi (i = 1±3) are Poisson's ratio and shear modulus, with the subscripts in equations (1) and (2) indicating the material number. On the analogy of a bimaterial model, stress ®elds for the three-phase model depend on the elastic mismatches through ai and bi (i = 1±2) (see Appendix A for proof). Previous studies [9±12] have demonstrated that the Dundurs' parameter a is the more important one and b does not signi®cantly aect the elastic ®elds of the model, thus, Young's moduli for the three materials were altered as a focus of interest in the following from the observation that Poisson's ratio for many materials is similar while Young's moduli values are more variable and are believed to be a more important parameter. By assuming Poisson's ratios for the three materials to be equal here, a1 and a2 become a function of the ratios of the shear or Young's moduli according to equation (1) and negative a indicates the material to be penetrated is stier than the material hosting the main crack under the present de®nition in this study. Values of Dundurs' parameters for many material combinations can be found in the literature [14]. 2.3. BEM solution The models presented above were analyzed by the boundary element method (BEM) because it has many advantages in dealing with linear elastostatic problems. It lowers the problem by one dimension through discretizing only the boundary and leads to an ecient numerical solution with reduced data preparation costs. The BEM formulation is deduced through dierent considerations such as virtual work principles or the weighted residual method used in the ®nite element formulation or Betti's reciprocal work theorems [17]. The equilibrium equation for an elastic problem in the absence of body force and initial stresses can be written as sij,j 0
3
where sij,j indicates the stress tensor sij dierentiated with respect to the coordinate xj. Substitution of the displacement±strain and stress±strain re-
LIU et al.: DEBONDING MECHANISMS
lations in elasticity into equation (3) leads to the Navier equations with the displacement as unknowns. Fundamental stress and displacement solutions due to a unit force traction in an in®nite or semi-in®nite body which satisfy the Navier equations have been obtained [17]. We are concerned with an elastic problem with a boundary S over which tractions and displacement are prescribed. The displacement at a point P within S is obtained in the form of an integral equation by applying Betti's theorems of reciprocal work to equation (3) as follows [18]: Cij P uj P Uij P,Qtj Q S
ÿ Tij P,Quj Q dS
4
where tj(Q) and uj(Q) are, respectively, the traction and displacement at Q along S; Uij P,Q and Tij P,Q are known fundamental solutions given elsewhere [17]. Within the fundamental solution domain, Uij P,Q stands for the xi-axis displacement at a point P, due to a unit load along the xj-axis acting at a point Q, while Tij P,Q is the surface force traction along the xi-axis at P, due to a unit load along the xj-axis acting at Q. In equation (4), Cij dij inside the boundary S and is equal to 1/2dij at a smooth boundary, while for surfaces with corners or edges Cij is frequently determined by removal of rigid movement in practice. The above formulation forms the basis of the direct boundary element method used in this study. In general, analytical solution of equation (4) is not possible and numerical implementation is necessary. Supposing that the boundary S is discretized into N elements, equation (4) is then expressed in the following form: N X Cij P uj P Uij P,Qtj Q n1 Sn
ÿ Tij P,Quj Q dS:
5
For a two-dimensional problem, the left-hand side of equation (5) represents the summation of line integrals over each line element and contains singular terms when the point or node P coincides with Q. Application of equation (5) to all element nodes P yields a system of linear equations Hu Gt
6
where u and t are displacement vector and surface force vector over S, respectively; H and G are the associated matrices obtained by numerically integrating equation (5). With suitable arrangement of known and unknown tractions and displacements along S, equation (6) can be solved for the unknown boundary values. For crack competition problems shown in Figs 1(a) and (c), plane strain energy release rates
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for imaginary de¯ected and penetrated cracks were calculated in the following manner. For a de¯ected crack at the interface, the stress ®elds near the interface crack tip are de®ned by [19±21] K1 iK2 r ie 7a sy itxy p l 2pr where K1 and K2 are the stress intensity factors for an interface crack and l is the total crack length employed to normalize the oscillatory term; e is a bimaterial constant which is de®ned as 1 k1 =m1 1=m2 e ln 7b k2 =m2 1=m1 2p where ki 3 ÿ 4i for the plane strain condition and ki 3 ÿ i = 1 ÿ i (i = 1, 2) for the plane stress condition. An extrapolation method was used to determine stress intensity factors for an interface crack through utilizing solutions near a crack tip obtained by numerical analysis [22]. From the BEM surface force solutions which are related to the normal stress and shear stress on the interface near a crack tip, the stress intensity factors for an interface crack may be determined by the following extrapolation method [22]: p K1 lim 2pr sy cos Q txy sin Q 8a rÿ 40
p K2 lim 2pr txy cos Q ÿ sy sin Q
8b
r : Q e log 2a
8c
rÿ 40
K1 and K2 in equations (8a)±(b) are related to the energy release rate, Gd, for an interface crack by [20, 21] 1 k1 1 k2 1 Gd K 21 m1 m2 16 cosh2 ep K 22 :
9
The stress intensity factor for the penetrated crack in Material 2 was calculated using the same method as that stated above. Stress intensity factors obtained could be transformed into the energy release rate by equation (9) with the results being de®ned as Gd and Gp for a de¯ected and penetrated crack, respectively. An assessment of the competition between penetration of the interface and de¯ection is addressed through calculating the ratios of energy release rates for an in®nitesimal de¯ected crack and a penetrated crack. Then, a necessary condition to enable a crack to de¯ect along the interface instead of penetration into the next material is given by [9]
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LIU et al.: DEBONDING MECHANISMS
Fig. 2. Normal stress distributions along the interface between Materials 2 and 3.
R c
Gd Gi c > Gc Gp
10
where Gd and Gp are the energy release rates for an in®nitesimal de¯ected crack and penetrated crack, respectively; Gc is the mode I toughness of the material to be penetrated; Gi(c) is the toughness of the interface and may depend on the phase angle of loading de®ned as tanÿ1 K2 =K1 . If the inequality in equation (10) is reversed, then the main crack will have a tendency to penetrate into Material 2. The convergence study of R values and their comparison with previous results [9, 10] are presented in Appendix B. 3. ANALYTICAL RESULTS AND DISCUSSION
Fig. 3. Shear stress distributions along the interface between Materials 2 and 3.
mechanism demands that, before the main crack tip meets the interface, the maximum decohesion tensile or shear stress along the interface be reached before the strength of Material 3 is reached. It also indicates that the latter higher shear stress has more potential to initiate shear-type ®ber debonding ahead of the main crack tip as envisioned by Cook and Gordon [3]. Recently, a shear-type debonding which occurred symmetrically with respect to the main crack tip has been observed in model composite experiments [23]. 3.2. Eect of elastic mismatch and interphase thickness on R [Cases (a) and (c)] Since the eect of the geometric parameter w1 on the stress ®elds for the de¯ected and penetrated
3.1. Stress distributions along the interface of Materials 2 and 3 [Cases (a) and (b)] The stress solution was obtained directly over the three sub-region boundaries and some typical results are presented here for the initial geometrical ratio noted in Section 2.1. Since the stress distributions along the interface of Materials 2 and 3 scale with the applied stress, the applied stress s is set to unity. Figure 2 shows the eect of a1 on the normal stress distribution along the interface at x w2 . Larger positive a1 values yielded larger normal stresses near the symmetrical crack plane. The interfacial shear stress also concentrated with a smaller value than the normal stress as depicted in Fig. 3. As the main crack tip advanced closer to the next interface, higher normal stresses were observed as shown in Fig. 4. The shear stress concentrated in similar fashion initially, while another larger concentrated peak with an opposite direction appeared as the main crack tip propagated closer to the interface (a = 0.04). Such a shear stress distribution was not found in homogeneous media. The debonding
Fig. 4. Eect of the distance between the main crack tip and the interface on normal and shear stress distributions along the interface between Materials 2 and 3.
LIU et al.: DEBONDING MECHANISMS
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Fig. 5. Relationship between R and a1 for w2 =w3 0:01.
Fig. 6. Relationship between R and a1 for w2 =w3 0:1.
crack is the same, w1 will not aect the energy release rate ratio R. The relevant geometric parameters aecting R were w2 and w3, so the ratio of w2/w3 was used below. Figures 5±7 present the relationship between R and a1 for thickness ratios w2/ w3=0.01, 0.1 and 0.2, respectively. Note that the range for R presented in Figs 5±7 was dierent. Also included is the eect of a2 on R. The curves for a2=0 in Figs 5±7 were close to the solution for in®nite bimaterials where b = 0. As in bimaterial systems [9±12], R reaches a minimum for a1=0 and increases for large negative and positive a1 (Figs 5± 7). The value of R showed a stronger dependence on large negative values of a1, with only a weak dependence on large positive a1 values. In addition, large negative a2 always produced large R, and vice versa (Figs 5±7). However, the ratio of energy release rates, R, depended not only on the elastic mismatch between material combinations to be
penetrated or de¯ected, but was also dependent on material combinations for the interphase (i.e. Material 2) and Material 3 (Figs 5±7). The thickness ratio between interphase and far-side bonding material also signi®cantly in¯uenced R for large negative a1, though no dependence was observed for large positive values of a1. Such a tendency was most strongly pronounced for the thickness ratio w2/w3=0.01 (Fig. 5). For the thickness ratios utilized between interphase and Material 3 (w2/ w3=0.01, 0.1 and 0.2), a1 only slightly in¯uenced R for a1>0.4, 0.0 and ÿ0.2, respectively. Due to the relative energy release rate ratio depending not only on the combinations of Material 1 and Material 2 but also those between Material 2 and Material 3, together with the eect of varying the interphase as shown in Figs 5±7, suitable selection of constituent moduli and interphase thickness provides the potential for development of
Fig. 7. Relationship between R and a1 for w2 =w3 0:2.
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LIU et al.: DEBONDING MECHANISMS
Fig. 8. Relationship between R and a2 for w2 =w3 0:01.
composites with desired debonding properties. It is apparent that adjustment of the interphase modulus allows a range of R values to be achieved, thus either lowering or enhancing interface toughness requirements for crack de¯ection. On the other hand, control of the interphase thickness will allow greater freedom in meeting such requirements. The analytical results presented in Figs 5±7 predict that increasing the ratio of w2/w3 will lead to smaller R for a2<0 but larger R for a2>0. In addition, R values may be systematically obtained for other material combinations by interpolating the present results. It was also noted from the presented analytical results that results for in®nite bimaterials, close to those of a2=0 in Figs 5±7, may not be applied directly to the interphase debonding and penetration diagram as the ratio of energy release rates, R, is fairly sensitive to large negative a1 values. For large negative a1, the interphase should be treated separately from Materials 1 and 3, particularly when the thickness ratio, w2/w3, is small. However, results for the three-phase material system converge to that of in®nite bimaterials for the case of w2 =w3 r0:2 and a1 > ÿ0:2. It would be expected that w2/w3>0.2 should produce better agreement between the two systems. Figure 8 presents R values for the thickness ratio w2 =w3 0:01 when the main crack reaches the interface between Materials 2 and 3. Note that the range for R presented in Fig. 8 was smaller than that in Figs 5±7. For material combinations in the range ÿ0:25
further weaker dependence of R on a1 was expected. However, when the stress criterion and Cook and Gordon mechanisms are involved, a1 plays a more important role in enhancing the stress concentration for certain cases as demonstrated in Figs 2±4. 3.3. Discussion on experimental situations Some fracture behavior evolving during tensile testing of unidirectionally-aligned SCS-6 (Textron, Lowell, Massachusetts) ®ber-reinforced Ti-15-3 alloy composite was shown and discussed below by using the analytical R results. The SCS-6 ®ber was commercially fabricated by chemical vapor deposition (CVD) of SiC on an amorphous carbon mono®lament core with 3 mm outermost carbon coating. The reaction layer was formed between the outermost carbon coating layer and the matrix, which was reported to crack ®rst in fracture tests [24±26]. However, as shown below, the SCS-6 ®ber along with the carbon coating may fracture before reaction layer cracking occurs for the following reasons: the weakening of ®ber strength by ®ber nudity, ®ber strength degradation during fabrication, initial defects induced during the CVD process, and scattered strength distribution of the ®ber. Besides, the fracture sequence following the ®ber breaks is unclear. Detailed tensile tests were thus done in a scanning electron microscope (SEM) to carry out direct observation focusing on the sequence. Specimens were gripped using friction grips and mounted on a motor-driven testing jig with a load cell. The composite panel containing six ®ber ply layers was fabricated by hot-isostatic-pressing. The nominal volume fraction of the SCS-6 ®ber was 0.39. The reaction layer between the outermost carbon coating layer and the matrix had a
LIU et al.: DEBONDING MECHANISMS
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Fig. 10. SEM photo showing the debonded interface and protruded ®ber during the pushout test.
mond paste up to 0.25 mm ®nish. Tensile loading was applied step by step and the tests were frequently interrupted for direct in-situ observation. A typical observed damage process of the composite is shown in Fig. 9. At an applied stress of 255.4 MPa, SiC ®ber cracks were observed [Fig. 9(a)]. As the loading continued to increase to 362.6 MPa, the outermost carbon coating was fractured [Fig. 9(b)]. With further increase in loading, the interface debonding between the outermost carbon coating and the reaction layer was detected at a loading level of 405 MPa [Fig. 9(c)]. To infer the interfacial debonding property, ®ber pushout tests were carried out and the interface between the matrix and carbon coating was found to be debonded (Fig. 10). A typical load±displacement curve obtained is shown in Fig. 11. The details of the pushout equipment and experimental procedure have been reported in Ref. [27]. The load±displacement curve was ®tted by using Liang and Hutchinson's formulation of the pushout
Fig. 9. SEM photos showing the fracture evolution process of SCS-6 ®ber-reinforced Ti-15-3 composites under tensile testing. Photos (a)±(c) correspond to three loading levels. Also shown is the loading direction.
mean thickness of 0.7±1 mm [see Fig. 9(b)]. Smooth rectangular type specimens 200 mm long, 1.89 mm wide and 1.15 mm thick were tested. One surface of the specimen was polished for observation with dia-
Fig. 11. Load±®ber displacement curve obtained from the pushout test.
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LIU et al.: DEBONDING MECHANISMS
test [28]. The pushout phase in Fig. 11 corresponds to frictional sliding which is characterized by Coulomb friction and gives a friction of m = 0.14 (equation (2.20) in Ref. [28]). The maximum load at the end of debonding was used to calculate the debonding toughness Gi by equation (2.17) in Ref. [28]. A value of Gi 4:1 N=m was obtained from Fig. 11. Many ®bers in the composite were tested and Gi was found to be in the range 2±5 N/ m. In addition, since the interface between SiC ®ber and carbon coating appeared to bond well in the pushout test, it was estimated that the toughness value Gi at this interface was at least larger than 2± 5 N/m. Assuming the SiC ®ber, carbon coating, and Ti-15-3 alloy and the reaction layer [Fig. 9(b)] as Materials 1, 2 and 3 corresponding to Fig. 1, respectively. w3 is roughly 110 mm to give a ®ber volume fraction of 0.39 so that the thickness ratio w2/w3 is about 0.03. From the scattered values of Young's moduli and Poisson's ratios of the SiC ®ber, carbon coating and matrix reported in the literature, a1=0±0.3 and a2=0.3± 0.54 were obtained. Based on plane strain toughnesses of the SiC ®ber of 4±5 MPa m1/2 [29], carbon coating 01.0 MPa m1/2 [30], the mode I fracture energies should be about G1=40.0±59.0 N/m for Material 1 and G2=2.5±4.3 N/m for Material 2. Therefore, the ratio of Gi/G2 should be larger that 0.46±2.11 when the main crack tip arrived at the interface between Materials 1 and 2; here, Gi stands for the interface toughness between SiC (CVD) and carbon coating. On the other hand, R was of the order of 00.28 and almost independent of the interphase thickness from the analytical results shown in Figs 5 and 6, so R Gi =G2 . These results suggest that the crack tip reaching the interface between the carbon coating and reaction
layer meets the necessary condition, which is in agreement with the experimental observation. Despite the complexity of the problem and unknown factors in materials and geometries, the modeling and analytical results are believed to be useful to explain the experimental ®ndings and provide insight into the fracture behavior in the presence of an interphase in a composite. Besides, if the theoretical results and experimental evidence presented in this study are employed, toughness upper or lower bounds of related materials and interfaces meeting the crack de¯ection and penetration conditions can be evaluated inversely.
4. CONCLUSIONS
Debonding possibilities in a three-phase model were numerically analyzed by the boundary element method, with the relative tendency of a crack tip in contact with interfaces to de¯ect or penetrate being focused on. The following conclusions were obtained: 1. The elastic ®elds of the three-phase model were shown to depend on two pairs of Dundurs' parameters de®ned between any two materials in the three-phase model. When the main crack tip penetrated into the interphase before reaching the next interface, the normal and shear stresses along the interface experienced concentration. The magnitude and position of the concentrated stresses were strongly in¯uenced by the relative distance between the crack tip and the interface. A higher shear stress was found when the main crack tip propagates closer to the interface and was thus likely to initiate shear-type debonding before the main crack tip met the next interface. 2. The ratio of energy release rates for in®nitesimal de¯ecting and penetrated cracks, de®ned as R Gd =Gp , was computed to predict relative tendencies to de¯ect and penetrate when a main crack reached the two interphase interfaces. Particular attention was given to the eect of material combinations between the interphase and two side-bonded materials, and also to the thickness ratio of the interphase to that of the bonded material without the main crack. With the present de®nition that a negative Dundurs' parameter a indicates the material to be penetrated is stier than the material with the main crack, R showed a strong dependence on large negative a values. When the crack penetrates into the interphase and arrives at the second interface, the dependence of R on the elastic property of the far-side bonding material seemed to be weak for most material combinations of materials hosting the main crack tip. The eect of thickness ratios in both cases was also provided.
LIU et al.: DEBONDING MECHANISMS
3. The analytical results were applied to experimental situations and provided insight into the fracture sequence observed during tensile tests of Ti15-3 alloy reinforced with unidirectionally aligned SCS-6 ®ber.
AcknowledgementsÐThe authors are grateful to I. J. Davies at National Aerospace Laboratory, Science and Technology Agency, Japan, for useful suggestions regarding the manuscript. REFERENCES 1. Evans, A. G., Mater. Sci. Engng, 1989, A107, 227. 2. Naslain, R., in High-Temperature Ceramic±Matrix Composites II: Manufacturing and Materials Development. American Ceramic Society, Ohio, 1995, pp. 23±39. 3. Cook, J. and Gordon, J. E., Proc. R. Soc. Lond., 1967, A229, 508. 4. Zak, A. K. and Williams, M. L., ASME J. appl. Mech., 1963, 30, 142. 5. Cook, T. S. and Erdogan, F., Int. J. Engng Sci., 1972, 10, 677. 6. Erdogan, F. and Arin, K., Int. J. Fract., 1975, 11, 191. 7. Goree, J. G. and Venezia, W. A., Int. J. Engng Sci., 1977, 15, 1. 8. He, M.-Y. and Hutchinson, J. W., ASME J. appl. Mech., 1989, 56, 270. 9. He, M.-Y. and Hutchinson, J. W., Int. J. Solids Struct., 1989, 25, 1053. 10. He, M.-Y., Evans, A. G. and Hutchinson, J. W., Int. J. Solids Struct., 1994, 31, 3443. 11. Martinez, D. and Gupta, V., J. Mech. Phys. Solids, 1994, 42, 1247. 12. Tullock, D., Reimanis, I. E., Graham, A. L. and Petrovic, J. J., Acta metall. mater., 1994, 42, 3245. 13. Gupta, V., Argon, A. S. and Suo, Z., J. appl. Mech., 1992, 59, S79. 14. Gupta, V., Yuan, J. and Martinez, D., J. Am. Ceram. Soc., 1993, 76, 305. 15. Dundurs, J., J. Composite Mater., 1967, 1, 310. 16. Dundurs, J., J. appl. Mech., 1969, 36, 650. 17. Brebbia, C. A., Telles, T. C. F. and Wrobel, L. C., Boundary Element Techniques: Theory and Applications in Engineering. Springer, Berlin, 1984. 18. Cruse, T. A., Boundary Element Analysis in Computational Fracture Mechanics. Kluwer, Dordrecht, 1988. 19. Rice, J. R. and Sih, G. C., Trans. ASME J. appl. Mech., 1965, 32, 418. 20. Rice, J. R., Trans. ASME J. appl. Mech., 1988, 55, 98. 21. Sun, C. T. and Jih, C. J., Engng Fract. Mech., 1987, 28, 13. 22. Yuuki, R. and Cho, S. B., Engng Fract. Mech., 1989, 34, 179. 23. Goto, K., Kagawa, Y., Nojima, K. and Iba, H., Mater. Sci. Engng, 1996, A212, 69. 24. Guo, S. Q., Kagawa, Y. and Honda, K., Metall. Mater. Trans., 1996, 27A, 2843. 25. Majumdar, B. S. and Newaz, G. M., Mater. Sci. Engng, 1995, A200, 114. 26. Thomin, S. H., Noel, P. A. and Dunand, D. C., Metall. Mater. Trans., 1996, 26A, 883. 27. Honda, K. and Kagawa, Y., Acta metall. mater., 1995, 43, 1477. 28. Liang, C. and Hutchinson, J. W., Mech. Mater., 1993, 14, 207.
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29. Chan, K. S., Metall. Trans., 1993, 24A, 1531. 30. Sato, C. et al., Strength and toughness of carbon. Annual Report of Japan Nuclear Research Institute, 1983. 31. Sundgren, J.-E. and Hentzell, H. T. G., J. Vac. Sci. Technol., 1986, A4, 2259. 32. Ceramic Source, American Ceramics Society, T13, 1993.
APPENDIX A On the basis of some well-known results from plane elasticity, the dependence of elastic ®elds for the models shown in Fig. 1 on the elastic constants can be found without diculty. The interfaces indicated in Fig. 1 are all well bonded, thus the traction and displacement components at the interfaces must satisfy the following equations: Tx iTy i ÿ Tx iTy i1
A1
ux iuy i ux iuy i1 i 1,2
A2
where values of T and u express the traction and displacement components at the interface, respectively, with the subscript i indicating the material number. Equations (A1) and (A2) require that the complex potentials fi(z), ci(z) (i = 1±3), which govern elastic ®elds, meet the next conditions following the argument for two-phase material systems [15]: 8 G1 k2 G1 ÿ 1 > 0 > < f1 G k 1 f2 G k 1 zf2 c2 1 1 1 1 A3 > > G1 k1 ÿ k2 f G1 k1 1 zf 0 c 0 c : zf 1 2 2 1 2 G1 k1 1 G1 k1 1 8 G2 k3 G2 ÿ 1 0 > > < f2 G k 1 f3 G k 1 zf3 c3 2 2 2 2 A4 > > zf 0 c 0 c G2 k2 ÿ k3 f G2 k2 1 zf : 2 3 3 2 3 G2 k2 1 G2 k2 1 where ki 3 ÿ 4i and 3 ÿ i = 1 i (i = 1±3) for plane strain and plane stress, respectively, and Gi mi1 =mi (i = 1, 2). Here, ni and mi (i = 1±3) are Poisson's ratio and shear modulus, with the subscript i indicating the material number. Equations (A3) and (A4) may be rewritten in terms of Dundurs' parameters as 8 1 ÿ b1 a1 ÿ b1 0 > > < f1 1 a f2 1 a zf2 c2 1 1 A5 > > 0 c a1 b1 f 1 b1 zf 0 c : zf 1 2 2 1 2 1 a1 1 a1 8 1 ÿ b2 a2 ÿ b2 0 > > < f2 1 a f3 1 a zf3 c3 2 2 > a2 b2 1 b2 0 > 0 : zf2 c2 f zf3 c3 1 a2 3 1 a2
A6
where the non-dimensional Dundurs' parameters ai and bi are explained in Section 2.2. By0 substituting equation (A6) c into equation (A5), f1 and zf 1 0 1 can be expressed as c where the associlinear combinations of f3 and zf 3 3 ated coecients are the elastic constant combination of Materials 1 and 3 appearing as if Materials 1 and 3 were bonded. Thus, two pairs of ai and bi de®ned above are independent parameters to determine the complex potentials fi(z), ci(z) (i = 1±3). Obviously, a similar argument can be generalized to systems with more than three phases or layers.
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Fig. B1. BEM meshes utilized in the current analysis for w2 =w3 0:1. De¯ected and penetrated cracks appeared separately in actual calculations. APPENDIX B The BEM code used has been shown to be highly ecient and accurate in solving two-dimensional bimaterial problems [22]. In the present analysis, Kelvin's solution was used as the fundamental solution. The whole solution domain shown in Fig. 1 was divided into three subregions, allowing the stress and displacement components at the interface crack and penetrated crack surface to be calculated directly. Figure B1 shows the typical boundary element discretization for the present analysis. Symmetry about the main crack surface was imposed, which implies the doubly de¯ected pattern was assumed to occur. The singly de¯ected pattern was not studied as previous work [11, 12] has shown only small numerical dierences to exist over a wide range of material combinations between the two, whilst the experimentally observed evidence pointed mostly to the doubly cracking de¯ection [29]. Quadratic line elements were used in the BEM division. Particularly ®ne elements were positioned near the interface crack tip, whilst neighboring elements were placed with suitable length proportions smaller than 2.0. The total node and element numbers were 391 and 195, respectively, for w2/ w3=0.1. The mesh density was kept constant for dierent w2/w3 (=0.01, 0.1 and 0.2), while the number of nodes and elements was adjusted. Past experience had shown this element density and length to be sucient in obtaining accurate solutions [22]. The size ratio for two adjacent BEM elements were ®xed between 1.0 and 2.0 whilst de¯ected and penetrated crack length increments were changed in order to study the accuracy and convergence behavior of energy release
rate ratios, R, for an in®nitesimal de¯ected crack over a penetrated crack. Such a check procedure is essential as analytical results are known to be very sensitive to BEM mesh discretization [12, 22]. For each de¯ected and pene-
Fig. B2. Relationship between relative energy release ratio, R, and crack increment, Da, normalized by w1 w2 w3 . The convergent incremental amount for an in®nitesimal crack was determined to be Da=w3 0:2 10ÿ4 through comparison with previous analytical results [9, 10] included in this ®gure.
LIU et al.: DEBONDING MECHANISMS trated crack increment, energy release rates were calculated using equations (7a)±(b), (8a)±(c) and (9). The convergent de¯ected and penetrated crack lengths should be small enough compared to all other geometric length quantities, including the main crack length. The eect of changing crack increments on the value of R is shown in Fig. B2 for w2/w3=0.01 and a = 0. It is observed that R approaches an asymptotic value as the crack increment is reduced continuously to zero. Further reduction in crack increment resulted in only little change in the numerical solution at a crack increment of approximately Da=w3 0:2 10ÿ4 for w2/w3=0.01. For such a crack
5247
increment, R was obtained over a wide range of a values (b = 0). In addition, excellent agreement with He et al. [9, 10] concerning two in®nite bodies was obtained for the range ÿ0:55RaR0:6, into which the majority of engineering material combinations falls (see Figs 5±8). Outside this range, fairly small deviation was noted. It was, therefore, concluded that the present BEM division, numerical procedures and the crack increment produced accurate convergent results. The incremental crack size noted above was used for R calculation presented in this study.
Acta mater. Vol. 46, No. 15, pp. 5237±5247, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain S1359-6454(98)00222-5 1359-6454/98 $19.00 + 0.00
DEBONDING MECHANISMS IN THE PRESENCE OF AN INTERPHASE IN COMPOSITES Y.-F. LIU{, Y. TANAKA and C. MASUDA National Research Institute for Metals, 1-2-1 Sengen, Tsukuba 305, Japan (Received 2 January 1998; accepted 19 June 1998) AbstractÐThe debonding mechanism of an interphase in layered and ®ber-reinforced composites was investigated with an emphasis on crack de¯ection and penetration. A single-edged notched model consisting of three phases was analyzed by the boundary element method. The dependence of elastic ®elds involved in the model on elastic constant combinations was revealed. Interfacial debonding other than the competition between crack de¯ection and penetration may be initiated due to the high tensile and shear stress concentration before a main crack propagates to meet the next interface. The ratio of energy release rates for in®nitesimal de¯ecting and penetrated cracks, de®ned as R Gd =Gp , was computed when a main crack reached an interphase interface. The results obtained modi®ed those for two-phase models. Discussion regarding fracture sequences observed in Ti-15-3 alloy reinforced with unidirectionally aligned SCS-6 ®ber was made. # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
Interphase layers with thicknesses ranging from sub-micron to millimeter order occur in ®lm±substrate structures, welded structures, laminates comprising of metal and/or ceramics, and ®berreinforced systems. Although there exist various fracture modes due to the presence of an interphase, debonding at the interface is an important fracture mechanism to achieve high toughness materials or to control the crack propagation path [1, 2]. Early eorts to investigate the debonding mechanism date back to Cook and Gordon [3]. They considered possible interfacial debonding occurrence before a main crack tip met the interface lying ahead of a main crack. Many other investigators focused on the situation where a main crack momentarily impinges on the interface and thus the crack de¯ection and penetration behavior is addressed [4±14]. Local crack tip stresses [4, 7, 13, 14] and dierences of fracture energy along de¯ected and penetrated cracks [8±12] were employed to determine the crack path. He and Hutchinson [9] were the ®rst to pay particular attention to the competition between crack penetration and de¯ection and they solved several important crack con®gurations which approached a bimaterial interface from dierent angles to the interface. He et al. further considered the in¯uence of thermal stresses on the competition between crack penetration and de¯ection [10]. Although work by He and Hutchinson treated the competition within in®nite media, Tullock et al. [12] accounted for ®nite-sized geometrical and boundary {To whom all correspondence should be addressed.
condition eects and calculated competition conditions using the boundary element method (BEM) in a bimaterial double-edged notch (DEN) specimen. In the manner of He and Hutchinson's solution procedures [9], the eect of anisotropy for dissimilar materials on the orthotropic level with respect to competition conditions was also studied [11]. However, dissimilar systems with an elastic constant jump at the interface were not considered in the Cook and Gordon mechanisms and quantitative results were not available in their treatment. Furthermore, when the interphase is taken into account, a three-phase material model becomes necessary where previous analytical results involving crack de¯ection and penetration mechanisms may not be applicable. Such a model related to the above fracture mechanisms was investigated in this study. The dependence of elastic ®elds on the elastic constants, the in¯uence of material properties and the thickness of the interphase on the mechanisms for crack de¯ection and penetration were analyzed. The results were applied to several material combinations found in ®ber-reinforced composites. 2. ANALYSIS
2.1. Problem statement The model in the present work was a singleedged notch (SEN)-like specimen composed of three-phase materials and subject to uniform tensile loading as shown in Fig. 1. The three materials, numbered 1, 2 and 3, respectively, were assumed to be isotropic and elastic with the elastic properties being given in Fig. 1. Of particular interest here is
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2.2. Dependence of elastic ®elds on the elastic constants It is known that stress ®elds for a bimaterial system depend on the material property mismatch in the form of Dundurs' parameters, a and b [15, 16]. For the three-phase model shown in Fig. 1, two sets of Dundurs' parameters, namely, a1 and b1 for Materials 1 and 2, and a2 and b2 for Materials 2 and 3 are de®ned as [16] mi 1 ÿ i1 ÿ mi1 1 ÿ i mi 1 ÿ i1 mi1 1 ÿ i
1
mi 1 ÿ 2i1 ÿ mi1 1 ÿ 2i mi 1 ÿ i1 mi1 1 ÿ i
2
ai
bi
Fig. 1. A three-phase material SEN-like model analyzed in this study. Also shown are material and geometric de®nitions.
the case in which the main crack propagates into the next material perpendicular to the dissimilar material interface. The thickness values of Materials 1, 2 and 3 were initially set at w1=100, w2=1 and w3=100 without a speci®c unit for an elastic problem, while w2 was changed later to obtain three ratios of w2/w3=0.01, 0.1 and 0.2 to simulate dierent physical backgrounds. Such a model corresponds to a precracked layered specimen and may be considered a two-dimensional approximation to the crack frontal situation present in many ®ber-reinforced composite systems, with the crack front just in contact with the cylindrical ®ber. This specimen is experimentally feasible. Three cases were considered: (a) the relative tendency to de¯ect or penetrate when a main crack reached the interface between Materials 1 and 2, (b) the interface between Materials 1 and 2 survived and the main crack penetrated into Material 2, and (c) competition between de¯ection and penetration when the crack reached the interface between Materials 2 and 3. For Case (b), the main crack tip experiences only Mode I loading, so the main crack will tend to propagate straight ahead when no local microstructural changes such as crystal anisotropy or porosity in the interphase (Material 2) are present. For Cases (a) and (c), the relative tendency for the main crack to de¯ect or penetrate was addressed by the energy criterion [9]. All the problems were analyzed under elastostatic plane strain conditions for the stated geometric ratios and various elastic mismatches.
where ni and mi (i = 1±3) are Poisson's ratio and shear modulus, with the subscripts in equations (1) and (2) indicating the material number. On the analogy of a bimaterial model, stress ®elds for the three-phase model depend on the elastic mismatches through ai and bi (i = 1±2) (see Appendix A for proof). Previous studies [9±12] have demonstrated that the Dundurs' parameter a is the more important one and b does not signi®cantly aect the elastic ®elds of the model, thus, Young's moduli for the three materials were altered as a focus of interest in the following from the observation that Poisson's ratio for many materials is similar while Young's moduli values are more variable and are believed to be a more important parameter. By assuming Poisson's ratios for the three materials to be equal here, a1 and a2 become a function of the ratios of the shear or Young's moduli according to equation (1) and negative a indicates the material to be penetrated is stier than the material hosting the main crack under the present de®nition in this study. Values of Dundurs' parameters for many material combinations can be found in the literature [14]. 2.3. BEM solution The models presented above were analyzed by the boundary element method (BEM) because it has many advantages in dealing with linear elastostatic problems. It lowers the problem by one dimension through discretizing only the boundary and leads to an ecient numerical solution with reduced data preparation costs. The BEM formulation is deduced through dierent considerations such as virtual work principles or the weighted residual method used in the ®nite element formulation or Betti's reciprocal work theorems [17]. The equilibrium equation for an elastic problem in the absence of body force and initial stresses can be written as sij,j 0
3
where sij,j indicates the stress tensor sij dierentiated with respect to the coordinate xj. Substitution of the displacement±strain and stress±strain re-
LIU et al.: DEBONDING MECHANISMS
lations in elasticity into equation (3) leads to the Navier equations with the displacement as unknowns. Fundamental stress and displacement solutions due to a unit force traction in an in®nite or semi-in®nite body which satisfy the Navier equations have been obtained [17]. We are concerned with an elastic problem with a boundary S over which tractions and displacement are prescribed. The displacement at a point P within S is obtained in the form of an integral equation by applying Betti's theorems of reciprocal work to equation (3) as follows [18]: Cij P uj P Uij P,Qtj Q S
ÿ Tij P,Quj Q dS
4
where tj(Q) and uj(Q) are, respectively, the traction and displacement at Q along S; Uij P,Q and Tij P,Q are known fundamental solutions given elsewhere [17]. Within the fundamental solution domain, Uij P,Q stands for the xi-axis displacement at a point P, due to a unit load along the xj-axis acting at a point Q, while Tij P,Q is the surface force traction along the xi-axis at P, due to a unit load along the xj-axis acting at Q. In equation (4), Cij dij inside the boundary S and is equal to 1/2dij at a smooth boundary, while for surfaces with corners or edges Cij is frequently determined by removal of rigid movement in practice. The above formulation forms the basis of the direct boundary element method used in this study. In general, analytical solution of equation (4) is not possible and numerical implementation is necessary. Supposing that the boundary S is discretized into N elements, equation (4) is then expressed in the following form: N X Cij P uj P Uij P,Qtj Q n1 Sn
ÿ Tij P,Quj Q dS:
5
For a two-dimensional problem, the left-hand side of equation (5) represents the summation of line integrals over each line element and contains singular terms when the point or node P coincides with Q. Application of equation (5) to all element nodes P yields a system of linear equations Hu Gt
6
where u and t are displacement vector and surface force vector over S, respectively; H and G are the associated matrices obtained by numerically integrating equation (5). With suitable arrangement of known and unknown tractions and displacements along S, equation (6) can be solved for the unknown boundary values. For crack competition problems shown in Figs 1(a) and (c), plane strain energy release rates
5239
for imaginary de¯ected and penetrated cracks were calculated in the following manner. For a de¯ected crack at the interface, the stress ®elds near the interface crack tip are de®ned by [19±21] K1 iK2 r ie 7a sy itxy p l 2pr where K1 and K2 are the stress intensity factors for an interface crack and l is the total crack length employed to normalize the oscillatory term; e is a bimaterial constant which is de®ned as 1 k1 =m1 1=m2 e ln 7b k2 =m2 1=m1 2p where ki 3 ÿ 4i for the plane strain condition and ki 3 ÿ i = 1 ÿ i (i = 1, 2) for the plane stress condition. An extrapolation method was used to determine stress intensity factors for an interface crack through utilizing solutions near a crack tip obtained by numerical analysis [22]. From the BEM surface force solutions which are related to the normal stress and shear stress on the interface near a crack tip, the stress intensity factors for an interface crack may be determined by the following extrapolation method [22]: p K1 lim 2pr sy cos Q txy sin Q 8a rÿ 40
p K2 lim 2pr txy cos Q ÿ sy sin Q
8b
r : Q e log 2a
8c
rÿ 40
K1 and K2 in equations (8a)±(b) are related to the energy release rate, Gd, for an interface crack by [20, 21] 1 k1 1 k2 1 Gd K 21 m1 m2 16 cosh2 ep K 22 :
9
The stress intensity factor for the penetrated crack in Material 2 was calculated using the same method as that stated above. Stress intensity factors obtained could be transformed into the energy release rate by equation (9) with the results being de®ned as Gd and Gp for a de¯ected and penetrated crack, respectively. An assessment of the competition between penetration of the interface and de¯ection is addressed through calculating the ratios of energy release rates for an in®nitesimal de¯ected crack and a penetrated crack. Then, a necessary condition to enable a crack to de¯ect along the interface instead of penetration into the next material is given by [9]
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LIU et al.: DEBONDING MECHANISMS
Fig. 2. Normal stress distributions along the interface between Materials 2 and 3.
R c
Gd Gi c > Gc Gp
10
where Gd and Gp are the energy release rates for an in®nitesimal de¯ected crack and penetrated crack, respectively; Gc is the mode I toughness of the material to be penetrated; Gi(c) is the toughness of the interface and may depend on the phase angle of loading de®ned as tanÿ1 K2 =K1 . If the inequality in equation (10) is reversed, then the main crack will have a tendency to penetrate into Material 2. The convergence study of R values and their comparison with previous results [9, 10] are presented in Appendix B. 3. ANALYTICAL RESULTS AND DISCUSSION
Fig. 3. Shear stress distributions along the interface between Materials 2 and 3.
mechanism demands that, before the main crack tip meets the interface, the maximum decohesion tensile or shear stress along the interface be reached before the strength of Material 3 is reached. It also indicates that the latter higher shear stress has more potential to initiate shear-type ®ber debonding ahead of the main crack tip as envisioned by Cook and Gordon [3]. Recently, a shear-type debonding which occurred symmetrically with respect to the main crack tip has been observed in model composite experiments [23]. 3.2. Eect of elastic mismatch and interphase thickness on R [Cases (a) and (c)] Since the eect of the geometric parameter w1 on the stress ®elds for the de¯ected and penetrated
3.1. Stress distributions along the interface of Materials 2 and 3 [Cases (a) and (b)] The stress solution was obtained directly over the three sub-region boundaries and some typical results are presented here for the initial geometrical ratio noted in Section 2.1. Since the stress distributions along the interface of Materials 2 and 3 scale with the applied stress, the applied stress s is set to unity. Figure 2 shows the eect of a1 on the normal stress distribution along the interface at x w2 . Larger positive a1 values yielded larger normal stresses near the symmetrical crack plane. The interfacial shear stress also concentrated with a smaller value than the normal stress as depicted in Fig. 3. As the main crack tip advanced closer to the next interface, higher normal stresses were observed as shown in Fig. 4. The shear stress concentrated in similar fashion initially, while another larger concentrated peak with an opposite direction appeared as the main crack tip propagated closer to the interface (a = 0.04). Such a shear stress distribution was not found in homogeneous media. The debonding
Fig. 4. Eect of the distance between the main crack tip and the interface on normal and shear stress distributions along the interface between Materials 2 and 3.
LIU et al.: DEBONDING MECHANISMS
5241
Fig. 5. Relationship between R and a1 for w2 =w3 0:01.
Fig. 6. Relationship between R and a1 for w2 =w3 0:1.
crack is the same, w1 will not aect the energy release rate ratio R. The relevant geometric parameters aecting R were w2 and w3, so the ratio of w2/w3 was used below. Figures 5±7 present the relationship between R and a1 for thickness ratios w2/ w3=0.01, 0.1 and 0.2, respectively. Note that the range for R presented in Figs 5±7 was dierent. Also included is the eect of a2 on R. The curves for a2=0 in Figs 5±7 were close to the solution for in®nite bimaterials where b = 0. As in bimaterial systems [9±12], R reaches a minimum for a1=0 and increases for large negative and positive a1 (Figs 5± 7). The value of R showed a stronger dependence on large negative values of a1, with only a weak dependence on large positive a1 values. In addition, large negative a2 always produced large R, and vice versa (Figs 5±7). However, the ratio of energy release rates, R, depended not only on the elastic mismatch between material combinations to be
penetrated or de¯ected, but was also dependent on material combinations for the interphase (i.e. Material 2) and Material 3 (Figs 5±7). The thickness ratio between interphase and far-side bonding material also signi®cantly in¯uenced R for large negative a1, though no dependence was observed for large positive values of a1. Such a tendency was most strongly pronounced for the thickness ratio w2/w3=0.01 (Fig. 5). For the thickness ratios utilized between interphase and Material 3 (w2/ w3=0.01, 0.1 and 0.2), a1 only slightly in¯uenced R for a1>0.4, 0.0 and ÿ0.2, respectively. Due to the relative energy release rate ratio depending not only on the combinations of Material 1 and Material 2 but also those between Material 2 and Material 3, together with the eect of varying the interphase as shown in Figs 5±7, suitable selection of constituent moduli and interphase thickness provides the potential for development of
Fig. 7. Relationship between R and a1 for w2 =w3 0:2.
5242
LIU et al.: DEBONDING MECHANISMS
Fig. 8. Relationship between R and a2 for w2 =w3 0:01.
composites with desired debonding properties. It is apparent that adjustment of the interphase modulus allows a range of R values to be achieved, thus either lowering or enhancing interface toughness requirements for crack de¯ection. On the other hand, control of the interphase thickness will allow greater freedom in meeting such requirements. The analytical results presented in Figs 5±7 predict that increasing the ratio of w2/w3 will lead to smaller R for a2<0 but larger R for a2>0. In addition, R values may be systematically obtained for other material combinations by interpolating the present results. It was also noted from the presented analytical results that results for in®nite bimaterials, close to those of a2=0 in Figs 5±7, may not be applied directly to the interphase debonding and penetration diagram as the ratio of energy release rates, R, is fairly sensitive to large negative a1 values. For large negative a1, the interphase should be treated separately from Materials 1 and 3, particularly when the thickness ratio, w2/w3, is small. However, results for the three-phase material system converge to that of in®nite bimaterials for the case of w2 =w3 r0:2 and a1 > ÿ0:2. It would be expected that w2/w3>0.2 should produce better agreement between the two systems. Figure 8 presents R values for the thickness ratio w2 =w3 0:01 when the main crack reaches the interface between Materials 2 and 3. Note that the range for R presented in Fig. 8 was smaller than that in Figs 5±7. For material combinations in the range ÿ0:25
further weaker dependence of R on a1 was expected. However, when the stress criterion and Cook and Gordon mechanisms are involved, a1 plays a more important role in enhancing the stress concentration for certain cases as demonstrated in Figs 2±4. 3.3. Discussion on experimental situations Some fracture behavior evolving during tensile testing of unidirectionally-aligned SCS-6 (Textron, Lowell, Massachusetts) ®ber-reinforced Ti-15-3 alloy composite was shown and discussed below by using the analytical R results. The SCS-6 ®ber was commercially fabricated by chemical vapor deposition (CVD) of SiC on an amorphous carbon mono®lament core with 3 mm outermost carbon coating. The reaction layer was formed between the outermost carbon coating layer and the matrix, which was reported to crack ®rst in fracture tests [24±26]. However, as shown below, the SCS-6 ®ber along with the carbon coating may fracture before reaction layer cracking occurs for the following reasons: the weakening of ®ber strength by ®ber nudity, ®ber strength degradation during fabrication, initial defects induced during the CVD process, and scattered strength distribution of the ®ber. Besides, the fracture sequence following the ®ber breaks is unclear. Detailed tensile tests were thus done in a scanning electron microscope (SEM) to carry out direct observation focusing on the sequence. Specimens were gripped using friction grips and mounted on a motor-driven testing jig with a load cell. The composite panel containing six ®ber ply layers was fabricated by hot-isostatic-pressing. The nominal volume fraction of the SCS-6 ®ber was 0.39. The reaction layer between the outermost carbon coating layer and the matrix had a
LIU et al.: DEBONDING MECHANISMS
5243
Fig. 10. SEM photo showing the debonded interface and protruded ®ber during the pushout test.
mond paste up to 0.25 mm ®nish. Tensile loading was applied step by step and the tests were frequently interrupted for direct in-situ observation. A typical observed damage process of the composite is shown in Fig. 9. At an applied stress of 255.4 MPa, SiC ®ber cracks were observed [Fig. 9(a)]. As the loading continued to increase to 362.6 MPa, the outermost carbon coating was fractured [Fig. 9(b)]. With further increase in loading, the interface debonding between the outermost carbon coating and the reaction layer was detected at a loading level of 405 MPa [Fig. 9(c)]. To infer the interfacial debonding property, ®ber pushout tests were carried out and the interface between the matrix and carbon coating was found to be debonded (Fig. 10). A typical load±displacement curve obtained is shown in Fig. 11. The details of the pushout equipment and experimental procedure have been reported in Ref. [27]. The load±displacement curve was ®tted by using Liang and Hutchinson's formulation of the pushout
Fig. 9. SEM photos showing the fracture evolution process of SCS-6 ®ber-reinforced Ti-15-3 composites under tensile testing. Photos (a)±(c) correspond to three loading levels. Also shown is the loading direction.
mean thickness of 0.7±1 mm [see Fig. 9(b)]. Smooth rectangular type specimens 200 mm long, 1.89 mm wide and 1.15 mm thick were tested. One surface of the specimen was polished for observation with dia-
Fig. 11. Load±®ber displacement curve obtained from the pushout test.
5244
LIU et al.: DEBONDING MECHANISMS
test [28]. The pushout phase in Fig. 11 corresponds to frictional sliding which is characterized by Coulomb friction and gives a friction of m = 0.14 (equation (2.20) in Ref. [28]). The maximum load at the end of debonding was used to calculate the debonding toughness Gi by equation (2.17) in Ref. [28]. A value of Gi 4:1 N=m was obtained from Fig. 11. Many ®bers in the composite were tested and Gi was found to be in the range 2±5 N/ m. In addition, since the interface between SiC ®ber and carbon coating appeared to bond well in the pushout test, it was estimated that the toughness value Gi at this interface was at least larger than 2± 5 N/m. Assuming the SiC ®ber, carbon coating, and Ti-15-3 alloy and the reaction layer [Fig. 9(b)] as Materials 1, 2 and 3 corresponding to Fig. 1, respectively. w3 is roughly 110 mm to give a ®ber volume fraction of 0.39 so that the thickness ratio w2/w3 is about 0.03. From the scattered values of Young's moduli and Poisson's ratios of the SiC ®ber, carbon coating and matrix reported in the literature, a1=0±0.3 and a2=0.3± 0.54 were obtained. Based on plane strain toughnesses of the SiC ®ber of 4±5 MPa m1/2 [29], carbon coating 01.0 MPa m1/2 [30], the mode I fracture energies should be about G1=40.0±59.0 N/m for Material 1 and G2=2.5±4.3 N/m for Material 2. Therefore, the ratio of Gi/G2 should be larger that 0.46±2.11 when the main crack tip arrived at the interface between Materials 1 and 2; here, Gi stands for the interface toughness between SiC (CVD) and carbon coating. On the other hand, R was of the order of 00.28 and almost independent of the interphase thickness from the analytical results shown in Figs 5 and 6, so R
layer meets the necessary condition, which is in agreement with the experimental observation. Despite the complexity of the problem and unknown factors in materials and geometries, the modeling and analytical results are believed to be useful to explain the experimental ®ndings and provide insight into the fracture behavior in the presence of an interphase in a composite. Besides, if the theoretical results and experimental evidence presented in this study are employed, toughness upper or lower bounds of related materials and interfaces meeting the crack de¯ection and penetration conditions can be evaluated inversely.
4. CONCLUSIONS
Debonding possibilities in a three-phase model were numerically analyzed by the boundary element method, with the relative tendency of a crack tip in contact with interfaces to de¯ect or penetrate being focused on. The following conclusions were obtained: 1. The elastic ®elds of the three-phase model were shown to depend on two pairs of Dundurs' parameters de®ned between any two materials in the three-phase model. When the main crack tip penetrated into the interphase before reaching the next interface, the normal and shear stresses along the interface experienced concentration. The magnitude and position of the concentrated stresses were strongly in¯uenced by the relative distance between the crack tip and the interface. A higher shear stress was found when the main crack tip propagates closer to the interface and was thus likely to initiate shear-type debonding before the main crack tip met the next interface. 2. The ratio of energy release rates for in®nitesimal de¯ecting and penetrated cracks, de®ned as R Gd =Gp , was computed to predict relative tendencies to de¯ect and penetrate when a main crack reached the two interphase interfaces. Particular attention was given to the eect of material combinations between the interphase and two side-bonded materials, and also to the thickness ratio of the interphase to that of the bonded material without the main crack. With the present de®nition that a negative Dundurs' parameter a indicates the material to be penetrated is stier than the material with the main crack, R showed a strong dependence on large negative a values. When the crack penetrates into the interphase and arrives at the second interface, the dependence of R on the elastic property of the far-side bonding material seemed to be weak for most material combinations of materials hosting the main crack tip. The eect of thickness ratios in both cases was also provided.
LIU et al.: DEBONDING MECHANISMS
3. The analytical results were applied to experimental situations and provided insight into the fracture sequence observed during tensile tests of Ti15-3 alloy reinforced with unidirectionally aligned SCS-6 ®ber.
AcknowledgementsÐThe authors are grateful to I. J. Davies at National Aerospace Laboratory, Science and Technology Agency, Japan, for useful suggestions regarding the manuscript. REFERENCES 1. Evans, A. G., Mater. Sci. Engng, 1989, A107, 227. 2. Naslain, R., in High-Temperature Ceramic±Matrix Composites II: Manufacturing and Materials Development. American Ceramic Society, Ohio, 1995, pp. 23±39. 3. Cook, J. and Gordon, J. E., Proc. R. Soc. Lond., 1967, A229, 508. 4. Zak, A. K. and Williams, M. L., ASME J. appl. Mech., 1963, 30, 142. 5. Cook, T. S. and Erdogan, F., Int. J. Engng Sci., 1972, 10, 677. 6. Erdogan, F. and Arin, K., Int. J. Fract., 1975, 11, 191. 7. Goree, J. G. and Venezia, W. A., Int. J. Engng Sci., 1977, 15, 1. 8. He, M.-Y. and Hutchinson, J. W., ASME J. appl. Mech., 1989, 56, 270. 9. He, M.-Y. and Hutchinson, J. W., Int. J. Solids Struct., 1989, 25, 1053. 10. He, M.-Y., Evans, A. G. and Hutchinson, J. W., Int. J. Solids Struct., 1994, 31, 3443. 11. Martinez, D. and Gupta, V., J. Mech. Phys. Solids, 1994, 42, 1247. 12. Tullock, D., Reimanis, I. E., Graham, A. L. and Petrovic, J. J., Acta metall. mater., 1994, 42, 3245. 13. Gupta, V., Argon, A. S. and Suo, Z., J. appl. Mech., 1992, 59, S79. 14. Gupta, V., Yuan, J. and Martinez, D., J. Am. Ceram. Soc., 1993, 76, 305. 15. Dundurs, J., J. Composite Mater., 1967, 1, 310. 16. Dundurs, J., J. appl. Mech., 1969, 36, 650. 17. Brebbia, C. A., Telles, T. C. F. and Wrobel, L. C., Boundary Element Techniques: Theory and Applications in Engineering. Springer, Berlin, 1984. 18. Cruse, T. A., Boundary Element Analysis in Computational Fracture Mechanics. Kluwer, Dordrecht, 1988. 19. Rice, J. R. and Sih, G. C., Trans. ASME J. appl. Mech., 1965, 32, 418. 20. Rice, J. R., Trans. ASME J. appl. Mech., 1988, 55, 98. 21. Sun, C. T. and Jih, C. J., Engng Fract. Mech., 1987, 28, 13. 22. Yuuki, R. and Cho, S. B., Engng Fract. Mech., 1989, 34, 179. 23. Goto, K., Kagawa, Y., Nojima, K. and Iba, H., Mater. Sci. Engng, 1996, A212, 69. 24. Guo, S. Q., Kagawa, Y. and Honda, K., Metall. Mater. Trans., 1996, 27A, 2843. 25. Majumdar, B. S. and Newaz, G. M., Mater. Sci. Engng, 1995, A200, 114. 26. Thomin, S. H., Noel, P. A. and Dunand, D. C., Metall. Mater. Trans., 1996, 26A, 883. 27. Honda, K. and Kagawa, Y., Acta metall. mater., 1995, 43, 1477. 28. Liang, C. and Hutchinson, J. W., Mech. Mater., 1993, 14, 207.
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29. Chan, K. S., Metall. Trans., 1993, 24A, 1531. 30. Sato, C. et al., Strength and toughness of carbon. Annual Report of Japan Nuclear Research Institute, 1983. 31. Sundgren, J.-E. and Hentzell, H. T. G., J. Vac. Sci. Technol., 1986, A4, 2259. 32. Ceramic Source, American Ceramics Society, T13, 1993.
APPENDIX A On the basis of some well-known results from plane elasticity, the dependence of elastic ®elds for the models shown in Fig. 1 on the elastic constants can be found without diculty. The interfaces indicated in Fig. 1 are all well bonded, thus the traction and displacement components at the interfaces must satisfy the following equations: Tx iTy i ÿ Tx iTy i1
A1
ux iuy i ux iuy i1 i 1,2
A2
where values of T and u express the traction and displacement components at the interface, respectively, with the subscript i indicating the material number. Equations (A1) and (A2) require that the complex potentials fi(z), ci(z) (i = 1±3), which govern elastic ®elds, meet the next conditions following the argument for two-phase material systems [15]: 8 G1 k2 G1 ÿ 1 > 0 > < f1 G k 1 f2 G k 1 zf2 c2 1 1 1 1 A3 > > G1 k1 ÿ k2 f G1 k1 1 zf 0 c 0 c : zf 1 2 2 1 2 G1 k1 1 G1 k1 1 8 G2 k3 G2 ÿ 1 0 > > < f2 G k 1 f3 G k 1 zf3 c3 2 2 2 2 A4 > > zf 0 c 0 c G2 k2 ÿ k3 f G2 k2 1 zf : 2 3 3 2 3 G2 k2 1 G2 k2 1 where ki 3 ÿ 4i and 3 ÿ i = 1 i (i = 1±3) for plane strain and plane stress, respectively, and Gi mi1 =mi (i = 1, 2). Here, ni and mi (i = 1±3) are Poisson's ratio and shear modulus, with the subscript i indicating the material number. Equations (A3) and (A4) may be rewritten in terms of Dundurs' parameters as 8 1 ÿ b1 a1 ÿ b1 0 > > < f1 1 a f2 1 a zf2 c2 1 1 A5 > > 0 c a1 b1 f 1 b1 zf 0 c : zf 1 2 2 1 2 1 a1 1 a1 8 1 ÿ b2 a2 ÿ b2 0 > > < f2 1 a f3 1 a zf3 c3 2 2 > a2 b2 1 b2 0 > 0 : zf2 c2 f zf3 c3 1 a2 3 1 a2
A6
where the non-dimensional Dundurs' parameters ai and bi are explained in Section 2.2. By0 substituting equation (A6) c into equation (A5), f1 and zf 1 0 1 can be expressed as c where the associlinear combinations of f3 and zf 3 3 ated coecients are the elastic constant combination of Materials 1 and 3 appearing as if Materials 1 and 3 were bonded. Thus, two pairs of ai and bi de®ned above are independent parameters to determine the complex potentials fi(z), ci(z) (i = 1±3). Obviously, a similar argument can be generalized to systems with more than three phases or layers.
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Fig. B1. BEM meshes utilized in the current analysis for w2 =w3 0:1. De¯ected and penetrated cracks appeared separately in actual calculations. APPENDIX B The BEM code used has been shown to be highly ecient and accurate in solving two-dimensional bimaterial problems [22]. In the present analysis, Kelvin's solution was used as the fundamental solution. The whole solution domain shown in Fig. 1 was divided into three subregions, allowing the stress and displacement components at the interface crack and penetrated crack surface to be calculated directly. Figure B1 shows the typical boundary element discretization for the present analysis. Symmetry about the main crack surface was imposed, which implies the doubly de¯ected pattern was assumed to occur. The singly de¯ected pattern was not studied as previous work [11, 12] has shown only small numerical dierences to exist over a wide range of material combinations between the two, whilst the experimentally observed evidence pointed mostly to the doubly cracking de¯ection [29]. Quadratic line elements were used in the BEM division. Particularly ®ne elements were positioned near the interface crack tip, whilst neighboring elements were placed with suitable length proportions smaller than 2.0. The total node and element numbers were 391 and 195, respectively, for w2/ w3=0.1. The mesh density was kept constant for dierent w2/w3 (=0.01, 0.1 and 0.2), while the number of nodes and elements was adjusted. Past experience had shown this element density and length to be sucient in obtaining accurate solutions [22]. The size ratio for two adjacent BEM elements were ®xed between 1.0 and 2.0 whilst de¯ected and penetrated crack length increments were changed in order to study the accuracy and convergence behavior of energy release
rate ratios, R, for an in®nitesimal de¯ected crack over a penetrated crack. Such a check procedure is essential as analytical results are known to be very sensitive to BEM mesh discretization [12, 22]. For each de¯ected and pene-
Fig. B2. Relationship between relative energy release ratio, R, and crack increment, Da, normalized by w1 w2 w3 . The convergent incremental amount for an in®nitesimal crack was determined to be Da=w3 0:2 10ÿ4 through comparison with previous analytical results [9, 10] included in this ®gure.
LIU et al.: DEBONDING MECHANISMS trated crack increment, energy release rates were calculated using equations (7a)±(b), (8a)±(c) and (9). The convergent de¯ected and penetrated crack lengths should be small enough compared to all other geometric length quantities, including the main crack length. The eect of changing crack increments on the value of R is shown in Fig. B2 for w2/w3=0.01 and a = 0. It is observed that R approaches an asymptotic value as the crack increment is reduced continuously to zero. Further reduction in crack increment resulted in only little change in the numerical solution at a crack increment of approximately Da=w3 0:2 10ÿ4 for w2/w3=0.01. For such a crack
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increment, R was obtained over a wide range of a values (b = 0). In addition, excellent agreement with He et al. [9, 10] concerning two in®nite bodies was obtained for the range ÿ0:55RaR0:6, into which the majority of engineering material combinations falls (see Figs 5±8). Outside this range, fairly small deviation was noted. It was, therefore, concluded that the present BEM division, numerical procedures and the crack increment produced accurate convergent results. The incremental crack size noted above was used for R calculation presented in this study.