134 Engineered interfaces in fiber reinforced composites "0 10 20 30 40 50 (mm) Fig. 4.24. Plot of partial debond stress, uz, as a function of debond length, e, for untreated Sic fiber-glass matrix composite. After Kim et al. (1991). In light of the foregoing discussion concerning the functional partitioning of the partial debond stress, the characteristic debond stresses can be evaluated. The initial debond stress, ao, is obtained for an infinitesimal debond length where the frictional stress component is zero, i.e., 60 = aele,o . (4.101) The maximal debond stress, 6:. is determined immediately before the load instability (Karbhari and Wilkins, 1990; Kim et al., 1991) of the partial debond stress, a:, when the debond length becomes t = L - zmax: % oe + (8 - at){ 1 - exp[-A(L - ~max)]} . (4.102) Details of the instability conditions of the debond process and Zmax are discussed in Section 4.3.4. Further, the solution for the initial frictional pull-out stress, ofr. upon complete debonding is determined when the debond length, f?, reaches the embedded length, L, and the crack tip debond stress, at, is zero: (4.103) In Eq. (4.103), it is assumed that the influence of the instantaneous fiber displacement relative to the matrix due to the sudden load drop after instability is negligible. Chapter 4. Micromechanics of sfi-ess transfer 135 4.3.4. Instability of debond process The instability condition requires that the derivative of the partial debond stress with respect to the remaining bond length (z = L - e) is equal to or less than zero, i.e., do$'dzdO (Kim et al., 1991). Therefore, the fiber debond process becomes unstable if (L - C) is smaller than a critical bond length, z,,,, where the slopes of the curves become zero in Figs. 4.23 and 4.24. At these bond lengths, the partial debond stress, a:, corresponds to the maximum debond stress, CT;. The zmax value is determined from Eq. (4.102) as 1 o(i5 - 0;) ((Ti - (Tt) + (a - 0;) (4.104) Numerical treatment of Eq. (4.104) gives z,,, values for the different composite systems as shown in Table 4.3. It is worth emphasizing that for the Sic fiber-glass matrix composites, z,,, values are very small relative to L,,, values, irrespective of the fiber surface treatments and when compared to other epoxy matrix based composites. To show clearly how and to what extent the parameter, zmax, varies with the properties of the interface and the composite constituents, a simple fiber pull-out model by Karbhari and Wilkins (1990) is chosen here. This model is developed based on the assumption of a constant friction shear stress, zfr, in the context of the shear strength criterion for interface debonding. In this model, the partial debond stress may be written as where the frictionless debond stress, (TO, is given by (4.105) (4.106) Eq. (4.106) is essentially similar to the solution of the debond stress derived earlier by Takaku and Arridge (1973). The above instability condition for the partial debond stress of Eq. (4.105) gives a rather simple equation for zmax as where p4 is a complex function of o! and y, and is given by (4.107) (4.108) 136 Engineered interfaces in jiber reinforced composites whose approximate solution for b >> a is identical to /3, given in Eq. (4.3). Eq. (4.107) suggests that the ratio of the bond strength at the bonded region to that at the debonded region, q,/Zfr, and the Young’s modulus ratio, CL = Em/& , are key material properties that determine zmax and thus control the stability of the debond process. It should be noted here that in the early work of Lawrence (1972), Laws et al. (1973) and later Gopalaratnam and Shah (1987) the maximum debond stress is found to be dependent on these properties. Eq. (4.107) has a limiting value zmax = 0 when q, N zfr and y N 0 in which the debond process becomes totally stable as in some ceramic matrix composites (e.g. Sic fiber-glass matrix composites (Butler et al., 1990)). From the discussion presented above, it is clear that the stability of the debond process can be evaluated by a single parameter, zmax, which is the shortest (remaining) bond length needed to maintain the debond process stable, and is a constant for a given composite system. Therefore, three different interface debond processes are identified in the following: totally unstable, partially stable and totally stable debond processes. The schematic plots of the applied stress versus displacement curves are illustrated in Fig. 4.25 for these debond processes. (i) If L <zmax, the debond process is totally unstable and the initial debond leads immediately to complete debonding (i.e. GO = ni). Therefore, the corresponding stress-displacement curve shows a monotonic increase in stress until debonding is initiated, followed by an instantaneous load drop (Fig. 4.25(a)). Totally unstable debonding may also occur when the frictional resistance in the debonded region is negligible (i.e. either zero residual clamping stress, 40, or negligible coefficient of friction p) such that zmaX approaches an infinite value as can be envisaged from Eq. (4.107). However, this situation seems most unlikely to occur in practical composites. (ii) If L > z,,,, which is the most common case where practical fiber pull-out tests are performed, the stress increases linearly until debond initiates. Then, the debond crack propagates in a macroscopically stable manner, leading to a non-linear increase in the debond stress, though ‘stick-slips’ are normally observed in the rising stress-displacement curve (Fig. 4.25(b)). Stable debonding proceeds until the (4 (b) (4 Fig. 4.25. Schematic presentations of applied stress versus displacement (0-6) relationship in fiber pull- out test: (a) totally unstable, (b) partially stable and (c) totally stable debond processes. After Kim et al. (1992). Chapter 4. Micromechanics of Stress trunsfer 137 debond length reaches e = L - z,,,, followed by unstable debonding leading to complete debonding. Therefore, this debond process is partially stable. (iii) In the extreme case of z,,, value approaching zero, as in some ceramic matrix composites, the debond process is always stable until complete debonding independent of embedded fiber length, L. The rising portion of the debond stress versus displacement curve (Fig. 4.25(c)) is typically linear without apparent ‘stick- slips’ and there is no appreciable load drop after complete debonding (Bright et al., 1991). This is because the interface is in principle frictionally bonded and there is little chemical bonding. That is Gi,, or Tb is very small. Therefore, the linear increase in stress represents primarily the frictional shear stress transfer across the interface without virtual debonding until the frictional resistance over the entire embedded fiber length is overcome. The maximum debond stress, cri, is then approximately equal to the initial frictional pull-out stress, qr, because the frictionless debond stress, op, is negligible (due to small Gi, or Q,). The concept of z,,, with regard to the issue of the stability of the debond process has practical implications for real composites reinforced with short fibers. There is a minimum fiber length required to maintain stable debonding and thus to achieve maximum benefits of crack-tip bridging between fracture surfaces without the danger of catastrophic failure. It should also be mentioned that in practical fiber pull-out experiments the stability for interface debonding deviates significantly from what has been discussed above, and is most often impaired by adverse testing conditions (e.g. soft testing machine, long free fiber length, etc.). Therefore. debonding could become unstable even for L > z,,, and in composites with zmay = 0. Moreover, when L is very short, as is the normal case in the microdebond test, the precipitous load drop after complete debonding may be aggravated by the release of the strain energy stored in the stretched fiber. The load drops to zero if the fiber is completely pulled out from the matrix. Alternatively, if the fiber is regripped by the clamping pressure exerted by the surrounding matrix material frictional pull- out of the fiber is possible to resume. Another important parameter related to the fiber length in the fiber pull-out test is the maximum embedded fiber length, L,,,, above which the fiber breaks instead of being completely debonded or pulled out. L,,, value for a given composite system can be evaluated by equating 02 of Eq. (4.102) to the fiber tensile strength, CJTS, (which is measured on a gauge length identical to the embedded fiber length used in fiber pull-out test), Le., (4.109) where (J[ is the crack tip debond stress determined for bond length z,,, = (L ~ t). L,,, values calculated for a constant fiber tensile strength CJTS = 4.8, 1.97 and 2.3 GPa for carbon fiber, steel fiber and Sic fiber, respectively, are included in Table 4.3. These predictions are approximately the same as the experimental L,,, values, e.g., the predictions for L,,, = 49.3 and 23.4 mm compare with experimental values L,,, = 5 1 .0 and 21.7 mm, respectively, for the untreated and acid treated Sic fibers 138 Engineered inlerfaces in Jiber reinforced composites (Fig. 4.28). It is worth noting that the L,,, value decreases significantly when the fiber surface is treated to improve the interfacial bonding (and thus the interface fracture toughness, Gic), e.g. acid treated Sic fibers versus untreated fibers. This observation is analogous to what is expected from the fiber fragmentation test of single fiber composites: the stronger the interface bond the shorter is the fiber fragment length at the critical stage (see Section 4.2). 4.3.5. Characterization of interface properties Microcomposite tests including fiber pull-out tests are aimed at generating useful information regarding the interface quality in absolute terms, or at least in comparative terms between different composite systems. In this regard, theoretical models should provide a systematic means for data reduction to determine the relevant properties with reasonable accuracy from the experimental results. The data reduction scheme must not rely on the trial and error method. Although there are several methods of micromechanical analysis available, little attempt in the past has been put into providing such a means in a unified format. A systematic procedure is presented here to generate the fiber pull-out parameters and ultimately the relevant fiber-matrix interface properties. In single fiber pull-out experiments, the most useful data that are readily obtained from the load-deflection records are the maximum debond stress, 02, and the initial frictional pull-out stress, ofr, as a function of L. If the debond process is carefully monitored for a large embedded fiber length, L, the initial debond stress, 00, can also be determined directly in the average sense, depending on the composite system. Most important properties to be calculated are the fracture toughness, Gi,, at the bonded region, and the coefficient of friction, p, and the residual clamping stress, 40, at the debonded region, by evaluating the pull-out parameters of, i and r~. There are several steps to be followed for this purpose. (i) Firstly, ofr versus L data allow the initial slope at L = 0 to be determined based on Eq. (4.103), (4.110) (ii) Secondly, the gradient can be taken from the linear region of the stress drop Ao(= 02 - ofr) versus L plots for large L where the crack tip debond stress is almost constant and independent of L, Le., d ln(Ao) dL A , where the difference between the stresses obtained immediately the load instability is given by Ao = o: - ofr = {of + Tj[exp(-;lz,,,) - 11) exp[-A(L - zmax)] (4.111) before and after (4.112) Chapter 4. Micromechanics of stress transfer 139 (iii) Thirdly, combining Eqs. (4.1 10) and (4.11 1) allows 2 and 8 (and thus p and qo from Eqs. (4.23) and (4.24)) to be determined. Alternatively, the asymptotic debond stress, 5, can be directly estimated at a long embedded length through linear regression analysis of the maximum debond stress, 0;. Once ;2. and are known, Eq. (4.102) may be used to evaluate the optimum value of Gi, (and also for zmax) that would give the best fit to the 0; versus L experimental results. In this procedure theoretical values for the maximum debond stress, o:, have to be obtained at instability. Alternatively, data for the initial debond stress, GO, versus L, if available from experiments, can be directly evaluated to determine Gi, based on the debond criterion of Eq. (4.99) for infinitesimal debond length. Application of this procedure to obtain Gic, 11 and 40 have been demonstrated in fiber pull-out for several fiber composites materials (Kim et al., 1992, Zhou et al., 1993). Having determined the relevant interface properties (Table 4.3), the maximal debond stress, a:, and the initial frictional pull-out stress, ofr, are compared with experimental data in Figs. 4.26-4.28 for three different composite systems of carbon fiber-epoxy matrix, steel fiber-epoxy matrix and Sic fiber-glass matrix. In general, there is very good agreement between theories and experiments over the whole range of the embedded fiber length, L, for all the composite systems considered. A new methodology has also been proposed recently by Zhou et al. (1994) to determine systematically the longest embedded fiber length for instability, zmax, without iteration and curve fitting of Eq. (4.102). 4.3.6. Multiple~fiber composite model From the review of the theoretical studies of the fiber pull-out test as discussed in Section 4.3.1, it is identified that most micromechanics theories are developed based on a shear-lag model of single fiber composites where the cylindrical surface of the matrix is invariably assumed to be stress free. Although this assumption is required to obtain the final solutions in closed form for the stress distributions it often leads to an unacceptably high applied stress required to initiate/propagate interface debonding when the radial dimension of the matrix is similar to that of the fiber (Le. for a high fiber volume fraction, F), This in turn implies that the application of the conventional models to practical composites is limited to those with a very small Vi where any effects of interactions between neighboring fibers are completely neglected. Therefore, a three-cylinder composite model is developed (Kim et al., 1994b) to simulate the response of practical composites of large vf and thus to accommodate the limitation of the shear-lag model of single fiber microcomposite test properly. Both the micromechanics analysis and the FE method are employed in parallel for fully bonded interface to validate the results obtained from each model. To analyze the stress transfer in the fiber pull-out test of a multiple fiber composite, the specimen is treated as a three-cylinder composite (Zhou and Mai, 1992) where a fiber is located at the center of a coaxial shell of the matrix, which, in turn, is surrounded by a trans-isotropic composite medium with an outer radius B, 140 8 0 Engineered interfaces in fiber reinforced eomposites n B n 9 ' 100 ' 200 ' 300 ' 400 ' 5bO L (pm) Fig. 4.26. Comparisons between experiments and theory of (a) maximum debond stress, c$, and (b) initial frictional pull-out stress for carbon fiber-epoxy matrix composites. After Kim et al. (1992). as schematically illustrated in Fig. 4.29. The radii of the fiber and matrix, a and b, are related to the fiber volume fraction vf = a2/b2, which is the same as that of the composite medium. When the fiber is subjected to an external stress, 0, at the loaded end (z = 0) while the matrix and composite medium are fixed at the embedded end (z = L), stress transfers from the fiber to the matrix and in turn from the matrix to the composite medium via the IFSSs, zi(a,z) and zi(b,z), respectively. For the cylindrical coordinates of the three-cylinder composite, the basic governing equations are essentially the same as those for the single fiber composite. However, the equilibrium equations between the external and the internal stresses have to be modified to take into account the presence of the composite medium. Eq. (4.87) is now replaced by: 141 (4 0.8 0.6 s a 0.4 0.2 b n t r / I I I. I. I. I. Fig. 4.27. Comparisons between experiments and theory of (a) maximum debond stress, cri, and (b) initial frictional pull-out stress for steel fiber-epoxy matrix composites: (0) uncoated fibers; (0) release agent coated fibers. After Kim et al. (1992). 1 1 0 = @z) +,a;,(z) +-<(z) , (4.1 13) 1 vi1 1 (4.114) where y, = b2/(B2 - b2), and B is the outer radius of the composite medium. The subscript c refers to the composite medium. In addition to Eq. (4.12) for the relationship between FAS and IFSS, equilibrium between IFSSs and MAS requires 142 Engineered interfaces in jiber reinforced composites n 3 n m (3'2 1 n t "0 10 20 30 40 50 60 (b) L (mm) Fig. 4.28. Comparisons between experiments and theory of (a) maximum debond stress, CJ:, and (b) initial frictional pull-out stress for Sic fiber-glass matrix composites: (0) untreated fibers; (0) acid treated fibers. After Kim et al. (1992). (4.115) Based on Lame's solution, the relationship between the IFSSs is taken as q(b,z) =-Ti(.,.) by2 , (4.116) where y2 = a2/(B2 - a2). The additional radial stress, q2(arz), acting at the fiber- matrix interface, which is caused by Poisson contraction of the fiber when subjected UYl Chapter 4. Micromechanics of stress transfer I43 Fig. 4.29. Schematic illustration of fiber pull-out test on a three cylinder composite. After Kim et al. (l994b). to an axial tension, is obtained from the continuity of tangential strain at the interface (4.117) where cq = E,/Ec and kl = 1 + 2y - v, + a1 (1 + 2yl + vc). Eq. (4.1 17) replaces ql (a, z) given by Eq. (4.18) applied for the single fiber composite model. Combining Eqs. (4.12) and (4.1 13) to (4.117) yields a differential equation for the FAS (4.118) The coefficients A3 and A4 are complex functions of the elastic properties and geometric factors of the constituents and are given in Appendix D. The solution for Eq. (4.1 18) is subjcctcd to the following boundary conditions assuming an unbonded cross-section of the embedded fiber end rq0) = 0, cr',(L) = 0 . (4.1 19) Therefore, the solutions for the FAS, MAS, MSS and IFSSs normalized with the applied stress 0, are obtained: @+ 1) sinh[fi(L -z)] +%sinh(&z) d sinh (&L) A3 ' $(z) - [...]... displacement between fiber and matrix to increase gradually There are Engineered interfaces in fiber reinforced composites 158 (a) (b) Fig 4.41 Schematic drawings of loading and unloading processes measuring the relative displacements6 and 6, in (a) fiber pull-out and (b) fiber push-out models under cyclic loading After Zhou et al (1993) two types of relative displacements of particular interest in this analysis:... the debonded interface, which are distinct in each loading 1 56 Engineered interfaces in jiber reinforced composites "0 Fig 4.39 Comparisons of initial debond stress, uo, and maximum debond stress, ai, between fiber pullout and fiber push-out as a function of embedded fiber length, L, for (a) release agent coated steel fiberepoxy matrix composites and (b) untreated S i c fiber- glass matrix composites. .. ratio, b / a , showing the criterion for debonding initiation at embedded fiber end o r loaded fiber end Engineered interfaces in $her reinforced composites 150 fundamental limitation of generating interface properties only valid in the comparative sense for given conditions that seldom represent those of practical composites of large fiber 6 In this regard, the use of multiple fiber composite specimens.. .Engineered interfaces in fiber reinforced composites 144 @ + 1) sinh[&(L-z)] - Y2 sinh (&L) +2sinh(&z) 1 Finite element analysis (FEA) is also developed in parallel to validate the results generated from the micromechanical model Both the composites containing single and multiple fibers are considered for the present FEA The geometry, the loading method and the boundary... the loaded end, allowing debond initiation from the embedded fiber end in preference to Engineered interfaces in fiber reinforced composites 1 46 1.21 z a 5 0.4 a, 3 5 0.2 C CI 0 ' (b) 0 .6 0.8 Axial distance, z/L 0.2 0.4 1 Fig 4.31 Distributions of (a) fiber axial stress and (b) interface shear stress along the axial direction obtained from micromechanics analysis for different fiber volume fractions,... between the IFSS and the resultant interfacial radial stress given by Eq (4.29), which are now replaced by: 2b Fig 4. 36 Schematic drawing of the partially debonded fiber in fiber push-out test Engineered interfaces in jiber reinforced composites 152 (4.127) In the same procedure as that employed for the fiber pull-out test, the solutions for stress distributions are obtained in the bonded region, which are... used in these models 4.3.7 Two-M1ay debonding phenomenon In the light of the discussion presented in Section 4.3 .6, it is seen that the surrounding composite medium in the three-cylinder composite model acts as a stiff annulus to suppress the development of IFSS at the embedded fiber end by constraining the radial boundary of the matrix cylinder This ensures that regardless Engineered interfaces in fiber. .. Schematic illustrations of the finite element models of (a) single fiber pull-out specimen and (b) a three cylinder composite After Kim et al (1994b) increasing 6 It is also interesting to note that the single fiber composite model predicts that the IFSS obtained at the loaded end remains almost constant regardless of 6 The pronounced effect of fiber 6 is further manifested in Figs 4.33 and 4.34, where... The results presented in Section 4.3 .6 suggest that the shear lag models based on a single fiber composite is inadequate for modelling a composite with a high fiber 6 From the experimental viewpoint, to measure the relevant fiber- matrix interface properties, the fiber volume fraction in single fiber pull-out tests is always very low (i.e 6 < 0.01) This effectively means that testing with these specimens... always expected to initiate at the loaded fiber end for all 6, if the shear strength criterion is employed for the interface debonding However, for the single fiber composite model, IFSS at the embedded fiber end increases rapidly whereas that obtained at the loaded fiber end either remains almost constant (Fig 4.33) or decreases with increasing 6 (Fig 4.34) Therefore, there is a critical fiber volume fraction . are obtained: @+ 1) sinh[fi(L -z)] +%sinh(&z) d sinh (&L) A3 ' $(z) - 144 Engineered interfaces in fiber reinforced composites @+ 1) sinh[&(L-z)] +2sinh(&z). showing the criterion for debonding initiation at embedded fiber end or loaded fiber end. 150 Engineered interfaces in $her reinforced composites fundamental limitation of generating interface. exceeds that of the loaded end, allowing debond initiation from the embedded fiber end in preference to 1 46 Engineered interfaces in fiber reinforced composites 1.2 1 z 5 0.4 3 5 0.2