- Email: [email protected]
Micromechanics of fibre fragmentation in model epoxy composites reinforced with α-alumina fibres
Composites Part A 29A (1998) 1353–1362 1359-835X/98/$ - see front matter q 1998 Published by Elsevier Science Ltd. All rights reserved
PII: S1359-835X(98)00058-X
Micromechanics of fibre fragmentation in model epoxy composites reinforced with a-alumina fibres
R.B. Yallee and R.J. Young* Manchester Materials Science Centre, UMIST/University of Manchester, Grosvenor Street, Manchester M1 7HS, UK (Received 25 November 1997; revised 4 March 1998; accepted 4 March 1998)
The micromechanics of fibre fragmentation in a-alumina/epoxy model composites have been investigated. Both sized and desized Nextel 610 a-alumina fibres embedded in room- and high-temperature (808C) cured epoxy resin matrices have been used. The technique of luminescence spectroscopy has been used to map the strain along the fibres during tensile loading of the matrices, and the distribution of interfacial shear stress has been derived by using a force-balance consideration. The experimental data were modelled with shear-lag analyses that account for both the elastic load transfer and friction at a debonded interface. The effects of fibre sizing, matrix curing temperature and fibre surface roughness upon the interfacial shear strength have also been determined. q 1998 Published by Elsevier Science Ltd. All rights reserved. (Keywords: C. micromechanics; B. fragmentation; a-alumina fibres; epoxy matrix; luminescence spectroscopy)
INTRODUCTION The fragmentation test is used widely to characterise fibre/ matrix interfacial adhesion in polymer-matrix composites1. In the test, a tensile load is applied to a dog-bone specimen consisting of a single brittle fibre embedded totally in a resin matrix. As the load is increased, the fibre breaks into fragments. The test is completed when the fibre breakage reaches the saturation level. After the test is completed, the fragment length is measured by optical microscopy2,3 or by acoustic emission4. The interfacial shear strength of the composite, t s, can be determined from the balance of forces which leads to the Kelly–Tyson equation5 ts ¼
rf jpf lc
(1)
where r f is the fibre radius, l c is the fibre critical length and jpf is fibre tensile strength at the critical length. This equation assumes a constant shear stress at the interface which is generally not the case, as will be demonstrated in this present investigation. Recently, luminescence spectroscopy has been used to study the deformation micromechanics of a-alumina/glass model composites6 and monitor the fragmentation process in a-alumina/epoxy model composites7. The stress- and strain-sensitive fluorescence R 2 line was used to map the * Corresponding author.
stress or strain distribution along PRD-166 alumina– zirconia fibres in the matrices at various levels of applied load. The corresponding interfacial shear stress, t i, distribution was derived by using Ef rf def (2) ti ¼ 2 dx where r f is fibre radius, E f is Young’s modulus of the fibre, and de f/dx is the differential of fibre strain with distance, x, along the fibre. In this present study, luminescence spectroscopy has been used to determine the distribution of strain along Nextel 610 a-alumina fibres in an epoxy matrix during fragmentation testing. The distribution of interfacial shear stress is then derived from the distribution of fibre strain. Both room- and high-temperature-cured epoxy resin matrices have been employed in order to study the effect of thermal stresses. The effect of fibre sizing on interfacial adhesion has been investigated by using both sized and desized fibres. The experimental fibre strain distributions have been compared with the prediction from total bonding8 and debonding5,9 shear-lag models. Nextel fibres are intended primarily to be used for reinforcement in metal- or ceramic-matrix composites rather than epoxy resins. However, the strong fluorescence spectra and well-defined stress-induced band shifts that are found for a-alumina fibres7 allow detailed measurements of fibre deformation to be undertaken which enable the
1353
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young Table 1 Materials properties of the Nextel fibres and the epoxy resin Properties
Sized Nextel
Desized Nextel
10
380 6 20 ( ) 11.9 6 0.7 (11) – –
Epoxy resin Cold cured
Young’s modulus (GPa) Diameter (mm) Shear yield stress (MPa) Poisson’s ratio
380 ( ) 12 (10)
11
Hot cured
12
3.0 (12) – 45 (12)
3.1 ( ) – 41 (12) 0.35 (13)
fundamental mechanisms of fibre reinforcement to be investigated with a high degree of precision by using model epoxy-matrix testpieces.
EXPERIMENTAL Materials and composite fabrication Nextel 610 a-alumina ( . 99%) fibres were used in this study. The as-received Nextel 610 fibres have a poly(vinyl alcohol) (PVA) sizing which can be removed by heating the fibres to 7008C. The manufacturer’s values of diameter and tensile modulus10 of the sized Nextel fibres are given in Table 1. The measured diameter and tensile modulus of desized Nextel fibres are also included in Table 111. An epoxy resin system that could be cured either at room temperature or elevated temperature was used as a matrix for the model composites, so that the effects of residual stresses could be investigated7. For the cold-cured epoxy, the resin mixture was cured at room temperature, 22 6 28C, for 7 days. The hot-cured epoxy was prepared by curing the resin at room temperature for 24 h and then at 808C for 8 h. The mechanical properties of the epoxy resin matrix are also given in Table 1. The fabrication of the single-fibre composite specimens has been described in detail elsewhere7.
Figure 1 Variation of the wavenumber of the fluorescence R 2 band for sized Nextel fibres with (a) applied strain (arrows indicate fibre failure) and (b) applied tensile stress
Luminescence spectroscopy Fluorescence spectra were obtained from Nextel 610 fibres and single-fibre composites during deformation by means of an unmodified Raman microprobe system7. The 632.8 nm red line of a 15 mW He–Ne laser was used to excite the fluorescence. Fluorescence spectra were obtained from fibres, bonded to the surface of poly(methyl methacrylate) strips, during deformation in a small four-point bending rig7. Spectra were also obtained from fibres during deformation in air in a specially constructed micrometer-driven straining rig with a calibrated ,2 N load cell. The fibre diameter was determined by using scanning electron microscopy (SEM) and a calibrated grid to determine the exact microscope magnification. Single-fibre composite specimens were deformed on a Minimat straining rig7. The strain in the matrix was measured with a resistance strain gauge. Spectra were recorded along the length of a fibre in a specimen (after 10– 15 s exposure time) at various levels of applied matrix strain.
1354
RESULTS AND DISCUSSION Strain and stress dependencies of fluorescence R 2 band for Nextel 610 fibres The dependence of the peak position of the fluorescence R 2 band from sized Nextel fibres on applied strain and tensile stress was determined by using the four-point bending and micrometer straining test rigs, respectively7. It can be seen from Figure 1a and Figure 1b that approximately linear shifts of the R 2 band position with both applied strain and tensile stress are obtained. Similar results were found for desized Nextel fibres11. The strain- or stress-induced shift is due to microscopic deformation of the a-alumina crystal lattice during macroscopic deformation of the fibre14. Table 2 gives the strain and stress dependences of the R 2 band position for sized Nextel and PRD-166 alumina– zirconia7 fibres. The Young’s modulus of the fibres can be estimated from the ratio of values of the tensile strain- and
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young Table 2 Strain and stress dependences of the fluorescence R 2 line Strain dependence (cm ¹1/%)
Sized Nextel PRD-166
Compressive
Tensile
Stress dependence (cm ¹1/GPa) Tensile
12.2 6 0.6 10.6 6 0.5
13.3 6 0.6 10.9 6 0.5
3.4 6 0.2 2.9 6 0.2
Figure 3 (a) Measured and theoretical (using eqn (3) with R/r f ¼ 10) variation of axial strain along a sized Nextel fibre in a cold-cured composite at different matrix strain levels. (b) Derived (using eqn (2)) variation of interfacial shear stress with distance along the fibre
Figure 2 (a) Measured and theoretical (using eqn (3) with R/r f ¼ 10) variation of axial strain along a desized Nextel fibre in a cold-cured composite at different matrix strain levels. (b) Derived (using eqn (2)) variation of interfacial shear stress with distance along the fibre
stress-dependent R 2 band-shift rates7. A tensile modulus of about 390 GPa for sized Nextel fibres is generated, which is comparable to 380 GPa quoted by the manufacturer10. The modulus of the Nextel 610 fibres is higher than that found for PRD-166 fibres (375 GPa)7. It can be seen that the fluorescence R 2 band-shift rate for the sized Nextel fibres is also higher than that for the PRD-166 fibres (see Table 2). This higher band-shift rate may be due the fact that the Nextel fibres have a higher modulus and are essentially pure a-alumina10 whereas the PRD-166 fibres contain ,20% zirconia7. Strain mapping and shear stress distributions at low matrix strains Nextel 610/cold-cured epoxy model composites. Figure 2a and Figure 3a show the distributions of strain with distance along the fibres at various levels of applied matrix strain for
the desized- and sized-Nextel/cold-cured epoxy single-fibre composite specimens. The fibre strain distribution was derived from the corresponding distribution of the straininduced fluorescence R 2 band shift, using the strain dependence given in Table 2 to convert the band shift to fibre strain. It can be seen that, before deformation (e m ¼ 0%), there is no strain in the fibres. As the applied matrix strain is increased, the fibre strain increases from the fibre ends to a maximum value along the central regions of the fibres. The fibre strain distributions in Figure 2a and Figure 3a are similar to those predicted by the shear-lag (Cox) model8, which assumes full bonding. The variation in tensile fibre strain, e f, with distance, x, along a fibre of length l is given by15: ÿ cosh nx=rf (3) ef ¼ em 1 ¹ cosh(ns) where n 2 ¼ E m/E f(1 þ n m) ln(R/r f), E f is the Young’s modulus of the fibre, E m is the Young’s modulus of the matrix, e m is the applied matrix strain, n m is the Poisson’s ratio of the matrix, r f is the radius of the fibre, R is the radius of the cylinder of resin around the fibre and s is the fibre aspect ratio, l/2r f. The experimental data in Figure 2a and Figure 3a have been fitted to the theoretical curves (solid lines) calculated from eqn (3) at each level of applied matrix strain. The
1355
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 4 Measured and theoretical (eqn (3)) variation of strain along the left end of the desized Nextel fibre in a cold-cured composite at e m ¼ 0.2% (best fit when R/r f ¼ 10)
average strain in the central region of the fibre has been used instead of the applied matrix strain because of slight discrepancies in measurement of applied matrix strain (60.05%). The value of R/r f determined from the specimen and fibre dimensions in the single-fibre composites is of the order of 100. However, the value determined empirically by comparing the theoretical (eqn (3)) and measured fibre strain distributions is much smaller. As can be seen in Figure 4, the best fit of the measured data is obtained when the ratio R/r f is 10. The ratio used is comparable to the value of 10 to 15 obtained in the case of aramid/cold-cured epoxy single-fibre composites16,17. Recently, Nairn18 investigated the accuracy of shear-lag analyses to model stress transfer in unidirectional composites and compared shear-lag predictions with finite element analysis (FEA) calculations of distributions of axial fibre stress and matrix shear stress at different fibre volume fractions. He demonstrated that shear-lag analysis made reasonable predictions of axial fibre stresses but was worse at predicting shear stresses. Nairn pointed out that the only way that shear-lag analysis can sensibly be used on singlefibre specimens is by treating the ratio R/r f or n in eqn (3) as an adjustable parameter instead of a defined constant. The ratio R/r f or n can be determined by comparison of shear-lag predictions and FEA calculations18 or by comparison with experimental results on stress transfer, such as those obtained with Raman spectroscopy16,17. In the present study excellent agreement was obtained with shear-lag analysis at low levels of applied matrix strain since both the fibre and matrix are in an elastic state and the fibre/matrix interface is intact. It can be seen from Figure 2a that at 0.4% matrix strain, the measured strain near the right end of the HT Nextel fibre is slightly lower than the predicted strain. This may be due to stress relaxation of the epoxy resin matrix during measurement and/or time dependence of the debonding process. At higher applied matrix strains, the fibre strain increases from the ends at a slower rate than predicted, due probably to interfacial debonding which was not considered in the full-bonding shear-lag (Cox) model8. The debonding occurred at e m ¼ 0.6% for the sized Nextel fibre (Figure 3a). It is noteworthy
1356
that the debonded region is more obvious at the right end than at the left end of the fibres. This behaviour was often observed. Since measurements were made by starting from the left end of the fibre and moving along to the right end, interfacial debonding may have occurred at the right end during static loading over a period of about an hour. Figure 2b and Figure 3b show the distributions of interfacial shear stress with distance along the fibre at various levels of applied matrix strain for desized- and sized-Nextel/cold-cured epoxy single-fibre composite specimens. The interfacial shear stress distributions were derived by differentiating cubic spline fits of the experimental data in Figure 2a and Figure 3a and substituting into eqn (2). It can be seen that at low levels of applied matrix strain the interfacial shear stress is a maximum at the fibre ends and decreases to zero at some distance along the fibre. The interfacial shear stress distributions are similar to those predicted by the full-bonding shear-lag (Cox) model8. At higher applied matrix strain levels, however, the maximum interfacial shear stress occurred at some distance away from the fibre ends due to interfacial debonding. This is clearly seen for the sized Nextel fibre at e m ¼ 0.6 and 0.8% (Figure 3b). Nextel 610/hot-cured epoxy composites. The distributions of axial fibre strain with distance along the fibre at various
Figure 5 (a) Measured and theoretical (using eqn (3) with R/r f ¼ 10) variation of axial strain along a desized Nextel fibre in a hot-cured composite at different matrix strain levels. (b) Derived (using eqn (2)) variation of interfacial shear stress with distance along the fibre
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 7 Measured and theoretical (eqn (3)) variation of strain along the left end of the desized Nextel fibre in a hot-cured composite at e m ¼ 0.2% (best fit when R/r f ¼ 10)
Figure 6 (a) Measured and theoretical (using eqn (3) with R/r f ¼ 10) variation of axial strain along a sized Nextel fibre in a hot-cured composite at different matrix strain levels. (b) Derived (using eqn (2)) variation of interfacial shear stress with distance along the fibre
levels of applied matrix strain for desized- and sized-Nextel/ hot-cured epoxy single-fibre composite specimens are shown in Figure 5a and Figure 6a, respectively. At the zero applied matrix strain, there is an initial compressive strain of about 0.34% along the central region of the fibre. This residual thermal strain developed during cooling of the composite from the cure temperature (808C) owing to the mismatch of thermal expansion coefficients between the fibre and the matrix, as discussed elsewhere7. As the applied matrix strain is increased the initial compressive strain is recovered and the fibre strain increases from the ends to a maximum value over the central region of the fibre as for the cold-cured specimens. The solid lines in Figure 5a and Figure 6a were calculated by using eqn (3). In this case, the average value of strain along the central region of the fibre rather than the applied matrix strain has been used because of the presence of the initial compressive strain in the fibre. At an applied matrix strain of less than 1.0%, the theoretical data had the best fit to the experimental data when the ratio R/r f was 10 (for both tension and compression), as in the case of the cold-cured composites. At higher matrix strains, the Cox shear-lag model with R/r f ¼ 10 does not fit the experimental data, probably due to interfacial debonding and/or matrix yielding which is not accounted for in the model. It should
be noted that plastic deformation at the interface was found to start to occur in a similar Kevlar/epoxy system for e m . 1.0%12. The measured fibre strain distribution for the left part of a desized Nextel fibre in hot-cured epoxy at e m ¼ 1.0% (shown in Figure 5a) is compared in Figure 7 with the shear-lag predictions (eqn (3)) for different values of R/r f. As can be seen, the best fit is obtained when R/r f ¼ 75. It can be seen therefore that for both the cold- and hot-cured specimens, there is a tendency for the ratio R/r f to increase (or n to decrease) as the matrix strain is increased. This can be taken to be an indication of a decrease in the efficiency of stress transfer across the fibre/matrix interface owing the onset of fibre/matrix debonding and/or matrix yielding. These observations are strictly relevant only to model single-fibre composites. It is clear that R/r f might be different for high-volume-fraction composites, although the observations of the increase of R/r f with matrix strain could be important for all types of composite. The distributions of interfacial shear stress with distance along the fibre at various levels of applied matrix strain for desized- and sized-Nextel/hot-cured epoxy single-fibre composite specimens are shown in Figure 5b and Figure 6b, respectively. The distributions of interfacial shear stress were derived by differentiating the cubic spline fits of the distributions of fibre strain shown in Figure 5a and Figure 6a, and applying eqn (2). It can be seen that the maximum interfacial shear stress occurs at the fibre ends and that the interfacial shear stress falls to zero along the middle of the fibre. This is similar to the behaviour predicted by the fullbonding (Cox) shear-lag model8. The interfacial shear stress distribution clearly deviates from the prediction at 1.2% matrix strain in the case of the sized Nextel fibre. It can be seen from Figure 6b that the maximum interfacial shear stress occurs away from the right end of the fibre due to interfacial debonding and/or matrix yielding. Fibre fragmentation Nextel 610/cold-cured epoxy composites. The distribution of fibre strain along a fragmented, desized Nextel fibre in a
1357
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 8 Variation of axial fibre strain with distance along a fragmented, desized Nextel fibre in the cold-cured epoxy resin matrix at 0.7 and 0.9% matrix strain
Figure 10 Variation of (a) fibre strain and (b) interfacial shear stress with distance along a fragmented, sized Nextel fibre in the cold-cured epoxy resin matrix at e m ¼ 1.2%
Figure 9 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along the desized Nextel fragment (2) in the cold-cured epoxy resin matrix at e m ¼ 0.9%
cold-cured epoxy matrix is shown in Figure 8. A single fibre break was obtained at 0.7% matrix strain and, when the matrix strain was increased further to 0.9%, three additional breaks occurred. Figure 9a and Figure 9b show the detailed fibre strain and interfacial shear stress distributions along fragment 2 in Figure 8 at 0.9% matrix strain. It can be seen that the fibre strain distribution along the fragment is almost triangular and the interfacial shear stress is relatively constant at about 25 MPa near the fragment ends. At higher
1358
matrix strain levels, similar strain and shear stress distributions were obtained. Such distributions imply debonding or matrix yielding over the entire fibre/matrix interface5,15 and they are similar to those predicted by Kelly and Tyson5 for the case of an elastic fibre in a plastic matrix. Figure 10a shows the distribution of fibre strain with distance along a fragmented, sized Nextel fibre in the coldcured epoxy matrix at e m ¼ 1.2%, where one break had occurred at the fibre centre. The distribution of interfacial shear stress along the fragmented sized Nextel fibre at 1.2% matrix strain is shown in Figure 10b. It can be seen that the distributions are different from those of the desized Nextel fragments (Figure 9b). Well-defined linear strain distributions and constant shear stresses occur over regions near the fragment ends. The linear strain distribution and region of constant shear stress of ,10 MPa correspond to a debonded interface 9. It can be seen that debonded lengths near the original fibre ends are longer than at the broken ends. This means that interfacial debonding probably occurred at the original ends, both during matrix loading to 1.2% and upon fibre fracture. The frictional shear stress is about 8 MPa. In the central region of each fragment, a different type of fibre strain distribution and interfacial shear stress are obtained than for the desized fibres. Here, good adhesion between the fibre and matrix appears to still hold and load transfer is elastic. The average value of maximum interfacial shear stress is about 25 MPa, which is lower than the matrix shear yield strength of 41 MPa12.
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 11 Variation of axial fibre strain with distance along a fragmented, desized Nextel fibre in the hot-cured epoxy resin matrix at 1.2% matrix strain
Figure 13 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along a desized Nextel fragment (5) in the hot-cured epoxy resin matrix at 1.5% matrix strain
Figure 12 Variation of axial fibre strain with distance along a fragmented, desized Nextel fibre in the hot-cured epoxy resin matrix at 1.5% matrix strain
Clearly, the full-bonding8 and complete-debonding models5,15 are not suitable for describing the axial fibre strain and interfacial shear stress distribution shown for the fragments in Figure 10. In such a case, a partial-debonding model9 is needed and this will be discussed further later. Nextel 610/hot-cured epoxy composites. Figures 11 and 12 show the distributions of fibre strain with distance along a fragmented, desized Nextel fibre in a hot-cured epoxy matrix at different levels of matrix strain. For this sample, the fibre broke into two unequal-length fragments at 1.2% matrix strain (Figure 11). When the matrix strain was increased to 1.5% (Figure 12), each fragment underwent further fracture. Figures 13 and 14 show the distributions of (a) fibre strain and (b) interfacial shear stress along the desized Nextel fragment 5 in Figure 12 at 1.5% and 3.0% matrix strain, respectively. At 1.5% matrix strain, there is an approximately linear strain distribution and a constant interfacial shear stress region extending over a distance of about 100 mm from the fragment ends. A constant shear stress of about 30 MPa is obtained at the fragment ends and it seems
that interfacial debonding was initiated when the fibre fragmented. When the matrix strain is increased to 3.0% (Figure 14a), the debonded region increases to 150 mm. Over the debonded region, the fibre strain now increases at a lower rate and the shear stress is constant at 15 MPa (Figure 14b). The frictional shear stress, t f, could be reduced at the higher matrix strain value by degradation of the interface by sliding of the ceramic fibre. A smoother polymer matrix surface would be obtained and consequently a lower value of t f measured. The maximum interfacial shear stress is about 43 MPa, which is similar to the matrix shear yield stress12, showing that the fibre is still well bonded in the central region. The fibre strain and shear stress distributions in Figures 13 and 14 tend to be symmetrical about the fragment centre unlike those of sized Nextel fragments in the cold-cured epoxy (Figure 10). Figures 15 and 16 show the distributions of (a) fibre strain and (b) interfacial shear stress along a sized Nextel fragment in a hot-cured epoxy matrix at 1.4% and 3.0% matrix strain, respectively. The distributions are similar to those for the desized Nextel fragment in the hot-cured epoxy matrix (Figures 13 and 14) and correspond to the prediction of the partial-debonding model9. A frictional shear stress of about 15 MPa and a maximum interfacial shear stress of about 40 MPa are also obtained at the 3.0% matrix strain. A notable difference in the transition zone between the
1359
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 14 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along a desized Nextel fragment (5) in the hot-cured epoxy resin matrix at 3.0% matrix strain
debonded and bonded region of the desized and sized Nextel fragment occurred at e m ¼ 3.0%. For the desized Nextel fragment, the transition zone is broad and extends from x ¼ 6 50 mm to x ¼ 6 150 mm (Figure 14). A considerably sharper transition zone occurs at x ¼ 6 75 mm for the sized Nextel fragment (Figure 16a). Partial-debonding model. The partial-debonding model proposed by Piggott9,15 can be used to describe the axial strain distribution along a fragment with a debonded interface that extends over a section of the fibre length. This model assumes that debonding occurs over a distance ml/2 from the fibre ends (with 0 , m , 1) and that perfect adhesion holds in the central region of the fibre. Stress is transferred elastically in the bonded region while in the debonded region stress is transferred via friction. For the debonded region, the interfacial shear stress is purely frictional and so: ti ¼ tf
(4)
The frictional shear stress, t f, is related to the radial pressure at the interface which arises from both the thermal radial stress and the Poisson contraction3. The fibre strain distribution for the debonded region is: 2t l ¹x (5) ef ¼ f rf Ef 2
1360
Figure 15 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along a sized Nextel fragment in the hot-cured epoxy resin matrix at 1.4% matrix strain
For the bonded region, the fibre strain distribution is: ÿ cosh nx=rf 2t sm ef ¼ e ¹ e ¹ f cosh[ns(1 ¹ m)] Ef
(6)
The solid lines in Figure 13a, Figure 14a, Figure 15a and Figure 16a are the best fits of the partial-debonding model and they were obtained when the ratio R/r f ¼ 75. It should be noted that the same R/r f value was used for the desized Nextel fibre in the hot-cured epoxy at 1.0% matrix strain as shown in Figure 7. The value is large in comparison to the ratio R/r f ¼ 10 used for the full-bonding case at low matrix strains and it may be taken as an indication of the decrease in efficiency of stress transfer across the interface at higher strains as discussed earlier. The partial-debonding model9,15 does not take into account thermal stresses. To account for the thermal residual strains in the fibre, the applied strain e in eqn (6) should be (e m þ e T), where e T is the compressive (negative) thermal strain in the fibre at e m ¼ 0%19. It can be seen from Figure 13a, Figure 14a, Figure 15a and Figure 16a that there is good agreement between the theoretical predictions and experimental results except in a region near the bonding/debonding transition.
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 17 Scanning electron micrograph of a desized Nextel fibre
Figure 16 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along a sized Nextel fragment in the hot-cured epoxy resin matrix at 3.0% matrix strain
Table 3 The maximum value of t i for a-alumina fibre/epoxy systems Maximum value of t i ( ; t s) (MPa) Desized Nextel Sized Nextel PRD-166 (unsized)
Cold-cured epoxy
Hot-cured epoxy
26 6 2 38 6 2 42 6 2
38 6 3 37 6 7 44 6 2
Interfacial shear strength The maximum measured values of t i for each of the four different Nextel 610 fibre/epoxy systems are given in Table 3 along with some earlier findings for the PRD-166/ epoxy system7. The effect of sizing and resin curing temperature on fibre/matrix interfacial adhesion can be clearly seen from Table 3. The maximum values can be used to estimate the interfacial shear strengths (assuming t s ; t i,max) for the different Nextel 610 fibre/epoxy systems. The interfacial shear strength of the sized-Nextel fibre/ cold-cured epoxy composite is higher than that of the desized-Nextel fibre/cold-cured epoxy composite. It seems that the PVA sizing promotes better adhesion between the Nextel 610 fibres and epoxy matrix. An increase in interfacial shear strength is also obtained when the desized Nextel fibre is embedded in the hot-cured epoxy matrix rather than in the cold-cured epoxy matrix. This could be due to the radial compressive stresses across the fibre/matrix
Figure 18 Scanning electron micrograph of a PRD-166 fibre
interface in the hot-cured system, making debonding more difficult. An increase in interfacial shear strength is not obtained for the sized Nextel fibres and similar values of shear strength are obtained with both the cold- and hotcured epoxy matrix. It appears that the radial compression does not have an effect upon the inherently stronger interface for the sized Nextel fibres. Alternatively, the size on the fibres may form an interphase which could control the mechanical behaviour of the interfacial regions. The interfacial shear strength of the Nextel 610 fibre/ epoxy system is lower than the shear yield stress of the resin (41–45 MPa) and thus the shear strength appears to be limited by debonding at the fibre/matrix interface. This is to be contrasted with earlier findings for the PRD-166 fibre/ epoxy system, whereby the higher level of interfacial shear strength appeared to be controlled by the shear yield strength of the epoxy resin matrix7. The PRD-166 and Nextel fibres differ in both composition and microstructure. When they are examined by SEM, the Nextel 610 fibres are seen to have relatively smooth surfaces (Figure 17) compared with the PRD-166 fibres which have much
1361
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young rougher surfaces (Figure 18). The grain size for the PRD166 fibres is about 0.5 mm whereas it is significantly smaller for the Nextel 610 fibres. One reason for the lower interfacial shear strength for the Nextel 610 fibres in the epoxy resin matrix may be simply due to a difference in surface roughness leading to better mechanical interlocking between PRD-166 fibres and the epoxy resin matrix.
ACKNOWLEDGEMENTS
CONCLUSIONS
REFERENCES
It has been demonstrated that luminescence spectroscopy can be used to study the micromechanics of load transfer in a-alumina fibre/epoxy composites during fragmentation testing. At low matrix strain levels before fibre fragmentation occurs, the fibre strain profiles along the Nextel 610 alumina fibres have been found to be in excellent agreement with those predicted by the Cox shear-lag model5. Deviations from the model are found to occur at higher levels of matrix strain as a consequence of the onset of matrix yielding and/or interfacial debonding. It has been found that a fibre fracture is accompanied by debonding of the fibre/matrix interface, and the extent of the initial debonded region is dependent upon the level of interfacial adhesion. A more pronounced debonded region is obtained for the desized Nextel fibre than for the sized Nextel fibre in the cold-cured epoxy matrix. It has also been shown that the experimental fibre strain distributions and shear stress profiles following fragmentation can be modelled well by using a partial-debonding model9,15, except near the boundary between the bonded and debonded regions of the interface.
1362
One of the authors (RJY) would like to thank the Royal Society for support in the form of the Wolfson Research Professorship in Materials Science. The work forms part of a large programme of research supported by the Engineering and Physical Sciences Research Council.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Herrera-Franco, P.J. and Drzal, L.T., Composites, 1992, 23, 2. Kalantar, J. and Drzal, L.T., J. Mater. Sci., 1990, 25, 4194. Wagner, H.D., Gallis, H.E. and Weisel, E., J. Mater. Sci., 1993, 28, 205. Netravali, A.N., Li, Z.-F., Sachse, W. and Wu, H.F., J. Mater. Sci., 1991, 26, 6631. Kelly, A. and Tyson, W.R., J. Mech. Phys. Sol., 1965, 13, 329. Yang, X. and Young, R.J., Acta Metall. Mater., 1995, 43, 2407. Yallee, R.B., Andrews, M.C. and Young, R.J., J. Mater. Sci., 1996, 31, 3349. Cox, H.L., Br. J. Appl. Phys., 1952, 3, 57. Piggott, M.R., Compos. Sci. Technol., 1991, 42, 57. 3M Datasheet, Document 11-6273-3. Yallee, R.B., Single-fibre composite micromechanics. Ph.D. thesis, University of Manchester Institute of Science and Technology, 1997. Andrews, M.C., Stress transfer in aramid/epoxy model composites. Ph.D. thesis, Victoria University of Manchester, 1994. Ciba-Geigy Datasheet, Ciba-Geigy, UK. Schawlow, A.L., in Advances in Quantum Electronics, ed. J.R. Singer. Columbia University Press, New York, 1961, p. 50. Lacroix, Th., Tilmans, B., Keunings, R., Desaeger, M. and Verpoest, I., Compos. Sci. and Technol., 1992, 43, 379. Galiotis, C., Compos. Sci. Technol., 1991, 42, 125. Andrews, M.C., Day, R.J., Patrikis, A.K. and Young, R.J., Composites, 1994, 25, 745. Nairn, J.A., Mech. Mater., 1997, 26, 63. Huang, Y. and Young, R.J., Composites, 1995, 26, 541.
PII: S1359-835X(98)00058-X
Micromechanics of fibre fragmentation in model epoxy composites reinforced with a-alumina fibres
R.B. Yallee and R.J. Young* Manchester Materials Science Centre, UMIST/University of Manchester, Grosvenor Street, Manchester M1 7HS, UK (Received 25 November 1997; revised 4 March 1998; accepted 4 March 1998)
The micromechanics of fibre fragmentation in a-alumina/epoxy model composites have been investigated. Both sized and desized Nextel 610 a-alumina fibres embedded in room- and high-temperature (808C) cured epoxy resin matrices have been used. The technique of luminescence spectroscopy has been used to map the strain along the fibres during tensile loading of the matrices, and the distribution of interfacial shear stress has been derived by using a force-balance consideration. The experimental data were modelled with shear-lag analyses that account for both the elastic load transfer and friction at a debonded interface. The effects of fibre sizing, matrix curing temperature and fibre surface roughness upon the interfacial shear strength have also been determined. q 1998 Published by Elsevier Science Ltd. All rights reserved. (Keywords: C. micromechanics; B. fragmentation; a-alumina fibres; epoxy matrix; luminescence spectroscopy)
INTRODUCTION The fragmentation test is used widely to characterise fibre/ matrix interfacial adhesion in polymer-matrix composites1. In the test, a tensile load is applied to a dog-bone specimen consisting of a single brittle fibre embedded totally in a resin matrix. As the load is increased, the fibre breaks into fragments. The test is completed when the fibre breakage reaches the saturation level. After the test is completed, the fragment length is measured by optical microscopy2,3 or by acoustic emission4. The interfacial shear strength of the composite, t s, can be determined from the balance of forces which leads to the Kelly–Tyson equation5 ts ¼
rf jpf lc
(1)
where r f is the fibre radius, l c is the fibre critical length and jpf is fibre tensile strength at the critical length. This equation assumes a constant shear stress at the interface which is generally not the case, as will be demonstrated in this present investigation. Recently, luminescence spectroscopy has been used to study the deformation micromechanics of a-alumina/glass model composites6 and monitor the fragmentation process in a-alumina/epoxy model composites7. The stress- and strain-sensitive fluorescence R 2 line was used to map the * Corresponding author.
stress or strain distribution along PRD-166 alumina– zirconia fibres in the matrices at various levels of applied load. The corresponding interfacial shear stress, t i, distribution was derived by using Ef rf def (2) ti ¼ 2 dx where r f is fibre radius, E f is Young’s modulus of the fibre, and de f/dx is the differential of fibre strain with distance, x, along the fibre. In this present study, luminescence spectroscopy has been used to determine the distribution of strain along Nextel 610 a-alumina fibres in an epoxy matrix during fragmentation testing. The distribution of interfacial shear stress is then derived from the distribution of fibre strain. Both room- and high-temperature-cured epoxy resin matrices have been employed in order to study the effect of thermal stresses. The effect of fibre sizing on interfacial adhesion has been investigated by using both sized and desized fibres. The experimental fibre strain distributions have been compared with the prediction from total bonding8 and debonding5,9 shear-lag models. Nextel fibres are intended primarily to be used for reinforcement in metal- or ceramic-matrix composites rather than epoxy resins. However, the strong fluorescence spectra and well-defined stress-induced band shifts that are found for a-alumina fibres7 allow detailed measurements of fibre deformation to be undertaken which enable the
1353
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young Table 1 Materials properties of the Nextel fibres and the epoxy resin Properties
Sized Nextel
Desized Nextel
10
380 6 20 ( ) 11.9 6 0.7 (11) – –
Epoxy resin Cold cured
Young’s modulus (GPa) Diameter (mm) Shear yield stress (MPa) Poisson’s ratio
380 ( ) 12 (10)
11
Hot cured
12
3.0 (12) – 45 (12)
3.1 ( ) – 41 (12) 0.35 (13)
fundamental mechanisms of fibre reinforcement to be investigated with a high degree of precision by using model epoxy-matrix testpieces.
EXPERIMENTAL Materials and composite fabrication Nextel 610 a-alumina ( . 99%) fibres were used in this study. The as-received Nextel 610 fibres have a poly(vinyl alcohol) (PVA) sizing which can be removed by heating the fibres to 7008C. The manufacturer’s values of diameter and tensile modulus10 of the sized Nextel fibres are given in Table 1. The measured diameter and tensile modulus of desized Nextel fibres are also included in Table 111. An epoxy resin system that could be cured either at room temperature or elevated temperature was used as a matrix for the model composites, so that the effects of residual stresses could be investigated7. For the cold-cured epoxy, the resin mixture was cured at room temperature, 22 6 28C, for 7 days. The hot-cured epoxy was prepared by curing the resin at room temperature for 24 h and then at 808C for 8 h. The mechanical properties of the epoxy resin matrix are also given in Table 1. The fabrication of the single-fibre composite specimens has been described in detail elsewhere7.
Figure 1 Variation of the wavenumber of the fluorescence R 2 band for sized Nextel fibres with (a) applied strain (arrows indicate fibre failure) and (b) applied tensile stress
Luminescence spectroscopy Fluorescence spectra were obtained from Nextel 610 fibres and single-fibre composites during deformation by means of an unmodified Raman microprobe system7. The 632.8 nm red line of a 15 mW He–Ne laser was used to excite the fluorescence. Fluorescence spectra were obtained from fibres, bonded to the surface of poly(methyl methacrylate) strips, during deformation in a small four-point bending rig7. Spectra were also obtained from fibres during deformation in air in a specially constructed micrometer-driven straining rig with a calibrated ,2 N load cell. The fibre diameter was determined by using scanning electron microscopy (SEM) and a calibrated grid to determine the exact microscope magnification. Single-fibre composite specimens were deformed on a Minimat straining rig7. The strain in the matrix was measured with a resistance strain gauge. Spectra were recorded along the length of a fibre in a specimen (after 10– 15 s exposure time) at various levels of applied matrix strain.
1354
RESULTS AND DISCUSSION Strain and stress dependencies of fluorescence R 2 band for Nextel 610 fibres The dependence of the peak position of the fluorescence R 2 band from sized Nextel fibres on applied strain and tensile stress was determined by using the four-point bending and micrometer straining test rigs, respectively7. It can be seen from Figure 1a and Figure 1b that approximately linear shifts of the R 2 band position with both applied strain and tensile stress are obtained. Similar results were found for desized Nextel fibres11. The strain- or stress-induced shift is due to microscopic deformation of the a-alumina crystal lattice during macroscopic deformation of the fibre14. Table 2 gives the strain and stress dependences of the R 2 band position for sized Nextel and PRD-166 alumina– zirconia7 fibres. The Young’s modulus of the fibres can be estimated from the ratio of values of the tensile strain- and
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young Table 2 Strain and stress dependences of the fluorescence R 2 line Strain dependence (cm ¹1/%)
Sized Nextel PRD-166
Compressive
Tensile
Stress dependence (cm ¹1/GPa) Tensile
12.2 6 0.6 10.6 6 0.5
13.3 6 0.6 10.9 6 0.5
3.4 6 0.2 2.9 6 0.2
Figure 3 (a) Measured and theoretical (using eqn (3) with R/r f ¼ 10) variation of axial strain along a sized Nextel fibre in a cold-cured composite at different matrix strain levels. (b) Derived (using eqn (2)) variation of interfacial shear stress with distance along the fibre
Figure 2 (a) Measured and theoretical (using eqn (3) with R/r f ¼ 10) variation of axial strain along a desized Nextel fibre in a cold-cured composite at different matrix strain levels. (b) Derived (using eqn (2)) variation of interfacial shear stress with distance along the fibre
stress-dependent R 2 band-shift rates7. A tensile modulus of about 390 GPa for sized Nextel fibres is generated, which is comparable to 380 GPa quoted by the manufacturer10. The modulus of the Nextel 610 fibres is higher than that found for PRD-166 fibres (375 GPa)7. It can be seen that the fluorescence R 2 band-shift rate for the sized Nextel fibres is also higher than that for the PRD-166 fibres (see Table 2). This higher band-shift rate may be due the fact that the Nextel fibres have a higher modulus and are essentially pure a-alumina10 whereas the PRD-166 fibres contain ,20% zirconia7. Strain mapping and shear stress distributions at low matrix strains Nextel 610/cold-cured epoxy model composites. Figure 2a and Figure 3a show the distributions of strain with distance along the fibres at various levels of applied matrix strain for
the desized- and sized-Nextel/cold-cured epoxy single-fibre composite specimens. The fibre strain distribution was derived from the corresponding distribution of the straininduced fluorescence R 2 band shift, using the strain dependence given in Table 2 to convert the band shift to fibre strain. It can be seen that, before deformation (e m ¼ 0%), there is no strain in the fibres. As the applied matrix strain is increased, the fibre strain increases from the fibre ends to a maximum value along the central regions of the fibres. The fibre strain distributions in Figure 2a and Figure 3a are similar to those predicted by the shear-lag (Cox) model8, which assumes full bonding. The variation in tensile fibre strain, e f, with distance, x, along a fibre of length l is given by15: ÿ cosh nx=rf (3) ef ¼ em 1 ¹ cosh(ns) where n 2 ¼ E m/E f(1 þ n m) ln(R/r f), E f is the Young’s modulus of the fibre, E m is the Young’s modulus of the matrix, e m is the applied matrix strain, n m is the Poisson’s ratio of the matrix, r f is the radius of the fibre, R is the radius of the cylinder of resin around the fibre and s is the fibre aspect ratio, l/2r f. The experimental data in Figure 2a and Figure 3a have been fitted to the theoretical curves (solid lines) calculated from eqn (3) at each level of applied matrix strain. The
1355
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 4 Measured and theoretical (eqn (3)) variation of strain along the left end of the desized Nextel fibre in a cold-cured composite at e m ¼ 0.2% (best fit when R/r f ¼ 10)
average strain in the central region of the fibre has been used instead of the applied matrix strain because of slight discrepancies in measurement of applied matrix strain (60.05%). The value of R/r f determined from the specimen and fibre dimensions in the single-fibre composites is of the order of 100. However, the value determined empirically by comparing the theoretical (eqn (3)) and measured fibre strain distributions is much smaller. As can be seen in Figure 4, the best fit of the measured data is obtained when the ratio R/r f is 10. The ratio used is comparable to the value of 10 to 15 obtained in the case of aramid/cold-cured epoxy single-fibre composites16,17. Recently, Nairn18 investigated the accuracy of shear-lag analyses to model stress transfer in unidirectional composites and compared shear-lag predictions with finite element analysis (FEA) calculations of distributions of axial fibre stress and matrix shear stress at different fibre volume fractions. He demonstrated that shear-lag analysis made reasonable predictions of axial fibre stresses but was worse at predicting shear stresses. Nairn pointed out that the only way that shear-lag analysis can sensibly be used on singlefibre specimens is by treating the ratio R/r f or n in eqn (3) as an adjustable parameter instead of a defined constant. The ratio R/r f or n can be determined by comparison of shear-lag predictions and FEA calculations18 or by comparison with experimental results on stress transfer, such as those obtained with Raman spectroscopy16,17. In the present study excellent agreement was obtained with shear-lag analysis at low levels of applied matrix strain since both the fibre and matrix are in an elastic state and the fibre/matrix interface is intact. It can be seen from Figure 2a that at 0.4% matrix strain, the measured strain near the right end of the HT Nextel fibre is slightly lower than the predicted strain. This may be due to stress relaxation of the epoxy resin matrix during measurement and/or time dependence of the debonding process. At higher applied matrix strains, the fibre strain increases from the ends at a slower rate than predicted, due probably to interfacial debonding which was not considered in the full-bonding shear-lag (Cox) model8. The debonding occurred at e m ¼ 0.6% for the sized Nextel fibre (Figure 3a). It is noteworthy
1356
that the debonded region is more obvious at the right end than at the left end of the fibres. This behaviour was often observed. Since measurements were made by starting from the left end of the fibre and moving along to the right end, interfacial debonding may have occurred at the right end during static loading over a period of about an hour. Figure 2b and Figure 3b show the distributions of interfacial shear stress with distance along the fibre at various levels of applied matrix strain for desized- and sized-Nextel/cold-cured epoxy single-fibre composite specimens. The interfacial shear stress distributions were derived by differentiating cubic spline fits of the experimental data in Figure 2a and Figure 3a and substituting into eqn (2). It can be seen that at low levels of applied matrix strain the interfacial shear stress is a maximum at the fibre ends and decreases to zero at some distance along the fibre. The interfacial shear stress distributions are similar to those predicted by the full-bonding shear-lag (Cox) model8. At higher applied matrix strain levels, however, the maximum interfacial shear stress occurred at some distance away from the fibre ends due to interfacial debonding. This is clearly seen for the sized Nextel fibre at e m ¼ 0.6 and 0.8% (Figure 3b). Nextel 610/hot-cured epoxy composites. The distributions of axial fibre strain with distance along the fibre at various
Figure 5 (a) Measured and theoretical (using eqn (3) with R/r f ¼ 10) variation of axial strain along a desized Nextel fibre in a hot-cured composite at different matrix strain levels. (b) Derived (using eqn (2)) variation of interfacial shear stress with distance along the fibre
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 7 Measured and theoretical (eqn (3)) variation of strain along the left end of the desized Nextel fibre in a hot-cured composite at e m ¼ 0.2% (best fit when R/r f ¼ 10)
Figure 6 (a) Measured and theoretical (using eqn (3) with R/r f ¼ 10) variation of axial strain along a sized Nextel fibre in a hot-cured composite at different matrix strain levels. (b) Derived (using eqn (2)) variation of interfacial shear stress with distance along the fibre
levels of applied matrix strain for desized- and sized-Nextel/ hot-cured epoxy single-fibre composite specimens are shown in Figure 5a and Figure 6a, respectively. At the zero applied matrix strain, there is an initial compressive strain of about 0.34% along the central region of the fibre. This residual thermal strain developed during cooling of the composite from the cure temperature (808C) owing to the mismatch of thermal expansion coefficients between the fibre and the matrix, as discussed elsewhere7. As the applied matrix strain is increased the initial compressive strain is recovered and the fibre strain increases from the ends to a maximum value over the central region of the fibre as for the cold-cured specimens. The solid lines in Figure 5a and Figure 6a were calculated by using eqn (3). In this case, the average value of strain along the central region of the fibre rather than the applied matrix strain has been used because of the presence of the initial compressive strain in the fibre. At an applied matrix strain of less than 1.0%, the theoretical data had the best fit to the experimental data when the ratio R/r f was 10 (for both tension and compression), as in the case of the cold-cured composites. At higher matrix strains, the Cox shear-lag model with R/r f ¼ 10 does not fit the experimental data, probably due to interfacial debonding and/or matrix yielding which is not accounted for in the model. It should
be noted that plastic deformation at the interface was found to start to occur in a similar Kevlar/epoxy system for e m . 1.0%12. The measured fibre strain distribution for the left part of a desized Nextel fibre in hot-cured epoxy at e m ¼ 1.0% (shown in Figure 5a) is compared in Figure 7 with the shear-lag predictions (eqn (3)) for different values of R/r f. As can be seen, the best fit is obtained when R/r f ¼ 75. It can be seen therefore that for both the cold- and hot-cured specimens, there is a tendency for the ratio R/r f to increase (or n to decrease) as the matrix strain is increased. This can be taken to be an indication of a decrease in the efficiency of stress transfer across the fibre/matrix interface owing the onset of fibre/matrix debonding and/or matrix yielding. These observations are strictly relevant only to model single-fibre composites. It is clear that R/r f might be different for high-volume-fraction composites, although the observations of the increase of R/r f with matrix strain could be important for all types of composite. The distributions of interfacial shear stress with distance along the fibre at various levels of applied matrix strain for desized- and sized-Nextel/hot-cured epoxy single-fibre composite specimens are shown in Figure 5b and Figure 6b, respectively. The distributions of interfacial shear stress were derived by differentiating the cubic spline fits of the distributions of fibre strain shown in Figure 5a and Figure 6a, and applying eqn (2). It can be seen that the maximum interfacial shear stress occurs at the fibre ends and that the interfacial shear stress falls to zero along the middle of the fibre. This is similar to the behaviour predicted by the fullbonding (Cox) shear-lag model8. The interfacial shear stress distribution clearly deviates from the prediction at 1.2% matrix strain in the case of the sized Nextel fibre. It can be seen from Figure 6b that the maximum interfacial shear stress occurs away from the right end of the fibre due to interfacial debonding and/or matrix yielding. Fibre fragmentation Nextel 610/cold-cured epoxy composites. The distribution of fibre strain along a fragmented, desized Nextel fibre in a
1357
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 8 Variation of axial fibre strain with distance along a fragmented, desized Nextel fibre in the cold-cured epoxy resin matrix at 0.7 and 0.9% matrix strain
Figure 10 Variation of (a) fibre strain and (b) interfacial shear stress with distance along a fragmented, sized Nextel fibre in the cold-cured epoxy resin matrix at e m ¼ 1.2%
Figure 9 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along the desized Nextel fragment (2) in the cold-cured epoxy resin matrix at e m ¼ 0.9%
cold-cured epoxy matrix is shown in Figure 8. A single fibre break was obtained at 0.7% matrix strain and, when the matrix strain was increased further to 0.9%, three additional breaks occurred. Figure 9a and Figure 9b show the detailed fibre strain and interfacial shear stress distributions along fragment 2 in Figure 8 at 0.9% matrix strain. It can be seen that the fibre strain distribution along the fragment is almost triangular and the interfacial shear stress is relatively constant at about 25 MPa near the fragment ends. At higher
1358
matrix strain levels, similar strain and shear stress distributions were obtained. Such distributions imply debonding or matrix yielding over the entire fibre/matrix interface5,15 and they are similar to those predicted by Kelly and Tyson5 for the case of an elastic fibre in a plastic matrix. Figure 10a shows the distribution of fibre strain with distance along a fragmented, sized Nextel fibre in the coldcured epoxy matrix at e m ¼ 1.2%, where one break had occurred at the fibre centre. The distribution of interfacial shear stress along the fragmented sized Nextel fibre at 1.2% matrix strain is shown in Figure 10b. It can be seen that the distributions are different from those of the desized Nextel fragments (Figure 9b). Well-defined linear strain distributions and constant shear stresses occur over regions near the fragment ends. The linear strain distribution and region of constant shear stress of ,10 MPa correspond to a debonded interface 9. It can be seen that debonded lengths near the original fibre ends are longer than at the broken ends. This means that interfacial debonding probably occurred at the original ends, both during matrix loading to 1.2% and upon fibre fracture. The frictional shear stress is about 8 MPa. In the central region of each fragment, a different type of fibre strain distribution and interfacial shear stress are obtained than for the desized fibres. Here, good adhesion between the fibre and matrix appears to still hold and load transfer is elastic. The average value of maximum interfacial shear stress is about 25 MPa, which is lower than the matrix shear yield strength of 41 MPa12.
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 11 Variation of axial fibre strain with distance along a fragmented, desized Nextel fibre in the hot-cured epoxy resin matrix at 1.2% matrix strain
Figure 13 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along a desized Nextel fragment (5) in the hot-cured epoxy resin matrix at 1.5% matrix strain
Figure 12 Variation of axial fibre strain with distance along a fragmented, desized Nextel fibre in the hot-cured epoxy resin matrix at 1.5% matrix strain
Clearly, the full-bonding8 and complete-debonding models5,15 are not suitable for describing the axial fibre strain and interfacial shear stress distribution shown for the fragments in Figure 10. In such a case, a partial-debonding model9 is needed and this will be discussed further later. Nextel 610/hot-cured epoxy composites. Figures 11 and 12 show the distributions of fibre strain with distance along a fragmented, desized Nextel fibre in a hot-cured epoxy matrix at different levels of matrix strain. For this sample, the fibre broke into two unequal-length fragments at 1.2% matrix strain (Figure 11). When the matrix strain was increased to 1.5% (Figure 12), each fragment underwent further fracture. Figures 13 and 14 show the distributions of (a) fibre strain and (b) interfacial shear stress along the desized Nextel fragment 5 in Figure 12 at 1.5% and 3.0% matrix strain, respectively. At 1.5% matrix strain, there is an approximately linear strain distribution and a constant interfacial shear stress region extending over a distance of about 100 mm from the fragment ends. A constant shear stress of about 30 MPa is obtained at the fragment ends and it seems
that interfacial debonding was initiated when the fibre fragmented. When the matrix strain is increased to 3.0% (Figure 14a), the debonded region increases to 150 mm. Over the debonded region, the fibre strain now increases at a lower rate and the shear stress is constant at 15 MPa (Figure 14b). The frictional shear stress, t f, could be reduced at the higher matrix strain value by degradation of the interface by sliding of the ceramic fibre. A smoother polymer matrix surface would be obtained and consequently a lower value of t f measured. The maximum interfacial shear stress is about 43 MPa, which is similar to the matrix shear yield stress12, showing that the fibre is still well bonded in the central region. The fibre strain and shear stress distributions in Figures 13 and 14 tend to be symmetrical about the fragment centre unlike those of sized Nextel fragments in the cold-cured epoxy (Figure 10). Figures 15 and 16 show the distributions of (a) fibre strain and (b) interfacial shear stress along a sized Nextel fragment in a hot-cured epoxy matrix at 1.4% and 3.0% matrix strain, respectively. The distributions are similar to those for the desized Nextel fragment in the hot-cured epoxy matrix (Figures 13 and 14) and correspond to the prediction of the partial-debonding model9. A frictional shear stress of about 15 MPa and a maximum interfacial shear stress of about 40 MPa are also obtained at the 3.0% matrix strain. A notable difference in the transition zone between the
1359
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 14 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along a desized Nextel fragment (5) in the hot-cured epoxy resin matrix at 3.0% matrix strain
debonded and bonded region of the desized and sized Nextel fragment occurred at e m ¼ 3.0%. For the desized Nextel fragment, the transition zone is broad and extends from x ¼ 6 50 mm to x ¼ 6 150 mm (Figure 14). A considerably sharper transition zone occurs at x ¼ 6 75 mm for the sized Nextel fragment (Figure 16a). Partial-debonding model. The partial-debonding model proposed by Piggott9,15 can be used to describe the axial strain distribution along a fragment with a debonded interface that extends over a section of the fibre length. This model assumes that debonding occurs over a distance ml/2 from the fibre ends (with 0 , m , 1) and that perfect adhesion holds in the central region of the fibre. Stress is transferred elastically in the bonded region while in the debonded region stress is transferred via friction. For the debonded region, the interfacial shear stress is purely frictional and so: ti ¼ tf
(4)
The frictional shear stress, t f, is related to the radial pressure at the interface which arises from both the thermal radial stress and the Poisson contraction3. The fibre strain distribution for the debonded region is: 2t l ¹x (5) ef ¼ f rf Ef 2
1360
Figure 15 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along a sized Nextel fragment in the hot-cured epoxy resin matrix at 1.4% matrix strain
For the bonded region, the fibre strain distribution is: ÿ cosh nx=rf 2t sm ef ¼ e ¹ e ¹ f cosh[ns(1 ¹ m)] Ef
(6)
The solid lines in Figure 13a, Figure 14a, Figure 15a and Figure 16a are the best fits of the partial-debonding model and they were obtained when the ratio R/r f ¼ 75. It should be noted that the same R/r f value was used for the desized Nextel fibre in the hot-cured epoxy at 1.0% matrix strain as shown in Figure 7. The value is large in comparison to the ratio R/r f ¼ 10 used for the full-bonding case at low matrix strains and it may be taken as an indication of the decrease in efficiency of stress transfer across the interface at higher strains as discussed earlier. The partial-debonding model9,15 does not take into account thermal stresses. To account for the thermal residual strains in the fibre, the applied strain e in eqn (6) should be (e m þ e T), where e T is the compressive (negative) thermal strain in the fibre at e m ¼ 0%19. It can be seen from Figure 13a, Figure 14a, Figure 15a and Figure 16a that there is good agreement between the theoretical predictions and experimental results except in a region near the bonding/debonding transition.
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young
Figure 17 Scanning electron micrograph of a desized Nextel fibre
Figure 16 Variation of (a) axial fibre strain and (b) interfacial shear stress with distance along a sized Nextel fragment in the hot-cured epoxy resin matrix at 3.0% matrix strain
Table 3 The maximum value of t i for a-alumina fibre/epoxy systems Maximum value of t i ( ; t s) (MPa) Desized Nextel Sized Nextel PRD-166 (unsized)
Cold-cured epoxy
Hot-cured epoxy
26 6 2 38 6 2 42 6 2
38 6 3 37 6 7 44 6 2
Interfacial shear strength The maximum measured values of t i for each of the four different Nextel 610 fibre/epoxy systems are given in Table 3 along with some earlier findings for the PRD-166/ epoxy system7. The effect of sizing and resin curing temperature on fibre/matrix interfacial adhesion can be clearly seen from Table 3. The maximum values can be used to estimate the interfacial shear strengths (assuming t s ; t i,max) for the different Nextel 610 fibre/epoxy systems. The interfacial shear strength of the sized-Nextel fibre/ cold-cured epoxy composite is higher than that of the desized-Nextel fibre/cold-cured epoxy composite. It seems that the PVA sizing promotes better adhesion between the Nextel 610 fibres and epoxy matrix. An increase in interfacial shear strength is also obtained when the desized Nextel fibre is embedded in the hot-cured epoxy matrix rather than in the cold-cured epoxy matrix. This could be due to the radial compressive stresses across the fibre/matrix
Figure 18 Scanning electron micrograph of a PRD-166 fibre
interface in the hot-cured system, making debonding more difficult. An increase in interfacial shear strength is not obtained for the sized Nextel fibres and similar values of shear strength are obtained with both the cold- and hotcured epoxy matrix. It appears that the radial compression does not have an effect upon the inherently stronger interface for the sized Nextel fibres. Alternatively, the size on the fibres may form an interphase which could control the mechanical behaviour of the interfacial regions. The interfacial shear strength of the Nextel 610 fibre/ epoxy system is lower than the shear yield stress of the resin (41–45 MPa) and thus the shear strength appears to be limited by debonding at the fibre/matrix interface. This is to be contrasted with earlier findings for the PRD-166 fibre/ epoxy system, whereby the higher level of interfacial shear strength appeared to be controlled by the shear yield strength of the epoxy resin matrix7. The PRD-166 and Nextel fibres differ in both composition and microstructure. When they are examined by SEM, the Nextel 610 fibres are seen to have relatively smooth surfaces (Figure 17) compared with the PRD-166 fibres which have much
1361
Micromechanics of fibre fragmentation: R. B. Yallee and R. J. Young rougher surfaces (Figure 18). The grain size for the PRD166 fibres is about 0.5 mm whereas it is significantly smaller for the Nextel 610 fibres. One reason for the lower interfacial shear strength for the Nextel 610 fibres in the epoxy resin matrix may be simply due to a difference in surface roughness leading to better mechanical interlocking between PRD-166 fibres and the epoxy resin matrix.
ACKNOWLEDGEMENTS
CONCLUSIONS
REFERENCES
It has been demonstrated that luminescence spectroscopy can be used to study the micromechanics of load transfer in a-alumina fibre/epoxy composites during fragmentation testing. At low matrix strain levels before fibre fragmentation occurs, the fibre strain profiles along the Nextel 610 alumina fibres have been found to be in excellent agreement with those predicted by the Cox shear-lag model5. Deviations from the model are found to occur at higher levels of matrix strain as a consequence of the onset of matrix yielding and/or interfacial debonding. It has been found that a fibre fracture is accompanied by debonding of the fibre/matrix interface, and the extent of the initial debonded region is dependent upon the level of interfacial adhesion. A more pronounced debonded region is obtained for the desized Nextel fibre than for the sized Nextel fibre in the cold-cured epoxy matrix. It has also been shown that the experimental fibre strain distributions and shear stress profiles following fragmentation can be modelled well by using a partial-debonding model9,15, except near the boundary between the bonded and debonded regions of the interface.
1362
One of the authors (RJY) would like to thank the Royal Society for support in the form of the Wolfson Research Professorship in Materials Science. The work forms part of a large programme of research supported by the Engineering and Physical Sciences Research Council.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Herrera-Franco, P.J. and Drzal, L.T., Composites, 1992, 23, 2. Kalantar, J. and Drzal, L.T., J. Mater. Sci., 1990, 25, 4194. Wagner, H.D., Gallis, H.E. and Weisel, E., J. Mater. Sci., 1993, 28, 205. Netravali, A.N., Li, Z.-F., Sachse, W. and Wu, H.F., J. Mater. Sci., 1991, 26, 6631. Kelly, A. and Tyson, W.R., J. Mech. Phys. Sol., 1965, 13, 329. Yang, X. and Young, R.J., Acta Metall. Mater., 1995, 43, 2407. Yallee, R.B., Andrews, M.C. and Young, R.J., J. Mater. Sci., 1996, 31, 3349. Cox, H.L., Br. J. Appl. Phys., 1952, 3, 57. Piggott, M.R., Compos. Sci. Technol., 1991, 42, 57. 3M Datasheet, Document 11-6273-3. Yallee, R.B., Single-fibre composite micromechanics. Ph.D. thesis, University of Manchester Institute of Science and Technology, 1997. Andrews, M.C., Stress transfer in aramid/epoxy model composites. Ph.D. thesis, Victoria University of Manchester, 1994. Ciba-Geigy Datasheet, Ciba-Geigy, UK. Schawlow, A.L., in Advances in Quantum Electronics, ed. J.R. Singer. Columbia University Press, New York, 1961, p. 50. Lacroix, Th., Tilmans, B., Keunings, R., Desaeger, M. and Verpoest, I., Compos. Sci. and Technol., 1992, 43, 379. Galiotis, C., Compos. Sci. Technol., 1991, 42, 125. Andrews, M.C., Day, R.J., Patrikis, A.K. and Young, R.J., Composites, 1994, 25, 745. Nairn, J.A., Mech. Mater., 1997, 26, 63. Huang, Y. and Young, R.J., Composites, 1995, 26, 541.