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matrix stress transfer through a discrete interphase: 2. High volume fraction systems
Composites Science and Technology 61 (2001) 565±578
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Fibre/matrix stress transfer through a discrete interphase: 2. High volume fraction systems R. Lane, S.A. Hayes, F.R. Jones * Department of Engineering Materials, Sheeld University, Sir Robert Had®eld Building, Mappin Street, Sheeld, S1 3JD, UK Received 8 June 2000; accepted 9 November 2000
Abstract The eect of plasticity on the reinforcing eciency of a broken ®bre, and the magnitude of the strain concentration experienced by the surrounding ®bres, has been assessed by the use of a 3-dimensional ®nite-element model. It was found that the occurrence of plasticity in the matrix markedly reduced the strain concentration in ®bres adjacent to a ®bre fracture. The eect of increasing the ®bre volume fraction on the level of strain concentration was examined and it was found that when deformation was elastic, at low applied strain, a higher ®bre volume fraction led to an increase in the strain concentration. However, when plastic deformation occurred, the strain concentration factor was signi®cantly lower and increasing the ®bre volume fraction had a negligible eect. The in¯uences of soft or sti interphases between ®bre and matrix were also studied and, during elastic deformation, these were found to be largely insigni®cant in determining the stress transfer processes, for the interphase thickness studied. At higher strains, the occurrence of plastic deformation in either the matrix or interphase was found to dominate the strain-transfer process and, therefore, the strain concentration. The reasons behind these dependencies are discussed and their eect on the failure of bulk composites considered. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: 3-Dimensional ®nite element model; Matrix plasticity; Carbon ®bres; Interphase; Stress concentration factor
1. Introduction Interphase formation is inevitable in the manufacture of composite materials and generally arises from the sizing resin. It has been shown by the authors [1±3] that the occurrence of plasticity in the interphase or surrounding matrix has a signi®cant eect on the stress transfer between matrix and ®bre in single-®bre model composites. Part 1 of this series [3] examined the strain transfer between matrix and ®bre, through an interphase, in a single-®bre model composite. This work showed that at low applied strains, when the strain transfer was elastic in nature, the interphase had limited in¯uence. However, at higher strains the phase with the lower yield strength deformed plastically, limiting the level of strain transferred. Plastic deformation in high volume fraction composite materials, incorporating an interphase, has not been widely studied. When an individual ®bre fractures within a continuous ®bre composite, the load it carried is transferred * Corresponding author. Tel.: +44-114-222-5477; fax: +44-114222-5943. E-mail address: f.r.jones@sheeld.ac.uk (F.R. Jones).
to adjacent unbroken ®bres, leaving it with an `ineective length' over which the strain redevelops. The surrounding ®bres experience an increase in strain over a `positively aected length', adjacent to the ®bre-break. The ineective length of the broken ®bre determines the positively aected length of the surrounding ®bres and thus the strain concentration. A short ineective length results in a high strain concentration in the adjacent ®bres, leading to an increased probability of further ®bre fracture and hence composite failure. Many authors have attempted to estimate the stress (or strain) concentration near a ®bre-break, using mathematical analyses, ®nite element models and Raman spectroscopy [4]. Most work quotes stress concentration factors, but the authors feel that strain concentration is more appropriate in the presence of plastic deformation. Hedgepeth and van Dyke [5] used a shearlag analysis to determine the stress concentrations in diering arrays of ®bres. They found that for a 2dimensional array and a 3-dimensional square array the maximum stress concentration factors were 1.33 and 1.146 respectively. However, their analysis predicted that the volume fraction of ®bres and the Young's modulus of the matrix would have no eect on the value
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(00)00229-3
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of the stress concentration factor, because the matrix was assumed to support only shear stress. Subsequently Fukuda and Kawata [6] provided a model for the prediction of the stress in the broken ®bre and the surrounding ®bres in a 2-dimensional array. They found that the Young's modulus of the matrix and the ®bre volume fraction had a signi®cant in¯uence on the magnitude of the stress concentration. Wagner and Eitan [7] also used a modi®ed shear-lag analysis, which took into account both volume fraction and the Young's modulus of the matrix. Their predictions were lower than those of Hedgepeth and van Dyke [5] when the ®bre spacing was large, but similar when the ®bres were in close contact. In studies using ®nite element analysis, Nedele and Wisnom [8,9] used a hexagonal array of ®bres and also a concentric cylinder model to predict the strain concentration factors. These were predicted to raise the probability of ®bre failure by a small amount, up to a maximum of 1%, depending on the precise specimen geometry and material properties. Thus, they concluded that stress concentrations were largely insigni®cant in determining the failure of composites. They also considered the eect of matrix plasticity, and found its contribution to be negligible. However, Fiedler and Schulte [10] used ®nite element analysis to predict values of 1.46 and 1.15 respectively for the stress concentrations in the neighbour and next-nearest neighbour ®bres of a square array. Thus, there is little consensus on the predicted levels of stress concentration. Experimental determination of the level of stress concentration has also given rise to many dierent results. Wagner et al. [11] carried out an experimental study using Raman spectroscopy, in which they found general agreement between their results and predictions from models that incorporated the eect of volume fraction and matrix Young's modulus. Chohan and Galiotis [4,12] measured values of 1.36 and 1.24 for 2-dimensional and a 3-dimensional array respectively, with the ®bres in close contact. Van den Heuvel et al. [13] also used Raman spectroscopy and measured a stress concentration factor of 1.26, with a ®bre spacing of 0.8 of a ®bre-diameter, for a 2-dimensional array. They reported that there was little consensus between experimental and theoretical values of stress concentration factors, and attributed this to the occurrence of plasticity in real systems. They also reported that at a ®bre spacing of approximately 10 ®bre-diameters, the stress concentration was insigni®cant. Many models have been proposed for the prediction of composites failure. Among the earliest of these was that proposed by Rosen [14], in which it was assumed that upon ®bre fracture the load was uniformly shared among the remaining ®bres, as the matrix would support no tensile load. Thus, no stress concentration factor was considered in the analysis. Zweben [15] extended this analysis by the inclusion of the stress concentrations
predicted by Hedgepeth and van Dyke [5], thus introducing a factor which meant the load was shared largely by the ®bres in the immediate vicinity of the broken one. His analysis gave a better correlation with experimental results than that of Rosen. The previously quoted results of Nedele and Wisnom [8,9], showing an insigni®cant stress concentration eect, led to its exclusion in the analysis of Wisnom and Green [16]. Their analysis did, however, allow for interaction between breaks within a given distance of each other in the ®bre direction. This approach predicted an increase in the extent of any damaged interfacial regions as breaks interacted and thus formed damage clusters. Further interaction between ®bre breaks and damage clusters would occur upon loading, ultimately resulting in catastrophic failure. Curtin and Takeda [17,18] presented a model that considered the decay in stress concentration experienced by successively more distant ®bres, allowing a more detailed description of the load sharing process to be obtained. Agreement between the predictions and experimental results was claimed to be reasonable, provided that the properties of the constituents and composite were carefully determined. However, the predictions from the model were always high and thus care would be needed in their use. Hence, the models for the prediction of composite failure are currently qualitative but require further re®nement if quantitative results are to be obtained. The majority of models used to predict composite failure, or the level of stress concentration around a ®bre-break assume that the matrix displays linear-elastic behaviour. By neglecting the non-linear stress±strain response of matrix resins, these models will inevitably have limited predictive capability. Hedgepeth and van Dyke [5] did, however, include the eect of matrix plasticity in their analysis, but found that at low loads the stress concentration factor was similar to that for an elastic matrix, a conclusion also reached by Nedele and Wisnom [8,9]. However, Hedgepeth and van Dyke predicted that as the load increased, the strain concentration would decrease in the presence of plasticity. Using ®nite element analysis, Feidler et al. [19] employed a 3dimensional ®bre array and included linear-elastic, linearelastic/perfectly plastic and also hypoelastic matrix properties. They found that the inclusion of plasticity led to a signi®cant decrease in the strain concentration factor. However, van den Heuvel et al. [20] used a 2dimensional ®bre array, which included digitised true stress/true strain curves for the resins employed in an experimental analysis [13]. They predicted that the stress concentration factor would increase with the occurrence of plasticity, because of an increase in the ineective length of the broken ®bre and commensurate increase in the positively eected length of the adjacent ®bres. Thus, the in¯uence of matrix plasticity on the strain concentration factor is at present unresolved, although
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previous work has suggested that its in¯uence is small, if not negligible. This study examines the eect of matrix or interphase plasticity on the strain distribution within a 3-dimensional composite structure. The incorporation of an interphase allows the performance of realistic composite structures to be examined. The implications of the ®ndings on the failure of bulk composite materials are considered. 2. Experimental 2.1. Model de®nition A 3-dimensional ®nite element model was developed using ANSYS 5.4, which incorporated four quarter ®bres positioned at the corners of a rectangular prism (Fig. 1). The model consisted of 9872 elements, the mesh density being concentrated near the ®bre-break. A cross-sectional view is shown in Fig. 2a, showing the mesh-density at the ®bre-break, while Fig. 2b shows the gradation in mesh-density within the model. The model included an interphase region between the ®bres and matrix, whose properties could be altered independently of the matrix material. The model was constructed with ®bres of radius 3.5 mm, to represent carbon ®bres, an interphase thickness of 0.2 mm and a length of 50 mm. The cross-sectional area of the matrix was varied to give ®bre-volume fractions of 38, 48 and 58%. The mesh consisted of 8-noded structural-solid brick elements (SOLID 45) in the ®bre and interphase, with 10-noded tetrahedral elements (SOLID 92) being used for the matrix. Initially the matrix was meshed using 4-noded tetrahedral (SOLID 72) elements, but a more accurate solution was attained using the 10-noded tetrahedral elements (SOLID 92), which are essentially the same but
Fig. 1. Schematic illustration of the ®nite-element model showing the presence of an interphase and the location of the ®bre-break.
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have additional mid-side nodes. The interface between the Solid 45 and Solid 92 elements left ¯oating mid-side nodes, however, this mesh produced a more acceptable solution in the matrix. The mid-side nodes were carefully monitored to ensure that no unacceptable motion occurred which could have in¯uenced the accuracy of the solution. The resulting mesh gave a solution that had stress continuity across the interface and improved matrix stress resolution. The length of the model was 50 mm and represented the ®bre from its end to its midpoint, the total fragment length, therefore, being 100 mm, after the application of symmetry conditions. A short fragment length was used to reduce the number of elements, thus ensuring stable and ecient solutions. The model was constrained on its four sides and the bottom surface using symmetric boundary conditions to prevent undue motion perpendicular to the plane. The
Fig. 2. Illustration of (a) the ®ve locations at which measurements of the strain concentration were taken in the neighbour (A) and nextnearest neighbour (B) ®bres; (b) the mesh-density gradation along the length of the model.
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top surface was then displaced to a strain of either 0.1 or 1%. At this strain, the ®bre was broken by eliminating the top layer of elements (approximately 1 mm) in one of the ®bres using the ANSYS `EKILL' command. This reduced their properties by a factor of 1106, thus allowing the redistribution of stress to be analysed and the eect of a ®bre-break on the composite to be assessed. The model is, therefore, representative of in®nitely long ®bres, one of which has fragmented into uniform 100 mm lengths. As such, the results are not directly comparable to those from models that assume a single break in an in®nitely long ®bre. 2.2. Materials The reinforcing ®bres were de®ned as type `A' carbon ®bres, including orthotropic properties, which are shown in Table 1. The mechanical properties of the resins used as the matrix and interphase were based on real epoxy resins, which had previously been used in an experimental study [3]. These two resins are termed 5050 and 6040 and their true-stress/true-strain curves are shown in Fig. 3. These were digitised into the ®nite element model, enabling a realistic calculation of the yield-characteristics, using a von Mises yield criterion inherent in the ®nite element code. Table 2 shows a summary of the elastic properties of the two resins. 2.3. Matrix/interphase systems. 2.3.1. The eect of matrix plasticity on the strain transfer characteristics An initial analysis was carried out in order to determine the eect of matrix plasticity in the 6040 system in comparison to an elastic system, with the same initial modulus of 3.48 GPa. The interphase was assigned the same properties as the matrix, and the two systems were studied at a ®xed volume fraction of 58%. 2.3.2. The eect of volume fraction on the strain transfer characteristics The eect of volume fraction on the strain transfer characteristics was determined using the two systems listed below, at 38, 48 and 58% volume fraction.
Table 1 Properties assigned to the type A carbon ®bres Property
Value
EA (GPa) A ET (GPa) T G (GPa)
220 0.2 14 0.25 35
1. 6040 as the matrix and interphase. 2. 5050 as the matrix and interphase. In this way, the eect of the volume fraction was initially studied in the absence of an interphase. 2.3.3. The eect of interphase properties on the strain transfer characteristics The eect of varying the interphase properties was determined using the two systems listed below, at 38, 48 and 58% volume fraction. 1. 5050 matrix, 6040 interphase (5050/6040). 2. 6040 matrix, 5050 interphase (6040/5050) By comparing the results of this study with those in which the interphase was absent, its eect could be determined. 3. Results 3.1. The eect of matrix plasticity on the strain transfer characteristics In order to examine the eect of matrix plasticity on strain redevelopment within the broken ®bre, in the vicinity of a ®bre-break, the axial tensile strain in the ®bre centre was examined. The model employed a volume fraction of 58%, incorporating either an elastic matrix/interphase or an elasto-plastic 6040 matrix/ interphase, both having an initial modulus of 3.48 GPa. The results are shown in Fig. 4, where the data in Fig. 4a were obtained at an applied strain of 0.1% and those in Fig. 4b at an applied strain of 1%. It can be seen that at an applied strain of 0.1% there is little dierence between the strain development pro®les for the elastic and 6040 systems. However, at an applied strain of 1% the rate of strain transfer in the 6040 system is signi®cantly lower than that for the elastic system. From these curves, the eciency of strain redevelopment (strain transfer eciency) was calculated by dividing the strain attained at the ®bre mid-point by the applied strain. The results are shown in Table 3. Due to the short length of the fragment, an accurate measurement of the ineective length could not be determined, as it exceeded that of the fragment. The eect of a ®bre-break on the surrounding ®bres was examined by studying the axial strain in the centre of the neighbour and next-nearest neighbour ®bres (Marked A and B respectively in Fig. 2a). The results are shown in Fig. 5, with Fig. 5a showing those at 0.1% applied strain and Fig. 5b those at 1% applied strain. From these strain pro®les the maximum strain concentration factor was calculated by recording the maximum strain in the curves and dividing it by the applied
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Fig. 3. True stress/true strain curves for the two resins used in this study. Table 2 Summary of the elastic properties of the two resins Resin system Property
5050
6040
Initial modulus (GPa) Average modulus to yield point (GPa) Yield stress (MPa) Yield strain (%) Assumed value of Poisson's ratio
1.76 0.79 35.16 4.40 0.36
3.48 0.84 53.50 6.37 0.36
Table 3 Strain transfer eciency to the broken ®bre for the elasto-plastic 6040 system and the elastic system, at a volume fraction of 0.58 Applied strain (%)
Elastic
6040
0.1 1
0.75 0.75
0.74 0.4
strain, the results are shown in Table 4. It can be seen that the strain concentration factors for the elastic system at 0.1 and 1% applied strain and the 6040 system at 0.1% applied strain are all very similar, with the neighbour ®bres experiencing a higher value than the nextnearest neighbours. However, at 1% applied strain, the maximum strain concentrations in the case of the elastoplastic 6040 system are signi®cantly lower in both neighbour and next-nearest neighbour ®bres. The results in Table 4 show the strain concentration factors at the ®bre centre, however the strain concentration factor will diminish as the radial distance from the
broken ®bre increases. Thus, it is anticipated that the maximum strain concentration factor experienced by the surrounding ®bres will occur at their surfaces, at the point closest to the broken ®bre, decaying around the circumference as the distance from the broken ®bre increases. This is an important consideration, as techniques such as Raman spectroscopy measure the strain concentration at the ®bres surface and thus any dierences will be potentially misleading. In order to examine this, the strain concentration factors were calculated for ®ve locations around the ®bre surface, for both the neighbour and next-nearest neighbour ®bres (see Fig. 2). The strain concentrations at each location in the neighbour ®bres are shown in Table 5, while those for the next-nearest neighbour ®bres are shown in Table 6. It is apparent that the measured strain concentration factor at location A1 for the neighbour and location B3 for the next-nearest neighbour are signi®cantly larger than those observed at the ®bre centres (Table 4). These values are seen to rapidly decay around the ®bres circumference. In the case of the neighbour ®bres (Table 5), the strain concentration is highest at location A1 for both the elastic and elasto-plastic systems, as the ®bre surface is closest to the broken ®bre at this point, and reduces around the circumference to location A5. The strain concentration factors for the elastic system at 0.1 and 1% applied strain and the 6040 system at 0.1% applied strain can be seen to be very similar at all locations around the ®bre. However, the 6040 system at 1% applied strain has signi®cantly lower strain concentration factors than the other systems, at all locations. The next-nearest neighbour ®bres (Table 6) have their maximum strain concentration at location B3 in each case, falling away symmetrically on either side. As
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Fig. 4. Tensile strain attained in the centre of the broken ®bre, for a volume fraction of 58% at: (a) 0.1% applied strain; (b) 1% applied strain.
Table 4 Maximum strain concentration factors in the ®bre-centre, for the elasto-plastic 6040 system and the elastic system, at a volume fraction of 0.58
Fibre position
0.1% Applied strain
1% Applied strain
Elastic
6040
Elastic
6040
1.15 1.05
1.14 1.05
1.07 1.03
Neighbour 1.14 Next-nearest neighbour 1.05
before, the strain concentration factors for the elastic systems at 0.1 and 1% applied strain and the 6040 system at 0.1% applied strain are similar at all locations, while the 6040 system at 1% applied strain again has signi®cantly lower values. 3.2. The eect of volume fraction on the strain transfer characteristics The eect of volume fraction on the strain transfer characteristics was initially examined in the absence of
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Fig. 5. Tensile strain attained in the centre of the surrounding ®bres, at a volume fraction of 58%, at: (a) 0.1% applied strain; (b) 1% applied strain.
Table 5 Maximum strain concentration values at the ®bre-surface for the neighbour ®bres at a volume fraction of 0.58 0.1% Applied strain
1% Applied strain
Location
Elastic
6040
Elastic
6040
A1 A2 A3 A4 A5
1.63 1.47 1.27 1.14 1.10
1.59 1.45 1.28 1.15 1.11
1.60 1.46 1.27 1.14 1.10
1.18 1.16 1.13 1.09 1.07
Table 6 Maximum strain concentration values at the ®bre-surface for the nextnearest neighbour ®bres at a volume fraction of 0.58 0.1% Applied strain
1% Applied strain
Location
Elastic
6040
Elastic
6040
B1 B2 B3 B4 B5
1.08 1.09 1.10 1.09 1.08
1.08 1.09 1.11 1.09 1.08
1.07 1.09 1.11 1.09 1.07
1.04 1.05 1.06 1.05 1.04
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an interphase. Two resin systems, the 6040 system and the 5050 system, were employed as the matrix, with the strain transfer characteristics for both systems being studied at volume fractions of 0.38, 0.48 and 0.58. The strain attained in the ®bre centre, for each system, at 0.1 and 1% applied strains were determined, with the results being presented in Figs. 6 and 7 respectively. Figs. 6a and 7a show the results from the 6040 system and Figs. 6b and 7b those from the 5050 system. The strain transfer eciencies for each system were calculated and are shown in Table 7. The results at 0.1% applied strain show that as the volume fraction increases the strain transfer eciency increases. However, the results obtained at 1% applied strain show little variation in the strain development process as volume fraction increases, for the individual systems. The maximum strain concentration in the centres of the neighbour and the next-nearest neighbour ®bres were determined for each system and are compared in Table 8. It can be seen that the strain concentrations, observed at 0.1% applied strain, increase as the volume fraction increases. However at 1% applied strain, the change in strain concentration with volume fraction is greatly reduced, with those observed for the next-nearest neighbour ®bres remaining constant. 3.3. The eect of interphase properties on the strain transfer characteristics In order to study the eect of interphase properties, two systems one with a 6040 matrix and a 5050 interphase (6040/5050) and the other with a 5050 matrix and a 6040 interphase (5050/6040) were examined at volume fractions of 0.38, 0.48 and 0.58. The in¯uence of the interphase could then be determined by comparison with the results obtained from the systems without an interphase (Section 3.2). The strain developed in the centre of the broken ®bre at applied strains of 0.1 and 1%, for each system, at each volume fraction, are shown in Figs. 8 and 9, with Figs. 8a and 9a showing those from the 6040/5050 system and Figs. 8b and 9b those from the 5050/6040 system. The strain transfer eciencies calculated from these curves are shown in Table 9. As with the systems without an interphase (Table 7) it can be seen that, while the strain transfer is elastic (at 0.1% applied strain) the strain transfer eciency depends on volume fraction, but that once the strain transfer is largely plastic in nature (at 1% applied strain) little dependence is observed. The strain concentration factors for the neighbour and next-nearest neighbour ®bres were determined as before and are shown in Table 10. It is again apparent that the strain concentration factors at 0.1% applied strain depend on volume fraction, while there is little dependence at 1% applied strain.
4. Discussion. 4.1. The eect of plasticity on the strain transfer characteristics The analysis of the strain attained in the centre of the broken ®bre, and the values of strain concentration in the surrounding ®bres shows that the two measurements display identical trends. While the strain transfer in the 6040 system is elastic (at 0.1% applied strain), there is no signi®cant dierence between it and the elastic system, the strain transfer being dominated by their similar elastic response. However, once plasticity becomes signi®cant (at 1% applied strain), the strain transfer eciency and the strain concentration in the 6040 system are reduced (Tables 3 and 4). This indicates that the reinforcing eciency of a broken ®bre will be signi®cantly reduced with respect to that predicted from an elastic analysis, and the surrounding ®bres will experience a reduced strain concentration as well. It must also be borne in mind that an applied strain of 1% is within the practical loading envelope of composite materials and a strain level at which ®bre fracture will be occurring. The increased likelihood of fracture of the surrounding ®bres, as a result of the strain concentrations induced around a ®bre break, depends on the strain they experience over and above that associated with the applied strain. At the ®bre surface, the results show that the highest strain concentration factor experienced by the ®bre occurs over a very limited area, and decays both circumferentially around the ®bre and volumetrically through the ®bre (Tables 5 and 6). It is also apparent that the highest strain concentration factors are greatly reduced upon the occurrence of plasticity. Therefore, the in¯uence of the maximum strain concentration (measured at the ®bre surface) on failure of the composite is likely to be limited, while analysis of the average strain concentration factor (measured in the ®bre centre) will give a more realistic estimate of the increased likelihood of ®bre failure. This ®nding has important implications for the experimental determination of strain concentration factors. 4.2. The eect of volume fraction on the strain transfer characteristics At 0.1% applied strain, the strain transfer eciency in a broken ®bre and the strain concentration in the neighbouring ®bres was found to be dependent on the volume fraction, but at 1% applied strain was largely independent (Tables 7 and 8). This indicates that the process of elastic strain transfer (0.1% applied strain) depends on the mechanical properties of the surrounding materials, with the in¯uence of the adjacent ®bres increasing as their volume fraction increases. However,
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Fig. 6. Tensile strain attained in the centre of the broken ®bres at an applied strain of 0.1% for: (a) 6040 system; (b) 5050 system.
Table 7 Strain transfer eciency to the broken ®bre, for the two systems without an interphase 6040
5050
Applied strain (%)
0.38 Vf
0.48 Vf
0.58 Vf
0.38 Vf
0.48 Vf
0.58 Vf
0.1 1
0.65 0.37
0.70 0.38
0.75 0.39
0.48 0.24
0.53 0.24
0.60 0.25
yield tends to isolate a broken ®bre from the surrounding material, leading to the rate of strain transfer depending primarily on the shear yield strength of the yielding material. These results agree with previous work by the authors in which single-®bre composites were studied [3]. The strain transfer eciencies and the strain concentration factors obtained from the 6040 system were higher than those for the 5050 system (Tables 7 and 8). This is because of the stier 6040 system dissipating energy less eectively than the softer 5050 system upon
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Fig. 7. Tensile strain attained in the centre of the broken ®bres at an applied strain of 1% for: (a) 6040 system; (b) 5050 system.
Table 8 Maximum strain concentration observed in the centre of ®bres adjacent to a ®bre-break, for the two systems without an interphase 0.1% Applied strain
1% Applied strain
Matrix/interphase system
0.38 Vf
0.48 Vf
0.58 Vf
0.38 Vf
0.48 Vf
0.58 Vf
6040 (Neighbour) 5050 (Neighbour) 6040 (Next-nearest) 5050 (Next-nearest)
1.11 1.08 1.04 1.02
1.13 1.10 1.04 1.03
1.15 1.12 1.05 1.03
1.07 1.04 1.03 1.01
1.07 1.05 1.03 1.01
1.07 1.05 1.03 1.01
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Fig. 8. Tensile strain attained in the centre of the broken ®bres at an applied strain of 0.1% for: (a) 6040/5050 system; (b) 5050/6040 system.
®bre fracture, resulting in higher energy transfer to the surrounding ®bres. This observation applies both before and after yield, because of the 6040 system has both a higher elastic modulus and shear yield strength compared to the 5050 system. 4.3. The eect of interphase properties on the strain transfer characteristics The strain transfer eciencies and strain concentrations, in the presence of an interphase, can be seen to
follow the same trends as those without an interphase. Both values depend on the volume fraction while the deformation is elastic and are independent once plasticity becomes signi®cant (Tables 9 and 10). By comparing Tables 7 and 9 and Tables 8 and 10, it can be seen that when the strain transfer is elastic in nature, the values for the 6040 and 6040/5050 system are very similar, as are those for the 5050 and 5050/6040 systems, with the values depending on the volume fraction. This suggests that while the strain transfer is elastic and thus dependent on the stiness of the composite as a whole,
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Fig. 9. Tensile strain attained in the centre of the broken ®bres at an applied strain of 1% for: (a) 6040/5050 system; (b) 5050/6040 system.
the matrix and ®bres dominate and the in¯uence of the interphase is small. This is because of the negligible volume of the interphase in this model. Previous results with single-®bre systems suggest that a thicker interphase would have a more pronounced in¯uence [3]. When the strain transfer is largely plastic in nature (1% applied strain), the dependence on volume fraction
is low (Tables 8 and 9). Again the 6040 system has the largest values, but the second largest are displayed by the 5050/6040 system, followed by the 6040/5050 system and ®nally the 5050 system (Tables 7 and 9). When considering the strain concentration factors (Tables 8 and 10), there is little dierence between any of the systems once plastic deformation has occurred, although
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Table 9 Strain transfer eciency to the broken ®bre, for the two systems with an interphase 6040/5050
5050/6040
Applied strain
0.38 Vf
0.48 Vf
0.58 Vf
0.38 Vf
0.48 Vf
0.58 Vf
0.1 1
0.62 0.27
0.67 0.27
0.72 0.27
0.49 0.31
0.55 0.31
0.63 0.32
Table 10 Maximum strain concentration observed in the centre of ®bres adjacent to a ®bre-break, for the two systems with an interphase 0.1% Applied strain
1% Applied strain
Matrix/interphase system
0.38 Vf
0.48 Vf
0.58 Vf
0.38 Vf
0.48 Vf
0.58 Vf
6040/5050 (Neighbour) 5050/6040 (Neighbour) 6040/5050 (Next-nearest) 5050/6040 (Next-nearest)
1.10 1.08 1.04 1.02
1.12 1.10 1.04 1.03
1.14 1.13 1.05 1.03
1.05 1.05 1.02 1.02
1.05 1.05 1.02 1.02
1.05 1.05 1.02 1.02
the 6040 system has marginally the highest value and the 5050 system marginally the lowest. These results suggest that the yielding material dominates the strain transfer process, resulting in the 5050 and 6040/5050 systems having the worst strain transfer processes as a signi®cant amount of energy is dissipated in the 5050 material, close to the broken ®bre. However, the matrix does have an in¯uence, leading to the performance of the 6040/5050 system being slightly superior. In the case of the 6040 system, which has a higher yield strength, less energy is dissipated by yield and, therefore, its performance is better. The 5050/6040 system has an intermediate strain transfer performance, as less yield occurs in the 6040 interphase than would have in a 5050 system, thus dissipating less energy. However, more energy is dissipated in the 5050 matrix than would be in a 6040 matrix, as some matrix yielding will occur, resulting in a reduced strain transfer eciency. The action of the interphase in determining the properties of the composite can therefore be summarised as follows. When plasticity occurs, the presence of a soft, yielding interphase will reduce the reinforcing eciency of a broken ®bre and will also limit the stress concentration in the surrounding ®bres. Conversely, a sti interphase will yield less and, therefore, the surrounding ®bres will experience a higher strain concentration factor than if the interphase was not present. In the presence of a sti interphase, the properties of the matrix are also signi®cant, as the matrix may yield prior to the interphase and thus dominate the strain transfer processes. 5. Conclusions Plasticity is observed to occur even when the applied strain is signi®cantly below the matrix/interphase yield
strain, because of the size of the strain concentration in the vicinity of a broken ®bre. Therefore, the assumption of linear-elasticity in the development of failure criteria is misleading. The occurrence of plasticity has a signi®cant eect on the strain distributions within a high volume fraction composite upon ®bre fracture. The eect of plasticity is to reduce the reinforcing eciency of the broken ®bre and also to reduce the strain concentration experienced by surrounding ®bres. The results suggest that by utilising plastic deformation in a thin interphase, within a polymer±matrix composite, its toughness can be altered without aecting the elastic properties. Thus, the choice of sizing resin must become a further design parameter in the selection and utilisation of composites. Acknowledgements The authors gratefully acknowledge the ®nancial support of the Advanced Composites Group and the Engineering and Physical Sciences Research Council. This work has been carried out in conjunction with the European Union Interphase Network. References [1] Tripathi D, Chen F, Jones FR. A comprehensive model to predict the stress ®elds in a single ®bre composite. J Compos Mater 1996;30:1514±38. [2] Lane R, Hayes SA, Jones FR. Modelling of the eciency of strain transfer across an interphase region in ®bre reinforced composites. Composite Interfaces 1999;6:425±40. [3] Hayes SA, Lane R, Jones FR. Fibre/matrix stress transfer through a discrete interphase. Composites (in press).
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[4] Chohan V, Galiotis C. Eects of interface, volume fraction and geometry on stress redistribution in polymer composites under tension. Comp Sci Tech 1997;57:1089±101. [5] Hedgepeth JM, van Dyke P. Local stress concentrations in imperfect ®lamentry composite materials. J Comp Mat 1967;1:294±309. [6] Fukuda H, Kawata K. On the stress concentration factor in ®brous composites. Fibre Sci Tech 1976;9:189±203. [7] Wagner HD, Eitan A. Stress concentration factors in twodimensional composites: eects of material and geometrical parameters. Comp Sci Tech 1993;46:353±62. [8] Nedele MR, Wisnom MR. Three-dimensional ®nite element analysis of the stress concentration at a single ®bre break. Comp Sci Tech 1994;51:517±24. [9] Nedele MR, Wisnom MR. Stress concentration factors around a broken ®bre in a unidirectional carbon ®bre-reinforced epoxy. Composites 1994;25:549±57. [10] Fiedler B, Schulte K. Stress concentrations in single and multiple ®bre model composites. Paper presented at Interfacial Phenomena in Composite Materials'97. September 1997, Eger, Hungary. [11] Wagner HD, Amer MS, Schadler LS. Fibre interactions in twodimensional composites by Raman spectroscopy. J Mat Sci 1996;31:1165±73. [12] Chohan V, Galiotis C. Interfacial measurements and fracture
[13]
[14] [15] [16] [17] [18] [19] [20]
characteristics of 2D microcomposites using remote laser Raman microscopy. Composites 1996;27A:881±8. Van den Heuvel PWJ, Peijs T, Young RJ. Failure phenomena in two-dimensional microcomposites: 2. A Raman spectroscopic study of the in¯uence of inter-®bre spacing on stress concentration. Comp Sci Tech 1997;57:899±911. Rosen BW. Tensile failure of ®brous composites. AIAA J 1964; 2:1985±91. Zweben C. Tensile failure of ®ber composites. AIAA J 1968; 6:2325±31. Wisnom MR, Green D. Tensile failure due to interaction between ®bre breaks. Composites 1995;26:499±508. Curtin WA, Takeda N. Tensile strength of ®ber-reinforced composites: I. Model and eects of local ®bre geometry. J Comp Mat 1998;32:2042±59. Curtin WA, Takeda N. Tensile strength of ®ber-reinforced composites: II. Application to polymer matrix systems. J Comp Mat 1998;32:2060±81. Fiedler B, Klisch A, Schulte K. Stress concentrations in multiple ®bre model composites. Composites 1998;29A:1013±19. Van den Heuvel PWJ, Wubbolts MK, Young RJ, Peijs T. Failure phenomena in two-dimensional multi-®bre model composite: 5. A ®nite element study. Composites 1998; 29A:1121±35.
www.elsevier.com/locate/compscitech
Fibre/matrix stress transfer through a discrete interphase: 2. High volume fraction systems R. Lane, S.A. Hayes, F.R. Jones * Department of Engineering Materials, Sheeld University, Sir Robert Had®eld Building, Mappin Street, Sheeld, S1 3JD, UK Received 8 June 2000; accepted 9 November 2000
Abstract The eect of plasticity on the reinforcing eciency of a broken ®bre, and the magnitude of the strain concentration experienced by the surrounding ®bres, has been assessed by the use of a 3-dimensional ®nite-element model. It was found that the occurrence of plasticity in the matrix markedly reduced the strain concentration in ®bres adjacent to a ®bre fracture. The eect of increasing the ®bre volume fraction on the level of strain concentration was examined and it was found that when deformation was elastic, at low applied strain, a higher ®bre volume fraction led to an increase in the strain concentration. However, when plastic deformation occurred, the strain concentration factor was signi®cantly lower and increasing the ®bre volume fraction had a negligible eect. The in¯uences of soft or sti interphases between ®bre and matrix were also studied and, during elastic deformation, these were found to be largely insigni®cant in determining the stress transfer processes, for the interphase thickness studied. At higher strains, the occurrence of plastic deformation in either the matrix or interphase was found to dominate the strain-transfer process and, therefore, the strain concentration. The reasons behind these dependencies are discussed and their eect on the failure of bulk composites considered. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: 3-Dimensional ®nite element model; Matrix plasticity; Carbon ®bres; Interphase; Stress concentration factor
1. Introduction Interphase formation is inevitable in the manufacture of composite materials and generally arises from the sizing resin. It has been shown by the authors [1±3] that the occurrence of plasticity in the interphase or surrounding matrix has a signi®cant eect on the stress transfer between matrix and ®bre in single-®bre model composites. Part 1 of this series [3] examined the strain transfer between matrix and ®bre, through an interphase, in a single-®bre model composite. This work showed that at low applied strains, when the strain transfer was elastic in nature, the interphase had limited in¯uence. However, at higher strains the phase with the lower yield strength deformed plastically, limiting the level of strain transferred. Plastic deformation in high volume fraction composite materials, incorporating an interphase, has not been widely studied. When an individual ®bre fractures within a continuous ®bre composite, the load it carried is transferred * Corresponding author. Tel.: +44-114-222-5477; fax: +44-114222-5943. E-mail address: f.r.jones@sheeld.ac.uk (F.R. Jones).
to adjacent unbroken ®bres, leaving it with an `ineective length' over which the strain redevelops. The surrounding ®bres experience an increase in strain over a `positively aected length', adjacent to the ®bre-break. The ineective length of the broken ®bre determines the positively aected length of the surrounding ®bres and thus the strain concentration. A short ineective length results in a high strain concentration in the adjacent ®bres, leading to an increased probability of further ®bre fracture and hence composite failure. Many authors have attempted to estimate the stress (or strain) concentration near a ®bre-break, using mathematical analyses, ®nite element models and Raman spectroscopy [4]. Most work quotes stress concentration factors, but the authors feel that strain concentration is more appropriate in the presence of plastic deformation. Hedgepeth and van Dyke [5] used a shearlag analysis to determine the stress concentrations in diering arrays of ®bres. They found that for a 2dimensional array and a 3-dimensional square array the maximum stress concentration factors were 1.33 and 1.146 respectively. However, their analysis predicted that the volume fraction of ®bres and the Young's modulus of the matrix would have no eect on the value
0266-3538/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(00)00229-3
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of the stress concentration factor, because the matrix was assumed to support only shear stress. Subsequently Fukuda and Kawata [6] provided a model for the prediction of the stress in the broken ®bre and the surrounding ®bres in a 2-dimensional array. They found that the Young's modulus of the matrix and the ®bre volume fraction had a signi®cant in¯uence on the magnitude of the stress concentration. Wagner and Eitan [7] also used a modi®ed shear-lag analysis, which took into account both volume fraction and the Young's modulus of the matrix. Their predictions were lower than those of Hedgepeth and van Dyke [5] when the ®bre spacing was large, but similar when the ®bres were in close contact. In studies using ®nite element analysis, Nedele and Wisnom [8,9] used a hexagonal array of ®bres and also a concentric cylinder model to predict the strain concentration factors. These were predicted to raise the probability of ®bre failure by a small amount, up to a maximum of 1%, depending on the precise specimen geometry and material properties. Thus, they concluded that stress concentrations were largely insigni®cant in determining the failure of composites. They also considered the eect of matrix plasticity, and found its contribution to be negligible. However, Fiedler and Schulte [10] used ®nite element analysis to predict values of 1.46 and 1.15 respectively for the stress concentrations in the neighbour and next-nearest neighbour ®bres of a square array. Thus, there is little consensus on the predicted levels of stress concentration. Experimental determination of the level of stress concentration has also given rise to many dierent results. Wagner et al. [11] carried out an experimental study using Raman spectroscopy, in which they found general agreement between their results and predictions from models that incorporated the eect of volume fraction and matrix Young's modulus. Chohan and Galiotis [4,12] measured values of 1.36 and 1.24 for 2-dimensional and a 3-dimensional array respectively, with the ®bres in close contact. Van den Heuvel et al. [13] also used Raman spectroscopy and measured a stress concentration factor of 1.26, with a ®bre spacing of 0.8 of a ®bre-diameter, for a 2-dimensional array. They reported that there was little consensus between experimental and theoretical values of stress concentration factors, and attributed this to the occurrence of plasticity in real systems. They also reported that at a ®bre spacing of approximately 10 ®bre-diameters, the stress concentration was insigni®cant. Many models have been proposed for the prediction of composites failure. Among the earliest of these was that proposed by Rosen [14], in which it was assumed that upon ®bre fracture the load was uniformly shared among the remaining ®bres, as the matrix would support no tensile load. Thus, no stress concentration factor was considered in the analysis. Zweben [15] extended this analysis by the inclusion of the stress concentrations
predicted by Hedgepeth and van Dyke [5], thus introducing a factor which meant the load was shared largely by the ®bres in the immediate vicinity of the broken one. His analysis gave a better correlation with experimental results than that of Rosen. The previously quoted results of Nedele and Wisnom [8,9], showing an insigni®cant stress concentration eect, led to its exclusion in the analysis of Wisnom and Green [16]. Their analysis did, however, allow for interaction between breaks within a given distance of each other in the ®bre direction. This approach predicted an increase in the extent of any damaged interfacial regions as breaks interacted and thus formed damage clusters. Further interaction between ®bre breaks and damage clusters would occur upon loading, ultimately resulting in catastrophic failure. Curtin and Takeda [17,18] presented a model that considered the decay in stress concentration experienced by successively more distant ®bres, allowing a more detailed description of the load sharing process to be obtained. Agreement between the predictions and experimental results was claimed to be reasonable, provided that the properties of the constituents and composite were carefully determined. However, the predictions from the model were always high and thus care would be needed in their use. Hence, the models for the prediction of composite failure are currently qualitative but require further re®nement if quantitative results are to be obtained. The majority of models used to predict composite failure, or the level of stress concentration around a ®bre-break assume that the matrix displays linear-elastic behaviour. By neglecting the non-linear stress±strain response of matrix resins, these models will inevitably have limited predictive capability. Hedgepeth and van Dyke [5] did, however, include the eect of matrix plasticity in their analysis, but found that at low loads the stress concentration factor was similar to that for an elastic matrix, a conclusion also reached by Nedele and Wisnom [8,9]. However, Hedgepeth and van Dyke predicted that as the load increased, the strain concentration would decrease in the presence of plasticity. Using ®nite element analysis, Feidler et al. [19] employed a 3dimensional ®bre array and included linear-elastic, linearelastic/perfectly plastic and also hypoelastic matrix properties. They found that the inclusion of plasticity led to a signi®cant decrease in the strain concentration factor. However, van den Heuvel et al. [20] used a 2dimensional ®bre array, which included digitised true stress/true strain curves for the resins employed in an experimental analysis [13]. They predicted that the stress concentration factor would increase with the occurrence of plasticity, because of an increase in the ineective length of the broken ®bre and commensurate increase in the positively eected length of the adjacent ®bres. Thus, the in¯uence of matrix plasticity on the strain concentration factor is at present unresolved, although
R. Lane et al. / Composites Science and Technology 61 (2001) 565±578
previous work has suggested that its in¯uence is small, if not negligible. This study examines the eect of matrix or interphase plasticity on the strain distribution within a 3-dimensional composite structure. The incorporation of an interphase allows the performance of realistic composite structures to be examined. The implications of the ®ndings on the failure of bulk composite materials are considered. 2. Experimental 2.1. Model de®nition A 3-dimensional ®nite element model was developed using ANSYS 5.4, which incorporated four quarter ®bres positioned at the corners of a rectangular prism (Fig. 1). The model consisted of 9872 elements, the mesh density being concentrated near the ®bre-break. A cross-sectional view is shown in Fig. 2a, showing the mesh-density at the ®bre-break, while Fig. 2b shows the gradation in mesh-density within the model. The model included an interphase region between the ®bres and matrix, whose properties could be altered independently of the matrix material. The model was constructed with ®bres of radius 3.5 mm, to represent carbon ®bres, an interphase thickness of 0.2 mm and a length of 50 mm. The cross-sectional area of the matrix was varied to give ®bre-volume fractions of 38, 48 and 58%. The mesh consisted of 8-noded structural-solid brick elements (SOLID 45) in the ®bre and interphase, with 10-noded tetrahedral elements (SOLID 92) being used for the matrix. Initially the matrix was meshed using 4-noded tetrahedral (SOLID 72) elements, but a more accurate solution was attained using the 10-noded tetrahedral elements (SOLID 92), which are essentially the same but
Fig. 1. Schematic illustration of the ®nite-element model showing the presence of an interphase and the location of the ®bre-break.
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have additional mid-side nodes. The interface between the Solid 45 and Solid 92 elements left ¯oating mid-side nodes, however, this mesh produced a more acceptable solution in the matrix. The mid-side nodes were carefully monitored to ensure that no unacceptable motion occurred which could have in¯uenced the accuracy of the solution. The resulting mesh gave a solution that had stress continuity across the interface and improved matrix stress resolution. The length of the model was 50 mm and represented the ®bre from its end to its midpoint, the total fragment length, therefore, being 100 mm, after the application of symmetry conditions. A short fragment length was used to reduce the number of elements, thus ensuring stable and ecient solutions. The model was constrained on its four sides and the bottom surface using symmetric boundary conditions to prevent undue motion perpendicular to the plane. The
Fig. 2. Illustration of (a) the ®ve locations at which measurements of the strain concentration were taken in the neighbour (A) and nextnearest neighbour (B) ®bres; (b) the mesh-density gradation along the length of the model.
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top surface was then displaced to a strain of either 0.1 or 1%. At this strain, the ®bre was broken by eliminating the top layer of elements (approximately 1 mm) in one of the ®bres using the ANSYS `EKILL' command. This reduced their properties by a factor of 1106, thus allowing the redistribution of stress to be analysed and the eect of a ®bre-break on the composite to be assessed. The model is, therefore, representative of in®nitely long ®bres, one of which has fragmented into uniform 100 mm lengths. As such, the results are not directly comparable to those from models that assume a single break in an in®nitely long ®bre. 2.2. Materials The reinforcing ®bres were de®ned as type `A' carbon ®bres, including orthotropic properties, which are shown in Table 1. The mechanical properties of the resins used as the matrix and interphase were based on real epoxy resins, which had previously been used in an experimental study [3]. These two resins are termed 5050 and 6040 and their true-stress/true-strain curves are shown in Fig. 3. These were digitised into the ®nite element model, enabling a realistic calculation of the yield-characteristics, using a von Mises yield criterion inherent in the ®nite element code. Table 2 shows a summary of the elastic properties of the two resins. 2.3. Matrix/interphase systems. 2.3.1. The eect of matrix plasticity on the strain transfer characteristics An initial analysis was carried out in order to determine the eect of matrix plasticity in the 6040 system in comparison to an elastic system, with the same initial modulus of 3.48 GPa. The interphase was assigned the same properties as the matrix, and the two systems were studied at a ®xed volume fraction of 58%. 2.3.2. The eect of volume fraction on the strain transfer characteristics The eect of volume fraction on the strain transfer characteristics was determined using the two systems listed below, at 38, 48 and 58% volume fraction.
Table 1 Properties assigned to the type A carbon ®bres Property
Value
EA (GPa) A ET (GPa) T G (GPa)
220 0.2 14 0.25 35
1. 6040 as the matrix and interphase. 2. 5050 as the matrix and interphase. In this way, the eect of the volume fraction was initially studied in the absence of an interphase. 2.3.3. The eect of interphase properties on the strain transfer characteristics The eect of varying the interphase properties was determined using the two systems listed below, at 38, 48 and 58% volume fraction. 1. 5050 matrix, 6040 interphase (5050/6040). 2. 6040 matrix, 5050 interphase (6040/5050) By comparing the results of this study with those in which the interphase was absent, its eect could be determined. 3. Results 3.1. The eect of matrix plasticity on the strain transfer characteristics In order to examine the eect of matrix plasticity on strain redevelopment within the broken ®bre, in the vicinity of a ®bre-break, the axial tensile strain in the ®bre centre was examined. The model employed a volume fraction of 58%, incorporating either an elastic matrix/interphase or an elasto-plastic 6040 matrix/ interphase, both having an initial modulus of 3.48 GPa. The results are shown in Fig. 4, where the data in Fig. 4a were obtained at an applied strain of 0.1% and those in Fig. 4b at an applied strain of 1%. It can be seen that at an applied strain of 0.1% there is little dierence between the strain development pro®les for the elastic and 6040 systems. However, at an applied strain of 1% the rate of strain transfer in the 6040 system is signi®cantly lower than that for the elastic system. From these curves, the eciency of strain redevelopment (strain transfer eciency) was calculated by dividing the strain attained at the ®bre mid-point by the applied strain. The results are shown in Table 3. Due to the short length of the fragment, an accurate measurement of the ineective length could not be determined, as it exceeded that of the fragment. The eect of a ®bre-break on the surrounding ®bres was examined by studying the axial strain in the centre of the neighbour and next-nearest neighbour ®bres (Marked A and B respectively in Fig. 2a). The results are shown in Fig. 5, with Fig. 5a showing those at 0.1% applied strain and Fig. 5b those at 1% applied strain. From these strain pro®les the maximum strain concentration factor was calculated by recording the maximum strain in the curves and dividing it by the applied
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Fig. 3. True stress/true strain curves for the two resins used in this study. Table 2 Summary of the elastic properties of the two resins Resin system Property
5050
6040
Initial modulus (GPa) Average modulus to yield point (GPa) Yield stress (MPa) Yield strain (%) Assumed value of Poisson's ratio
1.76 0.79 35.16 4.40 0.36
3.48 0.84 53.50 6.37 0.36
Table 3 Strain transfer eciency to the broken ®bre for the elasto-plastic 6040 system and the elastic system, at a volume fraction of 0.58 Applied strain (%)
Elastic
6040
0.1 1
0.75 0.75
0.74 0.4
strain, the results are shown in Table 4. It can be seen that the strain concentration factors for the elastic system at 0.1 and 1% applied strain and the 6040 system at 0.1% applied strain are all very similar, with the neighbour ®bres experiencing a higher value than the nextnearest neighbours. However, at 1% applied strain, the maximum strain concentrations in the case of the elastoplastic 6040 system are signi®cantly lower in both neighbour and next-nearest neighbour ®bres. The results in Table 4 show the strain concentration factors at the ®bre centre, however the strain concentration factor will diminish as the radial distance from the
broken ®bre increases. Thus, it is anticipated that the maximum strain concentration factor experienced by the surrounding ®bres will occur at their surfaces, at the point closest to the broken ®bre, decaying around the circumference as the distance from the broken ®bre increases. This is an important consideration, as techniques such as Raman spectroscopy measure the strain concentration at the ®bres surface and thus any dierences will be potentially misleading. In order to examine this, the strain concentration factors were calculated for ®ve locations around the ®bre surface, for both the neighbour and next-nearest neighbour ®bres (see Fig. 2). The strain concentrations at each location in the neighbour ®bres are shown in Table 5, while those for the next-nearest neighbour ®bres are shown in Table 6. It is apparent that the measured strain concentration factor at location A1 for the neighbour and location B3 for the next-nearest neighbour are signi®cantly larger than those observed at the ®bre centres (Table 4). These values are seen to rapidly decay around the ®bres circumference. In the case of the neighbour ®bres (Table 5), the strain concentration is highest at location A1 for both the elastic and elasto-plastic systems, as the ®bre surface is closest to the broken ®bre at this point, and reduces around the circumference to location A5. The strain concentration factors for the elastic system at 0.1 and 1% applied strain and the 6040 system at 0.1% applied strain can be seen to be very similar at all locations around the ®bre. However, the 6040 system at 1% applied strain has signi®cantly lower strain concentration factors than the other systems, at all locations. The next-nearest neighbour ®bres (Table 6) have their maximum strain concentration at location B3 in each case, falling away symmetrically on either side. As
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Fig. 4. Tensile strain attained in the centre of the broken ®bre, for a volume fraction of 58% at: (a) 0.1% applied strain; (b) 1% applied strain.
Table 4 Maximum strain concentration factors in the ®bre-centre, for the elasto-plastic 6040 system and the elastic system, at a volume fraction of 0.58
Fibre position
0.1% Applied strain
1% Applied strain
Elastic
6040
Elastic
6040
1.15 1.05
1.14 1.05
1.07 1.03
Neighbour 1.14 Next-nearest neighbour 1.05
before, the strain concentration factors for the elastic systems at 0.1 and 1% applied strain and the 6040 system at 0.1% applied strain are similar at all locations, while the 6040 system at 1% applied strain again has signi®cantly lower values. 3.2. The eect of volume fraction on the strain transfer characteristics The eect of volume fraction on the strain transfer characteristics was initially examined in the absence of
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Fig. 5. Tensile strain attained in the centre of the surrounding ®bres, at a volume fraction of 58%, at: (a) 0.1% applied strain; (b) 1% applied strain.
Table 5 Maximum strain concentration values at the ®bre-surface for the neighbour ®bres at a volume fraction of 0.58 0.1% Applied strain
1% Applied strain
Location
Elastic
6040
Elastic
6040
A1 A2 A3 A4 A5
1.63 1.47 1.27 1.14 1.10
1.59 1.45 1.28 1.15 1.11
1.60 1.46 1.27 1.14 1.10
1.18 1.16 1.13 1.09 1.07
Table 6 Maximum strain concentration values at the ®bre-surface for the nextnearest neighbour ®bres at a volume fraction of 0.58 0.1% Applied strain
1% Applied strain
Location
Elastic
6040
Elastic
6040
B1 B2 B3 B4 B5
1.08 1.09 1.10 1.09 1.08
1.08 1.09 1.11 1.09 1.08
1.07 1.09 1.11 1.09 1.07
1.04 1.05 1.06 1.05 1.04
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an interphase. Two resin systems, the 6040 system and the 5050 system, were employed as the matrix, with the strain transfer characteristics for both systems being studied at volume fractions of 0.38, 0.48 and 0.58. The strain attained in the ®bre centre, for each system, at 0.1 and 1% applied strains were determined, with the results being presented in Figs. 6 and 7 respectively. Figs. 6a and 7a show the results from the 6040 system and Figs. 6b and 7b those from the 5050 system. The strain transfer eciencies for each system were calculated and are shown in Table 7. The results at 0.1% applied strain show that as the volume fraction increases the strain transfer eciency increases. However, the results obtained at 1% applied strain show little variation in the strain development process as volume fraction increases, for the individual systems. The maximum strain concentration in the centres of the neighbour and the next-nearest neighbour ®bres were determined for each system and are compared in Table 8. It can be seen that the strain concentrations, observed at 0.1% applied strain, increase as the volume fraction increases. However at 1% applied strain, the change in strain concentration with volume fraction is greatly reduced, with those observed for the next-nearest neighbour ®bres remaining constant. 3.3. The eect of interphase properties on the strain transfer characteristics In order to study the eect of interphase properties, two systems one with a 6040 matrix and a 5050 interphase (6040/5050) and the other with a 5050 matrix and a 6040 interphase (5050/6040) were examined at volume fractions of 0.38, 0.48 and 0.58. The in¯uence of the interphase could then be determined by comparison with the results obtained from the systems without an interphase (Section 3.2). The strain developed in the centre of the broken ®bre at applied strains of 0.1 and 1%, for each system, at each volume fraction, are shown in Figs. 8 and 9, with Figs. 8a and 9a showing those from the 6040/5050 system and Figs. 8b and 9b those from the 5050/6040 system. The strain transfer eciencies calculated from these curves are shown in Table 9. As with the systems without an interphase (Table 7) it can be seen that, while the strain transfer is elastic (at 0.1% applied strain) the strain transfer eciency depends on volume fraction, but that once the strain transfer is largely plastic in nature (at 1% applied strain) little dependence is observed. The strain concentration factors for the neighbour and next-nearest neighbour ®bres were determined as before and are shown in Table 10. It is again apparent that the strain concentration factors at 0.1% applied strain depend on volume fraction, while there is little dependence at 1% applied strain.
4. Discussion. 4.1. The eect of plasticity on the strain transfer characteristics The analysis of the strain attained in the centre of the broken ®bre, and the values of strain concentration in the surrounding ®bres shows that the two measurements display identical trends. While the strain transfer in the 6040 system is elastic (at 0.1% applied strain), there is no signi®cant dierence between it and the elastic system, the strain transfer being dominated by their similar elastic response. However, once plasticity becomes signi®cant (at 1% applied strain), the strain transfer eciency and the strain concentration in the 6040 system are reduced (Tables 3 and 4). This indicates that the reinforcing eciency of a broken ®bre will be signi®cantly reduced with respect to that predicted from an elastic analysis, and the surrounding ®bres will experience a reduced strain concentration as well. It must also be borne in mind that an applied strain of 1% is within the practical loading envelope of composite materials and a strain level at which ®bre fracture will be occurring. The increased likelihood of fracture of the surrounding ®bres, as a result of the strain concentrations induced around a ®bre break, depends on the strain they experience over and above that associated with the applied strain. At the ®bre surface, the results show that the highest strain concentration factor experienced by the ®bre occurs over a very limited area, and decays both circumferentially around the ®bre and volumetrically through the ®bre (Tables 5 and 6). It is also apparent that the highest strain concentration factors are greatly reduced upon the occurrence of plasticity. Therefore, the in¯uence of the maximum strain concentration (measured at the ®bre surface) on failure of the composite is likely to be limited, while analysis of the average strain concentration factor (measured in the ®bre centre) will give a more realistic estimate of the increased likelihood of ®bre failure. This ®nding has important implications for the experimental determination of strain concentration factors. 4.2. The eect of volume fraction on the strain transfer characteristics At 0.1% applied strain, the strain transfer eciency in a broken ®bre and the strain concentration in the neighbouring ®bres was found to be dependent on the volume fraction, but at 1% applied strain was largely independent (Tables 7 and 8). This indicates that the process of elastic strain transfer (0.1% applied strain) depends on the mechanical properties of the surrounding materials, with the in¯uence of the adjacent ®bres increasing as their volume fraction increases. However,
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573
Fig. 6. Tensile strain attained in the centre of the broken ®bres at an applied strain of 0.1% for: (a) 6040 system; (b) 5050 system.
Table 7 Strain transfer eciency to the broken ®bre, for the two systems without an interphase 6040
5050
Applied strain (%)
0.38 Vf
0.48 Vf
0.58 Vf
0.38 Vf
0.48 Vf
0.58 Vf
0.1 1
0.65 0.37
0.70 0.38
0.75 0.39
0.48 0.24
0.53 0.24
0.60 0.25
yield tends to isolate a broken ®bre from the surrounding material, leading to the rate of strain transfer depending primarily on the shear yield strength of the yielding material. These results agree with previous work by the authors in which single-®bre composites were studied [3]. The strain transfer eciencies and the strain concentration factors obtained from the 6040 system were higher than those for the 5050 system (Tables 7 and 8). This is because of the stier 6040 system dissipating energy less eectively than the softer 5050 system upon
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Fig. 7. Tensile strain attained in the centre of the broken ®bres at an applied strain of 1% for: (a) 6040 system; (b) 5050 system.
Table 8 Maximum strain concentration observed in the centre of ®bres adjacent to a ®bre-break, for the two systems without an interphase 0.1% Applied strain
1% Applied strain
Matrix/interphase system
0.38 Vf
0.48 Vf
0.58 Vf
0.38 Vf
0.48 Vf
0.58 Vf
6040 (Neighbour) 5050 (Neighbour) 6040 (Next-nearest) 5050 (Next-nearest)
1.11 1.08 1.04 1.02
1.13 1.10 1.04 1.03
1.15 1.12 1.05 1.03
1.07 1.04 1.03 1.01
1.07 1.05 1.03 1.01
1.07 1.05 1.03 1.01
R. Lane et al. / Composites Science and Technology 61 (2001) 565±578
575
Fig. 8. Tensile strain attained in the centre of the broken ®bres at an applied strain of 0.1% for: (a) 6040/5050 system; (b) 5050/6040 system.
®bre fracture, resulting in higher energy transfer to the surrounding ®bres. This observation applies both before and after yield, because of the 6040 system has both a higher elastic modulus and shear yield strength compared to the 5050 system. 4.3. The eect of interphase properties on the strain transfer characteristics The strain transfer eciencies and strain concentrations, in the presence of an interphase, can be seen to
follow the same trends as those without an interphase. Both values depend on the volume fraction while the deformation is elastic and are independent once plasticity becomes signi®cant (Tables 9 and 10). By comparing Tables 7 and 9 and Tables 8 and 10, it can be seen that when the strain transfer is elastic in nature, the values for the 6040 and 6040/5050 system are very similar, as are those for the 5050 and 5050/6040 systems, with the values depending on the volume fraction. This suggests that while the strain transfer is elastic and thus dependent on the stiness of the composite as a whole,
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Fig. 9. Tensile strain attained in the centre of the broken ®bres at an applied strain of 1% for: (a) 6040/5050 system; (b) 5050/6040 system.
the matrix and ®bres dominate and the in¯uence of the interphase is small. This is because of the negligible volume of the interphase in this model. Previous results with single-®bre systems suggest that a thicker interphase would have a more pronounced in¯uence [3]. When the strain transfer is largely plastic in nature (1% applied strain), the dependence on volume fraction
is low (Tables 8 and 9). Again the 6040 system has the largest values, but the second largest are displayed by the 5050/6040 system, followed by the 6040/5050 system and ®nally the 5050 system (Tables 7 and 9). When considering the strain concentration factors (Tables 8 and 10), there is little dierence between any of the systems once plastic deformation has occurred, although
R. Lane et al. / Composites Science and Technology 61 (2001) 565±578
577
Table 9 Strain transfer eciency to the broken ®bre, for the two systems with an interphase 6040/5050
5050/6040
Applied strain
0.38 Vf
0.48 Vf
0.58 Vf
0.38 Vf
0.48 Vf
0.58 Vf
0.1 1
0.62 0.27
0.67 0.27
0.72 0.27
0.49 0.31
0.55 0.31
0.63 0.32
Table 10 Maximum strain concentration observed in the centre of ®bres adjacent to a ®bre-break, for the two systems with an interphase 0.1% Applied strain
1% Applied strain
Matrix/interphase system
0.38 Vf
0.48 Vf
0.58 Vf
0.38 Vf
0.48 Vf
0.58 Vf
6040/5050 (Neighbour) 5050/6040 (Neighbour) 6040/5050 (Next-nearest) 5050/6040 (Next-nearest)
1.10 1.08 1.04 1.02
1.12 1.10 1.04 1.03
1.14 1.13 1.05 1.03
1.05 1.05 1.02 1.02
1.05 1.05 1.02 1.02
1.05 1.05 1.02 1.02
the 6040 system has marginally the highest value and the 5050 system marginally the lowest. These results suggest that the yielding material dominates the strain transfer process, resulting in the 5050 and 6040/5050 systems having the worst strain transfer processes as a signi®cant amount of energy is dissipated in the 5050 material, close to the broken ®bre. However, the matrix does have an in¯uence, leading to the performance of the 6040/5050 system being slightly superior. In the case of the 6040 system, which has a higher yield strength, less energy is dissipated by yield and, therefore, its performance is better. The 5050/6040 system has an intermediate strain transfer performance, as less yield occurs in the 6040 interphase than would have in a 5050 system, thus dissipating less energy. However, more energy is dissipated in the 5050 matrix than would be in a 6040 matrix, as some matrix yielding will occur, resulting in a reduced strain transfer eciency. The action of the interphase in determining the properties of the composite can therefore be summarised as follows. When plasticity occurs, the presence of a soft, yielding interphase will reduce the reinforcing eciency of a broken ®bre and will also limit the stress concentration in the surrounding ®bres. Conversely, a sti interphase will yield less and, therefore, the surrounding ®bres will experience a higher strain concentration factor than if the interphase was not present. In the presence of a sti interphase, the properties of the matrix are also signi®cant, as the matrix may yield prior to the interphase and thus dominate the strain transfer processes. 5. Conclusions Plasticity is observed to occur even when the applied strain is signi®cantly below the matrix/interphase yield
strain, because of the size of the strain concentration in the vicinity of a broken ®bre. Therefore, the assumption of linear-elasticity in the development of failure criteria is misleading. The occurrence of plasticity has a signi®cant eect on the strain distributions within a high volume fraction composite upon ®bre fracture. The eect of plasticity is to reduce the reinforcing eciency of the broken ®bre and also to reduce the strain concentration experienced by surrounding ®bres. The results suggest that by utilising plastic deformation in a thin interphase, within a polymer±matrix composite, its toughness can be altered without aecting the elastic properties. Thus, the choice of sizing resin must become a further design parameter in the selection and utilisation of composites. Acknowledgements The authors gratefully acknowledge the ®nancial support of the Advanced Composites Group and the Engineering and Physical Sciences Research Council. This work has been carried out in conjunction with the European Union Interphase Network. References [1] Tripathi D, Chen F, Jones FR. A comprehensive model to predict the stress ®elds in a single ®bre composite. J Compos Mater 1996;30:1514±38. [2] Lane R, Hayes SA, Jones FR. Modelling of the eciency of strain transfer across an interphase region in ®bre reinforced composites. Composite Interfaces 1999;6:425±40. [3] Hayes SA, Lane R, Jones FR. Fibre/matrix stress transfer through a discrete interphase. Composites (in press).
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