Stress concentration factors around a broken fibre in a unidirectional carbon fibre-reinforced epoxy

An axisymmetric analysis was performed to investigate the case of a single broken fibre surrounded by matrix material and either perfect composite material or a system of concentric circular cylinders representing fibres and matrix separately, according to the assumed fibre volume fraction of the composite of 60%. The analysis shows that the stress concentration is dependent on the assumed fibre/matrix bonding conditions and on the assumed matrix behaviour. In all cases investigated it is shown that the stress concentration in the adjacent fibres is much lower than the 1.104 predicted by Hedgepeth and van Dyke. The positively affected length where there is an increase in stress is only about half the ineffective length of the broken fibre. Further away from the break the axial stress in the adjacent fibres actually drops below the nominal axial stress. This results in a very small enhanced probability of failure in the adjacent fibres.     

Statistical analysis of failure of unidirectionally fibre-reinforced composites with local load-sharing

A shear-lag model is presented in this paper for analysis of stress concentration of fibres adjacent to a crack in unidirectionally fibre-reinforced composites. The analytic expressions of stress concentration factors of fibres adjacent to the tip of a crack with r broken fibres are obtained by using this model. The maximum stress concentration factors of the fibre at the tip of the crack are calculated, and the numerical results show little difference with Hedgepeth's [1]. In addition, the concept of the effective stress concentration factor is used since shear-lag analysis overestimates stress concentration of fibres [2], and the affected length of a crack-tip fibre is defined by value of the stress concentration factor. The affected length has clear physical meaning and increases gradually with extension of the crack, which is in accord with the actual case. Finally, based on the previous investigation, statistical analysis of strength of unidirectionally fibre-reinforced composites is done by applying the method of the critical crack core model [3, 4]. The numerical results are close to the experimental results, and those obtained by using the effective stress concentration factors approach the experimental results most closely.     

Fibre strain mapping in aramid/epoxy microcomposites

Two-dimensional microcomposites, comprising regular arrays of Kevlar 49 fibres embedded in an epoxy resin, were fabricated and subjected to incremental tensile loading up to fracture. The strain along individual fibres at different levels of applied load was monitored using a laser Raman spectroscopic technique. A strain magnification of 80% maximum was measured in an intact fibre as a result of a fibre fracture in an adjacent fibre. The exact distribution of the strain increase within the transfer length of the broken fibre was obtained. The experimental stress concentration factor is compared with that derived from existing analytical models.     

Finite element analysis of the effect of fibre shape on stresses in an elastic fibre surrounded by a plastic matrix

The finite element (FE) method was used to calculate the axial and radial stress distributions as a function of axial distance, z, from the centre, and radius, r, in an elastic fibre surrounded by a plastic matrix. Plastic deformation of the matrix was considered to exert a uniform interfacial stress, , along half the length of the fibre. Axisymmetric models were created for uniform cylindrical, ellipsoidal, paraboloidal and conical fibres characterised by an axial ratio, q, and half length, L. Young's modulus for the material of the fibre and L were arbitrarily assigned values of unity, since they act as scaling factors; q also acts as a scaling factor but was assigned a value of 10 to create models with a fibrous appearance. For the cylindrical fibre, the axial stress increased linearly from the end towards the centre; the radial stress was more evenly distributed. At the other extreme, the conical fibre showed a uniform distribution of axial and radial stress. Results for ellipsoidal and paraboloidal fibres were intermediate between these two extremes. In general, the effect of taper is to lower peak stress at the fibre centre and make the stress distribution throughout the fibre more even. These results are in good agreement with recent analytical theories for the axial distribution of surface radial stress and axial stress along the fibre axis. However, FE models have the advantage of predicting full three-dimensional stress distributions.     

Stress disturbance due to broken fibres in metal matrix composites with non-uniform fibre spacing

Premature fracture of weaker fibres causes stress disturbances in composites. These disturbances are affected by non-uniformity of fibre spacing. In order to evaluate quantitatively how the disturbances in metal matrix composites are affected by the extent of non-uniformity of fibre spacing, a method of calculation is presented on the basis of two-dimensional shear lag analysis. Static tensile stress concentrations in the intact fibres to broken fibres, tensile stress distribution along the fibre axis in the broken and intact fibres and shear stresses between broken and intact fibres were calculated by the method presented, using some examples. It is shown quantitatively that the spacing between broken and intact fibres and that between intact and next fibres has a significant influence on tensile stress concentrations in intact fibres and also on the shear stresses between broken and intact fibres: the narrower the former spacing and the wider the latter spacing, the higher become both tensile and shear stress concentrations. This tendency is enhanced when the number of broken fibres is large and when the strain hardening of the matrix is high.     

Photoelastic study of the durability of interfacial bonding of carbon fibre-epoxy resin composites

The physical techniques of polarizing microscopy, including the quantitative measurements of small optical retardations, have been used to investigate elastic fields adjacent to short carbon fibres in epoxy resin composites. The elastic fields associated with shear stress distribution along the fibre-matrix interface have been employed to monitor the initiation of interface debonding during hot (100 °C) water uptake. By examining the development of stress birefringence during resin swelling in the resin adjacent to individual fibres, the differences in the durability of interfacial bonding and the fibre failure modes for differently coated fibres have been obtained. The results show that the state of self-stress in model composites, comprising a single carbon fibre in a film of epoxy resin, can, by immersion in hot distilled water, be enhanced to such an extent that the axial tension in the fibre can be sufficient to initiate fibre fracture. The results also show that, for fibres that have been given certain proprietary surface treatments, the fibre fractures by different failure modes.     

The effect of fibre alignment on composite strength: I. Single-fibre studies

Tests have been carried out in which a fibre was embedded normally in a polymer, and pulled at an angle to the normal to the polymer surface. It was found that some fibres, especially Kevlar, had to be pulled at quite large angles in order for the strengths to be much reduced. A criterion, the 50% angle, was developed. This is the angle of pull at which the fibre breaks at half its normal strength. 50% angles were: 20 ° for glass, 30–40 ° for carbon and 45 ° for Kevlar. Thus, although Kevlar fibres are weaker than glass, they should make stronger random-fibre composites as long as the fibres are long enough so that they are broken in the fracture process. Testing at 30 ° gives information about the strength of extremely small volumes of fibre material and indicates that AS4 carbon may have an intrinsic strength as high as 30 GPa.     

Tensile recoil measurement of compressive strength for polymeric high performance fibres

Recoil forces acting on the broken ends of a fibre after tensile failure are known to cause substantial damage to polymeric high performance fibres. This damage is the result of compressive stresses developed during snap-back, or recoil, whose magnitude exceeds the compressive strength of the fibre. An analysis describing the axial stress history experienced by a fibre following a tensile failure has been performed and the results have led to the development of a simple, single filament, recoil technique for measuring fibre compressive strength. A number of polymeric high performance fibres were examined using the technique and compressive strengths measured are in excellent agreement with values obtained from composite tests.     

Crack bridging and fibre pull-out in polyethylene fibre reinforced epoxy resins

An investigation has been undertaken of the stress distributions in high-performance polyethylene fibres bridging cracks in model epoxy composites. The axial fibre stress has been determined from stress-induced Raman band shifts and the effect of fibre surface treatment has been followed using untreated and plasma-treated polyethylene fibres. It is found that when the specimen is cracked, the fibres do not break and stress is transmitted from the matrix to the fibre across the fibre/matrix interface. A debond propagates along the fibre/matrix interface accompanied by friction along the debonded interface. The axial stress distributions in the fibres can be analysed using a partial-debonding model based upon shear-lag theory and it is found that the maximum interfacial shear stress at the bond/debond transition is a function of the debond length. The debonding process has been modelled successfully in terms of the interfacial fracture energy-based criterion developed by Hsueh for the propagation of a debond along a fibre/matrix interface accompanied by constant friction along the interface.     

Probabilistic aspects of strength of unidirectional fibre-reinforced composites with matrix failure

In the previous paper [1], the stress distribution and the expected number of successive fibre breakages around broken fibres were calculated. It showed the following results. The fracture process that the crack originates from one isolated broken fibre and propagates due to the stress redistribution following the fibre breakage is unlikely to occur in the real unidirectional fibre-reinforced composite material. The matrix-failure is considered to play an important role in the fracture process of real composite materials. In the present paper, the stress (or strain) distribution and the expected number of successive fibre breakages around broken fibres are calculated when the matrix-damaged regions exist. The stress (or strain) distribution is obtained based on the three-dimensional hexagonalarray shear-lag model. Uniform shear force is assumed to occur in the matrix-damaged region. The expected number of the successive fibre breakages is calculated on the assumption that the flaws in the fibre follow a Poisson process.     

Failure phenomena in two-dimensional multi-fibre microcomposites. Part 4: a Raman spectroscopic study on the influence of the matrix yield stress on stress concentrations

Raman spectroscopy was used to study the influence of the shear yield stress of the matrix on the stress situation in carbon/epoxy model composites containing a planar fibre array. The fibre used was a surface-treated high-modulus Tenax® HMS-40 carbon fibre showing good fibre/matrix adhesion. Three matrices were used, all consisting of a common epoxy resin and a mixture of a di-functional and a tri-functional aliphatic amine-based curing agent. By varying the ratio of the di-functional to the tri-functional curing agent, the shear yield stress of the matrix was varied. For all three matrices, it was found that in the area immediately neighbouring a fibre fracture, stress transfer takes place through a locally yielding matrix. More importantly, it was shown that the maximum interfacial shear stress approximately equals the shear yield stress of the bulk matrix. In addition, it was found that an increase in the shear yield stress of the matrix results in a decrease of both the ineffective length and the positively affected length. Further, the experimental results show that the shear yield stress of the matrix does not significantly influence the stress concentration in the fibres adjacent to a broken fibre.     

Fatigue behaviour of alumina-fibre-reinforced epoxy resin composite pipes under tensile and compressive loading conditions

Fatigue tests have been conducted on composites consisting of epoxy resin reinforced with alumina fibres (AFRP) under cyclic tensile and compressive loading conditions with the variation of fibre orientation. The behaviour of the stress/strain curve for a ±45° sample is different from those for the ±15 and ±25° composite specimens, whereas, the monotonic strength decreases with increase in fibre angle for all specimens, which satisfies the maximum stress failure criterion. Fatigue results show that the applied stress decreases with an increase in the number of cycles to failure under both loading conditions for all composite pipes, but for the ±45° sample the decrease was slow. The results of fatigue tests on a macroscopic level indicate that the matrix crack density slowly increased with increase in the normalized number of cycles to failure in all the specimens. The normalized apparent stiffness therefore falls with an increase of the normalized number of cycles to failure. However, the maximum stress decreased with the increase in the number of cycles to failure in the case of the ±45° pipe. Finally, it is observed that matrix cracking and delaminations are occurring in the ±45° sample whereas delamination and fibre buckling are appearing in the ±15 and ±25° samples.     

Fibre interactions in two-dimensional composites by micro-Raman spectroscopy

In the study of fracture processes in composite materials, the interactions between broken and intact fibres are of critical importance. Indeed, the redistribution of stress from a failed fibre to its unfailed adjacent neighbours, and the stress concentration induced in these, determine the extent to which a break in one fibre will cause more breaks in neighbouring fibres. The overall failure pattern is a direct function of the stress concentration factors. In this paper we use laser micro-Raman spectroscopy to study the extent of stress transfer and redistribution caused by fibre fracture in two-dimensional Kevlar 149 based microcomposites. The strain along the fibres was mapped at different levels of load, and specimens with different inter-fibre distances were used to study the fibre content effect. The experimental stress concentration factors were compared with values predicted from various theoretical models. The stress concentration factors generally agreed with those literature models that include interfibre distance and matrix effects. The overall failure pattern was found not to be a direct function of the stress concentration factors in this system, as fracture propagates from fibre to fibre even at large interfibre distances, and is apparently accompanied by relatively low values of the stress concentration factors. The critical cluster size, beyond which final fracture of the composite occurs in a catastrophic manner, was found to be larger than five adjacent fibre breaks in the present system, for all interfibre distances studied.Visiting Stein Fellow at Drexel University, July-September 1994.     

Effects of inter-fibre spacing and matrix cracks on stress amplification factors in carbon-fibre/epoxy matrix composites. Part I: planar array of fibres

When the loading on a composite is sufficient to cause fracture of an individual fibre, the resulting stress amplification in the adjacent intact fibres may be large enough to cause failure of these fibres. In this work, 3D elasto-plastic finite element analysis was used to investigate the effect of inter-fibre spacing on the stress amplification factor in a composite comprising a planar array of fibres. A Progressional Approach was used in the FE analysis to simulate the constituent non-linear processes associated with the generation of thermal residual stresses from fabrication, the fibre fracture event and the subsequent initiation and propagation of conical matrix cracks induced with incremental tensile loading. As the inter-fibre spacing increases, the effect of fibre fracture on the stress distribution in the neighbouring intact fibres is reduced, whereas the effect on the matrix material is increased, thereby inducing localised yielding. The presence of a conical-shaped matrix crack was found to increase both the stress amplification factor and the positively affected length in neighbouring fibres. For a large inter-fibre spacing, a longer matrix crack is required to obtain good agreement with LRS measurements of fibre stress.     

Compressive and tensile behaviour of carbon fibres

Attempts have been made to discuss the fibre axial tensile and compressive behaviour of several carbon fibres prepared from different precursors, and surface- and/or sizing-treatments. With all fibres, the number of breaks increased with increasing tensile or compressive strain, and remained constant beyond a certain strain. The constant number of breakages based on the precursor differed remarkably in compression, whereas those based on the surface- and/or sizing-treatments differed remarkably in tension. In tension, the PAN-based fibre could be broken with a somewhat greater ease than the pitch-based fibre, while in compression, the pitch-based fibre could be broken with somewhat greater ease than the PAN-based fibre. © 1998 Chapman & Hall     

Multi-scale fatigue behaviour modelling of Al–Al2O3 short fibre composites

Using very heterogeneous materials in structural parts submitted to cyclic loadings, this paper presents an elasto-plastic micromechanical model. After recalling the homogenisation principle based on a mean field theory, non-linear kinematic and isotropic strain hardening is introduced into the matrix. Validation is made on an Al–3.5%Cu/SiC particle composite, and an Al–Si7Mg/Al₂O₃ fibre composite is treated as a first application. Damage is introduced into the model using a fibre failure criterion. It is based on the evolution of the volume fraction of broken fibres as a function of the maximum principal stress in the fibre family. The damage law is identified by means of in situ tensile tests performed inside the scanning electronic microscope. The number of broken fibres is determined as a function of the applied load and the number of cycles. The model predicts the fatigue behaviour, the loss of stiffness, the volume fraction of broken fibres for different volume fractions, aspect ratios, distributions of orientation and distributions of strength of the fibres. The effect of the mechanical fatigue properties of the matrix is also studied.     

Visualization of debonding of fully and partially embedded glass fibres in epoxy resins

Very short glass fibres have been embedded in bars of epoxy resin and the debonding process was observed under the microscope as the polymer was stressed. In addition, fibre pull-out specimens have been similarly watched while the fibre was pulled. The interfaces of the fully embedded fibres failed across the fibre ends at strains of 0·2–0·3%, and circumferential failure started at about 0·6% strain. The low values for the end failures are compatible with models involving stress concentrations at the fibre ends. The circumferential failure value is in agreement with the results of earlier pull-out studies. In the case of the pull-out specimen, the behaviour was complex. Fibres with short embedded lengths debonded first across the embedded end. Failure of the cylindrical surface was too fast for the direction of crack propagation to be determined. Longer fibres first debonded at the fibre entry point and then arrested while debonding occurred across the embedded end. Final failure was again very fast. Long fibres debonded continuously, starting at the entry point. A slow fracture process appeared to be involved at least initially, so that the average shear stress on the region still bonded increased continuously throughout the process. Fast fracture occurred only in the very last stages of the process. These observations are compatible with the traditional theory, but reverse bonding is not ruled out for the shortest embedded lengths.     

Analysis of interfacial micromechanics of model composites using synchrotron microfocus X-ray diffraction

The deformation micromechanics of single-fibre embedded model composites of poly(p-phenylene benzobisoxazole) (PBO) and poly(p-phenylene terephthalamide) (PPTA) fibres, embedded in an epoxy resin have been examined using synchrotron microfocus X-ray diffraction. Single fibres (in air) were deformed and the c-spacing monitored to establish a calibration of crystal strain against applied stress. Subsequently, the variation in crystal strain along fibres, embedded in the resin matrix was mapped using synchrotron microfocus X-ray diffraction. Raman spectroscopy was then used to map molecular deformation on the same samples (recorded as shifts in the Raman band wavenumber) in order to provide a complementary stress data. A shear-lag analysis was conducted on the axial fibre stress data in order to calculate interfacial shear stress and identify different stress-transfer modes at fibre/resin interfaces. The results establish that the axial fibre stress distributions measured by synchrotron microfocus X-ray diffraction correlate well with those obtained using Raman spectroscopy. The interfacial shear stress data derived from the stress-transfer profiles also show a good degree of correlation.     

Weibull strength statistics for graphite fibres measured from the break progression in a model graphite/glass/epoxy microcomposite

Recent statistical theories for the failure of fibrous composities focus on the initiation and growth of clusters of broken fibres within the composite. These theoreis require the probability distribution for fibre strength at the length scale of micromechanical load transfer around a cluster of broken fibres. Such lengths are of the order of 10 to 150 fibre diameters, and thus the associated strengths have previously been unmeasurable by direct means. Using Weilbull/weakest-link rules, researchers have resorted to extrapolation of tension test results from gauge lengths two orders of magnitude longer. In this paper, a technique is developed to study the break progression of a single graphite fibre in an epoxy microcomposite tape, where the graphite fibre is flanked by two, proof-tested, glass fibres. These results are interpreted using a Weibull/Poisson model of the break progression, the number of breaks in the graphite fibre as a function of applied strain, which accounts for stress decay at the fibre ends. It is shown that such extrapolations of tension test data are too optimistic. In addition, different fibres from the same yarn cross-section, apparently have different flaw populations, unlike that which occurs at longer gauge lengths.     

Influence of spatial correlations on crack bridging by frictional fibres

The effect of fibre interaction on matrix cracking in a unidirectional fibre-reinforced composite is analyzed. It is assumed that the matrix material contains a crack in a plane perpendicular to the fibres. Fibres, remaining intact, debond from the matrix and then act as bridging ligaments in the crack wake. The debonding process is accompanied by frictional sliding governed by a Coulomb friction law. Fibres are considered to be randomly located in the transverse plane. The fibre axial stress and longitudinal displacement are expressed in terms of the solution to a model problem for a single fibre in an ambient stress field due to all other fibres and applied load. The stress field produced by the other fibres is described using an ensemble averaging procedure. The radial distribution function g(r) that provides a quantitative measure of the correlations between the positions of different fibres is evaluated numerically from the Percus-Yevick equation for hard disks. The dependence of the fibre axial stress on the relative fibre-matrix displacement is examined for different values of the volume fraction of fibres. The resulting stress-displacement law is compared with results for other choices of the function g(r) and with a law given by a concentric cylinder model.