Stress transfer in the fibre fragmentation test: Part III Effects of interface debonding and matrix yielding

The micromechanics of stress transfer is presented for the fibre fragmentation test of microcomposites containing debonded fibre–matrix interface and yielded matrix at the interface region. Results from the parametric study are discussed for carbon fibre composites containing epoxy and polyetheretherketone (PEEK) matrices, representing respectively typical brittle debonding and matrix yielding behaviour at the interface region. The stress transfer phenomena are characterized for the two interface failure processes. The sequence of interface failure and fibre fracture as a function of applied stress are also identified. Maximum debonded and yielded interface lengths are obtained above which a fibre will fracture into smaller lengths. There are also threshold fibre fragment lengths above which fibre will fracture without interface debonding or matrix yielding. The applied stresses for these conditions are governed by three strength properties of the composite constituents, namely interface shear bond strength, matrix shear yield strength and fibre tensile strength for given elastic constants of the fibre and matrix, and the geometric factors of the microcomposite. The ineffective length, a measure of the efficiency of stress transfer across the fibre–matrix interface, is shown to strongly depend on the extent to which these failure mechanisms take place at the interface region. This revised version was published online in November 2006 with corrections to the Cover Date.     

Multiple Data Set (MDS) weak-link scaling analysis of jute fibres

Jute technical fibres were tested in tension at 10 different gauge lengths between 6 mm and 300 mm (50 or 100 tests at long or short gauge lengths respectively). The Young’s modulus, strain to failure and ultimate tensile strengths were determined individually and then Weibull distribution parameters were estimated using the maximum likelihood method to quantify the variation. Single Data Set (SDS, standard) and Multiple Data Set (MDS) weak-link scaling (WLS) predictions were assessed using Anderson–Darling Goodness of Fit Numbers (GOFN). The use of MDS predictions provides better correlation with the experimental data than the standard weak-link scaling method. The authors recommend the use of MDS weak-link scaling for this problem with at least two points (but preferably three) with fibre lengths at two extremes (and, if used, the third point near the mean fibre length).     

Stress transfer in the fibre fragmentation test

An improved micromechanics model has been developed of the stress transfer for a single fibre embedded in a matrix subjected to uniaxial loading. Debond crack growth is analysed based on the shear strength criterion such that when the interfacial shear stress reaches the shear bond strength, debonding occurs; and the average strength concept based on Weibull statistics is considered for fibre fragmentation. The influences of the interfacial shear bond strength and the fibre strength on the stress distributions in the composite constituents are evaluated. Depending on the relative magnitudes of these two strength parameters and given the elastic constants and geometric factors, three distinct conditions of the fibre-matrix interface are properly identified which include full bonding, partial debonding and full frictional bonding. Also quantified are the necessary criteria which must be satisfied in order for each interface condition to be valid. Finally, the mean fibre fragment length is predicted as a function of applied strain using a model composite of carbon fibre-epoxy matrix. The parametric study suggests that the critical transfer length predicted when the applied strain (or stress) required for further fibre fragmentation approaches infinity, can be regarded as a material constant, which is the sum of the bonded and the debonded lengths for the model composite.     

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.     

A new data reduction scheme for the fragmentation testing of polymer composites

A new reduction scheme of fragmentation data for the derivation of interfacial mechanical properties in polymer composites is proposed. The scheme is based on a theoretical model that accounts for elastic load transfer and friction at the interface, as well as for the statistical nature of fibre strength. Interface mechanical behaviour is characterized by two independent parameters, namely the interface bond strength and interface frictional resistance. Derived values of the two interface properties are computed, such that they yield the best possible agreement between experimental and theoretical results for the evolution of fibre fragment aspect ratio and debonding ratio as a function of applied strain. Results are reported for carbon fibres embedded in an epoxy matrix, with different levels of fibre surface treatment.     

New evaluation technique of interfacial properties in GFRP

A new method was proposed to evaluate the mechanical properties at the interface between the fibres and the matrix in composites using an embedded single fibre coupon test. A mechanical parameter at the interface (called the interfacial transmissibility, ) was derived from the fibre strength and the apparent stress of the fibre immediately before the first fracture of embedded fibre, _fa. This parameter indicated the degree of the mechanical transmission from the matrix to the fibre through the interface. This avoided some complicated problems such as the stress distribution along fibre fragments and the critical fragment state in a typical single-fibre test. This new method was tried to determine the -values for a fibre glass/epoxy resin with different amounts of a coupling agent at the interface. In order to measure the stress at the first fracture, the fracture process was monitored with a video camera during the single fibre test. The stress values at the first fracture for many coupons were analysed as a function of the three-parameter Weibull distribution. The resulting average stress and its coefficient of variation indicated that the reliability of the measurement for the stress at the first fracture was not less than that obtained by the usual single-fibre test. The change of interfacial transmissibility with amount of the coupling agent revealed the existence of an optimal interface.     

Exact theory of fibre fragmentation in a single-filament composite

An exact theory is developed to describe the evolution of fibre fragmentation in a single-filament composite test as a function of the underlying fibre statistical strength and fibre/matrix interfacial shear stress, τ. The fragment distribution is a complicated function of fibre strength and τ because the stress around breaks which do occur recovers to the applied value, σ, over a length δ(σ) determined by τ. Therefore, no other breaks can occur within δ (σ) of an existing break. To account for this effect, the fibre fragment distribution is decomposed into two parts; fragments formed by breaks separated by more than δ (σ) at stress σ, and fragments smaller than δ (σ) which were formed at some prior stress σ′ < σ when a smaller δ(σ′) < δ(σ) prevailed. The distribution of fragments larger than δ (σ) is identical to that of a fibre with a unique non-statistical strength σ and is known exactly. The distribution of fragments smaller than δ(σ) can then be determined from the distribution of the longer fragments. Predictions of the theory are compared to simulations of fibre fragmentation for several common models of stress recovery around fibre breaks with excellent agreement obtained. The present theory can be utilized to thus derive both thein situ fibre strength at short gauge lengths ? δ and the τ from experimentally obtained fragment distributions, and an unambiguous inversion procedure is briefly discussed. The application of the theory to other multiple-cracking phenomena in composites is also discussed.     

Analysis of the single-fibre pull-out test by means of Raman spectroscopy: Part II. Micromechanics of deformation for an aramid/epoxy system

The single-fibre pull-out test has been analysed for Kevlar-49 fibres in a cold-cured epoxy resin by using both a conventional pull-out experiment and Raman spectroscopy. The interfacial shear strength (ISS) has been estimated from the pull-out force for fibres with a range of embedded lengths. Raman spectroscopy has been used to analyse the distribution of fibre strain in the pull-out test by mapping the variation of strain along an aramid fibre undergoing pull-out from the epoxy resin matrix. At low strains the behaviour follows elastic shear-lag analysis but, as the fibre strain is increased, debonding takes place at the fibre/matrix interface. It is found that this debond propagates along the interface until the entire fibre is debonded. The fibre is then pulled out of the resin matrix by a frictional pull-out process. It is shown that the conventional pull-out experiment produces only an apparent value of ISS and that through a partial-debonding model it is possible to use the interfacial parameters obtained from the Raman analysis to predict the data from the conventional test.     

A theory for the creep of steel fibre reinforced cement matrices under compression

The paper presents a theoretical model to predict the creep of cement matrices reinforced with randomly oriented discrete steel fibres. The theory considers the composite to be represented by an aligned steel fibre which is surrounded by a thick cylinder of the cement matrix. The fibre provides restraint to the flow component of creep of the matrix through the fibre-matrix interfacial bond strength. The delayed elastic strain component of creep is unaffected by the fibre. The fibre-matrix interfacial bond strength,, is shown to be primarily a function of the shrinkage of the cement matrix and the radial deformation caused by the sustained axial stress. In addition, the state of stress in the matrix at the interface is suggested to influence greatly the bond strength,. The validity of the theory is established by means of experimental data on concrete and mortar matrices reinforced with melt extract and hooked steel fibres, at sustained stress-strength ratios of 0.3 and 0.55. Finally an empirical expression is derived to determine the creep of steel fibre reinforced concrete, based on a knowledge of the creep in unreinforced matrices and fibre size and volume fraction.     

Study on the breaking strength of jute fibres using modified Weibull distribution

The two-parameter Weibull distribution does not always adequately describe the experimental bast fibre strength at different gauge lengths. For this reason, it was modified by incorporating the diameter variation of jute fibres in this paper. The fibre diameter was measured with an optical microscope. The three-parameter Weibull model was also used to compare with the modified model. It was found that as the fibre diameter variation increased the tensile strength of the jute fibre decreased. The strength predicted by modified Weibull distribution was more accurate than that of the two conventional models. In addition, the breaking strength of jute fibre was less sensitive to gauge length than that of cotton fibre because the breaking of jute filament involves ultimate cells breaking repeatedly and matrix cracking.     

Analysis of the effect of embedded fibre length on fibre debonding and pull-out from an elastic matrix

The nature and properties of the resistance to fibre-matrix interfacial debonding in composites composed of ductile fibres in a brittle or elastic matrix can be determined using the single-fibre pull-out test. The results of such tests on cementitious matrix specimens indicate a non-linear relationship between the debonding and/or pull-out load and the embedded length of the fibre. Several of the theories developed to explain the debonding process and enable estimation of the parameters representing the debonding resistance through an analysis of pull-out test results are reviewed in this first of a two-part paper. The application of these theories to experimental data for steel fibre-cementitious matrix pull-out specimens is examined in the second part.Nomenclature b _i effective thickness of the fibre—matrix interface - d _f diameter of the fibre - l _c embedded fibre length at which fibre fracture rather than pull-out occurs - l _d debonded fibre length - l _e embedded length of the fibre in a pull-out specimen - l _{e, min} a minimum embedded fibre length which equals (1/₂cosh^–1 (_ib,max/_ib,f)^1/2 - l _k minimum embedded fibre length required to support the debonding stress in the fibre - l _p maximum embedded fibre length at which complete debonding occurs instantaneously - q _{ib, max} elastic shear flow resistance to fibre-matrix interfacial debonding - q _{ib, f} frictional shear flow resistance to slipping at the fibre-matrix interface after the elastic bond has broken - r _f radius of the fibre - r _m effective radius of the matrix block in a pull-out test specimen - A _f cross-sectional area of the fibre - A _m cross-sectional area of the matrix block in a pull-out test specimen - C ₁ a constant representing the normal compressive stress at the fibre-matrix interface - C ₂ a constant representing the coefficient of friction between the fibre and the matrix at the interface - D length of the debonding plateau (see Fig. 5a) - E _m modulus of elasticity of the matrix - E _f modulus of elasticity of the fibre - G _i shear modulus of the fibre-matrix interface - G _m shear modulus of the matrix - P _f load applied to the fibre in a pull-out test - P _{f, max} maximum load applied to the fibre in a pull-out test - P _{f, ult} applied load at which fibre fracture occurs - P _{f, edb} load required to break the adhesional or elastic fibre-matrix interfacial bond in a pull-out test specimen - P _{f, edb} maximum value of P _{f, edb} (see Fig. 7) - P _{f, d} instantaneous decrease in applied fibre load when debonding is complete - P _f,r residual fibre load required to overcome initial frictional resistance to fibre pull-out - P _f, applied fibre load required to debond an infinitely long fibre with no frictional resistance to slipping at the fibre-matrix interface - an elastic constant ₁ = (2G _i/b _i r _f E _f)^1/2 ₂ = [(2 G _m/ln(r _m r _f))(1/A _f E _f–1/A _m E _m)]^1/2 ₃=[4 G _m/ln(r _m/r _f)r _f E _f]^1/2 ₄=[G _m(A _f E _f + A _m E _m/A _f E _f A _m E _m)]^1/2 - _i surface energy of the fibre-matrix interface - _f fibre extension or displacement in a pull-out test - slope of the linear portion of the P _{f, max} against ₂ l _e curve and is equal to _ib,fd _f/₂ - fibre-matrix misfit - coefficient of friction between the fibre and matrix at the interface - _f Poisson's ratio of the fibre - _m Poisson's ratio of the matrix - _{f, max} stress in the fibre at which interfacial debonding occurs in a pull-out test specimen, i.e. debonding stress - _{f, max} plateau debonding stress (see Fig. 5b) - _{f, po} stress in the fibre when fibre pull-out begins, i.e. immediately following the completion of interfacial debonding - _{f, ult} ultimate tensile strength of the fibre - _{i, n} normal compressive stress exerted by the matrix on the fibre across the interface - _{i, av} average shear stress at the fibre-matrix interface - _{i, max} maximum shear stress at the fibre-matrix interface - _{ib, av} average shear strength of the fibre-matrix interfacial bond - _{ib, max} maximum or adhesional shear strength of the fibre-matrix interfacial bond - _{ib, f} frictional resistance to slipping at the fibre-matrix interface after the elastic bond has broken     

黄麻纤维束力学性能的尺寸和应变率效应

为了探究黄麻纤维束的尺寸效应和应变率敏感性,利用C43电子式万能试验机和CEAST 9340落锤试验冲击系统分别在静动载条件下对黄麻纤维束进行测试,获得了杨氏模量、强度、峰值应变和韧性随标距和应变率的变化关系静载试验在1/600s-1应变率条件下进行,测试了6组不同标距(25、50、100、150、200和300mm)的试件;动载试验以应变率为变量,在4组不同的应变率(40、80、120和160s-1)条件下进行了测试,试件标距均为25mm。测试结果表明:随着试件标距增大,杨氏模量初始增大,当标距大于100mm时趋于稳定;强度、峰值应变和韧性均减小。随着应变率增大,杨氏模量和强度均增大;峰值应变初始减小后趋于稳定;韧性先减小后增大。鉴于植物纤维束材料较大的性能离散性,采用Weibull分布对试验数据进行拟合,获得了黄麻纤维束强度在不同试验条件(标距和应变率)下的分布规律。     

Fiber and interface strength distribution studies with the single-fiber composite test

Application of the single-fiber composite (sfc) tension test for fiber and interface strength determination is discussed. Fiber breaking and fiber/matrix debond propagation are modelled by Poisson processes. Fiber fragment length distribution as well as debond length dependency upon the applied stress are derived and their interrelation revealed.

Acoustic emission monitoring of the sfc during a test is utilized to obtain the dependency of mean fragment length upon stress and consequently on the Weibull distribution shape and scale parameters. Excellent agreement with data obtained by notoriously complicated conventional fiber tests is observed.     

A theory for the free shrinkage of steel fibre reinforced cement matrices

The paper presents a theoretical model to predict the free shrinkage of cement matrices reinforced with randomly oriented discrete steel fibres. The model is based on the consideration that the equivalent aligned length of a random fibre is responsible for restraining the shrinkage of a thick matrix cylinder of diameter equal to the fibre spacing, through the fibre-matrix interfacial bond strength. The validity of the model is established by means of extensive experimental data for different types of steel fibres in cement, mortar or concrete matrices. The theoretical model is also used to determine the values of coefficient of friction,, and the average bond strength,, of the fibre-matrix interface. It is shown that is a basic property of the matrix and fibre interface, which is affected by the surface roughness and mechanical deformation of the fibres., however, is greatly influenced by the shrinkage of the matrix and volume fraction of fibres. Finally, an empirical expression is derived to determine the shrinkage of steel fibre reinforced cement matrices based on the shrinkage of unreinforced matrices and fibre properties.     

Statistics of single‐fibre tensile strength incorporating spatial distribution of microcracks

The random distribution of single‐fibre tensile strength has been commonly characterized by the two‐parameter Weibull statistics. However, the calibrated Weibull model from one set of strength data at a given gauge length cannot accurately predicts the strength variation of the fibre at different gauge lengths. Instead of presuming the two‐parameter Weibull distribution or any other specific statistical distribution for the single‐fibre strength to begin with, this work proposes an approach to incorporating the appropriate spatial flaw distribution within a fibre and synchronizing multiple sets of tensile strength data to evaluate the single‐fibre strength distribution. The approach is examined and validated by published single‐fibre strength data sets of glass, ceramic and synthetic and natural carbon fibres. It is shown that the single‐fibre strength statistics does not necessarily always follow the two‐parameter Weibull distribution.     

Statistical analysis of the failure stresses of ceramic fibres: Dependence of the Weibull parameters on the gauge length, diameter variation and fluctuation of defect density

The strength distribution of ceramic fibres is commonly described using a two-parameter Weibull distribution function. This study shows that the determination of these parameters from around 30 tensile tests obtained at one single gauge length (L ₀) does not allow the strength distribution at other gauge lengths to be correctly predicted. The reliability in the Weibull parameter determination is lowered by variations in fibre diameter (D) and the insufficient number of fibres tested. An effective failure stress _E = · ( D · L ₀)^1/m is first introduced to take into account fibre diameter variations and to extract the two Weibull parameters from the 180 tests obtained at 6 gauge lengths. It is then shown that the linear size effect, which is expected from the standard Weibull model, is not appropriate to fit correctly this experimental strength distribution. The length dependence follows a power law (L ₀ ) leading to an effective failure stress _E = · ( D · L ₀ )^1/m . Diameter variations along the gauge length cannot be responsible for this non linear variation with the length, which is attributed to a large scale fluctuation of the density of defects. The value of can bring valuable information about fluctuations in the fibre processing conditions.     

Effect of friction between fiber and matrix on the fracture toughness of the composite interface

A method of evaluating the interfacial fracture toughness using a single-fibre composite test is proposed. In contrast with the existing techniques, the method takes into account the phenomenon of friction between the fibre and matrix in the debonding zone. A general mathematical solution of the problem and modelling of the friction phenomenon are presented. Finite-element analysis using a contact statement is utilized for numerical evaluation of the stress–strain state. The influence of the coefficient of friction and interfacial debonding length is analysed in detail. It is shown that the friction reduces the calculated value of the elastic strain energy release rate for a given debonding length, relative to that obtained when friction is neglected. The magnitude of the difference depends on the coefficient of friction, the elastic properties of the fibre and matrix, and the characteristics of the debonding mechanism. Experimental data on debonding in a series of glass-epoxy single-fibre composites are analysed using the proposed numerical technique to obtain the effects of fibre surface treatments and fibre strain-to-break on the interfacial fracture toughness. © Kluwer Academic Publishers     

Structure and properties of some vegetable fibres

The stress-strain curves for pineapple leaf fibre have been analysed. Ultimate tensile strength (UTS), initial modulus (YM), average modulus (AM) and elongation of fibres have been calculated as functions of fibre diameter test length and test speed. UTS, YM, and elongation lie in the range of 362 to 748 MN m^–2, 25 to 36 GN m^–2, and 2.0 to 2.8%, respectively for fibres of diameters ranging from 45 to 205m. UTS Was found to decrease with increasing test lengths in the range 15 to 65 mm. Various mechanical parameters show marginal changes with change in speed of testing in the range of 1 to 50 mm min^–1. The above results are explained on the basis of structural variables of the fibre. Scanning electron microscope studies of the fibres reveal that the failure of the fibres is mainly due to large defect content of the fibre bo1h along the fibre and through the cross-section, The crack is always initiated by the defective cells and further aggravated by the weak bonding material between the cells.     

Weibull analysis of strength-length relationships in single Nicalon SiC fibres

Nicalon SiC fibre is naturally brittle and offers high-temperature application in fibrous composites. Due to the randomly distributed flaws along the fibre, the statistical variability in single-fibre strength is obvious. In this paper, the effect of heat-cleaning procedures on Nicalon fibres has been investigated, and the statistical strength and variability of single Nicalon fibres have been characterized in tension and compared. Experimental results show that the strengths of single Nicalon fibres among the three types of heat-cleaning procedures are less than that of as-received unsized fibres by 22–30%. In addition, both the failure load and the failure stress of the fibres, for a given gauge length (50 mm) and yarn cross-section, can be well fitted to a two-parameter Weibull distribution. The effect of gauge length over the range from 10–175 mm, holding the strain rate constant, was also studied. The logarithmic strength-length plots show that the strength of single Nicalon fibres follows the weakest-link rule.