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This article provides a summary of some analytical methods that may be used to estimate the effective thermo-elastic constants of unidirectionally fiber reinforced composites. Both multi-phase and two-phase composites are considered. Matrix yielding effects and failure processes during uniaxial loading are also considered. Fiber failure and matrix cracking modes of damage are considered, and in addition compression strength, transverse compression strength, and in-plane longitudinal shear strength.

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... Finally the initiating fibres buckle and fracture, passing the loading on to adjacent fibres. When fibres run transversely to the applied force the composite will fail through fibre/matrix debonding and matrix shear cracking[119]. Typically laminar composites are relatively thin; this can result in Euler buckling of the coupon itself. ...

The tensile failure of unidirectional alumina fibre reinforced aluminium is studied in uniaxial loading along the fibre axis. The tensile strength is measured as a function of matrix yield strength, which is varied by varying the testing temperature, from RT to 600°C. Over the range of matrix yield strength (i.e., of temperature) examined, the fracture mode remains brittle. Batdorf’s (J Reinforced Plastics Compos 1982;1:153–164) simple ideal local load-sharing model describes well the observed behaviour, under the condition that it be adapted to account for the actual number of nearest neighbours characteristic of the fibre distribution in the composite. This is shown to be close to three, i.e., at variance with the usually assumed idealized hexagonal or square fibre arrangement patterns.

Models for the debonding of a fiber embedded in a brittle matrix are proposed and analyzed. Attention is restricted to systems having a residual compressive stress acting across the fiber/matrix interface. Debonding, as well as pullout after the fiber breaks, is accompanied by frictional sliding. Fiber—matrix interaction is modeled by a cylindrical cell with two sets of boundary conditions: one modeling an isolated fiber—matrix unit and the other a matrix containing an array of unidirectional fibers. The elastic properties of the fiber are taken to be transversely isotropic about the fiber axis, while the matrix is assumed to be isotropic. The debonding process is treated within the framework of fracture mechanics as a mode 2 crack. Two idealizations of friction are considered: a constant friction stress independent of normal compression across the interface, and Coulomb friction. Approximate closed form solutions to the model are presented. These are assessed using results from an accurate numerical analysis.

The initiation and growth of damage in composite materials are phenomena that precede the failure event where a material sample or component separates into two pieces. In fatigue, the damage grows slowly and leads to a gradual deterioration of mechanical properties. For the prediction of the fatigue behavior of unidirectional and laminated titanium composites, it is necessary to be able to take account of the effects on the thermoelastic constants of matrix cracking that is induced by fatigue stress cycling. The values of the thermoelastic constants for microcracked composites are determined by the way in which stress is transferred between fiber and matrix in unidirectional composites, and between neighboring plies in laminates, as a result of microcrack formation in the matrix. A summary is given of the recent progress that has been made at the National Physical Laboratory (NPL) on the development of analytical stress transfer models for unidirectional and laminated composites. The models are each based upon just a single assumption concerning the stress field that leads to stress and displacement solutions for which the Reissner energy function, used in a variational calculation, has stationary values. The use of the Reissner function allows both applied traction and displacement conditions to be imposed on the fiber/matrix interface and external boundary. In contrast to other applications of variational techniques, the models provide both the stress and displacement distributions at every point in the composite. Thus, complete solutions can be derived that satisfy exactly the equilibrium equations, the interface conditions, the stress-strain relationships apart from one, and the boundary conditions involving tractions. The remaining stress-strain relationship and boundary conditions involving displacements are satisfied in an average sense. For unidirectional composites, both perfectly bonded and frictionally slipping interfaces (such that the interfacial shear stress is a constant) are considered; while for laminates, consideration is restricted to interfaces that remain perfectly bonded in the presence of transverse cracks. The analytical stress transfer models are of high quality to the extent that they represent the best models that can be derived subject to the single assumption on which they are based. The models do, therefore, offer life prediction methodologies the opportunity of confidently making use of analytical methods that obey the important principles of mechanics. The estimation of effective stress intensity factors for bridged matrix cracks that occur in titanium composites subject to fatigue loading at modest load levels is an important element in the prediction of composite performance. Assuming that fibers remain intact, the paper makes use of the stress transfer models to predict the dependence of stress intensity factors (for long cracks) on the applied stress and crack separation, for bridged cracks in both unidirectional and laminated composites. The effects of crack interaction are determined, and the new model predictions are compared to those of shear-lag theory modified to take account of residual stresses arising from thermal expansion mismatch effects. It is also shown how predictions can be made for the dependence of the thermoelastic constants of unidirectional and laminated titanium matrix composites on the nonuniform spacings of matrix cracks that can be encountered in practice.

In this paper we compare three distinct asymptotic analyses for the probability distribution for strength of a certain class of fibrous materials. This class, which contains composite materials and twisted yarns and cables, is characterized by strong, mechanical interaction among neighboring fibers as is generated by a binding matrix or by transverse frictional forces. The model is the chain-of-bundles model, and we focus mainly on a planar geometry and a simple type of local load-sharing among fibers in a cross-section wherein the load of a failed fiber is shifted in equal portions to its two nearest flanking survivors. Through the study of typical examples we find that the results of the three methods all underscore the importance of the Weibull distribution for modeling composite strength, and are in excellent numerical agreement despite their differing analytical form. This is fortunate because only the two least accurate of the analyses show promise of being extended to more general three-dimensional geometries and local load-sharing rules of interest in applications. The paper lays the groundwork for considering such extension in future papers.

Predictions of the ultimate tensile strength of 3-dimensional fiber-reinforced composites as a function of the fiber statistical strength distribution and fiber geometry (square vs. hexagonal packing) are presented for materials in which the load transfer from broken to unbroken fibers is very localized. The predictions are obtained using a previously-developed simulation model adapted here for hexagonal fiber arrays. The model includes (1) the Hedgepeth and Van Dyke load transfer model to determine in-plane load transfer and (2) fiber slip in the longitudinal direction via a shear-lag model. Results show that, although the load transfer does depend on fiber geometry, the average composite tensile strength and the statistical distribution of strengths do not depend strongly on the fiber geometry. The size scaling of strength is then also shown to be nearly-independent of local fiber geometry. These results are physically reasonable since the critical clusters of fiber damage causing failure are observed to be larger than 15-20 fibers, so that the detailed local geometry at smaller length scales is not crucial to failure. Hence, analytic models developed previously for square fiber arrangements can be used with reasonable accuracy independent of fiber arrangements. Applications of the model to polymer matrix composites are discussed in a companion paper (Part II).

Asymptotic distributions are obtained for both the strength and the time to failure of a fibrous material for which mild bonding or friction exists between fibers. The analysis is based on the chain-of-bundles probability model, and equal load sharing is assumed for the nonfailed fiber elements in each bundle. Asymptotic results are obtained for the difficult but useful case where k, the number of bundles in the chain, grows very rapidly with respect to n, the number of fibers in each bundle. For both strength and time to failure, a classical extreme value distribution is found to be the asymptotic distribution, and the parameters are given in terms of certain fiber properties. The results apply to long, flexible fibrous structures such as yarns and cables.

A fracture mechanics analysis valid for long cracks is applied to three crack-bridging problems that arise in the composites area, namely, (1) matrix cracking in perfectly bonded unidirectional composites, (2) transverse cracking in perfectly bonded cross-ply laminates, and (3) matrix cracking in unidirectional composites in which frictional slip occurs at fiber matrix interfaces governed by the constant shear-stress shear-lag model of stress transfer. The fracture mechanics analysis is also applied to the three models of cracking in composites for the case in which the preexisting crack can have any length

This paper describes an experimental study of the compressive failure of T800/924C carbon-fibre/efoxy composite laminates. Undirectional laminates loaded parallel to the fibres have compressive strengths that are 70% of the tensile strength and fail by fibre-microbuckling. During microbuckling the fibre debonds from the matrix, and the fibres break in bending. Multidirectional []sm laminates were also tested in compression, and the critical failure mechanism observed was microbuckling of the 0° plies. The failure strain was almost the same as for the undirectional laminate, The failure strain was almost the same as for the unidirectional laminate, which indicated that the ±45° plies have no significant influence on the failure strength of the 0° plies.

A theory of the workhardening and Bauschinger effect in two-phase materials, combining dislocation mechanisms with a continuum model, is extended to high volume fraction of the hard phase by using the mean field theory of Paper I [Acta metall.31, 1795 (1983)]. Application of the extended model to available workhardening data for the simple experimental model system of copper with continuous tungsten fibres reveals a novel workhardening contribution: “elastic friction”. The contribution arises from the interaction of gliding dislocations with the complex spatially fluctuating pattern of internal stresses induced by the applied stress as a result of elastic heterogeneity. Elastic friction is taken into account in a simple model of the Bauschinger effect, the “modified Orowan-Wilson model”, which is substantiated by a new set of experiments on copper-tungsten with large tungsten volume fractions.

Following a review of statistical models of the failure of single fibers and bundles of these fibers, algebraic recurrence formulas are derived that generate expressions for the failure probabilities of bundles of classical fibers. One of these recurrence relations is suitable for the accurate numerical calculation of failure probabiliites of boundles consisting of up to 500 single fibers. It is shown how account can be taken of the effect of defect-free fibers having finite strength. Numerical results are compared to three asymptotic analytic approximations, two of which have been proposed int he statistical literature and are now applied to fiber problems for the first time.

It is shown how the results in Part I can be adapted to describe certain kinds of inelastic behaviour of arbitrary fibre composites. Uniaxial extension is discussed in some detail and bounds are stated for the main overall moduli and flow stress at any stage. Special attention is given to a composite in which the matrix behaves elastoplastically.

An analytic model for predicting the tensile strength of uniaxial fiber-reinforced composites is applied to several different graphite-fiber composites and shows good agreement with measured strengths. The model includes the effects of fiber statistical strength distribution, local load transfer from broken to unbroken fibers, interfacial shear strength, fiber pullout, and composite size. The model is applied in detail to AS-4 fiber/Epon828 matrix and T300 fiber/Epicote matrix composites, for which the requisite constituent and composite data are available in the literature. Data for the fiber strength at the critical length is obtained from single fiber composite experiments using the analysis of Curtin. For the AS-4/Epon828 material, the predicted tensile strength is 35% higher than the measured value if physical fiber volume fraction is used and 18% higher if a fiber volume fraction consistent with the measured composite Young's modulus is used. For the T300/Epicote system, the predicted strength is 35% higher if thermal stress effects in the s.f.c. test are neglected but only 20% higher if estimated thermal stresses are included. Strength predictions for an AS-4 fiber/thermoplastic matrix system and for two other T300 fiber-based composites are made using the same fiber strength data and comparable agreement (20%) is obtained. The theory predicts very modest changes in composite strength with increasing interfacial shear strengths, as observed experimentally. Overall, the model is in reasonable agreement with experiment when the constituent data is accurately determined, which establishes the model as the basic model for predicting uniaxial tensile strength in polymer matrix composites.

Composite materials are reviewed fromthe applied mechanics and engineering science point of view. Analyses are presented of composite materials properties such as: elasticity, thermal expansion, mositure swelling, viscoelasticity, conductivity (which includes, by mathematical analogy, dielectrics, magnetics, and diffusion) static strength, and fatigue failure.

An analysis is made of the effect of orientation of the fibres on the stiffness and strength of paper and other fibrous materials. It is shown that these effects may be represented completely by the first few coefficients of the distribution function for the fibres in respect of orientation, the first three Fourier coefficients for a planar matrix and the first fifteen spherical harmonics for a solid medium. For the planar case it is shown that all possible types of elastic behaviour may be represented by composition of four sets of parallel fibres in appropriate ratios. The means of transfer of load from fibre to fibre are considered and it is concluded that the effect of short fibres may be represented merely by use of a reduced value for their modulus of elasticity. The results of the analysis are applied to certain samples of resin bonded fibrous filled materials and moderately good agreement with experimental results is found.

This paper describes the behaviour of a carbon-fibre reinforced epoxy composite when deformed in compression under high hydrostatic confining pressures. The composite consisted of 36% by volume of continuous fibres of Modmur Type II embedded in Epikote 828 epoxy resin. When deformed under pressures of less than 100 MPa the composite failed by longitudinal splitting, but splitting was suppressed at higher pressures (up to 500 MPa) and failure was by kinking. The failure strength of the composite increased rapidly with increasing confining pressure, though the elastic modulus remained constant. This suggests that the pressure effects were introduced by fracture processes. Microscopical examination of the kinked structures showed that the carbon fibres in the kink bands were broken into many fairly uniform short lengths. A model for kinking in the composite is suggested which involves the buckling and fracture of the carbon fibres.

The theoretical stress-strain behaviour of a composite with a brittle matrix in which the fibre-matrix bond remains intact after the matrix has cracked, is described. From a consideration of the maximum shear stress at the fibre-matrix interface, the extent of fibre debonding and the crack spacing in a partially debonded composite are derived. The energetics of cracking and the conditions leading to an enhanced matrix failure strain are then discussed and, finally, the crack spacing expected in composites containing fibres isotropically arranged in two or in three dimensions is derived for the case of very thin and hence very flexible fibres.

A new computational technique, called the quadratic influence superposition (QIS) technique, is developed to study the stresses around arbitrary arrays of fiber breaks in a unidirectional composite loaded in simple tension, and consisting of elastic fibers in a matrix, which is either elastic-perfectly plastic or which can debond at the interface leaving residual friction. The method involves extending a recently developed break influence superposition (BIS) technique, where to model the behavior of damaged (yielded or debonded) matrix elements, we use special compensating shear stress profiles and develop the corresponding influence functions. The QIS technique appears to be at least an order of magnitude more efficient than other numerical schemes as the computation time is tied mainly to the amount of damage, and it is more accurate than a simpler version of this technique developed earlier. In illustrative examples, the method determines the Mode I fiber and matrix stress distributions around a “center crack” consisting of up to 31 contiguous fiber breaks. Incremental treatment is needed to establish the extent of the inelastic regions and the results, which achieve excellent agreement with exact shear lag analyses, clearly show that QIS calculated these correctly. Results show that the extent of the matrix damage region increases approximately linearly with applied load and nonlinearly with the number of breaks. The stress concentrations and overload profiles along nearby unbroken fibers are altered as compared to the fully elastic case with magnitudes reduced but length scales increased.

This paper presents solutions to the problem of a Mode I crack that is shielded by bridging tractions that can decay with the passage of either load cycles or time. The problem contains two competing time- or cycle-dependent processes, namely degradation of the shielding tractions (rate constant r1) and crack advance (rate constant r2), the latter being a function of the crack tip stress intensity factor range, Δktip. For given initial conditions and load level, the development of Δktip with crack length depends only on the ratio , rather than on each of r1 and r2 separately. Particular emphasis is placed on the roles of thresholds for crack advance or shielding degradation. It is shown that, depending on whether or not such thresholds exist, the problem can reduce asymptotically to one or another familiar, rate-independent problem, such as elastic/perfectly plastic bridging tractions in the limit (fatigue crack growth applications), or fracture in the presence of a viscous process zone when (monotonic loading applications).Solutions for general values of are found numerically by methods valid for general bridging and crack growth laws and for specimens of various common shapes. Crack propagation is comprises a history-dependent transient regime followed by a quasi-steady state regime that is insensitive to the initial conditions. Solutions are illustrated by specific application to bridging by linear springs that soften with passing load cycles or time. The springs may to represent the action of a repairing patch bonded over a crack in an alloy plate, or of bridging fibers in a metal—matrix composite or laminate.

The inelastic response of fiber-reinforced ceramic and metal matrix composites under fixed load at elevated temperature is due to the complementary effects of creep and damage in the constituents. After matrix cracking or tensile creep relaxation in a short time, subsequent deformation and failure are driven by shear stress relaxation in the matrix and at the fiber-matrix interface around broken fibers. The shear creep causes stress redistribution to unfailed fibers, causing further fiber breakage and shear relaxation, culminating in abrupt failure of the composite. This sequence of events is modeled both analytically and numerically within the Global Load Sharing (GLS) approximation previously utilized for quasi-static loading. Analytically, a unit cell model is used to obtain simple closed-form relationships for the time-dependent relaxation of the shear at the interface. This relaxing shear stress is then incorporated into a simulation model which follows the evolution of slip and fiber damage up to failure. The slip lengths and failure times are predicted vs matrix creep exponent n, fiber Weibull modulus m, applied load and, interestingly, physical specimen length. An analytic model for failure shows good agreement with the simulation results and so can be used for qualitative estimates of lifetime. Application to Ti-MMCs is discussed.

Solutions are presented for the effective shear modulus of two types of composite material models. The first type is that of a macroscopically isotropic composite medium containing spherical inclusions. The corresponding model employed is that involving three phases: the spherical inclusion, a spherical annulus of matrix material and an outer region of equivalent homogeneous material of unlimited extent. The corresponding two-dimensional, polar model is used to represent a transversely isotropic, fiber reinforced medium. In the latter case only the transverse effective shear modulus is obtained. The relative volumes of the inclusion phase to the matrix annulus phase in the three phase models are taken to be the given volume fractions of the inclusion phases in the composite materials at large. The results are found to differ from those of the well-known Kerner and Hermans formulae for the same models. The latter works are now understood to violate a continuity condition at the matrix to equivalent homogeneous medium interface. The present results are compared extensively with results from other related models. Conditions of linear elasticity are assumed.

Microbuckling in composite laminates is thought to initiate by the elastic bending of fibres, loaded by resin matrix material in shear. The fibres rotate and break in two places, forming a kink band. The fibres then rotate further until the matrix between the fibres fails, and the kink band and hence the laminate loses its load carrying capability. The present work investigates existing criteria for fibre microbuckling failure in a 0° unidirectional carbon-fibre-reinforced plastic (CFRP) laminate loaded in compression. From simple arguments, it is concluded that fibres undergoing bending cannot fail in tension on their convex side but rather that they fail in compression on their concave side. Inferences are made on which failure criterion should be used to predict unidirectional laminate failure when the failure mode is by 0° fibre microbuckling (or fibre kinking).

In this paper we describe a model to find the approximate equations for determining the in-plane shear modulus of a unidirectional fibre reinforced composite from the constituent material properties. Classical elasticity theory has been applied to the simplified model of a composite unit cell in which the concept of an interphase between fibre and matrix is taken into account. Thus the model considers that the composite material consists of three phases, that is the fibre, the matrix, and the interphase which is the part of the polymer matrix lying close to the fibre surface which possesses different physico-chemical properties from those of the main constituents. Thermal analysis was used for the determination of the thickness and volume fraction of the interphase. The theoretical results are compared with other theoretical expressions and with experimental data. The model introduced in this paper seems to be an improvement for the shear modulus.

A brief survey is given of recent and current theoretical studies in the area of micromechanics. Topics discussed include void collpase, transformation toughgening, fiber kinking and thermoelastic dissipation. The examples discussed illustrate the use of continuum-mechanical concepts to deduce information concerning consitutive and strength characteristics of metals, ceramics, composites, and rocks.

A computer model has been developed that follows the sequence of fibre failures in a thin layer of unidirectional fibre composite from the breaking strains of the individual fibres. No assumptions are made about the point of final failure, failure being reached in this computer model when the sequence of fibre fractures, at constant applied strain, becomes unstable.The model clearly shows the sequential development of fibre fractures in a composite layer and that only about four fibre fractures in one composite plane are necessary to initiate composite failure. This implies that elimination of the low strain fibres in the distribution could significantly improve composite strength. Predicted composite failure strains are compared with experimental values and the agreement is good.The effect of varying some of the parameters in the model are discussed and methods of refining and extending the model are presented.

This paper discusses tensile strength distributions for fibres (called classical fibres ) whose strength is independent of the rate of loading. Reasons are presented for expecting, in the absence of all other information, that the tensile strength of long classical libres from a common stationary source should obey the Weibull distribution. The statistical theory of the strength of bundles of classical fibres, as developed by Daniels is applied to infinite bundles composed of fibres which obey the Weibull distribution. It is found that the ratio of the tensile strength (units of force at break per initial unit area) of a bundle to the mean tensile strength of the constituent filaments decreases montonically with increasing dispersion in the strength of the constituent filaments. In general, the tensile strength of a large bundle has the same order of magnitude, but is less than the mean strength of the component filaments. Previous calculations have yielded this conclusion for fibres with a special time-dependence to their tensile strength ; here it is shown that the conclusion also applies to classical fibres.

A stress intensity approach is used to analyze tensile failure of brittle matrix composites that contain unidirectionally aligned fibers held in place by friction. In general, failure may initiate either by growth of a crack in the matrix, or by fracture of fibers that bridge the matrix crack. Subsequently, these failure processes may continue either unstably or stably with increasing applied stress. Solutions to the fracture mechanics analysis are obtained numerically in normalized form, with one microstructural variable, the normalized fiber strength. The analysis defines transitions between failure mechanisms and provides strength/crack-size relations for each mechanism. Explicit relations are derived for the matrix cracking stress (noncatastrophic failure mode), the condition for transition to a catastrophic failure mode, and the fracture toughness in a region of catastrophic failure, in terms of microstructural properties of the composite.

The objective of this work is to employ the variational model of a concentric cylinder to model accurately the fiber push-out (and re-push) test used for evaluation of interfacial properties. In this paper, the detailed stress distributions and load/displacement solutions obtained by considering a variety of interface boundary conditions (such as adhesively bonded, frictionally sliding and debonded) are compared with a numerical elasticity solution which uses the material properties of a polyester/epoxy system for which experimental data are also available. The present approach is shown accurately to predict the force versus debond length relationships in the progressive debonding portion of the push-out test while demonstrating a boundary layer effect in the behavior of fiber axial stress near the free surface and interfacial radial and shear stress in the slip region. The apparent debond toughness has been defined by using energy-balance arguments during a critical finite crack extension. The increase in crack-tip driving force with the length of the debond is attributed to crack-tip mode mixity while the magnitude of the release rate of shear energy is observed to be nearly constant. By using the critical shear-energy release rate criterion, the load/displacement response is then predicted during push-out and re-push tests. Finally, an estimate of the maximum debond stress, externally applied displacement and the critical debond length are made at the point of instability. All of these estimates are in good agreement with the experimental measurements.

This paper reviews the formulation of the problem of a bridged crack in an elastic medium as an integral equation, noting explicit forms for specimens of various common shapes. Numerical methods are provided for the convenient and efficient self-consistent solution of the integral equation when the bridging tractions, p, are a function of crack opening displacement, u, rather than an explicit function of position in the crack. Methods are presented for determining physically and computationally unstable crack configurations for various forms of p(u), including functions possessing a peak. Knowledge of both stable and unstable solutions is essential to demarking the transition from noncatastrophic (or ductile) failure to catastrophic (or brittle) failure.

Matrix fracture in brittle-matrix fiber composites is analyzed for composites that exhibit multiple matrix cracking prior to fiber failure and have purely frictional bonding between the fibers and matrix. The stress for matrix cracking is evaluated using a stress intensity approach, in which the influence of the fibers that bridge the matrix crack is represented by closure tractions at the crack surfaces. Long and short cracks are distinguished. Long cracks approach a steady-state configuration, for which the stress intensity analysis and a previous energy balance analysis are shown to predict identical dependence of matrix cracking stress on material properties. A numerical solution and an approximate analytical solution are obtained for smaller cracks and used to estimate the range of crack sizes over which the steady-state solution applies.

Static analyses similar to recently developed continuum mixture theories of wave propagation in structural composites are carried out in order to estimate the interfacial stresses that develop in composites subjected to various mechanical and thermal loadings. A variety of composite models are treated. These include the multi-cylindrical periodic fibres, the single cylindrical fibre and the single planar fibre reinforcement models. Results are compared with existing theoretical and experimental findings.

A fiber-reinforced ceramic subject to tensile stress in the fiber direction can undergo extensive matrix cracking normal to the fibers, while the fibers remain intact. In this paper, the critical conditions for the onset of widespread matrix cracking are studied analytically on the basis of fracture mechanics theory. Two distinct situations concerning the fiber-matrix interface are contemplated : (i) unbonded fibers initially held in the matrix by thermal or other strain mismatches, but susceptible to frictional slip, and (ii) fibers that initially are weakly bonded to the matrix, but may be debonded by the stresses near the tip of an advancing matrix crack. The results generalize those of the Aveston-Cooper-Kelly theory for case (i). Optimal thermal strain mismatches for maximum cracking strength are studied, and theoretical results are compared with experimental data for a SiC fiber, lithium-alumina-silicate glass matrix composite.

A model is developed for fatigue growth of matrix cracks in metals reinforced with aligned continuous elastic fibers. The mechanics of elastic cracks bridged by frictionally constrained fibers is used to develop the model, which provides estimates of the tip value of the stress intensity factor amplitude, ΔKTIP. It is found that when the applied load amplitude is held fixed during fatigue crack growth, ΔKTIP, and thus the rate of growth approach an asymptotic value independent of crack length. The residual strength after fatigue crack growth is also discussed. In some cases, the residual strength is unaffected by prior fatigue growth. But, in another regime, the matrix crack length allows fibers to begin breaking before the matrix crack grows. The strength is then inversely proportional to the square root of fatigue crack length.

Embedded single-fiber tests are often used to characterize the fiber/matrix interface, but their interpretation is usually limited by reliance on the qualitative view of the stresses provided by shear-lag analyses. This paper describes a new, three-dimensional, axisymmetric solution for the stresses around breaks in embedded fibers. The new solution is obtained using variational mechanics. It obeys equilibrium and traction boundary conditions exactly, obeys compatibility approximately, includes all components of the stresses, accounts for interacting fiber breaks, and includes residual thermal stresses. We apply the stress analysis to the single-fiber fragmentation test. In some sample calculations, we plot all components of stress at the fiber/matrix interface and give predictions for an “ideal” single-fiber fragmentation test. The stress analysis technique is readily adaptable to new problems such as the single-fiber pull-out test, the microdrop debond test, the description of interfacial fracture or yielding, and the effect of interfacial friction.

A new simple model for predicting the uniaxial stress–strain behavior of a unidirectional ceramic matrix composite, including stochastic matrix crack evolution, stochastic fiber damage and ultimate failure, is presented. The model demonstrates an important transition in composite behavior. “Brittle” (low failure strain) behavior occurs when the matrix cracking stresses are sufficiently high; the composite fails during the matrix cracking regime of deformation and at a strain that is controlled by the matrix flaw population and elastic properties. “Tough” (high failure strain) behavior occurs when the matrix cracking stresses are lower; matrix cracking is completed prior to failure and the failure strain of the composite is controlled by the fibers. In both cases, the failure strength is fiber-controlled. The model is applied to study SiC/SiC 500-fiber minicomposite deformation, using data recently obtained by Lissart and Lamon on two material types, “B” and “C”. Parameters for the matrix flaw population are used to fit the experimental stress–strain data but the failure is controlled by the measured fiber strength statistics. Excellent agreement is found for the “C” materials, which are in the transition regime between the brittle and tough limits and variations in fiber strength are postulated to be responsible for the wide range of behaviors found in the “B” materials. The fitted matrix flaw parameters are then used to predict the fiber/matrix interfacial sliding resistance and the values obtained are in excellent agreement with independent values determined from both unload/reload hysteresis loops and fiber pullout lengths. The new model provides a useful tool for understanding the interplay matrix and fiber flaw distributions and the overall dependence of stress–strain behavior on all the underlying constituent material properties.

The objective of this work is to study the effect of a frictional interface on the extent of interfacial debonding and internal stress distribution within a unidirectional ceramic matrix composite under thermo-mechanical loading. The configuration considered is a concentric cylinder with an annular crack in the axial plane of the matrix while the fiber-matrix interface is made to obey the Coulomb friction law. The present approach of employing a variational model is shown to satisfy all boundary and inequality conditions in the slip and stick regions and at the slip-stick transition point. The results are also shown to be in very good agreement with a numerical elasticity solution for points away from the matrix crack tip while some of the peculiarities near the crack tip associated with the numerical solution are identified.

A classical result due to Daniels is that the strength of a bundle of parallel fibres is asymptotically normally distributed. Extensions of this result are obtained and applied to a series-parallel model consisting of a long chain of bundles arranged in series. This model is of importance in studying the reliability of fibrous materials. Improved approximations are also obtained which reduce the error associated with Daniels' approximation both for the single bundle and for the series-parallel system.

Fiber reinforced composites strength by limit analysis methods

Composite mechanics disciplines are presented and described at their various levels of sophistication and attendant scales of application. Correlation with experimental data is used as the prime discriminator between alternative methods and level of sophistication. Major emphasis is placed on: (1) where composite mechanics has been; (2) what it has accomplished; (3) where it is headed, based on present research activities; and (4) at the risk of being presumptuous, where it should be headed. The discussion is developed using selected, but typical examples of each composite mechanics discipline identifying degree of success, with respect to correlation with experimental data, and problems remaining. The discussion is centered about fiber/resin composites drawn mainly from the author's research activities/experience spanning two decades at Lewis.