Shear-lag Theory Research Articles

Micromechanics characterization of sublaminate damage

A micromechanics analytical model is developed for characterizing the fracture behaviour of a fibre reinforced composite laminate containing a transverse matrix crack and longitudinal debonding along 0/90 interface. Both the matrix and the fibres are considered as linear elastic. A consistent shear lag theory is used to represent the stress-displacement relations. The governing equations, a set of differential-difference equations, are solved satisfying the boundary conditions appropriate to the damage configuration by making use of an eigenvalue technique. The properties of the constituents appear in the model explicitly. Displacements and stresses in the fibres and the matrix are obtained, and the growth of damage is investigated by using the point stress criterion. The investigation includes fibre stress distribution in zero degree plies, transverse crack and debonding intitiation as functions of laminate geometry, and the effect of fibre breaks in the zero degree ply on damage growth. The predicted damage growth patterns and the corresponding critical strains agree with the finite element and experimental results.

Steady-state cracking in ceramic matrix composites

The problem of steady-state cracking in which the crack extends under a constant load is investigated for two interface conditions: elastic (no slip) and frictional interface. The fiber, matrix and interface stresses in the cracked and uncracked regions of a unidirectional composite subjected to remote axial stress are obtained using an improved shear-lag theory that accounts for the axial and transverse load-carrying capacity of the matrix. The fiber axial stresses in the current problem coincide with those of previous models. However, the matrix shear and transverse normal stresses predicted in the present model for the interface conditions considered are different. The critical cracking conditions are established by using a generalized energy release rate formulation, and critical stresses are obtained in terms of microstructural properties of the constituents and the interface.

A simple development of the shear lag theory appropriate for composites with a relatively small modulus mismatch

The standard shear lag analysis is known to give unreliable predictions for composite stiffness and other properties when applied to systems with a relatively small ratio between the elastic moduli of the two constituents, such as metal matrix composites. This has long been thought to be at least partly due to neglect of the transfer of normal stresses across the fibre end jaces. In this paper, a simple postulate is made that this stress will approximate to the mean between the peak value in the fibre and the average value in the matrix remote from the interface. On this basis, simple analytical expressions are derived for composite stiffness and applied stress at the onset of yielding. It is shown by comparison with the Eshelby predictions and experimental data that the analysis appears reliable over the complete range of fibre aspect ratios and volume fractions when applied to metal matrix composites.

On the multiple fracture of low-elongation thin films deposited on high-elongation substrates

The multiple fracture of brittle, low-elongation thin films adhering to high-elongation plastic substrates subject to uniaxial strain is addressed. Results are presented for Permalloy on Kapton. It is observed that crack density in the film increases dramatically immediately after crack onset and subsequently approaches an asymptotic value with substrate strain. Thicker films experience crack onset earlier, and their asymptotic crack densities are lower than with thinner films. To provide some theoretical basis for the observed phenomena, the relevant literature is summarized. An empirical model is developed that fits the data obtained in this study and agrees with the results obtained by others as well. An attempt is made to mechanistically model the problem using a modified shear-lag theory developed initially for bulk composite materials. It is shown that the shear-lag model has potential application to this problem and needs further investigation.

Stress concentration factor of FRM by shear-lag theory and its statistical treatment.

The stress concentration factors in unidirectional fiber reinforced metal matrix composites with random distances between the fibers were simulated based on shear-lag analysis, on the assumption that the shear deformation of the matrix metal adjacent to a broken fiber could be approximately identified as a deformation of the elastic linear hardening plasticity. The results showed that the average value of the stress concentration factor gradually decreased as the applied stress at infinity increased, and gave almost the similar value as that obtained analytically in the case of the constant distances between the fibers. Further, it was shown that the increment in the stress concentration factor was expressed approximately by 2-parameter Weibull distribution, similar to the situation in which the matrix behaved elastically in the vicinity of broken fiber. However, the degree of the variation was extremely small in the former as compared with the latter.

ボロン繊維強化アルミニウムのシアラグ法に基づく強度特性のシミュレーション

The computer simulation method based on the shear-lag theory was proposed to predict the tensile strength of the unidirectional ceramic fiber reinforced metal composites, in which the elastic-linear hardening behavior of matrix was taken into consideration. The boron fiber reinforced aluminum composites were prepared by hot-pressing technique, and tested to check the tensile strength. The average strength of the composites (Vf = 11%) showed the good agreement between the predicted value and the experimental value. By using the simulation method, the size effect of the fiber reinforced metal was predicted. There was very small effect of inter-fiber spacing on the average tensile strength, but the average strength of composites decreased slightly with increase in number or reinforced fibers.

Mechanical properties and morphology of polypropylene composites. III. Short glass fiber reinforced elastomer modified polypropylene

AbstractThe impact strength and rigidity of polypropylene composites can be significantly improved by application of short glass fibers instead of mineral fillers in elastomer‐modified polypropylene. The properties of such composites are strongly dependent on the adhesive forces at the fiber‐matrix interface. Poor adhesion results in interfacial fracture solely by fiber‐matrix debonding, as evidenced by scanning electron microscopy on the fracture surfaces. This is accompanied by relatively low impact strengths. By contrast, increased adhesion leads to fracture not only by fiber‐matrix debonding, but also by crack propagation through the elastomeric phase at the fiber surface. This mechanism is thought to be responsible for a remarkable increase of the impact strength. Appropriate compositions of polypropylene, glass fiber, and elastomer resulted in composite properties similar to, or even better than, those of a typical acrylonitrile‐butadiene‐styrene copolymer. The lengths of the fibers recovered from the test specimens were somewhat smaller than the critical fiber lengths as calculated by simple shear lag theory. The properties of the present composites should thus be regarded as minima, rather than as potential maxima. This suggests that current composites may be suitable for engineering applications.

Stresses on and around fibres in a composite

The widely used shear-lag theory for the stresses and displacements in a short fibre composite is discussed and compared with more exact theories in the literature. A simple modification of shear-lag theory, using stress functions, is proposed that allows prediction of radial stresses in the fibre, while being easy to use for calculations. Predictions from the present theory are compared with those from the shear-lag theory and with some of the more exact theories. At low fibre concentrations, the present theory underestimates the radial stress at the fibre ends while predicting shear stresses comparable with those from shear lag. At high fibre concentrations both radial and shear stresses are overestimated.MST/635

Stresses on and around fibres in a composite

The widely used shear-lag theory for the stresses and displacements in a short fibre composite is discussed and compared with more exact theories in the literature. A simple modification of shear-lag theory, using stress functions, is proposed that allows prediction of radial stresses in the fibre, while being easy to use for calculations. Predictions from the present theory are compared with those from the shear-lag theory and with some of the more exact theories. At low fibre concentrations, the present theory underestimates the radial stress at the fibre ends while predicting shear stresses comparable with those from shear lag. At high fibre concentrations both radial and shear stresses are overestimated.MST/635

On the plastic deformation of fibre self-reinforced polymers

The plastic deformation of semi-crystalline polymers containing needle crystals has been investigated. Stress-strain curves exhibit two linear regions, the elastic (Hookeian) range and a second linear plastic part. The onset of the plastic part depends on pre-straining and the slopeα = dσ/de of the plastic part is dependent on the morphology of the samples. An interpretation is given in terms of the shear-lag theory of discontinuous uniaxially oriented fibres.

Establishment of Dynamic Failure Model and Examination on Dynamic Failure Process

The present paper proposes a new dynamic failure process simulation model considering the effect of matrix shear failure as well as random fiber breaks based on a shear-lag theory in which an additional time variable is also incorporated by taking the mass of fiber and matrix elements into account. An exact time-dependent stress redistribution process in a composite failure model is evaluated by means of a finite difference scheme based on an increment method for unidirectional carbon (C) and carbon (C) /glass (G) hybrid FRP models.It is observed that the time-dependent stress concentration factor evaluated by the simulation is in good accordance with the Hedgepeth's analytical solution. It is also clarified that the stress in carbon fiber is relieved to some extent due to the presence of neighbouring glass fibers in hybrid C/G hybrid FRP models.The stress wave propagation is successfully simulated by visualizing a time-dependent change in nodal displacements as well as pursuing a dynamic failure process of a composite failure model. These simulations are shown to be effective in characterizing different failure processes and stress wave propagation behaviors of unidirectional CFRP under varied higher strain rates.Finally, a simple damping term is taken into account in the dynamic failure process simulations. It is shown that a clear difference exists in the dynamic failure process and the resulting tensile stress-strain diagram of CFRP and C/G hybrid FRP under varied higher strain rates.

一方向ハイブリッド繊維強化材の動的引張破壊過程のシミュレーション

The authors proposed a new failure process simulation model considering interfacial debonding between fibers and matrix as well as random fiber breaking based on a shear-lag theory, in which not only fiber elements but also matrix elements are included. The failure model was applied to characterize not only the static tensile failure process but also the statistical nature of hybrid effect of several types of unidirectional hybrid FRP in the previous reports.The present paper proposes a new dynamic failure process simulation model in which an additional time variable is also incorporated by taking the mass of fiber and matrix elements into account. An exact time-dependent stress redistribution process in a new composite failure model is evaluated by means of a finite difference scheme based on an increment method. The numerical results are shown and discussed for two types of unidirectional carbon (C)/glass (G) hybrid FRP models.As the results, it is shown that a time-dependent stress concentration factor is exactly evaluated by the simulation which is in good accordance with Hedgepeth's solution and that the stress in a carbon fiber is relieved to some extent due to the presence of glass fiber in hybrid C/G FRP. It is also shown that a clear difference exists between the static and dynamic tensile failure patterns for both types of hybrid FRP. It is furthermore shown that the dynamic stress-strain behavior and the acoustic emission are successfully simulated by taking the stress drop and the released strain energy due to fiber and matrix failures into consideration, which gives a more realistic behavior than the static simulation.

On the strength of discontinuous silicon carbide reinforced aluminum composites

The incorporation of dSiC into aluminum matrices results in substantial improvements in yield strength relative to the matrix material, but little change in the proportional limit. The Orowan strengthening mechanism was shown to be insufficient to account for the increase in yield strength of the composite. An increased dislocation density of the matrix due to a difference in the thermal expansion of the SiC and the matrix does predict an increase in composite strength, but is unable to explain why the proportional limit of the composite remains similar to that of the matrix or why the strength of the SiCw/6061Al is anisotropic. The conventional shear lag theory predicts yield strength values that are less than those observed. This is particularly so for the SiCp/Al composites, where the yield strength of the composite is predicted to be equal to that of the matrix independent of SiC volume fraction. If, however, the shear lag theory is modified to take into account the tensile transfer of load, the predicted composite strength is of sufficient magnitude to explain the strengthening effect of both the SiC whisker and platelet material. In addition, the modified shear lag theory can be used to rationalize why the proportional limit of the composite is similar to that of the matrix.

Use of shear lag for composite microstress analysis - Linear array

The task of finding stress distributions in damaged composites analytically by means of the theory of elasticity is prohibitively difficult, so approximate techniques are generally employed. The most widely used is shear lag theory. This theory contains a parameter representing the shear coupling between fibers that has been difficult to evaluate in the past. The present paper evaluates this parameter for a linear array of fibers. In the limiting case of small fiber diameter to fiber spacing ratio and large matrix thickness to fiber diameter ratio an analytical solution is possible and is provided in this paper. For other geometries the solution is obtained experimentally. It is found that the assumption usually made for the parameter is rather far off for most geometries of practical interest. d d{ d2 D0 FG

Fiber Stress Concentrations in Kevlar/Epoxy Monolayers

A stress-based finite element analysis, formulated within the context of a shear-lag theory, is used to analyze a Kevlar 49/epoxy monolayer containing a single, preexisting fiber break. The highest and most localized stress concentration occurs when the epoxy matrix is idealized as linearly elastic and perfectly bonded to the fibers. This calculation suggests, however, that the fiber-matrix bond strength will be exceeded at relatively low load levels and fiber-matrix debonding will occur. Calculations which permit debond ing and also consider a residual frictional interface stress after the primary bond fails are presented. When there is no residual frictional stress, the fiber stress concentration is fully relieved and debonding occurs along the entire fiber length at loads only slightly higher (∼ 10%) than that needed to initiate bond failure. A residual frictional stress stabilizes debond growth. The fiber stress concentration factor decreases as the magnitude of the residual frictional stress decreases, while debond length and the length of fiber subject to the overstress increase.

Use of shear lag for composite microstress analysis Rectangular array

Abstract : A number of authors have employed shear lag theory to find the stress distribution near cracks in a uniaxially reinforced composite. Practical application of these equations has been hampered by the fact that they include a parameter h/d the magnitude of which is unknown. This parameter is evaluated for composites in which the fibers forms a rectangular array. (Author)

Experimental Determination of Stresses in Damaged Composites Using an Electric Analogue

Inadequate knowledge of the local stress distributions in damaged composites has been a major obstacle to progress in the understanding of damage accumulation and ultimate failure of such materials. Theoretical treatments of three-dimensional uniaxially reinforced composites are difficult, and direct experimental observations of stresses around interior flaws are not feasible. An experimental determination of stress distributions can be made using an electric analogue. A scaled model of the composite including the damage is made with the fibers replaced by conducting rods and the matrix replaced by an electrolyte. The resistivity ratio of rods to electrolyte is taken equal to the elastic modulus ratio of matrix to fiber. A tensile force applied in the fiber direction is modeled by applying a potential gradient in the rod direction. The displacement distribution in the composite is then modeled by the potential distribution in the analogue to an accuracy somewhat better than that given by shear lag theory. Thus stress distributions can be found by measuring potentials in the analogue with the aid of an electric probe.

Note on shear interaction between two fibers

Abstract : Application od shear lag theory to find the stress distribution in damaged composites has been handicapped by lack of knowledge concerning the effective shear stiffness of the matrix coupling adjacent fibers. This note gives an exact solution for a limiting case, namely two infinitely long rigid rods immersed in an infinite elastic matrix and displaced axially with respect to each other. (Authr)

Stress distribution of laminates including discontinuous layers

Discontinuity of fibres plays a critical role in the strength of fibre-reinforced composite materials. In this paper, the stress distribution of laminated plates of finite layers with some discontinuous layers is analysed using shear-lag theory. The strength of such plates is also predicted and the theoretical value is compared with experimental data.

Factors affecting stress transmission in crystalline polymers

AbstractThe paper is concerned with stress transmission in and around the molecular ties which connect lamellae in many semicrystalline polymers. Assuming a continuum model, finite element calculations have been performed to investigate stresses and displacements for a cylindrical tie embedded in a sheath of lamellar crystal. Two specific cases were studied: the single‐ended case where the tie enters the lamella from one side only, and the double‐ended case where it passes right through the lamellar crystal. Elastic constants for crystalline polyethylene were assumed, and a variety of different geometries were investigated, corresponding to single‐chain or multichain ties with sheaths of various diameters. The finite element study confirmed that the distance for stress diffusion from the tie into the lamellar sheath can be quite large, and that there are significant displacements due to tensile deformation of the tie within the crystal and shear deformation of the lamellar sheath. An interesting feature in the case of multichain ties is that tensile stress is concentrated around the outer surface of the tie. An analytical model is proposed, based upon shear lag theory. This model considers only the tensile stresses in the tie and the shear stresses in the sheath, and gives results which are fairly similar to those of the finite element calculations. Finally, the results of this study are applied to the problem of stress transmission in an oriented polymer having a lamellar structure. The main factors influencing stress transmission are: (i) the ratio G/E of the crystalline phase; (ii) the ratio of tie radius to lamellar crystal thickness; and (iii) the proportion of ties in the interface. By examining the range of values which (ii) and (iii) may take, it is concluded that the effective stiffness of intercrystalline ties can only be about 0.25 to 0.7 of the theoretical chain stiffness in most cases.