Interlaminar Normal Stress Research Articles

Transverse cracking and delamination in composite materials

An experimental investigation has been conducted on damage behaviour related with matrix cracking under static and fatigue loading using two stacking sequences of the (0/90 n /±45) s and (0/±45/90 n ) s laminates for n = 1, 3, and 6. The latter stacking sequence produces tensile interlaminar normal stress along free edges of a tensile coupon, whereas the other stacking sequence produces compression at the same edges. Emphasis in this study was on the interaction between transverse cracks and delamination, and effect of layer thickness on crack density and propagation. Transverse crack and delamination were periodically measured by interrupting testing until final failure. Experimental results indicate that the crack density and propagation are influenced by layer thickness, and delamination appears to be mainly governed by either, or a combination of, tensile interlaminar normal stress and size of transverse crack. The applied force at incipient delamination can be predicted by an existing theory with a certain limitation.

Reliability of the finite element method for calculating free edge stresses in composite laminates

The edge-stress problem for a [±45] s graphite/epoxy laminate was examined in detail. A review of the literature on this problem showed that the interlaminar normal stress σ z distributions along the interface between the +45° and -45° plies, obtained by various investigators, disagreed in magnitude and sign. In particular, a finite difference solution and a perturbation solution predicted a tensile σ z , whereas the finite element methods predicted a compressive stress. Since a stress singularity exists at the intersection of the interface and the free edge, the differences in magnitude of the peak stress were expected, but not the difference in the sign. This paper investigates the reliability of the displacement-formulated finite element method in analyzing the edge-stress problem. Analyses of two well-known elasticity problems, one involving a stress discontinuity and one a singularity, showed that the finite element analysis yields accurate stress distributions everywhere except in two elements closest to the stress discontinuity or singularity. Stress distributions for a [±45] s laminate showed the same behavior near the singularity as found in the well-known problems with exact solutions. The displacementformulated finite element method, therefore, appears to be a highly accurate technique for calculating interlaminar stresses in composite laminates. The disagreement among the numerical methods was attributed to the unsymmetric stress tensor at the singularity.

Three-Dimensional Analysis of [0/90]s and [90/0]s Laminates with a Central Circular Hole

Stress distributions were calculated near a circular hole in laminates, using a three dimensional finite element analysis. These stress distributions were presented three ways: through the thickness at the hole boundary, along radial lines at the 0/90 and 90/0 interfaces, and around the hole at these interfaces. The interlaminar normal stress, and the shear stress, distributions had very steep gradients near the hole boundary, suggesting interlaminar stress singularities. The largest compressive stress occurred at about 60 deg from the load axis. A simple procedure was introduced to calculate interlaminar stresses near the hole boundary. It used stresses calculated by an exact two dimensional analysis of a laminate with a hole as input to a quasi three dimensional model. It produced stresses that agreed closely with those from the three dimensional finite element model.

THERMAL STRESSES IN LAMINATED BEAMS

Abstract This paper considers laminated beams consisting of layers of different materials fastened together by thin adhesives. The stresses that result from subjecting the beam to temperature stimuli are calculated. The problem is treated by two-dimensional elasticity theory in conjunction with the variational theorem of complementary energy. A pair of governing differential equations is developed, and boundary conditions concerning stress-free surfaces and ends of the beam are satisfied. The calculation of the distributions of interlaminar normal and shear stresses shows that high stress intensity occurs in the end zones of the beam. Thus, the satisfaction of end conditions is of prime importance in the analysis of laminated structural elements. Delamination failure-when it occurs - will probably start at the ends of the beam, This agrees with observed failures of laminated structural elements subjected to stress-free end conditions.

Stresses in a three-dimensional unidirectional composite containing broken fibers

An approximate solution is developed for the determination of the inter-laminar normal and shear stresses in the vicinity of a crack in a three dimensional composite containing unidirectional linearly elastic fibers in an infinite linearly elastic matrix. In order to reduce the complexity of the formulation, certain assumptions are made as to the physically significant stresses to be retained. These simplifications reduce the partial differential equations of elasticity to differential-difference equations which are tractable using Fourier transform techniques. This “material modeling” approach is in contrast with solutions developed by considering each lamina as a homogeneous, orthotropic layer. The resulting solution does not contain the classical singular stress field for the fibers and the influence of broken fibers on unbroken fibers is felt by a change in stress concentration factors. The matrix stresses however, are unbounded as the fiber spacing vanishes and an equivalent fiber-matrix geometry is proposed which gives the correct singular behavior. The numerical solution is considered in detail and several specific examples are presented. The potential for damaged or debonded zones to be generated by an embedded crack is discussed, and stress concentration factors for fibers near the crack are given. Detailed comparisons are made between the present solution, the analogous two-dimensional problem, and corresponding shear-lag models.

Boundary layer effects in composite laminates

The mechanism of interlaminar load transfer for the fibre-reinforced laminate of finite-width is investigated. Two modes of interlaminar transfer are identified. The first mode characterised by the behaviour of the angle-ply laminate is shown to be dominated by the action of the interlaminar shear stress, τ xz, while the second mode reflects the action of the interlaminar normal stress and is associated with the bidirectional laminate. The behaviour of a general laminate configuration is examined. Finally, the influence of interlaminar stresses upon laminate strength is examined.

Effect of Stacking Sequence and Lay-Up Angle on Free Edge Stresses Around a Hole in a Laminated Plate Under Tension

One mode of failure in fiber reinforced laminated plates is delamination initiating at a free edge. Experimental results and theoretical stress investi gations found in the literature indicate that the interlaminar normal stress, σ z, is a factor in this mode of failure. In this paper, the stress behavior around a circular hole in a laminated plate is investigated using a three- dimensional finite element stress analysis. First, computed results for the tangential strain distribution around a circular hole for a (0/±45/0) s lamin ate under uniaxial loading are compared with laboratory data to provide a level of confidence in the analysis. Next, attention is focused on the inter laminar normal stress distribution around the free edge of the circular hole and the effects of changes in stacking sequence and lay-up angle. Initially, laminates with stacking sequences of (0/90) s are analyzed. Graphite/epoxy laminates of the type (02/±θ/∓θ) S, (±θ/∓θ/0 2) S, (902/±θ/∓θ) 2 , and (±θ/∓θ/902) S where θ is 30° , 45°, and 60° are also considered. Each lamin ate is subjected to a uniaxial stress loading distant to a hole.

Edge Effects in Angle-Ply Composite Laminates*

This paper presents the results of a zeroth-order solution for edge effects in angle-ply composite laminates obtained using perturbation tech niques and a limiting free body approach. The general solution for [±θ] laminates is applied to the special case of a [±45]s graphite/epoxy lamin ate. Interlaminar stress distributions are obtained as a function of the laminate thickness-to-width ratio h/b and compared to finite difference results. The solution predicts stable, continuous stress distributions, determines finite maximum tensile interlaminar normal stress σz, and provides mathe matical evidence for singular interlaminar shear stresses τ xz and τyz in [±45] graphite/epoxy laminates.

A multilayer hybrid-stress finite element analysis of a through-thickness edge crack in A ± 45° laminate

A solution is given for the three-dimensional stress field near a through-thickness edge crack in a thin ± 45° laminate having elastic ply moduli typical of graphite/epoxy. The stress distribution was obtained by a three-dimensional multilayer finite element analysis based on the hybrid stress model, formulated through the minimum complementary energy principle. The results indicate that the in-plane stresses of each individual ply follow the classical 1 √r stress singularity, but that the shape of isostress contours in the crack tip region is strongly distorted from predictions based on two-dimensional anisotropic fracture mechanics theory. The interlaminar shear stresses increase rapidly as the crack tip is approached, but are restricted to a local region around the crack tip and flanks. The interlaminar normal stress is assumed to be negligible in the formulation of the analysis.

Interlaminar Stresses of Uniformly Loaded Rectangular Composite Plates

An analytical solution is obtained for bending a rectangular composite plate with two sides simply-supported and other two sides free. The inter laminar shear stresses and the normal or "peel" stress, near and at the free edges of the plate, are predicted. Numerical example of a 4-layer (+θ/-θ) s laminate is given. The magnitudes of the interlaminar shear and normal stresses may be as high as 30% and 5% respectively, of the maximum bending stress of the laminate. It would play a dominant role in fatigue fracture of composite plates such as leaf-spring of automobiles, lever-arms of painting robots, and many orthopedic in-plant devices.

Dynamic Response of Laminated Composite Plates

HE classical method of separation of variables combined with the Mindlin-Goodman procedure1 is employed to analyze the dynamic response of an infinitely long simplysupported composite plate under a uniform dynamic pressure, P0, on the upper surface of the plate. Two cases of dynamic pressures are considered; 1) the magnitude of the uniform pressure remains unchanged, and 2) the magnitude of the uniform pressure increases linearly as a function of time and then remains constant. Numerical results for eight layer granite/epoxy and glass/expoxy composites with (0/0/0/ — 0) s and (#/ — 0/0/0)5 stacking sequences are evaluated. Dynamic response characterized in terms of normal displacement, bending stress, interlaminar shear stress, and interlaminar normal stress are evaluated and compared with the corresponding quantities obtained statically (i.e., a dynamic load factor is established). It is observed that the dynamic values are consistently twice the corresponding static values for Case 1, and varies from one to two times the corresponding static values for Case 2. Contents The equations of motion, including transverse shear deformation and rotary inertia, for an infinitely long laminated composite plate under cylindrical bending are given in Ref. 2. The solution to the governing equations in terms of separation of variables in conjunction with the Mindlin-Goodman procedure] is also presented in Ref. 2. In the present paper the interlaminar shear stress TXZ is determined by integrating the equation of motion (1) with respect to z using the condition Txz(±h/2,t) = 0 and the condition of continuity at the interfaces. Once rxz(k) is determined the interlaminar normal stress ozz (k) can also be determined by integrating the equations of motion

On the Significance of Effective Modulus Solutions for Fibrous Composites

The concept of effective moduli is widely used in the analysis and design of structural fibrous composites. In this approach, composite bodies are treated as laminated systems in which each layer is represented as a homogeneous anisotropic material. This approach, however, breaks down in the presence of non-uniform macroscopic stress fields. In this work, we attempt to examine the physical significance of effective modulus solutions in such cases by comparison with solutions in which the microstructure is recognized. While the meaning of a non-uniform stress field predicted by the effective modulus approach is generally ambiguous, we find, at least in the problems treated here, that the respective stresses averaged over dimensions comparable to fiber spacing agree quite well with corresponding exact averages. An example is presented which demonstrates that the effective modulus stress field cannot predict the correct physical response while the average stresses are at least qualitatively correct. Finally, an elementary approximate theory for the determination of average interlaminar normal stresses in the laminate free-edge problem is developed.

Effects of material and stacking sequence on behavior of composite plates with holes

Strain distributions to failure, tensile and compressive strain-concentration factors, and strength-reduction factors were determined for glass-, boron-, and graphite-epoxy plates with holes loaded in tension. Strain gages, photoelastic coatings and moire techniques were used. Ten variations of layup and stacking sequence were studied. The boron-epoxy composite was found to be the stiffest and strongest of the three. The graphite laminate with the highest stress concentration and the most linear strain response exhibited the highest strength-reduction factor. In all cases, the maximum strain at failure on the hole boundary was higher than the ultimate tensile-coupon strain. In general, it was found that, the higher the stress-concentration factor, the higher the strength-reduction factor. Thus, the [0/90/0/90]s layup with a stress-concentration factor of 4.82 had a strength-reduction factor of 3.18. At the other extreme, the most flexible layup [±45/±45]s with the lowest stress-concentration factor of 2.06 had the lowest strength-reduction factor of 1.10. Stacking sequences associated with the tensile interlaminar normal stress or high interlaminar shear stress near the boundary, resulted in laminates 10 to 20 percent weaker than corresponding alternate stacking sequences. Furthermore, it was found that stacking-sequence variations can alter the mode of failure from catastrophic to noncatastrophic.

On the Calculation of Interlaminar Normal Stress in Composite Laminate

In an effort to improve our understanding of delamination phenomena in composite bodies, an approximate method to define the distribution of the interlaminar normal stress, σ 2, along the central plane of a symmetric, finite-width, composite laminate is presented. The approach is based upon a modified version of a recent theory developed by Whitney and Sun and accounts for the influence of the pertinent material and geometric parameters on the shape of the distribution. The accuracy of the approach is demonstrated by comparison with an existing three-dimensional elasticity solution. Finally, an elementary expression for σz in practical bidirectional laminates is derived.

The Influence of Stacking Sequence on Laminate Strength

Based upon considerations relating to the nature of the inter laminar stresses in composite laminates, an approach is presented to predict the detailed stacking sequence of specific layer orientations which leads to optimum protection against delamination under uniaxial static and fatigue loadings. In particular, it is argued that the interlaminar normal stress, as well as the interlaminar shear stress, is instrumental in precipitating delamination and subsequent strength degradation. Correlation with existing experimental evidence of the stacking sequence phenomenon is presented.

Analysis of Laminated Circular Plates

An elasticity solution is obtained to the problem of a thin, laminated, isotropic circular plate under combined transverse and edge loading. The analysis yields an approximate, thin-plate theory which reduces to an exact, closed-form solution when the plate is loaded by edge forces and moments only. The results show that, under edge loading, the stress state is purely biaxial, there being no interlaminar normal or shearing stresses such as are usually present in bulk composite bodies. Further, an estimate of the edge-effect correction which is present when the applied stress distribution does not match the internal stress distribution, discloses no unusual effects due to the laminated structure. Under combined loading, a complex state of in-plane, normal and shearing stresses exists which is coupled through the property variations.