James G. Goree

Analysis of a unidirectional composite containing broken fibers and matrix damage

Abstract An analytical solution is developed for the determination of the stresses and displacements in a unidirectional fiber-reinforced composite containing an arbitrary number of broken fibers as well as longitudinal yielding and splitting of the matrix. The solution is developed using a “materials-modeling” approach which is based on a shear-lag stress transfer mechanism. The equilibrium equation in the axial direction gives a pair of integral equations which are solved numerically. Excellent agreement is shown to exist between the solution and experimental results for notched unidirectional boron/aluminum laminates without splitting. For brittle matrix composites (i.e. epoxy) equally good results are indicated for both matrix yielding and splitting. For yielding without splitting the fracture strength is found to depend on crack length while for large splitting it is crack length independent.

Mathematical modeling of damage in unidirectional composites

Abstract Solutions are developed for the two-dimensional region containing unidirectional fibers embedded in an elastic matrix with an initial flaw in the form of a transverse notch, a rectangular cut-out, and a circular hole. Subsequent damage due to the presence of the flaw is generated by remote stresses acting parallel to the fibers. This work is an extension of the paper by Goree and Gross [1] in which the flaw was taken in the form of a notch (crack) and the subsequent damage, due to loading, consisted of longitudinal matrix yielding and splitting at the end of the notch. The present study accounts for longitudinal matrix damage as in [1] and, in addition, includes transverse matrix and fiber damage in the vicinity of the flaw for the above three initial shapes. The fibers are taken as linearly elastic, the matrix material as elasticperfectly plastic and the classical shear-lag stress displacement assumptions are used. An ultimate stress failure criterion is used for both the fibers and the matrix; simple tension for the fibers and shear failure for the matrix. For ductile matrix composites (boron/aluminum) the present results indicate that both longitudinal matrix yielding and transverse notch extension must be included in order for the model to agree with experimental results. Interestingly, the extent of the transverse damage region at failure is shown to be approximately constant, independent of the initial flaw shape or length. Very little difference is found between the results for the three types of initial damage, i.e. the notch, rectangular cut-out and circular hole. In all cases, the presence of additional damage changes the nature of the stress distribution in the unbroken fibers. For the original Hedgepeth[2] problem of a notched laminate the stresses decay as the square root of the distance from the notch tip. Inclusion of longitudinal or transverse damage significantly reduces the maximum stress concentration in the unbroken fibers and gives a much more uniform stress state. It is shown that this behavior cannot be accounted for by introducing an effective notch length or crack tip damage zone with a square root behavior.

Stresses in a three-dimensional unidirectional composite containing broken fibers☆

Abstract 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.

Bonded elastic half-planes with an interface crack and a perpendicular intersecting crack that extends into the adjacent material—II☆

Abstract The solution is given for two bonded isotropic linearly elastic half-planes of different elastic properties having a crack along the interface as welt as a perpendicular crack in one of the half-planes which may intersect the interface crack. The appropriate integral equations are developed using displacement dislocations on the crack surfaces. Numerical results are presented for the stress intensity factors, strain energy release rate, stresses and displacements.

T-stress based fracture model for cracks in isotropic materials

Abstract A modified version of the T-stress based fracture model, developed by Cotterell and Rice, is proposed. In this version an experimentally determined Tcrit value is included in the model. The model is then applied to a branched crack in an isotropic material, and the direction of growth of the crack is predicted qualitatively. The branched crack problem is solved using the method of dislocations and a singular integral equation is obtained. The singular integral equation is solved using three different numerical techniques and their respective merits are discussed. The stress intensity factors and the T-stress in front of the branched crack tip are evaluated numerically. It is shown that the T-stress and the stress intensity factors are insensitive to the order of the singularity assumed at the reentrant wedge corner of the branched crack. For an uniaxial load and short kink length it is demonstrated that the kink will turn from its initial direction and realign with the main crack. However if the loading is biaxial then the direction of kink growth depends strongly on the applied transverse stress σxx. For a longer initial kink length the direction of kink growth depends on both the kink angle and loading.

Experimental evaluation of longitudinal splitting in unidirectional composites

An experimental study was conducted to determine the fracture behavior of center- notched, unidirectional graphite/epoxy laminates when subjected to tensile loading. The actual behavior is compared to the behavior predicted by a mathematical model based on classical shear-lag assumptions.

Crack Growth and Fracture of Continuous Fiber Metal Matrix Composites: Analysis and Experiments

It is reasonably well accepted that the standard procedures developed for isotropic homogeneous metals using linear elastic fracture mechanics models are not appropriate for either continuously or discontinuously reinforced metal matrix composites. For example, the ASTM plane strain fracture toughness test methods typically give widely different values of fracture toughness depending on the particular test specimen geometry as well as the fiber orientation. For unidirectional boron/aluminum composites one finds approximately a factor of two difference between the measured values of fracture toughness obtained from a center-notched test coupon and that given by a compact tension specimen. In particular for unidirectional composites, and to a slightly lesser degree for angle ply laminates, the dominant controlling mechanism for this behavior is matrix plasticity. A secondary toughening mechanism, resulting from the matrix plasticity, is stable transverse fiber failure. The present paper will focus on both the influence of the large plastic zone at the end of the notch and on the constraint that the fibers impose on the shape of this zone, as well as the transverse crack growth. First, a review of some particular experimental studies and methods of analysis for predicting crack growth and fracture of notched unidirectional metal matrix composites is given. Next, two mechanistic models for unidirectional composites with damage are presented. The first is an improved shear lag model that accounts for both of the above damage modes, and the second describes a recent extension of the shear lag concept in an attempt to include transverse stresses. A related finite width laminate model is then discussed, and it is indicated that an isotropic finite width correction factor is reasonably accurate for most center-notched test coupons.

A comparison of finite-difference and finite-element methods for calculating free edge stresses in composites

Abstract This study considers the accuracy of the finite difference method in the solution of linear elasticity problems that involve either a stress discontinuity or a stress singularity. Solutions to three elasticity problems are discussed in detail: a semi-infinite plane subjected to a uniform load over a portion of its boundary; a bimetallic plate under uniform tensile stress; and a long, midplane symmetric, fiber-reinforced laminate subjected to uniform axial strain. Finite difference solutions to the three problems are compared with finite element solutions to corresponding problems. For the first problem a comparison with the exact solution is also made. The finite difference formulations for the three problems are based on second order finite difference formulas that provide for variable spacings in two perpendicular directions. Forward and backward difference formulas are used near boundaries where their use eliminates the need for fictitious grid points. Moreover, forward and backward finite difference formulas are used to enforced continuity of interlaminar stress components for the third problem. The study shows that the finite difference method employed in this investigation provides solutions to the three elasticity problems considered that are as accurate as the corresponding finite element solutions. Furthermore, the finite difference method appears to give a solution for the laminate problem that characterizes the stress distributions near an interface corner in a more realistic manner than the finite element method.

Analysis of a unidirectional, symmetric buffer strip laminate with damage

Abstract A method of analysis capable of predicting accurately the fracture behavior of a unidirectional composite laminate containing symmetrically placed buffer strips is presented. The analysis is based on a materials modeling approach using the classical shear-lag assumption to describe the stress transfer between fibers. Explicit fiber and matrix properties of the three regions are retained and changes in the laminate behavior as a function of the relative material properties, buffer strip width and initial crack length are discussed. As an example, for a notch (broken fibers) in a graphite/epoxy laminate, the results show clearly the manner in which to select the most efficient combination of buffer strip properties necessary to arrest the crack. Ultimate failure of the laminate after crack arrest can occur under increasing load, either by continued crack extension through the buffer strip, or by fiber breakage in the undamaged half-plane. That is, for certain choices of relative material properties and width, the crack can jump the buffer strip. For some typical hybrid laminates it is found that a buffer strip spacing to width ratio of about four to one is the most efficient.

In-Plane Loading in an Elastic Matrix Containing Two Cylindrical Inclusions

A solution is presented for the stresses and displacements in an infinite elastic matrix containing two perfectly bonded rigid circular cylindrical inclusions of different radii, and of infinite length normal to the x-y plane. The matrix is subjected to in-plane stresses Sx and Sy at infinity as well as loading due to radial expan sion of the inclusions. Numerical results are presented and a com parison is made with the associated three dimensional problem of unequal rigid spheres.