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NASA NTRS1986

The crack problem in a reinforced cylindrical shell

In this paper a partially reinforced cylinder containing an axial through crack is considered. The reinforcement is assumed to be fully bonded to the main cylinder. The composite cylinder is thus modelled by a nonhomogeneous shell having a step change in the elastic properties at the z=0 plane, z being the axial coordinate. Using a Reissner type transverse shear theory the problem is reduced to a pair of singular integral equations. In the special case of a crack tip touching the bimaterial interface it is shown that the dominant parts of the kernels of the integral equations associated with both membrane loading and bending of the shell reduce to the generalized Cauchy kernel obtained for the corresponding plane stress case. The integral equations are solved and the stress intensity factors are given for various crack and shell dimensions. A bonded fiberglass reinforcement which may serve as a crack arrestor is used as an example.

Yahsi, O. S.↗

NASA NTRS1986

The crack problem in a reinforced cylindrical shell

A partially reinforced cylinder containing an axial through crack is considered. The reinforcement is assumed to be fully bonded to the main cylinder. The composite cylinder is thus modelled by a nonhomogeneous shell having a step change in the elastic properties at the z = 0 plane, z being the axial coordinate. Using a Reissner type transverse shear theory the problem is reduced to a pair of singular integral equations. In the special case of a crack tip touching the bimaterial interface it is shown that the dominant parts of the kernels of the integral equations associated with both membrane loading and bending of the shell reduce to the generalized Cauchy kernel obtained for the corresponding plane stress case. The integral equations are solved and the stress intensity factors are given for various crack and shell dimensions. A bonded fiberglass reinforcement which may serve as a crack arrestor is used as an example.

Yahsi, O. S.↗

NASA NTRS1986

Compliance matrices for cracked bodies

An algorithm is presented which can be used to develop compliance matrices for cracked bodies. The method relies on the numerical solution of singular integral equations with Cauchy-type kernels and provides an efficient and accurate procedure for relating applied loadings to crack opening displacements. The algorithm should be of interest to those performing repetitive calculations in the analysis of experimental results obtained from fracture specimens.

Ballarini, R.↗

NASA NTRS1985

A cylindrical shell with a stress-free end which contains an axial part-through or through crack

The interaction problem of a through or a part through crack with a stress free boundary in a semi-infinite cylindrical shell is considered. It is assumed that the crack lies in a meridional plane which is a plane of symmetry with respect to the external loads as well as the geometry. The circular boundary of the semi-infinite cylinder is assumed to be stress free. By using a transverse shear theory the problem is formulated in terms of a system of singular integral equations. The line spring model is used to treat the part through crack problem. In the case of a through crack the interaction between the perturbed stress fields due to the crack and the free boundary is quite strong and there is a considerable increase in the stress intensity factors caused by the interaction. On the other hand in the problem of a surface crack the interaction appears to be much weaker and consequently the magnification in the stress intensity factors is much less significant.

Erdogan, F.↗

NASA NTRS1985

An elastic strip with multiple cracks and applications to tapered specimens

In this paper an infinite elastic strip containing arbitrarily oriented cracks and subjected to uniform tension and a pair of concentrated forces is formulated in terms of a system of singular integral equations. Even though the technique is sufficiently general to solve new multiple crack problem, with the objective of applying the results to tapered specimens, only a certain symmetric crack geometry and loading conditions are considered. The stress intensity factors are calculated for edge cracks in the strip under uniform tension and for a 'compact' and a 'slender' tapered specimen (the latter simulating the double cantilever beam) under concentrated forces or crack surface wedge forces.

Liu, X.-H.↗

NASA NTRS1985

A clamped rectangular plate containing a crack

The general problem of a rectangular plate clamped along two parallel sides and containing a crack parallel to the clamps is considered. The problem is formulated in terms of a system of singular integral equations and the asymptotic behavior of the stress state near the corners is investigated. Numerical examples are considered for a clamped plate without a crack and with a centrally located crack, and the stress intensity factors and the stresses along the clamps are calculated.

Tang, R.↗

NASA NTRS1986

The surface crack problem in an orthotropic plate under bending and tension

The elasticity problem for an infinite orthotropic flat plate containing a series of through and part-through cracks and subjected to bending and tension loads is considered. The problem is formulated by using Reissner's plate bending theory and considering three dimensional materials orthotropy. The Line-spring model developed by Rice and Levy is used to formulate the surface crack problem in which a total of nine material constants has been used. The main purpose of this study is to determine the effect of material orthotropy on the stress intensity factors, to investigate the interaction between two asymmetrically arranged collinear cracks, and to provide extensive numerical results regarding the stress intensity factors. The problem is reduced to a system of singular integral equations which is solved by using the Gauss-Chebyshev quadrature formulas. The calculated results show that the material orthotropy does have a significant effect on the stress intensity factor.

Wu, B. H.↗

NASA NTRS1987

Macrocrack interaction with transverse array of microcracks

General formulation of a problem involving a macrocrack propagating through an area with microcracks is considered. The analysis is based on the simultaneous solution of a system of singular integral equations. Various methods described in the literature are discussed in detail and compared. The specific problem considered was a macrocrack approaching an infinite transverse array of microcracks. Results illustrate the effects of different loading types and can be used for interpretation of the microcrack toughening mechanisms. Numerical comparisons are made with data recently appearing in literature which demonstrates the importance of numerical accuracy. Reported results differ by a factor in certain cases as compared with data given in other literature.

Rubinstein, A. A.↗

NASA NTRS1987

Plates and shells containing a surface crack under general loading conditions

Various through and part-through crack problems in plates and shells are considered. The line-spring model of Rice and Levy is generalized to the skew-symmetric case to solve surface crack problems involving mixed-mode, coplanar crack growth. Compliance functions are introduced which are valid for crack depth to thickness ratios at least up to .95. This includes expressions for tension and bending as well as expressions for in-plane shear, out-of-plane shear, and twisting. Transverse shear deformation is taken into account in the plate and shell theories and this effect is shown to be important in comparing stress intensity factors obtained from the plate theory with three-dimensional solutions. Stress intensity factors for cylinders obtained by the line-spring model also compare well with three-dimensional solution. By using the line-spring approach, stress intensity factors can be obtained for the through crack and for part-through crack of any crack front shape, without recalculation integrals that take up the bulk of the computer time. Therefore, parameter studies involving crack length, crack depth, shell type, and shell curvature are made in some detail. The results will be useful in brittle fracture and in fatigue crack propagation studies. All problems considered are of the mixed boundary value type and are reducted to strongly singular integral equations which make use of the finite-part integrals of Hadamard. The equations are solved numerically in a manner that is very efficient.

Joseph, Paul F.↗

NASA NTRS1987

The surface crack problem in an orthotropic plate under bending and tension

The elasticity problem for an infinite orthotropic flat plate containing a series of through and part through cracks and subjected to bending and tension loads is considered. The problem is formulated by using Reissner's plate bending theory and considering three-dimensional material orthotropy. The Line-spring model developed by Rice and Levy is used to formulate the surface crack problem in which a total of nine material constants were used. The effects of material orthotropy on the stress intensity factors was determined, the interaction between two asymmetrically arranged collinear cracks was investigated, and extensive numerical results regarding the stress intensity factors are provided. The problem is reduced to a system of singular integral equations which is solved by using the Gauss-Chebyshev quadrature formulas. The calculated results show that the material orthotropy does have a significant effect on the stress intensity factor.

Wu, Bing-Hua↗

NASA NTRS1988

Macrocrack interaction with transverse array of microcracks

Rubinstein, A. A.↗

NASA NTRS1989

Finite element modeling of frictionally restrained composite interfaces

The use of special interface finite elements to model frictional restraint in composite interfaces is described. These elements simulate Coulomb friction at the interface, and are incorporated into a standard finite element analysis of a two-dimensional isolated fiber pullout test. Various interfacial characteristics, such as the distribution of stresses at the interface, the extent of slip and delamination, load diffusion from fiber to matrix, and the amount of fiber extraction or depression are studied for different friction coefficients. The results are compared to those obtained analytically using a singular integral equation approach, and those obtained by assuming a constant interface shear strength. The usefulness of these elements in micromechanical modeling of fiber-reinforced composite materials is highlighted.

Ballarini, Roberto↗

NASA NTRS1988

Cracking of a layered medium on an elastic foundation under thermal shock

The cladded pressure vessel under thermal shock conditions which is simulated by using two simpler models was studied. The first model (Model 1) assumes that, if the crack size is very small compared to the vessel thickness, the problem can be treated as a semi-infinite elastic medium bonded to a very thin layer of different material. However, if the crack size is of the same order as the vessel thickness, the curvature effects may not be negligible. In this case it is assumed that the relatively thin walled hollow cylinder with cladding can be treated as a composite beam on an elastic foundation (Model 2). In both models, the effect of surface cooling rate is studied by assuming the temperature boundary condition to be a ramp function. The calculated results include the transient temperature, thermal stresses in the uncracked medium and stress intensity factors which are presented as a function of time, and the duration of cooling ramp. The stress intensity factors are also presented as a function of the size and the location of the crack. The problem is solved for two bonded materials of different thermal and mechanical properties. The mathematical formulation results in two singular integral equations which are solved numerically. The results are given for two material pairs, namely an austenitic steel layer welded on a ferritic steel substrate, and a ceramic coating on ferritic steel. In the case of the yielded clad, the stress intensity factors for a crack under the clad are determined by using a plastic strip model and are compared with elastic clad results.

Rizk, Abd El-Fattah A.↗

NASA NTRS1989

Modeling growth of fatigue cracks which originate at rivet holes

When a structural component is subjected to repeated stress cycles, it can fail at stresses which are well below the tensile strength of the material. The processes leading to this failure are termed fatigue. Instances of fatigue failure in aircraft have become an increasing concern. The crack leading to failure often originate at rivet holes and then grow in response to stress cycles which occur during the operation of the aircraft. A necessary step to preventing failures in todays fleet of aging aircraft is to increase the frequency and quality of inspections; steps were already taken in this direction. There is also a need for modeling of fatigue crack growth in the aircraft structures so that improvements in design can be established and predictions of the life of the components can be made. The purpose is to provide a method to accurately predict the growth of fatigue cracks and to use this method to make predictions about the life of aircraft structural components. The method relies on the formulation and numerical solution of a singular integral equation(s) for an arbitrarily shaped crack(s) which propagate in response to the applied loading. Of special interest to the aging aircraft studies are cracks which originate at circular holes (i.e., rivet holes), but other crack geometries can be treated equally as well.

Mear, Mark E.↗

NASA NTRS1990

Green's functions for dislocations in bonded strips and related crack problems

Green's functions are derived for the plane elastostatics problem of a dislocation in a bimaterial strip. Using these fundamental solutions as kernels, various problems involving cracks in a bimaterial strip are analyzed using singular integral equations. For each problem considered, stress intensity factors are calculated for several combinations of the parameters which describe loading, geometry and material mismatch.

Ballarini, R.↗

NASA NTRS1991

The surface and through crack problems in layered orthotropic plates

An analytical method is developed for a relatively accurate calculation of Stress Intensity Factors in a laminated orthotropic plate containing a through or part-through crack. The laminated plate is assumed to be under bending or membrane loading and the mode 1 problem is considered. First three transverse shear deformation plate theories (Mindlin's displacement based first-order theory, Reissner's stress-based first-order theory, and a simple-higher order theory due to Reddy) are reviewed and examined for homogeneous, laminated and heterogeneous orthotropic plates. Based on a general linear laminated plate theory, a method by which the stress intensity factors can be obtained in orthotropic laminated and heterogeneous plates with a through crack is developed. Examples are given for both symmetrically and unsymmetrically laminated plates and the effects of various material properties on the stress intensity factors are studied. In order to implement the line-spring model which is used later to study the surface crack problem, the corresponding plane elasticity problem of a two-bonded orthotropic plated containing a crack perpendicular to the interface is also considered. Three different crack profiles: an internal crack, an edge crack, and a crack terminating at the interface are considered. The effect of the different material combinations, geometries, and material orthotropy on the stress intensity factors and on the power of stress singularity for a crack terminating at the interface is fully examined. The Line Spring model of Rice and Levy is used for the part-through crack problem. The surface crack is assumed to lie in one of the two-layered laminated orthotropic plates due to the limitation of the available plane strain results. All problems considered are of the mixed boundary value type and are reduced to Cauchy type of singular integral equations which are then solved numerically.

Erdogan, Fazil↗

NASA NTRS1991

Green's functions for dislocations in bonded strips and related crack problems

Ballarini, R.↗

NASA NTRS1992

Calculation of stress intensity factors in an isotropic multicracked plate. Part 1: Theoretical development

An essential part of describing the damage state and predicting the damage growth in a multicracked plate is the accurate calculation of stress intensity factors (SIF's). Here, a methodology and rigorous solution formulation for SIF's of a multicracked plate, with fully interacting cracks, subjected to a far-field arbitrary stress state is presented. The fundamental perturbation problem is derived, and the steps needed to formulate the system of singular integral equations whose solution gives rise to the evaluation of the SIF's are identified. This analytical derivation and numerical solution are obtained by using intelligent application of symbolic computations and automatic FORTRAN generation capabilities (described in the second part of this paper). As a result, a symbolic/FORTRAN package, named SYMFRAC, that is capable of providing accurate SIF's at each crack tip was developed and validated.

Binienda, W. K.↗