F. Erdogan

The Crack Problem for a Nonhomogeneous Plane

This study considers the plane elasticity problem for a nonhomogeneous medium containing a crack. It is assumed that the Poissons ratio of the medium is constant and the Youngs modulus E varies exponentially with the coordinate parallel to the crack. First the half plane problem is formulated and the solution is given for arbitrary tractions along the boundary. Then, the integral equation for the crack problem is derived. It is shown that the integral equation having the derivative of the crack surface displacement as the density function has a simple Cauchy-type kernel. Hence, its solution and the stresses around the crack tips have the conventional square-root singularity. The solution is given for various loading conditions. The results show that the effect of the Poissons ratio and consequently that of the thickness constraint on the stress intensity factors are rather negligible. 14 references.

Fracture Mechanics of Functionally Graded Materials.

Abstract In this paper, after a brief discussion of the elementary concepts of fracture mechanics in nonhomogeneous materials, a number of typical problem areas relating to the fracture of functionally gradient materials (FGMs) are identified. The main topics considered are the investigation of the nature of stress singularity near the tip of a crack fully embedded in a nonhomogeneous medium, the general problem of debonding of an FGM coating from a homogeneous substrate, the basic surface crack problem in FGMs and cracking perpendicular to the interfaces, periodic surface cracking and the associated problem of stress and energy relaxation, and the problem of stress concentration at and the initiation and growth of delamination cracks from the stress-free ends of FGM-coated homogeneous substrates under residual or thermal stresses. Each topic is very briefly reviewed, some sample results are presented and comparisons with the corresponding results obtained from homogeneous materials are made.

Stresses in bonded materials with a crack perpendicular to the interface

Abstract The problem of two elastic bonded half planes containing a crack perpendicular to the interface is considered. First the solution for the semi-infinite crack under concentrated wedge loading is given. Then the problem of a finite crack fully imbedded in one of the half planes or terminating at the interface is considered. The Mellin transform in conjunction with the dislocations is used to formulate the problem and to derive the integral equation. The integral equation is solved; the stress intensity factors, the crack surface displacement, and the stresses around the crack tip terminating at the interface are obtained. Noting that the power of the singularity for the interface tip of the crack is not − 1 2 , a tentative fracture criterion dealing with the initiation of fracture propagation is proposed.

Stress singularities in a two-material wedge

A method for the determination of stresses in a two-material wedge-shaped region is presented. The method is applicable for plane strain or plane stress problems and treats the general case where each region is a wedge of arbitrary angle. The results are obtained by the use of the Mellin transform and the theory of residues.The characteristic equation is investigated to determine the stress singularity resulting from certain combination of geometry and material properties. A formal solution is then presented for the case where the loading is in the form of a point dislocation along the interface. This solution is the Greens function for the more general mismatch problems and therefore has applications in solving other problems with compatible boundary conditions. The results obtained show that for small values of r the dominant effect is due to geometry and the secondary effect is caused by the choice of elastic constants of the materials.

The Surface Crack Problem for a Plate With Functionally Graded Properties

In this study the plane elasticity problem for a nonhomogeneous layer containing a crack perpendicular to the boundaries is considered. It is assumed that the Youngs modulus of the medium varies continuously in the thickness direction. The problem is solved under three different loading conditions, namely fixed grip, membrane loading, and bending applied to the layer away from the crack region. Mode I stress intensity factors are presented for embedded as well as edge cracks for various values of dimensionless parameters representing the size and the location of the crack and the material nonhomogeneity. Some sample results are also given for the crack-opening displacement and the stress distribution.

CRACK PROBLEMS IN FGM LAYERS UNDER THERMAL STRESSES

In this study an unconstrained elastic layer under statically self-equilibrating thermal or residual stresses is considered. The layer is assumed to be a functionally graded material (FGM), meaning that its thermo-mechanical properties are assumed to be continuous functions of the thickness coordinate. The layer contains an embedded or a surface crack perpendicular to its boundaries. Using superposition the problem is reduced to a perturbation problem in which the crack surface tractions are the only external forces. The dimensions, geometry, and loading conditions of the original problem are such that the perturbation problem may be approximated by a plane strain mode I crack problem for an infinite layer. After a general discussion of the thermal stress problem, the crack problem in the nonhomogeneous medium is formulated. With the application to graded coatings and interfacial zones in mind, the thickness variation of the thermo-mechanical properties is assumed to be monotonous. Thus, the functions suc...

The mixed mode crack problem in a nonhomogeneous elastic medium

Abstract A nonhomogeneous elastic medium containing a crack arbitrarily oriented with respect to the direction of property gradient is considered. The problem is solved under plane strain or generalized plane stress conditions. This is a highly simplified version of a class of physical problems that may arise in fracture mechanics studies of ceramic coatings, metal/ceramic composites and interfacial zones with continuously varying volume fractions or graded properties. The main results of the paper are the calculated mode I and II stress intensity factors. Among the questions studied are the effects of the material nonhomogeneity constant, the crack orientation, the loading conditions and the Poissons ratio on the stress intensity factors. The stress state near the crack tip and the crack opening displacement are also briefly discussed.

Stress Intensity Factors

In this work the concept of the stress intensity factor, the underlying mechanics problem leading to its emergence, and its physical relevance, particularly its relation to fracture mechanics are discussed. The reasons as to why it has become nearly an indispensable tool for studying such important phenomena as brittle fracture and fatigue or corrosion fatigue crack propagation in structural solids are considered. A brief discussion of some of the important methods of solution of elastic crack problems is given. Also, a number of related special mechanics problems are described. 24 references.

Two bonded half planes with a crack going through the interface

The plane problem of two bonded elastic half planes containing a finite crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations with generalized Cauchy kernels. Even though the system has three irregular points, it is shown that the unknown functions are algebraically related at the irregular point on the interface and the integral equations can be solved by a method developed previously. The system of integral equations is shown to yield the same characteristic equation as that for two bonded quarter planes in the general case of the through crack, and the characteristic equation for a crack tip terminating at the interface in the special case. The numerical results given in the paper include the stress intensity factors at the crack tips, the normal and shear components of the stress intensity factors at the singular point on the interface, and the crack surface displacements.

Interaction Between a Circular Inclusion and an Arbitrarily Oriented Crack

The plane interaction problem for a circular elastic inclusion imbedded into an elastic matrix which contains an arbitrarily oriented crack is considered. Using the existing solutions for the edge dislocations as Greens functions, first the general problem of a through crack in the form of an arbitrary smooth arc located in the matrix in the vicinity of the inclusion is formulated. The integral equations for the line crack are then obtained as a system of singular integral equations with simple Cauchy kernels. The singular behavior of the stresses around the crack tips is examined and the expressions for the stress intensity factors representing the strength of the stress singularities are obtained in terms of the asymptotic values of the density functions of the integral equations. The problem is solved for various typical crack orientations and the corresponding stress intensity factors are given.