Definitions in the case of multiple series equations and multiple integral equations are examined. In considering the solution of a given mixed boundary value problem perhaps the simplest technique is the direct application of the method of complex potentials provided the problem admits such potentials and the domain and the boundary conditions are suitable for such an application. The direct application of complex potentials is described with the aid of examples, taking into account a problem in potential theory, the case of periodic cuts, and an elasticity problem for a nonhomogeneous plane. The reduction to singular integral equations is discussed along with the numerical solution of singular integral equations of the first kind, integral equations with generalized Cauchy kernels, and singular integral equations of the second kind.

In this lecture after some introductory remarks, first certain concepts relating to the mixed boundary value problems in mechanics of materials and in potential theory will be defined and typical examples from crack and contact mechanics demonstrating unique applications of the singular integral equations will be described. Then the methods of solution of the singular integral equations based mostly on the solution by orthogonal polynomials will very briefly be outlined and some future research particularly in the field of fracture mechanics will be discussed.

In this study, the contact problems of thin films and cover plates are considered. In these problems, the loading consists of any one or combination of stresses caused by uniform temperature changes and temperature excursions, far field mechanical loading, and residual stresses resulting from film processing or welding. The primary interest in this study is in examining stress concentrations or singularities near the film ends for the purpose of addressing the question of crack initiation and propagation in the substrate or along the interface. The underlying contact mechanics problem is formulated by assuming that the film is a "membrane" and the substrate a graded elastic continuum, and is solved analytically by reducing it to an integral equation. The calculated results are the interfacial shear stress between the film and the graded substrate, the Mode II stress intensity factor at the end of the film, and the axial normal stress in the film. The results indicate that grading the material properties of the substrate helps to decrease the film stresses and the stress intensity factors at the free edges and to lower the axial normal stresses at the midsection where the film is most likely to crack.

Generally, the mixed boundary value problems in fracture and contact mechanics may be formulated in terms of integral equations. Through a careful asymptotic analysis of the kernels and by separating nonintegrable singular parts, the unique features of the unknown functions can then be recovered. In mechanics and potential theory, a characteristic feature of these singular kernels is the Cauchy singularity. In the absence of other nonintegrable kernels, Cauchy kernel would give a square-root or conventional singularity. On the other hand, if the kernels contain, in addition to a Cauchy singularity, other nonintegrable singular terms, the application of the complex function theory would show that the solution has a non-square-root or unconventional singularity. In this article, some typical examples from crack and contact mechanics demonstrating unique applications of such integral equations will be described. After some remarks on three-dimensional singularities, the key examples considered will include the generalized Cauchy kernels, membrane and sliding contact mechanics, coupled crack-contact problems, and crack and contact problems in graded materials.

The main objective of this study is to examine the three dimensional surface crack problems in functionally graded coatings subjected to mode I mechanical or transient thermal loading. The surface cracks are assumed to have a semi-elliptical crack front profile of arbitrary aspect ratio. The cracks are embedded in the functionally graded material (FGM) coating which is perfectly bonded to a homogeneous substrate. A three dimensional finite element method is used to solve the thermal and structural problems. Collapsed 20-node isoparametric elements are utilized to simulate the strain singularity around the crack front. The stress intensity factors are computed by using the displacement correlation technique. Four different coating types are considered in the analyses which have homogeneous, ceramic-rich (CR), metal-rich (MR) and linear variation (LN) material composition profiles. In the mechanical loading problems, the composite medium is assumed to be subjected to fixed-grip tension or three point bending. In the thermal analysis, a transient residual stress problem is considered. The stress intensity factors calculated for FGM plates are in good agreement with the previously published results on three dimensional surface cracks. The new results provided show that maximum stress intensity factors computed during transient thermal loading period for the FGM coatings are lower than those of the homogeneous ceramic ones.

Thermal barrier coatings generally consist of a metallic substrate which is the primary structural component, a metallic bond coat which serves as oxygen diffusion barrier, a very thin layer of thermally grown oxide and a ceramic top coat that provides the main thermal shielding. Homogeneous ceramic coatings as top coats appear to have certain undesirable features such as high residual and thermal stresses, generally low toughness and relatively poor bonding strength. The new concept of compositional grading of the top coat may help to overcome some of these shortcomings by eliminating the material property discontinuities. A common mode of failure in thermal barrier coatings seems to be the debonding of the top coat. In this study the related interface crack problem for a graded ceramic/metal top coat is considered. It is assumed that the thermophysical properties of the top coat continuously vary between that of the bond coat at the top coat-bond coat interface and that of the ceramic at and near the free surface. The main objective of the study is to examine the influence of the material nonhomogeneity parameters and relative dimensions on the stress intensity factors and the crack opening displacements.

Graded materials are multiphase composites with continuously varying thermophysical properties. The concept provides material scientists and engineers with an important tool to develop new materials tailored for some specific applications. One such application of this new class of materials is as top coats or interfacial regions in thermal barrier systems. A widely observed failure mode in these layered materials is known to be interfacial cracking that leads to spallation. In many cases it is the buckling instability of coating under mechanically or thermally induced compressive stresses that triggers spallation. Under in-plane loading since the linear elastic small deformation theory gives only a trivial solution, in this study the plane strain interface crack problem for a graded coating bonded to a homogeneous substrate is formulated by using a kinematically nonlinear continuum theory. Both the instability and the postbuckling problems are considered. The main objective of the study is the investigation of the influence of material nonhomogeneity, kinematic nonlinearity and plate approximation on the critical instability load and on such fracture mechanics parameters as strain energy release rate, stress intensity factors and crack opening displacements.

Many of the present and potential applications of graded materials involve contact problems. These are mostly load transfer components in elastic solids in the presence of friction. From a viewpoint of failure mechanics an important aspect of contact problems is surface cracking which is caused by friction forces. In this study the general crack/contact problem for a graded medium loaded by a sliding rigid stamp is considered. With applications to initiation and propagation of surface cracks in mind, the in-plane stress components on the surface of the graded medium, and the stress intensity factors at the tip of a surface crack are evaluated. The singular behavior of the coupled crack/contact problem is then examined, some examples are considered and sample results are given.

In this article the coupled problem of crack/contact mechanics in a nonhomogeneous medium is considered. The underlying physical problem to which the results might be applicable is the initiation and the subcritical growth of surface cracks in a graded medium loaded by a sliding rigid stamp in the presence of friction. The dimensions of the graded medium are assumed to be very large in comparison with the local length parameters of the crack/contact region. Thus in formulating the problem the graded medium is assumed to be semi-infinite. The objective of the study is to determine, in addition to contact stresses, the in-plane component of the surface stress and the stress intensity factors at the crack tip. These are the primary load factors controlling the initiation and the subsequent growth of surface cracks in the graded medium. The coupled mixed boundary value problem is solved and the results are presented for various combinations of friction coefficient, material nonhomogeneity constant and crack/contact length parameters.

: In this report the initiation and subcritical growth of surface cracks in homogeneous materials due to sliding contact are considered The load is applied through a rigid stamp with an arbitrary profile. The problem is formulated and solved under the assumptions of plane elasticity and Coulomb friction. The stress state on the surface of the half plane is analyzed in the absence of any cracks in order to determine the crack initiation force and angle. A surface crack is then introduced and the resulting coupled crack/contact problem is investigated. Extensive results are presented regarding mostly the mixed mode stress intensity factors which constitute the main driving force for the subcritically growing cracks.

: The collinear crack problem in an inhomogeneous orthotropic medium is considered under Mode I plane strain or plane stress loading conditions. It is shown that by introducing certain averaged orthotropy parameters, aside from a scaling parameter the results become only weakly dependent on the orthotropy constants. The main results of the study consist of the stress intensity factors at various crack tips as influenced by the material inhomogeneity parameter and by the relative size and position of the secondary cracks with respect to the dominant crack. Also considered is the crack/contact problem for the graded medium subjected to remote tension and bending through fixed grips. It is shown that the crack surface contact on the compression side of the loading has a magnifying effect on the stress intensity factor on the tension side.

The mixed mode crack problem in plane elasticity for a graded and oriented material is considered. The material property grading is intentional, whereas the property orientation or orthotropy is usually the consequence of material processing. It is assumed that the crack is located in a plane perpendicular to the direction of property grading and the principal axes of orthotropy are parallel and perpendicular to the crack. The four independent engineering constants E 11 , E 22 , G 12 , and ν 12 are replaced by a stiffness parameter, E = √E 11 E 22 , a stiffness ratio, δ = (E 11 /E 22 ) 1/4 , a Poisson's ratio, ν = √ν 12 ν 21 , and a shear parameter κ 0 = (E/2G 12 ) - ν. The corresponding mixed boundary value problem is reduced to a system of integral equations which is solved for various loading conditions and material parameters. The results presented consist of the strain energy release rate, the stress intensity factors and the crack opening displacements. It is found that generally the stress intensity factors increase with increasing material inhomogeneity parameter and shear parameter and with decreasing stiffness ratio.

In this study the problem of FGM coatings with an interface crack subjected to a mechanically or thermally induced compressive load parallel to the free surface is considered. First by using a nonlinear continuum theory the problem is reduced to an eigenvalue problem and the buckling instability load is evaluated analytically. The postbuckling problem is then examined numerically. The strain energy release rate and the stress intensity factors are directly calculated from special enriched crack tip elements.

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This feature article explores the concept of creating functionally graded metal-ceramic composite microstructures for thermal barrier coatings used in gas-turbine applications, From a thermomechanical perspective, this concept offers the possibility of significantly improving the life and reliability of thermal barrier coatings, However, prior research reveals that progress has been somewhat limited because of the oxidative instability exhibited by some metal-ceramic composite microstructures, The present study addresses some of the materials criteria and research issues associated with preparing chemically stable, yet mechanically durable, graded metal-ceramic microstructures for realistic application environments.

The fracture mechanics of molybdenum-quartz circular seals under residual stresses resulting from the cooling process are considered. For the purpose of studying the nature of stress singularity and probable crack initiation sites and directions, the problem is approximated by a plane strain problem. The interfacial zone between the two materials is modeled by either an ideal interface or a homogeneous shear layer. The circular seal is assumed to be an elastic inclusion with a nonlinearly tapered profile, one side of which is fully debonded. The other side of the inclusion is either fully bonded or partially debonded. For the fully bonded case, the stress singularity at the tip of the tapered inclusion is studied in detail. Results for interfacial shear stress, inclusion membrane stress, stress intensity factors as well as predictions for possible crack initiation directions based on the maximum cleavage stress are presented.

In this paper the fracture mechanics of orthotropic materials containing collinear interface cracks is considered. The primary objective is to study the influence of the thickness and the structure of the interfacial regions on the crack driving force. The interfacial region is assumed to be a relatively thin orthotropic elastic layer. The stress intensity factors or the strain energy release rates are assumed to be the main measure of the crack driving force. A relatively simple and efficient method is presented to solve the related elasticity problem. The results are obtained for a wide range of actual material combinations. In order to study the influence of the structure of the interfacial zone, the problem is also solved for isotropic and orthotropic materials bonded through a layer with hypothetically selected material properties. The results show that the effect of the thickness, the mechanical properties and the material orientation of the interfacial zone on the strain energy release rate could be very significant. An interesting and a rather useful result obtained from the collinear crack solutions is that the strain energy release rates for multiple cracks in bonded orthotropic materials with or without an interfacial layer may be predicted by using the results obtained for an isotropic homogeneous plane provided that the normalization factors are selected properly.

In this paper the mixed boundary value problem for a nonhomogeneous medium bonded to a rigid subspace is considered. The main objective is to investigate the techniques that would lead to analytically tractable solutions and to provide examples comparing the results of various kinds of material nonhomogeneities. The problem studied is a two dimensional diffusion problem in which the interface contains a plane crack. An elastic medium under antiplane shear loading is used to formulate the problem. However, the results may be interpreted in terms of any number of steady-state diffusion phenomena. The method used is essentially an inverse method in the sense that it provides the material constitutive behavior for which the mixed boundary value problem can be solved rather than solving the problem for a given material. Two different methods are described and some numerical examples are given.

In this paper, the plane problems of a single edge crack in two bonded elastic layers and in an elastic surface layer bonded to an elastic semi-infinite plane are analyzed by the body force method. The stress fields induced by a point force and a displacement discontinuity in two bonded elastic half-planes obtained by Hetenyi's solution, in other words, Green's function in closed forms, are used as fundamental solutions to solve those problems. The boundary conditions for stress free-edges of the layers and the crack surface are satisfied by superposing the distributed fundamental solutions and adjusting their densities. The stress intensity factors are systematically calculated for the various geometrical conditions and the various stiffness ratios of the layers.