Stress intensity factors for an interface crack between a functionally graded coating and a homogeneous substrate

In this paper we consider the problem of a functionally graded coating bonded to a homogeneous substrate with a partially insulated interface crack between the two materials subject to both thermal and mechanical loading. The problem is solved under the assumption of plane strain and generalized plane stress conditions. The heat conduction and the plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the temperature and the displacement fields in the medium as well as the crack tip stress intensity factors. A crack-closure algorithm recently developed by the authors is applied to handle the problem of having negative mode I stress intensity factors. The Finite Element Method was additionally used to model the crack problem and to compute the crack-tip stress intensity factors. The main objective of the paper is to study the effect of the material nonhomogeneity parameters, partial insulation of the crack surfaces and crack-closure on the crack tip stress intensity factors for the purpose of gaining better understanding of the thermo-mechanical behavior of graded coatings.     

Dynamic propagation of a finite crack in a micropolar elastic solid

Summary The dynamic propagation of a finite crack under mode-I loading in a micropolar elastic solid is investigated. By using an integral transform method, a pair of two-dimensional singular integral equations governing stress and couple stress is formulated in terms of displacement transverse to the crack, macro and micro rotations, and microinertia. These equations are solved numerically, and solutions for dynamic stress intensity and couple stress intensity factors are obtained by utilizing the values of the strengths of the square root singularities in macrorotation and the gradient of microrotation at the crack tips.     

Stress intensity factors around a crack parallel to a free surface of a half-plane

In this paper, stress intensity factors for a crack in a half-plane are considered. The crack is parallel to the stress-free surface of the half-plane and subjected to internal gas pressure. By using Fourier transforms, the mixed boundary value problem is reduced to the solution of a pair of dual integral equations. To solve the equations, the crack surface displacements are expanded in a series of functions which are zero outside the crack. The unknown coefficients in that series are solved with the aid of the Schmidt method. The stress intensity factors are calculated numerically and the results are compared with those given in other papers.     

A note on the impact response of a cracked orthotropic material

An approach involving potential functions is used to simplify the equations of motion for an Orthotropic material to an elementary differential equation whose solution may be used for a wide variety of problems in elastodynamics of Orthotropic materials. Solutions for the dynamic mode I and II stress intensity factors for a Griffith crack in an Orthotropic material are obtained. The new approach allows the stress intensity factors to be determined with a minimal use of integral transforms. The solutions for the stress intensity factors are shown to be the same as those obtained by previous researchers.     

Dynamic stress intensity factors due to scattering of harmonic waves by a crack in an infinite medium

An analytical method for calculating dynamic stress intensity factors in the mixed mode (combination of opening and sliding modes) using complex functions theory is presented. The crack is in infinite medium and subjected to the plane harmonic waves. The basis of the method is grounded on solving the two‐dimensional wave equations in the frequency domain and complex plane using mapping technique. In this domain, solution of the resulting partial differential equations is found in the series of the Hankel functions with unknown coefficients. Applying the boundary conditions of the crack, these coefficients are calculated. After solving the wave equations, the stress and displacement fields, also the J‐integrals are obtained. Finally using the J‐integrals, dynamic stress intensity factors are calculated. Numerical results including the values of dynamic stress intensity factors for a crack in an infinite medium subjected to the dilatation and shear harmonic waves are presented.     

Stress intensity factors for cracks at the vertex of a rounded V-notch

The method of singular integral equations was applied to determine the stress intensity factors for a system of cracks emanating from the vertex of an infinite rounded V-notch subjected to symmetric loading. The numerical values were obtained for two cases—the case of a single crack and the case of a system of two cracks of equal length. The influence of the rounding radius of the vertex of the notch and its opening angle on the stress intensity factors at the crack tips was analyzed. The solution obtained as a result has a general nature—the stress intensity factors at the crack tip are expressed as a function of the V-notch stress intensity factor and, hence, this solution could be treated as an asymptotic relation for finite bodies with deep V-notches subjected to symmetric loads.     

Determination of stress intensity factor solutions for cracks in finite-width functionally graded materials

A generalized method to determine the stress intensity factor equations for cracks in finite-width specimens of functionally graded materials (FGMs), based on force balance in regions ahead of the crack tip is provided. The method uses the Westergaard's stress distribution ahead of the crack in an infinite plate and is based on the requirement of isostrain deformation of layers of varying moduli ahead of the crack tip. It is shown that the modified Westergaard equation describes the normal stress distribution and the singular stress state ahead of the crack tip in a reasonably accurate manner. Based on this, closed-form analytical equations for the stress intensity factors of cracks in finite-width center cracked specimens were derived. Comparisons of the K values from the analytical equations with that obtained from FEM simulations indicate that the derived stress intensity factor equations for FGMs are reasonably accurate. For the finite-width center-cracked-tension (CCT) specimen, the errors are less than 10% for most of the crack lengths for materials with the outer layer modulus ratios varying from 0.2 to 5. The stress intensity factors were found to be sensitive to the absolute values of moduli of the layers, the modulus ratio of the outer layers as well as the nature of gradation including the increasing and the decreasing functional forms. The stress intensity factor equations are convenient for engineering estimates of stress intensity factors as well as in the experimental determinations of fracture toughness of FGMs.     

Scattering of the SH wave from a crack in a piezoelectric substrate bonded to a half-space of functionally graded materials

The scattering of the SH wave from a crack in a piezoelectric substrate which is bonded to a half-space of functionally graded materials (FGM) is studied. The governing equations along with permeable crack boundary, regularity and continuity conditions across the interface are reduced to a coupled set of Cauchy singular integral equations which are solved approximately by applying Chebyshev polynomials. Numerical results for the normalized dynamic stress intensity factors (NDSIF) and the normalized electric displacement intensity factors (NEDIF) are presented. The effects of the geometric and physical parameters, and the effects of the frequency and the angle of incidence on NDSIF and NEDIF are discussed.     

Interface fracture properties of a bimaterial ceramic composite

In this investigation, the interface fracture toughness is measured for a pair of ceramic clays which are joined together. The Brazilian disk specimen, which provides a wide range of mode mixity, is employed to measure these properties. Calibration equations relating the stress intensity factors to the applied load and geometry are determined by means of the finite element method and the M-integral. The effect of residual stresses is accounted for by employing a weight function to obtain the contribution to the stress intensity factors. Total stress intensity factors are obtained by superposition. These are employed to determine the critical interface energy release rate as a function of mode mixity from critical data obtained from tests carried out on the Brazilian disk specimens. An energy release rate fracture criterion is compared to the experimental results for .     

Stress solution for a crack at an arbitrary angle to an interface

Two-dimensional elasticity solution and the stress intensity factors are determined for a finite crack in one of the materials of a bimaterial composite. The crack has an arbitrary orientation and distance from the straight interface. The solution for general stress boundary conditions on the crack surface is presented in the form of coupled Fredholm integral equations of the second kind. Numerical values of the stress intensity factors are computed for various crack orientations, distances from the interface, and different combinations of material properties when the boundary conditions are uniform pressure and uniform shear stress.     

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.     

Stress intensity factors for a wide range of long-deep semi-elliptical surface cracks, partly through-wall cracks and fully through-wall cracks in tubular members

To calculate the rate of fatigue crack growth in tubular members, one approach is to make use of the fracture mechanics based Paris law. Stress intensity factors (SIF) of the cracked tubular members are prerequisite for such calculations. In this paper, stress intensity factors for circumferential deep semi-elliptical surface crack (a/t > 0.8), semi-elliptical partly through-wall crack and fully through-wall crack cracks in tubular members subjected to axial tension are presented. The work has produced a comprehensive set of equations for stress intensity factors as a function of a/T, c/πR and R/T for deep surface cracks. For the partly through-wall cracks and fully through-wall cracks, two sets of bounding stress intensity factor equations were produced based on which all stress intensity factors within the range of parameters can be obtained by interpolation.     

Parametric equations for T-butt weld toe stress intensity factors

This paper describes the generation of parametric equations for weld toe stress intensity factors. The methodology employed used a two-dimensional finite element analysis to evaluate the ‘crack opening’ stress distribution in the uncracked plane of T-butt geometries. This was then used as input into a dedicated weight function solution for the determination of stress intensity factors. The final parametric equations describe the stress intensity factor distributions for tension and bending as a function of plate thickness, weld attachment width, weld angle, weld root radius, crack length and crack shape. The equations are compared and validated against a wide spectrum of published values and appear by comparison accurate and wide ranging. The validation exercise uncovered situations where present design guidance is unconservative.     

A note on the effect of lateral bending stiffness of stringers attached to a plate with a crack

An infinite stringer which is assumed to be partially bonded to a plate through a layer of adhesive is considered. The stringer is assumed to have bending as well as longitudinal stiffness. The effect of the stringer's bending rigidity on the stress intensity factor at the tip of the crack is illustrated. Shear stress distribution between the plate and the stringer and the stress intensity factors will be obtained from the solution of a system of Fredholm integral equations which represent the continuity of displacements along the line of bond.     

Influence of initial stress on fracture of a halfspace containing a penny-shaped crack under radial shear

In this study, an axisymmetrical problem for a penny-shaped crack under radial shear is considered. The crack is located parallel to the surface of a halfspace, which is subjected to initial stress parallel to the crack plane. An approach proposed by Guz (1983) in the framework of the three-dimensional linearised solid mechanics is used. Analysis involves reducing the problem to a system of Fredholm integral equations of the second kind, where the solutions are identified with harmonic potential functions. The representations of the stress intensity factors K _I and K _II near the crack edges are obtained. These stress intensity factors are both influenced by the initial stress.     

Stress intensity factors for surface flaws in toughened glasses

An approximate method of deriving stress intensity factors as a function of angle for semicircular surface flaws is outlined. The method can be applied to a wide range of load distributions including non-polynomial distributions. The method is used to calculate stress intensity factors for thermally and chemically tempered glasses. The stress intensity factor equations developed are also used to calculate equivalent flaw sizes for some toughened glasses subjected to bending loads.     

Fracture parameters of a penny-shaped crack at the interface of a piezoelectric bi-material system

A penny-shaped crack at the interface of a piezoelectric bi-material system is considered. Analytical general solutions based on Hankel integral transforms are used to formulate the mixed-boundary value problem corresponding to an interfacial crack and the problem is reduced to a system of singular integral equations. The integral equations are further reduced to two systems of algebraic equations with the aid of Jacobi polynomials and Chebyshev polynomials. Thereafter, the exact expressions for the stress intensity factors and the electric displacement intensity factor at the tip of a crack are obtained. Selected numerical results are presented for various bi-material systems to portray the significant features of crack tip fracture parameters and their dependence on material properties, poling orientation and electric loading.     

Evaluation of the stress intensity factors for a crack approaching the bonded half-plates interface from isopachics

Summary The optical evaluation of the stress intensity factors from isopachic fringes is presented for a straight crack approaching the free boundary of a half-plate or the interface of two bonded plates. It is based on appropriate numerical approximation of the exact stress fields obtained by the method of singular integral equations. The proposed evaluation of the stress intensity factors is either by a numerical procedure or through the use of concise nomograms. Also, isopachic fringe patterns have been analytically constructed for a crack perpendicular to the interface and at various distances from it, to show the significant influence from the free boundary or the interface.     

An anisotropic layered material with a crack

Summary The problem of an anisotropic layered material which contains a plane crack in its interior is considered here. The problem is reduced to a set of Fredholm integral equations of the second kind which may be solved iteratively. Once these integral equations are solved, the crack tip stress intensity factors may be readily computed. Numerical results for some particular examples involving transversely isotropic materials are given here.     

On the dynamic behavior of a piezoelectric-elastic laminate with a crack in the piezoelectric material under electro-mechanical loads

The dynamic behavior of a piezoelectric-elastic laminate with a crack in the piezoelectric material under in-plane steady-state electro-mechanical loads is considered. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which are solved using Chebyshev polynomial expansions. Numerical results are provided to show the variation of both the dynamic stress intensity factors and electric displacement intensity factor with frequencies of the applied electro-mechanical loads. A phenomenon similar to “resonance” is observed when the applied loads act in some specific ranges of frequencies, and both the dynamic stress intensity factors and electric displacement intensity factor may increase significantly, which will lead to the failure of piezoelectric material. The effects of applied electric fields, crack geometry and elastic layer thickness on the phenomenon are also discussed.