The mode III axisymmetric crack problem in a non-homogeneous interfacial region between homogeneous half-spaces

In this study, the mode III axisymmetric crack problem in a non-homogeneous interfacial region between two homogeneous half-spaces is considered. The shear modulus of the interfacial layer is assumed to be μ₂(z)=μ₀ e_mz. It is also assumed that this shear modulus varies continuously between that of the two half-spaces. By using the Hankel transform technique the problem is reduced to a singular integral equation. The problem is solved for various material combinations, crack geometries and for three different sets of crack surface tractions, and the corresponding stress intensity factors are tabulated. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Reissner-Sagoci problem for a non-homogeneous half-space with a penny-shaped crack

The present paper examines the elastostatic problem related to the axisymmetric rotation of a rigid circular disc bonded to a non-homogeneous half-space containing a penny-shaped crack. The shear modulus of the half-space is assumed to vary with depth according to the relation (z) = ₁(z + c), c > 0 and ₁, are constants. Using Hankel transforms, the solution of the problem is reduced to integral equations and finally to simultaneous Fredholm integral equations of the second kind. By solving numerically the simultaneous Fredholm integral equations, results are obtained which are used to estimate the stress intensity factor at the crack tip and the torque required to rotate the disc through an angle ₀.     

The modelling of axisymmetric basal crack evolution in a borehole indentation problem

The evaluation of the load-displacement behaviour of end bearing caisson-type foundations located in brittle elastic geomaterials is a problem of importance to geomechanics and geotechnical engineering. The conventional study of this problem ignores the possible influences of fracture of the brittle geomaterial which accompanies the indentation process. This paper presents a computational approach to the problem of fracture extension from the edge of the basal plane of the borehole which is used to illustrate how the load-displacement exhibits a non-linear response due to the basal crack extension.     

A double receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, the axisymmetric problem of a frictionless double receding contact between a rigid stamp of axisymmetric profile, an elastic functionally graded layer and a homogeneous half space is considered. The graded layer is modelled as a nonhomogeneous medium with an isotropic stress-strain law. Assuming the double contact between the bodies to be frictionless, only compressive normal tractions can be transmitted in each contact area while the rest of the surface is free of tractions. Using an appropriate integral transform, the axisymmetric elasticity equations are converted analytically into a system of singular integral equations where the unknowns are the pressures and the radii of the receding contact area in the two contact zones. The global equilibrium conditions are supplemented to solve the problem. The singular integral equations are solved numerically using orthogonal Chebyshev polynomials. An iterative scheme based on the Newton-Raphson method is employed to obtain the receding contact radii and pressures that satisfy the equilibrium conditions. The main objectives of the paper are to study the effect of the nonhomogeneity parameter, the thickness of the graded layer and the magnitude of the applied load on the contact pressures, the radii of the receding contact zones and the indentation for the case of a spherical rigid punch.     

The elastostatic axisymmetric problem of a sphere containing a penny-shaped crack in a nonequatorial plane

The axisymmetric problem of a sphere containing a penny-shaped crack in a nonequitorial plane is solved with the use of Bousinesq stress functions. Two coordinate systems—oblate spheroidal for representing the crack surface and spherical polars for the spherical surface, translated along the z-axis with respect to each other—are used to satisfy boundary conditions. Integral representations and transformations of harmonic functions are used to relate stress functions in the two coordinate systems. This procedure-leads to a system of algebraic equations which is solved, for axisymmetric tractions on both the surfaces. Graphical results are presented for a specific loading case.     

The interaction between screw dislocations and an interfacial blunt crack and a sharp crack

The interaction between screw dislocations and an interfacial blunt crack and a sharp crack under loads at infinity is dealt with. Utilizing the Muskhelishvili complex variable method, the closed form solutions are derived. The stress intensity factor and critical stress intensity factor for dislocation emission are also calculated. The results show that the shielding effect increases with the increase of the shear modulus and the distance between the two cracks, but decreases with the increase of dislocation azimuth. The critical loads at infinity for dislocation emission increase with the increment of the emission angle and the distance.     

Antiplane shear crack problem in bonded materials with a graded interfacial zone

In this paper the interface crack problem for two elastic half spaces bonded through a nonhomogeneous interfacial zone is considered. It is assumed that the medium is under antiplane shear loading. The problem is solved for two different interfacial zone models that may approximate the actual diffusion bonded materials or homogeneous solids bonded through a functionally gradient material. Extensive results are obtained by varying the stiffness and the interfacial zone thickness to crack length ratios. Also, for various limiting cases the behaviour of the stress intensity factors and the strain energy release rates are studied.     

Transient responses of a special non-homogeneous magneto-electro-elastic hollow cylinder for a fully coupled axisymmetric plane strain problem

Summary By means of introducing a new dependent variable and the separation of variables technique as well as the superposition method, the axisymmetric plane strain dynamic problem of a special non-homogeneous magneto-electro-elastic hollow cylinder is finally transformed to two Volterra integral equations of the second kind with respect to two functions of time. At each time subinterval, a cubic Hermite polynomial is adopted to approximate the two undetermined functions, and the recursive formula is obtained.The integral equation is then solved successfully. The transient responses of displacements, stresses, electric and magnetic potentials are finally determined. Numerical results are presented at the end.     

The interaction between an interfacial crack and a microcrack under antiplane loading

This article provides a comprehensive theoretical treatment of the interaction between an arbitrarily located and oriented microdefect and a finite interfacial crack under antiplane loading. The analysis is based upon the use of an integral transform method and a self-consistent iterative superposition technique. The closed form expression for the resulting stress intensity factors at the interfacial crack are obtained by solving the appropriate singular integral equations, using Chebyshev polynomials. Typical examples are provided to show the effect of the location and orientation of the microdefect and the material combination upon the stress intensity factor of the interfacial crack.     

An axisymmetric external crack problem for an infinite medium with a cylindrical inclusion

Summary This paper deals with the determination of stresses in an infinite medium containing an external crack surrounding a cylindrical inclusion. The two media are assumed to be homogeneous, isotropic and elastic but with different elastic constants. The continuity of stresses and displacements is assumed at the common cylindrical surface due to perfect bonding. The problem is reduced to the solution of a Fredholm integral equation of the second kind. A closed-form expression is obtained for the stress-intensity factor. The integral equation is solved numerically and the results are used to obtain the numerical values of the stress-intensity factor which are displayed graphically.The authors thank the National Research Council of Canada for supporting this research through NRC Grant No. A-4177.     

3-D dynamic interaction between a penny-shaped crack and a thin interlayer joining two elastic half-spaces

Three-dimensional (3-D) elastodynamic interaction between a penny-shaped crack and a thin elastic interlayer joining two elastic half-spaces is investigated by an improved boundary integral equation method or boundary element method. The penny-shaped crack is embedded in one of the half-spaces, perpendicular to the interlayer and subjected to a time-harmonic tensile loading on its surfaces. Effective “spring-like” boundary conditions are applied to approximate the effects of the thin layer in the mathematical model. Integral representations for the displacement and the stress components are derived by using modified Green’s functions, which satisfy the “spring-like” boundary conditions identically. Then, application of the dynamic loading condition on the crack-surfaces results in a boundary integral equation (BIE) for the crack-opening-displacement over the crack-surfaces only. A solution procedure is developed for solving the BIE numerically. Numerical results for the mode-I dynamic stress intensity factor (SIF) are presented and discussed to show the variations of the mode-I dynamic SIF with the angular coordinate of the crack-front points, the dimensionless wave number, the material mismatch and the crack-layer distance.     

On the penny-shaped crack in a non-homogeneous interlayer under torsion

This paper presents a theoretical treatment of a penny-shaped crack in an interfacial zone, along the thickness of which the elastic modulus is assumed as ₂(z) = ( +bz) ^k , wherek represents the distribution parameter independent of material properties and interlayer thicknessh. The theoretical formulations governing the torsion deformation behavior of the material are based on the use of a dislocation density function and integral transform technique. The stress intensity factor is obtained by solving a singular integral equation. Numerical examples are given to show the effects of material properties, interlayer thickness, and especially the distribution parameterk on the stress intensity factor. In the numerical procedure, modified Bessel functions are used, and the rate of convergence depends greatly on the ratio ofh/c, wherec is the crack radius.     

The axisymmetric boussinesq problem for a semi-space in micropolar theory

A solution of the axisymmetric Boussinesq problem is obtained, for a homogeneous and Isotropic semi-space, from which are deduced simple formulae for the depth of penetration of the tip of a punch of arbitrary profile and the total load and torque which must be applied to the punch to achieve this penetration. Simple expressions are also deduced for the distribution of pressure and couple stress under the punch. Particular results are derived for cylindrical, conical and paraboloidal punch shapes.     

The axisymmetric Boussinesq problem for a semi-space in couple-stress theory

A solution of the axisymmetric Boussinesq problem is obtained, for a homogeneous and isotropic semi-space, from which are deduced simple formulae for the depth of penetration of the tip of a punch of arbitrary profile and the total load and torque which must be applied to the punch to achieve this penetration. Simple expressions are also deduced for the distribution of pressure and couple stress under the punch. Particular results are derived for cylindrical and conical punch shapes.     

The problem of a rigid punch on a non-homogeneous elastic half space

The elastostatic problem of a rigid punch on an elastic half space is considered. The medium is assumed to exhibit a non-homogeneity varying with depth. Using the Fourier Transform Technique, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically. The effect of non-homogeneity on the stress distribution under the punch and on the stress singularity is studied. The influence of Poisson's ratio on the results is also considered.     

Stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes

The stresses around a crack in an interfacial layer between two dissimilar elastic half-planes are obtained. The crack is parallel to the interfaces. The material constants of the layer vary continuously within a range from those of the upper half-plane to those of the lower half-plane. An internal gas pressure is applied to the surfaces of the crack. To derive the solution, the nonhomogeneous interfacial layer is divided into several homogeneous layers with different material properties. The boundary conditions are reduced to dual integral equations, which are solved by expanding the differences of the crack face displacements into a series. The unknown coefficients in the series are determined using the Schmidt method, and a stress intensity factor is calculated numerically for epoxy-aluminum composites.     

Modeling method for the crack problem of a functionally graded interfacial zone with arbitrary material properties

A new multi-layered model is developed for the fracture analysis of a functionally graded interfacial zone with arbitrary material properties. It is assumed that the interfacial zone is divided into sub-layers with the material properties of each sub-layer varying in a power-law function. The model is used to study the crack problem in the functionally graded interfacial zone between two homogeneous half-planes under a dynamic anti-plane load. Using Fourier–Laplace transforms and the transfer matrix method, the mixed boundary value problem is reduced to a Cauchy singular integral equation, which is solved numerically in the Laplace transform domain. Laplace numerical inversion transform is employed to obtain the stress intensity factors. The results show that the new model is general and effective for the crack problem of the functionally graded interfacial zone with arbitrary properties.     

The problem of a griffith crack in micropolar elasticity

In the linear theory of micropolar elasticity the problem of Griffith crack in a transverse field of constant uniaxial tension is studied. The problem is reduced to three Fredholm integral equations of the second kind. These equations, which have the same kernel, are solved numerically. The stress environment at the tips of the crack is found to depend on, apart from Poisson's ratio and a material length parameter τ, another parameter $\text{N}$ which characterises the coupling of the microstructure with the displacement field. $\text{N}$ does not occur in the analogous problem in couple stress theory. Classical results are recovered in the limits: (i) $\text{N}$ tends ot zero (τ fixed) and (ii) τ tends to zero ($\text{N}$ is of the same order of magnitude as τ or of lower order).