M.K. Kassir

The Reissner-Sagoci problem for a non-homogeneous solid☆

Abstract This note contains an analysis, exact within the context of the conventional theory of elasticity, of the state of stress and deformation induced in a non-homogeneous, isotropic, elastic half-space when a circular part of its bounding plane is subjected to axisymmetric twisting deformation. The shear modulus of the material is assumed to vary with the depth coordinate but Poissons ratio remains constant. For semiinfinite medium the method of Hankel transform is employed while in case of long circular cylinder of finite radius, a dual-series form of solution is adopted to reduce the mixed-boundary conditions to a standard Fredholm integral equation which is then solved by iteration.

Moving Griffith crack in an orthotropic material

Abstract In this paper, an integral transform technique is employed to solve the plane elastodynamic problem of a crack of fixed length propagating at a constant speed in a uniformly stressed medium. It is assumed that the crack is located in a plane of elastic symmetry of the material. The stresses and strains ahead of the crack tip are determined explicitly and the conditions governing the initial growth of the crack are investigated using two current theories of fracture (Maximum normal stress and Minimum strain-energy density). Based on these theories, it appears that, depending upon the particular orthotropy of the material, the crack may extend in a straight line for all velocities or may immediately branch out at low velocities (compared to the shear wave velocity of the material) or may start propagating along its initial position for small velocities and then, as the velocity increases, may curve and branch out.

Interaction functions of a rigid strip bonded to saturated elastic half-space

Abstract A comprehensive analytical solution is developed to generate the interaction functions of a rigid permeable strip in contact with a saturated porous elastic half-space. Linear hysteretic damping characteristics of the medium are included in the formulation. The mixed boundary-value problem of a welded (and smooth) surface footing is reduced to a set of coupled Fredholm integral equations of the first kind and a numerical solution is provided. Frequency-dependent interaction functions (stiffnesses and radiation damping coefficients) of a medium modelling a saturated dense sand are computed and exhibited graphically to reveal the influence of pore water, permeability effects and hysteretic damping characteristics of the medium. In the horizontal mode of response and without hysteretic damping, the pore water causes the effective stiffness to increase by 50–100% while it has negligible influence on the damping coefficient. With 5% hysteretic damping, the horizontal stillness changes sign and character and there is an increase of about 25% in its magnitude compared to the corresponding dry case. For the vertical and rocking modes of response, the impact of pore water on the interaction functions (both stiffnesses and damping coefficients) is much more pronounced. Without hysteretic damping, there are changes in sign in the stiffnesses of the medium, and the magnitudes of the interaction functions, relative to the corresponding dry case, could be as large as several folds, especially for the higher values of the dimensionless frequency. At 5% damping, the effective stiffnesses are increased by about 25–50% while the damping coefficients almost double. Concerning the influence of types of contact, the interaction functions are practically identical for smooth and welded contacts over the frequency range of interest. The results are useful in determining the response of surface structures to seismic excitation, in particular, where the ground water is at a level to impact such response.

Application of Papkovich-Neuber potentials to a crack problem

Abstract The problem of an elastic solid containing a semi-infinite plane crack subjected to concentrated shears parallel to the edge of the crack is considered in this paper. A closed form solution using four harmonic functions is found to satisfy the finite displacement and inverse square root stress singularity at the edge of the crack. Explicit expressions in terms of elementary functions are given for the distribution of stress and displacement in the solid. These are obtained by employing Fourier and Kontorovich-Lebedev integral transforms and certain singular solutions of Laplace equations in three dimensions. The variations of the intensity of the local stress field along the crack border are shown graphically.

Vertical vibration of a circular footing on a saturated half-space

Abstract The dynamic response of a circular footing experiencing oscillatory vertical motion on the surface of a liquid-filled, porous, elastic half-space is reduced to the solution of a Fredholm integral equation of the second kind. Frequency-dependent impedance functions, for a medium consisting of dense sand saturated by ground water, are computed and shown graphically to reveal the influence of dissipation of pore water and variations in the permeability coefficient and Poissons ratio of the medium. The presence of gound water in the elastic medium affects the magnitude and character of the influence functions over the frequency range of practical interest and should be included in determining the response of surface structures to dynamic loadings.

On the distribution of thermal stresses around an elliptical crack in an infinite solid

Abstract This article is concerned with the construction of harmonic functions from which the stresses and displacements in an infinite solid containing an elliptical crack and conducting heat under steady-state conditions may be obtained. Various solutions are presented when the crack surfaces are exposed to polynomial thermal environments. Detailed calculations are carried out for quadratic polynomials. Expressions for the stress-intensity factors which govern the stability behavior of cracks in some current fracture theories are deduced. The harmonic functions developed here may be used to solve various related mixed boundary problems for a half-space with an elliptical surface of separation. They immediately yield the stresses and displacements around an elliptical crack embedded in an infinite isothermal elastic solid when the crack surfaces are opened out and/or sheared by polynomial stresses.

A three-dimensional rectangular crack subjected to shear loading

Abstract In this paper a solution is derived to treat the three-dimensional elastostatic problem of a narrow rectangular crack embedded in an infinite elastic medium and subjected to equal and opposite shear stress distribution across its faces. Employing two-dimensional integral transforms and assuming a plane-strain solution across the width of the crack, the stress field ahead of the crack length is reduced to the solution of an integral equation of Fredholm type. A numerical solution of the integral equation and the corresponding mode II stress-intensity factor is obtained for several crack dimensions and Poisson’s ratios of the material.

Rotatory and horizontal vibrations of a circular surface footing on a saturated elastic half-space

The forced vibration of a liquid-filled, porous, elastic half-space produced by rotatory and horizontal oscillations of a circular surface footing is reduced to the solutions of Fredholm integral equations of the second kind. The circular footing is modeled by a pervious, weightless, rigid disk of negligible thickness. For a medium consisting of dense sand saturated by ground water, numerical solutions of the integral equations are obtained to reveal the variations of the impedance functions with the exciting frequency and the material parameters (permeability and Poissons ratio). Comparison with the response of the dry soil is also discussed. In the rocking oscillation case, the presence of the ground water is found to influence the magnitude and shape of the impedance functions and may need to be considered in applicable soil-structure interaction problems. In the horizontal vibration case, however, marginal influence of the pore water is found to affect the response of the medium.

Effect of crack shape and size on estimating the fracture strength and crack growth fatigue life of bridge cable steel wires

Macrofracture entails the creation of free surface. The process can be enhanced or impeded depending on the ways in which the material microstructures are designed to react against the operational conditions. Undesirable chemical reactions can be the main source of strength degradation for load-supporting large structural members. Assessment of the remaining strength of damaged cable wires for cable-stayed or suspension bridges can rely on a knowledge of strength and/or fatigue life depending on the prevailing stress amplitude and whether the cables are stayed or not. Although large-scale computer schemes are available for making detailed failure analyses, they are not conducive to retrieving technical information within a relatively short time. Computer simulation has not advanced to the stage where experimental validation could be spared. To this end, effective analytical/experimental methodologies are much in need of development. The fracture mechanics approach appears to have gained ground in recent t...

Thermal stresses in an elastic solid containing a plane crack

Abstract This paper investigates the three dimensional stress distribution that arise when a semi-infinite plane crack is opened out by the application of heat to arbitrary regions of its surfaces. The temperature and stress fields are determined by employing certain singular solutions of Laplaces equation in three dimensions and the techniques of Fourier and Kontorovich-Lebedev integral transforms. Explicit expressions are found for the stress intensity factors and their variations along the crack border are displayed graphically.