Penny-shaped crack in an elastic layer bonded to dissimilar half spaces

Two axially symmetric mixed boundary value problems in an elastic dissimilar layered medium are considered. It is assumed that an elastic layer is bonded to two semi-infinite half spaces along its plane surfaces, and contains a penny-shaped crack parallel to the interfaces. In the first problem the two half spaces are assumed to have the same elastic properties and the crack is located in the mid-plane of the layer. In the second problem we consider the case of three different materials and arbitrary crack location in the layer. The numerical examples are given for a constant pressure on the crack surface. The stress intensity factors are evaluated and are plotted as functions of the layer thickness-to-crack radius ratio or the relative distance of the crack from an interface.     

Point charge in the interior of an elastic dielectric half space

The axisymmetric boundary value problem of an elastic dielectric half space, having a point charge “4π?” at a distance “$\text{h}$” beneath its free surface, is solved by the methods of images and Hankel transforms. Expressions for the components of the displacement and polarization vectors, the stress and electric tensors throughout the (interior of the) half space are obtained and the surface energy density of deformation and polarization is determined. For the particular case of an elastic dielectric half space with a free surface and in the absence of charge, the results derived agree with those presented in [1]. Numerical results are presented on graphs depicting spatial and surface variations of various physical quantities indicating the effects of polarization.     

The contact problem of an orthotropic non-homogeneous elastic half space

The problem of a rigid punch on an elastic half plane with orthotropic and non-homogeneous material is considered. The axes of orthotropy are chosen to coincide with the Cartesian coordinate system in which one axis is parallel to the edge of the half plane and the other is perpendicular to it. Non-homogeneity is introduced in both directions of orthotropy as continuous functions along these directions. Using the Fourier Transform Technique, the mixed boundary value problem is reduced to a singular integral equation which is solved numerically. The formulation of the problem is obtained for a rigid punch with arbitrary shape.     

Fracture of a surface layer bonded to a half space

The plane problem of a cracked elastic surface layer bonded to an elastic half space is considered. The surface layer is assumed to contain a transverse crack whose surface is subjected to uniform compression. The problem is formulated in terms of a singular integral equation, the derivative of the crack surface displacement being the density function. By using appropriate quadrature formulas, the integral equation reduces to a system of linear algebraic equations. This system is solved; the stress intensity factors and the crack surface displacement for various crack geometries, namely for internal crack, edge crack, crack touching the interface, and completely broken layer cases, are obtained.     

Propagation of torsional waves in a viscoelastic layer over an inhomogeneous half space

This paper presents a theoretical study on propagation of torsional surface waves in a homogeneous viscoelastic isotropic layer with Voigt type viscosity over an inhomogeneous isotropic infinite half space. The non-homogeneity in half space is assumed to arise due to exponential variation in shear modulus and density. A closed-form solution has been obtained for the displacement in the layer as well as for a infinite half space. The dispersion and absorption relations for an torsional wave under the assumed geometry have been found. Numerical results are presented for propagation characteristics in terms of a number of non-dimensionalized parameters and have been produced graphically. This study investigates the effect of various parameters, namely non-homogeneity parameter, internal friction, the layer width and complex wave number on dissipation function and phase velocity of the torsional wave. Results in some special cases are also compared with existing solutions available from analytical methods, which show a close agreement.     

Diffraction of shear waves on cylindrical inclusions and cavities in an elastic half space

Dynamic problems of the steady-state oscillations of a half space with different types of cylindrical homogeneities (cavities, rigid and stiff inclusions) are examined. The boundary of the half space is assumed to be fixed or free of forces. A harmonic shear wave radiating from infinity or a concentrated harmonic source may be radiators of the exciting wave field. Integral representations of displacement amplitudes, which automatically satisfy fixity conditions on the boundary of the half space and radiation conditions at infinity are constructed. The edge problems are reduced to Fredholm integral equations of the second kind and to singular integral equations. Selection of additional conditions for the latter is substantiated. Some computer-generated results are presented.Translated from Problemy Prochnosti, No. 11, pp. 90–94, November, 1990.     

Transient response of an elastic half space to normal pressure acting over a circular area on an inclined plane

An earthquake source has been simulated as a simple finite source, i.e., normal pressure acting over an inclined fault plane. The transient response of the surface displacement of an elastic half space due to the above internal source is calculated. A series of transformations, followed by the traditional Cagniard–de Hoop technique, are used to compute the transient response. Various wave arrivals are discussed. Numerical computations bring out the special character of the finite source vis-à-vis the point source. The originality of the paper lies in the fact that for the first time an exact computation of the surface response due to an inclined finite source has been computed by Cagniard’s approach.     

Stress intensity near a crack in the composition of a half space and a layer under harmonic loading

We consider a three-dimensional problem for an elastic bimaterial body formed by a half space and a layer and containing a circular crack. The surface of the crack is subjected to the action of stationary torsional forces. The problem is reduced to the solution of a system of boundary integral equations of Helmholtz-potential type for the unknown jumps of displacements on the crack surfaces. The dynamic stress concentration in the vicinity of the crack contour is investigated. __________ Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 44, No. 2, pp. 27–32, March–April, 2008.     

Stress distribution due to a griffith crack at the interface of an elastic half plane and a rigid foundation

The stress field in a semi-infinite elastic media bonded to a rigid foundation and containing a crack at its interface is determined. Two specific problems are discussed-first when the elastic half space is under shear such that σ_xv ~ P as √(x² + y²) → ∞ and next when the crack is opened by equal and opposite pressure on each of its sides. The solution is obtained via Fourier transforms which lead to a simultaneous set of dual integral equations containing a trigonometric kernel. It is observed that in a small region near the ends of the crack, violent oscillations occur in both the stress and displacement components.     

A coupled BEM-stiffness matrix approach for analysis of shear deformable plates on elastic half space

In this paper, a new direct Boundary Element Method (BEM) is presented to solve plates on elastic half space (EHS). The considered BEM is based on the formulation of Vander Weeën for the shear deformable plate bending theory of Reissner. The considered EHS is the infinite EHS of Boussinesq–Mindlin or the finite EHS (with rigid end layer) of Steinbrenner. The multi-layered EHS is also considered. In the present formulation, the soil stiffness matrix is computed. Hence, this stiffness matrix is directly incorporated inside the developed BEM. Several numerical examples are considered and results are compared against previously published analytical and numerical methods to validate the present formulation.     

Pressure with friction of a perfectly rigid die upon an elastic half space with cracks

We study the interaction of a rigid die with a base of any shape and the surface of an elastic half space containing cracks in the presence of friction in the contact zone. The solution of the plane contact problem of the theory of elasticity is obtained by the method of singular integral equations. The detailed analysis of the problem is performed for the case where the base of the die is parabolic and a crack is rectilinear and appears on the surface of half space. We also investigate the effects of the friction coefficient, crack length, its orientation, and location on stress intensity factors KI and KII at the crack tip and the distribution of contact stresses under the die.     

Disturbance due to mechanical and thermal sources in a generalized thermo-microstretch elastic half space

Disturbances caused by impulsive concentrated mechanical and thermal sources in a homogeneous, isotropic generalized thermo-microstretch elastic medium are studied by the use of Laplace—Hankel transform techniques. The integral transforms are inverted using a numerical technique. Analytical expressions for displacement components, stress, couple stress, microstress and temperature field are derived for different models of generalized thermoelasticity and illustrated graphically. These results for stresses and displacements can be used in estimating the effects of a surface pressure wave. Stretch and micropolar effects on various expressions obtained analytically are also depicted graphically.     

Axially symmetric contact problem of pressing of an absolutely rigid ball into an elastic half space with inhomogeneous coating

We consider an axially symmetric contact problem of pressing of an absolutely rigid ball into an inhomogeneous half space formed by a homogeneous base and an inhomogeneous surface layer. The Poisson’s ratio of the layer is constant and its Young modulus is an exponential function of the distance from the surface of the half space. The solution of the problem of the theory of elasticity with continuous dependence of the Young modulus on the coordinate is compared with the solution of the problem in which the inhomogeneous layer is replaced with a package of homogeneous layers.     

Annular punch on an elastic layer overlying an elastic foundation

The problem solved here is the axisymmetric mixed boundary value problem of the isotropic homogeneous theory of elasticity, in which the normal displacement is specified inside an annular area $\text{a ≤ r ≤ b}$, the normal stress is zero in $\text{r < a, r \# b}$ and the shearing stress is zero on the whole face $\text{z = ?h}$, the upper face of the elastic layer; the continuity of the normal and radial displacements and the normal and shearing stresses is assumed at the interface $\text{z = 0}$ between the elastic layer and the elastic foundation having different elastic constants. The problem is reduced to the solution of a Fredholm integral equation of the first kind. The Fredholm integral equation is further put in terms of four simultaneous Fredholm integral equations of the second kind in four unknown functions. The iterative solution of these integral equations has been obtained for $\text{epsi = b/h ? 1}$, and $\text{λ = a/b ? 1}$ for the case of an annular cylindrical punch. The expressions for the normal stress $\text{σ}{}_{\mathrm{zz}}\text{(r, ?h)}$ for $\text{a ≤ r ≤ b}$ and the total load $\text{P}$ on the punch have been obtained.     

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.     

Axisymmetric contact problem for an elastic layer and an elastic foundation

This paper is concerned with the axisymmetric problem of an elastic layer lying on a semi-infinite foundation. The layer is pressed against the foundation by a uniform clamping pressure applied over its entire surface and a uniform vertical body force due to the effect of gravity. In addition, an axisymmetric vertical line load is applied to the layer. It is assumed that the contact between the layer and the foundation is frictionless and that only compresive normal tractions can be transmitted through the interface. The contact along the interface will be continuous if the value of the line load is less than a critical value. However, interface separation takes place if it exceeds this critical value. The problem is formulated and solved for the cases of tensile and compressive line loads. Numerical results for contact stress distributions are given for different material combinations.     

Stress in the vicinity of a griffith crack at the interface of a layer bonded to a half plane—an approximate method

Approximations to the stress field in the vicinity of a Griffith crack located at the interface of a layer bonded to a dissimilar half plane are determined. A systematic use of Fourier transforms reduces the problem to that of solving a set of simultaneous dual integral equations with trigonometric kernels and weighting functions. This latter problem is reduced to the solution of an uncoupled pair of singular integral equations. An approximate technique using Legendre polynomial expansions is discussed. The analysis shows that when a constant pressure is applied to the faces of the crack, the stress components have the distinctive oscillatory singularities at the crack tip. Expressions up to the order of $\text{h}{}^{\mathrm{?4}}$, where $\text{h}$ is the thickness of the layer and is much greater than 1, are derived for the stress components.