Non-homogeneous Half-space Research Articles

PROPAGATION OF TORSIONAL SURFACE WAVES IN A HOMOGENEOUS SUBSTRATUM OVER A HETEROGENEOUS HALF-SPACE

The paper discusses the propagation of torsional surface wave in a homogeneous substratum over a half-space with linearly varying rigidity and density. The study reveals that under assumed conditions, a torsional surface wave propagates in the medium. The velocities of torsional surface waves have been calculated numerically and are presented in a number of graphs. It is also observed that for a stratum over a homogeneous half-space, the velocity of torsional surface waves coincides with that of Love waves. For a non-homogeneous half-space it is observed that the velocity of torsional surface waves is always higher than that of Love waves propagated in a homogeneous layer over a homogeneous half-space. An attempt is also made to assess the possible propagation of torsional surface waves in a half-space with linearly varying rigidity and density, lacking a superficial layer. It is concluded that such a half-space allows two solutions for torsional waves while a homogeneous half-space has one.

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 μ(z) = μ1(z + c)α, c > 0 and μ1, α 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 ω0.

A flexible rectangular footing on aGibson soil: Required rigidity for full contact

The problem of a rectangular footing or a strip footing resting on a nonhomogeneous elastic half-space is studied in this paper. The medium is assumed to be isotropic with a shear modulus linearly increasing with depth G (z) = G 0+ mz and a constant Poisson's ratio equal to 1/3. An important feature of this model is that either the Winkler foundation or the elastic homogeneous half-space can be made special cases by letting G 0 or m equal zero, respectively some results are presented for these cases. In order to investigate the necessary conditions for a footing to be considered rigid, both rigid and flexible footings have been studied. Central concentrated (column) loading, in addition to the self-weight, is treated-being the most demanding in terms of the zero uplift requirement (under conditions of symmetry in the loading with respect to the plate geometry, imposed for reasons of mathematical feasibility). The contact is assumed to be tensionless. There are three important steps in this formulation. The fundamental solution of the nonhomogeneous half-space is separated into the fundamental solution of the homogeneous half-space and a function related to the nonhomogeneity of the half-space. The latter function is approximated by an analytically tractable expression. The contact region is discretized using an adaptive scheme that accounts for the possible edge and corner singularities. The latter scheme removes the burden of most of the numerical integration. A rigid strip footing and a rigid rectangular footing are treated first to ascertain the convergence of the solution procedure and to provide information requisite for the flexibility study. The title problem is transformed into the solution of three coupled two dimensional singular integral equations. The contact regions are found iteratively since the problem is nonlinear.

Reissner-Sagoci problem for a non-homogeneous half-space with surface constraint

This paper deals with the problem of twisting of a non-homogeneous, isotropic, half-space by rotating a circular part of its boundary surface (0⩽r⩽a, z = 0) through a given angle. A ring (a b, z = 0) is rigidly clamped. The shear modulus μ is assumed to vary with the cylindrical coordinates r, z by the power law (μ = μα,β r α z β). Such a dependence is of practical interest in the context of Soil Mechanics. The problem leads to a Fredholm integral equation of the second kind which is solved numerically, giving an evaluation of the influence of non-homogeneity on the torque at the surface and the stress intensity factor. The homogeneous case studied in [16] is recovered. Expressions for some quantities of physical importance such as the torque applied at the surface and stress intensity factor are obtained. It appears from our investigation that the influence of clamping dies out with increasing α and β. Quantitative evaluations are given and some curves for the relative increase, due to clamping, in the torque and in the stress intensity factor are presented.

The propagation of Love waves in a fluid-saturated porous anisotropic layer

The paper concerns the propagation of Love waves in a transverse-isotropic fluid-saturated porous layer overlaying an elastic nonhomogeneous half-space. Using the Biot's theory for the porous layer and the theory of elasticity for the lower medium, the dispersion equation has been derived. This complex transcendental equation that relates frequency, phase velocity, anisotropy factor of the layer and the inhomogeneity character of the half-space has been solved with the aid of successive approximation method. The effect of anisotropy and heterogeneity has been shown graphically.

Behaviour of an elastic non-homogeneous half-space. Part 2 - Circular and strip footings : Booker, J R; Balaam, N P; Davis, E H Int J Num Anal Meth GeomechV9, N4, July-Aug 1985, P369–381

Behaviour of an elastic non-homogeneous half-space. Part 2 - Circular and strip footings : Booker, J R; Balaam, N P; Davis, E H Int J Num Anal Meth GeomechV9, N4, July-Aug 1985, P369–381

Behaviour of an elastic non-homogeneous half-space. Part 1 - Line and point loads : Booker, J R; Balaam, N P; Davis, E H Int J Num Anal Meth GeomechV9, N4, July-Aug 1985, P353–367

Behaviour of an elastic non-homogeneous half-space. Part 1 - Line and point loads : Booker, J R; Balaam, N P; Davis, E H Int J Num Anal Meth GeomechV9, N4, July-Aug 1985, P353–367

Torsion of an elastic nonhomogeneous half-space by a circular die

Torsion of an elastic nonhomogeneous half-space by a circular die

Behaviour of an elastic non-homogeneous half space. Part 1 - line and point loads : Booker, J R; Balaam, N P; Davis, E H University of Sydney, School of Civil and Mining Engineering Research Report NR409, 24P

Behaviour of an elastic non-homogeneous half space. Part 1 - line and point loads : Booker, J R; Balaam, N P; Davis, E H University of Sydney, School of Civil and Mining Engineering Research Report NR409, 24P

An axisymmetrical problem of the theory of elasticity for a transversally-isotropic non-homogeneous half-space

The problem of the equilibrium of a transversally-isotropic half-space which is non-homogeneous is depth under the influence of perpendicular forces applied to the boundary is discussed. A method of solution, based on the integral transformation with respect to certain variables, and on a special approach to the system of ordinary differential equations which arises is proposed. The case where Young's and the shear moduli vary exponentially with depth, and Poisson's ratios are constant is considered as an example.

Solution of plane-periodic contact problem for nonhomogeneous half-space

Solution of plane-periodic contact problem for nonhomogeneous half-space

Asymmetric Contact Problem Including Wear for Nonhomogeneous Half Space

Contact problem with wear for asymmetric rigid die acting on a half space whose elastic modulus is a power function of depth is considered for the case when the die is rotating according to an arbitrary law. Zone of contact is taken to be a circle, and the wear is proportional to the work done by the tangential stresses obeying Coloumb’s law. Integral equation of the problem is derived and an exact solution of the equation is obtained in closed form. The case of inclined flat die is discussed as an illustrative example of the general method of solution that is proposed.

A contact problem for a nonhomogeneous half-space

A solution of a contact problem for a non-homogeneous half-space is given. The analysis is based on the assumption that the shearing displacement is prescribed within the band y = 0, ¦x¦<1, ¦z¦< ∞ and shearing stress is zero in the outside area |x|> 1, y = 0, ¦z¦<∞. The shear modulus of the non-homogeneous half-space is assumed in the following form μ = μ 0(1 + α 1 x) 2 (1 + β 1y) 2, where α 1, β 1 and μ 0 are constants. Here x, y and z are referred to a rectangular coordinates system.

Diffraction of antiplane shear waves by a griffith crack or a rigid strip lying at the interface of homogeneous and nonhomogeneous elastic half-spaces

Diffraction of antiplane shear waves by a griffith crack or a rigid strip lying at the interface of homogeneous and nonhomogeneous elastic half-spaces

Torsion by a circular die of a nonhomogeneous elastic layer bonded to a nonhomogeneous half-space

The title problem is equivalent to a torsional mixed boundary value problem in the theory of elasticity in which the only non-zero displacement component is specified inside the circular area r ⩽ l and the shearing stress σ θz , is zero in the area r > 1 on the face z = 0; the continuity of the displacement and the shearing stress σ θz is assumed at the interface z = h between the layer and the half-space having different modulii of rigidity. The modulii of rigidity are assumed to be continuous functions of 2. The problem is reduced to the solution of a Fredholm integral equation of the second kind whose iterative solution has been obtained valid for values of h > l, for a few particular cases. Expression for the torque T required to rotate the die through an angle ω has been obtained for the general as well as particular cases.

Discussion: Some results concerning displacements and stresses in a non-homogeneous elastic half-space

Discussion: Some results concerning displacements and stresses in a non-homogeneous elastic half-space

Transverse Propagation of Transient Electromagnetic Waves in a Non-homogeneous Half-space

The problem of a transient, transverse electromagnetic wave incident on a non-homogeneous half space is considered. Solutions are obtained by both the Laplace transform technique (LT) and the method of characteristics (MOC). The former method yields an infinite number of exact solutions in closed-form provided that the dielectric and permeability parameters are distributed as power laws in the spacial coordinate. A method for systematically generating these solutions is given. The method of characteristics, in numerical form, provides approximate solutions along the curved characteristics. Agreement between the two methods is excellent, except for a certain anomalous class of inhomogeneities. Finally, certain quasi-static solutions, involving a variety of inhomogeneities, are demonstrated.

The torsion of a non-homogeneous stratum with a hyperbolic variation of shear modulus

An exact formulation of the governing dual integral equations for the torsion of a non-homogeneous stratum due to a rigid circular body at its free surface is presented. The stratum varies in shear modulus according to the hyperbolic variation in a contemporary work [1]. It is shown that the unknown static stress distribution under the rigid body is governed by modified Bessel function of the first kind. By comparing the governing functions in the dual integral equations for five cases of elastic media: homogeneous half-space, and stratum, linearly non-homogeneous half-space and stratum and, finally, the present non-homogeneous stratum with hyperbolic variation, it is established that the surface shear modulus is the dominant parameter in the assessment of the stress and displacement fields in a non-homogeneous stratum where lateral variation of elastic properties is negligible.

Torsion of Nonhomogeneous Solid

A solution is given for the Reissner-Sagoci problem for a nonhomogeneous half-space and semi-infinite cylinder in each of which the shear modulus is assumed to vary with the depth. The formulation is reduced to Fredholm integral equations of the second kind. Numerical results have been presented for the physical quantities of interest. Some of these results indicate a marked difference from the corresponding response of a homogeneous isotropic solid. The analysis also reveals that the classical square root singularity in the shear stresses retains this order in the nonhomogeneous solid under consideration in contrast with a material possessing a modulus, in which the singularity is a function of a constant.

The deformation of a nonhomogeneous half-space under the action of a surface load

The deformation of a nonhomogeneous half-space under the action of a surface load