Non-homogeneous Elastic Half-space Research Articles

Propagation of love waves under the effect of parabolic irregularity in isotropic fluid-saturated porous medium

This study represents the Love wave propagation in a fluid-saturated transversely isotropic porous layer resting over a non-homogeneous elastic half-space. The irregularity considered is in the form of parabola at the interface of isotropic fluid saturated medium and non-homogeneous elastic half-space. The dispersion equation has been derived for the concerned problem by using Fourier Transform and inverse Fourier Transform method. The effect of different ratios of the depth of medium and the irregularity on the phase velocity of Love wave in the concerned problem are demonstrated graphically with the help of MATLAB, and it has been observed that the phase velocity is affected wave number, depth and shape of irregularity, and on medium width.

Simplified Method for the Lateral, Rotational, and Torsional Static Stiffness of Circular Footings on a Nonhomogeneous Elastic Half-Space Based on a Work-Equivalent Framework

Although there are many methods for assessing vertical stiffness of footings on the ground, simplified solutions to evaluate lateral, rotational, and torsional static stiffness are much more limited, particularly for nonhomogeneous profiles of shear modulus with depth. This paper addresses the topic by introducing a novel “work-equivalent” framework to develop new simplified design methods for estimating the stiffnesses of footings under multiple degrees-of-freedom loading for general nonhomogeneous soils. Furthermore, this framework provides a unified basis to analyze two existing design methods that have diverging results. 3D finite element analyses were carried out to investigate the soil–footing interaction for a range of continuously varying and multilayered nonhomogeneous soils, and to validate the new design approach.

Propagation of ultrasonic Love waves in nonhomogeneous elastic functionally graded materials

This paper presents a theoretical study of the propagation behavior of ultrasonic Love waves in nonhomogeneous functionally graded elastic materials, which is a vital problem in the mechanics of solids. The elastic properties (shear modulus) of a semi-infinite elastic half-space vary monotonically with the depth (distance from the surface of the material). The Direct Sturm–Liouville Problem that describes the propagation of Love waves in nonhomogeneous elastic functionally graded materials is formulated and solved by using two methods: i.e., (1) Finite Difference Method, and (2) Haskell-Thompson Transfer Matrix Method.The dispersion curves of phase and group velocity of surface Love waves in inhomogeneous elastic graded materials are evaluated. The integral formula for the group velocity of Love waves in nonhomogeneous elastic graded materials has been established. The effect of elastic non-homogeneities on the dispersion curves of Love waves is discussed. Two Love wave waveguide structures are analyzed: (1) a nonhomogeneous elastic surface layer deposited on a homogeneous elastic substrate, and (2) a semi-infinite nonhomogeneous elastic half-space. Obtained in this work, the phase and group velocity dispersion curves of Love waves propagating in the considered nonhomogeneous elastic waveguides have not previously been reported in the scientific literature. The results of this paper may give a deeper insight into the nature of Love waves propagation in elastic nonhomogeneous functionally graded materials, and can provide theoretical guidance for the design and optimization of Love wave based devices.

Group and Phase Velocity of Love Waves Propagating in Elastic Functionally Graded Materials

AbstractThis paper presents a theoretical study of the propagation behaviour of surface Love waves in nonhomogeneous functionally graded elastic materials, which is a vital problem in acoustics. The elastic properties (shear modulus) of a semi-infinite elastic half-space vary monotonically with the depth (distance from the surface of the material). Two Love wave waveguide structures are analyzed: 1) a nonhomogeneous elastic surface layer deposited on a homogeneous elastic substrate, and 2) a semi-infinite nonhomogeneous elastic half-space. The Direct Sturm-Liouville Problem that describes the propagation of Love waves in nonhomogeneous elastic functionally graded materials is formulated and solved 1) analytically in the case of the step profile, exponential profile and 1cosh2type profile, and 2) numerically in the case of the power type profiles (i.e. linear and quadratic), by using two numerical methods: i.e. a) Finite Difference Method, and b) Haskell-Thompson Transfer Matrix Method.The dispersion curves of phase and group velocity of surface Love waves in inhomogeneous elastic graded materials are evaluated. The integral formula for the group velocity of Love waves in nonhomogeneous elastic graded materials has been established. The results obtained in this paper can give a deeper insight into the nature of Love waves propagation in elastic nonhomogeneous functionally graded materials.

A frictionless contact problem for a flexible circular plate and an incompressible non-homogeneous elastic halfspace

In this paper we apply an energy method to examine the axisymmetric contact problem for a flexible circular plate in smooth contact an incompressible elastic halfspace, where the linear elastic shear modulus varies exponentially with depth. The approach adopted approximates the deflected shape of the plate by a power series expansion which satisfies the kinematics of deformation of the plate and the Kirchhoff boundary condition at the edge of the plate. The coefficients in the series are evaluated by making use of the principle of minimum potential energy. Results are obtained for the maximum deflection, the relative deflection and the maximum flexural moment in the circular plate. The results derived from the proposed procedure are compared with equivalent results derived from a computational procedure.

An adhesive contact problem for an incompressible non-homogeneous elastic halfspace

In this paper, we examine the axisymmetric adhesive contact problem for a rigid circular plate and an incompressible elastic halfspace where the linear elastic shear modulus varies exponentially with depth. The analytical solution of the mixed boundary value problem entails a set of coupled integral equations that cannot be solved easily by conventional integral transform techniques proposed in the literature. In this paper, we adopt a computational scheme where the contact normal and contact shear stress distributions are approximated by their discretized equivalents. The consideration of compatibility of deformations due to the indentation by a rigid indenter in adhesive contact gives a set of algebraic equations that yield the discretized equivalents of the contacts stresses and the axial stiffness of the medium.

Propagation of torsional surface wave in anisotropic poroelastic medium under initial stress

The present work deals with the possibility of propagation of torsional surface wave in fluid saturated poroelastic layer lying over nonhomogeneous elastic half space. Both the media are assumed to be under compressive initial stress. The half space has two types of inhomogeneity, viz; hyperbolic and quadratic. The dispersion equation for torsional wave in porous layer has been derived and observed that the presence of fluid in pores increases the velocity of the torsional surface wave but the phase velocity diminishes due to the presence of compressive initial stress in the porous layer. It is also observed that the velocity of the torsional surface wave increases due to the increase of initial stress in inhomogeneous half space. The inhomogeneity factor due to quadratic and hyperbolic variations in rigidity, density and initial stress of the medium decreases the phase velocity as it increases.

Effect of nonhomogeneity on the contact of an isotropic half-space and a rigid base with an axially symmetric recess

We study an axially symmetric frictionless contact problem between a nonhomogeneous elastic halfspace and a rigid base that has a small axisymmetric surface recess. We reduce the problem to solving Fredholm integral equations, solve these equations numerically, and establish a relationship between the applied pressure and the gap radius. We find and graph the effects of nonhomogeneity on the normal pressure, critical pressure and on the surface displacement.

Stiffness of a Flexible Circular Footing Embedded in an Elastic Half-Space

This paper addresses the problem of a circular footing of finite but nonzero stiffness embedded in an isotropic nonhomogeneous elastic half-space. The problem is solved by coupling the scaled boundary finite-element method with axisymmetric shell finite elements. The coupling of the two methods is validated by comparing computed solutions with analytical solutions for a flexible footing subjected to a uniformly distributed vertical load embedded in an elastic full-space. Based on the finding that the bending stiffness of the footing dictates the response for vertical and moment load cases, whereas the normal stiffness of the footing dominates the horizontal response, a convenient method for estimating dimensionless elastic stiffness coefficients is presented graphically. New results are presented for homogeneous and Gibson soil profiles and Poisson’s ratios of 0.2 and 0.499, to represent both sand and clay. An example demonstrating a practical application of these results is also provided.

The Reissner-Sagoci type problem for a non-homogeneous elastic cylinder embedded in an elastic non-homogeneous half-space

The problem considered is that of the torsion of a non-homogeneous elastic cylinder, which is embedded in a non-homogeneous elastic half-space (matrix) of different rigidity modulus. A rigid disc is bonded to the flat surface of the cylinder and torque is applied to the cylinder through a rigid disc. It is assumed that there is perfect bonding at the common cylindrical surface. Using integral transformation techniques the solution of the problem is reduced to dual integral equations. Later on the solution of the dual integral equations is transformed into the solution of a Fredholm integral equation of the second kind. Solving the Fredholm integral equation numerically the numerical results for torque and shear stress inside the cylinder are obtained and displayed graphically to demonstrate the effect of non-homogeneity of the elastic material on the torque and shear stress.

Elastic response of circular footings embedded in a non-homogeneous half-space

This study uses a semi-analytical technique (the scaled boundary finite-element method) to evaluate the load–displacement response of rigid circular footings embedded in a non-homogeneous elastic half-space. Results are presented in terms of dimensionless elastic stiffness coefficients. The footings are subjected to vertical, horizontal, moment and torsional loads. The cross-coupling between the horizontal and moment load case is also evaluated. The variation of soil shear modulus with depth may correspond to normally consolidated sand or clay, and the footing geometries considered cover a range of footings commonly encountered in engineering practice.

Elastic response of circular footings embedded in a non-homogeneous half-space

This study uses a semi-analytical technique (the scaled boundary finite-element method) to evaluate the load–displacement response of rigid circular footings embedded in a non-homogeneous elastic half-space. Results are presented in terms of dimensionless elastic stiffness coefficients. The footings are subjected to vertical, horizontal, moment and torsional loads. The cross-coupling between the horizontal and moment load case is also evaluated. The variation of soil shear modulus with depth may correspond to normally consolidated sand or clay, and the footing geometries considered cover a range of footings commonly encountered in engineering practice.

Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties

The contact problem for the impression of spherical indenter into a non-homogeneous (both layered and functionally graded) elastic half-space is considered. Analytical methods for solving this problem have been developed. It is assumed that the Lame coefficients vary arbitrarily with the half-space depth. The problem is reduced to dual integral equations. The developed methods make it possible to find the analytical asymptotically exact problem solution, suitable for a PC. The influence of the Lame coefficients variation upon the contact stresses and size of the contact zone with different radius of indenter as well as values of the impressing forces are studied. The effect of the non-homogeneity is examined. The developed method allows to construct analytical solutions with presupposed accuracy and gives the opportunity to do multiparametric and qualitative investigations of the problem with minimal computation time expenditure.

Axisymmetric mixed boundary value problems for an elastic halfspace with a periodic nonhomogeneity

This paper examines certain axisymmetric contact, crack and inclusion problems related to a nonhomogeneous elastic medium where the two elastic parameters are periodic functions of the axial variable z. A general formulation of the equations governing the axisymmetric deformations of the medium is presented. It is shown that the mixed boundary value problems can be reduced to a set of two ordinary differential equations and a Fredholm integral equation of the second kind. The ordinary differential equations are solved in a numerical fashion, and these solutions are used to evaluate the kernel function of the Fredholm integral equation. The resulting integral equations governing the contact problem are also solved via a numerical technique to obtain the load-displacement relationship for a rigid circular indentor in smooth contact with a periodically non-homogeneous elastic halfspace region. The procedures are extended to examine the problems associated with a penny-shaped crack and a rigid disc inclusion embedded in such a non-homogeneous elastic medium.

THE SETTLEMENT OF A RIGID CIRCULAR FOUNDATION RESTING ON A HALF‐SPACE EXHIBITING A NEAR SURFACE ELASTIC NON‐HOMOGENEITY

The present paper examines the elastostatic problem pertaining to the axisymmetric loading of a rigid circular foundation resting on the surface of a non-homogeneous elastic half-space. The non-homogeneity corresponds to a depth variation in the linear elastic shear modulus according to the exponential form G(z)=G1+G2e-ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The mixed boundary value problem associated with the indentation of the half-space by the rigid circular foundation is reduced to a Fredholm integral equation which is solved via a numerical technique. The numerical results presented in the paper illustrate the influence of the near-surface elastic non-homogeneity on the settlement of the foundation.

THE SETTLEMENT OF A RIGID CIRCULAR FOUNDATION RESTING ON A HALF-SPACE EXHIBITING A NEAR SURFACE ELASTIC NON-HOMOGENEITY

The present paper examines the elastostatic problem pertaining to the axisymmetric loading of a rigid circular foundation resting on the surface of a non-homogeneous elastic half-space. The non-homogeneity corresponds to a depth variation in the linear elastic shear modulus according to the exponential form G(z)=G1+G2e-ζz. The equations of elasticity governing this type of non-homogeneity are solved by employing a Hankel transform technique. The mixed boundary value problem associated with the indentation of the half-space by the rigid circular foundation is reduced to a Fredholm integral equation which is solved via a numerical technique. The numerical results presented in the paper illustrate the influence of the near-surface elastic non-homogeneity on the settlement of the foundation.

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.

Stress concentration due to underground excavation in cross-anisotropic and nonhomogeneous elastic halfspace

Materials occurring in nature are seldom isotropic and even less frequently homogeneous. The displacements and stresses in surface loaded cross-anisotropic and nonhomogeneous elastic halfspaces are significantly different from those in an isotropic and homogeneous halfspace. A similar effect is expected when prestressed halfspaces of a similar description are to be pierced with openings of various shapes located at various depths. The object of this paper is to assess this effect quantitatively. Both deep and shallow underground openings of circular, long elliptical and tall elliptical shapes are analysed. The medium containing these openings is discretized by a combination of finite and infinite elements to simulate near-field and far-field, respectively. The stress concentration factors at selected locations are calculated to demonstrate the effect of cross-anisotropy and nonhomogeneity of the media. The variation of surface displacements due to shallow excavation is also plotted. The popular belief that the effect of elastic anisotropy is no larger than the variation in rock strength is re-examined in light of the numerical results of this study.

Contact problem of an orthotropic non-homogeneous elastic half space : Bakirtas, I Int J Engng SciV22, N4, 1984, P347–359

Contact problem of an orthotropic non-homogeneous elastic half space : Bakirtas, I Int J Engng SciV22, N4, 1984, P347–359

Contact problem of an orthotropic non-homogeneous elastic half space : Bakirtas, I Int J Engng SciV22, N4, 1984, P347–359

Contact problem of an orthotropic non-homogeneous elastic half space : Bakirtas, I Int J Engng SciV22, N4, 1984, P347–359