Static deformation due to a long buried dip-slip fault in an isotropic half-space welded with an orthotropic half-space

Closed-form analytical expressions for the displacements and the stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space due to a dip-slip fault of finite width located at an arbitrary distance from the interface in the isotropic half-space are obtained. The Airy stress function approach is used to obtain the expressions for the stresses and the displacements. The case of a vertical dip-slip fault is considered in detail. The variations of the displacements with the distance from the fault and with depth have been shown graphically.     

An approximate secular equation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer

In this paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half-space is assumed to be welded. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate secular equation of fourth order in terms of the dimensionless thickness of the layer is derived. It is shown that this approximate secular equation has high accuracy. From the secular equation obtained, an approximate formula of third order for the velocity of Rayleigh waves is established.     

Normal point force and point electric charge in a piezoelectric transversely isotropic functionally graded half-space

Normal point force and point electric charge acting on surface of a transversely isotropic piezoelectric half-space with a functionally graded transversely isotropic piezoelectric coating are considered. Elastic moduli, piezoelectric constants and dielectric permeabilities of the coating vary with depth according to arbitrary functions. Analytical expressions for the elastic displacements and potential of the electrostatic field are derived for a fixed value of the depth coordinate. Asymptotical analysis of these formulas is derived for small and big values of a radial coordinate. An equilibrium of the half-space under the action of axisymmetric mechanical (normal and tangential) and electric loading is studied and a scheme of reducing the solutions of mixed boundary value problems to integral equations is obtained. As an illustration of the obtained solution, the PZT-5H piezoceramics with typical examples of functionally graded and homogeneous coatings are considered. The results include computations of the profiles of displacements and electric potential for different types of variation of electro-elastic properties in the coating.     

BEM–FEM coupling model for the dynamic analysis of piles and pile groups

This paper shows a BEM–FEM coupling model for the time harmonic dynamic analysis of piles and pile groups embedded in an elastic half-space. Piles are modelled using finite elements (FEM) as a beam according to the Bernoulli hypothesis, while the soil is modelled using boundary elements (BEM) as a continuum, semi-infinite, isotropic, homogeneous or zoned homogeneous, linear, viscoelastic medium. It is assumed that the soil continuity is not altered by the presence of the piles, and the tractions at the pile–soil interface are considered as a load applied within the half-space. The formulation is exposed in detail. In order to validate the model, selected numerical results of time harmonic impedances of different pile groups configurations are evaluated and contrasted with other reference values taken from the literature.     

A remark on the solution of the integral equation of planar cracks in three-dimensional elasticity

It is proposed that the classical integral equation of a planar crack under normal loading in three-dimensional isotropic elasticity is solved after an integration with respect to the Laplace operator. Then this equation coincides with the fundamental equation of contact problems for an isotropic elastic half-space. The problem of an elliptical crack under a constant normal loading is solved by the present approach.     

Tangential contact problem for a transversely isotropic elastic layer bonded to an elastic foundation

A system consisting of an elastic layer made of a transversely isotropic material bonded to an elastic half-space made of a different transversely isotropic material is considered. An arbitrary tangential displacement is prescribed over a domain S of the layer, while the rest of the layer’s surface is stress-free. The tangential contact problem consists of finding a complete field of stresses and displacements in this system. The generalized-images method developed by the author is used to get an elementary solution to the problem. It is also shown that an integral transform can be interpreted as a sum of generalized images. The case of a circular domain of contact is considered in detail. The results are valid for the case of isotropy as well.     

Spherical indentation of a transversely isotropic elastic half-space reinforced with a thin layer

The frictionless spherical indentation test is considered for a transversely isotropic elastic half-space reinforced with a thin layer whose flexural stiffness is negligible compared to its tensile stiffness. It is assumed that the deformation of the reinforcing layer can be treated as the generalized plane stress state. Closed-form analytical approximate equations for the maximum contact pressure, contact radius, and contact force are presented. The isotropic case is considered in detail.     

Slow vertical motions of an elliptic punch on an elastic half-space

The dynamic contact problem for a homogeneous, isotropic elastic half-space and a punch of an arbitrary base shape is studied. During the motion of the punch it is assumed that the contact area is fixed and there is no friction under the punch. An approximate solution to the problem is obtained under the assumption that the contact pressure under the punch is slightly varied during the time of travel of the Rayleigh wave along the distance equal to the diameter of the contact area. The solution of the problem of slow motions of a punch on an elastic half-space is reduced to the solution of the recurrent system of integral equations of the static contact problem. Asymptotic models of vertical motions of a flat-ended punch with an arbitrary base are constructed. The case of an elliptic punch is considered in detail.     

Elastoplastic contact/impact of rigidly supported composites

A general elastoplastic contact/impact model of a spherical object and a supported transversely isotropic composite layer or a half-space is presented. The main feature of the model is a contact law that is developed based on elastic–plastic and fully plastic indentation theory. For an impact event, the model parameters can easily be obtained analytically, computationally using finite elements (FE), and from experiments. The model results are compared to those from a nonlinear FE model developed in ABAQUS, and with limited experimental data showing excellent agreement. The model has also been tested for low-velocity impact, which compared very well with the FE results using ABAQUS/Explicit. The model is simple, yet it successfully captures a fairly complicated contact behavior.     

Numerical modeling of the V(z) curve for a thin-layer/substrate configuration

A numerical model is presented to calculate V(z) curves for a line-focus acoustic microscope and the specimen configuration of a thin isotropic elastic layer deposited on an isotropic elastic substrate. In this model, a Gaussian beam which is tracked through the lens into the coupling fluid, interacts with the thin-layer/substrate system. The numerical approach is based on the solution of singular integral equations by the boundary element method. The system of singular integral equations follows from the conditions at the interface of the coupling fluid and the thin layer and the interface of the thin layer and the substrate. An electrochemical reciprocity relation is used to express the voltage at the terminals of the microscope's transducer in terms of the calculated incident and back-scattered fields. V(z) curves are presented for various layer thicknesses and various combinations of the elastic constants of the layer and the substrate. The oscillations of the V(z) curves are related to the modes of wave propagation in a thin layer in contact with a solid half-space on one side and a fluid half-space on the other side. Calculated V(z) curves have also been compared with experimentally obtained curves, and good agreement is observed.     

Axisymmetric time-harmonic response of a transversely isotropic substrate–coating system

By virtue of a method of displacement potentials, an analytical treatment of the response of a transversely isotropic substrate–coating system subjected to axisymmetric time-harmonic excitations is presented. In determination of the corresponding elastic fields, infinite line integrals with singular complex kernels are encountered. Branch points, cuts, and poles along the path of integration are accounted for exactly, and the physical phenomena pertinent to wave propagation in the medium are also highlighted. For evaluation of the integrals at the singular points, an accurate analytical residual theory is presented. Comparisons with the existing numerical solutions for a two-layered transversely isotropic medium under static surface load, and a transversely isotropic half-space subjected to buried time-harmonic load are made to confirm the accuracy of the present solutions. Selected numerical results for displacements and stresses are presented to portray the dependence of the response of the substrate–coating system on the frequency of excitation and the role of coating layer.     

On propagation of shear waves in a multilayer medium including a fluid-saturated porous stratum

Summary The propagation of SH-type waves in a double surface layered medium with an intermediate transverse-isotropic fluid-saturated porous stratum has been examined. The top-most layer is assumed to be impermeable, elastic, homogeneous and isotropic, whereas the lower half-space is a nonhomogeneous one. The dispersion equation has been derived using the Biot's theory for the porous layer and the theory of elasticity for both the upper and the subjacent half-space.The corresponding dispersion equation for the simplified case where the dissipation caused by the relative fluid flow has been omitted has also been obtained. The numerical results obtained for this simplified case have been shown graphically.With 5 Figures     

Finite element method for orthotropic micropolar elasticity

The total potential energy for a body composed of an anisotropic micropolar linear elastic material is developed and used to formulate a displacement type finite element method of analysis. As an example of this formulation triangular plane stress (and plane couple stress) elements are used to analyze several problems. The program is verified by computing the stress concentration around a hole in an isotropic micropolar material for which an exact analytical solution exists. Several anisotropic material cases are presented which demonstrate the dependence of the stress concentration factor on the micropolar material parameters.     

Multiple 3D inhomogeneous inclusions in a half space under contact loading

A semi-analytic solution is given for multiple three-dimensional inhomogeneous inclusions of arbitrary shape in an isotropic half space under contact loading. The solution takes into account interactions between all the inhomogeneous inclusions as well as the interaction between the inhomogeneous inclusions and the loading indenter. In formulating the governing equations for the inhomogeneous inclusion problem, the inhomogeneous inclusions are treated as homogenous inclusions with initial eigenstrains plus unknown equivalent eigenstrains, according to Eshelby’s equivalent inclusion method. Such a treatment converts the original contact problem concerning an inhomogeneous half space into a homogeneous half-space contact problem, for which governing equations with unknown contact load distribution can be conveniently formulated. All the governing equations are solved iteratively using the Conjugate Gradient Method. The iterative process is performed until the convergence of the half-space surface displacements, which are the sum of the displacements due to the contact load and the inhomogeneous inclusions, is achieved. Finally, the obtained solution is applied to two example cases: a single inhomogeneity in a half space subjected to indentation and a stringer of inhomogeneities in an indented half-space. The validation of the solution is done by modeling a layer of film as an inhomogeneity and comparing the present solution with the analytic solution for elastic indentation of thin films. This general solution is expected to have wide applications in addressing engineering problems concerning inelastic deformation and material dissimilarity as well as contact loading.     

Tangential contact problems for several transversely isotropic elastic layers bonded to an elastic foundation

We consider a system consisting of n elastic layers made of different transversely isotropic materials bonded to each other and the last layer bonded to an elastic half-space made of a different transversely isotropic material. An arbitrary tangential displacement is prescribed over a domain S of the first layer, while the rest of the layer’s surface is stress free. The tangential contact problem consists of finding the complete stress and displacement fields in this system. The Generalized Images method developed by the author is used to get an elementary solution to the problem. We first consider the case of two layers and then generalize it for the case of n layers. The same problem is solved by the integral transform method, and it is shown that an integral transform can be interpreted as a sum of generalized images. The results are valid for the case of isotropy as well.     

Axisymmetrically loaded thin circular plate in adhesive contact with an elastic half-space

The interaction problem of adhesive contact between an axisymmetrically loaded thin circular plate and an isotropic or transversely isotropic elastic half-space is reduced to a problem of solving a pair of coupled integral equations for the unknown normal and shearing interfacial tractions. The system of integral equations is solved numerically and some results are presented.     

Stress concentration at irregular surfaces of anisotropic half-spaces

Summary An analytical method is presented to derive the stresses in anisotropic half-spaces with smooth and irregular surface morphologies. The half spaces can be subjected to body forces, surface tractions, and uniform far-field stresses. The general solution is expressed in terms of three analytical functions using the analytical function method of anisotropic elasticity. These three functions are then determined using a numerical conformal mapping technique and an integral equation method. Numerical examples are presented for the stress concentration at irregular surfaces induced by a uniform far-field horizontal stress. The elastic half-spaces are assumed to be transversely isotropic or isotropic, and their surface morphologies are constructed by the superposition of multiple long and symmetric ridges (mounds) and valleys (depressions). For isotropic media, the stress concentration depends only on the half-space surface geometry. It is found here that for anisotropic media, the half-space surface geometry, as well as the orientation of the planes of material anisotropy, have a great effect on the stress concentration. The degree of material anisotropy, on the other hand, has little influence on the stress concentration.     

Interface waves along an anisotropic imperfect interface between anisotropic solids

First and second order asymptotic boundary conditions are introduced to model a thin anisotropic layer between two generally anisotropic solids. Such boundary conditions can be used to describe wave interaction with a solid-solid imperfect anisotropic interface. The wave solutions for the second order boundary conditions satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. They couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. Interface imperfections are modeled by an interfacial multiphase orthotropic layer with effective elastic properties. This model determines the transfer matrix which includes interfacial stiffness and inertial and coupling terms. The present results are a generalization of previous work valid for either an isotropic viscoelastic layer or an orthotropic layer with a plane of symmetry coinciding with the wave incident plane. The problem of localization of interface waves is considered. It is shown that the conditions for the existence of such interface waves are less restrictive than those for Stoneley waves. The results are illustrated by calculation of the interface wave velocity as a function of normalized layer thickness and angle of propagation. The applicability of the asymptotic boundary conditions is analyzed by comparison with an exact solution for an interfacial anisotropic layer. It is shown that the asymptotic boundary conditions are applicable not only for small thickness-to-wavelength ratios, but for much broader frequency ranges than one might expect. The existence of symmetric and SH-type interface waves is also discussed.     

A potential problem arising from the strip-punch problem in elasticity

Three-dimensional contact problems in the classical theory of linear elasticity can often be regarded as mixed boundary-value problems of potential theory. In this paper we examinethe problem where contact between the indenting object (called a punch) and the elastic medium is maintained over an infinite strip. It is assumed that a rigid frictionless punch with a known profile has indented a homogeneous,isotropic and linearly elastic half-space. Applying the theory of Mathieu functions, an analytic solution of Laplace's equation is obtained through separation of variables in the elliptic cylinder coordinate system. Finally three examples are discussed where in each case the normal component of stress under the punch is numerically evaluated.     

Dilatation of a region in a two-phase elastic space

If two isotropic elastic half-spaces are in contact, with either smooth interface or perfect bond, and are deformed by a distribution of centres of dilatation over a volume V in one half-space, a solution, already known, is possible in terms of integrals over V. This note gives the derivation of simple relations between the field in the joined half-spaces and the field in a homogeneous space containing the same distribution.