Isotropic Elastic Layer Research Articles

Static deformation of a two-phase medium consisting of a rigid boundary elastic layer and an isotropic elastic half-space induced by a very long tensile fault

An analytical model is proposed to determine the static response of a multi-layered medium comprising of an underlying isotropic (ISO) elastic half-space that is in welded contact with an isotropic (ISO) elastic layer of uniform thickness H with a rigid boundary, induced by a tensile dislocation embedded in the layer. The integral expressions for stress fields are obtained by using the airy stress function. These integrals are calculated approximately, using Sneddon’s approach, by substituting the terms under the integral sign with a finite sum of exponential expressions. It is studied numerically and graphically how stresses vary with the change in the epicentral distance for different source locations embedded in the layer. The effect of the rigidity ratio is also analysed on the stress field. To examine the impact of the internal boundary, the stress fields in the layered and uniform half-spaces are compared graphically.

An approximate secular equation of Rayleigh-like waves in coated elastic half-space containing voids

Propagation of Rayleigh-like surface waves is studied in an isotropic elastic solid half-space coated with a thin isotropic elastic solid layer. The half-space and the thin coated layer are in welded contact with each other and contain a uniform distribution of small void pores. Effective boundary condition method is employed to derive an approximate secular equation of second-, third-, and fourth-orders in terms of dimensionless wavenumber. The corresponding secular equations are solved numerically to obtain the speed of propagating Rayleigh-like waves for a particular model. The computed results are presented graphically and compared with those obtained from exact secular equation. The fourth-order approximate secular equation is found to have high accuracy as it provides solutions that are in close vicinity of those obtained from the exact secular equation in the considered model. The presence of voids in the model is found to influence the speed of Rayleigh-like waves theoretically and verified numerically. By ignoring the presence of voids in the model, the secular equation is found to be in complete agreement to the earlier known results in the literature for the corresponding model.

Contact problems for a transversely isotropic layer

Two spatial, one axisymmetric and two plane contact problems are considered for a transversely isotropic elastic layer with one face subjected to sliding support. In the spatial and plane contact problems, the planes of isotropy may be either parallel or perpendicular to the layer faces. In the case of axial symmetry, the planes of isotropy are parallel to the layer faces. By using Fourier integral transforms, the contact problems are reduced to integral equations with respect to the contact pressure, the limiting cases of which are the well-known equations of the corresponding problems for an isotropic layer. For solving the spatial problems with unknown contact domains, the nonlinear boundary integral equations method is used, which make it possible to determine the contact pressure and the contact domain simultaneously. To extract the kernel principal part of the spatial problem integral equation when the isotropy planes are perpendicular to the layer faces, it is used the kernel of the integral equation of the corresponding contact problem for a transversely isotropic half-space obtained earlier without quadratures. The integral equation of the axially symmetric problem is reduced to a Fredholm integral equation of the second kind with the help of the method of pair equations, and the method of mechanical quadratures is used for numerical solutions. Plane problems are solved in a closed form based on special approximations of the kernel symbols. The approximations accuracy grows as anisotropy increases. Here, the anisotropy level can be characterized by the difference between ratio of a characteristic equation roots and unit because the unit value corresponds to the isotropic case. Mechanical characteristics as well as errors of the approximations are calculated for well-known transversely isotropic materials.

Tensile Fault Dislocation in an Irregular-Layered Elastic Half-Space

In the present paper, an analytical solution for the static deformation of a two dimensional model consisting of an infinite homogeneous isotropic elastic layer of uniform thickness placed over an irregular isotropic elastic half-space due to movement of a long tensile fault has been obtained. The rectangular shaped irregularity is assumed to be present in the lower half-space and assuming that the fault lies in the elastic layer at a finite depth say ’<i>h</i>’ to the upper surface of the layer. For numerical computation, the expressions of displacements and stresses are calculated by using Sneddon’s method and the effect of source depth and irregularity on the displacements and stresses has been investigated graphically.

The Effect of Mining Remnants on Elastic Strain Energy Arising in the Tremor-Inducing Layer

A vast majority of hard coal deposits in Poland have a multi-seam structure, hence the presence of mining remnants left from previous operations. The impact of those remnants (exploitation edges or residual pillars) can further intensify geomechanical phenomena occurring in the rock mass, leading to changes in the original state of stress. This applies to all layers within the rock strata, including thick and coherent ones (referred to as tremor-inducing layers) where the impacts of mining remnants are likely to trigger tremors, thus enhancing the rock bursts hazard. In the light of the geomechanical model of rock strata recalled in the study, it is assumed that homogeneous and isotropic elastic layers are found between the considered mining remnant (which is revealed as the stress distribution), and the rock medium modelled as a homogeneous and isotropic half-plane. Development of the state of stress in the bedded medium was brought down to the analysis of interacting elastic layers, where the biharmonic equation is satisfied for each layer and for each respective half-plane. This equation can be solved by the integral Fourrier transform method. The impacts of the exploitation edge and the residual pillar on the elastic strain energy in the tremor-inducing layer is illustrated by recalling the Burzyński’s stress criterion. Strain energy in the tremor-inducing layer was analysed for various deformation properties of the surrounding strata and for various methods of coal extraction from the seam underneath the tremor-inducing layer. The results of the study evidence that a change in deformation properties of strata in the vicinity of the tremor-inducing layer may affect the state of stress and strain energy, which impacts on the tremor hazard levels in the vicinity of mining remnants areas.

A MULTI-STAGE HOMOGENIZATION METHOD FOR DETERMINING THE EFFECTIVE ELASTIC PROPERTIES OF LAYERED MATERIALS

The paper presents a multi-stage homogenization method to determine the effective properties of layered materials which have n elastic isotropic layers or elastic transversely isotropic (in the direction of the layers) ones. The homogenization procedure uses (n-1) times an explicit analytical formula for the layered material with n layers. The normal stress and strain at the interface between two material layers are assumed to be continuous, i.e., no slip and no detachment. The results obtained from this method are compared with the existing analytical four-sub-matrix method and they show good agreements. The multi-stage homogenization method is a powerful tool that can be used to quickly identify the effective properties of the layered materials.

2D Green’s functions for an elastic layer on a rigid support loaded by an internal point force

The problem of a homogeneous isotropic elastic layer resting on a rigid base and loaded by an internal point force is analytically investigated under plane strain conditions. The displacement field is sought as the superposition of a fundamental solution (doublet state) and a homogeneous solution which allows satisfying the boundary conditions at the upper and lower boundaries of the layer. The displacement field is represented through convergent Fourier integral transforms. Once the closed form solution in the transformed domain is found, the displacement and stress fields are assessed by numerical inversion of transforms. Results concerning both the displacements and stresses for different positions of the load application point are reported and compared to FEM solution, finding very good agreement. The solutions obtained for horizontal and vertical point forces define the Green’s functions for the layer, which can be used to describe the mechanical interaction between the layer and an embedded body. As a striking application, the interaction between an elastic layer and an embedded laterally loaded wall or diaphragm is finally addressed, finding the contact pressure and, in turn, the internal forces in the diaphragm.

The influence of heterogeneity and initial stress on the propagation of Love-type wave in a transversely isotropic layer subjected to rotation.

In this paper, a mathematical model of Love-type wave propagation in a heterogeneous transversely isotropic elastic layer subjected to initial stress and rotation of the resting on a rigid foundation. Frequency equation of Love-type wave is obtained in closed form. The material constants and initial stress have been taken as space dependent and arbitrary functions of depth in the respective media. The dispersion equation is determined to study the effect of different types of parameters such as inhomogeneity, initial stress, rotation, wave number, the phase velocity on the Love-type wave propagation. The analytical solution has been obtained, we have used the separation of variables, method and the numerical solution using the bisection method implemented in MATLAB. We present a general dispersion relation to describe the impacts as the propagation of Love-type waves in the structures. Numerical results analyzing the dispersion equation are discussed and presented graphically. Moreover, the obtained dispersion relation is found in well agreement with the classical case in isotropic and transversely isotropic layer resting on a rigid foundation. Finally, some graphical presentations have been made to assess the effects of various parameters in the plane wave propagation in elastic media of different nature.

Dispersion of guided waves in stratified medium with a sandy layer

Dispersion analysis of acoustic guided waves, propagating in a stratified medium, containing an internal sandy layer in contact with the isotropic elastic substrate and the upper elastic isotropic layer, is developed by applying the modified Cauchy formalism coupled with the exponential fundamental matrices and the Thomson–Haskell transfer matrix method. The developed approach is suitable for analyzing dispersion properties of guided waves at the broad variation of the sandy parameter. Several phenomena, concerning anomalous dispersion of guided waves at varying sandy parameter, are observed, revealing the possibility of nondestructive evaluation of sandy layers within the upper crust.

Lamb waves in sandwich orthotropic elastic plates

In this paper, the propagation of Lamb waves in an orthotropic elastic sandwich plate is investigated. The surface layers and the core layer may be compressible or incompressible. The main aim of the paper is to derive explicit dispersion equations. First, the dispersion equations of symmetric modes and asymmetric modes are derived for compressible sandwich plates by using the effective boundary condition method along with the transfer matrices for orthotropic elastic layers. These dispersion equations recover the dispersion equations of Lamb waves propagating in compressible isotropic sandwich plates. The dispersion equations for incompressible sandwich plates are obtained by using the incompressible limit method. It is shown numerically that the thickness ratio and the stiff contrast affect strongly higher modes and the incompressibility has strong influence on symmetric modes. Remarkably, the obtained dispersion equations can be used to analyze the dispersion characteristic of Lamb waves propagating in sandwich plates with honeycomb core layers because the honeycomb core layers can be replaced effectively by transversely isotropic elastic layers by mean of homogenization.

Characterization and modelling of the sound reduction of hemp-clay walls in buildings

This paper focuses on the sound reduction of light earth at the wall scale. It aims at gaining knowledge and providing informations to professional building workers to help them in the acoustic optimization of biobased walls. Experimentations have been performed in controlled conditions using the intensimetry method, on 22 different samples to investigate thickness effect, wood frame effect, coating effect together with different light earth implementations (shuttering, spraying). These data were then confronted to two modelling approaches, the mass law estimate and a Transfer Matrix Method computation. It is shown that light earth performance can be satisfactorily predicted, whatever the wall configuration using the Transfer Matrix Method computation with isotropic elastic and porous layers. The study brings finally a good understanding of the physical phenomena occurring in biobased concretes, depending on the presence of coatings. For light earth, it also appears that mass law yields to a good approximation of the single-number rating for some configurations including at least one coating.

Comparison of ground deformation due to movement of a fault for different types of crack surface

An analytical solution of deformation of a homogeneous, isotropic elastic layer of uniform thickness overlying a homogeneous, isotropic viscoelastic half-space of Burger medium due to movement of an infinite fault for different types of crack surface has been studied. The deformation by this fault has been compared with the deformation by an infinite fault situated in viscoelastic half-space of Burger medium. The expressions of displacement, stress and strain have been obtained in aseismic period (period in between two major seismic events) by using Green’s function technique and Laplace transform by applying suitable boundary conditions and initial conditions. Finally these displacement, stress and strain components are numerically computed with suitable values of the model parameters and the results thus obtained are presented graphically. A detailed study of these expressions may give some idea about the amount of displacement due to different types of fault movement and the nature of stress-strain accumulation across the earthquake fault. A Comparison of these results has also been analysed and the effect of change of inclinations and velocities of the fault movement has been studied. Such theoretical models may be used for obtaining greater insight into the earthquake processes in seismically active region.

Plane Cracks in a Transversely Isotropic Layer

Problems of plane cracks of normal fracture (mathematical cuts) in the middle plane of a transversely isotropic elastic layer, the outer faces of which are under conditions of a sliding support, are considered. Isotropic planes are parallel or perpendicular to layer faces. Using the Fourier integral transform, the problems are reduced to integro-differential equations for crack opening, from which one can obtain the known equations of the corresponding problems for a transversely isotropic space and an isotropic layer by passing to the limit. A regular asymptotic method is applied for elliptical cracks; this method is effective for a relatively thick layer. It is shown the applicability domain of the method narrows with increasing anisotropy that is characterized by the roots of the characteristic equation (for an isotropic material, all roots are equal to unity). For strip-like cracks, closed solutions are obtained based on special approximations of the kernel symbols of integral equations, the relative errors of which decrease with increasing anisotropy. Calculations are made for known transversely isotropic materials.

Analysis of interfacial imperfections and electro-mechanical properties on elastic waves in porous piezo-composite bars

The main intention of this research work is to carry out an investigation regarding the mechanical and electrical characteristics of shear wave propagation in a porous piezoelectric material (PPEM) bounded by fiber-reinforced medium (FRM) and elastic half-space. The interface between two different media has been considered mechanically imperfect. The mechanical displacement and electric potential are obtained for respective medium by solving coupled electromechanical field equations. After that, the required expression of dispersion has been established using the relevant boundary conditions. For the purpose of validation, the obtained result is reduced to fairly match with the classical Love wave equation in an isotropic elastic layer and substrate (i.e., in the absence of upper surface, piezoelectricity, porosity, imperfection parameter) For the propose of numerical calculation, four sets of porous piezoelectric materials are considered namely, PZT-1, PZT-5H, PZT-7 and BaTiO3. Notable effects of various parameters involved in proposed structure such as piezoelectric, dielectric, reinforcement and interfacial imperfect on phase velocity with respect to frequency have been observed. In this regard, the numerical computation and graphical demonstration have been carried out. Moreover, first three different modes of SH-wave have also been reported.

Cracked elastic layer with surface elasticity under antiplane shear loading

A mode-III crack embedded in a homogeneous isotropic elastic layer of nanoscale finite thickness is studied in this article. The classical elasticity incorporating surface elasticity is employed to reduce a nonclassical mixed boundary value problem, where the layer interior obeys the traditional constitutive relation and the surfaces of the layer and the crack are dominated by the surface constitutive relation. Using the Fourier transform, we convert the problem to a hypersingular integro-differential equation for the out-of-plane displacement on the crack faces. By expanding the out-of-plane displacement as series of Chebyshev polynomials, the Galerkin method is invoked to reduce the singular integro-differential equation with Cauchy kernel to a set of algebraic linear equations for the unknown coefficients. An approximate solution is determined, and the influences of surface elasticity on the elastic field and stress intensity factor are examined and displayed graphically. It is shown that surface elasticity decreases the bulk stress and its intensity factor near the crack tips for positive surface shear modulus and gives rise to an opposite trend for a negative surface shear modulus.

Second-harmonic generation of two-dimensional elastic wave propagation in an infinite layered structure with nonlinear spring-type interfaces

The two-dimensional elastic wave propagation in an infinite layered structure with nonlinear interlayer interfaces is analyzed theoretically to investigate the second-harmonic generation due to interfacial nonlinearity. The structure consists of identical isotropic linear elastic layers that are bonded to each other by spring-type interfaces possessing identical linear normal and shear stiffnesses but different quadratic nonlinearity parameters. Explicit analytical expressions are derived for the second-harmonic amplitudes when a single monochromatic Bloch mode propagates in the structure in arbitrary directions by applying the transfer-matrix approach and the Bloch theorem to the governing equations linearized by a perturbation method. The second-harmonic generation by a single nonlinear interface and by multiple consecutive nonlinear interfaces are shown to be profoundly influenced by the band structure of the layered structure, the fundamental Bloch wave mode, and its propagation direction. In particular, the second harmonics generated at multiple consecutive interfaces are found to grow cumulatively with the propagation distance when the phase matching occurs between the Bloch modes at the fundamental and double frequencies.

Frequency-dependent AVO analysis using the scattering response of a layered reservoir

Sensitivity of reservoir properties to broadband seismic amplitudes can be weak, which makes interpretation ambiguous. Examples of challenging interpretation scenarios include distinguishing blocky reservoirs from fining sequences, low gas saturation from high gas saturation, and variable reservoir quality. Some of these rock and fluid changes might indicate stronger sensitivity to amplitudes over narrow frequency bands, which is a characteristic of frequency-dependent amplitude variation with offset (FAVO). We have developed a FAVO model for reservoir characterization, following a seismic scattering phenomenon through a set of isotropic elastic layers. The frequency dependency in our model comes from the time delays due to wave propagation within layers. The FAVO modeled response is a complex-valued amplitude varying with angle and frequency, and it is a function of the seismic velocities and thicknesses of individual layers, along with the conventional AVO response at all interfaces. Our FAVO seismic analysis consists of two main steps: (1) forward modeling using well logs to understand rock and fluid sensitivity to amplitudes to identify tuning frequencies with maximum amplitude excursions and (2) seismic analysis at tuning frequencies. With well-log models, we observed that the frequency-dependent tuning response is primarily dependent on the lithology stacking pattern of a reservoir; in the cases studied, the fluid and reservoir quality have secondary effects on the frequency dependence of the amplitudes. We evaluate synthetic models and field data from the Columbus Basin, Trinidad, to illustrate our frequency-dependent seismic analysis methods. For one of the sandstone reservoirs, a frequency-dependent attribute indicates better spatial resolution of the anomaly than a conventional amplitude extraction. FAVO attributes are complementary to conventional AVO attributes.

Green’s function technique to study the influence of heterogeneity on horizontally polarised shear-wave propagation due to a line source in composite layered structure

The present paper investigates Green’s function technique to study the impact of the non-linear heterogeneity on the propagation of the horizontally polarised shear (SH) wave in a composite layered structure due to the line source. The composite structure is comprised of two distinct isotropic homogeneous elastic layers of finite width lying over an isotropic heterogeneous semi-infinite elastic medium. A general quadratic heterogeneity (including the linear heterogeneity term) in the rigidity of the semi-infinite medium has been taken into account. An efficient analytical treatment involving Green’s function technique along with the application of Fourier transformation has been employed to establish the closed form of the dispersion equation for the propagating wave. As a special case of the problem, the closed form of the dispersion equation has been found in well agreement with the pre-established and classical results. The influence of linear and non-linear heterogeneities on the phase velocity of the horizontally polarised shear-wave has been analysed comparatively for numerous cases associated with the considered model and the corresponding single layer half-space model. It is reported that the phase velocity of the horizontally polarised shear-wave is more pronounced in the case of the corresponding single layer half-space structure as compared to the considered structure. Also, the impact of the general quadratic heterogeneity (supported by the linear heterogeneity) enhances the phase velocity most favourably as compared to the case of the simple quadratic heterogeneity, linear heterogeneity and homogeneity in the semi-infinite medium.

Excitation of Spatially Developed Lamb Waves by a System of Body and Surface Loads (Part 2)

The problem of excitation of spatially developed Lamb waves by a system of body and surface loads is formulated and completely solved by analytical methods. Using the direct and the inverse Hankel transforms, relations are derived for determining the amplitude factors of radially propagating non-axisymmetric Lamb waves excited by a system of body and surface loads in an arbitrary region of an isotropic elastic layer. Relations for calculating the displacement components of particles of the elastic layer in far field are obtained.

Рассеяние звуковых волн цилиндром с радиально-неоднородным упругим покрытием в плоском волноводе

В статье рассматривается задача рассеяния звуковых волн абсолютно жестким цилиндром с радиально-неоднородным изотропным упругим покрытием в плоском волноводе. Полагается, что волновод заполнен однородной идеальной жидкостью, одна его граница является абсолютно жесткой, а другая - акустически мягкой, законы неоднородности материала покрытия цилиндра описываются дифференцируемыми функциями, гармоническая звуковая волна возбуждается заданным распределением источников на сечении волновода. В случае установившихся колебаний распространение малых возмущений в идеальной жидкости описывается уравнением Гельмгольца. Колебания неоднородного изотропного упругого цилиндрического слоя описываются общими уравнениями движения сплошной среды. Для нахождения поля смещений в неоднородном покрытии построена краевая задача для системы обыкновенных дифференциальных уравнений второго порядка. Первичное поле возмущений представлено совокупностью собственных волн волновода. Давление рассеянного цилиндрическим телом поля ищется в виде потенциала простого слоя. Построена функция Грина для уравнения Гельмгольца, удовлетворяющая заданным граничным условиям на стенках волновода и условиям излучения на бесконечности. Функция плотности распределения источников ищется в виде разложения в ряд Фурье. Для нахождения коэффициентов этого разложения получена бесконечная линейная система уравнений. Проведено усечение бесконечной системы и ее решение найдено методом обратной матрицы. Получены аналитические выражения для рассеянного акустического поля в разных областях волновода.