Isotropic Half-space Research Articles

Investigation on the transmission of Love waves due to an impulsive line source in a heterogeneous double porous rock structure

The region around the reservoir sometimes encounters the seismic response of ground. The phenomena of Love waves propagation also caused by seismic activity which can further significantly influenced by an impulsive line source present in the rock structures of ground. The current study elucidates the propagation behavior of Love waves induced by an impulsive line source within the anisotropic layered heterogeneous double porous rock structures commonly found in proximity to the reservoir areas. The expression of the dispersion relation for the Love waves propagation due to an impulsive line source has been derived by using the Fourier transformation and Green’s function technique. The above-mentioned rock composite structure is consisted of two different portions: upper transversely isotropic double porous (TIDP) layer medium and lower heterogeneous isotropic double porous half-space (IDP) medium. In the aforesaid half-space medium, both types of heterogeneities viz. linear and quadratic have been considered. The impacts of heterogeneity parameters associated with the lower heterogeneous IDP half-space, porosity parameter (corresponds to both upper TIDP layer and lower IDP half-space) and anisotropic parameter of TIDP layer on Love waves phase velocity have also been studied. Some special outlines have also been depicted graphically.

Evanescent Acoustic Waves

A theoretical study of “geometric” SP-evanescent (head) waves propagating in an isotropic homogeneous half-space or half-plane with a free boundary shows that these waves can satisfy the condition of absence of effort on the boundary plane if and only if the Lamé parameter λ is vanishingly small, which makes the existence of head waves of this type practically impossible. The analysis is based on the Helmholtz representation for the displacement field in combination with the decomposition of the stress and strain tensor into spherical and deviatoric parts. The obtained result about the non-existence of this type of evanescent waves can find application in theoretical geophysics in the study of seismic wave fields in the vicinity of earthquake epicenters, as well as in non-destructive acoustic diagnostic methods.

Reflection coefficients for non-welded interface between orthorhombic half-spaces under initial stress

SUMMARY Complex oil and gas reservoirs represented by orthorhombic shale usually have the characteristics of strong anisotropy, wide distribution of non-welded interfaces (i.e. particular boundaries caused by horizontal bedding or fracture development), and ubiquitous initial stress effects. The knowledge of their seismic response characteristics plays an essential role in hydrocarbon exploration in stressed shale reservoirs. However, seismic wave reflection in the orthorhombic media under initial stress remains unclear. To address this issue, we used the theories of acoustoelasticity and elastic anisotropy to derive a stress-dependent effective elastic stiffness tensor. At the same time, we obtain the wave velocity and polarization characteristics directly by solving the Christoffel equation. According to the linear slip theory, we further constructed the exact reflection and transmission coefficient equations for the non-welded interface between isotropic and orthorhombic half-spaces under the initial stress. The effects of the non-welded interface on seismic wave velocity and reflection and transmission coefficients were systematically analysed. Meanwhile, we characterize the law of P-wave reflection coefficient effected by elastic coefficients. Our equations and results potentially lay an equation foundation for orthorhombic reservoirs in high-stress fields and are essential in orthorhombic reservoirs in high-stress geophysical exploration.

Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

Контактная задача для трансверсально изотропного слоя с трением

It is investigated the spatial contact problem of asymmetric interaction of two punches on a transversally isotropic layer with friction forces taken into account in an unknown contact domain. The planes of isotropy are parallel to the layer faces. The friction forces are taken into account along one coordinate axis. The lower layer face is subjected to sliding support. The layer material is characterized by five independent elastic parameters. The problem is reduced to an integral equation with respect to the contact pressure with the help of a double Fourier transformation together with the Coulomb law. The anisotropy degree is characterized be three dimensionless parameters arising in the kernel of the integral equation, two of which satisfy characteristic equation. In particular cases, the integral equation coincides with those well-known for the corresponding contact problems with friction for the isotropic layer and half-space. The B.A. Galanov method is used for numerical solutions. A system of the integral equation and integral inequality is considered. A rectangle is taken which a priori contains the unknown contact domain. By introducing special nonlinear operators, the system is reduced to only one nonlinear equation of the Hammerstein type which can be solved by the successive approximations method. The contact domain is determined by nodes at which the function required is positive. The punches are taken in the form of asymmetric elliptic paraboloids. The method allows us to investigate percolation, i.e. the process of junction of the discrete contact domains due to increasing the forces applied and settlements of the punches. Calculations are made for different friction coefficients, materials and relative thicknesses of the transversally isotropic layer.

THEPORE: A software package for modeling THErmo-PORo-elastic displacements

THEPORE (THErmo-POro-Elastic solutions) is an open source software to perform forward and inverse modeling of ground displacements induced by thermo-poro-elastic sources. The software, implemented in MATLAB, offers a library of analytical and semi-analytical solutions to compute ground displacements induced by thermo-poro-elastic deformation sources of different geometries, embedded in an elastic, homogeneous and isotropic half-space. The solutions have been verified against finite-element simulations. THEPORE includes also an inversion procedure of the deformation data to constrain the source parameters that better fit the observed signals.The software's functionality is showcased by inverting the GPS deformation data recorded on Vulcano Island at the onset of the 2021 unrest, in order to estimate the position and volume change of the source responsible for the observed deformations. The results encourage to consider THEPORE as a practical tool suitable for a fast preliminary estimation of the deformation source during a volcanic crisis.

On the existence of Rayleigh waves with full impedance boundary condition

The existence of Rayleigh waves (propagating in isotropic elastic half-spaces) with the tangential and normal impedance boundary conditions was investigated. It has been shown that for the tangential impedance boundary condition (TIBC), there always exists a unique Rayleigh wave, while for the normal impedance boundary condition (NIBC), there exists a domain (of impedance and material parameters) in which exactly one Rayleigh wave is possible and outside this domain a Rayleigh wave is impossible. In this paper, we consider the existence of Rayleigh waves with the full impedance boundary condition (FIBC) that contains both TIBC and NIBC. It is shown that the existence picture of Rayleigh waves for this general case is more complicated. It contains domain for which exactly one Rayleigh wave exists, domain where a Rayleigh wave is impossible, and domain for which all three possibilities may occur: two Rayleigh waves exist, one Rayleigh wave exists, and no Rayleigh wave exists at all. The obtained existence results recover the existence results established previously for the cases of TIBC and NIBC. The formulas for the Rayleigh wave velocity are derived. As these formulas are totally explicit, they are very useful in various practical applications, especially in the non-destructive evaluation of the mechanical properties of structures. In order to establish the existence results and derive formulas for the Rayleigh wave velocity, the complex function method, which is based on the Cauchy-type integrals, is employed.

Theoretical study of a tunable optical device on the base of magnetoactive cholesteric liquid crystals

We investigated the light reflection from half-space of cholesteric liquid crystal (CLC) being in external static magnetic field directed along the helix axis at normal light incidence. In this case there appear three photonic band gaps (PBGs) and these bands can be tuned in large intervals. We found the eigenpolarizations (EPs) at reflection of light from half-space and investigated their peculiarities. We have shown that in the absence both of absorption and of external magnetic field and at minimal influence of dielectric boundary (when the dielectric permittivity of isotropic half-space which borders with CLC half-space is equal to the CLC's mean dielectric permittivity) these polarizations are orthogonal. While, in the presence of external magnetic field they become quasi-orthogonal everywhere outside the main PBG (which exist also at the absence of external magnetic field), and they are significantly non-orthogonal within the main PBG. At some values of helix pitch they become significantly non-orthogonal practically everywhere. Moreover, in this case there exist some special points where the EPs degenerate. This system can apply as tunable narrow-band or broad-band filter and mirror, a highly tunable broad/narrow-band optical diode, and other devices.

Reflection and transmission of plane waves at interface between two transversely isotropic microstretch half-spaces

Reflection and transmission of plane waves at interface between two transversely isotropic microstretch half-spaces

Realizability of the Mode of Temperature Control of the Boundary of an Isotropic Half-Space with a Film Coating

Realizability of the Mode of Temperature Control of the Boundary of an Isotropic Half-Space with a Film Coating

Semi-analytical method for calculating ground vibrations from a tunnel in a homogeneous half-space with an irregular surface

This paper proposes a novel two-dimensional (2D) semi-analytical method for calculating ground vibrations from a tunnel situated in a homogeneous half-space with an irregular surface. The circular tunnel is conceptualized as an elastic solid, while the soil is modelled as an elastic, isotropic, and homogeneous half-space with an irregular surface. A virtual horizontal interface is introduced to divide the soil domain into an irregular region with an arbitrary-shaped surface and a half-space with a circular hollow. The wavefield scattered by the irregular surface is simulated by the boundary integral equation. Through the application of the discrete wavenumber approach and expansion of wavefield on the irregular surface, the boundary integral equation is transformed into a set of simultaneous matrix equations containing unknown expansion coefficients. By utilizing the transformation between cylindrical and plane waves, the boundary conditions on soil-tunnel interface are satisfied, yielding a solution for a harmonic point load acting on the tunnel in a half-space with an irregular surface. The proposed method can simulate topographies with arbitrary shapes and only requires the discretization of the irregular part of the ground surface, which provides a highly efficient tool to investigate the propagation characteristics of train-induced vibrations in complex topographies. The proposed method is verified through comparisons with the existing analytical methods and the finite element method. The dynamic responses of a tunnel embedded in a half-space with a Gaussian-shaped surface are meticulously investigated. A case study involving a site for the Shenzhen metro line 5 is performed. The numerical results demonstrate that the presence of irregular topography alters the distribution of ground vibrations, leading to significant differences in responses within the irregular region and certain disparities in the regular region. The effect of the location, size, and undulating direction of the irregular topography on ground vibrations is frequency-dependent.

Conical indentation over a transversely isotropic and layered elastic half-space

We propose a novel method for solving the static response of a conical indenter on a transversely isotropic and layered elastic half-space. The newly developed Fourier-Bessel series (FBS) system of vector functions, along with the unconditionally stable dual-variable and position method, is employed to derive the Green's function in the transversely isotropic and layered elastic half-space under a vertical ring load on the surface. To calculate the response at different field points on the surface, we apply discrete love numbers within the FBS vector system. The load densities in the discretized rings within the contact radius of the conical indenter are determined using the integral least-square method, along with a self-adaptive algorithm developed in this study. Finally, the relationship between the indentation depth (vertical displacement) and the applied load is obtained through force balance between the external load and the summed contact traction. The developed scheme is validated using existing exact solutions for the reduced homogeneous half-space case. Selected numerical results clearly demonstrate the effect of anisotropic material and layering on the indentation response. It is observed that, regardless of whether the structure is a stratified half-space or a layered structure with a rigid substrate, the material properties in the top layer have the most significant influence on the indentation behavior. In the case of a layered structure with an underlying elastic half-space, the material properties in the interlayer and bottom layer could also affect the indentation behaviors.

A 2D equivalent linear inversion model of bedrock motions in a layered transversely isotropic half-space

Inversion is the process that evaluates input motion on the bedrock from surface motions, primarily for use as input excitation for site seismic response or soil-structure interaction analyses. The paper presents a two-dimensional (2D) equivalent linear inversion model for bedrock motion in a multi-layered transversely isotropic (TI) half-space. Based on the exact dynamic stiffness matrices of TI soil layer and bedrock half-space, an inversion objective function is established from both horizontal and vertical surface acceleration records, and the optimization method is adopted to update the control parameters for the inversion process. Moreover, an equivalent linear method is used to account for the TI soil nonlinearity, thus realizing the nonlinear inversion for bedrock motion and incident angle. The inversion procedure is verified by using a single-layered TI soil profile model with three different materials. The numerical results demonstrate that the bedrock motion and the incident angle in the TI medium can be accurately inversed according to the surface motions and the TI site characteristics, indicating that the inversion method can be used for reliable engineering applications in the TI medium.

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.

The Propagation of Rayleigh-Type Waves in Nonlocal Inhomogeneous Transversely Isotropic Half-Space

This paper deals with propagation of Rayleigh-type waves in nonlocal inhomogeneous transversely isotropic half-space. From the characteristic equation of wave for the nonlocal inhomogeneous transversely isotropic medium, the existence of the number of surface waves depends on the inhomogeneity of the medium through the number of solutions satisfying the damping condition of the characteristic equation discussed. It has also been concluded that there exist cut-off frequency and escape frequency for the wave propagating in size-dependent materials based on the nonlocal theory. Dispersion equation for the propagation of Rayleigh-type surface waves at the free surface has been derived. Based on the obtained dispersion equation, the effects of the inhomogeneity of material and nonlocality parameter on the Rayleigh wave propagation are considered through some numerical examples.

Relaxed and Bonded Interfacial Conditions for the Axial Interaction of a Flexible Plate with a Transversely Isotropic Half-Space

Relaxed and Bonded Interfacial Conditions for the Axial Interaction of a Flexible Plate with a Transversely Isotropic Half-Space

Analytical analysis of the interaction between cavities and a crack in bonded isotropic and anisotropic half spaces under SH wave

Interaction between two cavities and a linear crack subjected to elastic shear horizontally (SH) wave is investigated analytically in this work. Two cavities are buried in two bounded half spaces respectively and the cross-section of each cavity has a different shape. The media of two half spaces are isotropic and anisotropic respectively, and the complex function method is applied to first solve the wave equation in each medium. Then, with the help of image principle, Green’s functions and wave field expressions in each separated half space are constructed. Subsequently, utilizing “conjunction” technique, a series of Fredholm integral equations containing undetermined forces at the interface are obtained and reduced in order to be solved. Dynamic stress distribution is mainly considered to discuss the interaction between the cavities and the crack. Finally, a parametric study on dynamic stress concentration factor (DSCF) and dynamic stress intensity factor (DSIF) presents the interaction is affected by several dimensionless factors simultaneously.

Development of governing partial differential equations of reinforcing thin films

In this paper, a novel interface model for elastic isotropic thin films perfectly bonded to the surrounding elastic media is introduced. Considering an average stress and displacement through the thin film thickness, the shear stress discontinuity across the interface is expressed in terms of spatial and time derivatives of displacement components. The derived interfacial equations are presented in a general form. As such, they are applicable to asymmetric elastodynamic problems of anisotropic solids with thin film interfaces. Employing the introduced interface model and Hankel transform technique, static Green's functions of an elastic isotropic half-space reinforced by a buried thin film are obtained. Results of some special cases, including axisymmetric loading, inextensible membrane, unreinforced half-space, and surface-coated half-space, are also presented. The robustness and effectiveness of the proposed interface model are shown by two solved numerical examples, and some elastic responses are plotted to show the impact of thin film stiffness. According to the numerical results, modeling elastic thin films as inextensible membranes can significantly alter the elastic responses.

Vertical vibration of a rigid disc over a transversely isotropic and layered poroelastic half-space

A novel semi-analytical method is proposed for solving the vertical vibration of a rigid disc over a transversely isotropic and layered poroelastic half-space. The fundamental solutions in the layered poroelastic half-space under a vertical patch load on the surface are solved in terms of the newly developed Fourier-Bessel series (FBS) system of vector functions and the unconditionally stable dual-variable and position method. The expansion coefficients in FBS system are discrete, as such the computational efficiency is greatly improved and the accuracy is better than the commonly used integral-transform methods. By virtue of the superposition method, the Green’s function due to vertical ring load is derived in the transform domain. By making use of the integral least-square scheme, the densities of a series of ring loads discretized within the disc area are determined. Finally, the dynamic vertical compliance is derived via the balance between the external applied force and the induced total contact traction. After verifying the reliability of the developed solution, selected numerical examples are presented to investigate the influence of material properties and excitation frequency on the forced vibration of the rigid disc.

Rayleigh wave propagation in centrosymmetric materials with micro-stiffness, flexoelectric and micro-inertia effects

A theoretical investigation of Rayleigh waves propagation in polarized media has been carried out using a reformulated flexoelectric theory for isotropic dielectrics with micro-inertia effect. Within this non-classical theory, the internal energy density is the functional of the strain tensor, dilatation gradient, deviatoric part of stretch gradient and rotation gradient tensors, polarization vector, and polarization gradient. The obtained system of governing equations additionally contains three material length-scale parameters to account the micro-stiffness effect, one material constant to capture the micro-inertia effect, two flexoelectric constants to describe the flexoelectric effect and three length scale parameters related to the polarization gradient. To solve the coupled governing equations, the method of Lamé-type potentials for mechanical displacement and electric polarization vectors is used. The influences of various factors such as micro-stiffness, flexoelectricity, electric quadrupoles and micro-inertia effects on the phase velocity of the Rayleigh waves in a homogeneous isotropic half-space are studied. It is found that above effects become significant with the increase of the wavenumber. This study can be important for the investigation of high frequency surface acoustic waves in dielectric materials.