Isotropic Elastic Half-space Research Articles

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.

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.

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.

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.

Improved modeling of anisotropic effects on seismic waves in layered transversely isotropic half-spaces: Implications for velocity profile inversion challenges

In this paper, we first derive the dynamic solution in a transversely isotropic (TI) elastic and layered half-space, induced by a time-harmonic vertical load on its surface. The solution is obtained via the efficient and unconditionally stable dual variable and position (DVP) matrix method along with the fast Fourier-Bessel series (FBS) expansion approach. In order to study the effect of material anisotropy, the degree of anisotropy, namely the ratio of the horizontal Young's modulus over vertical Young's modulus, is introduced as the key parameter, with a very reasonable assumption on other involved elastic constants in the layered TI half-space. This enables us to investigate the effect of material anisotropy on the wave Green's functions, dispersion curves, ellipticity, and polarity. By comparing with the effect of Poisson's ratio in the corresponding layered isotropic (ISO) half-space, we observe that nearly all the wave features with varying degrees of anisotropy in the TI layered half-space are similar to those with varying Poisson's ratio in the corresponding ISO half-space. This indicates the challenge or non-uniqueness in the inversion of TI velocity profiles using vertical surface loading only.

Numerical Analysis of Unsteady Vibrations of a Plate Resting on an Elastic Isotropic Half-Space

The paper is devoted to the numerical solution of the problem of vibrations of an infinite elastic plate resting on an elastic isotropic half-space using the analytical method based on the ray method with its numerical realization via the Maplesoft package. Unsteady oscillations are caused by the action of instantaneous loads on the plate, resulting in the appearance of two plane wave surfaces of strong discontinuity in the elastic half-space, behind the fronts of which, up to the contact boundary, the solution is constructed using ray series. The unknown functions entering the coefficients of the ray series and the equation of plate motion are determined from the boundary conditions of the contact interaction between the plate and the half-space. Previously, the approximate solution of this problem was obtained analytically without using mathematical packages, and the dynamic deflection of the plate involving only the first three terms of the ray series was written down. In this work, a two-layer medium with different properties was investigated using an algorithm developed to solve contact dynamic problems related to the occurrence and propagation of strong and weak discontinuity surfaces.

A framework for computing directivities for ultrasonic sources in generally anisotropic, multi-layered media

The knowledge of the directivity of ultrasonic sources is an important element in the design of non-destructive evaluation inspection. The commonly used directivity model of piezoelectric devices is the superposition of out-of-plane force monopoles on the stress-free surface of an isotropic elastic half-space. However, this does not cover a growing range of ultrasonic generation scenarios, as new ultrasonic technologies such as laser ultrasound (LU) emerges, and the need for inspection of materials such as carbon fibre reinforced polymers (CFRPs) rises. A directivity calculation framework based on the reciprocity theorem is developed in the current paper, which accommodates various combinations of source types, material properties, and boundary conditions, for example, a localised heating under a surface layer over an anisotropic half-space. The calculation is applicable to materials with weak anisotropy or wave modes that do not have cuspoidal wavefronts, and was validated with finite element models on key representative cases. The analytical solution for LU excitation in a CFRP sample also shows a good fit to the experimentally measured directivity of the quasi-longitudinal wave mode.

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.

Transient scattering of a Rayleigh wave by a cluster of subwavelength resonators—Towards asymptotic modeling of seismic surface metabarriers

A seismic metabarrier (intended for surface waves mitigation effect) is modeled as a cluster of single-degree-of-freedom resonator units deposited on the surface of an isotropic homogeneous elastic half-space. It is assumed that each resonator has a frictionless flat rigid base of diameter much smaller than the wavelength of an incoming surface wave. The slow-motions asymptotic method is applied for constructing the first-order asymptotic model of multiple time-dependent scattering of a pulsed Rayleigh wave with respect to the vertical displacements of the resonators (including their rigid bases and inertial elements) and the normal contact forces (time-dependent integral characteristics of the contact reactions beneath the resonator bases). Both the stationary and transient multiple scattering scenarios are considered. The variation of the amplitude reduction factor due to the model parameters variation is studied in detail.

Evaluation of the Fresnel approximation for the longitudinal particle displacement in shear wave beams

The longitudinal particle displacement in shear wave beams generated with disks that vibrate longitudinally at several hundred hertz is used to assess stiffness of soft tissue in vivo. Nonlinear elasticity of tissue, although less commonly used than linear stiffness, may be a more sensitive or specific biomarker of disease. A mathematical model for linear propagation of both transverse and longitudinal motion in a shear wave beam in an isotropic elastic half-space, and comparisons with measurements in tissue-mimicking phantoms, was published earlier this year [Archer et al., JASA 153, 1591 (2023)]. When including elastic nonlinearity in a propagation model, the complexity is reduced substantially by deriving an evolution equation based on the Fresnel approximation [Zabolotskaya, Sov. Phys. Acoust. 32, 296 (1986)]. For application to elastography, the parameter space in which the Fresnel approximation provides an accurate description of the longitudinal particle displacement must be established. In this presentation, the accuracy of the Fresnel approximation is assessed for linear shear wave beams, both unfocused and focused, by comparison with the existing mathematical model. Attention is devoted primarily to the longitudinal particle displacement in nearly incompressible media. [PGK is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]

Optimization of a punch shape with a doubly connected contact domain

The objective is to optimize the distribution of the normal pressure under a rigid punch (indenter) having a doubly connected contact domain close to a circular ring and interacting with a homogeneous isotropic elastic half-space. In the problem considered here, the required design variable is the punch shape. The functional to be minimized is the root-mean-square deviation of the present pressure distribution from some given distribution. An analytical technique is developed for solving the problem for the punches with doubly connected shape, by reducing to a sequence of similar problems for the circular ring punches using expansions of the simple layer potential. The method of expansion in terms of a small parameter is used. The simple layer potential expansion is proposed when mapping a doubly connected integration domain onto a circular ring by transforming the integration variables and transforming the coordinates of the pole of the kernel. As a result, a set of similar problems was obtained for a circular ring to determine the functions characterizing the normal pressure distribution over a non-circular ring contact domain under the punch, as well as the normal displacements, from where the optimal punch shape is determined.

Self-consistent approximations for the tangential-displacement correction to the incremental indentation stiffness

An axisymmetric problem of unilateral frictionless contact between a paraboloidal or conical indenter and a transversely isotropic elastic half-space is considered in the refined formulation by accounting for the tangential (radial) displacements of the surface points of the elastic body. Using the idea of self-consistent approximation, a system of two coupled relations for the main contact variables (contact force, indenter displacement, and contact radius) is derived. The concept of the true contact radius is discussed, and it has been argued that the incremental stiffness relation should be expressed in its terms. Correction factors for the force–displacement relation and the incremental indentation stiffness (as a function of the true contact radius) are evaluated in explicit form.

Dynamic loading in a transversely isotropic and layered elastic half-space

By introducing the Cartesian system of vector functions, we investigate the responses of a layered transversely isotropic elastic half-space induced by a dynamic load. The load is vertically applied in the layered half-space. It can be either time-harmonic or horizontally moving with a given velocity. It is demonstrated that, in each layer, in terms of the Cartesian vector functions, we need only a 4 × 4 system of equations for the expansion coefficients of the displacement and traction components, instead of a 6 × 6 system as usually presented. Furthermore, utilizing the vector functions, both the time-harmonic and moving-load cases can be solved in the same way. For the layered case, the previously introduced dual-variable and position method is further used to propagate the solutions from one layer to the other, with unconditional stability. The fast Fourier transform (FFT) is applied to obtain accurate results in the space/time domain efficiently by the numerical inverse Fourier transform, properly developed within the framework of the discrete convolution-FFT (DC-FFT) algorithm. The unique formulation and its corresponding sophisticated algorithm are first validated against existing solutions, and then applied to calculate the displacements and stresses in a layered transversely isotropic half-space. Finally, a typical three-layer flexible pavement structure under a moving circular load is analyzed in detail. Numerical results clearly show the significant effects of both material anisotropy and layering. The present formulation is concise and efficient for the calculation of all the displacements and stresses, which can be applied to the involved engineering practice. The results can also serve as benchmarks for future research in this direction.

Rigid circular footing on the surface of a transversely isotropic linear elastic half-space

It is sometimes convenient to treat a relatively rigid footing as a single element subjected to up to six force resultants: vertical load, horizontal load in two directions, overturning moment about two axes and torsion. Applications include calculations of initial, elastic settlement; foundation responses under seismic loading or for machine vibrations; and fixity of an offshore jack-up foundation under cyclic wave loading. Present codes of practice provide formulae that assume fully isotropic and linear elastic soil and in some case assume no friction between the footing and soil. This paper uses recent theoretical advances to develop solutions for stiffnesses of rigid circular footings on transversely isotropic linear elastic soil, including with interface friction. Closed-form solutions are developed for vertical load and overturning moment on a frictionless interface and for vertical load and torsion on a frictional interface. A boundary-element analysis is presented for the cases of lateral load and overturning moment on a frictional interface. By fitting expected forms to the results, new formulae are proposed for stiffnesses for these cases. The effects of interface friction on limiting loads are calculated. Practical applications are outlined, and some limitations are briefly discussed.

Contact tensions under the sole of rigid deep laying foundations and ground anchors

Purpose. To solve the problem of the allocation of contact vertical normal tensions along the sole of a rigid round stamp, located in an elastic isotropic half-space at a certain depth h  0. To compare the obtained solution with the well-known classical result for h = 0, to check the obtained results for adequacy. Methodology. Based on the analysis of the decision on the stress-strain state of the base, inside which there is a vertical arbitrary load distributed over the area of the circle, the necessary formulas are obtained to solve the problem. An algorithm for constructing an approximate solution has been developed, the essence of which is to use a combination of the boundary element method and the iteration process by S.N.Klepikov. For a number of depths, approximate solutions of the considered problem are obtained. Findings. The proposed algorithm for the approximate solution of the problem of indenting a round rigid stamp into the upper boundary of an elastic isotropic half-space has good agreement with the exact solution and can be used to solve contact problems. The outlines of the contact stress diagrams depend on the depth at which they are determined – the greater the depth, the flatter the outlines of the diagrams are, while starting from a certain depth, the diagrams of contact stresses practically coincide. The greater the depth is at which the stamp is located, the more force must be applied to obtain equal displacements of the stamp. Originality. The obtained research results significantly expand the possibilities of solving various problems of soil mechanics and foundation engineering, make it possible to obtain absolutely new results. In particular, a clear dependence of the contact stresses along the sole of a rigid round stamp on the depth at which it is located was identified. In addition, the presented data allow us to designate an absolutely new direction in the calculation of the foundations of ground anchors, namely, the calculation of their deformations. Practical value. For engineering practice, it is important that the greater the value of Poisson’s ratio of the base is, the greater the contact stresses are, other things being equal.

Point loads on the surface of a transversely isotropic linear elastic half-space

General solutions for the displacements, strains and changes in stress in a transversely isotropic linear elastic (TILE) half-space subject to a point load were published in 1998. They represent a massive advance on the nineteenth-century solutions by Boussinesq, Cerruti and others, which were for fully isotropic linear elastic half-spaces. The 1998 solutions were for loads at a general depth below the surface and were criticised for being difficult. The present paper deduces more accessible expressions for the special case of loads that are on the surface. This allows one clarification and one potential error in the original equations to be identified and avoided. With support from existing data of geological materials, agreement is found with a previous claim that a limit proposed in 1970 on one of the shear moduli for a TILE material is invalid. Corrected equations derived herein can be the basis of calculating ground stresses and movements due to linear responses to distributed surface loads, behaviours of flexible foundations and stiffnesses for rigid surface footings.

Longitudinal motion of focused shear wave beams in soft elastic media.

Shear waves are employed in medical ultrasound imaging because they reveal variations in viscoelastic properties of soft tissue. Frequencies below 1 kHz are required due to the substantially higher attenuation and lower propagation speeds than for compressional waves. Shear waves exhibiting particle motion in the direction of propagation, referred to as longitudinal shear waves, can be generated with longitudinal motion of a circular disk on the surface of a soft elastic medium. This approach permits imaging of the longitudinal shear wave with a conventional ultrasound transducer that is coaxial with the source of the shear wave. Presented here is a mathematical model describing the complete wave field generated by displacement at the surface of an isotropic elastic half-space. Numerical simulations are shown for longitudinal, transverse, torsional, and radial source polarizations, with emphasis on focused longitudinal shear waves. Predictions are consistent with measurements of light beams revealing that the longitudinal electric field component produces a smaller focal spot than the transverse field component [Dorn, Quabis, and Leuchs, Phys. Rev. Lett. 91, 233901 (2003)]. Simulations are compared with preliminary measurements of a focused longitudinal shear wave beam generated in a soft tissue phantom by longitudinal motion of a spherically concave piston.

Analytical prediction of ground movements due to circular shaft construction in clayey soils

With the rapid expansion of urbanization, shafts are needed to provide access to tunnelling projects or be utilized for permanent works. Their construction causes ground movements, affecting nearby foundations and buried infrastructure. This study presents an analytical equivalent volume loss approach for predicting three-dimensional ground movements resulting from a circular shaft construction: namely, ground losses in the elastic isotropic half space as a sequence of cavity contractions around the shaft are employed. First, the applicability and accuracy of proposed solutions are evaluated with published case studies and centrifuge tests. Thereafter, the method is employed to investigate the influence of the circular shaft construction on the surface and subsurface soil displacements. Considering design ground losses, this method allows for quick estimations of ground deformations and associated risk of serviceability losses.

ЗАМЕЧАНИЕ О ВЫЧИСЛЕНИИ СКОРОСТИ ВОЛНЫ РЭЛЕЯ И ПРОИЗВОДНОЙ ОПРЕДЕЛИТЕЛЯ РЭЛЕЯ В УПРУГИХ СРЕДАХ

There are many approximate and exact formulae to calculate surface wave velocity in an elastic medium. An analytical expression for Rayleigh wave velocity in volume wave velocity values has been obtained. A formula which determines the remainder in the excitation and diffraction of surface acoustic waves in a homogeneous isotropic elastic half-space involving solutions for the strain and stress fields in the form of quadratures is worked out. The values of the Rayleigh wave velocity and the derivative of the Rayleigh determinant for different media according to the reference data were obtained. The results can help in obtaining analytic expressions and reducing the calculation time of numerical solutions of the diffraction and excitation of acoustic waves.

Modeling Locations with Enhanced Earth’s Crust Deformation during Earthquake Preparation near the Kamchatka Peninsula

In seismically active regions of the Earth, to which the Kamchatka peninsula refers, pre-seismic anomalies are recorded in different geophysical fields. One of such fields is the acoustic emission of rocks, the anomalies of which are recorded 1–3 days before earthquakes at the distance of the first hundreds of kilometers from their epicenters. Results of joint acoustic-deformation measurements showed that growth of geoacoustic radiation intensity occurs during the increase in the level of deformations in rock masses by more than one order compared to the background values. Simulation studies of the areas with increased deformation are realized to understand the causes of anomalous acoustic-deformation disturbance occurrences before strong earthquakes. The model is based on the assumption that the Earth’s crust in the first approximation can be considered as a homogeneous isotropic elastic half-space, and an earthquake source can be considered as a displacements along a rectangular fault plane. Based on these assumptions, deformation regions of Earth’s crust were modeled during the preparations of two earthquakes with local magnitudes ML≈5 occurred on the Kamchatka Peninsula in 2007 and 2009. The simulation results were compared for the first time with the data of a laser strainmeter-interferometer installed at the Karymshina observation site (52.83∘ N, 158.13∘ E). It was shown that, during the preparation of the both earthquakes, the Karymshina observation site was within the region of shear deformations ≈10−7, which exceeded the tidal ones by an order. On the whole, simulation results corresponded to the results of the natural observations. Construction of an adequate model for the generation of acoustic-deformation disturbances before strong earthquakes is topical for the development of an early notification system on the threat of catastrophic natural events.