Showing papers on "Half-space published in 2018"

An extended polygon scaled boundary finite element method for the nonlinear dynamic analysis of saturated soil

Abstract: In this paper, the polygon scaled boundary finite element method is extended to analyze saturated soil based on the generalized Biot's dynamic consolidation theory. The displacement shape functions of the polygon element are obtained by elastic static theory while the pore pressure shape functions are constructed from steady-state seepage theory. A scaled boundary polygon equations for saturated soil is established by applying Galerkin method. Two sets of Gauss points are adopted, including Gauss points of line utilized to compute the shape functions and Gauss points of area employed to realize material nonlinearity. In order to verify and assess the reliability and accuracy of the presented method, a saturated elastic half space subjected to a uniform cyclic dynamic loading is simulated and the results are compared with the analytical solution. Moreover, a liquefaction analysis of a breakwater built on saturated sand soil with generalized plastic model is subsequently carried out. The results correspond well with those calculated by finite element method (FEM), which indicates the significant capability of the current method in solving nonlinear problems. The proposed method processes extraordinary mesh flexibility and fast reconstruction, which will make it a promising tool in liquefaction analysis.

A non-uniformly loaded anti-plane crack embedded in a half-space of a one-dimensional piezoelectric quasicrystal

Abstract: Closed-form representations are obtained using an extension of the classical continuous dislocation layer method combined with a method of images for the components of the phonon and phason stress and electric displacement fields around a generally loaded strip crack in a half-space of one-dimensional hexagonal piezoelectric quasicrystalline material parallel to its free surface. Representative numerical data are presented graphically.

Dynamic impedance functions for a rigid strip footing resting on a multi-layered transversely isotropic saturated half-space

Abstract: This paper is concerned with the dynamic impedance functions (force-displacement relationships) of a surface rigid strip footing resting on a multi-layered transversely isotropic (TI) saturated half-space. The rigid footing is perfectly bonded to the layered half-space and is subjected to time-harmonic vertical, horizontal and moment loadings. The half-space under consideration consists of a number of horizontal layers with different thicknesses and an underlying half-space, which are all governed by the Biot's poroelastodynamic theory. The surface of the half-space can be either fully permeable or impermeable. The dynamic interaction problem is solved by employing an indirect boundary element method (IBEM), which uses Green's functions for uniform strip loads acting on the surface of a multi-layered TI saturated half-space. The discretization of the method is restricted to the footing-subsoil interface because of the layered half-space kernel functions, and the accuracy of the method would not be affected by the thickness of the discrete layers because of the exact dynamic stiffness matrix. Comparison with the existing solutions for the TI elastic and isotropic saturated media is conducted to verify the method, which are special cases of the more general problems addressed. Selected numerical solutions are presented to portray the influence of material anisotropy, frequency of excitation, surfaced drainage condition and layering on the dynamic impedance functions. Numerical results show that the dynamic impedance functions for the TI material can be significantly different from those of the isotropic material. The variation of the TI parameters alters the resonant frequencies of the layer and further alters the dynamic interaction between the layer and the footing.

Love wave in a classical linear elastic half-space covered by a surface layer described by the couple stress theory

Abstract: A Love wave is derived for a new physical configuration in which a surface layer described by the couple stress theory covers a classical elasticity half-space. The dispersion equation is derived analytically when the thickness of the surface layer approaches zero. The correctness of the dispersion equation is confirmed via the second derivation path, namely the surface elasticity. The membrane with microstructure is described by the surface elasticity which significantly simplifies the derivation. New propagation features deduced from the dispersion curves are discussed.

Dynamic response of a circular inclusion embedded in inhomogeneous half-space

Abstract: Dynamic response of a shallow circular inclusion under incident SH wave in radially inhomogeneous half-space is researched by applying complex function theory and multipolar coordinate system. Considering that the mass density of the half-space varies along with the radius direction, the governing equation is expressed as a Helmholtz equation with a variable coefficient. Based on the conformal mapping method, the Helmholtz equation with a variable coefficient is transformed into its normalized form. Then, the expressions of incident wave, reflected wave and scattering wave are obtained, and the standing wave function is deduced by considering the circular inclusion subsequently. According to displacement and stress continuous condition of the inclusion, the undetermined coefficients in scattering wave and standing wave are solved. Finally, dynamic stress concentration factor around the inclusion is calculated and discussed. Numerical results demonstrate the validity of the method and influential factors of dynamic stress concentration factor.

Scattering by a Dielectric Sphere Buried in a Half-Space With a Slightly Rough Interface

Abstract: Analytical expressions for the scattering coefficients of a dielectric sphere buried under a rough interface are presented. The proposed method combines the small perturbation method (SPM) and the Mie solution by using the expansion of plane waves in terms of vector spherical functions (VSFs) and vice versa. First, using SPM, the zeroth- and the first-order perturbative scattered fields of a rough interface for illuminations from above and below are derived. Using these solutions, the field transmitted to the lower half-space is determined as a spectrum of down-going plane waves. The scattered fields from the sphere are then calculated using the vector Mie solution. Subsequently, the VSFs are expanded in terms of up-going plane waves. These plane waves illuminate the interface, and using SPM, the scattered fields in the upper and lower regions are determined as infinite summations of plane waves. The reflected plane waves are once again scattered by the sphere and the scenario repeats. By inspecting the form of the fields resulting from the few first interactions of the sphere and the rough interface, a recursive form is obtained for the scattered fields. This recursive form is then used to rewrite the system of equations in a form containing all interactions in a single-step formulation. Accordingly, the zeroth- and the first-order closed-form scattered fields are obtained. The derived expressions are analytically and numerically validated. Finally, the numerical results for the case of the rough interface with sinusoidal profile are presented and briefly discussed.

Three-dimensional coupled thermoelastodynamic stress and flux induced wave propagation for isotropic half-space with scalar potential functions

Abstract: An asymmetric three-dimensional thermoelastodynamic wave propagation with scalar potential functions is presented for an isotropic half-space, in such a way that the wave may be originated from an arbitrary either traction or heat flux applied on a patch at the free surface of the half-space. The displacements, stresses and temperature are presented within the framework of Biot’s coupled thermoelasticity formulations. By employing a complete representation for the displacement and temperature fields in terms of two scalar potential functions, the governing equations of coupled thermoelasticity are uncoupled into a sixth- and a second-order partial differential equation in cylindrical coordinate system. By virtue of Fourier expansion and Hankel integral transforms, the angular and radial variables are suppressed respectively, and a $$6{\mathrm{th}}$$ - and a $$2{\mathrm{nd}}$$ -order ordinary differential equation in terms of depth are received, which are solved readily, from which the displacement, stresses and temperature fields are derived in transformed space by satisfying both the regularity and boundary conditions. By applying the inverse Hankel integral transforms, the displacements and temperature are numerically evaluated to determine the solutions in the real space. The numerical evaluations are done for three specific cases of vertical and horizontal time-harmonic patch traction and a constant heat flux passing through a circular disc on the surface of the half-space. It has been previously proved that the potential functions used in this paper are applicable from elastostatics to thermoelastodynamics. Thus, the analytical solutions presented in this paper are verified by comparing the results of this study with two specific problems reported in the literature, which are an elastodynamic problem and an axisymmetric quasi-static thermoelastic problem. To show the accuracy of numerical results, the solution of this study is also compared with the solution for elastodynamics exists in the literature for surface excitation, where a very good agreement is achieved. The formulations presented in this study may be used as benchmark for other related researches and it may be implemented in the related boundary integral equations.

Study of Rayleigh type surface wave in the initial stressed irregular bottom of ocean due to point source

Abstract: Rayleigh type surface wave propagation in the irregular bottom of ocean model which is the interface of homogeneous liquid layer over laying an irregular boundary of homogeneous orthotropic half space under initial stresses has been discussed in this paper. Three different dispersion equations are obtained in the form of simple equation using and not using Perturbation technique. Some special cases have been considered. The effect of irregularity, initial stressed, point source, and depth of liquid layer on the propagation of Rayleigh waves has been analyzed and results of numerical discussion have been presented graphically for three different dispersion equations. Mainly the graphs are shown the variation of phase velocity with wave number in different cases.

In-plane dynamic Green's functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

Abstract: The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic (TI) half-space. The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces, which are then applied to the total system with the opposite sign. By adding solutions restricted in the loaded layer to solutions from the reaction forces, the global solutions in the wavenumber domain are obtained, and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform. The presented formulations can be reduced to the isotropic case developed by Wolf (1985), and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI half-space subjected to horizontally distributed loads which are special cases of the more general problem addressed. The deduced Green’s functions, in conjunction with boundary element methods, will lead to significant advances in the investigation of a variety of wave scattering, wave radiation and soil-structure interaction problems in a layered TI site. Selected numerical results are given to investigate the influence of material anisotropy, frequency of excitation, inclination angle and layered on the responses of displacement and stress, and some conclusions are drawn.

Rayleigh waves in a layered orthotropic elastic half-space with sliding contact:

Abstract: The presented paper is concerned with the propagation of Rayleigh waves in an orthotropic elastic half-space overlaid by an orthotropic elastic layer of arbitrary thickness. The layer and the half-...

Average scattered field from a random PEC cylinder buried below a slightly rough surface

Abstract: Scattering from a perfect electric conducting cylinder with random radius buried below a half space dielectric homogenous interface is studied. The cylindrical wave scattered by cylinder is expanded in terms of plane wave spectrum. Small perturbation method is used to study the interaction of each plane wave with the interface. The zeroth order term yields solution for a flat interface, whereas scattering from a rough surface is given by first-order term. Results are obtained for both TM and TE polarizations. Analytical expressions of the average scattered field are obtained and verified using numerical evaluation. Different scattering scenarios are simulated by varying the distribution of the radius. It is observed that average scattering cross section of an ensemble with normal/uniform distribution is almost equal to that of a cylinder with mean radius.

Generalized Magnetothermoelastic Interaction for a Rotating Half Space

Abstract: A generalized magnetothermoelasticity, in the context of Lord–Shulman theory, is employed to investigate the interaction of a homogeneous and isotropic perfect conducting half space with rotation. The Laplace transform for time variable is used to formulate a vector-matrix differential equation which is then solved by eigenvalue method. The continuous solution of displacement component while the discontinuous solutions of stress components, temperature distribution, induced magnetic and electric field have been analyzed in an approximate manner using asymptotic expansion for small time. The graphical representations also prove this continuity and discontinuity of the solutions.

An improved projection method for solving generalized variational inequality problems

Abstract: In this paper, we present a new algorithm for solving generalized variational inequality problems(GVIP for short) in finite-dimensional Euclidean space. In this method, our next iterate point is ob...

Stability estimates for the fault inverse problem

Abstract: We study in this paper stability estimates for the fault inverse problem. In this problem, faults are assumed to be planar open surfaces in a half space elastic medium with known Lame coefficients. A traction free condition is imposed on the boundary of the half space. Displacement fields present jumps across faults, called slips, while traction derivatives are continuous. It was proved in \cite{volkov2017reconstruction} that if the displacement field is known on an open set on the boundary of the half space, then the fault and the slip are uniquely determined. In this present paper, we study the stability of this uniqueness result with regard to the coefficients of the equation of the plane containing the fault. If the slip field is known we state and prove a Lipschitz stability result. In the more interesting case where the slip field is unknown, we state and prove another Lipschitz stability result under the additional assumption, which is still physically relevant, that the slip field is one directional.

Three-dimensional problem of Rayleigh waves propagating in a half-space with restrained boundary

Abstract: Baghramyan ave. 24/2, 0019, Yerevan, Armenia Alex Manoogian 1, 0025, Yerevan, Armenia Correspondence S. V. Sarkisyan, Alex Manoogian 1, 0025 Yerevan, Armenia Email: vas@ysu.am In this paper we obtain the dispersion equation in three-dimensional wave propagation problem in elastic half-space with elastically restrained surface. It is shown that, in the case of plane strain, the elastic restriction of the surface leads to decrease of the degree of the surface wave localization.

Interaction between a punch and an arbitrary crack or inclusion in a transversely isotropic half-space

Abstract: We consider the problem of an arbitrary shaped rigid punch pressed against the boundary of a transversely isotropic half-space and interacting with an arbitrary flat crack or inclusion, located in the plane parallel to the boundary. The set of governing integral equations is derived for the most general conditions, namely the presence of both normal and tangential stresses under the punch, as well as general loading of the crack faces. In order to verify correctness of the derivations, two different methods were used to obtain governing integral equations: generalized method of images and utilization of the reciprocal theorem. Both methods gave the same results. Axisymmetric coaxial case of interaction between a rigid inclusion and a flat circular punch both centered along the z-axis is considered as an illustrative example. Most of the final results are presented in terms of elementary functions.

Wave induced sliding at the interface between a layered elastic medium and a half-space

Abstract: The sliding interface between an unrestrained elastic half-space and a grounded layered half-space excited by an incident harmonic wave is investigated. The problem is formulated considering various possible boundary conditions and boundary inequalities at the sliding interface. The Coulomb friction model without distinction between the static and kinetic coefficients of friction is considered to govern the sliding condition. Three possible bands at the interface, namely slip, stick, and separation, are considered. The interface is assumed to be preloaded under normal and shear stresses. The solution is developed by modifying the problem of welded interface, which then is reduced to a set of algebraic equations. The effects of the incident angle, layer thickness, friction coefficient and externally applied stresses on the drift velocity of the unrestrained half-space are studied numerically for a pair of materials. It is shown that the sliding interface, and hence the drift velocity of unrestrained half-space is noticeably influenced by the layered medium. These results are expected to be useful for the development of a new kind of ultrasonic drive in future.

Control of steady-state thermal displacements and stresses in a plane-strained half-space

Abstract: Inverse thermoelasticity problems, formulated as a result of reduction of two-dimensional steady-state control problems for a component of the displacement vector or the stress tensor (or their combinations) in a given cross-section of a plane-strained half-space, are analyzed. The intensity of the inner heat sources was selected to be the control function. Solutions to the mentioned inverse problems are obtained in the space of continuous functions. The finiteness conditions for the found integral intensity of the inner heat sources are derived and analyzed. The key features of the obtained solution are analyzed numerically for certain types of thermal loadings and characterizing parameters.

Evaluation of the stress-strain state of half-space with cylindrical cavities

Abstract: A three-dimensional problem was solved in the theory of elasticity for an elastic uniform halfspace with cylindrical cavities parallel to each another and the half-space boundary. Stresses rapidly decaying to zero at big distances from the origin of coordinates are specified on the boundaries of cylindrical cavities and on the half-space boundary. The problem of solving such problems is topical. It is encountered in practice and solved using approximate methods. The approach used herein yields solutions of the stated problem with an a priori specified accuracy depending on the system order. In contrast to publications referred to in the paper, the focus, apart from another approach, was on analysing the stress state of half-space to study the mutual influence of cylindrical cavities and of the cavities with half-space boundary. The problem was solved using the generalised Fourier method for Lame equations in cylindrical coordinates linked to cylinders and Cartesian coordinates linked to half-space. For transition between basic solutions of Lame’s equations, special formulas were used for transition between local cylindrical systems of coordinates and between the Cartesian and cylindrical systems of coordinates. The truncation method was used to solve infinite systems of linear algebraic equations to which the problem was reduced. This yielded displacements and stresses in an elastic body. The numerical results were derived for the case of half-space and two cylinders under a load applied to the half-space boundary. Separate computations were performed for a load applied to the surface of the cylindrical cavity. In both cases the analysis of the stress-strain state indicates that space weakening due to cylindrical cavities or the half-space boundary gives rise to extremal stresses in these sites. The method can also be used for other boundary conditions.

Two-Parametric Analysis of Anti-Plane Shear Deformation of a Coated Elastic Half-Space

Abstract: Abstract The anti-plane shear deformation problem of a half-space coated by a soft or a stiff thin layer is considered. The two-term asymptotic analysis is developed motivated by the scaling for the displacement and stress components obtained from the exact solution of a model problem for a shear harmonic load. It is shown that for a rather high contrast in stiffness of the layer and the half-space Winkler-type behaviour appears for a relatively soft coating, while for a relatively stiff one, the equations of plate shear are valid. For low contrast, an alternative approximation is suggested based on the reduced continuity conditions and the fact that the applied load may be transmitted to the interface. In case of a stiff layer, a simpler problem for a homogeneous half-space with effective boundary condition is also formulated, modelling the effect of the coating, while for a relatively soft layer a uniformly valid approximate formula is introduced.

Quasi-Static Deformation of a Uniform Thermoelastic Half –Space Due to Seismic Sources and Heat Source

Abstract: This paper investigates the quasi-static plane deformation of an isotropic thermoelastic half-space due to buried seismic sources and heat source. Governing equations of thermo-elasticity are solved to obtain solutions for seismic sources in a thermoelastic half-space. The general solutions are acquired with the aid of Laplace and Fourier transforms and with the use of boundary conditions. The case of dip-slip line dislocation is studied in detail along with line heat source. Analytical solutions for two limiting cases: adiabatic and isothermal, are obtained. The solutions for displacement, stresses and temperature in space-time domain are obtained by using a numerical inversion procedure. The accuracy of the proposed method is verified through a comparison of the results obtained with the existing solutions for elastic medium. In addition, numerical results for displacements, stresses and temperature function, induced by a vertical dip-slip dislocation and line heat source, are presented graphically to illustrate the effect of inclusion of thermal effect in simulation of the problem.