Surface Of Half-space Research Articles

Reflection of coupled waves from the flat boundary surface of a nonlocal micropolar thermoelastic half-space containing voids

Reflection behavior of a set of coupled longitudinal/transverse waves incident obliquely at the flat boundary surface of a nonlocal micropolar thermoelastic half-space containing voids is investigated. Appropriate boundary conditions are setup for mechanically stress-free and thermally insulated boundary surface of the half-space. The equations yielding the reflection coefficients and the expressions of corresponding energy ratios of various reflected waves are presented. The nature of dependence of the reflection coefficients and their corresponding energy ratios against the angle of incidence is studied numerically for a specific model. Effect of several parameters on the reflection coefficients as well as on the corresponding energy ratios is noticed and depicted graphically. It is found that the sum of the energy ratios is approximately equal to unity at each angle of incidence. Some earlier known results are also recovered as special cases of the present formulation.

LAMB’S PROBLEM AND CHARACTERISTIC OF ENERGY TRANSMISSION IN UNSATURATED HALF-SPACE

Because unsaturated soil is widely distributed on the earth’s surface, when the traditional saturated two-phase medium theory is used for dynamic analysis, the results are often inconsistent with the actual situation. Aiming at this problem, this paper takes unsaturated elastic half-space as the research object, firstly based on continuum mechanics and porous media theory, and then considers the basic equations of mass conservation equation, momentum conservation equation, constitutive equation and effective stress principle of each phase in unsaturated porous media, and finally, we established a dynamic control equation in which skeleton displacement, pore water pressure and pore gas pressure are basically unknown quantities. Aiming at the dynamic response and energy transmission of the unsaturated half-space surface under the action of vertical concentrated harmonic loads, an axisymmetric calculation model of the classical Lamb problem in the frequency domain is established. The Helmholtz decomposition method is used and the displacement component of the skeleton uses the potential function Φ and Ψ to represent, and combined with the constitutive equation, the analytical solutions of physical quantities such as the displacement field and energy field of the half-space surface under different boundary conditions are obtained. Finally, influencing factors such as load parameters (excitation frequency) and material parameters (saturation, permeability coefficient) are analyzed and discussed through numerical examples. The results show that: (1) An increase in saturation or a decrease in excitation frequency will increase the surface displacement amplitude of the unsaturated half-space; (2) When the permeability coefficient drops to a critical value, the surface displacement amplitude will tend to a limit value, and the influence of permeability coefficient under permeable (gas) boundary and impermeable (gas) boundary conditions shows obvious difference.

Dynamical Green’s function for elastic half-space, and energy losses due to collision

This special issue celebrates 100 years since the birth of Moisey Isaakovich Kaganov. This date is a personal event for us, since Moisey Issakovich (or Musik, for his friends and close ones) is the father of one of us, and the grandfather of the other. In addition, we have both been his students. We received the problem discussed in this paper by succession. In 1949 Ilya Mikhailovich Lifshitz was interested in studying electrodynamic and elastic properties of solids, and this analysis required knowledge of the corresponding Green’s functions. He suggested to his two graduate students, Moisey Kaganov and Victor Tsukernik, to calculate the displacement vector caused by an instant point source acting at the surface of an elastic half-space. At that time they had chosen a different topic, and the problem hibernated until the early 1990s, when Moisey Issakovich suggested it as a subject for a Master’s thesis for one of us (ML) to be conducted under the supervision of the other one (IK). The results have been published in papers I. M. Kaganova and M. L. Litinskaia, Phys. Lett. A 200, 365 (1995)1 and I. M. Kaganova and M. L. Litinskaia, Phys. Lett. A 200, 375 (1995).2 The first paper discussed the derivation of the normal component of the displacement vector. We showed that the displacement can be calculated as an integral in the complex plane, and examined the displacement at the surface of the half-space and at the direction normal to the surface. We showed that the singularities of the displacement are linked to certain changes in the shape of the integration contour. In the second paper, we applied the expression for the normal displacement to the calculation of the elastic energy due to an external load, and found the amount of energy lost by a small ball incident onto an elastic half-space. In this publication, we expand the analysis by investigating the singularities of the displacement vector in an arbitrary point of the half-space, and briefly review our previous results.

Thermoelastic wave and thermal shock based on dipolar gradient elasticity and fractional-order generalized thermoelasticity

In the current work, the thermoelastic wave propagation and the thermal shock problems are studied based on the dipolar gradient elasticity and fractional order generalized thermoelasticity in order to take into account the microstructure effect and the thermal effect at a small scale. First, the fractional-order governing equations for coupled thermoelastic wave propagation are derived and the effects of microstructure parameters and fractional order on the dispersion and the attenuation are discussed; next, the transient problems by the thermal shock and the mechanical shock on the surface of half-space are studied. The Laplace transformation method and the state transfer equation method are used to solve this problem. The numerical results for the temperature, the displacement, the stress and the high order stress distributions are provided and shown graphically. The effects of the fractional order and the microstructure parameter on the thermoelastic behavior are investigated. Some interesting phenomenons and conclusions are presented at last.

Interior-stress fields produced by a general axisymmetric punch

This work is a supplement to the work of Sneddon on axisymmetric Boussinesq problem in 1965 in which the distributions of interior-stress fields are derived here for a punch with general profile. A novel set of mathematical procedures is introduced to process the basic elastic solutions (obtained by the method of Hankel transform, which was pioneered by Sneddon) and the solution of the dual integral equations. These processes then enable us to not only derive the general relationship of indentation depth D and total load P that acts on the punch but also explicitly obtain the general analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space. The usually known cases of punch profiles are reconsidered according to the general formulas derived in this study, and the deduced results are verified by comparing them with the classical results. Finally, these general formulas are also applied to evaluate the von Mises stresses for several new punch profiles.

Reflection at non-free boundary of a micropolar piezoelectric half-space

This paper presents a study on plane waves propagating in a micropolar piezoelectric material with transverse isotropy. The plane wave solutions of the governing field equations suggest that there exists three coupled plane waves propagating in a micropolar piezoelectric medium. Reflection of a plane wave incident at a non-free surface of a micropolar piezoelectric half-space is considered. The analytical expressions of the reflection coefficients and energy ratios are obtained. A quantitative example is set up to compute the speeds, reflection coefficients and energy ratios of reflected waves. The speeds, energy ratios and reflection coefficients are illustrated graphically against the propagation angle. The numerical results show that the boundary parameters connected to the mechanical nature of the non-free surface affect significantly the energy ratios and reflection coefficients.

Scattering by a Chiral Cylinder of Arbitrary Cross Section Above a Dielectric Half-Space

A simple numerical solution for electromagnetic scattering from a 2-D homogeneous chiral cylinder of arbitrary cross section placed above a dielectric half-space (DHS) is presented. A modified surface equivalence principle is used to obtain coupled electric field integral equations. For feasible application of method of moments, a perturbation method is used where a strip of finite width approximates the surface of the infinite half-space. The perturbed currents on the interface are expressed as a difference of the currents with the cylinder present and the exact solution (cylinder removed). The excitation vector now contains the reflected field from the DHS in addition to the incident field. Transverse magnetic (TM) and transverse electric (TE) excitations are treated. Computed results include perturbed currents on the dielectric interface and the scattered fields. Excellent agreement is observed for special cases available in the literature.

On spherical nanoinhomogeneity embedded in a half-space analyzed with Steigmann–Ogden surface and interface models

This paper determines the elastic fields in a positive half-space embedded with a spherical inhomogeneity under the context of Steigmann–Ogden surface/interface mechanical model. The half-space is loaded by an equal-biaxial far-field tension applied at infinity. While bulk domains are treated as linearly isotropic elastic solids, both the half-space plane boundary and the matrix/inhomogeneity interface are modeled by the Steigmann–Ogden theory. The well-developed method of Boussinesq displacement potentials is used to solve the elastostatic Navier’s equations of equilibrium. In view of the geometry and loading configurations, four sets of cylindrical and spherical harmonic potentials are carefully proposed to solve the problem. Implementation of the nonclassical Steigmann–Ogden boundary conditions at both the half-space free surface and the spherical interface results in a semianalytical solution in the form of infinite Legendre series. Extensive parametric studies are performed with respect to the surface and interface Steigmann–Ogden material parameters, shear moduli ratio between the spherical inhomogeneity and its surrounding matrix, and the inhomogeneity radius-to-depth ratio. Comparison and contrast with the well known Gurtin–Murdoch model reveals the significance of surface flexural rigidities at both boundaries of the considered mechanical model. When compared with the effects of matrix/inhomogeneity interface, the surface tension, Lamé constants and flexural rigidities at the half-space plane boundary are of secondary importance. This work is able to shed some lights on the modeling of self-organized adatoms and islands that is essential in the semiconductor industry.

Nonlinear and non-local analytical solution for Darcy–Forchheimer flow through a deformable porous inclusion within a semi-infinite elastic medium

Permeability is a nonlinear and non-local function of the intimate coupling between pore fluid flow and solid deformation in porous media. A class of related problems involves the effect of fluid injection or withdrawal on the transport properties of geomaterials. This paper presents an analytical solution for the nonlinear and non-local problem of fluid flow through a disk-shaped porous elastic inclusion below the free surface of an elastic half-space. The solution accounts for the following nonlinear mechanisms: (i) variations in the permeability coefficient due to the flow-induced deformation of the inclusion; (ii) the inertial losses of the pore fluid flow. The former and latter mechanisms are formulated using the Green's function for a dilatation centre in an elastic half-space and the Darcy–Forchheimer equation for fluid flow through porous media, respectively. An analytical perturbation solution to the considered problem is developed and validated against the numerical finite element solution to the same problem. The described nonlinear mechanisms are represented by two dimensionless parameter groups. The extreme values of these dimensionless groups govern the solution asymptotic behaviours mimicking the special-case solutions in which either mechanism is forced to vanish. The applied aspects of the solution are demonstrated through the wellbore performance index parameter that quantifies the subsurface rock ability to deliver the pore fluid toward or away from a wellbore in a porous reservoir. Unlike the linear models, the presented nonlinear solution captures the observed dependence of the performance index on the wellbore flow rate.

Modified lysmer’s analog model for two dimensional mat settlements under vertically uniform load

A two dimensional model of linearly elastic soil spring used for the settlement analysis of the flexible mat foundation is suggested in this study. The spring constants of the soils underneath the foundation were modeled assuming uniformly vertical load applied onto the foundation. The soil spring constants were back calculated using the three-dimensional finite element analysis with Midas GTS NX program. Variation of the soil spring constants was modeled as a two-dimensional polynomial function in terms of the normalized spatial distances between the center of foundation and the analytical points. The Lysmer's analog spring for soils underneath the rigid foundation was adopted and calibrated for the flexible foundation. For validations, the newly proposed soil spring model was incorporated into a two dimensional finite difference analysis for a square mat foundation at the surface of an elastic half-space consisting of soft clays. Comparative study was made for elastic soils where the shear wave velocity is 120~180 m/s and the Poisson's ratio varies at 0.3~0.5. The resulting foundation settlements from the two dimensional finite difference analysis with the proposed soil springs were found in good agreement with those obtained directly from three dimensional finite element analyses. Details of the applications and limitations of the modified Lysmer's analog springs were discussed in this study.

ПРИБЛИЖЕННОЕ ОПРЕДЕЛЕНИЕ ГЕОМЕТРИЧЕСКИХ ПАРАМЕТРОВ НАПЛЫВА ПРИ ВНЕДРЕНИИ ПАРАБОЛОИДНОГО ИНДЕНТОРА В ПОЛУПРОСТРАНСТВО

The problem of the indenter embedding into a half-space is formulated and solved. A concept of the elastic contact is used, which implies the uniformity of the stress-strain state in a surface layer. The process of the surface layer formation after mechanical treatment is studied. Based on the general theory of indenter penetration into a half-space, numerical and analytical solutions, determining the length and depth of penetration of a paraboloid into elastoplastic space, are obtained. According to the theory of indentation, the problem is reduced to the use of the method of approximate determination of geometric parameters of a bead, when a paraboloid is being embedded into elastoplastic space. An exact solution to the formulated problem allows one to obtain analytical dependences for calculating the velocities and to study the stress-strain state of the material in the half-space surface layer during the processing. The reliability of the obtained analytical solution is confirmed by numerical calculations without introducing additional hypotheses. Based on the analytical solution, the geometric parameters of the influx from the penetration depth are determined. Calculations can be performed at any embedding depth. Sag formation during the indenter embedding by 18 mm into plasticine specimens is considered as an example. It is shown that the rigid zone is insignificant or absent.

Steady-state growth of an interfacial crack by corrosion

The Wiener–Hopf technique is used to obtain a 2D steady-state solution for the progressive conversion of a pristine interface into a corroded interface between two dissimilar solids. The interface is of infinite extent, and comprises a semi-infinite pristine portion and a semi-infinite corroded portion. Fickian diffusion of the active species (solute) occurs in the upper half-space, with no diffusion in the lower half-space. Corrosion occurs by chemical reaction between the solute and the top surface of the lower half-space, and the path of solute diffusion involves 3 stages. The solute (i) leaves a solute-rich zone of disbonded and previously corroded interface, (ii) enters into and diffuses through the upper half-space, and (iii) leaves the upper half-space and enters the upstream pristine interface where it reacts with the surface of the lower half-space to produce the corrosion product. A steady state is established: the corrosion front moves at a constant velocity V which is dictated by the critical value of accumulated solute on the interface that is needed to form corrosion product and disbond the interface. The reaction zone directly ahead of the corrosion front has a characteristic length that depends upon the diffusion parameters and front velocity V. An asymptotic solution by the Wiener–Hopf analysis is obtained for the diffusion problem at large V. Scaling laws emerge and support the predictions of a much simpler 1D physical model.

Analytic formulas for complete and incomplete first-order TEM moments below ground in a conductive half-space

In the resistive limit, discrete conductors give rise to magnetic dipole fields. Magnetostatic modeling of time integrals, or moments, of transient electromagnetic (TEM) data therefore offers a means for fast approximate 3D modeling and inversion of TEM data sets. In our approximate inversion scheme, the net TEM moment response is represented as the combination of discrete conductor and uniform host responses. The inversion algorithm first estimates a homogeneous host conductivity, and then it subtracts the host response and fits the residual moment data by adjusting the conductivities of cells comprising a 3D rectangular mesh. To expedite calculation of the host response, we have derived analytic formulas for first-order TEM moments produced on and under the ground by a horizontal electric dipole on the surface of a homogeneous conductive half-space. We present analytic expressions for idealized all-time “complete” moments, or resistive limits, as well as for realizable finite-time “incomplete” moments. The moments produced by an arbitrary horizontal polygonal loop are determined by combining contributions from appropriately oriented electric dipoles. Downhole TEM moments computed with the new expressions reveal substantial differences between incomplete and complete moments when early time data are excluded and between step and impulse response incomplete moments. The role of the formulas in the first stage of our moment-based 3D inversion scheme is illustrated via analysis of downhole TEM data recorded at Santander, Peru. The host conductivity of best fit for early time B-field moments is 2.40 mS/m, consistent with apparent conductivities derived from ground TEM data recorded in the same area.

A Mathematical Model of Longitudinal Waves Incident at a Free Surface of a Pre-Stressed Dissipative Half-Space

The aim of this work is to study the behavior of reflection of a longitudinal wave at a free surface of dissipative half-space under the effects of compressive initial stresses. When a longitudinal wave is incident on the free surface of an elastic dissipative half-space, two damped waves (Primarywaves and secondary waves are reflected. Among of these waves, P-waves are affected by compressional initial stresses. The governing equation and corresponding closed-form solutions are derived based on Biot’s incremental deformation theory. The equations of motion are solved analytically and the influence of initial stress parameter on the reflection coefficient of P-wave incidents at the free surface of dissipative half-space is studied in detail. Numerical computations are performed for actual Earth crust and the results analyzing the incident of longitudinal waves are discussed and presented graphically. The analytical solutions and numerical results reveal that the compressive initial stress parameter has notable effects on the reflection coefficient of longitudinal wave incidents on the free surface of dissipative medium. In addition, it has been observed that the presence of compressive initial stresses increases the phase velocity of the longitudinal waves. To the authors’ best knowledge, effects of compressive initial stresses on the reflection coefficients of the incident longitudinal wave on a free surface of dissipative half-space have not been studied before. Since the actual Earth is subject to initial stresses due to different resources, understanding the influences of compressive initial stresses on the reflection coefficient of a longitudinal wave helps seismologists and earthquake engineers to get accurate results of the reflection coefficients of seismic waves propagation in the Earth. Thus, the present study would be useful for seismology and earthquake engineering fields and further study about the nature of seismic waves.

Surface Motion of a Half-Space Containing an Elliptical-Arc Canyon under Incident SH Waves

The existence of local terrain has a great influence on the scattering and diffraction of seismic waves. The wave function expansion method is a commonly used method for studying terrain effects, because it can reveal the physical process of wave scattering and verify the accuracy of numerical methods. An exact, analytical solution of two-dimensional scattering of plane SH (shear-horizontal) waves by an elliptical-arc canyon on the surface of the elastic half-space is proposed by using the wave function expansion method. The problem of transforming wave functions in multi-ellipse coordinate systems was solved by using the extra-domain Mathieu function addition theorem, and the steady-state solution of the SH wave scattering problem of elliptical-arc depression terrain was reduced to the solution of simple infinite algebra equations. The numerical results of the solution are obtained by truncating the infinite equation. The accuracy of the proposed solution is verified by comparing the results obtained when the elliptical arc-shaped depression is degraded into a semi-ellipsoidal depression or even a semi-circular depression with previous results. Complicated effects of the canyon depth-to-span ratio, elliptical axis ratio, and incident angle on ground motion are shown by the numerical results for typical cases.

Response of ramp-type heating in a monoclinic generalized thermoelastic material

PurposeThe two-dimensional deformation of a homogeneous, thermally conducting, monoclinic material has been studied by using Laplace and Fourier transforms technique. A linear temperature ramping function is used to more realistically model: thermal loading of the half-space surface. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating and loading. The displacements, stresses and temperature distribution so obtained in the physical domain are computed numerically and illustrated graphically. The comparison for Lord-Shulman (L-S), Green and Lindsay (G–L), Green and Naghdi (G–N) and Chandrasekharaiah and Tzou (CTU) theories have been shown graphically to estimate the effect of ramping parameter of heating for an insulated and temperature gradient boundaries.Design/methodology/approachThe design of the study is eigenvalue approachFindingsHomogeneous, thermally conducting monoclinic material has been taken under consideration to study the effect of linear temperature ramping parameter on temperature and normal displacement field. It is observed that magnitude of field quantities is large near the point of application of source for the non-dimensional values of time in all the four models. The numerical values for the field quantities are computed graphically for a wide range of values of finite pulse rise-time in the two situations t0 < t, t0 > t for generalized thermoelasticity theories.Originality/value(1) Governing equations for homogeneous, t0 thermally conducting, monoclinic material are described and solved. (2) Eigen value approach is used to solve the problem. (3) The effect of ramping parameter of heating has been studied for various models of the thermoelasticity to show the comparision between them.

An anisotropic layered poroelastic half-space subjected to a moving point load

The dynamic responses of an anisotropic layered poroelastic half-space subjected to a moving point load are studied based on the Stroh formalism and Fourier transform. The material in each layer of the half-space is anisotropic and poroelastic one. Utilizing the boundary conditions at the half-space surface (permeable or impermeable) and the continuity conditions between each layer, the three-dimensional (3D) solutions for the displacements, the pore pressure, the stresses and the fluid fluxes are obtained. Numerical examples show the validity and elegance of the present solution. For some special cases, our solutions agree with the known results. For the anisotropic case, the 3D dynamic Green's functions for the multi-layered poroelastic half-space show clearly the effect of material properties on the displacement and stress distributions.

Разработка и обоснование пространственной механической модели основания сооружений с приведенными расчетными характеристиками грунтов с учетом конечной жесткости фундаментных конструкций

Предложена и обоснована новая пространственная ме­ханическая динамическая модель оснований сооружений атомных станций.&#x0D; В настоящее время при выполнении практических расчётов широко применяется модель оснований соору­жений атомных станций, приведённая в [1]. В этой модели мгновенные жёсткости основания при вращательных движениях жёсткого невесомого штампа функционально зависят от массы сооружения и от моментов инерции масс сооружения относительно координатных осей, что противоречит физической сути понятия мгновенных жёст­костей основания при вращательных движениях жёсткого невесомого штампа на поверхности инерционного полу­пространства.&#x0D; В свете вышеизложенного разработка динамической механической модели основания, устраняющей упомянутые несоответствия, представляется важной задачей, имеющей несомненную научную и практическую ценность&#x0D; Для обоснования проектных решений фундаментных конструкций сооружений необходимо определить величи­ны внутренних усилий, возникающих в их сечениях. При учёте конечной жёсткости фундаментных конструкций с применением метода конечных элементов необходимо установить закон распределения квазистатических и мгновенных жёстокостей грунтов основания по подошве фундаментной плиты.&#x0D; Известно, что параметры модели основания по существу должны имитировать параметры реакции основания жёст­кого невесомого штампа при его единичных смещениях и скоростях (линейных и угловых) по и относительно коор­динатных осей.&#x0D; В работе определены и обоснованы параметры модели основания, закон распределения интегральных жёсткостей по поверхности контакта. На примере сопоставления расчётов систем реакторного отделения совместно с основаниями, параметры которых были установлены по [1; 2], обоснована достоверность предлагаемой модели.

Transient Close-Range Electromagnetic Field Coupling Between Loop Antennas at the Interface of Dissipative Half-Spaces

The problem of electromagnetic (EM) field short-range coupling between coplanar loop antennas located at the interface of two half-spaces with dielectric and conductive properties is solved analytically in the time domain (TD) via the Cagniard-DeHoop (CdH) technique under the diffusive approximation. It is demonstrated that the thus obtained closed-form TD analytical expressions can be applied to evaluate the close-range EM-field signal transfer between two loops lying on the surface of a dissipative half-space.

The critical forces in the weak ground under the influence of technogenic relief

The initial critical force that the ground is exposed to from the weight of a structure is one of the main characteristics of the “structure-ground environment” system. A significant part of grounds are classified as weak, and the strain modulus value does not exceed 5 MPa. These soils have a weak load-carrying capacity and they are significantly deformed by the weight of the structures and the static load of technogenic relief. The theory and methodology of determining the initial critical force and its maximal action depth for various two-dimensional technogenic land forms, the force diagrams of which are described by trapezes and triangles of various types, are considered. The models grounds considered herein are a homogeneous elastic half-space or horizontally layered strata on the surface of a homogeneous elastic half-space. The nomograms of the relations of the initial critical force and its maximum depth of action to the ground’s physical and mechanical properties are attained. The identified relations are quasi-linear and the value of the initial critical force increases with an increase in the ground’s physical and mechanical parameters. The presented nomograms significantly simplify the calculation of the initial critical force in the ground.