Semi-infinite Half-space Research Articles

Self-similar expansion of a nonideal dusty plasma

The self-similar expansion into vacuum of a nonideal dusty plasma filling semi-infinite half space has been investigated for both unmagnetized, as well as magnetized, cases. The nonideal contributions arising from the dust species have been modeled through van der Waals' equation of state, while the electrons and the ions have been assumed to be ideal. For the unmagnetized dusty plasma, the governing equations describing the self-similar expansion have been derived and solved numerically to obtain the expansion profiles. The effect of the nonideal contributions coming from the volume reduction coefficient (because of the finite size of the dust grains) and the intergrain cohesive forces have been analyzed. It is shown that a nonideal thermal plasma expands over a much larger distance than the isothermal ideal plasma. For the magnetized case, the analysis has been carried out by using an MHD model, wherein the plasma is frozen to the the magnetic field lines, and quasi-neutrality is maintained. The expansion dynamics is described by a set of self-similar nonlinear governing equations. A parameter study of the expansion process is undertaken and the contributions of the ambient magnetic field to the expansion is discussed. Explicit analytical solutions for the case of small, but finite, amplitude perturbations for both the unmagnetized and magnetized cases have also been obtained. Here, the self-similar equations can be exactly reduced to a quadrature, thus yielding explicit analytical solutions for the plasma profiles during the expansion.

Effects of soil–structure interaction on longitudinal seismic response of MSSS bridges

This paper presents the results of a comprehensive study on the effects of soil–structure interaction on longitudinal seismic response of existing bridges. FHWA's guidelines for footing foundation on semi-infinite elastic half-space are used to determine translational and rotational stiffnesses at the base of bridge abutments and piers. Similarly, stiffness and strength of abutment backfill soil are determined based on FHWA's procedures. Various stiffnesses at the abutments are then lumped (condensed) into one translational spring at the point of impact between the abutment and the deck. Translational springs at the abutments are bilinear with their yield strength in compression determined based on Mononobe–Okabe method. In tension it is equal to the friction force at the footing. Among the parameters considered is the case of damaged back wall, where it is assumed that due to shear failure at the juncture of the back wall and breast wall the abutment strength and stiffness, as well as mobilized abutment mass, have changed. Results indicate that soil–structure interaction (SSI) has a significant effect on the seismic response in the longitudinal direction. Abutment strength is the most critical parameter. Impact force, deck sliding, and SSI affects all plastic rotations at the base of columns. Thus, it is important that analytical models used in seismic evaluation of bridge systems explicitly consider SSI.

Two time-derivative Lorentz material (2TDLM) formulation of a Maxwellian absorbing layer matched to a lossy medium

A two time-derivative Lorentz material (2TDLM) is introduced to define polarization and magnetization fields that lead to an absorbing layer that can be matched to a lossy dielectric medium. The 2TDLM is a generalization of the successful uniaxial polarization and magnetization time-derivative Lorentz material (TDLM) which has been introduced as an absorbing boundary condition for simulation regions dealing with lossless materials. Expressions are derived to describe the propagation of an arbitrary plane wave in this 2TDLM Maxwellian absorbing material. They are used to study the scattering from a semi-infinite 2TDLM half-space of an arbitrary plane wave incident upon it from a lossy isotropic dielectric medium. Matching conditions are derived which produce reflectionless transmission through such an interface for any angle of incidence and frequency. Numerical tests are given which demonstrate the effectiveness of the resulting 2TDLM absorbing layer.

Electromagneto-thermoelastic plane waves with two relaxation times in a medium of perfect conductivity

The model of the two-dimensional equations of generalized magneto-thermoelasticity with two relaxation times in a perfectly conducting medium is established. The normal mode analysis is used to obtain the exact expressions for the temperature distribution, thermal stresses and the displacement components. The resulting formulation is applied to two different concrete problems. The first deals with a thick plate of perfect conductivity subjected to time-dependent heat source on each face, while the second concerns the case of a heated punch moving across the surface of a semi-infinite thermoelastic half-space of perfect conductivity subject to appropriate boundary conditions. Numerical computations for the horizontal component of the displacement are carried out and represented graphically for each problem. A comparison was made with the results obtained in the absence of a magnetic field.

Scattering of Electromagnetic Waves by Rough Interfaces and Inhomogeneous Layers

We consider a two-dimensional problem of scattering of a time-harmonic electromagnetic plane wave by an infinite inhomogeneous conducting or dielectric layer at the interface between semi-infinite homogeneous dielectric half-spaces. The magnetic permeability is assumed to be a fixed positive constant. The material properties of the media are characterized completely by an index of refraction, which is a bounded measurable function in the layer and takes positive constant values above and below the layer, corresponding to the homogeneous dielectric media. In this paper, we examine only the transverse magnetic (TM) polarization case. A radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as an equivalent mixed system of boundary and domain integral equations, consisting of second-kind integral equations over the layer and interfaces within the layer. Assumptions on the variation of the index of refraction in the layer are then imposed which prove to be sufficient, together with the radiation condition, to prove uniqueness of solution and nonexistence of guided wave modes. Recent, general results on the solvability of systems of second kind integral equations on unbounded domains establish existence of solution and continuous dependence in a weighted norm of the solution on the given data. The results obtained apply to the case of scattering by a rough interface between two dielectric media and to many other practical configurations.

Quantum radiation from moving dielectrics in two, three, and more spatial dimensions

Quantum radiation from a moving dielectric semi-infinite half-space is investigated in a scalar model in two, three, and arbitrary integer spatial dimensions. The dielectric medium is non-dispersive and non-absorbing. The Hamiltonian describing the field in the presence of the moving dielectric body is obtained from considerations of Lorentz invariance. The field is quantized by an expansion into photon eigenmodes that are composed of incoming, reflected, and transmitted plane waves. The treatment includes total internal reflection and evanescent modes which are specific to dimensions . The Schrodinger equation for the state of the field is solved to first order in the velocity of the dielectric by a generalized adiabatic approximation. Non-uniform motion is found to lead to the emission of photon pairs whose spectrum and angular distribution are analysed in detail. The radiated energy and the radiation-reaction force are studied with particular emphasis to their dependence on the dimension of the system. In addition, the energy is worked out for several test trajectories.

Reflection and refraction of plane waves at an interface between micropolar elastic solid and viscoelastic solid

An infinite medium consisting of two semi-infinite half-spaces in welded contact is considered. The half-space in which the plane harmonic attenuating P- or SV-wave is incident, is assumed to be a linearly viscoelastic solid. The second half-space, in which the waves are transmitted, is assumed to be a micropolar elastic solid. The amplitude ratios for the different reflected and refracted waves have been calculated theoretically. Numerical values of the amplitude ratios have been computed and plotted against the angle of emergence for different values of frequency ratio ( ω 2/ ω 2 0). The numerical computations reveal that the amplitude ratios of reflected and refracted waves depend on the angle of emergence and frequency of the incident wave.

The influence of layer thickness on the stress intensity factor of a crack lying in an (visco)elastic layer embedded in a different (visco)elastic medium (Plane strain, mode I analysis)

We consider here a problem in which a (visco)elastic composite is composed of two dissimilar materials. One of these (medium 2) is contained in a layer sandwiched between semi–infinite half spaces of medium 1. In order to study the possibility of fracture within the layer (medium 2) we consider the response to remote applied loads of a finite length crack in medium 2. A pure opening mode (plane strain) deformation is considered and we investigate how the stress intensity factor (SIF) varies with the thickness (2 ;h ) of the inside layer and the different properties of the two (visco)elastic media. Formulae for the SIF as a function of the thickness of the inside layer and the elastic constants are derived. Analytical results for h ; small and large are obtained as a function of the material properties and verified by numerical analysis.

Comments on “Flexural wave reduction using a compliant tube baffle” [J. Acoust. Soc. Am. 99(2), 691–699 (1996)] and “Signal pressure received by a hydrophone placed on a plate backed by a compliant baffle” [J. Acoust. Soc. Am. 89(2), 559–564 (1991)

The cited formulations for studying the baffling of flexural waves by a compliant tube grating [S. H. Ko, J. Acoust. Soc. Am. 99, 691–699 (1996)] and for determining the acoustic receive sensitivity of a hydrophone placed near a plate backed by a compliant tube grating [S. H. Ko and H. H. Schloemer, J. Acoust. Soc. Am. 89, 559–564 (1991)] are dependent on the validity of the lumped-fluid parameter approximations developed by Junger [M. C. Junger, J. Acoust. Soc. Am. 78, 1010–1012 (1985)] for a grating of closely spaced compliant tubes between two semi-infinite fluid half-spaces. If the fluid between the plate and the grating is great enough to minimize their interaction, the incorporation of the effective layer parameters developed by Junger should be reasonably valid. Unfortunately, the sample calculations in Ko (Ref. 4) and Ko and Schloemer (Ref. 5) have been made for the limiting case of the compliant tube grating in direct contact with the plates where the use of Junger’s approximations is inappropriate.

Modeling of Surface Modified Layers in the Presence of Surface Irregularities

Finite element calculations that examine the effects of surface modification on the deformation produced by pure rolling contact are presented. The model simulates the repeated, two-dimensional (line) contact of a cylinder that is rolling over a semi-infinite half space. The half space is treated as an elastic-linear-kinematic-hardening-plastic (ELKP) material with the cyclic flow properties of a hardened, HRC-62, bearing steel. Two different cases are examined: (i) a smooth half space is studied using a one-body model, and (ii) a half space with a 100 μm wide and 7 μm deep surface asperity is studied using a two-body model. In both cases, calculations are performed for a homogeneous body and a body with a shallow, surface modified layer. The surface modified layer is alternately: (a) stiffer, (b) harder, (c) softer, and (d) harder and stiffer as compared to the substrate. Consistent with the earlier studies of surface modification (Bhargava, 1987), the present findings indicate that the benefits of the mechanical property modifications are confined to the altered layer itself. This may explain the improvement in performance realized by relatively thin modified layers (≈5 μm).

The influence of layer thickness on the stress intensity factor of a crack lying in an elastic (viscoelastic) layer embedded in a different elastic (viscoelastic) medium (mode IIIanalysis

We consider here a problem in which an elastic (or viscoelastic) composite is composed of two dissimilar materials. One of these (medium 2) is contained in a layer sandwiched between semi-infinite half spaces of medium 1. In order to study the possibility of fracture within the layer (medium 2) we consider the response to remote applied loads of a finite length crack in medium 2. As a first step, mode 3 deformation is considered and we investigate how the stress intensity factor (SIF) varies with the thickness (2 h) of the inside layer and the different properties of the two elastic (viscoelastic) media. Formulase for the SIF as a function of the thickness of the inside layer of the elastic constants are derived. Analytical results for small and large h are obtained as a function of the material properties and verified by numerical analysis.

SAPSI-M: Computer Program for Analyzing Asphalt Concrete Pavements under Moving Arbitrary Loads

A new solution for the dynamic analysis of asphalt concrete pavements subjected to moving fluctuating loads has been developed. The method builds on an existing model for stationary loads, SAPSI, which uses the complex response method of transient analysis with a continuum solution in the horizontal direction and a finite-element solution in the vertical direction. The structural model is an n-layered damped-elastic medium. The subgrade can be modeled as either a rigid base or a semi-infinite half space. The loads are surface pressure loads, and the analysis is under axisymmetric conditions. The moving loads are modeled as a series of pulses with a duration equal to the time required for the wheel to pass a fixed point in the pavement. The proposed method of analysis in the new version of the program, SAPSI-M, is an improvement over the existing methods because (a) it treats moving fluctuating loads on an n-layered damped-elastic system; (b) it incorporates such important factors as wave propagation, inertia, and damping effects of the medium, as well as frequency-dependent asphalt concrete properties; and (c) it can handle any load configuration, thus making possible the modeling of multiple wheel configurations of truck axles and airplane landing gears. Theoretical results have shown that the effect of vehicle speed is significant, in part due to the frequency-dependent properties of the asphalt concrete. Comparison with field strain data from full-scale pavement tests has shown excellent agreement.

SAPSI-M: Computer Program for Analyzing Asphalt Concrete Pavements Under Moving Arbitrary Loads

A new solution for the dynamic analysis of asphalt concrete pavements subjected to moving fluctuating loads has been developed. The method builds on an existing model for stationary loads, SAPSI, which uses the complex response method of transient analysis with a continuum solution in the horizontal direction and a finite-element solution in the vertical direction. The structural model is an n-layered damped-elastic medium. The subgrade can be modeled as either a rigid base or a semi-infinite half space. The loads are surface pressure loads, and the analysis is under axisymmetric conditions. The moving loads are modeled as a series of pulses with a duration equal to the time required for the wheel to pass a fixed point in the pavement. The proposed method of analysis in the new version of the program, SAPSI-M, is an improvement over the existing methods because (a) it treats moving fluctuating loads on an n-layered damped-elastic system; (b) it incorporates such important factors as wave propagation, inertia, and damping effects of the medium, as well as frequency-dependent asphalt concrete properties; and (c) it can handle any load configuration, thus making possible the modeling of multiple wheel configurations of truck axles and airplane landing gears. Theoretical results have shown that the effect of vehicle speed is significant, in part due to the frequency-dependent properties of the asphalt concrete. Comparison with field strain data from full-scale pavement tests has shown excellent agreement.

Shakedown of Subgrade Soil Under Repeated Loading

A new model for predicting the performance of a subgrade during soil compaction by rollers as a function of contact pressure and the strength characteristics of soils was developed. The model can be used to develop improved methods of pavement design, considering that the accumulation of plastic strains may continue under additional load repetitions or may cease to increase with time, indicating adaptation or shakedown conditions. To develop the mechanistic model of a subgrade, the homogeneous semi-infinite elastoplastic half-space under repeated loading was considered. By using a Mohr-Coulomb failure criterion, the boundary loads for which shakedown conditions or the steady state will be attained were determined. The residual horizontal normal stresses in the half-space were calculated and were shown to be in agreement with the measured distribution with depth. The theoretical and experimental results as they apply to soil compaction and pavement design problems are discussed. The required compaction contact pressure and the thickness of the compacted zone are estimated. The structural pavement design approach, considering residual normal stresses in a subgrade, is presented.

Shakedown of Subgrade Soil under Repeated Loading

A new model for predicting the performance of a subgrade during soil compaction by rollers as a function of contact pressure and the strength characteristics of soils was developed. The model can be used to develop improved methods of pavement design, considering that the accumulation of plastic strains may continue under additional load repetitions or may cease to increase with time, indicating adaptation or shakedown conditions. To develop the mechanistic model of a subgrade, the homogeneous semi-infinite elastoplastic half-space under repeated loading was considered. By using a Mohr-Coulomb failure criterion, the boundary loads for which shakedown conditions or the steady state will be attained were determined. The residual horizontal normal stresses in the half-space were calculated and were shown to be in agreement with the measured distribution with depth. The theoretical and experimental results as they apply to soil compaction and pavement design problems are discussed. The required compaction contact pressure and the thickness of the compacted zone are estimated. The structural pavement design approach, considering residual normal stresses in a subgrade, is presented.

Contribution of Surface Irregularities to Rolling Contact Plasticity in Bearing Steels

A “two-body” elasto-plastic finite element model of two-dimensional rolling and rolling-plus-sliding has been developed to treat the effect of surface irregularities. The model consists of a smooth cylinder in contact with a semi-infinite half-space that is either smooth or fitted with one of two irregularities: a 0.4 μm deep groove, or a 7 μm deep groove. The model incorporates elastic-linear-kinematic-hardening-plastic (ELKP) and nonlinear-kinematic-hardening-plastic (NLKP) material constitutive relations appropriate for hardened bearing steel and the 440C grade. The calculated contact pressure distribution is Hertzian for smooth body contact, and it displays intense, stationary, pressure spikes superposed on the Hertzian pressure for contact with the grooved and ridged surface. The results obtained for the 0.4 μm deep groove are consistent with those reported by Elsharkawy and Hamrock (1991) for an EHD lubricated contact. The effect of translating the counterface on the half space, as opposed to indenting the counterface on the half-space with no translation, is studied. The stress and strain values near the surface are found to be similar for the two cases, whereas they are significantly different in the subsurface. Efforts have been made to identify the material constitutive relations which best describe the deformation characteristics of the bearing steels in the initial few cycles. ELKP material constitutive relations produce less net plastic deformation in the initial stages, for a given stress, than seen in experiments. NLKP model produces more plasticity than the ELKP model and shows promise for treating the net distortions in the early stages. Artificial indents were inserted on the running track of the cylindrical rolling elements and profilometer measurements of these indents were made, before and after rolling. These preliminary measurements show that substantial plastic deformation takes place in the process of rolling. The deformations of the groove calculated with the finite element model are compared to those measured experimentally.

Self-similar expansion of a warm dusty plasma—I. Unmagnetized case

The self-similar expansion into vacuum of a warm dusty plasma filling semi-infinite half-space is investigated by including the adiabatic equation of state for the dust fluid. Given the complex nature of the problem, for simplicity the analysis is restricted to a one-dimensional expansion of a collisionless plasma in which the dust particles have constant charge. Exact analytical solutions are obtained for the case of the cold dust fluid for positive as well as negative charging of the dust particles. For the warm case, numerical solutions are obtained by solving the relevant self-similar equations. It is shown that an isothermal negatively charged dust fluid expands over a larger distance than an adiabatic one before the density drops to nearly zero. The effects of several plasma parameters such as the density and the temperature ratios on the expansion profiles are discussed.

Self-similar expansion of a warm dusty plasma—II. Magnetized case

The self-similar transverse expansion into vacuum of a warm, magnetized dusty plasma filling semi-infinite half-space is investigated. The analysis is carried out using a MHD model wherein the plasma expansion preserves quasi-neutrality and the magnetic field lines are frozen to the plasma. As the nature of the study is quite complex, for simplicity the analysis is restricted to a one-dimensional expansion of a collisionless plasma in which the dust particles have constant charge. The self-similar governing equations obtained from the model equations are reduced to a quadrature. Analytical solutions for a number of cases of practical interest are obtained both for the isothermal as well as for the adiabatic expansions. The qualitative nature of the density and the velocity profiles during the expansion has been discussed. The magnetic field is found to constrain the transverse expansion of the plasma. For sufficiently strong magnetic fields, the dust density falls off to nearly zero at much shorter distances from the initial plasma-vacuum boundary whereas the expansion velocity is substantially reduced.

Techniques for Assessing the Nonlinear Resilient Response of Unbound Granular-Layered Structures

Abstract The resilient stress-deformation characteristics of aggregates that form granular pavement layers are reviewed. A standardized plate bearing test (PBT) is proposed that separates resilient and permanent deformation readings and detects material stress dependency. A nonlinear, stress-dependent model proposed by Ehrler is used to reproduce the observed deformation behavior of a granular layer subjected to repeated plate bearing tests at different applied stress levels. The use of equipment exploiting the relationship between resilient behavior and the natural frequency of vibration of a mass resting on a semi-infinite half space (the natural vibration method—NVM) is described. The results predict granular pavement stiffnesses when a nonlinear, stress-dependent model is employed, and these are very simply related to PBT results obtained on the same material. The NVM approach may be more practical than the modified PBT procedure proposed as it can be used more quickly.

Electromagnetic waves scattering by screens in an open planar waveguide

The general approach to the accurate calculation of the two-dimensional electromagnetic field scattered by a system of infinitely thin and extended perfectly conducting screens in a dielectric slab (waveguide) is suggested. The slab is situated between two semi-infinite homogeneous half-spaces of different electromagnetic properties and is illuminated by two-dimensional E - or H - polarized electromagnetic waves. The cross-section of the screens system is supposed to be arbitrary. The integral equations of the corresponding diffraction problem are derived and the numerical method of solving them is developed. The integral equations kernel contain the usual free-space Green function and the Sommerfeld-type integrals of two variables. The efficient method of the second kernel part evaluation is proposed. Some field calculation results for screens of rectilinear and elliptic cross-section are presented and discussed.