Three-dimensional Half-space Research Articles

Passing stiffness anisotropy in multilayers and its effects on nanoscale surface self-organization

A binary monolayer adsorbed on a solid surface can separate into distinct phases that further self-assemble into various two-dimensional patterns. The surface stresses in the two phases are different, causing an elastic field in the substrate. The self-organization minimizes the combined free energy of mixing, phase boundary, and elasticity. One can obtain diverse patterns by using substrates with various crystalline symmetries. Consider the pattern of a set of periodic stripes. The stripe orientation depends on the anisotropy in surface stress, substrate stiffness, and phase boundary energy. A more powerful and flexible way is to use a layered substrate. Surface properties designed for the applications of those patterns can be obtained by choosing appropriate materials and structures for the monolayer and the top layer of the substrate. The subsequent layers of the substrate provide the required stiffness anisotropy, the effect of which is passed to the monolayer patterns through the elastic field. We solve the elastic field in the anisotropic, heterogeneous, three-dimensional half-space by using the Eshelby–Stroh–Lekhnitskii formalism and the Fourier transformation. Depending on the thicknesses and the degrees of the stiffness anisotropy of the substrate layers, the lowest energy stripes can have tunable equilibrium size and orientation. We also discuss other possibilities of manipulating the phase patterns by engineering the elastic field.

Calculation of the Field of a Flaw in Three-Dimensional Half-Space

A general technique for calculating the field of an arbitrary flaw in three-dimensional half-space is proposed. The technique is based on the solution of a magnetostatic problem only on the surface of the flaw. The calculations are carried out for a uniform field normal to the metal surface.

Layer-stripping and parameter reconstruction for a hyperbolic equation in a three-dimensional inhomogeneous half-space

A time domain wave-splitting approach to the inverse problem for a hyperbolic differential equation in an inhomogeneous half-space is considered (the wave velocity is assumed to be constant). A layer stripping method is used to reconstruct the parameter in a non-iterative way, using the reflected field which is produced by an exterior step-function point source. Local continuation formulas of the field are used, and in particular the continuation formulas for the field in the vicinity of a discontinuous wavefront are derived. Numerical results for data continuation and the reconstruction of the dissipation coefficient are presented.

On the coupling of 3D bem and fem frame model applied to elastodynamic analysis

This work presents the coupling of framed structures approached by finite elements with three-dimensional bodies represented by the boundary element method. The coupling is particularly dedicated to analyse three-dimensional half space stiffened by piles and other composite domain problems, for which static and dynamic cases have been taken into account. Some numerical examples are analysed to point out the power and the accuracy of the proposed scheme.

Fatigue Crack Growth Behavior in Cemented Carbides and Ceramics due to Repeated Rolling Contact with Thermal Stresses by Frictional Heating.

This paper deals with the surface fatigue crack growth behavior in cemented carbides and ceramics under repeated rolling contact with frictional heating. In this analysis, a surface crack is modeled as an inclined planar surface crack in a three-dimensional half-space. Rolling contact is simulated as a long and narrow distributed load with both normal and shear components moving with constant veloccity. First, the three modes of stress intensity factors along the crack contour are analyzed. Next numerical calculations of the stress intensity factors are carried out for cemented carbides and ceramics. On the basis of these data, numerical results for crack growth contours are given based on a modified Paris power law. The threshold values are obtained by rotary and plane bending fatigue tests for WC-Co cemented carbides and Si3N4 ceramics. Morever, the thermal effects on the crack propagation life are considered for each material.

Peculiarities of Quantum Radiation in Three Dimensions from Moving Mirrors with High Refractive Index

In the perfect reflector limitn→∞, the photon pairs radiated by a moving three-dimensional half-space with nondispersive refractive index[formula]show peculiarities unheralded in 1D. The radiated power (per unit area) is different forn2=εandn2=μ(cases 1 and 2, respectively). The half-space is taken to oscillate harmonically either parallel or normally to its surface, and to interact with a massless scalar field; for normal motion in case 1 (2) the power is then identical to the electromagnetic power flowing into transverse-electric (transverse-magnetic) modes from a true dielectric withn2=ε.Even in the limit, the power radiated inwards into the perfectly reflecting material remains finite, except for sideways motion in case2. This paradoxical behaviour stems from the unfamiliar kinematics of such photon pairs, admitting peculiar and dimensionality-dependent conspiracies between vanishing matrix elements and diverging phase-space volumes. For outward radiation from normal oscillations, the perfect-reflector limits tally with those implied by results already known for the Maxwell field. For sideways oscillations, the outward power vanishes asn→∞, at different rates in cases 1 and 2; but it is always dominated by photons whose partners have travelled inwards. Appendices outline the opposite regime of dilute media withnclose to unity, and contrast 3D with 1D.

On the inversion of subsurface residual stresses from surface stress measurements

Matter removal is necessary in order to measure the stress tensor at an interior point of a solid by means of X-ray diffraction. This induces stress redistribution so that the measured stress is different from the original residual stress. This paper addresses the problem of reconstructing the original stress field from the measurements. A method of solution is described for the reconstruction problem, which leads to explicit inversion formulae for the case of some simple geometries. Closed-form expressions are obtained for the two- and three-dimensional half-space. For bodies of arbitrary shape, a numerical algorithm is proposed.

Discrete Wave‐Number Boundary‐Element Method for 3‐D Scattering Problems

The discrete wave-number boundary element method is extended to apply to three-dimensional elastodynamic problems. The method—which was first introduced and applied to two-dimensional problems—combines the direct boundary-element method with the discrete wave-number Green’s function. The advantage of this method is its accuracy and flexibility for boundary configurations. As a demonstration of the applicability of the method, the responses of a hemispherical canyon for incident SH, SV, and P waves in a three-dimensional half-space are presented in time domain as well as in frequency domain. Time histories of seismic motion along the surface in and around the canyon are studied for incident SH, SV, and P waves with the shape of a Ricker wavelet. Many of the physical phenomena—such as diffracted waves (called creeping waves) propagating inside the canyon and Rayleigh waves generated at the edges of the canyon and propagating outward, away from it—which have been observed in the response of the two-dimensional model (i.e., semicircular canyon in an elastic half-space), are also observed in the response of the three-dimensional model (i.e., hemispherical canyon in an elastic half-space). The mathematical formulation presented is general. Nevertheless, the axisymmetric geometry of the hemispherical canyon is exploited to develop a very efficient computational algorithm to solve the problem. Since a nonaxisymmetric scatterer may always be assumed to be embedded in a region that is defined by an axisymmetric imaginary boundary (such as a hemisphere), application of this algorithm can be extended to nonaxisymmetric scatterers.

Time-harmonic solutions of some dissipative problems for Maxwell’s equations in a three-dimensional half space

In constructing solutions of steady-state wave propagation problems, a common procedure is to assume that the frequency has a small imaginary part and, with an eventual solution in hand, to let this imaginary part go to zero − the principle of limiting absorption. There are three basic problems involved here. The first is to establish the principle of limiting absorption itself, i.e., to show in a rigorous manner that a steady-state solution can actually be obtained in this fashion

THREE-DIMENSIONAL ANALYSIS OF THERMAL EFFECTS ON SURFACE CRACK PROPAGATION IN ROLLING CONTACT

Abstract This analysis investigates the thermal effects on crack growth in rolling contact fatigue surface pitting. Here the prepitting crack is modeled as an inclined planar surface crack in a three-dimensional half-space. Rolling contact is simulated as a line load with both normal and shear components, moving with constant velocity over the surface of the half-space. A frictional heat input in the contact region is included to incorporate the thermal loading. The body force method for three-dimensional fracture mechanics is utilized to determine the three modes of stress intensity factors along the crack contour, which quantify crack growth. Numerical results for the stress intensity factors over a loading cycle are given for a semicircular crack. The effects of two thermal parameters, one representing heat input strength and the other representing velocity of the contact region, on the stress intensity factors are determined.

Radiation and Scattering of Acoustic Waves from Bodies of Arbitrary Shape in a Three-Dimensional Half Space

This paper is concerned with the application of the Boundary Integral Equation (BIE) method to acoustic radiation and scattering in a three-dimensional half space. Problems in this class include the radiation of sound from vibrating machines near reflecting surfaces and the scattering of sound from objects in air or water that are in close proximity to rigid or free surfaces. Written in the usual way, the Helmholtz integral formula involves integrations over all surfaces that border the acoustic domain. For half-space problems these integrations include the infinite plane that forms the half-space boundary. For half-space radiation and scattering problems, it is shown that the integrals on the infinite plane may be removed by using a modified Green’s function in the Helmholtz integral. Thus, all element discretization and numerical approximation are confined to the body or obstacle. This feature simplifies the engineering effort required to set up a problem and improves the accuracy of the results. Gone also is the question of where to terminate the discretization of the infinite plane. Examples of radiation and scattering near a rigid boundary are used as illustration of the method.

Electromagnetic waves in a three-dimensional half space with a dissipative boundary

We extend to dissipative problems the technique developed in [5, 12-141 for representing solutions of selfadjoint plane-boundary problems. It is clear that the method developed below is applicable to dissipative problems for an entire class of first-order symmetric-hyperbolic systems [4], but to avoid the stale format of strings of formal hypotheses we prefer to illustrate the method for a simple example of physical importance: the Cauchy problem for Maxwell’s equations in a half space with the Leontovich boundary condition. This problem is solved by a (C,) contractive semigroup for which we obtain an explicit representation in the form of superpositions of reflected planewave and surface modes (Theorem 4.16). To formulate the problem, let E = diag(el,,pI,) be the diagonal 6 x 6 matrix with diagonal elements E, ,U which are the electromagnetic constants of a homogeneous, isotropic, loss-free medium filling the half space R?, = {x E R3: x3 > -a}, a > 0. Let A(D), Dj= -iaj, j= 1, 2, 3, be the 6 X 6 matrix operator

Phase transitions in adsorbed polymer systems

The conformational properties of an adsorbed polymer obtained elsewhere by means of statistical mechanics are discussed. The polymer is modelled by a string of beads (monomer units) connected by freely rotating bonds and is free to move in a three-dimensional continuum half-space. The conformation properties: the number and the mean square end-to-end distance of beads on the wall, the density of beads of the wall, and the centre-of-mass of the polymer, are tabulated as a function of the adsorption energy parameter W.Like lattice models, this continuum theory predicts a phase transition at a critical value of W=Wc However, unlike lattice models, it is amenable to quantitative analysis. Numerical results for W are presented for real polymer–solvent/substrate systems assuming that the adsorption energy of a monomer unit is due to dispersion interactions.It is found that physically accessible temperature induced adsorption/desorption phase transitions are possible. Also suggested is the theoretical possibility of these phase transitions being induced by varying the dielectric properties of the solvent, e.g. by appropriate admixture.