Local spectrum analysis of field propagation in an anisotropic medium. Part II. Time-dependent fields

In Part I of this two-part investigation [J. Opt. Soc. Am. A 22, 1200 (2005)], we presented a theory for phase-space propagation of time-harmonic electromagnetic fields in an anisotropic medium characterized by a generic wave-number profile. In this Part II, these investigations are extended to transient fields, setting a general analytical framework for local analysis and modeling of radiation from time-dependent extended-source distributions. In this formulation the field is expressed as a superposition of pulsed-beam propagators that emanate from all space-time points in the source domain and in all directions. Using time-dependent quadratic-Lorentzian windows, we represent the field by a phase-space spectral distribution in which the propagating elements are pulsed beams, which are formulated by a transient plane-wave spectrum over the extended-source plane. By applying saddle-point asymptotics, we extract the beam phenomenology in the anisotropic environment resulting from short-pulsed processing. Finally, the general results are applied to the special case of uniaxial crystal and compared with a reference solution.     

A partially debonded ellipsoidal inclusion in an elastic medium. Part I: Stress and displacement fields

The paper calculates the elastic field of an ellipsoidal inclusion which has undergone an internal deformation and has debonded over a part of its boundary from the surrounding medium. The problem is reduced to the solution of a (singular) integral equation for the displacement discontinuity across the debond. The essential steps of a method of solving this equation are outlined. The elastic field is used in the companion paper (Part II) to calculate the stress intensity factors along the edge of debond.     

Mie scattering by an anisotropic object. Part I. Homogeneous sphere

Establishing a vector spherical harmonic expansion of the electromagnetic field propagating inside an arbitrary anisotropic medium, we extend Mie theory to the diffraction by an anisotropic sphere, with or without losses. The particular case of a uniaxial material leads to a simpler analysis. This work opens the way to the construction of a differential theory of diffraction by a three-dimensional object with arbitrary shape, filled by an arbitrary anisotropic material.     

The elastic field around an elliptical crack in an anisotropic medium under an applied stress of polynomial forms

A general form is presented for the stress disturbance caused by an elliptical crack in an anisotropic medium under an arbitrary stress field in the form of polynomials. Somigliana's dislocation method is employed for analysis as Willis [4] did, but a different integral process is taken. The results are expressed in the integral forms defined on the subspace of the surface of a unit sphere. The following theorem is proved: If the displacement discontinuity of the elliptical Somigliana's dislocation has the form where is a homogeneous polynomial of degree $\text{N}$ in the coordinates $\text{x′}{}_1$ and $\text{x′}{}_2$ and $\text{a}{}_1$ and $\text{a}{}_2$ are the semi-axes of the elliptical Somigliana's dislocation, the stresses on the plane of the Somigliana's dislocation are inhomogeneous polynomials in the coordinates, whose terms are of degree $\text{N, (N ? 2}$), $\text{(N ? 4),4.}$     

Collinear periodic cracks in an anisotropic medium

The problem of collinear periodic cracks in an anisotropic medium is examined in this paper. By means of Stroh formalism and the conformal mapping method, we obtain general periodic solutions for collinear cracks. The corresponding stress intensity factors, crack opening displacements and strain energy release rate are found.     

Scattering of radiation in an anisotropic turbulent medium

Scattering of laser radiation on density fluctuations in propagation of radiation through an anisotropic turbulent medium is analyzed. It is shown that the deviation angles in turbulent gas flows at atmospheric pressure equal ∼10⁻⁵–10⁻⁴ rad and can be detected by means of speckle photography. A statistical analysis of two-dimensional fields of deviation angles makes it possible to evaluate three-dimensional density correlation functions in a turbulent flow. It is shown that taking account of the turbulence anisotropy leads to distributions of the laser-radiation intensity over deviation angles that deviate substantially from the Gaussian distribution. Academic Scientific Complex “A. V. Luikov Institute of Heat and Mass Transfer of the National Academy of Sciences of Belarus,” Minsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 72, No. 1, pp. 96–101, January–February, 1999.     

Two-dimensional static deformation of an anisotropic medium

The problem of two-dimensional static deformation of a monoclinic elastic medium has been studied using the eigenvalue method, following a Fourier transform. We have obtained expressions for displacements and stresses for the medium in the transformed domain. As an application of the above theory, the particular case of a normal line-load acting inside an orthotropic elastic half-space has been considered in detail and closed form expressions for the displacements and stresses are obtained. Further, the results for the displacements for a transversely isotropic as well as for an isotropic medium have also been derived in the closed form. The use of matrix notation is straightforward and avoids unwieldy mathematical expressions. To examine the effect of anisotropy, variations of dimensionless displacements for an orthotropic, transversely isotropic and isotropic elastic medium have been compared numerically and it is found that anisotropy affects the deformation significantly.     

The overall sink strength of an inhomogeneous lossy medium. Part I: self-consistent estimates

Steady-state diffusion in a medium containing a continuous distribution of sinks is considered. The medium comprises a matrix, with sink strength $\text{k}{}_2{}^2$, which contains a random array of identical spherical inclusions, with sink strength $\text{k}{}_1{}^2$. On a macroscopic scale, the medium appears homogeneous, with uniform sink strength $\text{k}\text{?}{}^2$. This work is devoted to the estimation of the overall sink strength $\text{k}\text{?}{}^2$, in terms of $\text{k}{}_1{}^2\text{, k}{}_2{}^2$ and the statistics of the distribution of the inclusions. Part I discusses three distinct schemes of self-consistent type. One is based upon a simple embedding procedure and makes no explicit allowance for spatial correlations. The other two make use, in different ways, of the quasicristalline approximation (QCA). Part II develops variational principles which yield bounds for $\text{k}\text{?}{}^2$. The self-consistent estimates are interpreted relative to the variational formulation and explicit numerical results are presented.     

Electro-optic modulation in an anisotropic artificial Kerr medium

We describe the performance of intensity and phase modulators that use an aqueous suspension of polytetrafluoroethylene (PTFE) microparticles. In this medium, the electro-optic effect is caused by the reorientation of anisotropic microparticles in an applied electric field. The intensity modulator was constructed in the Kerr geometry by the use of a sample path length of 20 μm. The response time of the modulator is less than 25 ms, and the depth of modulation was measured to be 28 dB for a switching voltage of 134 V(rms). The switching voltage necessary to achieve a π-phase shift with the phase modulator is less than 30 (Vrms).     

Mechanical response of an anisotropic liquid-saturated porous medium

Summary.  An eigenvalue approach following Laplace transformation has been employed to study the mechanical response of an anisotropic liquid-saturated porous solid. Analytical solutions are obtained in the transformed domain. A numerical inversion technique is used for inverting the Laplace transforms and to get the results in the physical domain, numerically. The results are taken for two types of surface loadings: (i) Impulsive loading, (ii) continuous loading, for a particular model and are discussed graphically. Received July 30, 2001; revised December 9, 2002 Published online: May 20, 2003     

Limited weld residual stress measurements in fatigue crack propagation: Part I. Complete field representation through least-squares finite-element smoothing

An equilibrium‐based, weighted least‐squares algorithm is developed to reconstruct the complete residual stress field tensor throughout the specimen volume using a limited set of residual stress measurements and a priori boundary conditions. Using a set of residual stress data acquired by neutron diffraction over a limited region of an AA2024‐T351 friction‐stir‐welded specimen, the method is used to reconstruct the complete residual stress tensor field. Results indicate that (1) the algorithm is robust and effective in quantifying the complete residual stress field throughout the region of interest; (2) appropriate smoothing parameters are readily obtained so that the reconstructed, complete residual stress field accurately reflects the experimental measurements and (3) the resulting residual stress field is in self‐equilibrium.     

A finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Wave propagation in an asymmetric fibre-reinforced laminated plate. Part II

Summary The propagation of waves in a two-ply laminated plate is examined. Each lamina is composed of the same transversely isotropic, homogeneous, elastic material with the direction of transverse isotropy lying in the plane of the lamina. The two laminae are perfectly bonded to each other in such a way that their respective axes of transverse isotropy are orthogonal. The dispersion equation is derived for waves propagating at an arbitrary angle in the plane of the plate. The long and short wavelength behaviour is examined analytically and subsequently the equation is solved numerically. Results are plotted for the fundamental mode and first two harmonics. In order to gain further understanding of these dispersion curves in certain regions, the stress variation through the plate is examined.The corresponding dispersion curves are obtained for the ideal material which is assumed to be inextensible in the direction of transverse isotropy.With 12 Figures     

Boundary element analysis of moiré fields in anisotropic materials

A procedure for analyzing moiré fringe patterns using boundary elements is presented. The kernels of the boundary integrals are based on anisotropic elastic Green's functions developed for bimaterial problems. The interfacial boundary conditions are incorporated in the Green's functions so the interface does not require discretization. The bimaterial kernels are also appropriate for homogeneous problems as well as degenerate isotropic problems. The moiré fringe data provide full-field displacement information and are analyzed in a least-squares sense. The numerical procedure is shown to be a logical extension of the local collocation method developed for linear elastic fracture mechanics. An example is given to investigate convergence of the method, predictions of stress, and to investigate factors influencing the analysis. It is shown that moiré fields associated with both displacement components are needed for an accurate analysis.     

Vectorial theory of propagation in uniaxially anisotropic media.

We describe propagation in a uniaxially anisotropic medium by relying on a suitable plane-wave angular-spectrum representation of the electromagnetic field. We obtain paraxial expressions for both ordinary and extraordinary components that satisfy two decoupled parabolic equations. As an application, we obtain, for a particular input beam (a quasi-Gaussian beam), analytical results that allow us to identify some relevant features of propagation in uniaxial crystals.     

The onset of Brinkman ferroconvection in an anisotropic porous medium

Onset of ferroconvection in an anisotropic porous layer heated from below is investigated theoretically using modified Brinkman extended-Darcy equation with fluid viscosity different from effective viscosity. The isothermal bounding surfaces of a porous layer are considered to be either free or rigid-paramagnetic/ferromagnetic. The eigenvalue problem is solved exactly for free boundaries, while for realistic rigid-paramagnetic or rigid-ferromagnetic boundaries the critical stability parameters are obtained numerically using the Galerkin method. It is seen that the stability of the system depends on the nature of boundaries and rigid-paramagnetic boundaries are found to be preferred to the ferromagnetic ones as well as free boundaries in controlling ferroconvection in an anisotropic porous layer. It is observed that increase in the value of thermal anisotropy parameter and viscosity ratio is to delay the onset of ferroconvection, while increase in the value of mechanical anisotropy parameter and magnetic number is to hasten the onset of ferroconvection. Moreover, increasing the value of thermal anisotropy parameter and decreasing the value of mechanical anisotropy parameter is to narrow the convection cells.     

Dynamic crack propagation in steel line pipes. Part I: Experimental investigation

An experimental-theoretical research project on dynamic crack propagation on steel line pipes intended for arctic service was developed. The aim of this study is to characterize the dynamic material behavior, to propose line pipe qualification tests, and to identify the variables that govern the dynamic fracture process and their relationships. The first part of this research involves the experimental work and outlines a practical test to assess fitness for service of arctic pipelines. Burst tests of line pipes samples at low temperature were carried out and crack extension vs. time was measured. The axial crack fracture mechanism was quasi-cleavage in the center of the pipe wall with lateral shear lips. A pressure range for LBB (leak before break) condition was obtained. In Part II of this research a new model for the assessment of the dynamic material properties from the analysis of test results obtained in Part I is developed.