Root Singularity Research Articles

Analytic structure of two 1D-transport equations with nonlocal fluxes

We replace the flux term in Burger's equation by two simple alternates that contain contributions depending globally on the solution. In one case, the term is in the form of a hyperbolic equation where the characteristic speed is nonlocal, and in the other the term is in conservation form. In both cases, the nonanalytic is due to the presence of the Hilbert transform. The equations have a loose analogy to the motion of vortex sheets. In particular, they both form singularities in finite time in the absence of viscous effects. Our motivation then is to study the influence of viscosity. In one case, viscosity does not prevent singularity formation. In the other, we can prove solutions exist for all time, and determine the likely weak solution as viscosity vanishes. An interesting aspect of our work is that singularity formation can be viewed as the motion of singularities in the complex physical plane that reach the real axis in finite time. In one case, the singularity is a pole and causes the solution to blow up when it reaches the real axis. In the other, numerical solutions and an asymptotic analysis suggest that the weak solution contains a square root singularity that reaches the real axis in finite time, and then propagates along it. We hope our results will spur further interest in the role of singularities in the complex spatial plane in solutions to transport equations.

Lamellar inhomogeneities in a uniform stress field

Solutions for lamellar inhomogeneities in an infinitely extended isotropic solid subjected to a uniform stress field at infinity are obtained by using Eshelby's equivalent inclusion method. First, two limiting cases are studied: cracks (i.e. inhomogeneities with elastic moduli identically zero), and anticracks (i.e. inhomogeneities with infinitely large elastic moduli). Solutions are obtained for different modes of uniform loading in plane strain (biaxial load and in-plane shear), and anti-plane strain. It is observed that the stress field of an anticrack under biaxial load has an inverse square root singularity at the tip of the anticrack. The stress field arises from the contribution of a planar dislocation distribution and a dislocation dipole distribution. Finally the most general case of inhomogeneities with finite non-zero elastic moduli, which here are called quasicracks, is considered and new solutions are provided. Quasicracks are useful to model fibers in fiber-reinforced materials. Analytical solutions are provided for the stress field in the matrix, including the interface shear stress.

Characterization of orthogonal polynomials with respect to a functional

In this paper we study questions of existence, uniqueness and characterization of polynomials orthogonal with respect to a linear, not necessarily definite, functional L defined on the set of Laurent polynomials. First we characterize with the help of the function F L(z):= L( (y + z) (y − z) ) polynomials orthogonal with respect to L . Using this characterization, which has wide applications, we are able to settle the question of existence and uniqueness of orthogonal polynomials. The uniquely determined orthogonal polynomials will be called basic orthogonal polynomials. It is to be pointed out that, in contrast to the real case, there are natural numbers n μ such that there exist no polynomials of degree n μ which are orthogonal with respect to L , if L is indefinite and if we have “orthogonality-jumps” greater than 1. Furthermore the functional to which the basic orthogonal polynomials of the second kind are orthogonal is determined. Finally, we get explicit expressions for all basic orthogonal polynomials with respect to a “weight function” the support of which consists of several arcs of the unit circle, changes sign from arc to arc and has square root singularities at the boundary points of the arcs. These polynomials can be considered as the basic polynomials in describing and generating orthogonal polynomials with periodic reflection coefficients.

Electron spectral density in a disordered system of hard sphere scatterers

The spectral density of an electron propagating in a disordered system of hard sphere scatterers is studied by use of a self-consistent approximation for the self-energy. The spectral density at fixed wavenumber is found to be a single-peaked function of energy. The approximation yields a sharp wavenumber-dependent band edge. For large wavenumbers the spectral density is well approximated by a Lorentzian, but for small wavenumbers it is dominated by a characteristic square root singularity at the band edge.

Tunneling spectroscopy of normal metals with charge-density or spin-density waves.

Tunneling current-voltage characteristics (CVC) are calculated for symmetrical and nonsymmetrical junctions made up of metals with charge-density or spin-density waves and a distortion of the Fermi-surface nesting sections described by the order parameter \ensuremath{\Sigma}. For the symmetrical junction the CVC are odd functions of the bias voltage V and do not depend on the sign of \ensuremath{\Sigma}. The differential conductivities have root singularities at eV=\ensuremath{\Sigma} and jumps at eV=2\ensuremath{\Sigma}. For the nonsymmetrical junction the CVC depend on the sign of \ensuremath{\Sigma}. Relevant differential conductivities are nonsymmetrical, with one branch being smooth and another having a root singularity at eV=\ensuremath{\Sigma}. A qualitative agreement exists with the tunneling and point-contact spectroscopy measurements for layered dichalcogenides, ${\mathrm{NbSe}}_{3}$, and ${\mathrm{URu}}_{2}$${\mathrm{Si}}_{2}$.

The crack tip region in hydraulic fracturing

We present analytical tip region solutions for fracture width and pressure when a power law fluid drives a plane strain fracture in an impermeable linear elastic solid. Our main result is an intermediate asymptotic solution in which the tip region stress is dominated by a singularity which is particular to the hydraulic fracturing problem. Moreover this singularity is weaker than the inverse square root singularity of linear elastic fracture mechanics. We also show how the solution for a semi-infinite crack may be exploited to obtain a useful approximation for the finite case.

Asymptotic Stress Field of a Mode III Crack Growing Along an Elastic/Elastic Power-Law Creeping Bimaterial Interface

The asymptotic stress field near the tip of a crack subjected to antiplane shear loading is analysed. The crack is growing quasi-statically along an elastic/elastic power-law creeping bimaterial interface. We find there is a separable solution with the following characteristics: for n < 3, where n is the power-law creeping exponent, the asymptotic stress field is dominated by the elastic strain rates and has an inverse square root singularity, r−1/2, where r is the distance from the current crack tip. For n ≥ 3, the near-tip stress and strain fields has a singularity of the form r−1/(n−1). The strength of this field is completely specified by the current crack growth rate, besides material properties, and is otherwise independent of the applied load and of the prior crack growth history.

The standard SU (2)�U(1) model in an external magnetic field at finite temperature and nonzero chemical potential

The thermodynamic potential (TDP) of the boson sector of the Weinberg-Salam model in an external uniform magnetic field at finite temperature and nonzero chemical potential is calculated in one-loop approximation. In the high temperature limit with the use of Mellin summation technique explicit expressions for the thermodynamic potential are obtained to analyze its new minimum, different from the ordinary Higgs one. This new minimum is due to the root singularity in the oneloop expression for the TDP, originating as a consequence of the tachyonic mode in the spectrum of gauge boson in the magnetic field. It is demonstrated that in the high temperature limit the Higgs minimum possesses properties common with the ordinary superconductors and Abelian Higgs model. The influence of the magnetic field on the Higgs minimum is indirect and has a sufficiently complicated character. The presence of the non-zero chemical potential lowers the critical temperature leading to the disappearance of the Higgs minimum.

On the non-integrability of a family of Duffing-van der Pol oscillators

We investigate the non-integrability of a family of Duffing-van der Pol oscillators x+ alpha x(x2-1)+x+ beta x3= gamma cos omega t by studying the analytic properties of the dynamics in complex time. We find that the solutions of (*) have no worse than algebraic singularities at t*, with only (t-t*)1/2 terms present in their series expansions, unlike, for example, the alpha =0 Duffing case, where, typically, log(t-t*) terms arise. Still, when integrating (*) around long enough contours, a remarkably intricate pattern of square root singularities emerges, on different sheets, which appears to prevent solutions from ever returning to the original sheet. Such evidence of infinitely-sheeted solutions, termed the ISS property, has also been observed in a number of Hamiltonian systems and is illustrated here on a simple example of a single, first-order differential equation. We suggest that the ISS property is a necessary condition for non-integrability, i.e. non-existence of a complete set of analytic, single-valued constants of the motion, which would permit the complete integration of a dynamical system in terms of quadratures.

Using stokes expansion for natural convection inside a two-dimensional cavity

We consider the two-dimensional problem of steady natural convection in a circular cavity filled with viscous fluid and subjected to a cosine temperature variation on the boundary. The solution is expanded for low Grashof number and extended to 35 terms by computer. Analysis shows that convergence is limited by a square root singularity on the negative real axis of the Grashof number. An Euler transformation and extraction of the leading, secondary and tertiary singularities at infinity render the series accurate for all values of the Grashof number. The major conclusion of this investigation is that, while in previous works using the method of series extension-for example the slowly rotating pipe studied by Mansour (1985, J. Fluid Mech., 150, 1–24)- the limiting values of the quantities of interest for high values of the similarity parameters disagreed with other techniques, in the present work partial agreement with other investigations has been achieved.

Steady-state heat conduction problem of the interface crack between dissimilar anisotropic media

A solution is given for the steady-state heat conduction problem of the interface crack between dissimilar anisotropic media. Based on the Hilbert problem formulation and a special technique of analytical continuation, exact expressions are obtained for the temperature and temperature gradients for both the heat flux prescribed and temperature prescribed boundary conditions. It is found that the temperature gradients near the crack tip always possess the characteristic inverse square root singularity in rectilinearly anisotropic bodies provided the heat conductivity coefficients are positive definite and symmetric. Moreover, the temperatures or temperature gradients associated with the dissimilar media can be easily obtained from the corresponding problem associated with the homogeneous media by a simple substitution. Special examples are given to the homogeneous and isotropic materials and the solutions reduce to the results given in the literature. The strength of heat flux singularities related to the crack dimension is also discussed.

A contact problem with friction and adhesion for an elastic layer with stiffeners

The contact of an elastic layer with an infinite stiffener to which a uniform constant normal load and a concentrated tangential force are applied, is considered. In the neighbourhood of the point of application of this force, on the line of the contact of the stiffener with the layer a segment is separated out, on which the effect of the Coulomb friction is taken into account. Outside this segment the stiffener and the layer are under conditions of complete adhesion. The problem is reduced to a Prandtl-type integro-differential equation specified on two semi-infinite segments, for whose solution an analytical method is proposed. The method is based on reducing the equation to a vectorial Riemann problem and then to an algebraic Poincaré-Koch system. The latter admits of an explicit solution and also inversion through recurrent relations that are effective when using numerical computations. The length of the Coulomb friction zone and the contact tangential stresses in the adhesion zone are determined. Unlike Melan's problem [1] the contact stresses have no logarithmic singularity and are continuous everywhere in the contact area. The solution of the problem of the contact of a layer with a finite stiffener subject to a uniform pressure along the whole length and to an extension by forces concentrated at the tips is also obtained. The contact area is divided into an intermediate zone of adhesion and two zones of coulomb friction. The problem is reduced to a Prandtl-type integro-differential equation specified on the segment, and it is solved by analogy with the solution of the equation of the first problem. Such a formulation of the problem implies that the contact tangential stresses are bounded at the tips of the stiffener and are continuous at the points of the boundary between the zones of adhesion and Coulomb friction. When adhesion occurs along the entire line of contact the tangential stresses, in general, have a root singularity [2]. In the problem of the contact of a plane punch with a half-plane under conditions of friction and adhesion, the contact stresses at the tips of the punch have [3] a power singularity (that differs from a root one).

Continuously infinite commensurate-incommensurate phase transition of a two-dimensional competing Ising model

We consider the critical behavior of a two-dimensional competing axial Ising model including interactions up to third nearest neighbors in one direction. On the basis of a low-temperature analysis relating the transfer matrix of this model with the Hamiltonian of theS = 1/2XXZ chain, it is shown that the usual square root singularity dominating commensurate-incommensurate phase transitions of two-dimensional systems merges into a continuously infinite transition for certain relations among the coupling parameters. The conjectured equivalence between the maximum eigenstate of the transfer matrix associated with this model and the ground state of theXXZ chain is tested numerically for lattice widths up to 18 sites.

Photoinduced absorption from triplet excitations in poly(2-methoxy, 5-(2′-ethyl-hexyloxy)- p-phenylene vinylene) oriented by gel-processing in polyethylene

We use the technique of steady state photoinduced absorption to study the long-lived photoexcitations in the conjugated polymer poly(2-methoxy, 5-(2′-ethyl-hexyloxy)- p-phenylene vinylene) (MEH-PPV) as a dilute component in ultrahigh molecular weight polyethylene (PE) oriented by gel-processing. In contrast to the three states found previously in films of alkoxy derivatives of poly( para-phenylene vinylene), the photoinduced absorption spectrum of MEH-PPV/PE blends has only one subgap absorption with peak at 1.4 eV. The single peak in the photoinduced absorption spectrum and the relatively long lifetime, as determined from the dependence of the signal on the modulation frequency, are consistent with the identification of the photoexcitation as a long-lived triplet exciton. We conclude that the formation of charged bipolarons is suppressed in the aligned blends due to the increased order and dilution. By using a modified double-modulation technique to separate the triplet-to-excited-triplet-state (TT ∗) absorption from the intense luminescence, we are able to determined the lineshape of the TT ∗ absorption. We find an asymmetric lineshape which is well described as a broadened square root singularity, indicative of a system with a one-dimensional electronic structure. This lineshape lacks well-defined vibronic structure indicating little configurational relaxation on exciting from T to T ∗.

Fracture mechanics for piezoelectric ceramics

We Study cracks either in piezoelectrics, or on interfaces between piezoelectrics and other materials such as metal electrodes or polymer matrices. The projected applications include ferroelectric actuators operating statically or cyclically, over the major portion of the samples, in the linear regime of the constitutive curve, but the elevated field around defects causes the materials to undergo hysteresis locally. The fracture mechanics viewpoint is adopted—that is, except for a region localized at the crack tip, the materials are taken to be linearly piezoelectric. The problem thus breaks into two subproblems: (i) determining the macroscopic field regarding the crack tip as a physically structureless point, and (ii) considering the hysteresis and other irreversible processes near the crack tip at a relevant microscopic level. The first Subproblem, which prompts a phenomenological fracture theory, receives a thorough investigation in this paper. Griffith's energy accounting is extended to include energy change due to both deformation and polarization. Four modes of square root singularities are identified at the tip of a crack in a homogeneous piezoelectric. A new type of singularity is discovered around interface crack tips. Specifically, the singularities in general form two pairs: r1/2±iεand r1/2±iε, where ε. and k are real numbers depending on the constitutive constants. Also solved is a class of boundary value problems involving many cracks on the interface between half-spaces. Fracture mechanics are established for ferroelectric ceramics under smallscale hysteresis conditions, which facilitates the experimental study of fracture resistance and fatigue crack growth under combined mechanical and electrical loading. Both poled and unpoled fcrroelectrie ceramics are discussed.

The characteristics of viscoplastic effects in dynamic crack propagation in nuclear reactor pressure vessel steels

A small-scale yielding analysis is performed for steady state, rapid crack propagation in order to quantify the influence of viscoplasticity in A533B steel. The Bodner-Partom viscoplastic model is used in this analysis. Because the elastic strain rates dominate the viscoplastic strain rates near the crack tip, an inverse square root singularity in the stress exists at the tip. Estimates for the strength of the crack-tip field in terms of the strength of the remote elastic field and the material properties of the A533B steel are established. The plastic shielding is sufficient to reduce the strength of the crack-tip field to an order of magnitude smaller than that of the remote field. A finite-element analysis is also performed to establish the size of the zone of dominance for the crack-tip field. The crack-tip zone of dominance is typically several orders of magnitude smaller than the extent of the plastic zone. Aside from posing rather severe finite-element modeling problems, this large disparity brings into the question the relevance of the crack-tip region when compared to the size of the fracture process zone.

Constant-speed propagation of a penny-shaped interface crack under pure torsion

A transient problem of propagation of a penny-shaped crack in the plane interface between two different media is solved. The body is twisted by a torque at infinity that tends to shear the bond between the two media. A small initial flaw is assumed to expand uniformly in the plane of the interface at a speed less than the smaller of the SH wave speeds of the two media. The solution is carried out by a combination of two methods: a method of rotational superposition of two-dimensional solutions and the Smirnov-Sobolev method of self-similar potentials for twodimensional problems. The dynamic stress intensity factor and the crack tearing displacement are determined. It is found that the stress field near the crack tip has a square root singularity. The maximum crack-tearing displacement is found to occur at close to three-fourths of the way from the crack origin to the running crack tip.

Computation of SIF (stress intensity factor) of corner crack in interior wall of nozzle of nuclear vessel

The computation of the stress intensity factor (SIF) of a corner crack in the interior wall of a nozzle for a steel nuclear vessel is a complicated 3-D case. The finite element method is a powerful tool for this case. An improved 3-D collapsed isoparametric singular element with quarter-points is presented in this paper and the required inverse square root singularity in these vertical planes of a crack is derived. The method of construction for the corresponding transitional element that was extensively used in the 3-D case is also presented. The Williams formula as well as the least-squares linear fitting method was used to compute K I according to the displacements near the crack front. The SIF of a corner crack in the interior wall of a nozzle of a Reactor Pressure Vessel (RPV) of a typical 300-MW nuclear power plant was computed and verified by the 3-D photo-elastic test and the diffusion of light test. To analyse the effect of various factors on the SIF, the computations were carried out on 12 models.

The Energy Spectrum of Fronts: Time Evolution of Shocks in Burgers‚ Equation

Andrews and Hoskins used semigeostrophic theory to argue that the energy spectrum of a front should decay like the −8/3 power of the wavenumber. They note, however, that their inviscid analysis is restricted to the very moment of breaking; that is, to the instant t = tβ when the vorticity first becomes infinite. In this paper, Burgers' equation is used to investigate the postbreaking behavior of fronts. We find that for t > tβ, the front rapidly evolves to a jump discontinuity. Combining our analysis with the Eady wave/Burgers„ study of Blumen, we find that the energy spectrum is more accurately approximated by the −8/3 power of the wavenumber, rather than by the k−2 energy spectrum of a discontinuity, for less than two hours after the time of breaking. We also offer two corrections. Cai et al. improve a pseudospectral algorithm by fitting the spectrum of a jump discontinuity. This is not legitimate at t = tβ because the front initially forms with a cube root singularity and its spectral coefficients decay at a different rate. Whitham claims that for t > tβ, the characteristic equation has two roots. We show by explicit solution that there are actually three.

On nonintegrable systems with square root singularities in complex time

Dynamical arguments are presented which suggest that there are nonintegrable systems without clustering of singularities, infinite singularities, or singularities with an infinite number of branches in the complex t-plane. An example with only square root singularities is studied, for which strong numerical evidence is presented for nonintegrability and infinitely sheeted solutions.