Line-integral representations of the displacement and stress fields due to an arbitrary Volterra dislocation loop in a transversely isotropic elastic full space

AbstractTransversely isotropic materials or hexagonal crystals are commonly utilized in various engineering fields; however, dislocation solutions for such special materials have not been fully developed. In this paper, we present a comprehensive study on this important topic, where only Volterra dislocations of the translational type are considered. Based on the potential theory of linear elasticity, we extend the well-known Burgers displacement equation for an arbitrarily shaped dislocation loop in an isotropic elastic full space to the transversely isotropic case. Both the induced displacements and stresses are expressed uniformly in terms of simple and explicit line integrals along the dislocation loop. We introduce three quasi solid angles to describe the displacement discontinuities over the dislocation surface and extract a simple step function out of these angles to characterize the dependence of the displacements on the configuration of the dislocation surface. We also give a new explicit formula for calculating accurately and efficiently the traditional solid angle of an arbitrary polygonal dislocation loop. From the present line-integral representations, exact closed-form solutions in terms of elementary functions are further obtained in a unified way for the displacement and stress fields due to a straight dislocation segment of arbitrary orientation. The non-uniqueness of the elastic field solution due to an open dislocation segment is rigorously discussed and demonstrated. For a circular dislocation loop parallel to the plane of isotropy, a new explicit expression of the induced elastic field is presented in terms of complete elliptic integrals. Several numerical examples are also provided as illustration and verification of the derived dislocation solutions, which further show the importance of material anisotropy on the dislocation-induced elastic field, and reveal the non-uniqueness feature of the elastic field due to a straight dislocation segment

Line-integral representations for the elastic displacements, stresses and interaction energy of arbitrary dislocation loops in transversely isotropic bimaterials

AbstractThe elastic displacements, stresses and interaction energy of arbitrarily shaped dislocation loops with general Burgers vectors in transversely isotropic bimaterials (i.e. joined half-spaces) are expressed in terms of simple line integrals for the first time. These expressions are very similar to their isotropic full-space counterparts in the literature and can be easily incorporated into three-dimensional (3D) dislocation dynamics (DD) simulations for hexagonal crystals with interfaces/surfaces. All possible degenerate cases, e.g. isotropic bimaterials and isotropic half-space, are considered in detail. The singularities intrinsic to the classical continuum theory of dislocations are removed by spreading the Burgers vector anisotropically around every point on the dislocation line according to three particular spreading functions. This non-singular treatment guarantees the equivalence among different versions of the energy formulae and their consistency with the stress formula presented in this paper. Several numerical examples are provided as verification of the derived dislocation solutions, which further show significant influence of material anisotropy and bimaterial interface on the elastic fields and interaction energy of dislocation loops

An assessment of the strength of knots and splices used as eye terminations in a sailing environment

Research into knots, splices and other methods of forming an eye termination has been limited, despite the fact that they are essential and strongly affect the performance of a rope. The aim of this study was to carry out a comprehensive initial assessment of the breaking strength of eye terminations commonly used in a sailing environment, thereby providing direction for further work in the field. Supports for use in a regular tensile testing machine were specially developed to allow individual testing of each sample and a realistic spread of statistical data to be obtained. Over 180 break tests were carried out on four knots (the bowline, double bowline, figure-of-eight loop and perfection loop) and two splices (three-strand eye splice and braid-on-braid splice). The factors affecting their strength were investigated. A statistical approach to the analysis of the results was adopted. The type of knot was found to have a significant effect on the strength. This same effect was seen in both types of rope construction (three-strand and braid-on-braid). Conclusions were also drawn as to the effect of splice length, eye size, manufacturer and rope diameter on the breaking strength of splices. Areas of development and further investigation were identified

Does Treewidth Help in Modal Satisfiability?

Many tractable algorithms for solving the Constraint Satisfaction Problem (CSP) have been developed using the notion of the treewidth of some graph derived from the input CSP instance. In particular, the incidence graph of the CSP instance is one such graph. We introduce the notion of an incidence graph for modal logic formulae in a certain normal form. We investigate the parameterized complexity of modal satisfiability with the modal depth of the formula and the treewidth of the incidence graph as parameters. For various combinations of Euclidean, reflexive, symmetric and transitive models, we show either that modal satisfiability is FPT, or that it is W[1]-hard. In particular, modal satisfiability in general models is FPT, while it is W[1]-hard in transitive models. As might be expected, modal satisfiability in transitive and Euclidean models is FPT.Comment: Full version of the paper appearing in MFCS 2010. Change from v1: improved section 5 to avoid exponential blow-up in formula siz

The Energy-dependent Checkerboard Patterns in Cuprate Superconductors

Motivated by the recent scanning tunneling microscopy (STM) experiments [J. E. Hoffman {\it et al.}, Science {\bf 297}, 1148 (2002); K. McElroy {\it et al.}, Nature (to be published)], we investigate the real space local density of states (LDOS) induced by weak disorder in a d-wave superconductor. We first present the energy dependent LDOS images around a single weak defect at several energies, and then point out that the experimentally observed checkerboard pattern in the LDOS could be understood as a result of quasiparticle interferences by randomly distributed defects. It is also shown that the checkerboard pattern oriented along $45^0$ to the Cu-O bonds at low energies would transform to that oriented parallel to the Cu-O bonds at higher energies. This result is consistent with the experiments.Comment: 3 pages, 3 figure

Exclusion Statistics of Quasiparticles in Condensed States of Composite Fermion Excitations

The exclusion statistics of quasiparticles is found at any level of the hierarchy of condensed states of composite fermion excitations (for which experimental indications have recently been found). The hierarchy of condensed states of excitations in boson Jain states is introduced and the statistics of quasiparticles is found. The quantum Hall states of charged $\alpha$-anyons ($\alpha$ -- the exclusion statistics parameter) can be described as incompressible states of $(\alpha+2p)$-anyons ($2p$ -- an even number).Comment: 4 page

Quasiparticle excitation in and around the vortex core of underdoped YBa_2Cu_4O_8 studied by site-selective NMR

We report a site-selective ^{17}O spin-lattice relaxation rate T_1^{-1} in the vortex state of underdoped YBa_2Cu_4O_8. We found that T_1^{-1} at the planar sites exhibits an unusual nonmonotonic NMR frequency dependence. In the region well outside the vortex core, T_1^{-1} cannot be simply explained by the density of states of the Doppler-shifted quasiparticles in the d-wave superconductor. Based on T_1^{-1} in the vortex core region, we establish strong evidence that the local density of states within the vortex core is strongly reduced.Comment: 5 pages, 3 figure

IBM-1 description of the fission products 108,110,112^{108,110,112}Ru

IBM-1} calculations for the fission products $^{108,110,112}$Ru have been carried out. The even-even isotopes of Ru can be described as transitional nuclei situated between the U(5) (spherical vibrator) and SO(6) ($\gamma$-unstable rotor) symmetries of the Interacting Boson Model. At first, a Hamiltonian with only one- and two-body terms has been used. Excitation energies and $B$(E2) ratios of gamma transitions have been calculated. A satisfactory agreement has been obtained, with the exception of the odd-even staggering in the quasi-$\gamma$ bands of $^{110,112}$Ru. The observed pattern is rather similar to the one for a rigid triaxial rotor. A calculation based on a Hamiltonian with three-body terms was able to remove this discrepancy. The relation between the IBM and the triaxial rotor model was also examined.Comment: 22 pages, 8 figure

A weakly stable algorithm for general Toeplitz systems

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A. Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx = A^Tb, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm

Hamiltonian theory of gaps, masses and polarization in quantum Hall states: full disclosure

I furnish details of the hamiltonian theory of the FQHE developed with Murthy for the infrared, which I subsequently extended to all distances and apply it to Jain fractions \nu = p/(2ps + 1). The explicit operator description in terms of the CF allows one to answer quantitative and qualitative issues, some of which cannot even be posed otherwise. I compute activation gaps for several potentials, exhibit their particle hole symmetry, the profiles of charge density in states with a quasiparticles or hole, (all in closed form) and compare to results from trial wavefunctions and exact diagonalization. The Hartree-Fock approximation is used since much of the nonperturbative physics is built in at tree level. I compare the gaps to experiment and comment on the rough equality of normalized masses near half and quarter filling. I compute the critical fields at which the Hall system will jump from one quantized value of polarization to another, and the polarization and relaxation rates for half filling as a function of temperature and propose a Korringa like law. After providing some plausibility arguments, I explore the possibility of describing several magnetic phenomena in dirty systems with an effective potential, by extracting a free parameter describing the potential from one data point and then using it to predict all the others from that sample. This works to the accuracy typical of this theory (10 -20 percent). I explain why the CF behaves like free particle in some magnetic experiments when it is not, what exactly the CF is made of, what one means by its dipole moment, and how the comparison of theory to experiment must be modified to fit the peculiarities of the quantized Hall problem