Modification of Bertrand's theorem and extended Runge-Lenz vector

It is shown that for a particle with suitable angular moments in the screened Coulomb potential or isotropic harmonic potential, there still exist closed orbits rather than ellipse, characterized by the conserved aphelion and perihelion vectors, i.e. extended Runge-Lenz vector, which implies a higher dynamical symmetry than the geometrical symmetry O-3. The closeness of a planar orbit implies the radial and angular motional frequencies are commensurable.

The Complete Proof of the Virial Theorem in the Refined TFD Theory for all Electrons of an Atom in a Solid

The complete proof of the virial theorem in refined Thomas-Fermi-Dirac theory for all electrons of an atom in a solid is given.

Radiation effects on the immiscible polymer blend of nylon1010 and high-impact polystyrene (HIPS) I: Gel/dose curves, mathematical expectation theorem and thermal behaviour

This paper studies the radiation properties of the immiscible blend of nylon1010 and HIPS. The gel fraction increased with increasing radiation dose. The network was found mostly in nylon1010, the networks were also found in both nylon1010 and HIPS when the dose reaches 0.85 MGy or more. We used the Charleby-Pinner equation and the modified Zhang-Sun-Qian equation to simulate the relationship with the dose and the sol fraction. The latter equation fits well with these polymer blends and the relationship used by it showed better linearity than the one by the Charleby-Pinner equation. We also studied the conditions of formation of the network by the mathematical expectation theorem for the binary system. Thermal properties of polymer blend were observed by DSC curves. The crystallization temperature decreases with increasing dose because the cross-linking reaction inhibited the crystallization procession and destroyed the crystals. The melting temperature also reduced with increasing radiation dose. The dual melting peak gradually shifted to single peak and the high melting peak disappeared at high radiation dose. However, the radiation-induced crystallization was observed by the heat of fusion increasing at low radiation dose. On the other hand, the crystal will be damaged by radiation. A similar conclusion may be drawn by the DSC traces when the polymer blends were crystallized. When the radiation dose increases, the heat of fusion reduces dramatically and so does the heat of crystallization. (C) 1999 Elsevier Science Ltd. All rights reserved.

Evaluation of the Effectiveness of Representative Methods for Determining Young's Modulus and Hardness from Instrumented Indentation Data

The effectiveness of Oliver & Pharr's (O&P's) method, Cheng & Cheng's (C&C's) method, and a new method developed by our group for estimating Young's modulus and hardness based on instrumented indentation was evaluated for the case of yield stress to reduced Young's modulus ratio (sigma(y)/E-r) >= 4.55 x 10(-4) and hardening coefficient (n) <= 0.45. Dimensional theorem and finite element simulations were applied to produce reference results for this purpose. Both O&P's and C&C's methods overestimated the Young's modulus under some conditions, whereas the error can be controlled within +/- 16% if the formulation was modified with appropriate correction functions. Similar modification was not introduced to our method for determining Young's modulus, while the maximum error of results was around +/- 13%. The errors of hardness values obtained from all the three methods could be even larger and were irreducible with any correction scheme. It is therefore suggested that when hardness values of different materials are concerned, relative comparison of the data obtained from a single standard measurement technique would be more practically useful. It is noted that the ranges of error derived from the analysis could be different if different ranges of material parameters sigma(y)/E-r and n are considered.

Hydrodynamic interaction between two vertical cylinders in water waves

The hydrodynamic interaction between two vertical cylinders in water waves is investigated based on the linearized potential flow theory. One of the two cylinders is fixed at the bottom while the other is articulated at the bottom and oscillates with small amplitudes in the direction of the incident wave. Both the diffracted wave and the radiation wave are studied in the present paper. A simple analytical expression for the velocity potential on the surface of each cylinder is obtained by means of Graf's addition theorem. The wave-excited forces and moments on the cylinders, the added masses and the radiation damping coefficients of the oscillating cylinder are all expressed explicitly in series form. The coefficients of the series are determined by solving algebraic equations. Several numerical examples are given to illustrate the effects of various parameters, such as the separation distance, the relative size of the cylinders, and the incident angle, on the first-order and steady second-order forces, the added masses and radiation-damping coefficients as well as the response of the oscillating cylinder.

Dynamical symmetry of screened Coulomb potential and isotropic harmonic oscillator

It is shown that for the screened Coulomb potential and isotropic harmonic oscillator, there exists an infinite number of closed orbits for suitable angular momentum values. At the aphelion (perihelion) points of classical orbits, an extended Runge-Lenz vector for the screened Coulomb potential and an extended quadrupole tensor for the screened isotropic harmonic oscillator are still conserved. For the screened two-dimensional (2D) Coulomb potential and isotropic harmonic oscillator, the dynamical symmetries SO3 and SU(2) are still preserved at the aphelion (perihelion) points of classical orbits, respectively. For the screened 3D Coulomb potential, the dynamical symmetry SO4 is also preserved at the aphelion (perihelion) points of classical orbits. But for the screened 3D isotropic harmonic oscillator, the dynamical symmetry SU(2) is only preserved at the aphelion (perihelion) points of classical orbits in the eigencoordinate system. For the screened Coulomb potential and isotropic harmonic oscillator, only the energy (but not angular momentum) raising and lowering operators can be constructed from a factorization of the radial Schrodinger equation.

Semi-weight function method on computation of stress intensity factors in dissimilar materials

Semi-weight function method is developed to solve the plane problem of two bonded dissimilar materials containing a crack along the bond. From equilibrium equation, stress and strain relationship, conditions of continuity across interface and free crack surface, the stress and displacement fields were obtained. The eigenvalue of these fields is lambda. Semi-weight functions were obtained as virtual displacement and stress fields with eigenvalue-lambda. Integral expression of fracture parameters, K-I and K-II, were obtained from reciprocal work theorem with semi-weight functions and approximate displacement and stress values on any integral path around crack tip. The calculation results of applications show that the semi-weight function method is a simple, convenient and high precision calculation method.

Wavelike patterns in one-dimensional coupled map lattices

We investigate the existence of wavelike solution for the logistic coupled map lattices for which the spatiotemporal periodic patterns can be predicted by a simple two-dimensional mapping. The existence of such wavelike solutions is proved by the implicit function theorem with constraints. We also examine the stabilities of these wave solutions under perturbations of uniform small deformation type. We show that in some specific cases these perturbations are completely general. The technique used in this paper is also applicable to investigate other space-time regular patterns.

A unified model for dislocation nucleation, dislocation emission and dislocation free zone

In this paper, a unified model for dislocation nucleation, emission and dislocation free zone is proposed based on the Peierls framework. Three regions are identified ahead of the crack tip. The emitted dislocations, located away from the crack tip in the form of an inverse pileup, define the plastic zone. Between that zone and the cohesive zone immediately ahead of the crack tip, there is a dislocation free zone. With the stress field and the dislocation density field in the cohesive zone and plastic zone being, respectively, expressed in the first and second Chebyshev polynomial series, and the opening and slip displacements in trigonometric series, a set of nonlinear algebraic equations can be obtained and solved with the Newton-Raphson Method. The results of calculations for pure shearing and combined tension and shear loading after dislocation emission are given in detail. An approximate treatment of the dynamic effects of the dislocation emission is also developed in this paper, and the calculation results are in good agreement with those of molecular dynamics simulations.

Dislocation nucleation and emission from crack-tip

The problems of dislocation nucleation and emission from a crack tip are analysed based on Peierls model. The concept adopted here is essentially the same as that proposed by Rice. A slight modification is introduced here to identify the pure linear elastic response of material. A set of new governing equations is developed, which is different from that used by Beltz and Rice. The stress field and the dislocation density field can be expressed as the first and second Chebyshev polynomial series respectively. Then the opening and slip displacements can be expanded as the trigonometric series. The Newton-Raphson Method is used to solve a set of nonlinear algebraic equations. The new governing equations allow us to extend the analyses to the case of dislocation emission. The calculation results for pure shearing, pure tension and combined tension and shear loading are given in detail.

Near dipole-dipole effects on the propagation of few-cycle pulse in a dense two-level medium

The propagation behaviors, which include the carrier-envelope phase, the area evolution and the solitary pulse number of few-cycle pulses in a dense two-level medium, are investigated based on full-wave Maxwell-Bloch equations by taking Lorentz local field correction (LFC) into account. Several novel features are found: the difference of the carrier-envelope phase between the cases with and without LFC can go up to pi at some location; although the area of ultrashort solitary pulses is lager than 2 pi, the area of the effective Rabi frequency, which equals to that the Rabi frequency pluses the product of the strength of the near dipole-dipole (NDD) interaction and the polarization, is consistent with the standard area theorem and keeps 2 pi; the large area pulse penetrating into the medium produces several solitary pulses as usual, but the number of solitary pulses changes at certain condition. (C) 2005 Optical Society of America.