Boundary value problems of elasticity theory for plane domains with one-dimensional elastic reinforcements

This article is a translation of an article published in Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No 1, pp 103-114 Jan-Feb 1991.Many authors have examined problems related to the load transmission from an elastic rod to an elastic plane. It was assumed in the majority of investigationa that the stringer is a thin rectilinear rod transmitting only longitudinal forces while the rod contact with the plane is realized along a line. different modifications of sheet contact with a rectilinear tensile stringer considered as an inner stringer of finite length or as an infinite edge stringer were analyzed in [1, 2]. Problems about the reinforcement of holes in a plate by a thin rod of constant section that possesses bending and longitudinal stiffnesses were solved in [3]. The eccentricity of the connection between the shell middle surface and the rod was taken into account in [4] in a study of shells reinforced by thin curvilinear rods. Other models of the one-dimensional element connected to an elastic medium without taking account of its bending stiffness were analyzed in [5, 6]. Solutions of a number of problems with circular reinforcing elements are obtained in [7]. An isotropic finite or infinite, linearly elastic plate reinforced along part or all of the boundary and along certain internal lines by elastic curvilinear rods possessing variable longitudinal and bending stiffnesses, variable curvature and thickness, the eccentricity of the connection to the plate and with an arbitrary transverse section shape symmetric relative to the plate middle surface are studied in this paper. Boundary conditions on the line of plate contact with the inner or edge elastic rods are obtained for the reinforcement models generalizing [1, 2] by using the theory of elastic rods in the case of a plane state of stress. Existence and uniqueness theorems are proved for appropriate boundary value problems; the singularity of the stresses at angles and tips of the rods are proved. The relationships obtained carry over completely to the plane strain problem for an elastic cylinder reinforced by homogeneous cylindrical shells along the generator. Some of the results described here are represented in [8]

Instabilities for a relativistic electron beam interacting with a laser irradiated plasma

The effects of a radiation field (RF) on the unstable modes developed in relativistic electron beam--plasma interaction are investigated assuming that $\omega_{0} >\omega_{p}$, where $\omega_{0}$ is the frequency of the RF and $\omega_{p}$ is the plasma frequency. These unstable modes are parametrically coupled to each other due to the RF and are a mix between two--stream and parametric instabilities. The dispersion equations are derived by the linearization of the kinetic equations for a beam--plasma system as well as the Maxwell equations. In order to highlight the effect of the radiation field we present a comparison of our analytical and numerical results obtained for nonzero RF with those for vanishing RF. Assuming that the drift velocity $\mathbf{u}_{b}$ of the beam is parallel to the wave vector $\mathbf{k}$ of the excitations two particular transversal and parallel configurations of the polarization vector $\mathbf{E}_{0}$ of the RF with respect to $\mathbf{k}$ are considered in detail. It is shown that in both geometries resonant and nonresonant couplings between different modes are possible. The largest growth rates are expected at the transversal configuration when $\mathbf{E}_{0}$ is perpendicular to $\mathbf{k}$. In this case it is demonstrated that in general the spectrum of the unstable modes in $\omega$--$k$ plane is split into two distinct domains with long and short wavelengths, where the unstable modes are mainly sensitive to the beam or the RF parameters, respectively. In parallel configuration, $\mathbf{E}_{0} \parallel \mathbf{k}$, and at short wavelengths the growth rates of the unstable modes are sensitive to both beam and RF parameters remaining insensitive to the RF at long wavelengths.Comment: 23 pages, 5 figure

Stable periodic waves in coupled Kuramoto-Sivashinsky - Korteweg-de Vries equations

Periodic waves are investigated in a system composed of a Kuramoto-Sivashinsky - Korteweg-de Vries (KS-KdV) equation, which is linearly coupled to an extra linear dissipative equation. The model describes, e.g., a two-layer liquid film flowing down an inclined plane. It has been recently shown that the system supports stable solitary pulses. We demonstrate that a perturbation analysis, based on the balance equation for the field momentum, predicts the existence of stable cnoidal waves (CnWs) in the same system. It is found that the mean value U of the wave field u in the main subsystem, but not the mean value of the extra field, affects the stability of the periodic waves. Three different areas can be distinguished inside the stability region in the parameter plane (L,U), where L is the wave's period. In these areas, stable are, respectively, CnWs with positive velocity, constant solutions, and CnWs with negative velocity. Multistability, i.e., the coexistence of several attractors, including the waves with several maxima per period, appears at large value of L. The analytical predictions are completely confirmed by direct simulations. Stable waves are also found numerically in the limit of vanishing dispersion, when the KS-KdV equation goes over into the KS one.Comment: a latex text file and 16 eps files with figures. Journal of the Physical Society of Japan, in pres

Negative differential resistivity in superconductors with periodic arrays of pinning sites

We study theoretically the effects of heating on the magnetic flux moving in superconductors with a periodic array of pinning sites (PAPS). The voltage-current characteristic (VI-curve) of superconductors with a PAPS includes a region with negative differential resistivity (NDR) of S-type (i.e., S-shaped VI-curve), while the heating of the superconductor by moving flux lines produces NDR of N-type (i.e., with an N-shaped VI-curve). We analyze the instability of the uniform flux flow corresponding to different parts of the VI-curve with NDR. Especially, we focus on the appearance of the filamentary instability that corresponds to an S-type NDR, which is extremely unusual for superconductors. We argue that the simultaneous existence of NDR of both N- and S-type gives rise to the appearance of self-organized two-dimensional dynamical structures in the flux flow mode. We study the effect of the pinning site positional disorder on the NDR and show that moderate disorder does not change the predicted results, while strong disorder completely suppresses the S-type NDR.Comment: 10 pages, 1 table, 7 figure

Plasma instability and amplification of electromagnetic waves in low-dimensional electron systems

A general electrodynamic theory of a grating coupled two dimensional electron system (2DES) is developed. The 2DES is treated quantum mechanically, the grating is considered as a periodic system of thin metal strips or as an array of quantum wires, and the interaction of collective (plasma) excitations in the system with electromagnetic field is treated within the classical electrodynamics. It is assumed that a dc current flows in the 2DES. We consider a propagation of an electromagnetic wave through the structure, and obtain analytic dependencies of the transmission, reflection, absorption and emission coefficients on the frequency of light, drift velocity of 2D electrons, and other physical and geometrical parameters of the system. If the drift velocity of 2D electrons exceeds a threshold value, a current-driven plasma instability is developed in the system, and an incident far infrared radiation is amplified. We show that in the structure with a quantum wire grating the threshold velocity of the amplification can be essentially reduced, as compared to the commonly employed metal grating, down to experimentally achievable values. Physically this is due to a considerable enhancement of the grating coupler efficiency because of the resonant interaction of plasma modes in the 2DES and in the grating. We show that tunable far infrared emitters, amplifiers and generators can thus be created at realistic parameters of modern semiconductor heterostructures.Comment: 28 pages, 15 figures, submitted to Phys. Rev.