General Two-dimensional Theory Research Articles

Sliding and rocking response of rigid blocks due to horizontal excitations

To study the dynamic response of a rigid block standing unrestrained on a rigid foundation which shakes horizontally, four modes of motion can be identified, i.e., rest, slide, rock, and slide and rock. The occurrence of each of these four modes and the transition between any two modes depend on the parametric values specified, the initial conditions, and the magnitude of ground acceleration. In this paper, a general two-dimensional theory is presented for dealing with the various modes of a free-standing rigid block, considering in particular the impact occurring during the rocking motion. Through selection of proper values for the system parameters, the occurrence of each of the four modes and the transition between different modes are demonstrated in the numerical examples.

Von Neumann stability analysis of Biot's general two-dimensional theory of consolidation

Von Neumann stability analysis is performed for a Galerkin finite element formulation of Biot's consolidation equations on two-dimensional bilinear elements. Two dimensionless groups—the Time Factor and Void Factor—are identified and these quantities, along with the time-integration weighting, are used to explore the stability implications for variations in physical property and discretization parameters. The results show that the presence and persistence of stable spurious oscillations in the pore pressure are influenced by the ratio of time-step size to the square of the space-step for fixed time-integration weightings and physical property selections. In general, increasing the time-step or decreasing the mesh spacing has a smoothing effect on the discrete solution, however, special cases exist that violate this generality which can be readily identified through the Von Neumann approach. The analysis also reveals that explicitly dominated schemes are not stable for saturated media and only become possible through a decoupling of the equilibrium and continuity equations. In the case of unsaturated media, a break down in the Von Neumann results has been shown to occur due to the influence of boundary conditions on stability. © 1998 John Wiley & Sons, Ltd.

Erratum: Black holes of a general two-dimensional dilaton gravity theory

A general dilaton gravity theory in 1+1 spacetime dimensions, with a cosmological constant $\ensuremath{\lambda}$ and a new dimensionless parameter $\ensuremath{\omega}$, contains as special cases the constant curvature theory of Teitelboim and Jackiw, the theory equivalent to vacuum planar general relativity, the first-order string theory, and a two-dimensional purely geometrical theory. The equations of this general two-dimensional theory admit several different black holes with various types of singularities. The singularities can be spacelike, timelike, or null, and there are even cases without singularities. Evaluation of the ADM mass, as a charge density integral, is possible in some situations by carefully subtracting the black hole solution from the corresponding linear dilation at infinity.

Black holes of a general two-dimensional dilaton gravity theory.

A general dilaton gravity theory in 1+1 spacetime dimensions with a cosmological constant $\lambda$ and a new dimensionless parameter $\omega$, contains as special cases the constant curvature theory of Teitelboim and Jackiw, the theory equivalent to vacuum planar General Relativity, the first order string theory, and a two-dimensional purely geometrical theory. The equations of this general two-dimensional theory admit several different black holes with various types of singularities. The singularities can be spacelike, timelike or null, and there are even cases without singularities. Evaluation of the ADM mass, as a charge density integral, is possible in some situations, by carefully subtrating the black hole solution from the corresponding linear dilaton at infinity.

A transverse shear deformation theory for homogeneous monoclinic plates

A general two-dimensional theory suitable for the static and/or dynamic analysis of a transverse shear deformable plate, constructed of a homogeneous, monoclinic, linearly elastic material and subjected to any type of shear tractions at its lateral planes, is presented. Developed on the basis of Hamilton's principle, in conjunction with the method of Lagrange multipliers, this new theory accounts for an unlimited number of choices of through-thickness displacement distributions, while, starting with the smallest possible number of independent displacement components (five, for a shear deformation theory), it is capable of further operating with as many degrees of freedom as desired. For the particular case of a theory operating with five degrees of freedom, special attention is given to displacement expansions producing symmetric, through thicknes, distributions of transverse shear strain. For the cylindrical bending problem of a specially orthotropic plate, the governing equations of that five-degrees-of-freedom theory are solved and for three different choices of symmetric, through tickness, transverse shear deformation, numerical results are obtained and compared with corresponding results based on the exact three-dimensional solution existing in the literature. The comparisons made show clearly, that the multiple options offered by the new theory, by either suitably altering the displacement expansions or gradually increasing the degrees of freedom involved, will be found useful in future studies dealing with the static and/or dynamic analysis of homogeneous plates.

A general laminated plate theory accounting for continuity of displacements and transverse shear stresses at material interfaces

A general two-dimensional theory, suitable for the static and/or the dynamic analysis of transverse shear deformable laminated plates, is presented. This displacement-based theory is capable of satisfying continuity of both displacements and transverse shear stresses at the plate material interfaces. The derivation of its governing differential equations is based on the application of Hamilton's principle in conjunction with the method of Lagrange multipliers. Moreover, this new theory is capable of accounting for unlimited multiple choices of continuous displacement distributions, through the plate thickness, while, starting with the smallest possible number of independent displacement components (five, for a shear deformation theory), it is capable of further operating with as many degrees of freedom as desired. With such a double-infinite freedom, it is concluded that for the analysis of the particular laminated plate considered one may start with the solution of the governing equations of the 5-degrees-of-freedom theory derived for relatively simple choices of through-the-thickness displacement distributions. Then, either increasing the number of the degrees of freedom or reforming, suitably, the aforementioned displacement distributions, one may iteratively improve the efficiency of the theory until a sufficient degree of accuracy is achieved for the results obtained.

General two-dimensional theory of laminated cylindrical shells

The theory accounts for a desired degree of approximation of the displacements through the thickness, thus accounting for any discontinuities in their derivatives at the interface of laminae. Geometric nonlinearity in the sense of the von Karman strains is also included. Navier-type solutions of the linear theory are presented for simply supported boundary conditions

Phase Conjugation by Degenerate Four-wave Mixing a General Two-dimensional Theory

A general two-dimensional theory of degenerate four-wave mixing is presented for arbitrary relative angle of incidence between the two sets of counterpropagating waves and for arbitrary polarizations of the waves. Coupled vector differential equations are derived and solved analytically under certain restricted conditions. It is shown that a linearly polarized object wave leads to a linearly polarized image wave but with a rotation of the polarization axis.

Dynamic response of a clamped/free hollow circular cylinder under travelling torsional impact loads

Impact-induced vibrations in the casing of a gas centrifuge due to a sudden failure of the spinning rotor (crash) can cause structural disintegrity of the casing. In order to study the influence of the rotor failure behaviour and the impact load histories on the dynamic response of the casing, a simple crash model is proposed in this paper to analyse the transient torsional response due to tangential components of the impact loads. The casing is modeled as a linear-elastic hollow circular cylinder, clamped at the lower end and free at the upper end. The rotor is thought to break up in identical sections in a sequence determined by its fracture behaviour. Each section is assumed to cause an axi-symmetric load distribution at the inner surface of the casing. Therefore the problem is essentially reduced to the analysis of a clamped/free cylinder under travelling torsional impact loads. The problem is solved by representing the impact loads as local pulses acting over the length of the sections. A perturbation method is used to show that the general two-dimensional theory of axi-symmetric torsional wave propagation in circular cylinders, for the problem under consideration, may be approximated by the elementary one-dimensional theory. Solutions are obtained according to the usual modal expansion approach. Measurements of transient torsional responses are shown to be in good agreement with the calculated responses by choosing a suitable shape of the pulses. The effects of travelling velocity and pulse shape are investigated. Finally the transfer of kinetic energy in the rotor to vibrational energy of torsion in the casing is studied.

Shaping the force law in two-dimensional particle-mesh models

A general two-dimensional theory is presented for the choice of charge-sharing schemes in particle-mesh algorithms. An ordered hierarchy of schemes is obtained in which the lowest orders are represented by the familiar nearest-grid-point (NGP) and cloud-in-cell (CIC) schemes. Of the higher order nine-point charge-sharing schemes the triangular-shaped and gaussian-shaped cloud are favored. The theory is also given for the shaping of the short-range force law by the introduction of potential-correction coefficients which modify the multiplying factor used in the calculation ofthe potential by Fourier transform methods. Empirical results are given demonstrating the correctness of the theory and showing that the angular anisotropy of the force law can be reduced from 50% for NGP or CIC to less than the 0.5% by the above methods.

On the beta falloff in junction transistors

It is shown that the βαI c -2law, experimentally verified for junction transistors at large collector currents, may be predicted on the basis of a general two-dimensional theory, valid for any injection level, developed by the authors in a previous paper.