A theory of dislocations and disclinations in elastic plates

The problem of the stress state of a thin elastic plate, containing dislocations and disclinations, is considered using Kirchhoff's theory. The problem of the equilibrium of a multiply connected plate with Volterra dislocations with specified characteristics is formulated. The problem of the flexure of an annular slab resulting from a screw dislocation and a twisting disclination is solved. The solutions of problems of concentrated (isolated) dislocations and disclinations in an unbounded plate as well as the dipoles of dislocations and disinclinations are found. It is shown that a screw dislocation in a thin plate is equivalent to the superposition of two orthogonal dipoles of torsional disclinations. By taking the limit from a discrete set of defects to their continuous distribution, a theory of thin plates with distributed dislocations and disclinations is constructed. Solutions of problems of the flexure of circular and elliptic plates with continuously distributed disclinations are obtained. An analogy is established between the problem of the flexure of a plate with defects and the plane problem of the theory of elasticity with mass forces, and also between a plane problem with dislocations and disclinations and the problem of the flexure of a plate with specified distributed loads.     

The linear theory of dislocations and disclinations in elastic shells

A stress state of a thin linearly elastic shell containing both isolated as well as continuously distributed dislocations and disclinations is considered using the classical Kirchhoff–Love model. A variational formulation of the problem of the equilibrium of both a multiply connected shell with Volterra dislocations as well as shells containing dislocations and disclinations distributed with a known density is given. The mathematical equivalence between the boundary-value problem of the stress state of a shell caused by distributed dislocations and disclinations and the boundary-value problem of the equilibrium of a shell under the action of specified distributed loads is established. A number of problems on dislocations and disclinations in a closed spherical shell is solved. The problem of infinitesimally deformations of a surface when there are distributed dislocations is formulated.     

Topological part of the interaction of phonons with dislocations and disclinations

Dnepropetrovsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 3, pp. 428–446, June, 1990.     

Topological interaction of phonons with dislocations and disclinations. II. The scattering problem

Dnepropetrovsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 86, No. 1, pp. 81–97, January, 1991.     

Radiative transfer in a two-dimensional rectangular annulus medium

The radiative transfer equation in a two-dimensional rectangular annulus medium is solved numerically. The numerical method is based on a finite difference scheme and a product quadrature discrete-ordinate scheme. The discretized equation of transfer is solved iteratively to give the radiation intensity. The medium is assumed to absorb, emit, and anisotropically scatter radiation. It is exposed to diffusely emitting and diffusely reflecting boundaries. The results of the total intensity for various radiative parameters are presented. The method can be modified easily to solve the rectangular medium without the annulus. Our results in this case compare very well with those of Crosbie et al. [1], Thynell et al. [2], and Wu [3].     

Persistence of a two-dimensional super-Brownian motion in a catalytic medium

Summary. The super-Brownian motion X ^ϱ in a super-Brownian medium ϱ constructed in [DF97a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ([DF97a]) and three ([DF97b]). Here we fill the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d≤3), given the catalyst, the variance of the process is finite `uniformly in time'. This is in contrast to the `classical' super-Brownian motion where this holds only in high dimensions (d≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction. Received: 21 November 1996 / In revised form: 31 March 1997     

Modeling the field of a cable in a two-dimensional nonhomogeneous medium

In this article we model the electromagnetic field of a cable on the surface of a two-dimensional nonhomogeneous medium used for electromagnetic sounding. The model of the medium represents the sea. Calculations are reported describing the coast effect on the electromagnetic field. An iterative field calculation method is proposed, which reduces the computing effort. The effect of sub-seafloor nonhomogeneities on the electromagnetic field measured at the seafloor is investigated numerically.     

Stability theory for a two-dimensional channel

A scheme for deriving conditions for the nonlinear stability of an ideal or viscous incompressible steady flow in a two-dimensional channel that is periodic in one direction is described. A lower bound for the main factor ensuring the stability of the Reynolds–Kolmogorov sinusoidal flow with no-slip conditions (short wavelength stability) is improved. A condition for the stability of a vortex strip modeling Richtmyer–Meshkov fluid vortices (long wavelength stability) is presented.     

Continuous procedure of stochastic approximation in a semi-Markov medium

Using the Lyapunov function for an averaged system, we establish conditions for the convergence of the procedure of stochastic approximation

Parametric study of simultaneous transient conduction and radiation in a two-dimensional participating medium

The lattice Boltzmann method (LBM) has been used to solve the energy equation of a transient conduction–radiation heat transfer problem in a two dimensional Cartesian enclosure filled with an emitting, absorbing and scattering media. The control volume finite element method (CVFEM) is used to obtain the radiative information. Based on this new nonlinear hybrid algorithm, the effects of various influencing parameters on the transient thermal response such as the scattering albedo, the conduction–radiation parameter, and the wall emissivity are studied on the distributions of temperature, radiative heat fluxes. Numerical results are presented as compared with other published works and they are found to be in satisfactory agreement. The advantages of the proposed numerical approach include, among others, simple implementation on a computer, accurate CPU time and capability of stable simulation. Therfore, the method can capture fundamental behaviours in thermal flows of engineering interest, in addition it will have computaional advantages when the geometry is more complex.     

On a version of the nonlinear equations of the nonlocal moment theory of plasticity of materials with mobile disclinations

Using the methods of nonequilibrium thermodynamics, continuum mechanics, differential geometry, and the continuous theory of disclinations, we obtain a closed system of differential equations that makes it possible to determine the unknown plastic fields, the defect densities connected with them, and the stresses caused in the body. The connection between the kinetic potentials and the load surface of the system is established.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 62–69.     

The stability of a metastable shock wave in a viscoelastic medium under two-dimensional perturbations

The unsteady motions of a viscoelastic medium are considered, taking account of a small anisotropy and a small non-linearity. The behaviour of a metastable, quasi-transverse shock wave when it interacts with non-one-dimensional perturbations is investigated numerically. The stability of this wave under non-one-dimensional perturbations of large amplitude is demonstrated.     

Derivation of a rod theory for biphase materials with dislocations at the interface

Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of SO(3). First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of Γ-convergence.     

Emission of screw dislocations from the surface of a thin defect into an isotropic medium

By using the method of jump functions, we investigate the quasistatic stress field and stress concentration near a thin strip at the interface between materials during emission of screw dislocations from it into the medium. We construct analytic solutions in the cases of finite and semiinfinite cracks and perfectly rigid films. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 67–75, April–June, 1998.     

Continuous compression waves in the two-dimensional Riemann problem

The interaction between a plane shock wave in a plate and a wedge is considered within the framework of the nondissipative compressible fluid dynamic equations. The wedge is filled with a material that may differ from that of the plate. Based on the numerical solution of the original equations, self-similar solutions are obtained for several versions of the problem with an iron plate and a wedge filled with aluminum and for the interaction of a shock wave in air with a rigid wedge. The behavior of the solids at high pressures is approximately described by a two-term equation of state. In all the problems, a two-dimensional continuous compression wave develops as a wave reflected from the wedge or as a wave adjacent to the reflected shock. In contrast to a gradient catastrophe typical of one-dimensional continuous compression waves, the spatial gradient of a two-dimensional compression wave decreases over time due to the self-similarity of the solution. It is conjectured that a phenomenon opposite to the gradient catastrophe can occur in an actual flow with dissipative processes like viscosity and heat conduction. Specifically, an initial shock wave is transformed over time into a continuous compression wave of the same amplitude.     

Mathematical modeling of the vertical magnetic dipole field in a two-dimensional nonhomogeneous medium

We consider a quasi-three-dimensional problem of remote marine sounding by a high-power stationary source located on land. A transition from the three-dimensional problem to a family of parametric two-dimensional problems is performed. The solution of the remote marine sounding problem is obtained with high accuracy after solving about 20 two-dimensional problems. The integral equations are solved by the modified integral current method, which has proved highly efficient for field computations inside a strongly conducting anomaly. The electric field amplitude is observed to increase with depth. The width of the coastal current channel is estimated by analyzing the vertical magnetic field component.