Localised knife waves in a structured interface

We consider a Mode III lattice with an interface layer where the dynamic crack growth is caused by a localised sinusoidal wave. In the wave–fracture scenario, the ‘feeding wave’ (here also called the knife wave) delivers energy to the moving crack front, while the dissipative waves carry a part of this energy away from the front. The questions addressed here are:

• What are the conditions of existence of the localised knife wave?

• What is the lower bound of the amplitude of the feeding wave, which supports the crack propagation, for a given deformational fracture criterion?

• How does the crack speed depend on the amplitude of the feeding wave?

• What are the dissipative waves? How much energy is irradiated by these waves and what is the total dissipation?

• What are the conditions of existence of the steady-state regime for the propagating crack?

We consider analytically two established regimes: the steady-state regime, where the motion of neighbouring masses (along the interface) differs only by a constant shift in time, and an alternating-strain regime, where the corresponding amplitudes differ by sign. We also present the numerical simulation results for a model of a high-contrast interface structure. Along with the energy of the feeding and dissipative waves, an energy radiated to the bulk of the lattice is identified.

Stress state of a cross-base prism under torsion

The stress state of a cross-base prism under torsion is analyzed. The lower- and upper bounds for stresses are found by solving an infinite system of linear algebraic equations and using Koyalovich’s limitants method. It is proved that the infinite system is regular and that its solution exists and is unique. The asymptotic behavior of the unknowns is established. The convergence of the series for stresses is accelerated Translated from Prikladnaya Mekhanika, Vol. 44, No. 11, pp. 79–93, November 2008.     

Transport and deposition of colliding particles in turbulent channel flows

The purpose of this paper is twofold: (i) to present statistical models that describe particle–turbulence interactions as well as particle–particle collisions and (ii) to gain a better understanding of the effect of inter-particle collisions on transport, deposition, and preferential concentration of heavy particles in turbulent channel flows. The models presented are based on a kinetic equation for the probability density function of the particle velocity distribution in anisotropic turbulent flow. The model predictions compare reasonable well with numerical simulations and properly reproduce the crucial trends of computations.     

Three-dimensional, three-component wall-PIV

This paper describes a new time-resolved three-dimensional, three-component (3D-3C) measurement technique called wall-PIV. It was developed to assess near wall flow fields and shear rates near non-planar surfaces. The method is based on light absorption according to Beer–Lambert’s law. The fluid containing a molecular dye and seeded with buoyant particles is illuminated by a monochromatic, diffuse light. Due to the dye, the depth of view is limited to the near wall layer. The three-dimensional particle positions can be reconstructed by the intensities of the particle’s projection on an image sensor. The flow estimation is performed by a new algorithm, based on learned particle trajectories. Possible sources of measurement errors related to the wall-PIV technique are analyzed. The accuracy analysis was based on single particle experiments and a three-dimensional artificial data set simulating a rotating sphere.     

Localization and Propagation in Random Lattices

We analyze the motion of a particle on random lattices. Scatterers of two different types are independently distributed among the vertices of such a lattice. A particle hops from a vertex to one of its neighboring vertices. The choice of neighbor is completely determined by the type of scatterer at the current vertex. It is shown that on Poisson and vectorizable random triangular lattices the particle will either propagate along some unbounded strip or be trapped inside a closed strip. We also characterize the structure of a localization zone contained within a closed strip. Another result shows that for a general class of random lattices the orbit of a particle will be bounded with probability one.     

Szegő Difference Equations, Transfer Matrices¶and Orthogonal Polynomials on the Unit Circle

We develop the theory of orthogonal polynomials on the unit circle based on the Szegő recurrence relations written in matrix form. The orthogonality measure and C-function arise in exactly the same way as Weyl's function in the Weyl approach to second order linear differential equations on the half-line. The main object under consideration is the transfer matrix which is a key ingredient in the modern theory of one-dimensional Schr?dinger operators (discrete and continuous), and the notion of subordinacy from the Gilbert–Pearson theory. We study the relations between transfer matrices and the structure of orthogonality measures. The theory is illustrated by the Szegő equations with reflection coefficients having bounded variation. Received: 26 February 2001 / Accepted: 28 May 2001     

Matrix Models: Geometry of Moduli Spaces and Exact Solutions

We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points and matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich–Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satisfied by various generalized Kontsevich matrix models, we derive time transformations that establish exact relations between different models appearing in mathematical physics. We solve the Hermitian one-matrix model using the moment technique in the genus expansion and construct a recursive procedure for solving this model in the double scaling limit.     

The Stability of Laminated Elastomer Coatings Under Surface Loading

The stability of laminated elastomer coatings at normal and high temperatures is considered in a three-dimensional formulation. The problem is formulated and the basic characteristic equations are derived. Graphs of the physicomechanical parameters of some elastomers against temperature are experimentally obtained. Specific examples for a three-layer coating are considered     

Conformational flexibility of pyrimidine ring in adenine and related compounds

The structural non-rigidity of aromatic pyrimidine rings in adenine and selected related compounds has been investigated by quantum chemical non-empirical methods of calculation at the MP2 and DFT levels of theory. The results of the calculations demonstrate that the pyrimidine ring possesses a notable degree of conformational flexibility despite its aromatic character. The transition of the heterocycle from a planar equilibrium geometry to a non-planar structure with an endocyclic torsion angle of ±20° results in an energy increase of less than 2.8 kcal/mol. An analysis of the population of ground and excited vibrational levels for the lowest out-of-plane vibration of the ring indicates that in adenine 45% of the molecules have a non-planar pyrimidine ring with relevant torsion angle up to ±17° at any moment of time.     

Gaussian spectral rules for second order finite-difference schemes

Earlier the authors suggested an algorithm of grid optimization for a second order finite-difference approximation of a two-point problem. The algorithm yields exponential superconvergence of the Neumann-to-Dirichlet map (or the boundary impedance). Here we extend that approach to PDEs with piecewise-constant coefficients and rectangular homogeneous subdomains. Examples of the numerical solution of the 2-dimensional oscillatory Helmholtz equation exhibit exponential convergence at prescribed points, where the cost per grid node is close to that of the standard five-point finite-difference scheme. Our scheme demonstrates high accuracy with slightly more than two grid points per wavelength and reduces the grid size by more than three orders as compared with the standard scheme.