Numerical simulation of the rheological properties of granular composites by using a structural approach

Composites with an elastomeric matrix containing rigid particles of diameter 10–1000 μm are studied. One of possible mechanisms of the rheological behavior of such filled systems, related to the origination and growth of vacuoles near the rigid inclusions in a viscous matrix, is considered. For simulating the mechanism of formation of rheological properties of the filled elastomers, we use a structural cell in the form of an elastomeric cylinder, whose height and diameter are equal in magnitude, with a rigid spherical inclusion at its center. Deformation of the cells is examined with the observance of boundary conditions providing the preservation of their close packing. The inclusion is assumed to be rigid, and the matrix properties are described by equations of the linear hereditary viscoelasticity theory. The formation of vacuoles is described by using the approach suggesting that an initial debonding begins to propagate when the energy accumulated in the extended matrix reaches a value sufficient to create a new interface. The heterogeneity of the composite is simulated by taking into account the variability of the local filler concentration. Creep curves obtained for composite cells with different content of the solid phase are presented. Comparisons between the numerical and experimental results show a satisfactory agreement. Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 895–906, November–December, 2008.     

Positivity of the self-diffusion matrix of interacting Brownian particles with hard core

We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle. We will do this for all activities, z>0, of Gibbs measures; in particular, for large z– the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian balls. Received: 22 September 1997 / Revised version: 15 January 1998     

A fractal model of reinforcement of elastoplastic nanocomposites

Within the framework of fractal analysis and percolation theory, an alternative model of reinforcement of filled polymers is offered. Practically, this model can be used only to describe the reinforcement of nanocomposites, because, according to the treatment considered, a pronounced reinforcement can be reached only at ratios of filler particle diameter to the statistical segment length of about 10 and less. A theoretical calculation showed a good qualitative and quantitative agreement with experiments. The type of reinforcement mechanism of composites is determined by the type of the space (fractal or Euclidean) in which the structure of the polymeric matrix is formed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 797–802, November–Decem ber, 2006.     

Measure-valued limits of interacting particle systems with <Emphasis Type="Italic">k</Emphasis>-nary interactions

 Results on existence, uniqueness, non-explosion and stochastic monotonicity are obtained for one-dimensional Markov processes having non-local pseudo-differential generators with symbols of polynomial growth. It is proven that the processes of this kind can be obtained as the limits of random evolutions of systems of identical indistinguishable particles with k-nary interaction. Received: 24 May 2002 / Revised version: 19 February 2003 / Published online: 12 May 2003 Mathematics Subject Classification (2000): 60K35, 60J75, 60J80 Key words or phrases: Interacting particles – k-nary interaction – Measure-valued processes – One-dimensional Feller processes with polynomially growing symbols – Duality – Stochastic monotonicity – Heat kernel     

Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the j th component of the effective potential for n particles equals the partial derivative with respect to the coordinate of the jth particle of “magnetic potential energy” of n particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for d-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy). Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 691–698, May 1997.     

Functional Inequalities for Particle Systems on Polish Spaces

Various Poincaré–Sobolev type inequalities are studied for a reaction–diffusion model of particle systems on Polish spaces. The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which we call reaction–diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on the product spaces Eⁿ (n≥1) which determine the motion of particles, and the reaction part induced by a Q-process on ℤ₊ and a sequence of reference probability measures, where the Q-process determines the variation of the number of particles and the reference measures describe the locations of newly produced particles. We prove that the validity of Poincaré and weak Poincaré inequalities are essentially due to the pure reaction part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently, for the corresponding Q-process. But under a mild condition, stronger inequalities rely on both parts: the reaction–diffusion Dirichlet form satisfies a super Poincaré inequality (e.g., the log-Sobolev inequality) if and only if so do both the corresponding Q-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples are presented to illustrate the main results. Mathematics Subject Classifications (2000) 4FD0F, 60H10. Feng-Yu Wang: Supported in part by the DFG through the Forschergruppe “Spectral Analysis, Asymptotic Distributions and Stochastic Dynamics”, the BiBoS Research Centre, NNSFC(10121101), and RFDP(20040027009).     

Strength and deformation characteristics of poly(ethylene terephtalate)/elastomer blends

Changes in the toughness of crystalline poly(ethylene terephtalate) upon addition of two elastomers — ethylene-propylene-diene terpolymer and ethylene-α-octene copolymer — are investigated. Blends with increasing elastomer content (up to 30 wt.%) were obtained and modified by γ-radiation up to a 300-kGy absorbed dose. The interrelation between the toughness and certain tensile characteristics (elastic modulus, yield stress, ultimate stress, elongation at break, and specific fracture energy) of the blends, with various structural features of components (crystallinity and the degree of cross-linking) and morphology of the blends, is discussed. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 4, pp. 521–534, July–August, 2008.     

Particle Decays and Stability on the de Sitter Universe

We study particle decay in de Sitter space–time as given by first-order perturbation theory in a Lagrangian interacting quantum field theory. We study in detail the adiabatic limit of the perturbative amplitude and compute the “phase space” coefficient exactly in the case of two equal particles produced in the disintegration. We show that for fields with masses above a critical mass m _c there is no such thing as particle stability, so that decays forbidden in flat space–time do occur here. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. Particles with mass lower than critical have a completely different behavior: the masses of their decay products must obey quantification rules, and their lifetime is zero.     

On exact solutions for quantum particles with spin <Emphasis Type="Italic">S</Emphasis> = 0, 1/2, 1 and de Sitter event horizon

Exact wave solutions for particles with spin 0, 1/2 and 1 in the static coordinates of the de Sitter space–time model are examined in detail. Firstly, for scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient R_ej{R_{\epsilon j}} on the background of the de Sitter space–time model is analyzed. It is shown that the determination of R_ej{R_{\epsilon j}} requires an additional constrain on quantum numbers er/ (^h/_2p) c >> j{\epsilon \rho / \hbar c \gg j}, where ρ is a curvature radius. When taken into account of this condition, the R_ej{R_{\epsilon j}} vanishes identically. It is claimed that the calculation of the reflection coefficient R_ej{R_{\epsilon j}} is not required at all because there is no barrier in an effective potential curve on the background of the de Sitter space–time. The same conclusion holds for arbitrary particles with higher spins, it is demonstrated explicitly with the help of exact solutions for electromagnetic and Dirac fields.     

Thermocapillary motion of deformable fluid particles in tubes

The boundary integral technique is used to study the effect of deformation on the steady, creeping, thermocapillary migration of a fluid particle under conditions of axisymmetry, negligible thermal convection and an insulated tube wall. The spherical radius of the fluid particle (i.e. the radius as if the particle were a sphere, a ′= (3V _p ^′ /4π)^1/3, V _p ^′ is the particle volume) and that of the tube are denoted, respectively, by a′and b′. For small capillary numberCa = 0.05, only for a large fluid particle (a′/b′ = 0.8) is deformation significant. Fora′/b′= 0.8, hydrodynamic stresses squeeze the particle, reduce the interaction of the particle with the wall and thereby increase the terminal velocity. For small particles a′/b′< 0.8 and Ca = 0.05 the fluid particles translate as spheres, due to the fact that the fluid particle is too far away from the wall to be subject to distending hydrodynamic stresses. The deformable particle moves faster than a spherical one in the thermocapillary migration. The increase in velocity with capillary number is larger for thermocapillary motion than for buoyancy.     

Particles of variable size

A problem of all particle methods is that they produce large vacuum regions when they are applied to a free gas flow, for example. With the approach recently proposed by the author [Numer. Math. (1997) 76: 111–142], this difficulty can be avoided. One can let the particles adapt their size to the local state of the fluid. How, is described in the present article. The diameter as an additional degree of freedom strongly improves the performance of the numerical methods based on this particle model. Received November 22, 1996 / Revised version received March 30, 1998     

A Central Limit Theorem for a Localized Version of the SK Model

In this note, we consider a SK (Sherrington–Kirkpatrick)-type model on ℤ ^d for d≥1, weighted by a function allowing to any single spin to interact with a small proportion of the other ones. In the thermodynamical limit, we investigate the equivalence of this model with the usual SK spin system, through the study of the fluctuations of the free energy. This author’s research partially supported by CAPES.     

Failure of particulate reinforced polymers

In this presentation, a review is given on the main effects of mineral particulate fillers (with an aspect ratio of about unity) on the deformation and fracture of amorphous and semicrystalline thermoplastic and thermosetting polymers. Elastomeric modifiers, polymer blends, and filled elastomers are not considered here. Fillers are generally used to reduce cost as well as the thermal sensitivity of mechanical properties of the matrix material and to improve, if possible, the strength and toughness. The addition of particulate fillers influences all stages of the fabrication and use of the resulting composites. We focus on the effects of a stiff second phase on elastic moduli, matrix structure, and on deformation, creep, and failure mechanisms. As the main mechanisms, particle-matrix debonding, void formation, and matrix microshear yielding are identified. Toughness is less sensitive to the quality of adhesion since particle-matrix debonding and formation of voids can be tolerated. If well controlled, debonding contributes to deformation (formation of voids should be well distributed in space and time). Reference is also made to the surprising and positive effect of CaCO₃ particles on the toughness and impact resistance of HDPE, which increases at small interparticle distances due to interfacial effects on lamellar growth in the ligament area. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 305–316, March–April, 2000.     

A representation of the Belavkin equation via Feynman path integrals

 The Belavkin equation, describing the continuous measurement of the position of a quantum particle, is studied. A rigorous representation of its solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the complex Cameron-Martin space is given. Received: 7 January 2002 / Revised version: 20 June 2002 / Published online: 19 December 2002 Mathematics Subject Classification (2000): 81, 81S40, 60H15 Key words or phrases: Belavkin equation – Continuous measurement – Quantum theory – Oscillatory integrals – Feynman path integrals     

Curvature and absorption

The problem of finding a distribution of the sources of particles (or radiation) in a bounded domain D by the outputting flow through the boundary δD is considered. It is assumed that the domain D is filled with a medium whose absorption, scattering diagram, and Riemannian metric are known. Under certain assumptions on these characteristics of the medium, the uniqueness of a solution and its stability are proved. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 187–189.     

Continued fraction of e2 with confluent hypergeometric functions

The tanh-type, tan-type, and e-type Hurwitz continued fractions have been generalized by the author. In this paper, we study a generalized form of e²-type Hurwitz continued fractions by using confluent hypergeometric functions. We also obtain a similar type of Tasoev continued fractions. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 513–531, October–December, 2006.     

A Generalization of a Theorem of Liao

Let X be a metrizable space and let φ：R× X → X be a continuous flow on X. For any given {φt}-invariant Borel probability measure, this paper presents a {φt}-invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the contiImous flow （X, {φt}）. The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non-uniformly hyperbolic systems and skew-product flows.     

Numerical simulation of sheath plasma by the particle method taking account of coulomb collisions

We study the problem of the behavior of a plasma bounded longitudinally by an absorbing sheath. This model contains charged particles (electrons and ions) moving subject to a self-consistent electrostatic field. New particle pairs are generated in the region of a distributed source. As a numerical model we used the electrostatic “particle-in-cell” method supplemented by the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using the Monte Carlo method. We give a mathematical statement of the problem. The computations were carried out using the direct implicit method with the “explicit limit” time step. The results of numerical simulation of this system are given. We consider the formation and evoluiton of potential structures (multiple weak nonmonotonic double layers). Five figures. Bibliography: 35 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 75–89.     

Hypersonic gas flows around two-dimensional ogival bodies in the presence of foreign particles

The motion of solid particles in a hypersonic two-phase flow of a nonviscous, compressed layer in a vicinity of the point of a two-dimensional ogive is investigated. An analytical solution is obtained, which provides the means, as distinct from a numerical solution, for writing equations for particle trajectories and finding coefficients of particle, sedimentation. Bibliography: 3 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 74–79.