On the Euler flow in ℝ2

We give a global existence and uniqueness theorem for the Euler flow in ² for suitable initial velocity fields, possibly diverging at infinity.     

Extremum theorems for the analysis of elastic plastic structures containing unstable elements

Summary This paper deals with incremental extremum principles in an associated flow-law theory of plasticity allowing for unstable, work-softening behavior, in contrast to the traditional assumptions. The theorem for the strain rates, corresponding to the minimum potential energy principle of elasticity, is proved to hold under appropriate condition which limits the unstabilizing effects present in the considered situation of the system. The theorem for the stress rates, corresponding to the complementary energy principle, turns out not to hold in general. The theorem for the plastic strain rates is shown to have the same range of validity as that for strain rates.

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Sommario Si esaminano i teoremi di minimo d'ambito differenziale in una teoria della plasticità con leggi di scorrimento associate, che ammetta, a differenza degli assunti tradizionali, il comportamento instabile o ad incrudimento negativo. Si dimostra che il teorema per le deformazioni incrementali, corrispondente al principio della minima energia potenziale totale dell'elasticità, è valido sotto un'opportuna condizione che limita gli effetti instabilizzanti presenti nel sistema nella situazione considerata. Il teorema negli sforzi incrementali, corrispondente al principio dell'energia complementare totale, risulta non più valido in generale. Si mostra che il teorema negli incrementi di deformazione plastica ha lo stesso campo di validità del teorema negli incrementi di deformazione totale.

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This research was conducted at the Faculté Polytechnique de Mons, Belgium, with the financial support of a NATO fellowship. The author wishes to thank Professor M. Save for the kind hospitality, the C.N.R. (National Research Council) for the award of the fellowship.     

The stress state of an elastic orthotropic medium with a penny-shaped crack

The static-equilibrium problem for an elastic orthotropic space with a circular (penny-shaped) crack is solved. The stress state of an elastic medium is represented as a superposition of the principal and perturbed states. To solve the problem, Willis approach is used, which is based on the triple Fourier transform in spatial variables, the Fourier-transformed Greens function for an anisotropic material, and Cauchys residue theorem. The contour integrals obtained are evaluated using Gauss quadrature formulas. The results for particular cases are compared with those obtained by other authors. The influence of anisotropy on the stress intensity factors is studied.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 76–83, December 2004.     

Asymmetric dynamic instability of axisymmetric polar dimpling of thin shallow spherical shells

If the parameter ², which measures the thickness-to-rise of the shell, is small, the axisymmetric polar dimpling of shallow spherical shell due to quadratic pressure distribution is dynamic instability, i.e., a small perturbation can change it to an asymmetric polar dimple mode. In two cases, the problem can be reduced to an eigenvalue problem Tw_n=c_n w_n, where T can approximately be reduced to a Sturm-Liouville operator if ²1. The existence of at least one real eigenvalue of T, which means that the axisymmetric polar dimpling is dynamically unstable, is proved by spectral theorem or Hilbert theorem. Furthermore, an eigenfunction, which represents one of the asymmetric modes of the unstable dimple shell, belonging to an eigenvalue of T is found.The Project Supported by National Natural Science Foundation of China. This paper was accepted to present at ICTAM 88(Grenoble).     

THE CRITERION ALGORITHM OF RELATION OF IMPLICATION BETWEEN PERIODIC ORBITS(Ⅱ)

In recent years,there is a wide interest in Sarkovskii’s theorem and related study.According to Sarkovskii’s theorem,if the continuous self-map f of the closed interval has a3-periodic orbit,then f must has an n-periodic orbit for any positive integer n.But f can nothave all n-periodic orbits for some n.Example.LetEvidently f has only one kind of3-periodic orbit in the two kinds of3-periodic orbits,whichexplains that it isn’t far enough to uncover the relation between periodic orbits by theinformation which Sarkovskii’s theorem has offered.In this paper,we raise the concept oftype of periodic orbits,and give a feasible algorithm which decldes the relation ofimplication between the two kinds of periodic orbits.     

Maximal Elements and Equilibrium of Abstract Economy

ＩｎｔｒｏｄｕｃｔｉｏｎＧｅｏｒｇｅＸｉａｎＺｈｉＹｕａｎａｎｄＥ .Ｔａｒａｆｄａｒｉｎ [1 ]ｐｒｏｖｅｄｔｈｅｆｏｌｌｏｗｉｎｇｅｘｉｓｔｅｎｃｅｔｈｅｏｒｅｍｏｆｍａｘｉｍａｌｅｌｅｍｅｎｔｓｆｏｒａｎｅｗｔｙｐｅｏｆｐｒｅｆｅｒｅｎｃｅｃｏｒｒｅｓｐｏｎｄｅｎｃｅｓｗｈｉｃｈａｒｅμ＿ｍａｊｏｒｉｚｅｄ .ＴｈｅｏｒｅｍＡ　ＬｅｔＸｂｅａｎｏｎｅｍｐｔｙｃｏｎｖｅｘｓｕｂｓｅｔｏｆａＨａｕｓｄｏｒｆｆｌｏｃａｌｌｙｃｏｎｖｅｘｔｏｐｏｌｏｇｉｃａｌｖｅｃｔｏｒｓｐａｃｅａｎｄｌｅｔＤｂｅａｎ…     

On Ericksen’s Theorem for Unconstrained Hyperelastic Materials

After reviewing the proof of Ericksens theorem for the set of unconstrained, homogeneous, isotropic, hyperelastic materials, Ericksens result is sharpened to cover the subset of materials that possess a natural configuration and satisfy the empirical inequalities. Mathematics Subject Classifications (2000) 74B20.     

Symmetrization properties of parabolic equations in symmetric domains

Symmetry properties of positive solutions of a Dirichlet problem for a strongly nonlinear parabolic partial differential equation in a symmetric domainD R ⁿ are considered. It is assumed that the domainD and the equation are invariant with respect to a group {Q} of transformations ofD. In examples {Q} consists of reflections or rotations. The main result of the paper is the theorem which states that any compact inC(D) negatively invariant set which consists of positive functions consists ofQ-symmetric functions. Examples of negatively invariant sets are (in autonomous case) equilibrium points, omega-limit sets, alpha-limit sets, unstable sets of invariant sets, and global attractors. Application of the main theorem to equilibrium points gives the Gidas-Ni-Nirenberg theorem. Applying the theorem to omega-limit sets, we obtain the asymptotical symmetrization property. That means that a bounded solutionu(t) asr approaches subspace of symmetric functions. One more result concerns properties of eigenfunctions of linearizations of the equations at positive equilibrium points. It is proved that all unstable eigenfunctions are symmetric.     

An iteration method for integral equations arising from axisymmetric loading problems

Let the concentrated forces and the centers of pressure with unknown density functions x(ξ) and y(ξ) respectively be distributed along the axis z outside the solid, then one can reduce an axismmetric loading problem of solids of revolution to two simultaneous Fredholm integral equations. An iteration method for solving such equations is duscussed. A lemma equivalent to E. Rakotch’s contractive mapping theorem and a theorem concerning the convergent proof of the iteration method are presented.     

Periodic Solutions of Two-Dimensional Forced Systems: The Massera Theorem and Its Extension

We assume that all solutions of a two-dimensional, periodically forced differential system (of period T) can be continued for all future time. If there exists one solution that is future bounded, then there exists a solution of period T (Theorem 3.4). This is the Massera theorem. To extend the Massera theorem, we assume that there exists a future bounded solution that is also bounded away from a known T-periodic solution . We prove that either there is another periodic solution of period qT for some integer q 1 or all compact motions that remain a finite distance from have a well-defined irrational rotation number about (Theorem 4.3).     

The axisymmetric problem with initial conditions for the equations of motion of an ideal incompressible fluid

The present paper presents a proof of the existence and uniqueness theorem for the solution of the axisymmetric problem with initial conditions for the Euler equations in the case of an incompressible fluid. We consider the case of the nonporous wall, and also the transpiration problem in the formulation given in [1]. Global unique solvability is proved for assumptions only on the smoothness of the conditions and for all values of the time t. The existence theorem for a small time segment in the case of a nonporous wall has been proved for the general three-dimensional problem in [2, 3]. For the proof we use a method analogous to that developed in [1] for planar flows. The a priori estimate of the vorticity which is used in the present study was obtained previously in [4],The author wishes to thank V. I. Yudovich for continued interest in the study and many valuable suggestions.     

REPRESENTATION THEOREM AND ONE-ITERATION THEOREM FOR FREDHOLM INTEGRAL EQUATION OF THE FIRST KIND Ax=y

In this paper,two theorems are presented.The representation theorem states:if the Fredholm integral equation of the first kind Ax=y,with bounded L_2 kernel,has a uniquesolution(?),then(?),whereThe one-iteration theorem states:(?)can be achieved in one iteration by(?)=x_0 g_0 A*(y-Ax_0)iff one of the following conditions is satisfied.     

Time domain radiation and scattering by thin wires

Direct time domain solutions for radiation from and scattering by thin conducting wires are considered. The problem is formulated in terms of two coupled integrodifferential equations derived from the retarded potentials, the continuity equation, and the boundary conditions for the wire. Solution of these equations is effected using the method of moments resulting in a set of simultaneous time iterative matrix equations. A time domain reciprocity theorem suitable for thin wire objects is also presented. The theorem demonstrates the relationship between reciprocity and the adjoint operator for the problem. Two moment solutions are presented for straight wires: (1) a point tested solution, and (2) a pulse tested solution. The radiation field is found by direct application of the reciprocity theorem. All solutions are presented as algorithms suitable for computation. The algorithms are time iterative by nature, and inexpensive in terms of computer time. Computed results are presented for the straight wire scatterer excited by a plane wave with unit step time dependence. The end fire effects predicted by traveling wave theory are observed in the scattered field results. The source of these effects is shown to be the end of the wire last intercepted by the incident field. Additional results are presented for the straight wire antenna excited by a unit step voltage applied at an arbitrary driving point. The computed driving point current is compared to the results derived for the case of an infinite wire antenna over the time interval that the comparison is valid.     

Multiple and maximal periodicity in linear elastostatics

Properties of elastic states in which the strain is periodic in an arbitrary number of directions are investigated. It is shown that, even though the corresponding displacements might not, in a non-trivial sense, be periodic, they do satisfy a semi-periodicity condition. Other general results, including a version of Betti's reciprocal theorem and a theorem of work and energy are derived and discussed. Problems involving periodicity in a maximal number of directions are examined in greater detail. Additional restrictions on the displacement corresponding to maximally periodic strains are derived and the uniqueness and periodicity of solutions to maximally periodic and slightly maximally periodic boundary value problems are discussed.     

Fast randomization of an electron gas by trapped electro-acoustic waves

Summary This paper presents a simplified theoretical model for a phenomenon of fast randomization of an electron gas, investigated experimentally in this laboratory. It is assumed that a certain region in constricted electrical discharges, called plasma sac can become the siege of trapped electro-acoustic waves generated by a beam-plasma interaction. The interaction between those waves and the electron gas is treated by a pseudo collisional method, leading to a generalization of Boltzmann's H theorem. The velocity distribution which is essentially Maxwellian with a significant deficiency in the fast electron tail, is deduced.     

The Rellich-Kondrachov theorem for unbounded domains

Summary The full Kondrachov compactness theorem for Sobolev imbeddings of the type W ₀ ^m,p (G) W ₀ ^j,r (G) on bounded domains G in R ⁿ is extended to a large class of unbounded domains with reasonable n- 1 dimensional boundaries. A Poincaré inequality is obtained for such domains and a compactness theorem for traces of functions in W ₀ ^m,p (G) on lower dimensional hyperplanes is also proved.     

Shakedown analysis of shell structures of kinematic hardening materials

It is of great practical importance to analyze the shakedown of shell structures under cyclic loading, especially of those made of strain hardening materials.In this paper, some further understanding of the shakedown theorem for kinematic hardening materials has been made, and it is applied to analyze the shakedown of shell structures. Though the residual stress of a real state is related to plastic strain, the time-independent residual stress field as we will show in the theorem may be unrelated to the time-independent kinematically admissible plastic strain field . For the engineering application, it will be much more convenient to point this out clearly and definitely, otherwise it will be very difficult. Also we have proposed a new method of proving this theorem. The above theorem is applied to the shakedown analysis of a cylindrical shell with hemispherical ends. According to the elastic solution, various possible residual stress and plastic strain fields, the shakedown analysis of the structure can be reduced to a mathematical programming problem. The results of calculation show that the shakedown load of strain hardening materials is about 30–40% higher than that of ideal plastic materials. So it is very important to consider the hardening of materials in the shakedown analysis, for it can greatly increase the structure design capacity, and meanwhile provide a scientific basis to improve the design of shell structures.     

Bifurcation of poloidal field in the flow induced by a radial electric current

A study is made of the steady-state flow of a viscous conducting incompressible fluid in a half-space, induced by an electric current spreading from a point source on a solid surface. There is a critical finite value of the current at which the velocity on the axis becomes infinite. It is shown that for a high-conductivity medium bifurcation of a new MHD regime with nonzero poloidal field and rotation takes place for arbitrarily large values of the current. The axisymmetric MUD dynamo effect detected does not contradict the Cowling-Braginskii antidynamo theorem, since the conditions of the theorem are not fulfilled. In the case of low conductivity the paradox can be resolved using the model of an infinitely narrow turbulent jet. Here, too, self-excitation of the field and rotation are detected and their strengthening leads to suppression of the turbulence and relaminarization of the jet.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 4–11, March–April, 1993.     

Third-order systems: Periodicity conditions

Recently a third-order existence theorem has been proven to establish the sufficient conditions of periodicity for the most general third-order ordinary differential equation

x+f(t,x,x^′,x^″)=0