Investigation of the Stability Problems of Elastic Bodies Using the Method of Mathematical Theory of Elasticity

In this paper,using the equilibrium equations and boundary conditionsof elastic stability problem of Новожилов and the method of mathematicaltheory of elasticity,we solve some elastic stability problems,which werestudied byищлинскииandвоицеховская,and obtained more reason-able results than theirs.     

Model of Self‐Organization of an Ensemble of Cracks Radiating Sound

A model of selforganization of cracks arising in a rock specimen (granite) compressed by a press is proposed. The model is based on the assumption of acoustic wave interaction between the cracks. To construct the model of selforganization of cracks, solutions of the Fokker–Planck equation are used. The experimentally observed spontaneous increase in the activity of acoustic emission, spatial and temporal clusterization, and formation of a fractal structure in rock specimens under constant and slowly varying loads are explained.     

Homogenization of the nonlinear Kelvin–Voigt model of viscoelasticity and of the Prager model of plasticity

Denoting by the stress tensor, by the linearized strain tensor, by A the elasticity tensor, and assuming that is a convex potential, the inclusion accounts for nonlinear viscoelasticity, and encompasses both the linear Kelvin–Voigt model of solid-type viscoelasticity and the Prager model of rigid plasticity with linear kinematic strain-hardening. This relation is assumed to represent the constitutive behavior of a space-distributed system, and is here coupled with the dynamical equation. An initial- and boundary-value problem is formulated, and the existence and uniqueness of the solution are proved via classical techniques based on compactness and monotonicity. A composite material is then considered, in which the function and the tensor A rapidly oscillate in space. A two-scale model is derived via Nguetseng’s notion of two-scale convergence. This provides a detailed account of the mesoscopic state of the system. Any dependence on the fine-scale variable is then eliminated, and the existence of a solution of a new single-scale macroscopic model is proved. The final outcome is at variance with the nonlinear extension of the generalized Kelvin–Voigt model, which is based on an apparently unjustified mean-field-type hypothesis.     

Estimation of overall properties of random polycrystals with the use of invariant decompositions of Hooke’s tensor

In the paper the theoretical analysis of bounds and self-consistent estimates of overall properties of linear random polycrystals composed of arbitrarily anisotropic grains is presented. In the study two invariant decompositions of Hooke’s tensors are used. The applied method enables derivation of novel expressions for estimates of the bulk and shear moduli, which depend on invariants of local stiffness tensor. With use of these expressions the materials are considered for which at the local level constraints are imposed on deformation or some stresses are unsustained.     

Modeling of the Front of Melting and Destruction of a Melt Film During Gas‐Laser Cutting of Metals

The products of laser cutting of metals on an automated laser setup are investigated. Results of model experiments are presented, where soft wax was used instead of metal transforming into the melt; soft wax filled a narrow flat slot between two glass plates and was removed by a heated air stream. The physical processes of melting of the liquidwax film, its destruction, and entrainment by the gas jet being assumed to be analogous to the processes of metalmelt spraying inside the cut in fullscale experiments, the characteristic size of drops formed thereby is evaluated. The modeling results are in qualitative agreement with the results of fullscale experiments. It is shown that the quality of laser cutting of metals directly depends on the character of spraying of the liquid melt and the process of its removal.     

Group foliation of the Lamé equations of the classical dynamical theory of elasticity

We perform the group foliation of the system of Lamé equations of the classical dynamical theory of elasticity for an infinite subgroup contained in a normal divisor of the main group. The resolving system of this foliation includes the following two classical systems of mathematical physics: the system of equations of vortex-free acoustics and the system of Maxwell equations, which allows one to use wider groups to obtain exact solutions of the Lamé equations. We obtain a first-order conformal-invariant system, which describes shear waves in a three-dimensional elastic medium. We also give examples of partially invariant solutions.     

Influence of small harmonic terms on eigenvalues of monodromy matrix of piecewise-linear oscillators

In this paper we consider the problem which can appear at the determination of the dynamical stability of the responses of oscillators with discontinuous or steep derivative of the restoring characteristic obtained in the frequency domain. For that purpose, a simple one degree-of-freedom system with piecewise-linear force-displacement relationship subjected to a harmonic excitation is analysed. Stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the Floquet-Liapounov theorem. Confirmation of the results obtained in the frequency domain is done by comparing with the results obtained in the time domain by the method of piecing the exact solutions. Determination of the dynamical stability can be made more reliable by using the proposed plot of maximum modulus of the eigenvalues of the monodromy matrix in dependence of non-dimensional frequency and the number of harmonics included in the supposed approximate solution.     

The exact theory of bending of prismatic shells — an application of transfer matrices

Conclusion In this paper a new application of transfer matrices has been made in connection with the exact theory of bending of prismatic shells. It is shown that use of transfer matrices reduces the number of unknowns from 8 n to four, where n is the total number of walls, for a given integer m. This simplification is specially applicable to structures with open or simply connected closed sections.     

Table of Contents of Volume 36 – 2001

Volume Contents

Table of Contents of Volume 36 – 2001     

Table of Contents of Volume 35 – 2000

Volume Contents

Table of Contents of Volume 35 – 2000     

Reliability Analysis of Structural Members Composed of Several Random Components,Making Use of Theory of Random Processes∗

This paper presents a method of reliability analysis of structural members, considering both the load and the resistance to be composed of several random components. The analysis is based on the theory of linear statistical approximation, extended to random processes. To illustrate application of the present theory, numerical calculations are carried out on a singly reinforced concrete beam with rectangular cross section and on a steel compression member with random initial deflections     

Application of the method of direct separation of motions to the parametric stabilization of an elastic wire

The paper considers the application of the method of direct separation of motions to the investigation of distributed systems. An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol. 11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment, an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system.     

A Topological Proof of Stability of N-Front Solutions of the FitzHugh–Nagumo Equations

Consideration is devoted to traveling N-front wave solutions of the FitzHugh–Nagumo equations of the bistable type. Especially, stability of the N-front wave is proven. In the proof, the eigenvalue problem for the N-front wave bifurcating from coexisting simple front and back waves is regarded as a bifurcation problem for projectivised eigenvalue equations, and a topological index is employed to detect eigenvalues.     

Adjoint problems of mechanics of continuous media in gas‐laser cutting of metals

A mathematical model of gaslaser cutting of metal plates in an inert gas is proposed. The formation and flow of the liquid metal melt film at the cutting front is considered within the framework of incompressible boundarylayer equations. Based on the resultant analytical solution, a local law of energy conservation on the cutting surface is derived, which takes into account the meltfilm thickness and the temperature dependence of thermophysical parameters of the metal. The problem of the cutting shape and depth is solved in the twodimensional formulation. A comparison with experimental data is made in terms of the cutting depth and maximum cutting velocity for carbon and alloy steel.     

The Calculation of Semi-circular Corrugated Tube-An Application of the General Solution of Ring Shell

In this paper,the deformation and stress distribution in semi-circularcorrugated tube under axial force are calculated by means of the generalsolutions of circular ring shell given in previous paper[1].     

Hausdorff dimension of set generated by exceptional oscillations of a class of N-parameter Ganssian processes

A class of N-parameter Gaussian processes are introduced,which are more general than the N-parameter Wiener process.The definition of the set generated by exceptional oscillations of a class of these processes is given,and then the Hansdorff di- mension of this set is defined.The Hausdorff dimensions of these processes are studied and an exact representative for them is given,which is similar to that for the two-parameter Wiener process by Zacharie(2001).Moreover,the time set considered is a hyperrectangle which is more general than a hyper-square used by Zacharie(2001).For this more gen- eral case,a Fernique-type inequality is established and then using this inequality and the Slepian lemma,a Lévy's continuity modulus theorem is shown.Independence of incre- ments is required for showing the representative of the Hausdorff dimension by Zacharie (2001).This property is absent for the processes introduced here,so we have to find a different way.     

Representation of classifier distributions in terms of hypergeometric functions

This paper derives alternative analytical expressions for classifier product distributions in terms of Gauss hypergeometric function, 2F1, by considering feed distribution defined in terms of Gates-Gaudin-Schumann function and efficiency curve defined in terms of a logistic function. It is shown that classifier distributions under dispersed conditions of classification pivot at a common size and the distributions are difference similar. The paper also addresses an inverse problem of classifier distributions wherein the feed distribution and efficiency curve are identified from the measured product distributions without needing to know the solid flow split of particles to any of the product streams.     

Stability for basic system of equations of atmospheric rhotion

The topological characteristics for the basic system of equations of atmo- spheric motion were analyzed with the help of method provided by stratification theory.It was proved that in the local rectangular coordinate system the basic system of equations of atmospheric motion is stable in infinitely differentiable function class.In the sense of local solution,the necessary and sufficient conditions by which the typical problem for determining solution is well posed were also given.Such problems as something about“speculating future from past”in atmospheric dynamics and how to amend the condi- tions for determining solution as well as the choice of underlying surface when involving the practical application were further discussed.It is also pointed out that under the usual conditions,three motion equations and continuity equation in the basic system of equations determine entirely the property of this system of equations.     

Stability of Filtration of a Gas—Liquid Mixture

The stability of steady regimes of filtration of a gas—liquid mixture at pressure lower than the saturation pressure is studied for the case of a nonmonotonic dependence of the relative phase permeability of the liquid on the gas saturation. It is shown that periodic self–oscillations can appear, and their evolution leads to deterministic chaos due to the appearance and destruction of quasiperiodic motions.     

On The Reissner Theory of Bending of Flastic Plates

The Reissner equations of elastic plates are rederived on the bases of the incomplete generalizedvariational principle of Complementary energy.The Stress function to is naturally obtained from thevariational Calculation in the form of Lagrange multiplier.The stucture of solutions of the Reissnerequations is thus defined.On the bases of these discussions,a simplified theory has been put forward,in which the equations of equilibrium involving the shearing influence can be reduced into a fourthorder differential equation similar to those of the Classical theory of plates.