The dynamic stability of an elastic column

Lyapunov's second method is used to investigate the stability of the rectilinear equilibrium modes of a non-linearly elastic thin rod (column) compressed at its end. Stability here is implied relative to certain integral characteristics, of the type of norms in Sobolev spaces; the analysis is carried out for all values of the problem parameter except the bifurcation values.

The realm of problems connected with the Lagrange-Dirichlet equilibrium stability theorem and its converse involves specific difficulties when considered in the infinite-dimensional case: stability in infinite-dimensional systems is investigated relative to certain integral characteristics such as norms /1/, and as the latter may be chosen with a certain degree of arbitrariness, different choices may result in different stability results. On the other hand, there is no relaxation of any of the difficulties encountered in the case of a finite number of degrees of freedom.

We shall consider a certain natural mechanical system with a finite number of degrees of freedom. If the first non-trivial form of the potential energy expansion is positive-definite, the equilibrium position is stable. A similar statement has been proved for infinitely many dimensions as well /1–3/, using Lyapunov's direct method, and the total energy may play the role of the Lyapunov function.

The situation with respect to instability is more complex. In the finite-dimensional case, if the first non-trivial form of the potential energy expansion may take negative values, instability may be demonstrated in many cases by means of a function proposed by Chetayev in /4/. A general theorem has been proved /1/ for instability in infinitely many dimensions, relying on an analogue of Chetayev's function. Such functions have also been used /5, 6/ to prove the instability of equilibrium in specific linear systems with an infinite number of degrees of freedom.

However, Chetayev's functions /4/ are not suitable tools to prove the instability of equilibrium in most non-linear systems. Another “Chetayev function”, which is actually a perturbed form of Chetayev's original function from /4/, has been proposed /7/, and it has been used to prove instability when the equilibrium position is an isolated critical point of the first non-trivial form of the potential energy expansion.

The majority of problems concerning the onset of instability of equilibrium configurations of elastic systems have been considered from a quasistatic point of view (see, e.g., /8, 9/). Problems of elastic stability and instability were considered in a dynamical setting in /2, 5/, where stability was investigated by Lyapunov's direct method. However, most of the results obtained in this branch of the field concern linear systems, and there are extremely few publications dealing with the onset of instability in non-linear elastic systems using Lyapunov's direct method. This is because in an unstable elastic system the quadratic part of the potential energy may change sign, and therefore the analogues of Chetayev's function from /4/ are not usually suitable for solving these problems. Dynamic instability has been studied or a specific non-linearly elastic system /10/, with the fact of instability established by using an analogue of the Chetayev function from /7/.

This paper presents one more example of a study of dynamic instability crried out for a non-linearly elastic system by Lyapunov's direct method.     

Mathematical models and algorithms for studying strength and stability of shell structures

In this paper, we describe severalmathematical models of deformation of reinforced shell structures, including those that account for various properties of the material. For the structures composed of orthotropic and isotropicmaterials, we consider linearly elastic and physically nonlinear problems, as well as the problems of creep. All models are constructed on the basis of the functional of total potential energy of the shell deformation. Geometric nonlinearity and transverse shears are taken into account. Strengthening ribs are introduced both in the discrete way and by the method of structural anisotropy. Three different algorithms of studying the strength and stability of shells are presented each of which is most effective for a specific range of problems.     

Multifibre polymer composites: Prediction of deformation and thermophysical properties

Conclusions A theoretical and experimental investigation was carried out to examine the possibilities of a structural approach for prediction of elastic constants, creep functions and thermophysical characteristics of hybrid polymer composites reinforced with anisotropic fibres of several types. The theoretical solutions were obtained by generalizing the self-consistent method for the case of a three phase model. The effects of brittle fibre breakdown under tension in the direction of reinforcement of a unidirectional hybrid composite were studied under conditions of a short-term loading and a long-term creep. It has been shown that a creep of viscoelastic fibres plays a principal role in creep of the hybrid composite. It is just this creep that significantly increases the fibre damage during creep of the composite.A variant of the solution has been proposed for predicting the thermorheologically complex behavior of hybrid composites containing not only elastic but also viscoelastic thermorheologically simple components with different temperature-time shift factors. The peculiarities of thermal expansion of hybrid composites and the possibilities for a purposeful control of thermal expansion coefficients by hybridization were studied. The considered thermal interval included a region of transition of the polymer matrix from a glass state into a viscoelastic one.The control tests were performed for specimens of organic/glass, organic/carbon, glass/carbon and organic/boron polymer composites with different ratios of fibre volume contents. On the whole, the obtained accuracy of predicting the characteristics of the examined hybrid composites may be considered as acceptable for engineering applications.Published in Mekhanika Kompozitnykh Materialov, Vol. 30, No. 3, pp. 299–313, May–June, 1994.     

Unconditional Nonlinear Stability in Temperature-Dependent Viscosity Flow in a Porous Medium

The equations of flow in porous media attributable to Forchheimer are considered. In particular, the problem of thermal convection in such a medium is addressed when the viscosity varies with temperature. It is shown that nonlinear stability may be achieved naturally for all initial data by working with L ³ or L ⁴ norms. It is also shown that L ² theory is not sufficient for such unconditional stability. Previous work has established nonlinear stability for vanishingly small initial data thresholds, but we believe this is the first analysis that addresses the important physical problem of unconditional stability. It is shown how to extend the nonlinear analysis for a viscosity linear in temperature to the cases when the viscosity may be quadratic or when penetrative convection is allowed in the layer.     

Theory of plasticity and creep taking account of microstrains

A relationship between the theories of plasticity and creep of the type /1, 2/ and theories based on the concept of slip is set up. A most logical structure is proposed for the constitutive equations of the theory which is convenient for engineering calculations.

It has been shown /3/ that the theory of slip /4/ results from the theories /1, 2/. However, it remains unclear whether a deeper connection exists between these theories. Moreover, the connection between creep theories constructed using the approach in /1, 2/ and creep theories based on the slip concept was not generally examined. A survey of the development of polycrystalline strain theory /5/ yields a complete representation of the state of matters in plasticity and creep theories.     

On a Malkin — Massera theorem : PMM vol. 43, no. 6, 1979 pp. 975–979

The results obtained in [1, 2] are complemented by an assertion on asymptotic stability uniform with respect to t₀, x₀, and also are extended to the problems of asymptotic stability with respect to a part of the variables and of optimal stabilization with respect to a part of the variables.     

消振器的数学原理

在工程技术中往往采用消振器来消除自激振荡,使设备或机器不受损坏.本文给出了一个消振器的数学模式 我们讨论了如何适当选取方程组(*)的参数c₁,k₁,k₂,使其零解是全局渐近稳定的,得到了方程组(*)的零解全局渐近稳定的若干定理.     

Vibrocreep of polymer materials

The vibrocreep of low-density polyethylene (LDP) in uniaxial tension has been investigated in the presence of vibration in the direction of action of the constant load. The material was deformed under nonisothermal conditions owing to heating caused by the dissipation of vibrational energy. Superimposing vibrations leads to a considerable increase in creep rate. It is shown that this increase can not be explained solely in terms of the rise in temperature due to heating of the material; there is also a dynamic creep acceleration effect. Avariant of the vibrocreep approximation with allowance for the dynamic and temperature creep acceleration effects is proposed.Mekhanika Polimerov, Vol. 4, No. 3, pp. 413–420, 1968     

On the stability of mixed convective motions in a vertical layer with wave-like boundaries

The stability of the stationary and oscillatory convective motions which develop in a vertical layer with periodically curved boundaries is studied for the case of longitudinal fluid injection. The amplitude of the boundary undulations and the flow of fluid along the layer are both assumed to be small, and methods of perturbation theory are used. The characteristic properties of the incremental spectrum of the spatially periodic motions are studied and the most dangerous types of perturbations as well as the forms of the stability regions are determined.

Theoretical investigations of the effect of spatial inhomogeneity of the boundary conditions on the stability of convection were sparse, and they deal mainly with horizontal layers of fluid /1–3/. Stationary, spatially periodic motions in a vertical layer with curved boundaries were investigated in /4/ for the case of free convection (when the flow was closed), and their stability was investigated in /5/. It was established that the presence of a small but finite flow of fluid along the layer leads to an increase in the number of different modes of flow, and to the appearance of non-stationary convective motions in the region near the threshold.     

Creep Stability of Thin-Walled Composite Structures

The studies on the long-term stability of composite plates and shells under limited creep carried out mainly by the research associates of the Institute of Polymer Mechanics are reviewed. The statement of the stability problems is discussed, according to which a viscoelastic structural member can be regarded as stable if a disturbance in the form of a small initial deflection asymptotically tends with time to a small constant value. In the case of stability, as evidenced by experiments, the increase in the axisymmetric components of the initial deflection, dominating in the early stage, die down with time. On the contrary, the amplitudes of nonaxisymmetric initial imperfections grow at an increasing velocity. Analytical investigations show that the initial imperfections, when expanded into Fourier series, have a spectrum of short- and long-term critical forces. The deflection components having a critical force exceeding the external load are damped out, whereas those having a smaller critical force increase infinitely. The accelerated growth in the deflection, after a time, leads to transient buckling of the shell into a new stable equilibrium form. The problems of optimization of the structure and geometry of thin-walled composite constructions, with constraints on their long-term stability and critical time, are discussed.     

Discharge of gas into vacuum with temperature at the boundary subject to exponential law

A two-dimensional self-similar problem of discharge of a heat conducting gas Into vacuum is analyzed. The temperature at the boundary of gas and vacuum is assumed to change as an exponential function of time. The coefficient of thermal conductivity depends exponentially on temperature and density. The initial gas density is assumed to be finite and constant. With definite values of exponents this problem is self-similar i.e. the system of partial differential equations can be reduced to the solution of a system of ordinary equations.

The self-modeling properties of solutions of this kind of problems has been noted earlier in [1 and 2]. The problem analyzed here is a particular case of the problem of piston motion considered in [3]. In this problem, however, there appears at the boundary of gas and vacuum a new singular point which does not occur in the piston problem.

A numerical solution of the boundary value problem defined by a system of ordinary equations is made difficult by the presence in the latter of singular points, and of discontinuities in the sought solution. These difficulties have been overcome by a qualitative analysis of the behavior of integral curves, and by the selection of a suitable method of numerical integration.

It is shown in this work that, depending on the initial parameters of the problem, there may exist two kinds of solutions. This had been noted earlier in [1, 3 and 4]. Examples of these are presented here. The degeneration of the solution into a trivial one, when the thermal conductivity coefficient is either invariant of density, or increases with increasing density, is pointed out.     

On some problems with unknown boundaries for the heat conduction equation PMM, vol. 31, no. 6, 1967, pp. 1009–1020

A certain class of problems with unknown boundaries are considered herein in connection with the problem, posed by Barenblatt and Ishlinskii [1], on the impact of a viscoplastic rod on a rigid obstacle, which was the fundamental model for typification of this class. The presence of singularities in the unknown functions (the desired solution of the heat conduction equation has a discontinuous point, the derivatives of the unknown boundary are unbounded), and the nonmonotonous behavior of the unknown boundary are characteristic of the considered problems^*.

A theorem on the of the solution of these problems is established, functional equations are derived for the unknown boundaries (equivalent to an initial value problem), and some properties of the solution are discussed (in more detail in the case of the above-mentioned problem of impact of a rod).     

Instability of Statical Solutions of Steel 2D Frames in High Temperature

This work is intended as an attempt to stability analysis of static equations of flat steel frames in fire. High temperature during fire adversely affects on structural steel elements by activating irreversible processes (plasticity, creep). Rheological property can be described in simplified form by power term. Steel properties is represented by physical equations which contains three terms: linear (elasticity), nonlinear and rheological (related to creep). (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the problem op n-stationary centers

Among the various problems of celestial mechanics related to the n-body problem, a certain amount of interest attaches to the specific situation wherein a passive gravitational point mass M moves under the assumption that the relative disposition of the other active gravitational masses experiences no large changes.

If the attracting masses are regarded as points and if changes in the relative disposition of the attracting bodies are neglected, one arrives at the problem of the motion of the point M in a field produced by n-stationary attracting centers (the point M here represents the (n+l)-th body).

In addition to the problem of central motion (n = 1), soluble dynamics problems of this category include Euler's case [1] of two (n= 2) stationary Newtonian attracting centers.

This problem, which for a long time was of solely theoretical Interest as an example of an integrable Liouville system [2], has recently been attracting attention in connection with the mechanics of artificial satellites, particularly after it was shown that the potential of an oblate spheroid can be approximated by the potential of two specifically chosen stationary Newtonian attracting centers [3 and 4].

The solution of the problem for n-attracting centers for n ≥ 3 is unknown, except for a single special case of three centers pointed out by Lagrange and considered In greater detail by J.A. Serre [5].

We shall concern ourselves here with problems on the existence of periodic trajectories in the case of n-attracting centers. An arbitrary and not necessarily Newtonian gravitational law will be assumed.

Our analysis is based on the theory of quasiindices of singular force field points as set forth in [60].     

基于有限元的金属基短纤维复合材料MMC的一种统计蠕变模型

在有限元分析的基础上建立了一个单向应力状态下金属基短纤维复合材料(MMC)的统计蠕变模型.首先建立细胞模型并进行有限元分析,得到了单向应力状态下材料细观尺寸及载荷方向对宏观蠕变响应的影响规律.通过在细胞模型中增加一界面层(考虑材料特性和厚度)来研究基体和纤维的界面对MMC宏观蠕变响应的影响.基于细胞模型的数值结果,提出了一适用于纤维平面随机分布的随机统计模型,该模型考虑了纤维的断裂.通过试验获得纤维的统计分布规律.分析结果表明随机统计模型可以满意地描述试验结果.进一步讨论了材料细观尺寸,纤维的断裂特性以及界面层的材料特性和厚度对MMC宏观蠕变响应的影响.     

Critical time of a long multilayer viscoelastic shell

The buckling in stability of a long multilayer linearly viscoelastic shell, composed of different materials and loaded with a uniformly distributed external pressure of given intensity, is investigated. By neglecting the influence of fastening of its end faces, the initial problem is reduced to an analysis of the loss of load-carrying capacity of a ring of unit width separated from the shell. The new problem is solved by using a mixed-type variational method, allowing for the geometric nonlinearity, together with the Rayleigh-Ritz method. The creep kernels are taken exponential with equal indices of creep. As an example, a three-layer ring with a structure symmetric about its midsurface is considered, and the effect of its physicomechanical and geometrical parameters, as well as of wave formation, on the critical time of buckling in stability of the ring is determined. It is found that, by selecting appropriate materials, more efficient multilayer shell-type structural members can be created. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 5, pp. 617–628, September–October, 2007.     

Nonlinear systems arising from nonisothermal, non-Newtonian Hele-Shaw flows in the presence of body forces and sources

In this paper, we give a formal derivation of several systems of equations for injection moulding. This is done starting from the basic equations for nonisothermal, non-Newtonian flows in a three-dimensional domain. We derive systems for both (T⁰, p⁰) and (T¹, p¹) in the presence of body forces and sources. We find that body forces and sources have a nonlinear effect on the systems. We also derive a nonlinear “Darcy law”. Our formulation includes not only the pressure gradient, but also body forces and sources, which play the role of a nonlinearity. Later, we prove the existence of weak solutions to certain boundary value problems and initial-boundary value problems associated with the resulting equations for (T⁰,p⁰) but in a more general mathematical setting.     

Dimensional stability in truncated moment problems

An approach to truncated moment problems is developed, via the Riesz functional and an assumed dimensional stability of its associated Hilbert spaces. Although equivalent to a concept of “flatness” introduced by R. Curto and L. Fialkow, the dimensional stability discussed in this paper has a different geometric aspect and leads to statements parallel to those of the quoted authors, as well as to some newer ones, obtained by simpler arguments. A stability equation, giving a local characterization of the dimensional stability, is also presented.     

A Mehrotra-Type Predictor-Corrector Algorithm for P*(κ) Linear Complementarity Problems

Mehrotra-type predictor-corrector algorithm,as one of most efficient interior point methods,has become the backbones of most optimization packages.Salahi et al.proposed a cut strategy based algorithm for linear optimization that enjoyed polynomial complexity and maintained its efficiency in practice.We extend their algorithm to P_*（κ）linear complementarity problems.The way of choosing corrector direction for our algorithm is different from theirs. The new algorithm has been proved to have an ο(（1+4κ）（17+19κ） √（1+2κn）^3/2log[（x⁰）^Ts⁰/ε] worst case iteration complexity bound.An numerical experiment verifies the feasibility of the new algorithm.     

On some contact problems for composite half-spaces PMM vol. 31, no. 6, 1967, pp. 1001–1008

Solutions are presented herein of some contact problems connected with the torsion of a composite half-space. In the general case the problem of the torsion of a composite elastic half-space is examined by means of the rotation of a stiff finite cylinder welded into a vertical recess of this half-space. Moreover, the following particular problems on the torsion of such a half-space are considered.

1. 1) A composite half-space with a vertical elastic infinite core, twisted by means of the rotation of a stiff stamp affixed to the upper endplate of the elastic core.

2. 2) A half-space with a vertical cylindrical infinite hole, twisted by means of the rotation of a stiff finite cylinder welded into the upper part of this hole.

In the general case the solution of the problem reduces to the solution of an integral equation of the second kind on a half-line. The question of the solvability of this fundamental integral equation is investigated, and it is shown that its solution may be constructed by successive approximations.

Let us note that the problem of the torsion of a homogeneous half space and of an elastic layer by means of rotation of a stiff stamp has been considered by Rostovtsev [1], Reissner and Sagoci [2], Ufliand [3], Florence [4], Grilitskii [5] and others.

The problem of the torsion of a circular cylindrical rod and the half-space welded to it which are subject to a torque applied to the free endface of the rod has been considered by Grilitskii and Kizyma[6].

The torsion of an elastic half-space with a vertical cylindrical inclusion of some other material by the rotation of a stiff stamp on the surface of this half-space has been considered in [7], wherein it has been assumed that the stamp is symmetrically disposed relative to the axis of the inclusion and lies simultaneously on both materials.