On the definitions of failure and critical states in hypoplasticity

A granular body is said to be at failure or in a critical state if the stress state does not change while the body is continuously deformed. Within the framework of hypoplasticity, such states, generally called stationary states,are conventionally defined by the condition that an objective (the Jaumann) stress rate vanishes. However, not all stationary states attained under monotonic deformation lie within the scope of this definition. Simple shear is an example. In fact, stationary states are characterized by zero material time derivative of the stress tensor rather than zero Jaumann rate. In the present paper, we give a generalized definition of stationarity by the condition of zero material time derivative of the stress tensor. The new definition extends the set of possible stationary states and includes those which are not covered by the previous definition. Stationary states are analysed quantitatively using calibrated hypoplastic equations. It is shown numerically that, if the norm of the spin tensor is of the same order as, or smaller than, the norm of the stretching tensor, the old definition approximates all possible sationary states with sufficient accuracy.      

An approximate solution of some two-dimensional problems of nonstationary filtration

A method of approximate solution of the two-dimensional problems of nonstationary filtration of a gas is proposed. It is based on the application of the method of conformai mapping and the method of successive change of stationary states. The nonstationary influx of a gas into the well is computed by way of example.     

Bifurcations of the Steady Motion of Three-Axle Pneumowheel Vehicles

Manifolds of stationary states are constructed. They consist of main and secondary branches. When the angle of turn of the front wheels and the vehicle velocity change, bifurcations of birth and confluence of stationary states are possible. The phase-plane portraits are given     

Bifurcating electric wires

Summary The stationary states of a string through which an electric current is sent and which is placed in an axial magnetic field, are investigated. Using methods of constrained variational principles, it is shown that, in case the string is inextensible, only those stationary states which have least total potential energy are stable.     

On Transitions to Stationary States in One‐Dimensional Nonlinear Wave Equations

. We consider the long‐time asymptotics of solutions to one‐dimensional nonlinear wave equations, which are infinite‐dimensional Hamiltonian systems. We assume that the nonlinear term is concentrated at a finite segment of the line. We prove long‐time convergence to stationary states for all finite‐energy solutions in the Fréchet topology defined by local energy seminorms. This means that the set of stationary states is a point attractor for the systems in the Fréchet topology. The investigation is inspired by N. Bohr's postulate on the transitions between stationary states in quantum systems.     

The onset of convection in a viscoelastic liquid saturated anisotropic porous layer

The linear stability of a viscoelastic liquid saturated horizontal anisotropic porous layer heated from below and cooled from above is investigated by considering the Oldroyd type liquid. A generalized Darcy model, which takes into account the viscoelastic properties, the mechanical and thermal anisotropy is employed as momentum equation. The critical Rayleigh number, wavenumber, for stationary and oscillatory states and frequency of oscillation are determined analytically. It is shown that oscillatory instabilities can set in before stationary modes are exhibited. The effect of the viscoelastic parameter, the mechanical and thermal anisotropy parameters and specific heat ratio on the linear stability of the system is analyzed and presented graphically.     

Experiments on waves induced in the hollow core of vortices

Recent experimental findings with respect to wave activity in the core of a hollow disk-shaped vortex generated in a cylindrical container with a flat disk rotating at the bottom are reported herein. For relatively low viscosity fluid, several stationary states of the core exist within a certain range of disk speeds. This range becomes narrow as the wave number grows. Between the stationary states, mixed, time dependent states were found to occur. Their interval of endurance decrease with the wave number. The disk speed at which the static state first appears and ends increases linearly with the original height. The phase velocity of the stationary waves rises with the angular velocity of the disk. For a liquid with an intermediate value of viscosity, abrupt transitions from one equilibrium state to the other is taking place. Steady vortex core patterns with wave numbers from one to eleven are observed. States were the basic pattern is subharmonically modulated and states where a wave packet encircles periodically the core are also encountered. Hysterisis is clearly evident. The core of a highly viscous liquid is stable. A suppressed form of instability appears at very large liquid heights and disk speeds.     

The Dynamics of Finite-Dimensional Systems Under Nonconservative Position Forces

General theorems on the stability of stationary states of mechanical systems subjected to nonconservative position forces are presented. Specific mechanical problems on gyroscopic systems, a double-link pendulum with a follower force and elastically fixed upper tip, multilink pneumowheel vehicles, a monorail car, and rail-guided vehicles are analyzed. Methods for investigation of divergent bifurcations and catastrophes of stationary states are described     

Prediction of the response of non-linear oscillators under stochastic parametric and external excitations

The method of equivalent external excitation is derived to predict the stationary variances of the states of non-linear oscillators subjected to both stochastic parametric and external excitations. The oscillator is interpreted as one which is excited solely by an external zero-mean stochastic process. The Fokker-Planck-Kolmogorov equation is then applied to solve for the density functions and match the stationary variances of the states. Four examples which include polynomial, non-polynomial, and Duffing type non-linear oscillators are used to illustrate this approach. The validity of the present approach is compared with some exact solutions and with Monte Carlo simulations.     

On Global Attraction to Stationary States for Wave Equations with Concentrated Nonlinearities

The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as \(t\rightarrow \pm \infty \) to stationary states. The attraction is caused by nonlinear energy radiation.     

Adjoint approach of the spatial sensitivity to disturbances of internal flows with free surface

A local adjoint technique is developed in order to determine the most sensitive location to perturbations of steady states near bifurcation points in the case of confined flows with free‐surface boundary. Transitions to stationary or periodic flows are studied. The method is validated by comparison of its results with those given by a time approach. It is then applied to the stability study and the feedback control of thermocapillary flows in liquid bridge. Copyright © 2005 John Wiley & Sons, Ltd.     

Material instabilities in inelastic saturated porous media under dynamic loadings

Instabilities in inelastic saturated porous media are investigated here for general three-dimensional states under dynamic loadings using a perturbation approach.Under quasi-static conditions, unbounded growth of perturbations is related to the emergence of stationary discontinuities under drained or undrained conditions, while under dynamic conditions, unbounded growth is related either to the emergence of stationary discontinuities (and these are set by drained conditions) or to the appearance of the flutter phenomenon (acceleration waves).For associative behaviour unbounded growth always corresponds to localization under drained conditions and the onset of growth of perturbations occurs here only through divergence growth. It is only for non-associative flow that unbounded growth may correspond to undrained localization in quasi-static conditions and to flutter under dynamic conditions.     

On a variational principle in thermodynamics

The dynamic principle is proposed that a thermodynamic system evolves in time so that the total energy balance including the energy drawn from the environment becomes stationary for all admissible variations of thermodynamic states. It is shown that this principle allows to obtain variational characterizations of contact Hamiltonian equations (even in presence of ports), reaction equations and doubly nonlinear reaction–diffusion equations. Further, examples are discussed which support this principle.     

Asymptotic analysis of stationary distribution of the front of a two-stage consecutive exothermic reaction in a condensed medium

An approximate theory of the stationary distribution of the plane front of a two-stage exothermic consecutive chemical reaction in a condensed medium is developed in the article. The method of joined asymptotic expansions is used in constructing the solutions. The ratio of the sum of the activation energies of the reactions to the final adiabatic combustion temperature is a parameter of the expansion. The characteristic limiting states of the stationary distribution of the wave corresponding to different values of the parameters figuring in the problem are shown. Approximate analytical expressions for the wave velocity and distribution of concentrations are obtained for each of the states.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 75–87, January–February, 1973.     

Electron broadening of isolated lines with stationary non-equilibrium level populations

It is shown that a quantum kinetic theory approach to line broadening, extended to stationary non-equilibrium states, yields corrections to the standard electron impact widths of isolated lines that depend on the population of the radiator internal levels. A consistent classical limit from a general quantum treatment of the perturbing electrons also introduces corrections to the isolated line widths. Both effects are essential in preserving detailed-balance relations. Preliminary analysis indicates that these corrections may resolve existing discrepancies between theoretical and experimental widths of isolated lines. An experimental test of the results is proposed.     

One‐shot pseudo‐time method for aerodynamic shape optimization using the Navier–Stokes equations

This paper presents a numerical method for aerodynamic shape optimization problems in compressible viscous flow. It is based on simultaneous pseudo‐time stepping in which stationary states are obtained by solving the pseudo‐stationary system of equations representing the state, costate and design equations. The main advantages of this method are that it blends in nicely with previously existing pseudo‐time‐stepping methods for the state and the costate equations, that it requires no additional globalization in the design space, and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP methods. For design examples of 2D problems, the overall cost of computation can be reduced to less than 2 times the forward simulation runs. Copyright © 2011 John Wiley & Sons, Ltd.     

Superstatistics: theory and applications

Superstatistics is a superposition of two different statistics relevant to driven nonequilibrium systems with a stationary state and intensive parameter fluctuations. It contains Tsallis statistics as a special case. After briefly summarizing some of the theoretical aspects, we describe recent applications of this concept to three different physical problems, namely a) fully developed hydrodynamic turbulence, b) pattern formation in thermal convection states, and c) the statistics of cosmic rays.Received: 13 March 2003, Accepted: 27 May 2003, Published online: 9 December 2003PACS: 05.40.-a     

Traveling Monotonic Fronts in the Discrete Nagumo Equation

We give an alternative proof of the theorem, which states that no propagation failure occurs for the discrete Nagumo equation with “translationally invariant” stationary monotonic fronts. The theorem was recently proved with the use of the invariant manifolds for lattice differential equations by Hupkes, Pelinovsky, and Sandstede. The alternative proof relies on the analysis of the advance-delay operator associated with the translationally invariant stationary front. This operator exhibits an infinite-dimensional kernel spanned by Fourier harmonics of front’s translations, which are accounted when the stationary front is continued into the traveling one.     

On Nonlinear Stability of Isotropic Models in Stellar Dynamics

Consider an aggregation of mass particles in space which attract each other according to Newton's law of attraction. This system can be described by distribution functions satisfying the Vlasov equation and the Poisson laws. We obtain the nonlinear stability of certain stationary states, including those obeying modified Emden's laws. A priori estimates of the energy-Casimir functions around stationary states are established for distribution functions for which the L^5/3L^{5/3}-norms of the density functions are uniformly bounded. A uniform bound on the kinetic energy of the system readily implies that these norms of the density functions are indeed uniformly bounded. In this way we prove nonlinear stability.     

On the dynamics of compressor surge

Investigations concerned with the stability of stationary states and the possibility of self-excited oscillation (surge) occurring in systems with a centrifugal compressor (or a centrifugal pump) lead, for a simplified model, to an analysis of a set of two first-order differential equations. The paper presents such an analysis for the case when the machine characteristic can be expressed by a continuous unique curve as well as for that when the characteristic is neither a unique nor even a smooth curve. It is shown which of the singular points is the saddle point and in the case of the latter type of characteristic, which point can be taken for the saddle; this approach is believed to make practical analyses more straightforward.