Numerical analysis method for studying local forms of stability loss of bearing layers of three-layered shells using mixed forms

A revised formulation of linearized stability problems of three-layered shells with a sofi filler has been presented. The form of stability loss of the rigid layers is mixed in the shells when the moment precritical stress-strain state (SSS) is reached and is localized in the principal moment SSS zones. If the filler thickness is much greater than the thickness of the rigid layers, the size of the bulges and thickness of the filler have the same order of magnitude. Thus, a very fine grid must be used for a numerical solution of the stability loss equations, which poses considerable computational difficulties. A numerical analysis method is proposed for the local forms of mixed mode stability loss of the rigid layers of a three-layered shell. Using this method, the solution of equations for the precritical SSS by the finite element scheme is found but an analytical solution of reduced stability loss equations is presented for estimating the critical load. This solution is an asymptotic approximation for local modes of stability loss implemented into design.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 1, pp. 88–100, January–February, 1995.     

Mean-square asymptotic stability of solutions of systems of stochastic differential equations with random operators

We obtain conditions of asymptotic behavior of trivial solutions of systems of stochastic differential equations with random operators.Translated from Ukrainskii Matematicheskii Zhunal, Vol. 47, No. 7, pp. 990–1001, July, 1995.     

Dynamic compressive strength of epoxy composites

The strength of laminated and unidirectionally reinforced composite materials was investigated in conditions of dynamic uniaxial compression with a strain rate of 50–1000 sec^–1 using the split Hopkinson pressure bar method. It was shown that in conditions of dynamic compression, glass/epoxy, aramid/epoxy, and carbon/epoxy composites exhibit elastic-brittle behavior with anisotropy of the strength and elastic properties. The effect of the strain rate on the strength characteristics of fiberglass-reinforced plastics was demonstrated.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6, pp. 776–782, November–December, 1995.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October, 1995).     

Multicritical phenomena in flow of viscoelastic liquids. 1. Zaremba-Fromm-De Witt liquid

It has been shown that multicritical phenomena caused by nonlinearity of viscosity and high elasticity, and forced anisotropy at finite shear rates take place during flow of viscoelastic polymer melts which are isotropic in the resting state. The sign of the low-frequency asymptotic values of the dynamic viscosity and elasticity measured during steady flow is a criterion of the appearance of instability. These arguments are illustrated by the solution and analysis of the complex reaction to low-amplitude, periodic shear of a steady-flowing, very simple viscoelastic liquid — ZFD liquid. It was shown that the instability of viscoelastic liquids for a given steady shear rate is due to the effect of perturbations lasting for no less than some limiting value and its manifestations are caused by superposition of different types of instability — multicritical phenomena.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 555–572, July–August, 1995.The study was conducted based on Topic 93,177 of the Latvian Science Council.     

Estimation of solutions of nonautonomous systems

We study the problem of using the direct Lyapunov method to get estimates for solutions of a system of ordinary differential system in general form. Theorems on asymptotic stability and behavior of solutions are proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 242–248, February, 1995.     

On solutions with generalized power asymptotics to systems of differential equations

In the paper we study methods for constructing particular solutions with nonexponential asymptotic behavior to a system of ordinary differential equations with infinitely differentiable right-hand sides. We construct the corresponding formal solutions in the form of generalized power series whose first terms are particular solutions to the so-called truncated system. We prove that these series are asymptotic expansions of real solutions to the complete system. We discuss the complex nature of the functions that are represented by these series in the analytic case.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 851–861, December, 1995.     

An investigation of stresses in composite shells on the basis of methods of asymptotic synthesis due to various local actions

It is assumed that the shells are infinite or semi-infinitely long. The fundamental differential equations of twodimensional shell theory are employed in order to obtain a solution of the problem. Also, effective methods of an asymptotic synthesis (MAS) are suggested to avoid difficulties in calculations. For these methods various approximate equations are used, which are: Vlasov-Donnell's, of the semi-membrane theory, of the edge effect theory, the bending or planning states. Solutions of the local thermoelastic problem were obtained in the form of Fourier's series along the contour and in the form of Fourier's integral to the axis direction. Convenient expressions for stresses are obtained by using the MAS, which permits us to reduce the calculation to a calculation of the elementary functions and a very rapidly convergent series.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October 1995.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6, pp. 783–791, November–December, 1995.     

Asymptotic analysis of three-dimensional dynamic equations of a thin plate : PMM vol. 38, n 6, 1974, pp. 1072–1078

Two-dimensional dynamic equations of thin plate vibrations are obtained from the three-dimensional dynamic equations of elasticity theory on the basis of an asymptotic method [1 – 3], Such an approach permits establishing the limits of applicability of the two-dimensional dynamic equations and the corresponding boundary and initial conditions, and indicating the means of obtaining refined results.The question of the construction of an inner state of stress of a thin plate under dynamic conditions is examined herein. The possibility of considering states of stress with distinct variability in time and in the coordinates and with a distinct relationship between the displacement intensities, is taken into account.     

Conditions of technical stability of nonlinear evolution systems

We derive general criteria of technical stability on finite and infinite time intervals and criteria of technical asymptotic stability for a class of physical processes described by nonlinear evolution equations in a Hilbert space. The behavior of the solutions of these equations is characterized in terms of Gâteaux differentials. The comparison method based on the Lyapunov second method is applied.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 95–103, 1988.     

Model of a layered plate considering nonideal contact between layers

We propose a mathematical model for doing calculations for layered plates, allowing for both rigid and sliding contact in the presence of frictional forces between the sliding layers. The model takes into account the distribution of tangential and normal displacements across the thickness of the sliding layered stack, and also the distribution of transverse normal stresses. The strain tensor is obtained using the Cauchy relations; the stress tensor is obtained based on Hooke's law. Tne Lagrange variational principle allows us to obtain the resolvent system of differential equations and the corresponding boundary conditions. The spatial model for deformation of a layered plate has a number of special features compared with familiar models. The system of differential equations has operators no higher than second order. It is described relative to displacements on the faces of the stack. This is convenient in solving problems involving sliding of layers with and without friction.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 671–676, September–October 1995.     

Analysis of the anisotropic properties of honeycomb sandwich plates

A mathematical model has been described and an approach proposed to the diagnostics of the anisotropic properties of the relative elasticity constants of three-layer (sandwich) plates with a honeycomb filler. Compact analytical relationships between the anisotropy coefficients and the geometrical dimensions of the cells have been obtained. A practical approach has been proposed for the selection of rational parameters of the honeycomb structure, at which its anisotropic properties would be in the chosen region of the functional space. The conditions have been formulated at which the constructive anisotropy, inherent to honeycomb structures, degenerates to isotropy for some relative elastic constants.Paper to be presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 4, pp. 482–487, July–August, 1995.     

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

This article is concerned with the asymptotic accuracy of the Computational Singular Perturbation (CSP) method developed by Lam and Goussis [The CSP method for simplifying kinetics, Int. J. Chem. Kin. 26 (1994) 461–486] to reduce the dimensionality of a system of chemical kinetics equations. The method, which is generally applicable to multiple-time scale problems arising in a broad array of scientific disciplines, exploits the presence of disparate time scales to model the dynamics by an evolution equation on a lower-dimensional slow manifold. In this article it is shown that the successive applications of the CSP algorithm generate, order by order, the asymptotic expansion of a slow manifold. The results are illustrated on the Michaelis–Menten–Henri equations of enzyme kinetics.     

Some properties of random evolutions

We study asymptotic properties of matrix-valued random evolutions and consider an example of evolutions of this type.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1333–1337, October, 1995.     

Asymptotic solutions of coupled dynamic problems of thermoelasticity for thin bodies of anisotropic inhomogeneous-in-plan materials

Two-dimensional recurrence resolvents for an inhomogeneous thin body (plates of variable thickness and shells) are derived by an asymptotic method based on the three-dimensional equations of the coupled dynamic problem of the thermoelasticity of an anisotropic body, which are solved in the case of anisotropy, having, at each point, one plane of symmetry perpendicular to the transverse axis. Recurrence formulae are derived in a general formulation for determining the components of the stress tensor, the strain vector and the function of the change in the temperature field, when different boundary conditions of dynamic problems of the theory of coupled thermoelasticity and thermal conductivity are given on the end surfaces of a thin body. An algorithm for determining the analytical and numerical (necessary) solutions of these boundary-value problems with an arbitrarily specified accuracy is developed.     

Integration of a class of systems of differential equations by using a contour integral

For systems ofq linear differential equations ofnth order with polynomial matrix coefficients, a fundamental family of formal solutions defined in a certain sector of a complex plane is constructed by using the Laplace contour integral. For large positive values of an independent variable, the asymptotic representations of indicated solutions are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 30–38, January, 1995.     

Asymptotic expansions associated with the jackknife functional. I

We obtain an asymptotic expansion of the functional of the jackknife method, which is used for the estimation of the variance of observational errors in a nonlinear regression model.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 443–451, April, 1995.     

Linear continuous operators for the Stieltjes moment problem in Gelfand–Shilov spaces

We obtain linear continuous operators providing a solution to the Stieltjes moment problem in the framework of Gelfand–Shilov spaces of rapidly decreasing smooth functions. The construction rests on an interpolation procedure due to R. Estrada for general rapidly decreasing smooth functions, and adapted by S.-Y. Chung, D. Kim and Y. Yeom to the case of Gelfand–Shilov spaces. It requires a linear continuous version of the so-called Borel–Ritt–Gevrey theorem in asymptotic theory.     

Dynamic Robust Output Min–Max Control for Discrete Uncertain Systems

Min–max control is a robust control, which guarantees stability in the presence of matched uncertainties. The basic min–max control is a static state feedback law. Recently, the applicability conditions of discrete static min–max control through the output have been derived. In this paper, the results for output static min–max control are further extended to a class of output dynamic min–max controllers, and a general parametrization of all such controllers is derived. The dynamic output min–max control is shown to exist in many circumstances under which the output static min–max control does not exist, and usually allows for broader bounds on uncertainties. Another family of robust output min–max controllers, constructed from an asymptotic observer which is insensitive to uncertainties and a state min–max control, is derived. The latter is shown to be a particular case of the dynamic min–max control when the nominal system has no zeros at the origin. In the case where the insensitive observer exists, it is shown that the observer-controller has the same stability properties as those of the full state feedback min–max control.     

Technical stability conditions for some dynamic systems with distributed and concentrated parameters

Sufficient conditions are derived for technical stability on a finite and an infinite time interval and for asymptotic technical stability (both local and global) of a class of processes described by a system of complex ordinary nonlinear differential equations, delayed dynamic processes, and some three-wave interacting processes. The Lyapunov direct method is used, combined with the comparison method and Gâteaux generalized differentiation.Institute of Mechanics of the Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 113–123, 1989.     

Asymptotics of Bounded-at-Infinity Solutions of the Principal Resonance Equation

We consider a system of six nonlinear differential equations obtained by averaging fast forced oscillations. The main result consist in the construction of the asymptotics at infinity for the general solution with bounded amplitudes. We show that the structure of asymptotic series depends on the parameters so that the coefficients of the series vary in jumps on the resonance set.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 85–97.Original Russian Text Copyright © 2005 by L. A. Kalyakin, Yu. Yu. Bagderina.