Refined stability theory for sandwich shells with transversally stiff core. 2. Linearized equations of neutral equilibrium

Neutral equilibrium equations of the refined theory of stability for sandwich shells with a transversally stiff core are constructed and used for studying local mixed forms of stability loss (FSL), as well as admitting different variants of simplification, depending on the type of precritical state and realized FSL. The generalized Reissner variational principle used for deriving the stability equations allows us to refine transverse shear stresses in the core as compared to [1]. A method for a highly accurate definition of these stresses is proposed. Namely, after the integration of three-dimensional equilibrium equations over the transverse coordinate, the number of free constants and the number of static conditions to be satisfied are equalized according to the actual stress distribution across the thickness.Science and Technology Center for Study of Dynamics and Strength. Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 6, pp. 786–795, November–December, 1997.     

Refined stability theory for sandwich shells with transversally stiff core. 3. Simplest one-dimensional problems

The applicability and accuracy of stability equations of the refined theory for sandwich shells with a transversally stiff core proposed in [1] are investigated. The model problem of calculating the critical loads and stress fields in the core at mixed forms of the loss of stability is solved for an infinitely wide sandwich plate with an orthotropic core and composite load-carrying layers subjected to in-plane edge loads. The case of pure bending of the plate is considered in detail. The results obtained by variation of the physical-mechanical parameters are compared with the solutions of the three-dimensional theory for the core [2]. It is shown that the version of the refined theory [1] is more accurate than the other two-dimensional theories.For Pt. 2 see [1].Center for Study of Dynamics and Stability, Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 1, pp. 57–65, January–Feburary, 1998.     

Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms

A variant of the refined geometric nonlinear theory is suggested for nonshallow shells with a transversely soft core of medium thickness with regard to modifications of metric characteristics across the core thickness. The kinematic relations for the core are derived by sequential integration of the initial three-dimensional equations of elasticity theory along the transverse coordinate. The equations are preliminarily simplified by the assumption that the tangential stress components are equal to zero. With the example of sandwich plates, it is shown that these equations allow us to investigate synphasic, antiphasic, mixed flexural, and mixed flexural-shear buckling forms of load-bearing layers and the core depending on the precritical stress-strain state. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 95–108, January–February, 2000.     

Application of the technical theory to sandwich shells with a solid polymeric core

The possibilities of using the technical theory for analyzing cylindrical sandwich shells with a core of low-modulus polymeric material are considered. It is shown to be necessary to make assumptions concerning the distribution of the deformations over the elements of the three-layer section and to take account of the shear strains in the core, the flexural rigidity in the longitudinal direction, and the Poisson ratio in determining the forces and moments. The theoretical conclusions have been experimentally confirmed by static tests on a model.All-Union Structural Engineering Correspondence Institute, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 298–304, March–April, 1973.     

Nonlinear equations of a timoshenko-type theory of laminated anisotropic shells

In recent years analysis of the stress—strain state of shell structures made out of composite materials has been based on refined shell theories which take into account strains in the direction normal to the reference surface. There are several approaches to the formulation of the refined theories. One can point to shell theories developed on the basis of variational principles (e.g., [1, 2]) as well as theories created with the help of iterational processes (e.g., [3–6]). A resolving system of nonlinear equations for laminated anisotropic shells has been derived in the proposed research based on the Reissner variational principle [7, 8]. A similar linear theory which takes into account the strain e₃₃ also has been developed in [1]. If the shear stiffnesses of the layers differ greatly from each other in the transverse direction, then one can treat the shell structure as a single-layer shell of nonuniform structure. In this case it is advisable to solve a problem of the type of a uniform shell with minimal stiffnesses.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 501–507, May–June, 1979.     

Transient dynamic response of sandwich shells with transversely compressible core

The present study provides an investigation of the effect of the transverse core compressibility on the dynamic buckling response of sandwich structures. The study utilizes a previous v. Kármán type higher-order model for shallow sandwich shells. An analytical solution is obtained by means of an extended Galerkin procedure in conjunction with an explicit fourth order Runge-Kutta algorithm to solve the transient problem. In an example analysis, it is observed that the transverse core compressibility can have strong effects even on the global response of sandwich structures. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress-strain state and stability of composite sandwich shells with a scaling zone between the core and facings

A finite element model is presented for analyzing the strength and stability of sandwich shells of arbitrary configuration with an adhesion failure zone between the core and one of the facings. The model is based on the assumptions that both facings are laminated Timoshenko-type composite shells, only transverse shear stresses in the core and normal stresses in the thickness direction have nonzero values, a free slip in the tangential plane in the adhesion failure zone and unilateral contact along the normal are possible, and the prebuckling state in the stability problem is linear. Biquadratic nine-node approximations for all functions and numerical integration were used. The displacements and rotation angles of the normals toward the facings as well as stresses in the core are taken as global degrees of freedom. The algebraic problem is solved using a special step-by-step procedure of determining the contact area in the scaling zone and employing unilateral constraints for some of the unknowns. Numerical examples are also given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 29, No. 5, pp. 640–652, September–October, 1993.     

Refined linear theory of elastic anisotropic multilayer shells. Part 2

On the basis of the refined linear theory of elastic anisotropic multilayer shells of arbitrary shape derived in [1] it is established that a number of theorems of the linear theory of elasticity have analogues in the theory of multilayer anisotropic shells.For Part 1 see [1].Institute of Fluid Mechanics, Bucharest. Translated from Mekhanika Polimerov, No. 1, pp. 100–109, January–February, 1976.     

Elastoplastic stability of sandwich plates with a light core

The elastoplastic stability of sandwich plates with a light core is theoretically investigated. Certain specific problems are considered.Tashkent Lenin State University. Translated from Mekhanika Polimerov, No. 3, pp. 568–571, May–June, 1971.     

Implicit-explicit schemes for flow equations with stiff source terms

In this paper, we design stable and accurate numerical schemes for conservation laws with stiff source terms. A prime example and the main motivation for our study is the reactive Euler equations of gas dynamics. Furthermore, we consider widely studied scalar model equations. We device one-step IMEX (implicit-explicit) schemes for these equations that treats the convection terms explicitly and the source terms implicitly.For the non-linear scalar equation, we use a novel choice of initial data for the resulting Newton solver and obtain correct propagation speeds, even in the difficult case of rarefaction initial data. For the reactive Euler equations, we choose the numerical diffusion suitably in order to obtain correct wave speeds on under-resolved meshes.We prove that our implicit-explicit scheme converges in the scalar case and present a large number of numerical experiments to validate our scheme in both the scalar case as well as the case of reactive Euler equations.Furthermore, we discuss fundamental differences between the reactive Euler equations and the scalar model equation that must be accounted for when designing a scheme.     

Nonlinear stability of discontinuous Galerkin methods for delay differential equations

The present paper is devoted to a study of nonlinear stability of discontinuous Galerkin methods for delay differential equations. Some concepts, such as global and analogously asymptotical stability are introduced. We derive that discontinuous Galerkin methods lead to global and analogously asymptotical stability for delay differential equations. And these nonlinear stability properties reveal to the reader the relation between the perturbations of the numerical solution and that of the initial value or the systems.     

Stability and asymptotic stability of -methods for nonlinear stiff Volterra functional differential equations in Banach spaces

This paper is concerned with the stability and asymptotic stability of θ-methods for the initial value problems of nonlinear stiff Volterra functional differential equations in Banach spaces. A series of new stability and asymptotic stability results of θ-methods are obtained.     

Nonlinear stability of general linear methods for neutral delay differential equations

This paper is concerned with the numerical solution of neutral delay differential equations (NDDEs). We focus on the stability of general linear methods with piecewise linear interpolation. The new concepts of GS(p)$GS\left(p\right)$-stability, GAS(p)$GAS\left(p\right)$-stability and weak GAS(p)$GAS\left(p\right)$-stability are introduced. These stability properties for (k,p,0)$(k,p,0)$-algebraically stable general linear methods (GLMs) are further investigated. Some extant results are unified.     

Solution of the equations of the relativistic theory of gravitation for equilibrium massive bodies with allowance for their equations of state

The solutions of the equations of the relativistic theory of gravitation that describe the equilibrium state of a spherically symmetric isolated massive body are analyzed. It is shown that if the mass of the body is greater than the critical value equilibrium states do not exist; the minimum sizes of such bodies are always greater than the Schwarzschild sizes. We investigate the equilibrium sizes, the structure of the exterior gravitational field, and the distributions of the interior pressures and densities in the case of characteristic astrophysical objects such as the earth, Jupiter, the sun, neutron stars, and white dwarfs. The results agree satisfactorily with observations.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 1, pp. 122–139, January, 1993.     

Two-dimensional vortex sheets for the nonisentropic Euler equations: Nonlinear stability

We show the short-time existence and nonlinear stability of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on the weakly linear stability result of Morando and Trebeschi (2008) [20]. The missing normal derivatives are compensated through the equations of the linearized vorticity and entropy when deriving higher-order energy estimates. The proof of the resolution for this nonlinear problem follows from certain a priori tame estimates on the effective linear problem in the usual Sobolev spaces and a suitable Nash–Moser iteration scheme.     

Static and dynamic stability for geometrically nonlinear governing equations of elastic thin shallow shells

In this study, the governing equations for large deflection of elastic thin shallow shells are deduced into an algebraic cubic equation to determine the unknown coefficient of the assumed deflection by applying Galerkin's method in combination with the algebraic polynomial and Fourier series. For the dynamic problem, the coefficient is replaced by an unknown function of time; after the same process is applied, the governing equations are deduced to be a nonlinear ODE of order two called the Duffing equation, and its analytical solution is known. The combination of the algebraic polynomial and Fourier series gives very rapid convergence in the asymptotic solutions.     

Dynamic stability of cylindrical orthotropic shells with an elastic core under a longitudinal periodic load

The dynamic instability of a cylindrical orthotropic shell with an elastic core subjected to a longitudinal periodic load is considered. Equations are obtained for determining the regions of dynamic instability for different core models.