Victor L. Berdichevsky

Theory of anisotropic thin-walled closed-cross-section beams

A variationally and asymptotically consistent theory is developed in order to derive the governing equations of anisotropic thin-walled beams with closed sections. The theory is based on an asymptotic analysis of two-dimensional shell theory. Closed-form expressions for the beam-stiffness coefficients, stress and displacement fields are provided. The influence of material anisotropy on the displacement field is identified. A comparison with the displacement fields obtained by other analytical developments is performed. The stiffness coefficients and static response are also compared with finite element predictions, closed-form solutions and test data.

Asymptotic theory for static behavior of elastic anisotropic I-beams

Abstract End effects for prismatic anisotropic beams with thin-walled, open cross-sections are analyzed by the variational-asymptotic method. The decay rates for disturbances at the ends of prismatic beams are evaluated, and the most influential end disturbances are incorporated into a refined beam theory. Thus, the foundations of Vlasovs theory, as well as restrictions on its applicability, are obtained from the variational-asymptotic point of view. Vlasovs theory is proved to be asymptotically correct for isotropic I-beams. The asymptotically correct generalization of Vlasovs theory for static behavior of anisotropic beams is presented. In light of this development, various published generalizations of Vlasovs theory for thin-walled anisotropic beams are discussed. Comparisons with a numerical 3-D analysis are provided, showing that the present approach gives the closest agreement of all published theories. The procedure can be applied to any thin-walled beam with open cross-sections.

The JPL proton fluence model: an update

Abstract The development of new technologies and the miniaturization of sensors bring new requirements for our ability to predict and forecast hazardous space weather conditions. Of particular importance are protons in the energy range from 10s to 100s of MeV which cause electronic part and solar cell degradation, and pose a hazard to biological systems in space and to personnel in polar orbit. Sporadic high-energy solar particle events are a main contributor to the fluences and fluxes of such protons. A statistical model, JPL 1991 (J. Geophys. Res. 98 (1991) 13,281), was developed to specify fluences for spacecraft design and is now widely used. Several major solar proton events have occurred since that model was developed and one objective of this paper is to see if changes need to be made in the model due to these recent events. Another objective is to review the methods used in JPL 1991 in the light of new understandings and to compare the JPL methods with those used in other models. We conclude that the method used in developing JPL 1991 model is valid and that the solar events occurring since then are completely consistent with the 1991 model. Since no changes are needed we suggest that the name of the model be changed to “the JPL fluence model”.

STATISTICAL MECHANICS OF VORTEX LINES

A rotatable capacitance sensor preferably constructed of a generally cylindrical body mounted on bearings to a carriage assembly. The body has a number of parallel, spaced apart conductive plates having edges extending through the exterior body surface, and alternate ones of the plates are connected via a bearing to a conductor in one part of the carriage assembly, and the remaining ones of the plates are connected via a bearing to a second conductor in another part of the carriage assembly.

On Saint-Venant¿s principle in the dynamics of elastic beams

Abstract In dynamics, Saint-Venant’s principle of exponential decay of stress resulting from a self-equilibrating load is not valid. For a beam type structure, a self-equilibrated load may penetrate well inside the beam. Although this effect has been known for a long time, at least since Lamb’s paper [Proc. Roy. Soc. Lon. Ser. A 93 (1916) 114], it was not clear how to characterize it quantitatively. In this paper we propose a “probabilistic approach” to evaluate the magnitude of the penetrating stress state. The key point is that, in engineering problems, the distribution of the self-equilibrated load is usually not known. By assigning to the self-equilibrated load some probabilistic measure one can find probabilistic characteristics of the penetrating stress state. We develop this reasoning for the simplest case: longitudinal vibrations of a two-dimensional semi-infinite, elastic isotropic homogeneous strip, excited by a periodic load at the end. We show the frequency range where Saint-Venant’s principle can be used with good accuracy, and thus, one-dimensional classical beam theory still can be applied. We characterize also the increase in this range which is achieved in the refined plate theory proposed by Berdichevsky and Le [J. Appl. Math. Mech. (PMM) 42 (1) (1978) 140].

Micromechanics of diffusional creep

Abstract In polycrystalline materials at high temperatures and low stresses, creep occurs mostly by the diffusion of vacancies through the grain bodies and over the grain boundaries. A continuum theory of vacancy motion is considered to analyze diffusional creep on a microscopical level. A linear version of such a theory was formulated by Nabarro, Herring, Coble and Lifshitz. We revise this theory from the perspectives of continuum mechanics and present it in a thermodynamically consistent nonlinear form. A certain difficulty, which one has to overcome in this endeavor, is the absence of Lagrangian coordinates in diffusional creep, the major building block of any theory in continuum mechanics. A linearized version of the theory is studied for the case of bulk diffusion. We consider the derivation of macro constitutive equations using the homogenization technique. It is shown that macroequations are nonlocal in time and nonlocality is essential in primary creep. For secondary creep polycrystals behave as a viscoelastic body. For secondary creep, a variational principle is found which determines microfields and macromoduli in stress-strain rate constitutive equations. A two-dimensional honeycomb microstructure and single crystal deformation are studied numerically by a finite element method.

Theory of Elastic Plates and Shells

Consider the surface \(\mathop \Omega \limits^ \circ \) in three-dimensional space and, at each point on the surface, erect a segment of length h directed along the normal to the surface; the centers of the segments are on \(\mathop \Omega \limits^ \circ .\) The segments cover some three-dimensional region, \(\mathop V\limits^ \circ \) (Fig. 14.1). If h is much smaller than the minimum curvature radius of the surface \(\mathop \Omega \limits^ \circ ,\) R, and the characteristic size of the surface \(\mathop \Omega \limits^ \circ \), L

Dynamics of Elastic Bodies

The extrapolation to dynamics of the minimization principles formulated above encounters difficulties, the essence of which can be observed for systems with one degree of freedom.

Thermodynamics of Duffing’s Oscillator

We study the averaged characteristics of the response of Duffings oscillator to harmonic excitation. We show that, as in classical thermodynamics, response characteristics are potential functions of excitation characteristics

Introduction to stochastic variational problems

The lectures provide an introduction to the Chapters on stochastic variational problems from the author’s book Variational Principles of Continuum Mechanics, Springer, 2009.